Proceedings of the 19th International Meshing Roundtable

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Proceedings of the 19th International Meshing Roundtable

Suzanne Shontz (Ed.)

Proceedings of the 19th International Meshing Roundtable

ABC

Suzanne Shontz Pennsylvania State University 343J IST Building University Park, PA 16802 Tel.: 814-865-0193 E-mail: [email protected]

ISBN 978-3-642-15413-3

e-ISBN 978-3-642-15414-0

DOI 10.1007/978-3-642-15414-0 Library of Congress Control Number: 2010935732 c 2010 Springer-Verlag Berlin Heidelberg This work is subject to copyright. All rights are reserved, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilm or in any other way, and storage in data banks. Duplication of this publication or parts thereof is permitted only under the provisions of the German Copyright Law of September 9, 1965, in its current version, and permission for use must always be obtained from Springer. Violations are liable to prosecution under the German Copyright Law. The use of general descriptive names, registered names, trademarks, etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use. Typesetting: Data supplied by the authors Production & Cover Design: Scientific Publishing Services Pvt. Ltd., Chennai, India Printed on acid-free paper 987654321 springer.com

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Preface

The papers in this volume were selected for presentation at the 19th International Meshing Roundtable (IMR), held October 3–6, 2010 in Chattanooga, Tennessee, USA. The conference was started by Sandia National Laboratories in 1992 as a small meeting of organizations striving to establish a common focus for research and development in the field of mesh generation. Now after 19 consecutive years, the International Meshing Roundtable has become recognized as an international focal point annually attended by researchers and developers from dozens of countries around the world. The 19th International Meshing Roundtable consists of technical presentations from contributed papers, research notes, keynote and invited talks, short course presentations, and a poster session and competition. The Program Committee would like to express its appreciation to all who participate to make the IMR a successful and enriching experience. The papers in these proceedings were selected by the Program Committee from among numerous submissions. Based on input from peer reviews, the committee selected these papers for their perceived quality, originality, and appropriateness to the theme of the International Meshing Roundtable. We would like to thank all who submitted papers. We would also like to thank the colleagues who provided reviews of the submitted papers. The names of the reviewers are acknowledged in the following pages. We extend special thanks to Jacqueline Hunter for her time and effort to make the 19th IMR another outstanding conference.

August 2010

19th IMR Program Committee


Contents

Session 1A, Surface Meshing Hexagon-Based All-Quadrilateral Mesh Generation with Guaranteed Angle Bounds . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Xinghua Liang, Yongjie Zhang

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Q-TRAN: A New Approach to Transform Triangular Meshes into Quadrilateral Meshes Locally . . . . . . . . . . . . . . . . . . . Mohamed S. Ebeida, Kaan Karamete, Eric Mestreau, Saikat Dey

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Mesh Construction with Prescribed Properties Near Boundary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Boris Azarenok

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A Transfinite Meshing Approach for Body-In-White Analyses . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Kirk Beatty, Nilanjan Mukherjee

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Session 2A, Optimization Introducing the Target-Matrix Paradigm for Mesh Optimization via Node-Movement . . . . . . . . . . . . . . . . . . . . . . . . . . . Patrick Knupp An Analytical Framework for Quadrilateral Surface Mesh Improvement with an Underlying Triangulated Surface Definition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Kiran Shivanna, Nicole Grosland, Vincent Magnotta

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85

Efficient Solution of Elliptic Partial Differential Equations via Effective Combination of Mesh Quality Metrics, Preconditioners, and Sparse Linear Solvers . . . . . . . . . . . . . . . . . . 103 Jibum Kim, Shankar Prasad Sastry, Suzanne M. Shontz

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Contents

Virtual Control Volumes for Two-Dimensional Unstructured Elliptic Smoothing . . . . . . . . . . . . . . . . . . . . . . . . . . . . 121 Steve L. Karman Jr.

Session 2B, Surface Reconstruction and Repair Reconstructing High-Order Surfaces for Meshing . . . . . . . . . . . . 143 Xiangmin Jiao, Duo Wang Simple Method for Constructing NURBS Surfaces from Unorganized Points . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 161 Nallig Leal, Esmeide Leal, John William Branch Sealing Faceted Surfaces to Achieve Watertight CAD Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 177 Brandon M. Smith, Timothy J. Tautges, Paul P.H. Wilson A Metric for Automatic Hole Characterization . . . . . . . . . . . . . . 195 German Sanchez T., John W. Branch, Pedro Atencio

Session 3A, Hex Meshing Receding Front Method: A New Approach Applied to Generate Hexahedral Meshes of Outer Domains . . . . . . . . . . . . . 209 Xevi Roca, Eloi Ruiz-Gironés, Josep Sarrate EBMesh: An Embedded Boundary Meshing Tool . . . . . . . . . . . . 227 Hong-Jun Kim, Timothy J. Tautges Sharp Feature Preservation in Octree-Based Hexahedral Mesh Generation for CAD Assembly Models . . . . . . . . . . . . . . . . 243 Jin Qian, Yongjie Zhang Pen-Based User Interface for Geometric Decomposition for Hexahedral Mesh Generation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 263 Jean Hsiang-Chun Lu, Inho Song, William Roshan Quadros, Kenji Shimada

Session 4A, Research Notes Session 4B, Research Notes Session 5A, Professional Development Session 5B, Research Notes

Contents

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Session 6A, Adaptive Particle Systems for Adaptive, Isotropic Meshing of CAD Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 279 Jonathan R. Bronson, Joshua A. Levine, Ross T. Whitaker A Study on Using Hierarchical Basis Error Estimates in Anisotropic Mesh Adaptation for the Finite Element Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 297 Lennard Kamenski Bisection-Based Triangulations of Nested Hypercubic Meshes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 315 Kenneth Weiss, Leila De Floriani Optimizing Voronoi Diagrams for Polygonal Finite Element Computations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 335 Daniel Sieger, Pierre Alliez, Mario Botsch

Session 6B, Applications Creating Geometry and Mesh Models for Nuclear Reactor Core Geometries Using a Lattice Hierarchy-Based Approach . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 351 Timothy J. Tautges, Rajeev Jain Multi-tissue Mesh Generation for Brain Images . . . . . . . . . . . . . . 367 Yixun Liu, Panagiotis Foteinos, Andrey Chernikov, Nikos Chrisochoides A Toolkit for Parallel Overset Grid Assembly Targeting Large-Scale Moving Body Aerodynamic Simulations . . . . . . . . . 385 George Zagaris, Michael T. Campbell, Daniel J. Bodony, Eric Shaffer, Mark D. Brandyberry A Dimension-Independent Data Structure for Simplicial Complexes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 403 Leila De Floriani, Annie Hui, Daniele Panozzo, David Canino Index of Authors and Co-authors . . . . . . . . . . . . . . . . . . . . . . . . . . . . 421 Paper Reviewers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 423 Committee Members . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 425

Hexagon-Based All-Quadrilateral Mesh Generation with Guaranteed Angle Bounds Xinghua Liang and Yongjie Zhang Department of Mechanical Engineering, Carnegie Mellon University, Pittsburgh, PA 15213, USA Tel.: (412) 268-5332; Fax: (412) 268-3348 [email protected]

Summary. In this paper, we present a novel hexagon-based mesh generation method which creates all-quadrilateral (all-quad) meshes with guaranteed angle bounds and feature preservation for arbitrary planar domains. Given any planar curves, an adaptive hexagon-tree structure is constructed by using the curvature of the boundaries and narrow regions. Then a buffer zone and a hexagonal core mesh are created by removing elements outside or around the boundary. To guarantee the mesh quality, boundary edges of the core mesh are adjusted to improve their formed angles facing the boundary, and two layers of quad elements are inserted in the buffer zone. For any curve with sharp features, a corresponding smooth curve is firstly constructed and meshed, and then another layer of elements is inserted to match the smooth curve with the original one. It is proved that for any planar smooth curve all the element angles are within [60◦ -ε, 120◦ +ε] (ε ≤ 5◦ ). We also prove that the scaled Jacobians defined by two edge vectors are in the range of [sin(60◦ -ε), sin90◦ ], or [0.82, 1.0]. Furthermore, the same angle range can be guaranteed for curves with sharp features, with the exception of small angles in the input curve. In addition, all the elements in the final mesh are grouped into five types, and most elements only need a few flops to construct the stiffness matrix for finite element analysis. This will significantly reduce the computational time and the required memory during the stiffness matrix construction. We have applied our algorithm to a set of complicated geometries, including the Lake Superior map, and a three-component air foil with sharp features. Keywords: Quadrilateral mesh, guaranteed angle bound, hexagon-tree, sharp feature, narrow region.

Corresponding author.

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X. Liang and Y. Zhang

1 Introduction A fully automatic and robust mesh generator is always desired by the community of finite element analysis and computer graphics. Although provably good-quality triangular mesh generation methods have been well developed for planar and curved surfaces, many analysts prefer to use quadrilateral (quad) elements because of their superior performance in various applications. Only a few algorithms exist in literature addressing provable angle bounds for all-quad mesh generation [4, 1], but none of them guarantee a good angle range with both lower and upper bounds. As early as in 1991, Bishop [5] had theoretically proved that any planar n-gon can be meshed by O(n) quads with all the angles bounded between 45◦ -ε and 135◦+ε, where ε is a small perturbation. However, there was no algorithm that can generate such guaranteed-quality all-quad meshes until in 2009 a quadtree-based method [11] was firstly developed to construct all-quad meshes for any given point clouds or planar smooth curves, with a guaranteed angle range of [45◦ ε, 135◦ +ε] (ε ≤ 5◦ ). Inspired by [17, 18], we find that an adaptive hexagonal subdivision scheme can also be used for all-quad mesh generation, with the advantages of not creating any hanging nodes and providing better angle bounds as compared to using the quadtree. In this paper, we improve the algorithm described in [11] by using the hexagon-tree structure, which results in a better guaranteed angle range, [60◦ -ε, 120◦ +ε] (ε ≤ 5◦ ), for any complicated smooth curves (e.g., the Lake Superior map in Fig. 1). Moreover, this angle range can also be guaranteed even for geometries with sharp features, except small angles in the input geometry.

Fig. 1. The Lake Superior map. (a) An all-quad mesh with all angles ∈ [57◦ , 122◦ ]; and (b-c) Zoom-in pictures of (a).

Hexagon-Based All-Quad Mesh Generation

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For any planar smooth curves, six steps are developed to construct all-quad meshes with guaranteed angle bounds. Firstly, each curve is decomposed into a set of line segments based on its curvature. Secondly, a strongly balanced adaptive hexagon-tree structure is constructed without creating any hanging nodes. The element size is controlled by the boundary curvature and narrow regions. Thirdly, elements outside and around the boundary are removed to create a hexagonal core mesh and a buffer zone. Next, a group of templates are designed to adjust the boundary edges and improve their formed angles facing the boundary in the core mesh. Then angular bisectors are used to construct the first buffer layer. Finally, the points generated in the first layer are projected to the boundary and form the second buffer layer. It is proved that for any planar smooth curves, all the angles in the constructed mesh are within [60◦ -ε, 120◦+ε] (ε ≤ 5◦ ), and all the scaled Jacobians defined by two edge vectors are within [sin(60◦ -ε), sin90◦], or [0.82, 1.0]. If a given curve contains sharp features, the corresponding smooth curve is firstly constructed and guaranteed-quality quads are created through the above six steps. Then four cases are introduced to match the smooth curve to the original one while preserving the angle range, with the exception of small angles in the input curve. We have tested our algorithm with several complicated geometries, including the Lake Superior map, and a three-component air foil with sharp features. Our algorithm efficiently deals with curves in large-scale size, and generates meshes with guaranteed angle bounds while minimizing the number of elements. Additionally, we categorize all the quad elements in the generated meshes into five element types. In finite element analysis, the stiffness matrices of element types 1-4 can be precomputed which can significantly reduce the computational time, save memory, and provide more precise results. The remainder of this paper is organized as follows: Section 2 reviews the related work. Section 3 explains the detailed algorithm for guaranteed-quality mesh generation of smooth curves. Section 4 talks about sharp feature and boundary layer. Section 5 shows some results. Section 6 introduces five element types and explains the advantages of our algorithm in terms of stiffness matrix construction. Finally, Section 7 presents our conclusion and discussion.

2 Previous Work Quadrilateral Mesh Generation: For unstructured quad mesh generation [13], there are three direct methods: advancing front, domain decomposition, and grid-based. For practical purposes, the advancing front method generally gives high quality and reliable mesh generation. In [25], two triangles are firstly created using the traditional advancing front method and then combined to a single quad. A paving algorithm places a complete row of quads next to the front toward the interior [6]. The robustness of the paving algorithm is enhanced in [20] through creating individual quads rather than a complete row. However, the closure algorithms for the interior are still

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unstable. Heuristic decisions are made which usually generate elements with poor quality. Domain decomposition can be achieved by various techniques. Tam and Armstrong [19] introduced medial axis decomposition. Quadros et al. [15] introduced an algorithm that couples medial axis decomposition with an advancing front method. In general, these methods produce high quality meshes but they are not robust and may require a great deal of user interaction especially when the domain has non-manifold boundaries. For grid-based methods, Baehmann et al. [2] modified a balanced quadtree to generate a quad mesh for an arbitrary domain. Schneiders et al. [16] used an isomorphism technique to conform an adaptive octree structure to the object boundaries. Zhang et al. developed an octree-based isocontouring method to generate adaptive quadrilateral and hexahedral meshes for manifold and non-manifold domains [14, 21, 22, 24]. Grid-based algorithms are robust but often generate poor quality elements at the boundary. Quality Improvement: In finite element analysis, elements with small angles and angles close to 180◦ inside the mesh usually lead to ill-conditioned linear systems. Therefore, a post-procedure is crucial to improving the overall quality of the elements. Smoothing and clean-up methods are the two main categories of mesh improvement. Smoothing methods relocate vertices without changing the connectivity. Although simple and easy to implement, Laplacian smoothing is heuristic and unstable and sometimes inverts or degrades the local elements. Optimization-based smoothing methods were then proposed [7, 9], which relocate each node to the optimum location based on the local gradient of the surrounding element quality. Optimization-based methods provide much better mesh quality but they need excessive amounts of computations. For this reason such methods are usually combined with a Laplacian smoothing technique [8]. Surface feature preservation represents another challenging problem. Methods based on local curvature [3] and volume preserving geometric flows [23] were developed to identify and preserve the main surface features. Clean-up methods for quad meshes [10] were utilized to improve the node valence and topology. Pillowing [12] was used to ensure that any two adjacent quads share at most one edge. However, none of the above meshing and quality improvement techniques can guarantee any bounds for the element angles. Last year, we developed a quadtree-based method which creates all-quad meshes for any planar smooth curves with a guaranteed angle range of [45◦ -ε, 135◦ +ε] (ε ≤ 5◦ ). In this paper, we will present a novel hexagon-based approach that constructs allquad meshes with a better guaranteed angle range, [60◦ -ε, 120◦ +ε].

3 Guaranteed-Quality Meshing of Smooth Curves Given a planar domain Ω and closed smooth curves C represented by cubic splines, we aim to generate an all-quad mesh for the regions enclosed by C with a good guaranteed angle range. Six steps are designed as shown in


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Fig. 2, including (1) curve decomposition, (2) adaptive hexagon-tree construction, (3) buffer zone clearance, (4) template implementation, (5) first buffer layer construction, and (6) second buffer layer construction. 3.1

Curve Decomposition

Given any closed smooth curves C represented by cubic splines, we firstly decompose C into a set of piecewise-linear line segments, considering the curve local curvature. Two requirements need to be satisfied: the angle ε formed by two neighboring lines is ≤ 5◦ ; and, the approximation error δ of each line segment is less than a given threshold. As a result, we obtain non-uniform sampling points X, see Fig. 2(a).

(a)

(d)

(b)

(e)

(c)

(f)

Fig. 2. Flow chart of all-quad mesh generation. (a) Curve decomposition; (b) Adaptive hexagon-tree construction; (c) Buffer zone clearance; (d) Template implementation; (e-f) First and second buffer layer construction.

3.2

Adaptive Hexagon-Tree Construction

Instead of using the quadtree, here we construct an adaptive hexagon-tree because it does not induce any hanging nodes and provides a better angle range. The hexagon-tree construction is based on the sampling points X obtained from curve decomposition, see Fig. 2(b). Firstly, we define the size function as si = min(dij ), where dij is the distance between two points i and j (i, j ∈ X and i = j). Then a large hexagon Ω is generated to enclose all the points X, as shown in Fig. 3(a). Ω corresponds to the root of the hexagontree, marked as Level 0. Then each cell c in the hexagon-tree is recursively

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refined using two refinement templates in Fig. 3(b-c), unless the cell size is smaller than the size function of any point inside this cell. After that, each pair of semi-hexagons is grouped into one hexagon if possible, see Fig. 3(d). For each refinement or grouping, we update the corresponding cell levels. Similar to [18], hexagons are always marked in even levels and semi-hexagons are always marked in odd levels. We also apply the strongly balanced criterion to the hexagon-tree, which means the level difference around each cell c is at most one. For those cells not satisfying this criterion, we split and group them as shown in Fig. 3(e). As a final step, we split each hexagon to two semi-hexagons to obtain all-quad meshes. Local mesh refinement and mesh coarsening are supported in our algorithm by using refinement (Fig. 3(b-c)) and grouping (Fig. 3(d)). Narrow region is another important feature of the input geometry C. A region is defined as narrow if it contains no more than two elements in one direction after the adaptive hexagon-tree construction. The detected narrow region will be refined until all directions contain more than two elements, which guarantees the correct topology during mesh generation. 3.3

Buffer Zone Clearance

In order to conform the hexagon-tree mesh to the boundary, we delete elements outside or near the boundary curves so that later we have enough spaces to generate quads with provable good angles. We call such a procedure buffer zone clearance. Similar definitions in [11] are utilized here, see Fig. 4(a). A buffer zone is a zonal area that serves the purpose of keeping the quad mesh distant from boundaries. A boundary edge is an edge contained in only one element of the hexagonal core mesh, e.g., AB, BC and CD. One boundary edge consists of two boundary points, two neighboring boundary

(a)

(b)

(d)

(c)

(e)

Fig. 3. Operations used in adaptive hexagon-tree construction. (a) Hexagonal bounding box; (b-c) Two refinement templates; (d) Grouping; and (e) Strongly balanced tree construction. The numbers represent the corresponding cell levels. Hexagons are in even levels, and semi-hexagons are in odd levels.


(a)

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(b)

Fig. 4. Definitions and Criterion 3 in Buffer Zone Clearance. (a) The red curve is the boundary. A, B, C and D are boundary points. AB, BC and CD are boundary edges. α and β are boundary angles. ψ is a boundary edge angle; and (b) An example of applying Criterion 3.

edges form a boundary angle, and a boundary edge angle is the angle formed from the boundary edge to the boundary which cannot cross other edges. E.g., α and β are two boundary angles, and ψ is a boundary edge angle. Three criteria are applied to adaptive hexagon-tree construction, and then a hexagonal core mesh and a buffer zone are created, see Fig. 2(c). The buffer zone is filled with quads with all angles within [60◦ -ε, 120◦ +ε]. These three criteria are: 1) Any elements outside the regions to be meshed are deleted; 2) Any elements intersecting with the given curves C are deleted; 3) If the shortest distance from any vertex to the boundary is less than a pre-defined threshold εs , all elements sharing this vertex are deleted. Here we choose εs = 12 max(si ), where si is the size of the ith element sharing this vertex. Fig. 4(b) shows an example of applying Criterion 3. 3.4

Template Implementation

After buffer zone clearance, the boundary angles of the core mesh are among {60◦ , 120◦ , 180◦ , 240◦ , 300◦}, and the boundary edge angles are within [0◦ , 180◦]. In order to generate good-quality elements around the boundary, we design the following three operations to improve the boundary angles and boundary edge angles. Boundary angle improvement: For any boundary angle 60◦ , we delete the larger element sharing this boundary angle. For any boundary angle 300◦ , we delete the element containing this boundary angle. Finally, for any two neighboring boundary angles 240◦ -240◦, we delete the element containing them. Now the boundary angles are among {120◦, 180◦ , 240◦}. Self-intersection avoidance: In the following first buffer layer construction, we will use angular bisectors of the boundary angles to construct new elements. To avoid element self-intersection, we check all the possible combinations among the improved boundary angles {120◦ , 180◦ , 240◦ }. For a pair of boundary angles, there are 32 =9 combinations in total. However, 120◦ 240◦ , 180◦-180◦ and 180◦ -240◦ never cause self-intersections as shown in

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Fig 5, and 240◦ -240◦ has been removed in Operation 1 (boundary angle improvement). Considering symmetry and complementary of the boundary angles, only 120◦ -120◦ and 120◦ -180◦ are left. For each of them, we divide it into two groups since we will implement different templates on them. Therefore, there are a total of four combinations which may introduce self-intersections: 120◦ -120◦-120◦, 120◦ -120◦-(180◦ or 240◦ ), 120◦ -180◦-120◦, and 120◦ -180◦(180◦ or 240◦ ). The developed templates for these four combinations are shown in Fig. 6. In Fig. 6(d), if |CC | > 2|BC|, we add a parallelogram ABCD to avoid self-intersections. From Fig. 6, it is obvious to observe that all the newly generated edges are parallel to the existing ones. Therefore, the boundary angles and boundary edge angles are still among {120◦, 180◦ , 240◦} and [0◦ , 180◦], respectively. Note that the boundary angle pair 120◦-120◦ is eliminated, see Fig. 6(a-b).

(a) 120◦ -240◦

(b) 180◦ -180◦

(c) 180◦ -240◦

Fig. 5. Three combinations that never cause self-intersection in our algorithm. Dash lines are angular bisectors.

(a) 120◦ -120◦ -120◦

(b) 120◦ -120◦ -(180◦ or 240◦ )

(c) 120◦ -180◦ -120◦

(d) 120◦ -180◦ -(180◦ or 240◦ )

Fig. 6. Four templates to avoid self-intersections


(a) 120◦

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(b) 240◦

Fig. 7. Boundary edge angle improvement

Boundary edge angle improvement: To improve the boundary edge angles, we first check two boundary angles 120◦ and 240◦ . For each boundary angle 120◦ as shown in Fig. 7(a), if |AA | > |BB | or |CC | > |BB |, we add a parallelogram ABCD. Similarly, for each boundary angle 240◦ as shown in Fig. 7(b), if |AA | < |BB | or |CC | < |BB |, we add a parallelogram BCDE. We keep doing this until these two cases vanish at the boundary of the core mesh. After that, Operation 2 (self-intersection avoidance) may be applied again to remove self-intersections. Since the newly generated edges are parallel to the existing ones, the boundary angle at Point C is also 120◦ , see Fig. 7(a). Then we have |CC | ≥ |DD | & |CC | ≥ |EE |. The worst boundary edge angle happens when |CC | = |DD | or |CC | = |EE |, which results in a 60◦ angle for the boundary edge CD or CE. Similarly for the boundary angle 240◦ in Fig. 7(b), the worst boundary edge angle is also 60◦ . The boundary angle 180◦ performs as a straight line, which finally connects to either the boundary angle 120◦ or the boundary angle 240◦ , so its worst boundary edge angle is the same, 60◦ . Therefore after template implementation, the boundary edge angle range is improved from [0◦ , 180◦ ] to [0◦ , 60◦ ]. Remark: In summary, all the element angles in the hexagonal core mesh are either 60◦ or 120◦. After template implementation, all the boundary angles are among {120◦, 180◦ , 240◦ }, and all the boundary edge angles are within [0◦ , 60◦ ]. 3.5

First Buffer Layer Construction

To fill the buffer zone with guaranteed-quality quads, we firstly generate the first buffer layer. For each boundary point in the core mesh, we use angular bisectors to calculate a corresponding point inside the buffer zone called the first buffer point. Then, each pair of neighboring boundary points and their first buffer points construct a quad. All these new quads form the first buffer layer as shown in Fig. 2(e). Here is the algorithm to calculate the first buffer points. As shown in Fig. 8(a), the boundary point B is shared by two boundary edges AB and BC with corresponding boundary edge angles. We choose the larger boundary edge angle ψ, and draw its angular bisector. We also draw an angular

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bisector of the boundary angle at B. These two angular bisectors intersect at B which is the first buffer point of B. If the larger boundary edge angle ψ is small (e.g. ≤ 10◦ ), then we just project point B to the boundary, and choose the middle as B . Using all the boundary points and their first buffer points, we form the first buffer layer inside the buffer zone. Lemma 1: All the element angles in the first buffer layer are within [60◦ ε, 120◦+ε], where ε ≤ 5◦ . After the first buffer layer construction, the boundary edge angle range becomes [0◦ , 30◦ ]. Proof: To simplify the proof, we first assume that the boundary curve around each boundary edge is a straight line. In Fig. 8(a), A and B are the first buffer points corresponding to the boundary points A and B, the boundary angles at A and B are 2α and 2β, and the boundary edge angle of AB is ψ. Suppose A and B are on the angular bisector of ψ. After template implementation, the boundary edge angle ψ ∈ [0◦ , 60◦ ], and the boundary angle combinations of 120◦ -120◦ and 240◦-240◦ are eliminated. Therefore suppose α ≤ β, we have 2α ∈ {120◦ , 180◦} or α ∈ [60◦ , 90◦ ], 2β ∈ {180◦ , 240◦ } or β ∈ [90◦ , 120◦ ], and ϕ = ψ/2 ∈ [0◦ , 30◦ ]. Then, we can get θ = 180◦ − α − ϕ ∈ [60◦ , 120◦ ], and γ = ϕ + (180◦ − β) ∈ [60◦ , 120◦ ]. In summary, all the angles of the quad ABB A are within [60◦ , 120◦ ]. However, considering that the boundary curve has a small perturbation ε ≤ 5◦ , we relax the angle range to [60◦ -ε, 120◦+ε]. Finally, after the first buffer layer construction, the new boundary edge angle range becomes half of the original one, [0◦ , 30◦ ]. 3.6

Second Buffer Layer Construction

The second buffer layer construction is straightforward. As shown in Fig. 8(b), we project all the second buffer points to the boundary and obtain

(a)

(b)

Fig. 8. Two buffer layers. (a) First buffer layer; and (b) Second buffer layer


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the corresponding second buffer points. Then the second buffer layer is generated, see Fig. 2(f). We use Lemma 2 to prove its angle bounds. Lemma 2: All the element angles in the second buffer layer are within [60◦ ε, 120◦+ε], where ε ≤ 5◦ . Proof: We still assume that the boundary curve is a straight line around each boundary edge. In Fig. 8(b), A and B are the first buffer points, A and B are the second buffer points. Obviously, γ = θ = 90◦ . From Lemma 1, we have the boundary edge angle ψ ∈ [0◦ , 30◦ ]. Therefore, α = 90◦ − ψ ∈ [60◦ , 90◦ ] and β = 90◦ + ψ ∈ [90◦ , 120◦ ]. Considering the boundary curve may have a small perturbation ≤ 5◦ , the angle range is relaxed to [60◦ -ε, 120◦ +ε]. Remark: After applying the designed six steps in Fig. 2, the element angles in the core mesh, the first and the second buffer layers are all within [60◦ ε, 120◦+ε] (ε ≤ 5◦ ). As proven in [11], the scaled Jacobian defined by two edge vectors [21, 23] is a sine function of the element angle. For example, if we denote the scaled Jacobian as J, the two normalized edge vectors as e1 and e2 , and the angle formed by them as θ, then J = |e1 × e2 | = |e1 ||e2 |sinθ = sinθ because |e1 | = |e2 | = 1. Therefore our algorithm also guarantees that all the scaled Jacobians in the constructed mesh are in the range of [sin(60◦ ε), sin90◦] (ε ≤ 5◦ ), or [0.82, 1.0].

4 Sharp Feature and Boundary Layer If the input geometry contains sharp features, we need an extra buffer layer, called sharp feature layer, to preserve the sharp features and also guarantee the angle bounds. For each input curve C, we first generate a smooth curve C inside the region to be meshed with a distance d. To achieve this, we consider three cases as shown in Fig. 9: (1) smooth parts of the curve; (2) sharp feature with an angle ∈ (0◦ , 180◦ ]; and (3) sharp feature with an angle ∈ (180◦ , 360◦ ). For the smooth part, we simply move the curve inside with a distance d. Then, for each sharp angle ∈ (0◦ , 180◦), as shown in Fig. 9(b), we draw the angular bisector of this angle, and choose a point O such that |OC| = (r + d)/cos(ψ/2). Taking O as the circle center, we obtain an arc with the radius r between the two projection lines OA and OE. Here the radius r can be arbitrary. For each sharp angle ∈ (180◦ , 360◦ ), as shown in Fig. 9(c), we also draw the angular bisector of this angle, but choose a point O such that |OC| = (r − d)/cos(ψ/2). Taking O as the circle center, we again obtain an arc with the radius r between the two projection lines OA and OE . However, to guarantee the arc is inside the region to be meshed, we require |OC| < r, or r < d/(1 − cos(ψ/2)).

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(a)

(b)

(c)

Fig. 9. Smooth curve generation from the curves with sharp features. (a) Smooth parts; (b) Sharp angles ∈ (0◦ , 180◦ ); and (c) Sharp angles ∈ (180◦ , 360◦ ).

As soon as the smooth curve C is constructed, it can be meshed using the six steps in Section 3. Now we need to connect the resulting mesh to the original curve C. For each smooth part, as shown in Fig. 9(a), we directly project all the second buffer points to the curve C, and construct corresponding quads. For each part with a sharp angle, as shown in Fig. 9(b-c), we draw lines from the center O to each second buffer point. These lines intersect with the curve C at a set of boundary points. Then these boundary points and the second buffer points are used to form quads. By doing that, the curve C is completely meshed with guaranteed-quality quads, except elements near the sharp features. Then the final step is to improve these elements. Noticing that we enforce a second buffer point C to locate on the angular bisector of the sharp angle. We also have the following Lemma 3. Lemma 3: For smooth parts and sharp angles within [120◦, 180◦ ) or (180◦ , 240◦ ], the constructed quads in the sharp feature region are always ∈ [60◦ , 120◦]. Proof: For the smooth parts, all the element angles are obviously close to 90◦ , as shown in Fig. 9(a). For any sharp angle ∈ [120◦ , 180◦), quads containing the sharp point have the worst angle range, see Fig. 9(b). Thus we only need to prove the angle range of BCC B is good. Since CC is the angular bisector, α ∈ [60◦ , 90◦ ). B and C are on the arc, so β = γ = 90◦ + ϕ/2. Then θ = 180◦ − ϕ − α ∈ (90◦ − ϕ, 120◦ − ϕ]. From Section 3.1, ϕ = ε ≤ 5◦ . Therefore, all the four element angles of BCC B are within [60◦ , 120◦ ). Similarly, for any sharp angle ∈ (180◦ , 240◦], as shown in Fig. 9(c), we only need to prove the angle range of BCC B is good. β ∈ (90◦ , 120◦ ] because CC is the angular bisector. Since B and C are on the arc, α = θ = 90◦ − ϕ/2. Then γ = 180◦ − β + ϕ ∈ [60◦ + ϕ, 90◦ + ϕ], where ϕ = ε ≤ 5◦ . Therefore, all the four element angles of BCC B are within (60◦ , 120◦ ].


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For sharp angles within (0◦ , 120◦ ) or (240◦ , 360◦ ), the angle bounds of the constructed quads may not be good. Here we explain how to improve the angle bounds for these two cases. When the sharp angle α ∈ (0◦ , 120◦) as shown in Fig. 10(a), for each boundary point X (X = {A, B, C, G, H, I}), if ∠X XE > 120◦, we rotate the edge X X such that ∠X XE = 120◦ . Finally, we delete the edge E E and add two other edges E D and E F such that β = θ = 120◦ , γ = 120◦ − α when α ∈ (0◦ , 60◦ ], or γ = α, β = θ = 180◦ − α when α ∈ (60◦ , 120◦ ). In this way, we guarantee all the angles ∈ [60◦ , 120◦ ] except the sharp angle α in the input curve. With the sharp angle ∈ (240◦ , 360◦ ), we use a different method as shown in Fig. 10(b). Here the green region and the blue region are symmetric w.r.t. the angular bisector AA . The green region is generated by connecting O with each second buffer point, and the blue region is improved by inserting another layer of elements locally. In the blue region, Point B is removed, and two new points B and C are added such that β = β = β = 120◦ and θ = 60◦ . Before we reach a rectangle element, we continue to add new points D and E using the angular bisectors of ∠C C C and ∠D D D. If η or ζ < 60◦ , we adjust D or E to make it equal to 60◦ . Then the edge F F is rotated to have ∠F F E = 120◦ . Finally, two new points F and G are added to split the original quad EE F F to three new quads, where F F is an angular bisector, E F //EF and F G //F F . Then we use Lemma 4 to prove the angle bounds of quad elements in these two cases.

(a)

(b)

Fig. 10. Quality improvement of elements around sharp features. (a) Sharp angle ∈ (0◦ , 120◦ ); and (b) Sharp angle ∈ (240◦ , 360◦ ).

14


Lemma 4: For sharp angles within (0◦ , 120◦ ) or (240◦ , 360◦), all the element angles in the sharp feature layer are within [60◦ -ε, 120◦+ε] (ε ≤ 5◦ ), except small sharp angles in the given curve. Proof: For any sharp angle α ∈ (0◦ , 120◦ ) as shown in Fig. 10(a), it is obvious that all the elements, except the element containing the sharp corner E, have the angle range ∈ [60◦ , 120◦ ] because we enforced the maximum element angle to be ≤ 120◦ . The only exception is the sharp angle when it is less than 60◦ . Since this bad angle is required by the input geometry, we just keep it. For any sharp angle ∈ (240◦ , 360◦ ) as shown in Fig. 10(b), we firstly check the elements around the sharp corner A. We can easily obtain β = β = β = 120◦, γ = θ = 90◦ − ϕ/2 and θ = α = 90◦ − ψ/2. Then α = 360◦ − γ − θ − β = 60◦ + ϕ, and γ = 360◦ − θ − α − β = 60◦ + ψ. Moreover, α = ∠A AC − α = (120◦ − α , 180◦ − α ) ∈ (60◦ − ϕ, 120◦ − ϕ), γ = 360◦ − α − β − θ ∈ (60◦ + ψ, 120◦ + ψ], and θ = 60◦ . From Section 3.1, ϕ = ψ = ε ≤ 5◦ ; therefore, the angle range of the three elements sharing Point B is [60◦ − ε, 120◦ + ε]. It is obvious that the angle range of the other elements is [60◦ , 120◦ ]. Therefore, all the angles generated in this case are within [60◦ -ε, 120◦ +ε]. Remark: In summary, all the element angles in the sharp feature layer are within [60◦ -ε, 120◦+ε], where ε ≤ 5◦ , except small sharp angles in the given curve. As an example, we use the Delta Wing to test our algorithm for sharp feature preservation. The meshing results are shown in Figs. 11 & 12. All the element angles are of provable good quality. The lower angle bound in Fig. 11 is required by the small sharp angle (31.75◦ ) in the given curve. Additionally, multiple boundary layers are required in Computer Fluid Dynamics (CFD) simulations. Our algorithm can conveniently construct boundary layers by splitting the elements in the second buffer layer or the sharp feature layer, see Fig. 12. Only three templates are needed, as shown in Fig. 12(e-g). The black points are boundary points, the red points are new points, and the blue quads are elements constructed for the boundary layer. Obviously, all the angles are preserved in Fig. 12(f-g), but an angle is split in Fig. 12(e), resulting in a reduction of the angle range to [30◦ − ε, 150◦ + ε]. Fig. 12(e) is needed only if there are sharp features > 240◦ , and the angle range will be reduced only once no matter how many boundary layers are constructed.


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Fig. 11. An interior mesh of Delta wing with sharp features. (a) All-quad mesh with all angles ∈ [31.75◦ , 120◦ ]. The lower bound corresponds to the small angle in the geometry; and (b-d) Zoom-in pictures of (a).

Fig. 12. An exterior mesh of Delta wing with sharp features. (a) All-quad mesh with all angles ∈ [60◦ , 120◦ ] before boundary layer generation, and [42◦ , 147◦ ] after generating a boundary layer; and (b-d) Zoom-in pictures of (a); (e-g) Three boundary layer templates with one, two and three boundary points. Black points are boundary points, and red points are new points.

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Fig. 13. The Mouse mesh. (a) An all-quad mesh with all angles ∈ [60◦ , 120◦ ]; and (b-e) Zoom-in pictures of (a).

Fig. 14. A three-component air foil with sharp features. (a) An all-quad mesh with all angles ∈ [56◦ , 120◦ ]; and (b-f) Zoom-in pictures of (a).

5 Results We have applied our algorithm to three models: the Mouse in Fig. 13, the Lake Superior map in Fig. 1, and a three-component air foil with sharp features in


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Table 1. Mesh statistics of three testing models Dataset

Mouse

Sampling Point 971 Mesh Size (vertex, element) (24702, 22554) Angle Range [min, max] [60◦ , 120◦ ] Scaled Jacobian [worst, best] [0.86, 1.0] Aspect Ratio [best, worst] [1.0, 31.6] Time (s) 26

Lake Superior

Air Foil

3921 (63927, 59124) [57◦ , 122◦ ] [0.83, 1.0] [1.0, 25.8] 213

329 (5695, 5331) [56◦ , 120◦ ] [0.83, 1.0] [1.0, 308.0] 4

Fig. 14. Our results were computed on a PC equipped with a 2.4 GHz Intel Q6600 CPU and 4GB of Memory. Statistics of these three meshes are given in Table 1. The sampling points are points obtained from the curve decomposition. The aspect ratio is defined as the longest edge over the shortest edge in a single element. We can observe that our algorithm generates meshes with all the angles ∈ [60◦ -ε, 120◦+ε] (ε ≤ 5◦ ) and the scaled Jacobians ∈ [sin(60◦ -ε), sin90◦ ], or [0.82, 1.0]. Moreover, we can conclude from Table 1 that the run time of our algorithm correlates with the geometry complexity, e.g., the Lake Superior example takes more time than others due to its complicated boundaries. Among the six steps in our algorithm, adaptive hexagon-tree construction and buffer zone clearance are the two most time-consuming steps, since they need to compare each element with its surrounding sampling points.

6 Five Element Types We can observe that most elements in the resulting meshes have similar shapes. To take advantage of that, we categorize all the elements into five types as listed in Fig. 15. The percentage of each element type in the three meshes is listed in Table 2. For element types 1-4, we can use the precomputed stiffness matrix templates to reduce the computational time and the required memory for stiffness matrix construction in finite element analysis. Type 1 is an isosceles trapezoid with a 60◦ angle, where the top and bottom edge lengths can be multiples of a, see Fig. 15(a). When n = 1, it is the dominant element type, which occupies about 50% in the whole mesh. This is

(a) Type 1

(b) Type 2

(c) Type 3

Fig. 15. Five element types

(d) Type 4

(e) Type 5

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X. Liang and Y. Zhang Table 2. Element percentage and flops for stiffness matrix construction

Element type Mouse Lake Superior Air Foil Type Type Type Type Type

1 2 3 4 5

56.5% 5.5% 8.9% 24.3% 4.8%

62.8% 4.9% 7.7% 20.0% 4.6%

63.0% 2.6% 3.8% 22.9% 7.7%

Flops (1st )

Flops

68 16 54 16 347 347 274 274 5,476∗ /12,376∗ ∗ 5,476∗ /12,376∗ ∗

Note: The 5th , 6th columns are the flops for the first computation and the rest, respectively. Types 1-4 are calculated analytically using the corresponding equations in Appendix; ∗ Using 4 Gaussian integration points;

∗∗ Using 9 Gaussian integration points.

because during the adaptive hexagon-tree construction, this kind of elements is widely used as the transitional element, and each hexagonal element is finally split into two of this type. For other elements with n = 2, 3, 4, they are only generated during template implementation. Type 2 is a parallelogram with a 60◦ angle, where the top and bottom edge lengths are also multiples of a, see Fig. 15(b). This type is generated during template implementation, and only occupies a small portion of the whole mesh. We can firstly calculate the element stiffness matrices for all the shapes of Types 1-2 and store them as templates. When forming the global stiffness matrix, for each element of Types 1-2, we can directly get the values from corresponding stiffness matrix templates instead of recalculating them using the general method. Therefore, we can significantly save the computational time and memory. Type 3 is a trapezoid with a 60◦ angle and a 120◦ angle. This type is mainly generated during the first buffer layer construction where we use angular bisectors for the boundary angles 120◦ and 240◦ . Type 4 is a trapezoid with two 90◦ angles. Like Type 3, this type is mainly induced by the projection during the second buffer layer construction. Different from Types 1-2, the stiffness matrix of Types 3-4 cannot be stored with certain calculated values. However, the formula used in calculating the stiffness matrix can be derived analytically and further simplified. Type 5 is a general element. It mainly exists in the first buffer layer, but also has a small amount in the second buffer layer. Since this element type is irregular, we can only use the general way to calculate the stiffness matrix for each element, which takes lots of time. However, as shown in Table 2, elements of Type 5 generally occupy a small percentage (< 10%) in the final meshes. Remark: As an instance, we treat all the five element types as 4-node bilinear quad elements in the finite element analysis for a Laplace problem, and count the flops used to form the element stiffness matrix for each element type. The results are listed in Table 2, and the derived expressions for each type of stiffness matrix can be found in the appendix. Compared to Type 5 which needs at least 5476 flops for each element, Types 1 and 2 only need 16 flops to


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generate an element stiffness matrix since the values of their stiffness matrices can be saved as templates, and Types 3 and 4 only need 347 and 274 flops for each element, respectively. We can conclude that using Types 1-4 can save a great amount of computational flops (> 80%) in constructing the stiffness matrix. Usually, Gaussian integration is utilized to calculate the stiffness matrix of a general element, it is inevitable to bring numerical errors. However, since we have analytical expressions of the stiffness matrices for Types 1-4, the resulting stiffness matrices are precise. This is another advantage of our algorithm. Finally, besides speeding up the computation of forming the stiffness matrix, our algorithm can also save lots of memory. As discussed above, the element stiffness matrix of Types 1 and 2 can be stored as templates after the first time calculation, we do not need to store the stiffness matrix for each element. From Table 2, we can see that Types 1-2 elements occupy at least 50% of the final mesh, which means we can at least save half memory during the stiffness matrix construction.

7 Conclusions and Discussion In this paper, we present a novel hexagon-based meshing algorithm, which creates all-quad meshes with provable angle bounds for arbitrary planar curves. It is proved that all the elements are quads with angles ∈ [60◦ ε, 120◦+ε] (ε ≤ 5◦ ) and the scaled Jacobians ∈ [sin(60◦ -ε), sin90◦] or [0.82, 1.0], except small angles in the input geometry. By categorizing elements into five types, we can significantly reduce the computational time, save memory, and provide more precise results during stiffness matrix construction. We have applied our algorithm to a set of complicated geometries, including the Lake Superior map and an air foil with sharp features. Our algorithm provides a theoretical basis of guaranteed-quality all-quad mesh generation. To guarantee the angle range, sometimes the algorithm needs aggressive refinement required by narrow regions, boundary curvatures and sharp features. This restriction will force more elements in the final result, which may not be practical for some applications. Moreover, we did not apply any quality improvement techniques to our generated meshes because it would make the guarantee of angle bounds unpredictable. For practical purposes, the application of smoothing and optimization would further improve the overall mesh quality. Finally, since we use the hexagon-tree in our algorithm, there is no 90◦ angles in the core mesh, and only 50 ∼ 60% vertices are regular (with valence number 4). However, most elements near the boundary are close to rectangles, which is suitable for CFD simulations. Compared to the quadtree method in [11], the hexagon-tree approach results in better angle bounds and no hanging nodes. For the same geometry, these two algorithms generate meshes with similar mesh sizes. However, the hexagon-tree cannot generate square elements in the core mesh while the

20


quadtree can. During template implementation, the templates based on the hexagon-tree are much simpler than those in the quadtree method, because the combination of the boundary angles {n × 60◦ |n = 1, 2, ..., 5} is less than that of {n × 45◦ |n = 1, 2, ..., 7}. Moreover, we use a totally different method to preserve sharp features in this paper, the angle bounds are strictly proven. In the current algorithm, the generated interior and exterior meshes do not conform to each other. In the future, we will extend our algorithm to interior and exterior mesh generation with conformal boundaries and guaranteed angle bounds.

Acknowledgements We would like to thank Deborah Gruner for proofreading the paper. This research was supported in part by ONR grant N00014-08-1-0653, which is gratefully acknowledged.

Appendix The following stiffness matrices are derived based on a 4-node bilinear quad element for the Laplace equation. All of them are symmetric. The stiffness matrix for Types 1-4 are denoted as M1 , M2 , M3 and M4 , respectively. ⎡ ⎤ m11 sym ⎢ ⎥ 1 ⎢ m21 m22 ⎥, where: M1 = 3√ ⎦ 3 ⎣ m31 m32 m33 m41 m42 m43 m44 = m22 = 3 − 3n + 5n2 log((1 + n)/n), = [−3 + 12n − 10n2 log((1 + n)/n)]/2, = m42 = −3 − 6n + 5n(1 + n)log((1 + n)/n), = m32 = [3 + 6n − 10n(1 + n)log((1 + n)/n)]/2, = m44 = −6 − 3n + 5(1 + n)2 log((1 + n)/n), = [15 + 12n − 10(1 + n)2 log((1 + n)/n)]/2 (n = 1, 2, 3, 4). ⎡ 4−3n+4n2 ⎤ √ sym 6 3n 2 ⎢ 1−2n2 ⎥ 4+3n+4n √ √ ⎢ ⎥ 3 3n 6 3n ⎥ (n = 1, 2, 3, 4) M2 = ⎢ 2 2 2 ⎢ −2+n ⎥ 1−2n 4−3n+4n √ √ ⎣ 3√3n ⎦ 3 3n 6 3n m11 m21 m31 m41 m33 m43

2 1−2n √ 3 3n

M3 =

√ 1 3 3b(a−c)3

2 −2−3n−2n √ 6 3n

⎡

2 −2+n √ 3 3n

2 4+3n+4n √ 6 3n

⎤ m11 sym ⎢ m21 m22 ⎥ ⎢ ⎥, where: ⎣ m31 m32 m33 ⎦ m41 m42 m43 m44

m11 = 3(a − c)[a2 (a − b − 3c) − c(3b2 + 3bc + c2 ) + a(b2 + 3bc + 3c2 )] − c2 Δ, m21 = 12 [3b(−a + 2b + c)(a2 − c2 ) + 2acΔ],


21

m22 = −3(a − c)[a2 (a − 2b − 3c) + (3a − c)(b2 + bc + c2 )] − a2 Δ, m31 = −3(a − c)[a2 (a − b − 2c) + b2 c + a(b2 + bc + c2 )] − acΔ, m32 = 12 {3(a − c)(a2 (2a − 3b − 4c) − bc(2b + c) + 2a(3b2 + 2bc + c2 )] + a2 Δ}, m33 = 3b(a − c)[a2 + bc − a(3b + c)] − a2 Δ, m41 = 12 {3(a − c)[a2 (b + 2c) − 2a(b2 + 2bc + 2c2) + c(6b2 + 3bc + 2c2 )] + 2c2 Δ}, m42 = −3(a − c)[a2 c + a(b − 2c)(b + c) + c(b2 + bc + c2 )] − acΔ, m43 = 12 [3b(−a + 2b + c)(a2 − c2 ) + 2acΔ], m44 = 3b(a − c)[a(b + c) − c(3b + c)] − c2 Δ, Δ = (2a2 − 3ab + 6b2 − 4ac + 3bc + 2c2 )log(c/a). ⎡ ⎤ m11 sym ⎢ m21 m22 ⎥ 1 ⎢ ⎥, where: M4 = 6b(a−c) 3 ⎣ m31 m32 m33 ⎦ m41 m42 m43 m44 m11 = 3(a − c)[(a − 3c)(a2 + b2 ) + c2 (3a − c)] − 2c2 Δ, m21 = 3b2 (a2 − c2 ) + 2acΔ, m22 = 3(a − c)[a2 (a − 3c) + (3a − c)(b2 + c2 )] + 2a2 Δ, m31 = −3(a − c)[a2 (a − 2c) + b2 c + a(b2 + c2 )] − 2acΔ, m32 = 3(a − c)[a2 (a − 2c) − b2 c + a(3b2 + c2 )] + 2a2 Δ, m33 = −3b2 (3a2 − 4ac + c2 ) − 2a2 Δ, m41 = 3(a − c)[c(a2 + 3b2 + c2 ) − a(b2 + 2c2 )] + 2acΔ, m42 = −3(a − c)[ab2 + (a2 + b2 )c − c2 (2a − c)] − 2acΔ, m43 = 3b2 (a2 + c2 ) + 2acΔ, m44 = 3b2 (a − 3c)(a − c) − 2c2 Δ, Δ = [3b2 + (a − c)2 ]log(c/a).

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Q-TRAN: A New Approach to Transform Triangular Meshes into Quadrilateral Meshes Locally Mohamed S. Ebeida1 , Kaan Karamete2, Eric Mestreau2 , and Saikat Dey2 1 2

Org. 1414, Applied Math and Applications, Sandia National Laboratories [email protected] Code 7130, Physical Acoustics Branch, Naval Research Laboratories [email protected], [email protected], [email protected]

Summary. Q-Tran is a new indirect algorithm to transform triangular tessellation of bounded three-dimensional surfaces into all-quadrilateral meshes. The proposed method is simple, fast and produces quadrilaterals with provablygood quality and hence it does not require a smoothing post-processing step. The method is capable of identifying and recovering structured regions in the input tessellation. The number of generated quadrilaterals tends to be almost the same as the number of the triangles in the input tessellation. Q-Tran preserves the vertices of the input tessellation and hence the geometry is preserved even for highly curved surfaces. Several examples of Q-Tran are presented to demonstrate the efficiency of the proposed method. Keywords: Mesh generation, all-quadrilateral, indirect methods, curved surfaces.

1 Introduction Automatic mesh generation is one of the main procedures in Finite Element Analysis (FEA) as well as many other fields. Hence, a robust automatic mesh generator becomes an indispensable tool in such applications. Nowadays, the generation of triangular meshes on 3D surfaces by either the Delaunay triangulation [1, 2, 3, 4] or the advancing front technique [5, 6, 7, 8] are considered matured. However, in many applications, quadrilaterals are preferable to triangular meshes due to their superior performance for various applications such as sheet metal forming and crash simulations. A smaller set of literature exists for quadrilateral meshing. In the last decade, different approaches have been proposed in the area of unstructured all-quadrilateral mesh generation. Virtually, all existing quadrilateral meshing algorithms can be grouped into two main categories, namely, the direct and the indirect approaches.

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In the indirect methods, quadrilaterals are formed based on a triangular tessellation called a background mesh. The simplest approach under this category is to divide each triangular face into three quadrilaterals by inserting a vertex into at the center of each face as well as splitting all of its three edges. This method is fast, robust and preserves the angle bounds of the background mesh. However it triples the number of the faces of the input tessellation and introduce a large number of irregular vertices, which are usually not favored in many applications. An irregular vertex is an internal vertex with valence number other than four. An alternative method is to combine adjacent pairs of triangles to form a single quadrilateral [9, 10]. The drawback of this method that in order to generate a mesh with good quality, many triangular faces in the initial tessellation might be left in the output mesh. To minimize the number of these remaining triangles, several heuristic procedures suggested to control the order in which triangles are combined, and in some cases split during the conversion[11, 12, 13]. Velho proposed a local transformation method that utilizes clustering of triangular pairs and produce all-quadrilateral meshes[14]. Owen et al. [15] developed an advancing front algorithm, Q-MORPH, based on a background mesh and the techniques used in the paving algorithm [16]. Q-MORPH generates an all-quadrilateral mesh with well aligned rows of elements and a fewer number of irregular vertices. However, this method needs an initial front to start the quadrilateral conversion and the quality of the generated quadrilaterals depends heavily on that front. This might be a problem for applying Q-MORPH to closed 3D triangular tessellation that is not associated with sharp features. Miyazaki et al. proposed a method to generate a suitable initial front in these cases[17]. However, the Q-Morph transformation is still not local, since each triangular face cannot be transformed unless it becomes adjacent to the advancing front. Moreover, Q-Morph does not preserve the vertices of the input tessellation which may alter the associated geometry for a curved input surface. On the other hand, in the direct methods, quadrilaterals are directly created over the problem domain. These methods are generally classified into one of three main categories. The first depends on some form of domain decomposition [18, 19, 20]. The second is based on the advancing front method [21, 16, 22]. The third extracts a quadrilateral surface using a Cartesian grid [23] or packing of circles [24]. In general, direct methods are slower and less robust compared to indirect methods. The main objective of this study is to present a new indirect quadrilateral mesh generation scheme over 3D curved surface while preserving the vertices of the input triangular tessellation. The proposed method is partially similar to the technique proposed by Velho [14]; however, Q-Tran produces fewer quadrilaterals and fewer irregular vertices in general.

Q-TRAN: Transform Triangular Meshes into Quadrilateral Meshes

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2 Outline of the Q-Tran Algorithm The Q-Tran algorithm is briefly outlined in the following steps: 1. Initial edge classification: Each edge in the input triangular tessellation is classified as one of the following six types (see Figure 1) • A boundary edge, BE, is either an edge adjacent to a single triangular face or associated with a sharp feature or specified by the user. • An anisotropic diagonal edge, ADE, is a non-boundary edge adjacent to two triangular faces such that its length is greater than the length of the remaining four edges and at least one of the two triangle is anisotropic. A triangular face is considered anisotropic if its aspect ratio exceeds 3.0 otherwise it is considered to be isotropic. • An isotropic diagonal edge, IDE, is an edge that does not fall into any of the two categories mentioned above and adjacent to two isotropic triangular faces such that its length is greater than the length of the remaining four edges and none of these edges is a boundary edge. • A corner edge, CE, is an edge that does not fall into any of the three categories mentioned above and adjacent to two isotropic triangular faces with at least one of these two has two boundary edges. • A regular edge, RE, is any other edge that does not fall into any of the four categories mentioned above. • A special regular edge, SRE, is a regular edge with two adjacent triangles that has two aligned boundary edges. 2. Reclassification of regular and corner edges: • A regular edge that shares a vertex with an anisotropic diagonal edge is reclassified as a boundary edge. • A special regular edge is reclassified as a boundary edge. • A corner edge is reclassified into an anisotropic diagonal edge if its two adjacent faces has four boundary edges. • A corner or a regular edge is reclassified into a boundary edge if it would result in violating the angle bounds of the input tessellation during the conversion process. 3. Check for optimal solution: If the number of diagonal edges is half the number of the faces in the input tessellation, an optimal solution exists. Retrieve it by merging the two adjacent triangles to each diagonal edge into a single quad and exit, otherwise proceed to the next step. 4. Creation of new vertices: • Create a vertex, edge vertex, at the center of non-regular edge. • Create a vertex, regular face vertex, at the center of each triangular face with three regular edges or with two boundary edges. • Create a vertex, boundary face vertex, at the center of each triangular face with three boundary edges. 5. Quadrilateral formation: • For each boundary face vertex, generate three quadrilaterals by splitting the associated boundary face into three quadrilaterals.

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(a) ADE

(b) ADE

(d) CE

(c) IDE

(e) SRE

Fig. 1. Classification of the edges of the input tessellation

•

For each anisotropic diagonal edge, transform the adjacent two triangles into four quadrilaterals. • For each regular edge, generate a quadrilateral using the two vertices of that edge and the two created vertices in the adjacent faces. • For each corner edge generate the quadrilaterals as shown in Figure 3(b). 6. Topology clean-up: • Minimize the number of irregular vertices using face collapse of the quadrilaterals with two opposite tri-valence vertices that are created during the Q-Tran algorithm. Do not execute this step if the quality will deteriorate due this heuristic operation. This operation is illustrated in Figure 2.

3 Quality of the Generated Mesh In this section, we study the quality of the quadrilaterals generated during the Q-Tran algorithm in terms of preserving the angle bounds of the input tessellation. As mentioned in the previous section, there are six cases associated with the generation of the quadrilaterals within Q-Tran. The first three types, Group A, are illustrated in Figure 3(a). In these cases, the angle bounds of the input tessellation are preserved automatically. The proof is trivial and follows the simple fact: x+y ≤ max(x, y) (1) 2 where x and y in Equation 1 represents two adjacent angles in the triangular tessellation while x+y represents the corresponding angle in the quadrilateral 2 tessellation. min(x, y) ≤


(a) Input

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(b) Output

Fig. 2. Topology clean-up using face collapse to reduce the number of irregular vertices. A quadrilateral face is collapsed converting two irregular vertices into a regular one. The triangular tessellation is shown using dotted lines in both figures.

For the second group, the quality of the input tessellation is preserved by construction, where an edge from Group B would be reclassified into an edge from Group A if it causes a violation to the angle bounds of the input tessellation. However, we note that throughout the test problems that we have used in this paper we have never encountered this case. Further investigations are required to prove that this case can ever exists.

4 Analysis of Q-Tran Performance In order to test the performance of Q-Tran, we generated a sequence of triangular tessellations covering a planar hexagon. A triangular tessellation, Mi , i = 2, 3, ..., 7, is obtained by isotropic refinement of the previous tessellation, Mi−1 , where each triangular face is split into four triangular faces. Q-Tran is then utilized to convert each tessellation in this sequence into an all-quadrilateral mesh. Note that all the internal edges in any tessellation here are regular edges, which means that random Topology clean-up will be applied everywhere in the mesh at the end of the algorithm. This should represent a worst case scenario for Q-Tran with regard to the time of execution, given that the algorithm should have the same performance for planar and curved surfaces. The results of this test are summarized in Table 1. As we notice from these results, random clean-up operations might increase the relative number of the generated quadrilaterals in some cases to be as much as twice the number of the faces in the input tessellation. All of the tests performed in this section were performed using a 32-bit operating system and 2.0 GHz (T7250) processor with 2.0 GB of memory. Q-Tran was implemented in a generic function so the performance might vary based on the utilized datastructure.

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(a) Group A

(b) Group B Fig. 3. Quadrilateral formation in Q-Tran. The quality of the quadrilaterals generated in Group A is guaranteed automatically while the quality of the quadrilaterals generated in Group B is guaranteed by construction. Table 1. Performance of Q-Tran for the hexagon problem Model Name Hexagon-3 Hexagon-4 Hexagon-5 Hexagon-6 Hexagon-7 Hexagon-8 Hexagon-9

N umT 96 384 1,536 6,144 24,576 98,304 393,219

N umQ N umQ /N umT Time (seconds) Rate (N umT /min.) 158 1.65 0.451 12,700 668 1.74 0.486 47,407 2,920 1.90 0.557 165,457 11,856 1.93 1.359 271,258 47,458 1.93 4.206 350,584 104,976 1.07 13.219 446,194 405,914 1.03 54.880 429,904

The performance of Q-Tran is mainly affected by the relative number different edges categories in the input tessellation. For example, if all the edges are classified as boundary edges, then the number of the faces of the input tessellation will be quadrupled. On the contrary, if the number of diagonal edges in the mesh is half the number of the faces of the input tessellation, then the latter will be reduced by half in the output mesh. For most cases, where the relative ratio of internal to boundary edges is relatively large, Q-Tran tends to


(a)

(b)

(c)

(d)

(e)

(f)

(g)

(h)

29

Fig. 4. The first four meshes in a sequence utilized to test the performance of Q-Tran. The upper figures show the triangular tessellation of a Hexagon while the lower figures show the corresponding quadrilateral meshes. Table 2. Performance of Q-Tran for various problems Model Name hook ref-plane siklone-1 siklone-2 siklone-3 topology-1 topology-2 vase-1 vase-2 vase-3 tori-1 tori-2 tori-3

N umT 1,014 3,406 29,428 117,716 470,864 35,000 140,0000 19,460 77,844 311,360 12,000 48,000 192,016

N umQ N umQ /N umT Time (seconds) Rate (N umT /min.) 1,212 1.20 3.350 18,161 3,776 1.11 0.933 219,035 31,749 1.08 6.809 259,315 123,892 1.05 18.855 374,593 487,156 1.03 70.674 399,748 41,993 1.20 7.244 289,895 171,785 1.23 23.989 350,160 27,508 1.41 6.28 185,923 107,546 1.38 14.346 325,570 419,060 1.35 53.685 347,985 12,129 1.01 2.769 260,021 49,788 1.04 7.412 388,559 197,252 1.03 28.422 405,353

almost preserve the number of the faces in the input tessellation. With regard to the required computational time, Q-Tran tends to be much faster as the number of diagonal edges relatively increase, since fewer clean-up operations would be required. As shown in the next section we have applied Q-Tran to a wide range of different types meshes. Some of these meshes have some structured patches with anisotropic faces, others are decomposed of highly curved surfaces with no boundary edges at all, ... etc. The performance of Q-Tran for these meshes is summarized in Table 2. To have an overall average estimation of the Q-Tran performance, the results of all the test performed in this section are illustrated in Figure 5. These results shows that the number of the quadrilaterals in most cases tends to be almost the same as the number of the triangles in the input. This may increase

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(a) Input

(b) Output

Fig. 5. Q-Tran performance. The relation between the number of the faces in the input and the output meshes is illustrated in the right figure while the execution time for the various test problems is illustrated in the left figure. The inclined line in each of the two figures is used as a reference.

in the worst cases to be as much as twice that number. On average, Q-Tran seems to convert a triangular tessellation into an all-quadrilaterals meshes at an average rate of 375,000 triangles per minute using a 2.0 GHz processor. Again, this is considered a rough estimation since we utilized a generic implementation These results might be further improved if a datastructure-specific implementation is to be utilized because Q-Tran depends to a large extent on the speed of adjacency queries.

5 Example Problems Six examples, shown in Figures 6-11, demonstrates various features of the Q-Tran algorithm. The first example shown in Figure 6 demonstrates the capability of Q-Tran to handle highly curved surfaces with no boundary edges. This example illustrates the ability of Q-Tran to detect and preserve structured regions in the input tessellation. Figure 7, on the other hand, shows the quality of the quadrilaterals generated for a Cad model with non-manifold boundary edges. This example includes almost all the types of edge classifications within Q-Tran. The ability to handle a triangular mesh with some anisotropic faces is illustrated in Figures 8. As this figure show, the final quadrilateral mesh preserves the directionality of the stretched faces. Figures 9 and 10 show that the vertices distribution in the final quadrilateral mesh is almost the same as in the input tessellation. This is demonstrated using a highly curved surface as well as a simple planar surface with variable density distribution. The final example, in Figure 11, demonstrates that efficiency of recovering the structured regions from the input tessellation. As illustrated in that figure Q-Tran was capable of recovering all of the structured regions in this example leaving behind a very small number of irregular vertices.


(a) Input: 35,000 triangles

(b) Output: 41,993 quadrilaterals

Fig. 6. Topology Model (execution time = 7.244 seconds)



Fig. 7. Hook Model (execution time = 3.35 seconds)



Fig. 8. Vase Model (execution time = 6.28 seconds)

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Fig. 9. Siklone Model (execution time = 6.809 seconds)



Fig. 10. Refined-Plane Model (execution time = 0.933 seconds)



Fig. 11. Tori Model (execution time = 2.769 seconds)


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6 Conclusion The Q-Tran algorithm is an indirect quadrilateral meshing algorithm that utilizes edge classification to transform triangles into quadrilaterals locally. It generates an all-quadrilateral mesh with provably-good quality. The resulting quadrilaterals, in general, follow the boundaries of the domain. The Q-Tran algorithm is capable of detecting and recovering the structured regions in the input tessellation. It can handle isotropic and anisotropic cases with almost the same efficiency. Compared to the Q-Morph algorithm, Q-Tran can be implemented in parallel, preserves the vertices of the input tessellation and does not require an initial front. Moreover, the quality of the generated quadrilaterals is guaranteed, hence, no smoothing is required as a post processing step. Improvements include minimizing of the irregular vertices as well as controlling the directionality in the isotropic structured regions.

Acknowledgments This research was performed under contract to US Naval Research Laboratory as part of the Computational Research and Engineering Acquisition Tools and Environments (CREATE) program of the DoD High Performance Computer Modernization Program. We would like to thank Dave Morris, and Frank Su for the “Topology” and the “Vase” models. These models are courtesy of 3dvia (http://www.3dvia.com). We would also like to thank Herbert Edelsbrunner, (http://www.cs.duke.edu/~edels/Tubes/), for the “Tori” model.

References 1. Chew, L.P.: Constrained Delaunay triangulations. Algorithmica 4, 97–108 (1989) 2. Dey, T.K., Bajaj, C.L., Sugihara, K.: On good triangulations in three dimensions. Int. J. Comput. Geom. & App. 2, 75–95 (1992) 3. Miller, G.L., Talmor, D., Teng, S.-H., Walkington, N.: A Delaunay based numerical method for three dimensions: generation, formulation, and partition. In: 27th Annual ACM Symposium on the Theory of Computing, pp. 683–692 (1995) 4. Cohen-Steiner, D., Colin, E., Yvinec, M.: Conforming Delaunay triangulations in 3D. In: 18th Annual Symposium on Computational Geometry, pp. 199–208 (2002) 5. George, P.L., Seveno, E.: The advancing front mesh generation method revisited. Int. J. Numer. Meth. Engng. 37, 3605–3619 (1994) 6. Mavriplis, D.J.: An advancing front Delaunay triangulation algorithm designed for robustness. J. of Comput. Phys. 117, 90–101 (1995) 7. Lohner, R.: Extensions and improvements of the advancing front grid generation technique. Commun. Numer. Meth. Engng. 12, 683–702 (1996)

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8. Lau, T.S., Lo, S.H.: Finite element mesh generation over analytical surfaces. Comput. Struct. 59, 301–309 (1996) 9. Heighwayl, E.A.: A mesh generator for automatically subdividing irregular polygons into quadrilaterals. IEEE Transactions on Magnetics Mag-19, 2535–2538 (1983) 10. Itoh, T., Inoue, K., Yamada, A., Shimada, K., Furuhata, T.: Automated conversion of 2D triangular Mesh into quadrilateral mesh with directionality control. In: 7th International Meshing Roundtable, pp. 77–86 (1998) 11. Lo, S.H.: Generating quadrilateral elements on plane and over curved surfaces. Comput. Struct. 31, 421–426 (1989) 12. Johnston, B.P., Sullivan Jr., J.M., Kwasnik, A.: Automatic conversion of triangular finite element meshes to quadrilateral elements. Int. J. Numer. Meth. Engng. 31, 67–84 (1991) 13. Lee, C.K., Lo, S.H.: A new scheme for the generation of a graded quadrilateral mesh. Comput. Struct. 52, 847–857 (1994) 14. Velho, L.: Quadrilateral meshing using 4-8 clustering. In: CILANCE 2000, pp. 61–64 (2000) 15. Owen, S.J., et al.: Q-Morph: An indirect approach to advancing front quad meshing. Int. J. Numer. Meth. Engng. 44, 1317–1340 (1999) 16. Blacker, T.D., Stephenson, M.B.: Paving: A new approach to automated quadrilateral mesh generation. Int. J. Numer. Meth. Engng. 32, 811–847 (1991) 17. Miyazaki, R., Harada, K.: Transformation of a closed 3D triangular mesh to a quadrilateral mesh based on feature edges. Int. J. Comput. Sci. Network Security. 9, 30–36 (2009) 18. Baehmann, P.L., Wittchen, S.L., Shephard, M.S., Grice, K.R., Yerry, M.A.: Robust geometrically based, automatic two-dimensional mesh generation. Int. J. Numer. Meth. Engng. 24, 1043–1078 (1987) 19. Tam, T.K.H., Armstrong, C.G.: 2D finite element mesh generation by medial axis subdivision. Adv. Engng. Software 13, 313–324 (1991) 20. Joe, B.: Quadrilateral mesh generation in polygonal regions. Comput. Aid. Des. 27(3), 209–222 (1991) 21. Zhu, J.Z., Zienkiewicz, O.C., Hinton, E., Wu, J.: A new approach to the development of automatic quadrilateral mesh generation. Int. J. Numer. Meth. Engng. 32, 849–866 (1991) 22. White, D.R., Kinney, P.: Redesign of the paving algorithm: Robustness enhancements through element by element meshing. In: 6th International Meshing Roundtable, pp. 323–335 (1997) 23. Zhang, Y., Bajaj, C.: Adaptive and quality quadrilateral/hexahedral meshing from volumetric Data. Comput. Meth. in Appl. Mech. Engng. 195, 942–960 (2006) 24. Bern, M., Eppstein, D.: Quadrilateral meshing by circle packing. Int. J. Comp. Geom. & Appl. 10(4), 347–360 (2000)

Mesh Construction with Prescribed Properties Near Boundary Boris Azarenok Dorodnicyn Computing Center of Russian Academy of Sciences, Moscow, Russia [email protected]

Summary. A quasi-conformal mapping of the parametric domain onto the underlying physical domain is used to generate a 2D structured mesh with required properties: grid line orthogonality and prescribed mesh point clustering near the domain boundary. The functions implementing the mapping are sought by solving the Dirichlet problem for the elliptic partial differential equations. An additional control for the cell shape is executed by introducing a local mapping. Example of the mesh near the airfoil is presented. Keywords: structured mesh; elliptic grid generation; control metric.

1 Introduction When constructing a structured (block-structured) mesh on a two-dimensional domain with aim to simulate viscous gas flow near a body, there are additional requirements imposed on the mesh. It is grid line orthogonality and prescribed strong mesh point clustering near the streamline body. If the outside domain boundary is in “infinity” and may have an arbitrary shape, then one may use algorithms based on solving hyperbolic partial differential equations (PDEs) (cf. [1, 2]). If the entire boundary is stringently defined, such methods are badly applicable due to the boundary value problem (BVP) is ill-possed. Besides, the hyperbolic PDEs transfer singularities of the solution (i.e., breaks on the boundary) along characteristics. Quasilinear hyperbolic PDEs may produce a non-smooth and even discontinuous solution even with smooth boundary conditions [3]. For the rigidly defined boundary it is convenient to utilize elliptic PDEs of the second order (cf. [4, 5, 6]) producing meshes at a quasi-conformal mapping. Here it is sought a mapping of the domain P in the parametric plane with a given square mesh onto the domain Ω in the physical plane where the grid is required. We call P a parametric domain and Ω is called a physical domain. If to employ a homeomorphic mapping F : P→Ω then the image of the square mesh on P is an unfolded grid on Ω. For the Laplace equations,

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the Radó theorem [7] asserts that the harmonic mapping of a simply connected bounded domain onto a simply connected bounded convex domain is univalent subject to a given homeomorphism between the boundaries. To satisfy the conditions of the Rad´ o theorem for nonconvex physical domains, the inverted Laplace equations are applied (cf. [4, 6]). The use of the inverted Laplace equations provides a smooth and quasiuniform grid. However, the grid so generated is not always satisfactory in the sense that points may not be clustered to where they are needed and grid orthogonality near the domain boundary is not supported. Besides, some problem arises at discrete level that leads to mesh folding (cf. [8, 9, 10, 11]). In [5], with purpose to implement grid line control, a substitution of variables was implemented for the inverted Laplace equations. In [12], the source terms were used in those Eqs. and, in [16], an algorithm of specifying coefficients in source terms was suggested to impose grid line orthogonality and prescribed mesh point clustering near the domain boundary. An additional algebraic transformation was applied in [13], additional local mapping was executed in [14, 10]. It is not possible to obtain orthogonal mesh in the entire domain with a curvilinear boundary, except simple canonical domains (disk, ring or their sector, etc.) because the BVP is ill-possed [15]. In complicated domains, the mesh, generated by using a conformal mapping, is not applicable for mathematical simulation. On the other hand, one may obtain a mesh with orthogonal grid lines of one family towards the boundary and prescribed mesh point clustering near the domain boundary. To this end, in the source terms of the Poisson equations of [12] it is specified coefficients at the exponent by solving an inverse problem [16]. This process is rather cumbersome and practical calculation demonstrates that grid line may approach the boundary at an angle substantially differing from 90o [16, 2, 17]. We present the grid generation method based on solving the Dirichlet problem for the elliptic PDEs [10, 11]. Grid line control is imposed by introducing an additional local mapping which induces the control metric. We present the way of specifying the control metric with aim of obtaining grid line orthogonality and required mesh point clustering near the domain boundary. Example of the mesh near the NACA6409 airfoil is considered.

2 Grid Generation Using Mapping Let a structured mesh be required on a simple connected domain Ω represented as a curvilinear quadrangle in the plane of physical variables x, y. The mesh is defined by the nodal Cartesian coordinates ri,j = (x, y)i,j , i = 1, 2, . . . , N1 , j = 1, 2, . . . , N2 , subject to given nodal coordinates on the boundary ∂Ω ri,1 , ri,N2 , i=1, . . . , N1 ; r1,j , rN1 ,j j=1, . . . , N2 .

Mesh Construction with Prescribed Properties Near Boundary

37

To this end, the parametric domain P, rectangle with the sides N1 −1 and N2 −1 subdivided into unit squares, in the plane ξ, η is mapped onto the physical domain Ω. If a homeomorphic mapping F : P→Ω can be found, the image of the square mesh on the domain P will be an unfolded mesh on the domain Ω. This is true at least in continuous approach. Meantime at the discrete level, truncation error may case grid folding (cf. [11, 9, 8]). As grid lines on Ω it is utilized the equipotential lines of the functions ξ(x, y) and η(x, y). The functions F=x(ξ)=(x(ξ, η), y(ξ, η)), of class C2 , executing the mapping x(ξ) : P→Ω (see Fig. 1), are sought by solving the Dirichlet problem for the elliptic PDEs [10, 11] ˜ 22 (P − Q) + G ˜ 12 (S − R) (1) L(x) = g22 xξξ − 2g12xξη + g11 xηη − xξ G ˜ 11 (R − S) + G ˜ 12 (Q − P ) = 0, −xη G L(y) = 0,

˜ ˜ ˜ ˜ ˜ kl = √Gkl , P = g11 ∂ G12 − g12 ∂ G11 , Q = 1 g11 ∂ G22 − g22 ∂ G11 , G ∂η ∂η 2 ∂ξ ∂ξ det G

˜ 22 ˜ 12 ˜ 22 ˜ 11 ∂G ∂G 1 ∂G ∂G S = g12 − g22 , R= g11 − g22 . ∂ξ ∂ξ 2 ∂η ∂η The components of the metric tensor gkl , induced by the mapping x(ξ), are g11 = x2ξ + yξ2 , g12 = xξ xη + yξ yη , g22 = x2η + yη2 . In Eqs. (1), we use one more metric G. The control metric G is induced by a local mapping X(ξ) of a subdomain P1 ⊂P to a domain in the plane of new variables X, Y . At discrete level it may be, for instance, four adjacent cells in the parametric plane ξ, η, forming a 9-point stencil to approximate Y

6

η

y

-

X

X(ξ)

x(ξ)

P

ξ

Ω

-

x

Fig. 1. Mapping x(ξ) : P→Ω is sought by solving problem (1),(2). Control metric G is specified by local mapping X(ξ), e.g., 4 cells (dotted) in plane ξ, η onto 4 cells in plane X, Y .

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the system (1), onto four adjacent cells in the plane X, Y (see Fig. 1). The components of the control metric tensor Gkl are G11 = Xξ2 + Yξ2 , G12 = Xξ Xη + Yξ Yη , G22 = Xη2 + Yη2 . The boundary conditions specify the boundary correspondence x = xb (ξ, η) ,

(ξ, η) ∈ ∂P.

y = yb (ξ, η),

(2)

The mapping x(ξ) is quasi-conformal, i.e., it transforms an infinitesimal circle of P to an infinitesimal ellips of Ω. In some particular cases, the mapping x(ξ) : P→Ω is a composition of two sequential mappings. First, we construct the mapping X(ξ) of the domain P onto the domain C in the plane X, Y (see Fig. 2). Therefore, a mesh (or a part of the mesh) is constructed on the domain C. Next the image of this mesh is obtained on the domain Ω at the mapping x(X) : C→Ω. If Gkl is the identity tensor, then Eqs. (1) turn to the inverted Laplace equations of [4] and x(ξ) : P→Ω is the inverse harmonic mapping. By the Rad´ o theorem [7], the direct harmonic mapping ξ(x) : Ω→P, specified by the Laplace equations, is univalent. PDEs (1) are the Euler equations to the universal functional suggested in [14]. The property of universality implies that by specifying the control metric Y

C

X

x(X) y

X(ξ)

Ω

η

x(ξ)

P

x ξ Fig. 2. In particular cases, first mapping X(ξ) : P→C is used and then mapping x(X) : C→Ω is considered

Mesh Construction with Prescribed Properties Near Boundary

39

G any given smooth homeomorphic mapping F : P→Ω may be reproduced by solving the problem (1),(2) and, at discrete level, any given unfolded mesh may by constructed. In addition, Eqs. (1) are invariant to nonsingular transformations of the coordinates x, y and X, Y . These two properties of Eqs. (1) allow to obtain a grid with required properties subject to such a grid exists.

3 Discretization and Numerical Solution Eqs. (1) are approximated to second order at the interior mesh node (i, j) on the domain P via the difference relations on the 9-point square stencil of spacing 1. The derivatives of the function x(ξ, η) are approximated as follows xξ ≈ [xξ ]i,j = 0.5(xi+1,j − xi−1,j ), xη ≈ [xη ]i,j = 0.5(xi,j+1 − xi,j−1 ), xξξ ≈[xξξ ]i,j =xi+1,j − 2xi,j + xi−1,j , xηη ≈[xηη ]i,j =xi,j+1 − 2xi,j + xi,j−1 ,

(3)

xξη ≈ [xξη ]i,j = 0.25(xi+1,j+1 − xi+1,j−1 − xi−1,j+1 + xi−1,j−1 ),

and the components of the metric tensor gkl are approximated as follows [g11 ]i,j = [xξ ]2i,j + [yξ ]2i,j , [g22 ]i,j = [xη ]2i,j + [yη ]2i,j , [g12 ]i,j = [xξ ]i,j [xη ]i,j + [yξ ]i,j [yη ]i,j . The components of the control metric tensor Gkl are approximated similarly. Substituting these relations into Eqs. (1) gives approximation of the operators L(x) and L(y) denoted [L(x)]i,j and [L(y)]i,j , respectively. The resulting discretized equations with boundary conditions (2) are solved by employing the iterative procedure. Let nodal coordinates be known at the lth iteration. Then at the l+1th iteration they are updated by formulae [5] l xl+1 i,j = xi,j + τ

[L(x)]i,j [L(y)]i,j l+1 l , yi,j = yi,j +τ . (4) 2([g11 ]i,j + [g22 ]i,j ) 2([g11 ]i,j + [g22 ]i,j )

Here [.] denotes approximation of the underlying expression at the node (i, j) at the lth iteration, iterative parameter 0 0.6 and ≤ > 0.8 and ≤

|J| U 31 0.1 19 0.2 39 0.4 136 0.6 267 0.8 911 1.0 6161

L 3 3 5 45 118 314 7076

Pa 0 0 2 23 105 339 7095

Pr 0 0 4 22 104 311 7123

Max κ ≥1.0 and ≤1.5 >1.5 and ≤1.0 >2.0 and ≤3.0 >3.0 and ≤5.0 >5.0 and ≤7.5 >7.5 and ≤10.0 >10.0

U 4610 1184 927 556 136 48 103

L 6565 648 236 72 23 11 9

Pa 6445 856 205 50 7 1 0

Pr 6512 785 201 60 3 3 0

Table 5. Mesh quality statistics for phalanx bone Mesh Min U -0.123 L 0.849 Pa 0.848 Pr 0.847

Scaled |J| Max Mean Std dev 1.000 0.904 0.143 1.000 0.993 0.016 1.000 0.993 0.016 1.000 0.993 0.016

Min 1.000 1.000 1.000 1.000

Max ∞ 1.296 1.255 1.242

κ Mean Std dev ∞ ∞ 1.075 0.080 1.071 0.066 1.071 0.063

Table 6. Mesh quality statistics for femur

Mesh Min U -0.808 L 0.516 Pa 0.582 Pr 0.622


Min 1.003 1.000 1.000 1.000

Max ∞ 2.256 1.787 1.642

κ Mean Std dev ∞ ∞ 1.211 0.138 1.212 0.135 1.211 0.132

difference in the normal orientation at the nodes of a quadrilateral element. Hence, an untangling function (Equation 5) that minimizes the difference in node normal orientations has been proposed.

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Table 7. Mesh quality statistics for vertebra (posterior)

Mesh Min U -0.849 L -0.634 Pa 0.149 Pr 0.138


Min 1.000 1.000 1.000 1.000

κ Max Mean Std dev ∞ ∞ ∞ ∞ ∞ ∞ 7.854 1.266 0.382 8.819 1.258 0.405

During the shape improvement, different criteria are used for defining feasible/valid region for a given node. Escobar et al. [13] use the parametric space defined by the projection of sub mesh (elements sharing a given node) onto the best fit plane as the valid region. Garimella et al.[14] use an imaginary plane containing the opposing edge of the node of interest and perpendicular to the normal at the node. If the node crosses the imaginary plane during node movement, the element is considered invalid. In the proposed method, the element is considered invalid if the node movement makes the angles as computed in Equation 6 greater than the user defined value (θ in Equation 5). A simple two dimensional problem (Fig. 6) is solved to compare the two techniques. It is seen from Fig. 6e that the two techniques converge to the same solution, but the optimization path computed is different. The projection-based optimization direction has a much straighter path to the optimal location (Vertex V2 in Fig. 6e) than the parametric method. Three examples with varying degree of curvature change on the underlying surfaces are chosen to test the mesh improvement techniques. Curvature change is a good indicator of the uniformity of the node distribution on the unsmooth mesh and with it the initial mesh quality. In all the three cases, the unsmooth meshes contain elements with negative scaled Jacobian thus implying low mesh quality. Laplacian smoothing improves the overall quality of the meshes shown (Fig. 7c, Fig. 8c and Fig. 9c) as compared to the unsmooth meshes but does not guarantee a valid mesh in all examples. However, the parameterization-based (Fig. 7d, Fig. 8d and Fig. 9d) and projection-based (Fig. 7e, Fig. 8e and Fig. 9e) methods perform better resulting in improved mesh quality (Table 5, 6 and 7) as compared to Laplacian smoothing. For example, negative Jacobians that exist for the posterior portion of sheep vertebrae mesh obtained by Laplacian smoothing are eliminated with the techniques proposed in the paper. The results obtained between projection and parameterization-based smoothing are comparable with both the methods resulting in valid meshes. Laplacian smoothing works well for regions on the surface with low rate of change of curvature as is the case with phalanx bone (Fig. 7). Hence, mesh quality statistics obtained through Laplacian smoothing compare favorably with projection-based and parameterization-based methods (Table 2 and Table 5). For the regions with high curvature (Fig. 8b inset), Laplacian smoothing loses the geometric fidelity (Fig. 8c inset) as compared to

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parametric (Fig. 8d inset) and projection (Fig. 8e inset) based methods. One more drawback of Laplacian smoothing is that it tends to fold the mesh (Fig. 9c top-inset) in the regions of concave boundaries (Fig. 9b top-inset). The parametric (Fig. 9d top-inset) and projection-based methods (Fig. 9e top-inset) avoid mesh folding thus improving the mesh quality. Hence, mesh quality statistics for optimization methods show dramatic improvement compared with Laplacian smoothing for sheep vertebra (Table 4). Even though parametric and projection-based methods yield valid meshes, the location of nodes in a converged solution is not identical because, the optimization paths for the two methods are not identical (Fig. 6e). The contour of the objective function of an individual node is convex but the local objective function is not [19]. The local objective function can contain multiple local minima and hence the difference in the location of nodes. Fig. 7f, Fig. 8f and Fig. 9f show the overlaid meshes from parametric and projection methods for all the three examples. If the underlying surface definition contains non-triangular elements, the elements can be triangulated and the proposed method applied. If the triangulation is not allowed, the procedure described in the literature can be used [14]. For planar and volumetric meshes, the optimization path is a straight line and can easily be stored as a vector. For non-planar meshes, the optimization path is made of segments of lines that lie on the underlying surface. The implementation of global optimization becomes difficult for non-planar meshes because, the optimization paths for all the nodes have to be stored and the intermediate locations of the nodes for fractional step lengths have to be computed. Here local optimization is used because; every node is moved to its best possible location one by one. If the node moves out of a cell, the optimization starts afresh thus simplifying the implementation.

4 Conclusion Enhancements/new techniques for non-planar quadrilateral surface mesh improvement are presented. A modified untangling function for quadrilateral meshes based on node normals is proposed. Two techniques, (1) parameterization-based and (2) projection-based method for optimization direction computation are presented. The parameterization-based method that improves the surface mesh quality while maintaining the mesh fidelity has been enhanced and presented herein. An analytical framework for, (1) mesh quality metrics and (2) optimization direction computation in physical and parametric space has been presented. A new projection-based optimization direction computation has been devised and implemented and its performance is comparable to that of the parametric method. The proposed methods have been successfully tested on quadrilateral surface meshes with underlying triangulated surfaces. The results are superior to those obtained by the Laplacian smoothing method.

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References 1. Blacker, T.D., Meyers, R.J.: Seams and wedges in plastering: A 3D hexahedral mesh generation algorithm. Engineering with Computers 2(9), 83–93 (1993) 2. Grosland, N.M., et al.: IA-FEMesh: An open-source, interactive, multiblock approach to anatomic finite element model development. Computer Methods and Programs in Biomedicine 94(1), 96–107 (2009) 3. Knupp, P.: Hexahedral and tetrahedral mesh untangling. Engineering With Computers 17(3), 261–268 (2001) 4. Knupp, P.M.: A method for hexahedral mesh shape optimization. International Journal for Numerical Methods in Engineering 58(2), 319–332 (2003) 5. Freitag, L.A., Ollivier-Gooch, C.: Tetrahedral mesh improvementd using swapping and smoothing. International Journal for Numerical Methods in Engineering 40(21), 3979–4002 (1997) 6. Field, D.: Laplacian smoothing and delaunay triangulations. Communications in Applied Numerical Methods 4, 709–712 (1988) 7. Knupp, P.M.: Achieving finite element mesh quality via optimization of the Jacobian matrix norm and associated quantities. Part I- a framework for surface mesh optimization. International Journal for Numerical Methods in Engineering 48, 401–420 (2000) 8. Knupp, P.M.: Achieving finite element mesh quality via optimization of the Jacobian matrix norm and associated quantities. Part II- a framework for volume mesh optimization and the condition number of the Jacobian matrix. International Journal for Numerical Methods in Engineering 48, 1165–1185 (2000) 9. Chen, Z., Tristano, J.R., Kwok, W.: Construction of an objective function for optimization based smoothing. Engineering with Computers 20, 184–192 (2004) 10. Yin, J., Teodosiu, C.: Constrained mesh optimization on the boundary. Engineering with Computers 24, 231–240 (2008) 11. Frey, P.J., Borouchaki, H.: Geometric surface mesh optimization. Computing and Visualization in Science 1(3), 113–121 (1998) 12. Zhang, Y., Bajaj, C., Xu, G.: Surface smoothing and quality improvement of quadrilateral/hexahedral meshes with geometric flow. Communications in Numerical Methods in Engineering 25(1), 1–18 (2007) 13. Escobar, J.M., Montero, G., Montenegro, R., Rodriguez, E.: An algebraic method for smoothing surface triangulations on a local parametric space. International Journal for Numerical Methods in Engineering 66, 740–760 (2006) 14. Garimella, R., Shashkov, M., Knupp, P.: Triangular and quadrilateral surface mesh quality optimization using local parametrization. Computer Methods in Applied Mechanics and Engineering 193, 913–928 (2004) 15. Garimella, R.V., Shashkov, M.J.: Polygonal surface mesh optimization. Engineering with Computers 20, 265–272 (2004) 16. Pbay, P., Thompson, D., Shepherd, J., Knupp, P., Lisle, C., Magnotta, V., Grosland, N.: New Applications of the Verdict Library for Standardized Mesh Verification Pre, Post, and End-to-End Processing. In: Proceedings of the 16th International Meshing Roundtable (2007) 17. ANSYS Inc. Theory reference, Release 9.0, Ch. 13 (2004) 18. Knupp, P.M.: Algebraic mesh quality metrics for unstructured initial meshes. Finite Elements in Analysis and Design 39(3), 217–241 (2003) 19. Diachin, L.F., Knupp, P.M., Munson, T., Shontz, S.: A comparison of two optimization methods for mesh quality improvement. Engineering with Computers 22(2), 61–74 (2006)

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Appendix 1. Jacobian computation (x − l1 ) JP2 = y

− l1 0

,

JP3 =

−x −y

(l1 − x) −y

(14)

2. Determinant of Jacobian and Frobenius norm |JP1 | = |JP2 | = |JP3 | = |J| = l1 y 2

JP1 = l1 2 + l3 2 ,

2

JP2 = l1 2 + l2 2 ,

(15) 2

JP3 = l2 2 + l3 2

(16)

3. Untangling function derivatives ∂fPi ∂αPi abs(αPi ) = ( + 1) ∂X ∂X αPi

(17)

∂αPi ∂nPi ∂|JPi | =n· |JPi | + (n · nPi − cos(θ)) (18) ∂X ∂X ∂X ⎛ ⎞ ⎛ ⎞ ⎛ ⎞ 0 Z1 − Z2 Y2 − Y1 ∂nPi ∂n ∂n P P i i ⎠, 0 = ⎝ Z2 − Z1 ⎠ , =⎝ = ⎝ X1 − X2 ⎠ ∂X ∂Y ∂Z Y1 − Y2 Y1 − Y2 0 (19) 4. Shape improvement function derivatives |JPi | ∂κPi = ∂X

∂JPi 2 X

+

∂|JPi | 2 X JPi 2

2|JPi |

∂|JPi | l1 (X − X1 ) − x(X2 − X1 ) = ∂X y 2

2

2

(20)

(21)

∂ JP1 ∂ JP2 ∂ JP3 = 2(X−X1 ), = 2(X−X2 ), = 2(2X−(X1+X2 )) ∂X ∂X ∂X (22) Similar equations can be written for coordinates Y and Z.

Efficient Solution of Elliptic Partial Differential Equations via Effective Combination of Mesh Quality Metrics, Preconditioners, and Sparse Linear Solvers Jibum Kim, Shankar Prasad Sastry, and Suzanne M. Shontz Department of Computer Science and Engineering The Pennsylvania State University University Park, PA 16802 [email protected], [email protected], [email protected]

Abstract. In this paper, we study the effect the choice of mesh quality metric, preconditioner, and sparse linear solver have on the numerical solution of elliptic partial differential equations (PDEs). We smoothe meshes on several geometric domains using various quality metrics and solve the associated elliptic PDEs using the finite element method. The resulting linear systems are solved using various combinations of preconditioners and sparse linear solvers. We use the inverse mean ratio and vertex condition number metrics in addition to interpolation-based, scale-variant and scale-invariant metrics. We employ the Jacobi, incomplete LU, and SSOR preconditioners and the conjugate gradient, minimum residual, generalized minimum residual, and bi-conjugate gradient stabilized solvers. We focus on determining the most efficient quality metric/preconditioner/linear solver combination for the numerical solution of various elliptic PDEs.

1 Introduction Discretization methods, such as the finite element (FE) method, are commonly used in the numerical solution of partial differential equations (PDEs). The accuracy of the computed PDE solution depends on the degree of the approximation scheme, the number of elements in the mesh [1], and the quality of the mesh [2, 3]. In addition, the stability and convergence of the finite element method is affected by poor quality elements [4]. Analytical studies have been performed at the intersection of meshing and linear solvers. For example, mathematical connections between mesh geometry, interpolation errors, and stiffness matrix conditioning for triangular and tetrahedral finite element meshes have been studied [5]. A mesh and solver co-adaptation strategy for anisotropic problems has been developed [6]. Relationships between the spectral condition number of the stiffness matrix and mesh geometry for second-order elliptic problems have been determined [7].

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Several computational studies have been performed which examined the connections between finite element meshes and linear solvers in various contexts. For example, the effect of unstructured meshes on the preconditioned conjugate gradient solver performance for the solution of the Laplace and Poisson equations has been examined [8, 9]. In [10], the relative performance of multigrid methods for unstructured meshes was studied on fluid flow and radiation diffusion problems. Trade-offs associated with the cost of mesh improvement in terms of solution efficiency has been examined for fluids [11, 12]. In this paper, we examine the connections between geometry, mesh smoothing, and solution convergence for elliptic PDEs via an engineering approach. In particular, we seek answers to the following questions pertaining to the solution of an elliptic PDE on a given geometric domain. Which mesh quality metric should be used to smoothe the initial mesh? What is the most efficient combination of mesh quality metric, preconditioner, and solver for solving an elliptic PDE? What is the effect of modifying the PDE coefficients and boundary conditions on the answers to the above questions? Our goal is to determine the best quality metric/preconditioner/linear solver combination which results in a small condition number of the preconditioned matrix and fast solver convergence for a given PDE, geometric domain, and initial mesh. To answer the above questions, we use Mesquite [13], a mesh quality improvement toolkit, and PETSc [14], a linear solver toolkit, to perform a numerical study investigating the performance of several mesh quality metrics, preconditioners, and sparse linear solvers on the solution of various elliptic PDEs of interest. The quality metric/preconditioner/linear solver combinations are compared on the basis of efficiency in solving several elliptic PDEs on realistic unstructured tetrahedral finite element meshes. We use Mesquite and PETSc in their native state with the default parameters. Only these two toolkits are employed so that differences in solver implementations, data structures, and other such factors would not influence the results.

2 Finite Element Solution of Elliptic PDEs We consider the solution of second-order elliptic PDEs using the finite element (FE) method. An elliptic PDE on a geometric domain, Ω, is defined as −Δu + au = f on Ω,

(1)

where a and f are given functions. If a = 0, (1) reduces to Poisson’s equation. We consider both Dirichlet and generalized Neumann boundary conditions on the boundary, ∂Ω. The FE method [15] is used to discretize the domain, Ω, and to discretize the PDE resulting in the linear system Aξ = b.

(2)

Triangular and tetrahedral meshes are used to discretize the domain, Ω, in 2D and 3D, respectively. The approximate solution, ξ, of u can be computed

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105

by solving (2). The matrix A is given by A=K+M +N , where K is the stiffness matrix; M is the mass matrix, and N is a matrix containing boundary information. For elliptic PDEs, K is a symmetric positive definite or nonnegative definite matrix, and M is a symmetric positive definite matrix [7]. The vector b is the sum of two vectors, F and G. For a two-dimensional → geometric domain with generalized Neumann boundary conditions, − n · (∇ u)+du=g, on ∂Ω, where d and g are given functions. Furthermore, the K, M , N matrices and the F and G vectors can be computed as follows [15]: " " Ki,j = (φi · φj ) dx dy; Mi,j = (aφi · φj ) dx dy; Ω

Ω

"

" (dφi · φj ) ds; Fi =

Ni,j = ∂Ω

" (f φi ) dx dy; Gi =

Ω

(gφi ) ds, ∂Ω

where φi and φj are piecewise linear basis functions.

3 Mesh Quality Metrics Table 1 provides the notation used to define the following mesh quality metrics: inverse mean ratio (IMR) [16], vertex condition number (VCN) [17], an interpolation-based, scale-variant metric (SV) [5], and an interpolationbased, scale-invariant metric (SI) [5]. Table 2 defines IMR, VCN, SV, and SI. For IMR and VCN, a lower value indicates a higher quality element. For SV and SI, a higher value indicates a higher quality element.

Table 1. Notation used in the definition of mesh quality metrics in Table 2. The above notation is for the 3D case. Similar quantities can be defined in 2D. Notation a, b, c, and d C = [b⎛− a; c − a;⎞ d − a] 1 12 12 ⎜ 0 √3 √3 ⎟ W =⎝ 2 √6 ⎠ 0 0 √23 Ar1, Ar2 , Ar3 , and Ar4 lij Vol Area s1 , s2 , and s3

Definition Position vectors for vertices in a tetrahedral element Jacobian of a tetrahedral element Incidence matrix for an equilateral tetrahedron Areas of the triangular faces in a tetrahedral element Length of the edge common to triangular face i and j in a tetrahedral element Volume of a tetrahedral element Area of a triangular element Edge lengths of a triangular element

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J. Kim, S.P. Sastry, and S.M. Shontz Table 2. The mesh quality metric definitions Quality Metric

Formula

IMR

% % & &2 %CW −1 %2 / (3 &det(CW −1 )& 3 ) [16] F

VCN

C F C −1 F [17]

SV (2D)

Area/s1 s2 s3 [5]

SV (3D)

(Vol

SI (2D)

Area/(s1 s2 s3 ) 3 [5]

SI (3D)

' 3 ' 4 2 Vol ( 4i=1 Ari )/( 1≤i<j≤4 Ari Arj lij ) [5]

'4 i=1

Ari )/(

' 1≤i<j≤4

2 Ari Arj lij ) [5]

2

4 Mesh Optimization We denote the elements of a mesh and the number of mesh elements by E and |E|, respectively. The overall quality of the mesh, Q, is a function of the individual element qualities, qi , where qi is the quality of the ith element in the mesh. The mesh quality depends on both the choice of qi , which is described in Section 3, and the function used to combine them. For the IMR and VCN quality metrics, we define the overall mesh quality, Q, as the sum of squares of the individual elements’ qualities: Q=

|E|

qi2 .

(3)

i=1

For the SV and SI quality metrics, we define the overall mesh quality, Q, as the sum of the squares of the reciprocal of the individual elements’ qualities: Q=

|E| 1 . q2 i=1 i

(4)

We use the Fletcher-Reeves nonlinear conjugate gradient method [18] in Mesquite [13] to minimize Q (defined by either (3) or (4)). Mesquite employs a line search version of the nonlinear conjugate gradient method.

5 Iterative Linear Solvers Four iterative Krylov subspace methods are employed to solve the preconditioned linear system. The conjugate gradient (CG) solver is a well-known


107

iterative method for solving systems with symmetric positive definite matrices. It produces a sequence of orthogonal vectors on successive iterations. Let P be the preconditioner (to be described in Section 6). The convergence rate of the preconditioned CG method depends upon the condition number, κ, of P −1 A. In the 2-norm, κ can be approximated as % % % %−1 κ2 (P −1 A) = %P −1 A%2 %P −1 A%2 ≈ λmax (P −1 A)/λmin (P −1 A), where λmax and λmin are the maximum and minimum eigenvalues of P −1 A, respectively. The fastest convergence occurs when eigenvalues are clustered around the same non-null value and, hence, κ is near 1. The minimal residual (MINRES) algorithm solves linear systems with symmetric indefinite matrices. It generates a sequence of orthogonal vectors and attempts to minimize the residual in each iteration. Similar to CG, a small condition number for P −1 A and clustering of eigenvalues around the same non-null value results in fast convergence. For nonsymmetric matrices, the generalized minimal residual (GMRES) method is one of the most widely used iterative solvers. Similar to CG and MINRES, GMRES computes orthogonal vectors on each iteration; however, the entire sequence needs to be stored. Therefore, the version of GMRES which restarts GMRES every m steps, i.e., GMRES(m), is used in practice. It is known that a large value of m is effective in decreasing the convergence time; however, the optimal value of m depends upon the problem [19]. The biconjugate gradient stabilized (Bi-CGSTAB) method is a biorthogonalization technique, which generates two sets of biorthogonal vectors instead of producing long orthogonal vectors. Bi-CGSTAB is known to have comparable or even faster convergence than other biorthogonalization methods such as the conjugate gradient squared method. However, Bi-CGSTAB sometimes shows an irregular convergence rate similar to other biorthogonalization methods [20]. Bi-CGSTAB and GMRES are the most widely-used iterative methods for solving systems based on nonsymmetric matrices.

6 Preconditioners The objective of introducing a preconditioner, P , into the solution of a linear system is to make the system easier to solve, whereby reducing the convergence time of the iterative solver. The reader is referred to [21] (and the references therein) for further information on iterative solvers and preconditioners. In this paper, four preconditioners are employed. The first is the Jacobi preconditioner, which is simply the diagonal of A. The second is the symmetric successive over relaxation (SSOR) preconditioner. SSOR is similar to Jacobi but decomposes A into L (the strictly lower triangular part), D (the diagonal), and U (the strictly upper triangular part), i.e., A=L+D+U . The SSOR preconditioner is given by

108


P = (D − ωL)D−1 (D − ωU ), where ω represents the relaxation coefficient. The incomplete LU (ILU) preconditioner with level zero fill-in (ILU(0)) and level one fill-in (ILU(1)) are used as the third and fourth preconditioners, respectively. The basic idea of the ILU preconditioner is to determine ˜ and upper triangular (U ˜ ) matrices such that the matrix L ˜U ˜ -A satlower (L) isfies certain constraints [21]. The ILU preconditioner works well for many problems, but it fails when it encounters negative or zero pivots.

7 Numerical Experiments Three experiments are performed to investigate the following questions. (1) Which mesh quality metric should be used to smoothe the initial mesh? (2) What is the most efficient combination of mesh quality metric, preconditioner, and solver for solving an elliptic PDE? (3) What is the effect of modifying the PDE coefficients and boundary conditions on the answers to the above questions? Table 3 summarizes the experiments and corresponding PDE problems to be solved. In Experiments 7.1 and 7.2, we focus on solving Poisson’s equation with Dirichlet boundary conditions in order to obtain insight on the above questions and then extend the results to other PDE problems in Experiment 7.3. The machine employed for this study is equipped with an Intel Xeon x550 processor (2.67 GHz) and 24GB of RAM [22].

Table 3. Listing of numerical experiments. The letters (a) through (e) represent the five PDE problems under consideration. Exp. Exp. No. Name. 7.1 Choice of metric 7.2 Best combination 7.3 Modification of PDE coefficients and boundary conditions

PDE Problems (a) −Δu=1 on Ω, u=0 on ∂Ω −Δu=1 on Ω, u=0 on ∂Ω (b) −Δu+100u=1 on Ω, u=0 on ∂Ω → (c) −Δu+100u=1 on Ω, − n · (∇ u)+100u=1 on ∂Ω → − (d) −Δu=1 on Ω, n · (∇ u)+u=1 on ∂Ω → (e) −Δu=1 on Ω, − n · (∇ u)+100u=1 on ∂Ω

Geometric Domains. The four geometric domains considered in our experiments are shown in Figure 1. Circle and gear are 2D problems, whereas bevel and drill are 3D problems. Triangle [23] and Tetgen [24] were used to generate initial meshes. Half the interior vertices in each mesh were perturbed to create test meshes that were further from optimal. Properties of the test meshes and the corresponding finite element matrices are shown in Table 4. Here, nnz is the number of non-zeros in matrix A.


(a) Circle mesh

(b) Gear mesh

(c) Bevel mesh

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(d) Drill mesh

Fig. 1. Coarse initial meshes on the circle, gear, bevel, and drill geometric domains indicative of the actual initial meshes

Table 4. Properties of meshes on geometric domains and matrix A mesh # vertices # elements Circle (2D) 508,173 1,014,894 Gear (2D) 499,842 995,142 Bevel (3D) 492,003 3,001,591 Drill (3D) 500,437 3,084,942

nnz 3,554,395 3,489,810 7,583,137 7,757,509

Mesh Smoothing. In order to improve the quality of each mesh, a local implementation of the nonlinear conjugate gradient mesh smoothing method (described in Section 4) was used in conjunction with the mesh quality metrics (described in Section 3). The minimum, average, and maximum mesh qualities before and after smoothing for the circle, gear, bevel, and drill are shown in Table 5. For our experiments, accurate mesh smoothing corresponds to five iterations of smoothing, as the mesh quality did not improve significantly after five iterations. Similarly, one iteration of smoothing was employed for inaccurate mesh smoothing, as a significant improvement in the mesh quality was observed after just one iteration. The results in Table 5 indicate that, in addition to an improvement in the average mesh quality, the quality of the worst mesh elements improves as accurate mesh smoothing is applied. Finite Element Solution. The FE method described in Section 2 is used to discretize the domain, Ω, and to generate a linear system of the form Aξ=b. PETSc [14] is used to generate the preconditioners, P , and to solve the linear system, P −1 Aξ=P −1 b. We employ the solvers and preconditioners described in Sections 5 and 6, respectively, to solve the preconditioned linear system. Table 6 enumerates the 16 preconditioner-solver combinations used in our experiments. The default parameters for each preconditioner and solver were employed except for the GMRES restart value. We employed a restart value of 100 which was the most effective in preliminary experiments. This is consistent with the fact that larger m values often result in decreased solver time [19].

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Table 5. The quality of the initial, inaccurately smoothed, and accurately smoothed meshes for the circle, gear, bevel, and drill geometric domains (a) 2D circle and gear meshes Metric Smoothing IMR

VCN

SV

SI

Initial Inaccurate Accurate Initial Inaccurate Accurate Initial Inaccurate Accurate Initial Inaccurate Accurate

min 1.000 1.000 1.000 1.000 1.001 1.000 5.80e-4 4.21e-4 7.18e-4 1.107 1.113 1.117

Circle avg 1.399 1.123 1.039 1.393 1.113 1.039 3.35e-3 2.87e-3 2.72e-3 1.372 1.198 1.149

max 1916 53.527 4.727 479 10.726 2.920 4.595 0.220 0.028 1709 40.526 3.686

min 1.000 1.000 1.000 1.000 1.000 1.000 2.24e-3 8.30e-4 7.66e-4 1.115 1.108 1.109

Gear avg max 1.570 339025 1.066 153.772 1.043 6.343 1.503 67806 1.063 30.196 1.042 3.535 5.41e-3 1194 3.60e-3 0.736 3.53e-3 0.055 1.589 299877 1.159 87.734 1.145 3.972

(b) 3D bevel and drill meshes Metric Smoothing IMR

VCN

SV

SI

Initial Inaccurate Accurate Initial Inaccurate Accurate Initial Inaccurate Accurate Initial Inaccurate Accurate

min 1.000 1.000 1.000 1.069 1.056 1.031 0.262 0.199 0.200 1.000 1.000 1.000

Bevel avg 1.501 1.320 1.233 1.927 1.367 1.262 3.492 2.505 2.210 2.018 1.471 1.316

max 1235 63.656 26.459 1537 39.244 27.157 65652 322 110.355 38421 243.199 67.963

min 1.000 1.000 1.000 1.067 1.039 1.030 0.208 0.204 0.206 1.000 1.000 1.000

Drill avg max 1.487 2132 1.311 15.568 1.226 12.334 1.887 3767 1.362 5.272 1.258 3.756 0.710 32477 0.518 29.588 0.457 6.492 1.967 86880 1.458 272.944 1.305 17.744

The default stopping criteria in PETSc were employed. For example, the absolute tolerance, abstol, and the relative tolerance, rtol, are set to 1e-50 and 1e-05, respectively. The maximum number of iterations for solving the preconditioned linear system is set to 10,000. When the preconditioned linear system is solved, ξ0 is set to the default value of 0. The preconditioned linear system converges on the ith iteration if the following inequality is satisfied:

ri < max(rtol r0 , abstol), where ri is the residual at the i

th

iteration and r0 is the initial residual.

(5)


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Table 6. The sixteen combinations of preconditioners and solvers. For example, 10 refers to using the ILU(0) preconditioner with the GMRES solver. Preconditioner Jacobi SSOR ILU(0) ILU(1)

Solver CG GMRES MINRES 1 2 3 5 6 7 9 10 11 13 14 15

BI-CGSTAB 4 8 12 16

Timing. In our experiments, the total time is defined as the sum of the smoothing time and the solver time. The smoothing time is the time to achieve an accurately smoothed mesh as described in Section 4 except for our preliminary experiments which also included inaccurate mesh smoothing. The solver time is the time the solver takes to satisfy (5) and includes the time to generate P and to solve P −1 Aξ=P −1 b. When the linear system diverges or does not converge, we report failure. Inaccurate Mesh Smoothing. For our preliminary experiments, we consider solving Poisson’s equation with Dirichlet boundary conditions, i.e., problem (a) in Table 3, on inaccurately and accurately smoothed meshes for a given domain. We consider inaccurate mesh smoothing, as engineers often perform inaccurate mesh smoothing in practice. Experimental results for inaccurate and accurate mesh smoothing are shown in Table 7. For this experiment, the SSOR preconditioner with the GMRES solver is employed. These results are representative of the results obtained when other preconditioner-solver combinations are used. Table 7 shows the smoothing time and the solver time when employing various amounts of mesh smoothing for the three mesh quality metrics described in Section 3. We observe the following rank ordering with respect to smoothing time: IMR < VCN < SV < SI. The ranking is in order of fastest to slowest. This rank ordering also holds true when we consider the smoothing and solver time together. The rank ordering between different quality metrics will be discussed in more detail in Section 7.1. Table 7 also shows that accurately smoothed meshes result in matrices with a lower condition number than those of inaccurately smoothed meshes. This results in a lower solver time for accurately smoothed meshes. However, in terms of the total time, inaccurately smoothed meshes result in faster total time than do accurately smoothed meshes for the SV and SI quality metrics because the smoothing time is very high for these two metrics. Because accurately smoothed meshes were observed to result in lower total time than inaccurately smoothed meshes, only accurately smoothed meshes are considered from here onwards.

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Table 7. Minimum and maximum eigenvalues, condition number, and timing (sec) as a function of quality metric and amount of mesh smoothing on a circle domain. Smoothing represents the number of smoothing iterations performed. NOS (no smoothing) describes the initial mesh. For this table, the SSOR preconditioner with the GMRES solver was employed. Mesh quality metrics NOS IMR VCN SV SI Smoothing 0 1 5 1 5 1 5 1 5 λmin (P −1 A) 1.8e-05 2.4e-5 2.7e-05 2.5e-05 2.7e-05 2.4e-05 2.7e-05 2.4e-05 2.7e-05 λmax (P −1 A) 1 1 1 1 1 1 1 1 1 κ(P −1 A) 55252 40822 36853 40347 36845 41979 37013 41411 36914 Smoothing 0 8 41 13 67 98 517 142 755 time Solver time 1448 905 630 951 771 1037 773 898 612 Total time 1448 913 671 964 838 1135 1290 1040 1367

7.1

Choice of Mesh Quality Metric

The goal of this experiment is to determine the best mesh quality metric for solving Poisson’s equation with Dirichlet boundary conditions based on efficiency and accuracy. Efficiency of the Solution. The time required to smoothe the test meshes using various quality metrics is shown in Table 8. If we consider the smoothing time alone, the following rank ordering is seen: IMR < VCN < SV < SI. This is because numerical computation of the IMR metric for mesh optimization is highly optimized in Mesquite. Numerical computation of the other mesh quality metrics, i.e., VCN, SV, and SI, are not as optimized and are less efficient to compute. The computation of SV and SI is also more expensive than the other two quality metrics.

Table 8. Mesh smoothing time (sec) for various mesh quality metrics Mesh Circle Gear Bevel Drill

Mesh quality metrics IMR VCN SV SI 41 67 517 755 39 65 523 784 146 208 270 366 159 213 277 377

Table 9 shows typical timing results for smoothing the bevel mesh according to the various quality metrics. In terms of the solver time, there is a significant difference between solving linear systems on smoothed versus nonsmoothed (NOS) meshes. After smoothing, the condition number of


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Table 9. Minimum and maximum eigenvalues, condition number, and timing (sec) for the matrix P −1 A for the bevel meshes before and after smoothing. The ILU(0) preconditioner with the GMRES solver was used to solve the resulting linear system. Mesh quality metrics λmin (P −1 A) λmax (P −1 A) κ(P −1 A) Smoothing time Solver time Total

NOS IMR 0.005 0.0063 18.2 1.8 3216 285 0 159 3250 50 3250 209

VCN SV SI 0.0064 0.0063 0.0064 1.8 1.8 1.8 285 285 283 213 277 377 48 54 42 261 331 419

P −1 A decreases significantly, and the minimum and maximum eigenvalues move closer to 1. This results in faster convergence of the linear solution on the smoothed meshes. However, in terms of solver time, as long as the same preconditioner and solver are employed, there is little difference when various mesh quality metrics are used to smoothe the mesh, as the metrics essentially all try to generate equilateral elements, and there are no poorly shaped elements after smoothing. The element shapes in the mesh highly affect both the eigenvalues and the condition number of A. Equilateral elements result in a smaller condition number than do poorly shaped elements. Here, poorly shaped means elements with very small (near 0◦ ) or very large (near 180◦ ) dihedral angles [5]. Thus, when the preconditioner and solver are fixed, the choice of mesh quality metric does not significantly affect the condition number or the solver time. Further information on the relationship between eigenvalues and the condition number can be found in [5]. For the total time, the rank ordering amongst mesh quality metrics is: IMR < VCN < SV < SI. A similar trend was observed for meshes on the other domains when solving Poisson’s equation with Dirichlet boundary conditions, i.e., problem (a) in Table 3. Accuracy of the Solution. The exact solution of Poisson’s equation with Dirichlet boundary conditions (i.e., problem (a) in Table 3) on the unit circle is given by: u = (1 − x2 − y 2 )/4, where x and y denote the x and y vertex coordinates, respectively. We compare our FE solution, uh , with the exact solution to verify the accuracy of our solution. The discretization error, e, between the FE solution and the exact solution is defined as e = u − uh ∞ . Our results show that for all mesh quality metrics, e ≈ 1e − 4, for the unit circle domain in Table 4. Furthermore, the choice of metric does not affect the solution accuracy for meshes of this size (i.e., approximately 1M elements). 7.2

Best Combination of Metric, Preconditioner and Solver

In Section 7.1, we observed that IMR was the most efficient quality metric for smoothing when solving Poisson’s equation with Dirichlet boundary

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conditions (i.e., problem (a) in Table 3). In this experiment, we seek to determine the most efficient combination of mesh quality metric, preconditioner, and solver for this problem. The solver time as a function of the different mesh quality metrics for various preconditioner-solver combinations is presented in Table 10 for the circle, gear, bevel, and drill domains. The ’*’ entries in these tables correspond to preconditioner-solver combinations which either diverge or do not converge. In most cases, the best combinations, which have the fastest solver time, are the SSOR, ILU(0), and ILU(1) preconditioners with the CG solver (5, 9, and 13 in Table 6). The ILU(0) or ILU(1) preconditioners with the MINRES solver (7 and 11 in Table 6) show comparable performance. We observe that the best preconditioner varies with respect to the geometric domain and the mesh quality metric. However, the best solvers do not change in most cases. The CG and MINRES solvers consistently show better performance than do GMRES and Bi-CGSTAB. The main reason is that both CG and MINRES take advantage of the symmetry properties of P −1 A, because the solvers are designed for symmetric matrices. Table 10 also shows the number of iterations required to converge, which is an implementation-independent metric. In terms of the number of iterations required to converge, the ILU(1) preconditioner with the MINRES solver (15 in Table 6) outperforms other combinations. In many cases, the MINRES solver requires fewer iterations to converge than does the CG solver although the CG solver has a faster solver time than the CG solver. Note also that the ILU(0) preconditioner has a faster solver time than the ILU(1) preconditioner although the ILU(0) requires more iterations to converge. The least effective combination observed is the Jacobi preconditioner with the Bi-CGSTAB solver (4 in Table 6). In most cases, this combination diverges because the residual norm does not decrease. The Bi-CGSTAB solver shows irregular convergence behavior as was discussed in Section 5. The least efficient preconditioner-solver combination is the slowest combination which satisfies (5). In most cases, the least efficient combination is the Jacobi preconditioner with the GMRES solver (2 in Table 6). This combination sometimes does not converge due to the iteration limit. In terms of the solver time and the number of iterations required to converge, the choice of quality metric matters most when a poor combination of preconditioner and solver are used. For example, if we compare the solver time on the circle mesh for the ILU(0) preconditioner with the GMRES solver (2 in Table 6) amongst quality metrics (i.e., IMR and SV) in Table 10(a), the solver time for SV is 40% higher than it is for IMR. However, if a better combination is chosen (e.g., 5, 9, or 13 in Table 6), the difference between mesh quality metrics is not that significant. This demonstrates that the choice of preconditioner and solver is more significant than the choice of mesh quality metric in the solution of Poisson’s equation with Dirichlet boundary conditions.


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Table 10. Linear solver time (sec) and number of iterations required to converge for problem (a) as a function of mesh quality metric for the 16 preconditioner-solver combinations (see Table 6) on the four geometric domains. A ’*’ denotes failure. For each quality metric, the numbers in the top and bottom rows represent the linear solver time and number of iterations to convergence, respectively. (a) Circle. (1-16 denote the preconditioner-solver combinations.) Metric

1 381 1305 377 1303 446 1441 350 1307

IMR VCN SV SI

2 3 * 212 * 921 * 195 * 843 * 262 * 1101 * 187 * 860

4 * * * * * * * *

5 166 593 179 594 165 609 155 594

6 631 2052 772 2349 773 2322 613 1934

7 137 357 154 402 165 392 139 384

8 134 361 144 361 156 407 160 361

9 127 530 128 530 146 541 139 531

10 631 1684 708 2120 911 2885 795 2010

11 174 398 155 381 157 387 216 376

12 * * * * * * * *

13 79 284 75 284 87 284 87 284

14 252 759 263 747 350 757 252 756

15 98 206 114 210 198 212 86 208

16 83 197 98 211 99 204 107 212

(b) Gear. (1-16 denote the preconditioner-solver combinations.) Metric

1 111 606 129 606 118 623 121 604

IMR VCN SV SI

2 913 2411 963 2360 542 2198 757 2352

3 4 142 106 430 428 93 * 381 * 101 * 426 * 102 * 426 *

5 6 62 236 257 475 66 164 257 475 61 143 257 477 65 171 257 475

7 8 67 82 171 153 62 82 169 152 67 33 176 166 81 70 181 153

9 10 71 191 233 426 54 220 233 427 53 161 239 440 49 172 233 427

11 73 146 53 163 61 162 70 154

12 64 174 56 165 * * 68 176

13 59 134 43 134 40 134 43 134

14 102 156 86 156 64 156 54 156

15 73 87 69 85 42 93 44 94

(c) Bevel. (1-16 denote the preconditioner-solver combinations.) Metric IMR VCN SV SI

1 79 190 113 190 64 190 66 191

2 263 343 239 342 169 342 140 343

34 117 * 126 * 106 * 126 * 63 * 130 * 79 * 131 *

5 59 72 64 72 39 72 37 72

6 76 88 65 88 60 88 54 88

7 54 54 91 53 47 52 57 52

8 51 48 71 48 47 48 42 48

9 44 62 37 62 32 62 32 62

10 50 67 48 67 54 67 42 68

11 42 47 39 46 41 47 42 47

12 69 49 77 49 44 49 49 49

13 63 43 59 43 49 43 46 43

14 51 39 55 39 46 39 45 39

15 76 27 61 27 52 27 55 28

16 169 35 85 35 62 36 135 35

(d) Drill. (1-16 denote the preconditioner-solver combinations.) Metric IMR VCN SV SI

1 169 282 110 281 132 281 120 282

2 260 496 260 495 222 495 386 496

3 101 170 95 167 91 179 140 181

4 * * * * * * * *

5 51 103 70 103 61 103 57 103

6 7 8 84 65 66 118 75 62 81 67 94 118 77 62 71 98 64 118 75 62 85 90 49 118 76 62

9 49 93 52 93 63 93 58 93

10 72 100 106 100 76 99 80 100

11 74 64 66 58 60 68 58 57

12 13 14 15 16 128 75 90 53 97 70 49 47 31 41 66 72 64 102 81 69 49 47 32 41 150 118 88 58 73 69 49 47 30 41 53 59 56 67 84 69 49 47 32 41

16 91 113 47 90 45 87 54 104

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Table 11 shows the solver time and total time for the most and least efficient preconditioner-solver combinations as a function of geometric domain and mesh quality metric. Combinations which did not converge according to (5) are not considered here. In terms of the solver time, the best mesh quality metric varies with respect to the geometric domain, and there is no clear winner. The results also show the importance of choosing an appropriate combination of preconditioner and solver. For example on the gear domain, the least efficient combination (2 in Table 6) is 94% slower than it is for the most efficient combination (13 in Table 6). We also observe that the rank ordering for mesh quality metrics described in Section 7.1 is consistent with the total time rank ordering. The table shows that IMR with the most efficient (best) combination of preconditioner and solver outperforms the other mesh quality metrics. The total time for IMR is 88% less than it is for SI, for the most efficient (i.e., ILU(1) with CG) combination. For the least efficient combination (i.e., Jacobi with GMRES), IMR is 51% faster than that of the least efficient quality metric, i.e., SI.

Table 11. Solver time (sec) and total time (sec) for problem (a) as a function of the best (most efficient) and worst (least efficient) combination of preconditioner and solver for various mesh quality metrics. Only combinations satisfying (5) are considered. (a) Solver time Mesh Best worst Best Gear worst Best Bevel worst Best Drill worst Circle

Mesh quality metrics IMR VCN SV SI 79 75 87 87 631 772 911 795 59 43 40 43 913 963 542 757 42 37 32 32 263 239 169 140 49 52 61 49 260 260 222 386

(b) Total time Mesh Best worst Best Gear worst Best Bevel worst Best Drill worst Circle

Mesh quality metrics IMR VCN SV SI 120 142 604 842 672 839 1428 1550 98 108 563 827 952 1028 1065 1541 188 245 302 398 409 447 439 506 208 265 338 426 419 473 499 763


7.3

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Modifying the PDE Coefficients and Boundary Conditions

In this experiment, we modify both the PDE coefficients and boundary conditions and investigate whether or not the previous conclusions hold true for other elliptic PDE problems (i.e., problems (b)-(e) in Table 3). We first investigate the effect that including the mass matrix, M , has on the best combination of mesh quality metric, preconditioner, and solver for problems (b) and (c) in Table 3. Problem (b) and (c) represent an elliptic PDE with a mass matrix and Dirichlet and generalized Neumann boundary conditions, respectively. Table 12 shows the solver time and the number of iterations required to converge for these two problems. For this experiment, the IMR metric on the bevel domain is employed. Tables 12(a) and 12(b) represent the linear solver time and the number of iterations required to converge for problems (b) and (c), respectively. Similar trends are observed when other mesh quality metrics are employed. Interestingly, in terms of the solver time, the worst combination of preconditioner and solver is different from the previous results obtained for Poisson’s equation with Dirichlet boundary conditions. For Poisson’s equation with Dirichlet boundary conditions, the worst combination was the Jacobi preconditioner with the Bi-CGSTAB or GMRES solver (4 and 2 in Table 6). However, after we add the mass matrix with a=100 in (1), the worst combinations are the ILU(1) preconditioner with the CG or GMRES solver (13 and 14 in Table 6). In most cases, the best combination is the ILU(0) preconditioner with the MINRES solver (11 in Table 6). Different from Poisson’s equation, the solver time of the ILU(1) preconditioner with the CG or MINRES solver on problem (b) is up to 82% larger than it is for the most efficient combination of preconditioner and solver. This trend also occurs for other values of the PDE coefficients, such as a = 10, 50 in (1). This example shows that the best and worst combinations of the preconditioner and solver vary due to the addition of the mass matrix. However, in terms of the number of iterations required to converge, the worst combination (13 and 14 in Table 6) requires only two iterations to converge, mainly due to the slow generation time of the ILU(1) preconditioner for these cases. The observed trends are similar for problems (b) and (c). In terms of the smoothing time, the rank ordering amongst different quality metrics remains the same, i.e., IMR < VCN < SV < SI. In terms of the solver time, the choice of quality metric does not affect the results significantly. The most efficient combination of the mesh quality metric, preconditioner, and solver for solution of these elliptic PDEs is the IMR quality metric with the ILU(0) preconditioner and the MINRES solver. This is consistent with those obtained when solving Poisson’s equation with Dirichlet boundary conditions. However, the least efficient combination is now the ILU(1) preconditioner with the GMRES solver (14 in Table 6). Second, problems (d) and (e) in Table 3, are solved to see the effect of modifying the boundary conditions. Table 12 shows the solver time and the number of iterations required to converge for the IMR quality metric on the bevel domain. The other mesh quality metrics show similar trends. Tables 12(a) and

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Table 12. Linear solver time (sec) and number of iterations required to converge as a function of PDE coefficients and boundary conditions on the bevel domain. A ’*’ denotes failure. For this experiment, the IMR quality metric is used. The keys for the PDE problems and the preconditioner-solver combinations are given in Tables 3 and 6, respectively.

PDE problem (b) (c) PDE problem (d) (e)

1 155 185 83 188

1 3 10 6 16

2 193 335 179 349

Preconditioner-Solver 2 34 5 6 7 8 9 10 5 3 3 11 3 4 4 6 4 9 66 5534 3 3 10 9 8 8555 7 8 15 10 9 9955 8 8

Combinations 11 12 13 14 15 5 4 14 32 17 2 2 2 2 2 8 8 19 17 18 4 6 5 5 3

16 17 2 21 3

Preconditioner-Solver Combinations 3 4 5 6 7 8 9 10 11 12 13 74 * 44 162 93 41 57 64 52 85 96 134 * 77 96 58 48 71 90 50 57 37 105 112 47 55 61 52 42 42 62 61 62 131 133 72 88 54 48 60 72 46 51 34

14 98 36 45 34

15 50 27 59 23

16 58 30 71 32

12(b) represent the solver time and the number of iterations required to solve problems (d) and (e) in Table 3, respectively. Experimental results show that the most efficient mesh quality metric is still IMR if smoothing and solver time are considered together. The rank ordering amongst the various quality metrics is as follows: IMR < VCN < SV < SI. The least efficient combination of preconditioner and solver is the Jacobi preconditioner with the GMRES solver (2 in Table 6), which is consistent with previous results. There are multiple best combinations, e.g., the SSOR preconditioner with the CG solver (5 in Table 6). The solver time of the best combination is up to 77% faster than it is for the least efficient combination. However, the best combinations are similar to those seen for the previous PDE problems.

8 Conclusions and Future Work For all elliptic PDEs considered, the IMR metric is the most efficient. The least efficient mesh quality metric is SI. The best preconditioner-solver combination varies with respect to the PDE. For Poisson’s equation, the most efficient preconditioner-solver combinations, which have the fastest solver time, are the SSOR, ILU(0), and ILU(1) preconditioners with the CG solver. The ILU(0) or ILU(1) preconditioner with the MINRES solver shows comparable performance. The least effective combination observed is the Jacobi preconditioner with the Bi-CGSTAB solver, which always diverged. In many cases, the least efficient combination is the Jacobi preconditioner with the GMRES solver. In terms of the total time, IMR is 88% faster than it is for SI if the


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most efficient preconditioner-solver combination is employed. For the least efficient combination, IMR is still 51% faster than it is for SI. Modifying the coefficients and boundary conditions of the elliptic PDEs effects the efficiency ranking of the preconditioner-solver combinations. The efficiency rankings are more sensitive to modifications in the PDE coefficients than to modifications in the boundary conditions. Unlike the solution of Poisson’s equation with Dirichlet boundary conditions, the ILU(1) preconditioner with the CG or MINRES solver shows inefficient performance, in terms of the solver time, on elliptic PDEs involving a mass matrix. In addition, the solver time of the ILU(1) preconditioner with the CG or MINRES solver is up to 82% greater than it is for the most efficient combination of preconditioner and solver. However, the rank ordering amongst quality metrics is the same as it is for Poisson’s equation. For future research, we will investigate the relationship between the choice of mesh quality metric and the efficient solution of parabolic and hyperbolic PDEs on anisotropic unstructured meshes.

Acknowledgements The authors would like to thank Anirban Chatterjee and Padma Raghavan for interesting the third author in this area of research and Nicholas Voshell for helpful discussions. This work was funded in part by NSF grant CNS 0720749 and an Institute for Cyberscience grant from The Pennsylvania State University. This work was supported in part through instrumentation funded by the National Science Foundation through grant OCI-0821527.

References 1. Babuska, I., Suri, M.: The p and h-p versions of the finite element method, basic principles, and properties. SIAM Rev. 35, 579–632 (1994) 2. Berzins, M.: Solution-based mesh quality for triangular and tetrahedral meshes. In: Proc. of the 6th International Meshing Roundtable, Sandia National Laboratories, pp. 427–436 (1997) 3. Berzins, M.: Mesh quality - Geometry, error estimates, or both? In: Proc. of the 7th International Meshing Roundtable, Sandia National Laboratories, pp. 229–237 (1998) 4. Fried, E.: Condition of finite element matrices generated from nonuniform meshes. AIAA Journal 10, 219–221 (1972) 5. Shewchuk, J.: What is a good linear element? Interpolation, conditioning, and quality measures. In: Proc. of the 11th International Meshing Roundtable, Sandia National Laboratories, pp. 115–126 (2002) 6. Du, Q., Huang, Z., Wang, D.: Mesh and solver co-adaptation in finite element methods for anisotropic problems. Numer. Methods Partial Differential Equations 21, 859–874 (2005) 7. Du, Q., Wang, D., Zhu, L.: On mesh geometry and stiffness matrix conditioning for general finite element spaces. SIAM J. Numer. Anal. 47(2), 1421–1444 (2009)

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8. Ramage, A., Wathen, A.: On preconditioning for finite element equations on irregular grids. SIAM J. Matrix Anal. Appl. 15, 909–921 (1994) 9. Chatterjee, A., Shontz, S.M., Raghavan, P.: Relating Mesh Quality Metrics to Sparse Linear Solver Performance. In: SIAM Computational Science and Engineering, Costa Mesa, CA (2007) 10. Mavripilis, D.: An assessment of linear versus nonlinear multigrid methods for unstructured mesh solvers. J. Comput. Phys. 175, 302–325 (2002) 11. Batdorf, M., Freitag, L., Ollivier-Gooch, C.: Computational study of the effect of unstructured mesh quality on solution efficiency. In: Proc. of the 13th CFD Conference. AIAA, Reston (1997) 12. Freitag, L., Ollivier-Gooch, C.: A cost/benefit analysis of simplicial mesh improvement techniques as measured by solution efficiency. Internat. J. Comput. Geom. Appl. 10, 361–382 (2000) 13. Brewer, M., Freitag Diachin, L., Knupp, P., Leurent, T., Melander, D.: The Mesquite Mesh Quality Improvement Toolkit. In: Proc. of the 12th International Meshing Roundtable, Sandia National Laboratories, pp. 239–250 (2003) 14. Balay, S., Buschelman, K., Gropp, W.D., Kaushik, D., Knepley, M.G., McInnes, L.C., Smith, B.F., Zhang, H.: PETSc Webpage (2009), http://www.mcs.anl.gov/petsc 15. Becker, E.B., Carey, G.F., Oden, J.T.: Finite Elements: An Introduction. Prentice-Hall, Englewood Cliffs (1981) 16. Munson, T.: Mesh Shape-Quality Optimization Using the Inverse Mean-Ratio Metric. Mathematical Programming 110, 561–590 (2007) 17. Knupp, P.: Achieving finite element mesh quality via optimization of the Jacobian matrix norm and associated quantities, Part II - A framework for volume mesh optimization and the condition number of the Jacobian matrix. Internat. J. Numer. Methods Engrg. 48, 1165–1185 (2000) 18. Nocedal, J., Wright, S.: Numerical Optimization, 2nd edn. Springer, Heidelberg (2006) 19. Baker, A.H., Jessup, E.R., Kolev, T.V.: A simple strategy for varying the restart parameter in GMRES(m). J. Comput. Appl. Math. 230, 751–761 (2009) 20. Barrett, R., Berry, M., Chan, T.F., Demmel, J., Donato, J., Dongarra, J., Eijkhout, V., Pozo, R., Romine, C., Van der Vorst, H.: Templates for the Solution of Linear Systems: Building Blocks for Iterative Methods, 2nd edn. SIAM, Philadelphia (1994) 21. Saad, Y.: Iterative Methods for Sparse Linear Systems, 2nd edn. Society for Industrial and Applied Mathematics (2003) 22. Cyberstar Webpage, http://www.ics.psu.edu/research/cyberstar/index.html 23. Shewchuk, J.R.: Triangle: Engineering a 2D Quality Mesh Generator and Delaunay Triangulator. In: Lin, M.C., Manocha, D. (eds.) FCRC-WS 1996 and WACG 1996. LNCS, vol. 1148, pp. 203–222. Springer, Heidelberg (1996) 24. Si, H.: TetGen - A Quality Tetrahedral Mesh Generator and Three-Dimensional Delaunay Triangulator, http://tetgen.berlios.de/

Virtual Control Volumes for Two-Dimensional Unstructured Elliptic Smoothing Steve L. Karman Jr. SimCenter: National Center for Computational Engineering University of Tennessee at Chattanooga Chattanooga TN, USA [email protected]

Summary. A two-dimensional unstructured elliptic smoothing method is described where the Winslow equations are discretized using a finite volume approach. Virtual control volumes for each node are constructed with element shapes that are nearly ideal. Green-Gauss theorem is used to formulate gradients over an element or a collection of elements for a node, which ultimately leads to a coupled non-linear system of equations. Modifications enable the scheme to reproduce results similar to structured mesh schemes. Results are included that demonstrate basic mesh smoothing and boundary motion. In addition, layers of quadrilateral elements can be added to selected boundaries and the interior point positions are determined via elliptic smoothing. Keywords: Unstructured elliptic smoothing, Winslow smoothing, Finite-volume control volume, Mesh quality improvement.

1 Introduction Elliptic smoothing has been routinely used with structured meshes for many years [1]. It is a very mature technology that allows mesh generation practitioners to create smooth structured meshes. Forcing functions have been developed that can influence the grid spacing and grid line angularity to meet the specific needs of the analysis tool [2, 3, 4]. This technology has been mostly unavailable to the unstructured mesh community, due to the lack of a global mapping from physical space to computational space. Knupp described an approach for smoothing two-dimensional unstructured meshes using a finite difference method where the local logical space was created using trigonometric functions [5]. Finite difference formulae were then derived using Taylor series expansions about the central node. Results in his article show quadrilateral meshes, although the derivation includes an arbitrary number of neighboring nodes, which could be part of triangles and quadrilaterals. Knupp also refers to a proprietary publication by Tipton that is an extension of the method to three dimensions using a finite element approach [6]. Karman et. al described a finite-volume method in two and three dimensions for unstructured meshes comprised of mixed elements [7]. In the finite-volume approach elliptic smoothing of unstructured meshes was possible

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if an existing valid mesh is available to serve as the computational domain. Virtual control volumes were then introduced where a local computational space for each node is constructed that is independent from a neighboring node’s computational space [8]. These virtual control volumes are the key component of the method and the current paper describes in detail the construction and use of these virtual control volumes for meshes comprised of quadrilaterals and triangles. Aspects of the discretization strategy are explored to allow for the same type of grid control afforded to the structured community. Results for smoothing of static and dynamic problems are included. The cases include hybrid meshes where the quadrilateral elements are added to existing meshes with the points positioned using elliptic smoothing.

2 Winslow Smoothing on Structured Meshes Most elliptic smoothing methods utilize the Winslow equations as the governing equations for mesh distribution [1]. These equations are derived from a Laplacian operator or Poisson operator applied to the computational coordinates. The Laplacian operator form is shown in equations (1). When satisfied, these relationships describe a smooth distribution of computational coordinates (ξ, η) in physical space (X, Y).

∇ 2ξ = ξxx + ξyy = 0 ∇ 2η = ηxx + ηyy = 0

(1)

Theoretically the Laplace equations satisfy the max-min property, which states that the parameter on the interior of the domain will not exceed the values on the boundary. For mesh generation this means the grid lines will not cross. The known variables are the computational coordinates ξ and η. The unknown variables are x and y. So a transformation is necessary to cast the equations in computational space, resulting in the Winslow equations given in equations (2).

αxξξ − 2 βxξη + γxηη = 0 αyξξ − 2 βyξη + γyηη = 0 α = xη2 + yη2

(2)

β = x ξ x η + yξ y η γ = xξ2 + yξ2 Winslow smoothing is not the same as Laplacian smoothing. Many researchers will perform Laplacian smoothing, which is based on the Laplacian operator in equation (1) cast in computational space, shown in equation (3).

∇2 x = xξξ + xηη = 0 ∇2 y = yξξ + yηη = 0

(3)

Virtual Control Volumes for Two-Dimensional Unstructured Elliptic Smoothing

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Laplacian smoothing for structured meshes is essentially an averaging of surrounding nodes physical coordinates. The same is true when implemented for unstructured meshes. It does not contain the cross derivative term of the Winslow equations and can result in grid crossing, especially for non-convex domains. Finite Difference For structured meshes the Winslow equations are typically solved on an equally spaced Cartesian grid in computational space. The stencil typically used to discretize the first and second derivative terms is shown in Figure 1. The numerical formulae for the first and second derivatives for x in ξ and η directions is shown in equations (4). These are centered difference formulae for the central node. Notice that the derivatives in the ξ direction use only nodes on a constant j line and the derivatives in the η direction using only nodes on a constant i line, while the cross derivative formula uses only the corner nodes. Figure 2 shows the resulting mesh for a spike case where the center of the bottom boundary protrudes into the middle of the domain. The boundary nodes are fixed in position. Winslow smoothing produces a mesh without crossing at the spike. This illustrates that the max-min property of the underlying Laplacian operator is being enforced.

xξ = xη = xξξ = xηη = xξη =

x i+1, j − x i−1, j 2Δξ x i, j +1 − x i, j −1 2Δη x i+1, j − 2x i, j + x i−1, j Δξ 2 x i, j +1 − 2x i, j + x i, j −1

(4)

Δη2 x i+1, j +1 − x i+1, j −1 − x i−1, j +1 + x i−1, j −1 ΔξΔη

3 Virtual Control Volumes for Unstructured Meshes In order to solve the Winslow equations on unstructured meshes a computational domain must be created. A global computational domain is possible if one uses an existing valid mesh. However, this will not provide any real benefit, since a valid mesh already exists. Instead, it is possible to construct a computational domain for each node. These “virtual control volumes” were first introduced in References [8] and [9]. Under the premise that a local computational domain comprised of nearest neighbors is necessary to assemble the global system of equations, it is

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possible to construct nearly ideal computational stencils for each node. This does assume that the valid global mesh connectivity does exist. These stencils are comprised of the elements connected to the central node. However, instead of using the physical coordinates for each node in the stencil, computational coordinates are constructed. These computational elements can be created to have nearly ideal shape quality or can be manipulated to reflect a desired stretching or clustering, corresponding to a viscous layer distribution or an adaptation spacing field.

Fig. 1. Finite difference stencil

Fig. 2. Structured mesh Winslow solution for spike case

The creation of these computational stencils in two dimensions is relatively straightforward. The stencil for each node will have an arbitrary number of surrounding elements and neighboring nodes; information that is contained in the global mesh connectivity. The process involves distributing directly connected nodes on the unit circle and then positioning any secondarily connected nodes to produced ideal element shapes. Directly connected nodes are those nodes that share an edge with the central node. Secondarily connected nodes do not share an edge with the central nodes, such as the opposite corner of a quadrilateral element. Figure 3 shows a region of a hybrid mesh containing triangular and quadrilateral elements. Figure 4 shows a sample virtual control volume for a node at the edge of the quadrilateral layers (any of the 4 nodes in the center region of the plot with the same element connectivity). The central node is labeled ‘0’. The surrounding nodes are numbered 1 through 7, in order. The directly connected nodes are 1, 2, 3, 4, and 6. The secondarily connected nodes are 5 and 7. The first directly connected node is positioned on the unit circle in a direction that corresponds to the direction of the physical node, although this is not necessary. It could be positioned at computational coordinate (1, 0). The remaining directly connected nodes are then distributed on the unit


Fig. 3. Physical mesh containing triangles and quadrilaterals

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Fig. 4. Sample virtual control volume for node at edge of quadrilateral layers in Figure 3

circle according to the angle distributions, θt, and θq. These angles are determined based on the number of triangles (nt) and the number of quadrilaterals (nq) connected to the central node, given in equations (5). The goal is to produce angles and computational coordinates that reflect the desired element shapes in physical space.

if nt = 0 else if nq = 0 else if nq = 1 and nt > 1 else if nt = 1 and nq > 1 else if nt > 1 and nq > 1

2π nq 2π θt = nt 3π π θt = , θq = 2nt 2 π 3π θt = , θq = 2 2nq

θq =

θt =

π nt

, θq =

(5)

π nq

The secondarily connected nodes are then placed opposite the central node using a vector emanating from the central node in the direction of the midpoint of the diagonal edge of the connected nodes, shown in Figure 4 as edge 4 – 6. That vector is copied to the midpoint of the diagonal edge and the secondarily connected node, node 5 in the figure, is positioned at the end of the vector. The length of that vector is the minimum of the original vector length and 3 2 times the length of the diagonal edge. This will result in perfect squares for the quadrilateral elements when the number of quadrilaterals is equal to 1, 2 or 4. Otherwise, the outer portion of the quadrilateral will be an equilateral triangle. Examples of

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virtual control volumes for all-triangular stencils and all-quadrilateral stencils are shown in Figure 5.

Fig. 5. Examples of virtual control volumes for different numbers of triangles and an allquadrilateral stencil

Triangles in these virtual control volume will be equilateral only when the triangle angle, θt, is equal to π/3, which occurs with 6 triangles in an all-triangular stencil or 3 triangles in a hybrid stencil, such as the one shown in Figure 4. Changing the lengths of the edges emanating from the central node does not alter the angle relationships. These are the “best” control volumes possible and the physical mesh will attempt to mimic the shapes of these virtual elements. These two-dimensional control volumes are essentially equivalent to the local logical space described by Knupp [5]. The adjacent nodes are ordered in a counter-clockwise fashion and the computational coordinates are distributed on a unit circle. Knupp’s finite difference formulae include cosine and sine functions, which can be computationally prohibitive. The current approach computes and stores computational coordinates. The method could be extended to include general polygons instead of just triangles and quadrilaterals.

5 Finite Volume Formulation of Winslow Equations One approach to solving the Winslow equations is to consider the coefficients, α, β, γ as constant. Then integrate the first equation from (2) over ξ-η space.

α ∫∫ xξξ dΩ − 2 β ∫∫ xξη dΩ + γ

∫∫ xηη dΩ = 0

(6)

Ω represents the area in two dimensions. Examine the first integral and define such that

v = xξ iˆ + 0 ˆj ∇ • v = xξξ

v

(7)


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The divergence theorem can be used to convert the double integral to a line integral.

∫∫

∇ • F dΩ =

∫ F •nˆdΓ

(8)

Γ represents the boundary in two dimensions. Therefore, the first integral becomes

∫∫ xξξ dΩ = ∫ xξ nˆξ dΓ

(9)

A similar operation is used to convert the other double integrals terms. The resulting set of integral equations become

α ∫ xξ nˆ ξ dΓ − 2 β ∫ xη nˆ ξ dΓ + γ

∫ x nˆ dΓ = 0 α ∫ y nˆ dΓ − 2 β ∫ y nˆ dΓ + γ ∫ y nˆ dΓ = 0 ξ

ξ

η ξ

η η

(10)

η η

The numerical representation of these surface integrals is given in equations (11). The summation is over the surrounding elements. This is illustrated in Figure 6 for a hybrid node surrounded by 3 triangles and 2 quadrilaterals for a total of 5 elements. ne

ne

ne

i=1

i=1

i=1

ne

ne

ne

i=1

i=1

i=1

α ∑[xξ ]i (tξ ) i − 2 β∑[xη ]i (tξ ) i + γ ∑[xη ]i (tη ) i = 0 α ∑[yξ ]i (tξ ) i − 2 β∑[yη ]i (tξ ) i + γ ∑[yη ]i (tη ) i = 0

(11)

The normal vectors, t , correspond to the normal vectors of the external edge of the surrounding elements. The first derivative terms in (11) are replaced with a formula for the corresponding component of a gradient calculation of each

Fig. 6. Control volume and external area vectors for a hybrid node

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element using a Green-Gauss formulation. For instance, the derivative of x with respect to ξ is given by (12).

(x l + x r ) (nξ ) j 2 j =1 ns

∑ xξ =

(12)

A

where A represents the area of the element. The subscripts l and r correspond to the left and right node numbers for edge j. The ξ component of the normal vector for side j is (nξ ) j . The summation is carried out over the number of sides of the element. A similar expression can be developed for the η derivative. Substituting for all the first derivative terms will result in a system of equations for node 0 using nearest neighbor node information. Combining the equations from all nodes in the mesh will result in a sparse matrix linear system of equations that can be solved for new physical coordinates (x,y) using techniques such as point-implicit scheme with under-relaxation. Since the coefficients α, β and γ were assumed frozen, but are actually functions of the physical locations, an outer iteration loop is required to update these coefficient values. When a point-implicit method is used an inner iteration loop is used to converge the linear system. Full convergence of the inner iterations is not necessary. Convergence of the outer iterations is required. Quadrilateral Case A comparison of the finite volume approach with the structured, finite-difference approach is instructive. The structured mesh from Figure 2 was converted to an unstructured collection of quadrilaterals and solved using the finite volume methods described above. The control volume for an interior node is shown in Figure 7. In this case, the first directly connected node is positioned at computational coordinate (1, 0). The resulting unstructured Winslow solution in Figure 8 is different from the structured mesh solution. There is grid crossing at the top of the spike and some kinks in the mesh near the corners on the bottom boundary. An examination of first derivatives using the finite volume approach reveals differences with the finite difference formulae. Assuming unit edge lengths in the control volume, equation (13) shows influence from the corner nodes 2, 4, 6 and 8 in computing the derivative in the ξ direction, whereas the structured finite difference formula uses only information corresponding to nodes 3 and 7.

xξ =

(x 4 + 2x 3 + x 2 ) − (x 6 + 2x 7 + x 8 ) 8

(13)

The finite volume numerical formula for the surface integral of term 1 in equation (10) is shown below, equation (14). Clearly there is influence from all nodes on


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the top and bottom edges of the control volume. This is in direct contrast to the finite difference formula for xξξ in equations (4).

Fig. 7. Control volume for central node in Finite Fig. 8. Result of Winslow smoothing using Volume method Finite Volume method

∫ xξ nˆξ dΓ =

x 4 − 2x 5 + x 6 x − 2x1 + x 8 + x 3 − 2x 0 + x 7 + 2 2 2

Fig. 9. Control volumes for modified FiniteVolume method

(14)

Fig. 10. Winslow solution using modified control volumes

This is a common problem associated with finite-volume solutions to diffusion type equations, such as the Laplace equation. If the structured mesh result is considered “correct” and serves as a guide then the finite volume scheme must be modified to exclude those corner nodes where appropriate. In order to replicate the

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structured mesh result the control volumes used with the terms involving α and γ (the first and third integral) must be replaced with the collection of triangles shown in blue in Figure 9. This is sometimes referred to as “cutting the corner” and it ignores the influence of the opposite node in the quadrilateral. The first derivatives for the central node, used to compute α, β, and γ, are also computed using this modified stencil. The finite volume result for the second integral correctly uses only the corner nodes, so the full stencil comprised of the 4 quadrilaterals is still used for the β term. The resulting smoothed mesh using this hybrid stencil is shown in Figure 10, which is identical to the structured mesh result. If the “cut the corner” stencil (blue triangles) is used for all three integrals the result is still valid, shown in Figure 11, but different from the structured mesh result. The main difference occurs at the spike. Clearly the discretization strategy used for the β term is influencing the result. In fact, for the stencils comprised of the four blue triangles shown in Figure 9 the β term computes to exactly zero. This was verified by numerically ignoring this term and the result was identical to the Fig. 11. Unstructured Winslow solution using all triangle mesh shown in Figure 11. stencils It is interesting to note that using the hybrid stencil (cut-the-corner for the α and γ terms and full stencil for the β term), but rotating the entire stencil by 45 degrees will also produce the result shown in Figure 11. Apparently the hybrid stencil is not invariant to rotation. This is not a desirable property. For nodes with only four quadrilaterals it is easy to align the stencils with the computational coordinates. For nodes with triangles and quadrilaterals or nodes with more than 4 quadrilaterals it may be difficult to devise a strategy to align the stencil in the computational domain. Triangular Case Diagonalizing, in a symmetric manner, the quadrilaterals from the initial Stransfinite interpolation mesh for Figure 2 produces the mesh in Figure 12. Using the original finite volume scheme described above produces the result in Figure 13.


Fig. 12. Quadrilateral mesh diagonalized to produce all triangles

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Fig. 13. Winslow smoothed triangle mesh for spike

Everything appears valid until one examines the region near the top of the spike, shown in Figure 14. The first interior node above the spike has been moved below the boundary, resulting in grid crossing. The majority of the interior nodes contain six triangles. The virtual control volume for those nodes is shown in Figure 15.

Fig. 14. Magnified view of mesh near spike Fig. 15. Virtual control volume for most of the interior nodes of the triangular spike mesh

Rotating the stencil by 45 degrees does not change the outcome, so the original finite volume scheme appears to be invariant to rotation. All derivatives are computed using a consistent Green-Gauss derived formulation. As the stencil is rotated the gradient components are altered, but the magnitude of the derivative stays constant.

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5 Augmented Stencil for Triangles Since including the opposite corner node of quadrilaterals in the β term was helpful, it is possible to expand the stencil for triangles to bring in more information. Figure 16 shows an expanded stencil that includes the first neighboring triangle of each directly connected triangle. Also shown in bold red is a quadrilateral comprised of a directly connected triangle and the augmented neighbor. This will serve as the “quadrilateral” in the hybrid stencil described earlier. This is similar to Knupp’s use of the opposite node of the quadrilateral when the number of surrounding neighbor nodes was insufficient to determine the required finitedifference derivatives. The α and γ terms will use only the directly connected triangles and the β term will use the augmented quadrilaterals. For nodes containing a mix of triangles and quadrilaterals in the original stencil, only the triangles will be augmented with neighboring triangles.

Fig. 16. Augmented stencil for nodes with triangular elements

Fig. 17. Triangle mesh with augmented stencils

The smoothed mesh using the augmented triangle stencils is shown in Figure 17. This is a valid mesh with no grid crossing. However, it is not symmetric from left to right, as expected. The cause is believed to be the augmented stencils near the boundary. One of the directly connected triangles of these nodes cannot be augmented with a neighbor because a neighbor does not exist. Instead the stencil will be unbalanced and the “first” node of the stencil may differ from node to node, resulting in arbitrary alignment of the unbalanced-augmented stencil with respect to the computational coordinate directions. Rotating all the stencils by 45 degrees results in a valid mesh, however the asymmetry still exists but is flipped from right to left. If aligning node “1” with the physical edge direction is used it requires constant adjustments to the stencil and still does not produce a completely symmetric mesh.


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Augmenting the stencils with adjacent triangles has provided valid meshes without grid crossing for the spike case. The sensitivity to the rotation of the control volume is bothersome. This is not a desirable trait and further research is required to fully understand the causes and devise an appropriate modification to the discretization strategy. The goal should be to devise a strategy that accurately models all three terms of the equations and is invariant to rotation in computational space. An important point to be made, though, is the augmented stencil approach does provide meshes without grid crossing. And this method can be turned on or off as needed. Even without the augmented stencil the method seems to work extremely well for most cases. The exceptions are cases with extreme sharp edges. If the computational mesh for the structured case were rotated through any angle the original finite-difference formulae would no longer be valid. Taylor series expansions could be re-derived for the new orientation or a least squares method could be used to compute the derivatives. But modified difference formulae have already been used for a structured mesh. The modified scheme described in the previous section was carefully constructed for the all-quadrilateral mesh in order to reproduce the structured mesh result. All stencils were comprised of exactly four quadrilaterals that were carefully aligned with the ξ and η directions. The finite-volume formulation is equivalent to performing a Taylor series expansion about the central node if the higher order terms are ignored. When those stencils were rotated 45 degrees the results were different, but still valid. This demonstrates that the discretization strategy for structured meshes is also not invariant to rotation.

6 Results All cases shown below are computed without any manipulation of the control volumes to account for viscous-type spacing or for solution based adaptation. Some cases are computed using the original finite volume approach outlined in Section 4. For those cases, no augmented stencils were used for the triangles and the “cutthe-corner” approach was used for quadrilaterals. Thus the scheme is invariant to scaling, translation and rotation of the virtual control volumes. Other cases used the augmented stencils to obtain valid meshes where sharp corners exist. Rotating Circle The first case is a simple geometry consisting of an outer circular boundary and an inner circular boundary. The original mesh was created using Pointwise [10]. The mesh contains 2,415 nodes and 4,710 triangles. The inner and outer boundaries are comprised of two sections with 31 equally spaced points each for a total of 61 point for each circle. Views of the original mesh can be seen in Figure 18 and Figure 20. The original finite volume scheme without augmented stencils was used for this case. The Winslow smoothed mesh is shown in Figure 19 and Figure 21. The differences between the meshes are slightly more evident in the magnified views. The original mesh is certainly valid and contains high quality triangles. Pointwise uses a Delaunay method to triangulate the points and the distribution is very smooth. The Winslow smoothed mesh does not attempt to enforce the

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Delaunay property. Winslow smoothing attempts to make the physical mesh match the “ideal” element shapes in the virtual control volumes. No attempt is made to declare one mesh better or worse than the other. This merely shows that Winslow smoothing can be used with existing tessellations where the boundaries are held fixed. In cases where the initial distribution of points was less smooth the Winslow method will show more variation from the Delaunay-based result.

Fig. 18. Original mesh for circle case

Fig. 19. Winslow smoothed mesh

Fig. 20. Magnified view of original mesh near Fig. 21. Winslow smoothed mesh near inner boundary inner boundary

One of the major benefits Winslow smoothing provides is the ability to perform mesh movement. So for the next demonstration the inner circle is rotated 90


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degrees counterclockwise. The inner boundary points are held fixed at the new location and the outer boundary points remain in their original position. The global mesh is shown in Figure 22. The twisting of the interior mesh is evident in the global view, but is less evident in the magnified view, Figure 23. The shearing of the mesh is minimal for this rotation angle. Additional rotation of the inner boundary is certainly possible and would increase the shearing, unless the points on the boundaries were allowed to float.

Fig. 22. Global view of rotated circle mesh

Fig. 23. Magnified view of inner boundary for rotated circle mesh

The next modification to this rotated mesh is shown in Figure 24. Ten layers of quadrilateral elements were added to the mesh connectivity at the inner boundary. The smoothed quadrilateral elements show evidence of the shearing exhibited by the triangular elements in Figure 22. The basic method was employed without any attempt at controlling mesh spacing or grid line angularity. Winslow is attempting to make the physical mesh match the virtual control volume shapes, which are nearly ideal. When the inner boundary is rotated back to the original position the shearing of the quadrilateral elements disappears, shown in Figure 25. The sequence of smoothing, rotating and adding layers of quadrilateral elements could have been performed in any order. The same mesh would be produced had the rotation never taken place. And the creation of the quadrilateral elements does not require one to distribute the points in the normal direction away from the boundary. The physical points were actually initialized to be coincident with the original boundary points. Winslow smoothing produced the final point distribution, attempting to make the quadrilateral elements perfect squares.

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Fig. 24. Quadrilateral layers added to rotated mesh

Fig. 25. Hybrid mesh rotated back to original position

NACA0012 Airfoil The second case is the famous NACA0012 airfoil. The original triangular mesh, created in Pointwise, contains 4,392 points and 8,424 triangles. The outer boundary is in two sections with 61 points per section, equally spaced. The airfoil boundary is also defined in two sections, upper and lower, with 121 points per section. The points are clustered toward the leading and trailing edges in a symmetric manner from top to bottom. The augmented stencils were used to smooth this triangular mesh. The global mesh after Winslow smoothing is shown in Figure 26. A magnified view of the region near the airfoil is shown in Figure 27. Ten quadrilateral element layers were added to the connectivity near the airfoil boundary. The β term is discretized with the full quadrilateral and augmented triangle formulations. The smoothed mesh is shown in Figure 28. At first it might seem like this is not the expected outcome. But in fact it is because no grid spacing control is being applied. The virtual control volumes for the quadrilateral elements are perfect squares and the physical mesh is attempting to mimic that shape for the quadrilaterals. So where there is clustering at the leading and trailing edge the local quadrilaterals (squares) are smaller. In the middle of the airfoil, shown in Figure 28, the point spacing along the surface is larger and the neighboring quadrilaterals are nearly perfect squares there, as well. A view of the trailing edge region is shown in Figure 29. Finally, this hybrid mesh is translated one unit in the positive X direction and one unit in the positive Y and rotated 90 degrees. The smoothed mesh is shown in Figure 30. There was some difficulty encountered converging the solution due to


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Fig. 26. Global smoothed mesh for NACA0012 airfoil

Fig. 27. Magnified view of triangular mesh in region near airfoil

Fig. 28. Smoothed mesh with 10 quadrilateral layers added near airfoil

Fig. 29. Magnified view of hybrid mesh near trailing edge of airfoil

Fig. 30. Airfoil translated in X and Y and rotated 90 degrees

Fig. 31. Magnified view of mesh near the trailing edge

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the skewing of the triangles in the far field. The augmented triangles were turned off and the scheme was able to converge. The full quadrilateral formula was used for the β term in order to accurately handle the sharp trailing edge. Figure 31 shows the magnified view of the trailing edge region. There are no negative areas or skewed elements in the mesh. 30P30N Multi-Element Airfoil The final case is a 3-element airfoil section known as 30P30N. The triangular mesh was created using Pointwise and it consisted of 22,165 points and 42,743 triangles. The outer boundary was defined in four sections with 21 points per section, equally spaced. The slat section was divided into two sections with the sharp points of the section providing the demarcation. One section contained 200 points while the other contained 150 points. The main element was defined in three sections and had three sharp points that defined the sharp trailing edge and the cove region. The largest section contained 600 points and the other two contained 100 points and 50 points. The flap was divided into 3 sections; an upper section with 200 points, a lower section with 200 points and blunt trailing edge section with 7 points. The distributions on each section were clustered toward the sharp points. This initial mesh was smoothed using the augmented triangle stencils. Figure 32 shows the smoothed mesh with the three elements. The slat region is shown in more detail in Figure 33 and the flap is detailed in Figure 34. The gap region between the main airfoil element and the flap is magnified in Figure 35.

Fig. 32. Three-element airfoil mesh

Fig. 33. Magnified view of leading edge slat

The flap was rotated an additional 20 degrees downward and the slat an additional 10 degrees downward. The smoothed meshes are shown in Figure 36 and Figure 37, compared to the meshes in Figure 33 and Figure 34. Some excessive stretching was encountered at the sharp trailing edge of the main airfoil section, Figure 38. The augmented stencils were turned off to produce the mesh in Figure 39, which shows slightly less skewing.


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With the flap rotating the elements in the gap are stretched. The elliptic smoothing is attempting to restore the elements to the ideal shapes provided in the virtual control volumes. The augmented stencil and the β term carried the affect of the skewing around the trailing edge and on to the topside of the airfoil section. Turning off the augmented stencil lessened the effect.

Fig. 34. Magnified view of trailing edge flap

Fig. 35. View of gap between main element and trailing edge flap

Fig. 36. Magnified view of slat rotated 10 degrees downward

Fig. 37. Magnified view of flap rotated 20 degrees downward

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Fig. 38. Main element trailing edge using augmented triangle stencils

Fig. 39. Main element trailing edge without using augmented triangle stencils

Five quadrilateral element layers were added to the original, un-rotated triangular mesh from Figure 32. The smoothed mesh, computed without augmented triangle stencils and the “cut-the-corner” approach for the quadrilaterals, is shown in Figure 40. A magnified view of the gap between the slat and main element is shown in Figure 41. The mesh near the sharp tip of the slat is smooth and contains no grid crossing. The mesh for the backward-facing step of the cove region is shown in Figure 42. The pulling of the mesh out of the corner is typical for Winslow smoothing with structured meshes. Again, no control of grid spacing was attempted, so this is an expected result. The mesh in the gap between the main element and the flap, shown in Figure 43, shows some skewing at the sharp trailing edge. The quadrilaterals at the sharp corner are skewed, with a slightly inverted corner, but the areas are still positive. Grid spacing control is expected to lessen the skewing in this region since there will be more room between the airfoil sections for the isotropic (triangle) portion of the mesh.


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Fig. 40. Three-element airfoil with five quadrilateral layers added

Fig. 41. Magnified view of hybrid mesh in region between slat and main element

Fig. 42. Hybrid mesh in cove region of main airfoil section

Fig. 43. Hybrid mesh in gap between main element and flap

Conclusions The development of a two-dimensional unstructured elliptic smoothing method has been described. The Winslow equations are discretized using a finite volume approach. Individual control volumes for each node are constructed in a virtual space where the element shapes are nearly ideal. The baseline discretization uses Green-Gauss theorem to formulate gradients over an element or a collection of elements for a node. As coded this formulation would not reproduce the structured mesh elliptic smoothing behavior for a spike problem. A modification to the scheme for quadrilateral elements was successful at producing structured mesh results for the spike problem. The triangulated version of the spike mesh was

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smoothed resulting in one node near the spike outside the domain. A second modification to the finite-volume method incorporated additional triangles in the virtual control volume stencil that could serve as a basis for incorporating the same discretization used for quadrilateral elements, resulting in a valid mesh for the spike problem. Results were shown for three additional cases. For each case basic smoothing was demonstrated as well as boundary motion. The resulting meshes were smooth and contained no invalid elements. The final demonstration for each case was to add layers of quadrilateral elements to selected boundaries. The points were distributed via the Winslow smoothing. No mesh spacing control was attempted and the resulting meshes were smooth, without grid crossing or invalid elements. The three-dimensional extension has been completed and will be published in the near future.

References 1. Winslow, A.M.: Numerical Solution of the Quasilinear Poisson Equations in a Nonuniform Triangle Mesh. Journal of Computational Physics 2, 149–172 (1967) 2. Thompson, J.F., Thames, F.C., Mastin, C.W.: Boundary-Fitted Curvilinear Coordinate Systems for Solution of Partial Differential Equations on Fields Containing Any Number of Arbitrary Two-Dimensional bodies. In: NASA CR 2729 (July 1977) 3. Thomas, P.D., Middlecoff, J.F.: Direct Control of the Grid Point Distribution in Meshes Generated by Elliptic Equations. AIAA Journal 18, 652–656 (1979) 4. Sorenson, R.L.: A Computer Program to Generate Two-Dimensional Grids About Airfoils and Other Shapes by Use of Poisson’s Equations. NASA TM-81198 (1980) 5. Knupp, P.M.: Winslow Smoothing on Two-Dimensional Unstructured Meshes. Engr. With Computers 15, 263–268 (1999) 6. Tipton, R.: Grid Optimization by Equipotential Relaxation. Lawrence Livermore National Laboratory, 26pp (November 28, 1990) (manuscript) 7. Karman, S.L., Karman Jr., S.L., Anderson, W.K., Sahasrabudhe, M.: Mesh Generation Using Unstructured Computational Meshes and Elliptic Partial Differential Equation Smoothing. AIAA Journal 44(6), 1277–1286 (2006) 8. Sahasrabudhe, M.: Unstructured Mesh Generation and Manipulation Based on Elliptic Smoothing and Optimization. Ph.D. Dissertation, University of Tennessee at Chattanooga (August 2008) 9. Karman Jr., S.L., Sahasrabudhe, M.S.: Unstructured Elliptic Smoothing Revisited. AIAA-2009-1362, Orlando, FL (January 2009) 10. http://www.pointwise.com/

Reconstructing High-Order Surfaces for Meshing Xiangmin Jiao and Duo Wang Department of Applied Mathematics, Stony Brook University, Stony Brook, NY 11794 [email protected]

Summary. We consider the problem of reconstructing a high-order surface from a given surface mesh. This problem is important for many meshing operations, such as generating high-order finite elements, mesh refinement, mesh smoothing and mesh adaptation. We introduce two methods, called Weighted Averaging of Local Fittings (WALF) and Continuous Moving Frames (CMF). These methods are both based on weighted least squares polynomial fittings and guarantee C 0 continuity. Unlike existing methods for reconstructing surfaces, our methods are applicable to surface meshes composed of triangles and/or quadrilaterals, can achieve third and even higher order accuracy, and have integrated treatments for sharp features. We present the theoretical framework of our methods, experimental comparisons against other methods, and its applications in a number of meshing operations. Keywords: mesh generation; curves and surfaces; mesh adaptivity; high-order methods; accuracy.

1 Introduction Surface meshes and their manipulations are critical for meshing, numerical simulations, and many other related problems. Some example problems that involve manipulating surface meshes include mesh generation and mesh enhancement for finite element or finite volume computations [5], mesh smoothing in ALE methods [1], and mesh adaptation in moving boundary problems [11]. In many of these problems, a continuous CAD model may not be available. Instead, only a surface mesh, typically with piecewise linear or bilinear faces, is available. In this paper, we consider the problem of reconstructing a highly accurate, continuous geometric support from a given surface mesh. We refer to this problem as high-order surface reconstruction (or simply high-order reconstruction). Besides meshing, this reconstruction problem also arises in computer graphics [2] and geometric modeling [17]. In general, the high-order reconstruction should satisfy some (if not all) of the following requirements:


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Continuity: The reconstructed surface should be continuous to some degree (e.g., C 0 , C 1 , or C 2 continuous, depending on applications). Feature preservation: The reconstruction should preserve sharp features (such as ridges and corners) in the geometry. Geometric accuracy: The reconstruction should be accurate and asymptotically convergent to the exact surface to certain order under mesh refinement. Stability: The reconstruction should be numerically stable and must not be oscillatory under noise. Note that different applications may have emphasis on different aspects of the problem. For example, in computer graphics and geometric design, the visual effect may be the ultimate goal, so smoothness and feature preservation may be most important. Therefore, methods proposed for such applications tend to focus on the first two issues, and the numerical issues of asymptotic convergence and stability are mostly ignored. Our focus in this paper is on meshing for finite element or finite volume computations, for which these numerical issues are very important. Indeed, the low order of accuracy of the geometry would necessarily limit the accuracy of the solutions of differential equations, and numerical instabilities and excessive oscillations can have even more devastating effect to numerical simulations. In this paper, we present two methods, called Weighted Averaging of Local Fittings (WALF) and Continuous Moving Frames (CMF), for reconstructing a featurepreserving, high-order surface from a given surface mesh. Both methods are based on the assumptions that the vertices of the mesh accurately sample the surface, and the faces of the mesh correctly specify the topology of the surface, and utilize the numerical techniques of weighted least squares approximations and piecewise polynomial fittings. These methods apply to surface meshes composed of triangles and/or quadrilaterals, and also to curves (such as ridge curves on a surface). Unlike existing methods, which are typically only first or second order accurate, our methods can achieve third- and even higher order accuracy, while guaranteeing C 0 continuity. For its weighted least squares nature, these methods are also tolerant to noise. We present the theoretical framework of our methods. We also present experimental comparisons of our methods against some others, and its applications in a number of meshing operations. The remainder of the paper is organized as follows. Section 2 presents some background knowledge, including local polynomial fittings and weighted least squares approximations. Section 3 describes the new methods for high-order surface reconstruction and compares them with some others. Section 4 applies our methods with a number of meshing operations for triangle and quadrilateral meshes and meshes with sharp features. Section 5 concludes the paper with a discussion.

2 Preliminaries: Vertex-Based Polynomial Fittings Our high-order reconstruction is based on local polynomial fittings and weighted least squares approximations. We have successfully used these techniques previously to compute differential quantities of discrete surfaces (such as normals and

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curvatures) to high-order accuracy; see e.g. [13, 18]. However, due to their local nature, those approaches for computing differential quantities do not provide a continuous, global reconstruction of a surface. We hereafter briefly review these techniques and then adapt them to high-order reconstruction in the next section. For more details on the theoretical background, readers are referred to [13] and references therein. 2.1

Local Polynomial Fitting

Local polynomial fittings, also known as Taylor polynomials in numerical analysis [8], are based on the well-known Taylor series expansions about a point. We are primarily concerned with surfaces, so the local fitting is basically an interpolation or approximation to a neighborhood of a point P under a local parameterization (say, with parameters u and v), where P corresponds to u = 0 and v = 0. The polynomial fitting may be defined over the global xyz coordinate system or a local uvw coordinate system. In the former, the neighborhood of the surface is defined by the coordinate function f (u, v) = [x(u, v), y(u, v), z(u, v)]. In the latter, assuming the uv-plane is approximately parallel with the tangent plane of the surface at P , each point in the neighborhood of the point can be transformed into a point [u, v, f (u, v)] (by a simple translation and rotation), where f is known as the local height function. Let u denote [u, v]T . Let ϕ(u) denote a smooth bivariate function, which may be the local height function or the x, y, or z component of the coordinate function for j+k a parametric surface. Let cjk be a shorthand for ∂u∂ j ∂vk ϕ(0). Let d be the desired degree of the polynomial fitting, and it is typically small, say ≤ 6. If ϕ(u) has d + 1 continuous derivatives, it can be approximated to (d + 1)st order accuracy about the origin u0 = [0, 0]T by ϕ(u) =

d j+k=p

j+k=d+1 uj v k ∂ j+k u ˜j v˜k + ϕ(˜ u, v˜) , j k j!k! ∂u ∂v j!k! p=0 j,k≥0 j,k≥0 ( )* + ( )* + Taylor polynomial remainder

cjk

(1)

where 0 ≤ u ˜ ≤ u and 0 ≤ v˜ ≤ v. T Suppose we have a set of data points, say [ui , vi , ϕi ] for i = 1, . . . , m − 1, sampled from a neighborhood near P on the surface. Substituting each given point into (1), we obtain an approximate equation , d j+k=p uji vik cjk ≈ ϕi , (2) j!k! p=0 j,k≥0

which has n = (d + 1)(d + 2)/2 unknowns (i.e., cjk for 0 ≤ j + k ≤ d, j ≥ 0 and k ≥ 0), resulting in an m × n rectangular linear system. Note that one could force the polynomial to pass through point P by setting c00 = 0 and removing its corresponding equation, reducing to an (m − 1) × (n − 1) rectangular linear system. This may be beneficial if the points are known to interpolate a smooth surface.

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Let us denote the rectangular linear system obtained from (2) as V X ≈ F,

(3)

where X is an n-vector composed of cjk , and V is m × n, known as a generalized Vandermonde matrix. For a local height function, F is an m-vector composed of fi ; for a parametric surface, F is an m × 3 matrix, of which each column corresponds to a component of the coordinate function. The above formulations can be easily adapted to curves in 2-D or 3-D, by using the univariable instead of the bivariable version of Taylor series expansions. For a curve in 3-D, the parameterization has only one parameter (say u), and the local height function has two components. When applied to a surface mesh, the point P is typically a vertex, and its neighborhood are typically some k-ring neighborhood. Following [13], we allow k to have half-ring increments: • • •

The 1-ring neighbor faces of a vertex v are the faces incident on v, and the 1-ring neighbor vertices are the vertices of these faces. The 1.5-ring neighbor faces are the faces that share an edge with a 1-ring neighbor face, and the 1.5-ring neighbor vertices are the vertices of these faces. For an integer k ≥ 1, the (k + 1)-ring neighborhood of a vertex is the union of the 1-ring neighbors of its k-ring neighbor vertices, and the (k + 1.5)-ring neighborhood is the union of the 1.5-ring neighbors of the k-ring neighbor vertices.

Figure 1 illustrates these neighborhood definitions up to 2.5 rings. We typically choose k to be (d + 1)/2 (for non-noisy surface) or d/2 + 1 (for noisy surface), but may also enlarge k if there are fewer than 1.5 times of the required number of points in the k-ring. 2.2

Weighted Least Squares Approximation

Numerically, (3) can be solved using the framework of weighted linear least squares [7, p. 265], i.e., to minimize a weighted norm (or semi-norm), min V X − F Ω = min Ω(V X − F ) 2 , X X

(4)

where Ω is a weighting matrix. Typically, Ω is an m × m diagonal matrix, whose T ith diagonal entry ωi assigns a priority to the ith point [ui , vi ] by scaling the ith row of V . It is desirable to assign lower priorities to points that are farther away from the origin or whose normals differ substantially from the w direction of the local coordinate frame, such as that defined in (8). The formulation (4) is equivalent to the linear least squares problem V˜ X ≈ B, where V˜ = ΩV and B = ΩF .

(5)

In general, V˜ is m × n and m ≥ n. A technical difficulty is that this linear system may be very ill-conditioned (i.e., the singular values of V˜ may differ by orders of


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1 ring

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Fig. 1. Examples of 1-, 1.5-, 2-, and 2.5-ring vertices for typical vertex in triangle mesh. Each image depicts the neighborhood of the center black vertex.

magnitude) due to a variety of reasons, such as poor scaling, insufficient number of points, or degenerate arrangements of points [14]. The conditioning number of V˜ can be improved by using a scaling matrix S and changing the problem to min AY − B 2 , where A = V˜ S and Y = S −1 X. Y

(6)

˜ i denote the ith column of V˜ . The ith We chose S to be a diagonal matrix. Let v diagonal entry of S is chosen to be ˜ v i 2 , which approximately minimizes the condition number of V˜ S [7, p. 265]. 2.3

Accuracy and Stability of Least Squares Polynomial Fittings

The local least squares polynomial fitting provides us the theoretical foundation for high-order reconstruction of surfaces, established by the following proposition [13]: Proposition 1. Given a set of points [ui , vi , f˜i ] that interpolate a smooth height function f or approximate f with an error of O(hd+1 ). Assume the point distribution and the weighting matrix are independent of the mesh resolution, and the ˜ in (6) is bounded by some constant. condition number of the scaled matrix A = AS The degree-d weighted least squares fitting approximates cjk to O(hd−j−k+1 ). Here, h is a local measure of mesh resolution (such as average edge length of the k-ring neighborhood). We refer to readers to [13] for the proof of the proposition. Note that a necessary condition for the accuracy is that the condition number of the scaled matrix A must be bounded, but it is not automatically the case even if the number of points is greater than the number of unknown coefficients. We achieve

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well-conditioning by either expanding the neighborhood or reducing the degree of fitting if the condition number is determined to be large, and in turn guarantee both accuracy and stability.

3 Continuous, High-Order Surface Reconstruction The method described in the previous section applies locally at each individual vertex of a mesh. There was no coordination among the local fittings at different vertices, so the method does not reconstruct a continuous surface. To construct a continuous surface, there are at least three different options: 1. compute multiple local fittings at vertices and then compute a weighted averaging of these fittings; 2. enforce continuity of local coordinate frames and weights for local fittings; 3. introduce additional control points to define continuous/smooth surface patches. Most methods in the literature use the latter two options. For example, the moving least squares [15] uses the second option to construct a C ∞ surface from a point cloud. Walton’s method [17] adopted by Yams [4, 3] uses the third option. In this section, we describe two methods that belong to the first two categories, respectively. We will first focus on triangle meshes for smooth surfaces in this section, and will present the extension to quadrilateral meshes and for meshes with sharp features in the next section when describing their applications in meshing. 3.1

Weighted Averaging of Local Fittings (WALF)

A simple approach to construct a high-order surface is to compute a weighted average of the local fittings at vertices. We refer to this approach as Weighted Averaging of Local Fittings (WALF). To achieve continuity of the surface, the weights used by the weighted averaging must be continuous over the mesh. One such a choice is the barycentric coordinates of the vertices over each triangle. Consider a triangle composed of vertices xi , i = 1, 2, 3, and any point p in the triangle. For each vertex xi , we obtain a point q i for p from the local fitting in the local uvw coordinate frame at xi , by projecting p onto its uv-plane. Let ξi , i = 1, 2, '33 denote the barycentric coordinates of p within the triangle, with ξi ∈ [0, 1] and i=1 ξi = 1. We define q(u) =

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(7)

i=1

as the approximation to point p. Figure 2 shows a 2-D illustration of this approach, where ξi are the barycentric coordinates of point p within the edge x1 x2 . WALF constructs a C 0 continuous surface, as can be shown using the properties of finite-element basis functions: The barycentric coordinates at each vertex of a triangle corresponds to the shape function of the vertex within the triangle, and the shape function of the vertex in all elements forms a C 0 continuous basis function (i.e., the linear pyramid function for surfaces or the hat function for curves). Let φi


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q n1

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Fig. 2. 2-D illustration of weighted averaging of local fitting. The black curve indicates the exact curve. The blue and green curves indicate the fittings at vertices x1 and x2 , respectively. q is the WALF approximation of point p and is computed as a weighted average of the points q 1 and q 2 on the blue and green curves, respectively.

denote the basis function associated with the ith vertex of the mesh, and it is zero almost everywhere except within the triangles incident on the ith vertex. Therefore, q can be considered as a weighted average of the polynomials at all the vertices, q(u) =

n

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i=1

and then it is obvious that q is C ∞ within each triangle and C 0 over the whole mesh. The idea of WALF is intuitive, but the analysis of its accuracy is by no means straightforward. If the coordinate systems were the same at all vertices, then the analysis would have been easy, as q would have inherited the accuracy of qi . However in our case, the local fittings at the three vertices of a triangle are in different coordinate systems in general, and this discrepancy of coordinate systems can lead to additional error terms. Under the same assumptions as Proposition 1, we obtain the following property of WALF. Proposition 2. Given a mesh whose vertices approximate a smooth surface Γ with an error of O(hd+1 ), the distance between each point on the WALF reconstructed surface and its closest point on Γ is O(hd+1 + h6 ). Note that the above proposition gives an upper bound of the error, so the lower bound of the convergence rate is min(6, d + 1). The bound of h6 is due to the discrepancy of local coordinate systems at different vertices. The proof is given in the Appendix. 3.2

Continuous Moving Frames (CMF)

WALF is a simple and intuitive method, but its order of accuracy may be limited. We now present a method that can overcome this limitation by using local coordinate frames that move continuously from point to point. We refer to such a scheme as continuous moving frame (CMF). The basic idea is to use the finite-element basis

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functions to construct continuous moving frames and weights for local fittings. In particular, assume each vertex has an approximate normal direction at input. Conˆ i denote the unit vertex sider a triangle x1 x2 x3 and any point p in the triangle. Let n normal at the ith vertex. We compute a normal at p as % .% 3 3 % % % % ˆ= î % ˆ i% . n ξi n ξi n % % i=1

i=1

ˆ we construct a local uvw coordinate system along axes s ˆ, ˆt, and n, ˆ where Given n, ˆ ˆ and t form an orthonormal basis of the tangent plane. Within this local coordinate s frame, we formulate the weighted least squares as

ΩV X − ΩF 2 , where V again is the generalized Vandermonde matrix, and Ω is the weight matrix. In practice, the Vandermonde matrix for a point p should involve a small stencil in the neighborhood of the triangle. We use the union of the stencils of the three vertices of the triangle. Conceptually, it is helpful to consider the Vandermonde matrix involving all the points of the mesh, but the weight matrix Ω assigns a zero weight for each point that is not in the stencil. For the reconstructed surface to be smooth, it is important that Ω is continuous as the point p moves within the geometric support of the mesh. In addition, it is also important that Ω is invariant ˆ and ˆt). of rotation of tangent plane (i.e., be independent of the choice of s We define the weight as follows. For p within the triangle x1 x2 x3 , we first define a weight for each vertex (say jth vertex) in the mesh corresponding to xi as /

+ 2 2 ˆj ˆ Ti n ξi n e−xj −p /hi if vertex j is in stencil of ith vertex wij = (8) 0 otherwise where

ˆj ˆ Ti n n

+ =

ˆ Ti n ˆ j if n ˆj ≥ ˆ Ti n n 0 otherwise

(9)

for some small ≥ 0 and hi is a local mesh-resolution measure ' at the ith vertex. 3 Then for the weighting matrix, the weight for jth vertex is then i=1 wij . In the actual implementation, for simplicity we list the jth vertex separately for its appearance in the stencil of each vertex of the triangle, and include only the vertices whose weights are nonzeros in V and Ω. Similar to WALF, CMF constructs a C 0 continuous surface, because Ω, V , and F are all C 0 continuous, as long as the resulting linear system is well-conditioned. The accuracy of CMF follows that for weighted least squares approximation in [13], and we obtain the following properties of CMF. Proposition 3. Given a mesh whose vertices approximate a smooth surface Γ with an error of O(hd+1 ), the shortest distance from each point on the CMF reconstructed surface to Γ is O(hd+1 ).


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Relationship with Moving Least Squares. The idea of using moving frames is not new, and goes back to Élie Cartan for differential geometry. One of the incarnations of the idea of using moving frames for discrete surfaces is the so-called moving least squares (MLS) for point clouds [15]. CMF shares some similarities to MLS. In particular, they are both based on weighted least squares approximations within some local frames. However, they also differ in some fundamental ways. First, moving least squares uses global weighting functions that are exponential in the distance, and theoretically, MLS is C ∞ . However, because global weighting functions are too expensive to compute, practical implementations typically truncate small weights to zeros, leading to a loss of continuity. In contrast, CMF uses only a local support by construction. Second, MLS does not guarantee the order of accuracy, because its weights are global and purely based on Euclidean distance. Although its convergence was conjectured in [15], we have observed that MLS does not converge even for simple geometries such as a torus. In contrast, CMF uses the mesh connectivity as a clue in selecting the stencils, instead of based on Euclidean distance. Third, CMF can take into account the normals in the weighting function, to filter out points across sharp features. This allows CMF to handle surfaces with sharp features in a natural way, which is important for meshing operations. On the other hand, it is difficult to treat sharp features in the framework of MLS. Because of their local supports, CMF is more easily adapted to treat sharp features, as we describe in the next section. 3.3

Experimental Results

We report some experimental results of our two proposed methods, and compare them with some other methods. We first show the mesh convergence study of WALF and CMF. While it is typically unnecessary to use schemes with higher than third or fourth order accuracy, to demonstrate the capabilities and limitations of these two methods, we report results with polynomials of up to degree 6. We performed our experiment using a torus with in-radius of 0.7 and outer-radius of 1.3, with an unstructured triangular mesh. We considered three levels of mesh refinement. The coarsest mesh has 329 vertices and 658 triangles, whereas the finest mesh has 21,156 vertices and 42,312 triangles. In this test, we randomly generated 10 points on each face of the mesh, then project them onto high order surface using WALF or CMF. We compute the error as the shortest distance from each approximate point to the torus. Figure 3 shows the L∞ errors of WALF and CMF for the meshes. In the figure, the horizontal axis corresponds to the level of mesh refinement, and the vertical axis corresponds to the L∞ errors. In the legends, the “degree” indicates the degree of polynomial fittings used, and “linear” indicates the error for linear interpolation within triangles. We show the average convergence rates along the right of the plots for each curve, which was calculated as log (error3 /errorbase ) / log (h3 /hbase ), where errori denotes the L∞ error of all the randomly inserted points for the ith coarsest mesh, and hi is the maximum edge length of the corresponding mesh. We chose the base to be 0 for CMF and 1 for WALF, because the errors for WALF

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were large for degrees 5 and 6 for the coarsest mesh, leading to artificially too large convergence rates. From the figures, it is obvious that quadratic and higher-degree fittings produced far more accurate results than linear interpolation. Both WALF and CMF achieved a convergence rate of (d + 1) when d is odd and higher than (d + 2) when d is even. The superconvergence for even-degree fittings is likely due to statistical error cancellations of the leading error terms, which are of odd degrees. However, such error cancellations are not guaranteed when the points are very far from being symmetric, especially near boundaries or sharp features. Some conclusions can be drawn from our comparisons between WALF with CMF. In terms of accuracy, we note that WALF gave smaller errors (up to 50% smaller) than CMF for finer meshes, although they delivered very similar convergence rates. The reason for the smaller errors for WALF was probably that WALF uses a smaller stencil for each polynomial fitting. In terms of efficiency, WALF and CMF are comparable when approximating a single point. However, when approximating many points, WALF allows reusing the polynomial fittings computed at the vertices, and hence it can have smaller amortized cost. Furthermore, WALF is also more versatile, because one can change the polynomial at each vertex in WALF (such as adapting its degree of polynomial) without losing C 0 continuity, but the same cannot be done with CMF. Therefore, we favor WALF over CMF, unless some application requires guaranteed seventh or higher order accuracy. Besides WALF and CMF, some other methods have been developed for high order reconstructions and been used in the meshing community. One method that is worth noting is that proposed by Walton [17] and adopted by Frey for surface meshing [3]. One property of Walton’s method is that it achieves C 1 (or G1 ) continuity for the reconstructed mesh. However, there does not seem to be any analysis of the accuracy of Walton’s method in the literature. Figure 4 shows a comparison of the errors of Walton’s method as well as WALF using quadratic and cubic fittings, with linear interpolation as the baseline of comparison. The two figures show the errors in L∞ and L2 norms for the torus. From the result, it is evident that Walton’s method converges only linearly and is actually less accurate than linear 2

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interpolation for finer meshes. Our preliminary theoretical analysis also suggests that Walton’s method can be at most second order accurate (i.e., no better than linear interpolation), and its practical accuracy may be worse because its dependence on the tangent directions of the edges, which are in general unavailable and can be estimated only to low accuracy. Therefore, it is obvious that C 1 (and in fact even C ∞ ) continuity does not imply accuracy of the reconstruction, although they may produce smooth looking. On the other hand, as we will show in the next section, high-order methods with C 0 continuity typically produce errors that are too small to cause any noticeable discontinuities.

4 Applications to Meshing and Finite Elements The targeted applications for high-order surface reconstruction for this paper are meshing for finite element analysis. We hereafter further customize our framework for meshing and then apply the resulting techniques to meshing operations. 4.1

Quadrilateral Elements and Sharp Features

To be generally applicable to meshing, the reconstruction techniques should work not only for triangulations of smooth surfaces but also for meshes with quadrilateral elements (including quadrilateral meshes or quad-dominant meshes) and surfaces with sharp features. Generalization to Meshes Containing Quadrilateral. For a quadrilateral element, we need to use the finite element shape functions Ni . Let ξ and η be the two natural coordinates within a quadrilateral element, with 0 ≤ ξ ≤ 1 and 0 ≤ η ≤ 1, then the shape functions associated with the four vertices are N1 = (1 − ξ)(1 − η), N2 = ξ(1 − η), N3 = (1 − ξ)η, N4 = ξη.

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1-ring

1.5-ring

2-ring

2.5-ring

Fig. 5. Examples of 1-, 1.5-, 2-, and 2.5-rings of typical vertex in quadrilateral mesh. Each image depicts the neighborhood of the center black vertex.

A key issue is the selection of stencils for a quadrilateral mesh or quad-dominant mesh. The definition of n-ring neighbors in [13] tends to produce too many points for quadrilateral meshes. We redefine the neighborhood of a vertex as follows: • • •

The 0-ring of a vertex is the vertex itself ; The k-ring vertices of a vertex (where k = 1, 2, 3, . . .) is the set of vertices that are share an edge with a vertex in the (k-1)-ring; The (k + 0.5)-ring of a vertex (where k = 1, 2, 3, . . .) is the union of k-ring vertices and the vertices that share elements with an edge between two vertices in the k-ring.

For a triangle mesh, the above definition is equivalent to that in [13]. However, this definition is also well-suited to other types of surface meshes. Figure 5 shows the 1-, 1.5-, 2-, and 2.5-rings of a typical vertex in a quadrilateral mesh or a quad-dominant mesh. In general for degree-d fittings, we find it most effective to use a ring of (d + 1)/2 for a mesh without noise or a ring of d/2 + 1 or larger for meshes with noise. Treatment of Sharp Features. Sharp features, such as ridges and corners, are challenging problems in their own right. We have so far implemented a simple treatment for sharp features. First, we identify feature edges and vertices and connect the feature edges to form ridge curves using an algorithm such as that in [9, 10]. We treat the ridge edges as internal boundaries within the mesh and require the k-ring neighbors of vertices do not go across ridge curves. This is accomplished by virtually


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splitting the mesh along ridge curves in our mesh data structure. For ridge curves themselves, we separate them into sub-curves that do not have corners in their interior. For each sub-curve, we perform a high-order reconstruction using either WALF or CMF for curves. This treatment is sufficient for most meshing operations. 4.2

High-Order Finite Elements

An application of our method is to construct a high order (in particular, quadratic or cubic) finite element mesh from a given mesh with only linear elements and accurate vertex coordinates. This problem has practical relevance, because some mesh generators produce only a mesh with linear elements from an accurate CAD model, and it may be desirable to reconstruct high-order elements without having to access the CAD model. We formally stated the problem as follows: Given a mesh with linear elements, assume the vertices are sufficiently accurate (e.g., they are exact or are at least third or fourth-order accurate), construct a finite element mesh with quadratic or cubic elements with third or fourth order accuracy. When using high-order surface reconstruction, this problem can be solved in the following procedure: 1. For each element σ, loop through its edges. If there is not an element that is abutting σ and has an ID smaller than that of σ, assign a new node ID to each node on the edge; if a node ID has already been assigned in the adjacent element, retrieve the node ID from that adjacent element; 2. Loop through all elements to assign new nodes IDs for new nodes on faces; 3. Expand the array for nodal coordinates, and evaluate the position for each new vertex using high-order surface reconstruction. We have implemented this procedure for reconstructing quadratic and cubic elements from linear triangles or bilinear quadrilaterals using WALF and CMF. Figure 6 shows an example for meshes generated with quadratic and cubic elements for a torus. The high-order schemes produced notable improvements to the smoothness of the surface to linear approach, and the overall errors are significantly smaller. Note that in actual numerical simulations, not only the geometry but also some field variables defined on the mesh need to be reconstructed to high order. The same highorder reconstruction we presented can be used for that purpose, but it is beyond the scope of this paper. 4.3

Uniform Mesh Refinement

A problem related to generating a high-order mesh is a uniform refinement of a surface mesh. The problem may be stated as follows: given a coarse surface mesh with sufficiently accurate vertex coordinates, construct a finer surface mesh by subdividing the elements. Like the previous problem, uniform mesh refinement introduces additional nodes to edges and/or faces, but in addition it also introduces new edges to subdivide the elements. Figure 7 shows an example of refining a quadrilateral mesh with sharp features, and Figure 8 shows that for a triangular mesh of a dragon

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Fig. 6. Illustration of generating high-order finite elements from given mesh. Left: a coarse torus with linear elements; Middle: same mesh but with quadratic elements, visualized by decomposing each triangle into four triangles. Right: same mesh but with cubic elements, visualized by decomposing each triangle into nine triangles.

Fig. 7. Example of refining quadrilateral mesh by subdividing each element into nine quadrilaterals. In left, dark lines show original coarse mesh, and dashed lines show linear subdivision. Right images shows refined mesh using cubic fitting with WALF and feature treatments.

head. Note that if the new points are added onto the linear edges and faces, the refined mesh is not only inaccurate but also nonsmooth, as evident in the left image of the figures. This problem is resolved by using high order reconstructions. The right image of Figure 7 and bottom-right image of Figure 8 show a refinement mesh using WALF and feature treatments. The resulting meshes are much smoother and more accurate. This procedure can be useful for generating high-quality finer resolution meshes from a mesh without requiring access to the CAD model. 4.4

Mesh Smoothing and Mesh Adaptivity

More general meshing applications of our techniques are the smoothing and adaptivity of surface meshes. In these settings, not only new vertices may be added, existing vertices may also be moved. For these meshing operations, a common approach is to keep the original mesh during mesh smoothing/adaptation, and project new vertices onto the faceted, piecewise linear geometries (see e.g., [6]). Such an approach has only second order accuracy. Another approach was taken by Frey [3], who constructed a G1 continuous surface using Walton’s method [17]. However, our experiments have shown that Walton’s method is only about first order accurate, despite its G1 continuity. Other methods have been developed (such as [16]), but none could deliver high order accuracy.


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Fig. 8. Example of refining triangular mesh by subdividing each element into four triangles. Upper row shows a dragon mesh and the zoom in near the head. Lower left image shows a refined mesh using linear interpolation. Lower right image shows refined mesh using WALF with quadratic fitting and feature treatments.

Fig. 9. Example of applying high-order reconstruction in meshing smoothing. The left image shows the original torus mesh, and the right image shows the smoothed mesh.

Instead of using low-order methods, we propose to use high-order surface reconstructions. As an example, we integrate high-order surface reconstruction with the variational mesh smoothing framework described in [12]. To utilize the high-order surface, we first compute the motion of each point within the tangent plane, and then project the new point onto the high-order surface. Note that the use of the tangent plane is useful, because it introduces an error that is high order compared to the tangential displacement, so the projection onto the high-order surface involves only a high order adjustment. Figure 9 shows an example of a smoothed surface mesh using WALF method compared to the original mesh generated by the isosurface function in MATLAB. The maximum angles were 166.1 and 128.8 degrees before and after smoothing, respectively, and minimum angles were 0.65 and 23.8 degrees, respectively. The smoothing process significantly improved the mesh quality while preserving the geometry to high order.

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5 Conclusions and Discussions In this paper, we studied the problem of reconstructing a high-order surface from surface meshes, in the context of meshing for finite element computations. We presented two methods, namely WALF and CMF, which are based on weighted least squares approximations with piecewise polynomial fittings. Unlike the traditional methods used in meshing, these methods can deliver very high order accuracy, independent of the order of the element of the mesh. The end results are high-order reconstructions that are efficient, noise resistant, feature preserving, and well suited for meshing and finite element computations. We demonstrate the high accuracy of our method compared to some other methods, and discussed its application in the context of reconstructing high-order finite elements, mesh refinement, mesh smoothing, and mesh adaptivity. Between WALF and CMF, we favor WALF for its simplicity and versatility (for example, WALF can be interpolatory, but CMF cannot be without sacrificing numerical stability), unless one requires guaranteed seventh or even higher order accuracy. A property of our methods is that it enforces only C 0 continuity. Imposing only 0 C continuity allows us more freedom to achieve higher order accuracy. However, C 0 continuity may be deemed as a limitation of our method in some applications in computer-aided design and geometric modeling, especially when very coarse models with very few control points are used. In such cases, the loss of C 1 or C 2 continuity may lead to noticeable artifacts. We will investigate the reconstruction of C 1 surfaces with high-order accuracy in our future research.

Acknowledgement This work was supported by National Science Foundation under award number DMS-0809285. The first author is also supported by DOE NEUP program under contract #DE-AC07-05ID14517 and by DoD-ARO under contract #W911NF0910306.

References 1. Donea, J., et al.: Arbitrary Lagrangian-Eulerian methods. In: Stein, E., de Borst, R., Hughes, T.J. (eds.) Encyclopedia of Computational Mechanics, ch. 14, Wiley, Chichester (2004) 2. Fleishman, S., Cohen-Or, D., Silva, C.T.: Robust moving least-squares fitting with sharp features. ACM Transactions on Graphics (TOG) 24(3) (2005) 3. Frey, P.J.: About surface remeshing. In: Proc. 9th Int. Meshing Roundtable, pp. 123–136 (October 2000) 4. Frey, P.J.: Yams: A fully automatic adaptive isotropic surface remeshing procedure. Technical report, INRIA, RT-0252 (2001) 5. Frey, P.J., George, P.L.: Mesh Generation: Application to finite elements. Hermes (2000) 6. Garimella, R.: Triangular and quadrilateral surface mesh quality optimization using local parametrization. Computer Methods in Applied Mechanics and Engineering 193(9-11), 913–928 (2004)


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7. Golub, G.H., Van Loan, C.F.: Matrix Computations, 3rd edn. Johns Hopkins (1996) 8. Heath, M.T.: Scientific Computing: An Introductory Survey, 2nd edn. McGraw-Hill, New York (2002) 9. Jiao, X.: Volume and feature preservation in surface mesh optimization. In: Proc. 15th Int. Meshing Roundtable (2006) 10. Jiao, X., Bayyana, N.R.: Identification of C 1 and C 2 discontinuities for surface meshes in CAD. Comput. Aid. Des. 40, 160–175 (2008) 11. Jiao, X., Colombi, A., Ni, X., Hart, J.: Anisotropic mesh adaptation for evolving triangulated surfaces. Engineering with Computers 26, 363–376 (2010) 12. Jiao, X., Wang, D., Zha, H.: Simple and effective variational optimization of surface and volume triangulations. In: Proc. 17th Int. Meshing Roundtable, pp. 315–332 (2008) 13. Jiao, X., Zha, H.: Consistent computation of first- and second-order differential quantities for surface meshes. In: ACM Solid and Physical Modeling Symposium, pp. 159–170. ACM, New York (2008) 14. Lancaster, P., Salkauskas, K.: Curve and Surface Fitting: An Introduction. Academic Press, New York (1986) 15. Levin, D.: The approximation power of moving least-squares. Mathematics of Computation 67, 1517–1531 (1998) 16. Semenova, I.B., Savchenko, V.V., Hagiwara, I.: Two techniques to improve mesh quality and preserve surface characteristics. In: Proc. 13th Int. Meshing Roundtable, pp. 277– 288 (2004) 17. Walton, D.: A triangular G1 patch from boundary curves. Computer-Aided Design 28(2), 113–123 (1996) 18. Wang, D., Clark, B., Jiao, X.: An analysis and comparison of parameterization-based computation of differential quantities for discrete surfaces. Computer Aided Geometric Design 26(5), 510–527 (2009)

Appendix Proof of Proposition 2 We analyze the accuracy for triangles, but it helps to refer to Figure 2 for a 2-D illustration. Let q ∗i denote the intersection of the exact surface with the direction ni from a point p (i.e., q ∗i is the exact solution for q i in the fitting at vertex xi ). '3 ¯ denote the closest point to point q = Let q i=1 ξi q i on the exact surface. Let ' 3 ∗ ∗ ∗ ¯ be its closest point on the surface. Then, q = i=1 ξi q i and q ¯ ∗ ≤ q − q ∗ + q ∗ − q ¯ ∗ .

q − q¯ ≤ q − q For q − q ∗ , note that q − q ∗ ≤ are used, qi − q ∗i = O(hd+1 ), so

'3

i=1 ξi q i

(10)

− q ∗i . When dth degree fittings

q − q ∗ = O(hd+1 ).

(11)

¯ ∗ , q∗1 − q ∗2 = | cos θ1 | q ∗1 − p + | cos θ2 | q ∗2 − p , where θi is For q ∗ − q ˆ i . Note that cos θi = O(h), since by assumption n î the angle between q ∗1 q ∗2 and n

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is at least a first order approximation to the normal at xi , and a first order approximation to the normals at q ∗1 and q ∗2 , whereas the line segment q ∗1 q ∗2 is at least a first order approximation to a tangent direction at q∗1 and q ∗2 . Because p is a point on triangle x1 x2 x3 , whose edge length is O(h) by assumption, p − q ∗i = O(h2 ). Therefore, q∗1 − q∗2 = O(h3 ), and similarly for q∗1 − q ∗3 and q ∗2 − q ∗3 . Because q ∗ is a point on triangle q ∗1 q ∗2 q ∗3 , ¯ ∗ = O(h3 )2 = O(h6 ).

q∗ − q

(12)

Combining (10-12), we conclude that q−¯ q = O(hd+1 )+O(h6 ) = O(hd+1 +h6 ).

Simple Method for Constructing NURBS Surfaces from Unorganized Points Nallig Leal, Esmeide Leal, and John William Branch Universidad Autónoma del Caribe Calle 90 No 46 – 112 Tel. 57-5-3671286 Barranquilla, Colombia, Suramérica {nleal,esleal}@uac.edu.co Universidad Nacional de Colombia, Sede Medellín Medellín, Colombia, Suramérica Carrera 80 No 65-223 [email protected]

Summary. In this paper, a new method for constructing NURBS surfaces from scattered and unorganized points is presented. The method is easy to implement and allows fitting a NURBS surface to a scattered point cloud without constructing either NURBS patches networks or polygon meshes. Based on the projection of the points onto the regression plane of the cloud, the method detects both regions where the cloud lacks points, and regions where the cloud is too dense. Then, by inserting and removing points, the point cloud is regularized and approximated by a NURBS surface. In order to reduce the approximation error, an evolutionary strategy obtains the weights of the NURBS surface so that the distance between the point cloud and the NURBS is minimal. Finally, the points inserted outside the bounds of the point cloud are removed by trimming the NURBS surface. Numerical and graphical results are provided, demonstrating that the method produces valid surfaces with low fitting error. Keywords: Surface fitting, NURBS surfaces, Points regularization, PCA, Evolutionary Strategies.

1 Introduction 3D reconstruction is a process by which objects are reproduced in the computer memory, keeping its physical characteristics (dimensions, volume and shape). 3D reconstruction is a difficult task that covers, in general terms, five stages: data acquisition, registration, integration, segmentation, and surface fitting [1]. The approach presented in this paper deals with the surface fitting stage, in which the computational model of the object is obtained. NURBS is one of the most employed surface fitting models, provided that it is a standard representation of curves and surfaces [2] and is widely supported by modern standards like OpenGL and IGES, which are used for graphics and

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geometric data exchange [3]. In addition, the NURBS surface model has stability, flexibility, local modification properties and is robust to noise. Yet, the NURBS surface model has a disadvantage: the input data points should be mapped on a regular grid structure [4]. In the 3D reconstruction process, the registration and integration stages produce massive scattered and unorganized point clouds that cannot be mapped on a regular grid structure. Such point clouds cannot be fitted by a NURBS surface and are not suitable for usage in computer-aided design (CAD) systems [5]. In order to fit a NURBS surface to an unorganized and scattered point cloud, several approaches have been presented [4, 5, 6, 7, 8, 9]. Such approaches fit to the cloud a network of NURBS patches with some degree of continuity between them. The construction of the network requires constructing polygon meshes and complicated procedures of refinement, reparameterization and maintainability of the continuity between the patches, which is computationally expensive in terms of memory and processing. In this paper a new method for constructing NURBS surfaces from scattered and unorganized points is presented. In contrast with others, our method does not need to construct a network of NURBS patches. Furthermore, previous construction of polygon meshes, mesh refinement and data reparameterization are not required. Our method first detects the global bias of the point cloud fitting a regression plane to it by using weighted principal components analysis. Then, all the points are projected onto the plane and a two-dimensional regularity analysis of the point cloud is made. The analysis consists of detecting regions with low point density and regions with high point density. By inserting and removing points, based on the two-dimensional analysis, the point cloud is regularized. In order to reduce the fitting error, an evolutionary strategy obtains the weights of the control points that belong to the cloud. Finally, the points inserted outside the cloud boundary are removed by trimming the NURBS surface. The main contribution of our work is that we proposed an automatic, efficient and simple method for constructing a complete NURBS surface from a point cloud. The remainder of this paper is organized as follows. In section 2, related work dealing with reconstruction of NURBS surfaces from scattered and unorganized points is presented. In section 3, the fundamentals of NURBS surfaces are presented. In section 4, a short overview of evolutionary strategies is presented. In section 5, the stages of our method are explained. In section 6, results of our method are provided. In section 7, conclusions and future work are discussed.

2 Related Work Point clouds are considered the main information source in the 3D reconstruction process; unfortunately, such point clouds are not suitable for integration into CAD systems [5]. Constructing NURBS surfaces from point clouds would allow the incorporation of such information source in CAD systems.

Simple Method for Constructing NURBS Surfaces from Unorganized Points

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Several approaches have been proposed for solving the problem of constructing NURBS surfaces from scattered and unorganized points. Eck and Hoppe [6] solved the problem by generating a network of B-Spline patches. They first construct a triangular mesh over the cloud and project the points onto the mesh to obtain an initial parameterization. Then, a merging process from triangular mesh is carried out for constructing a quadrilateral mesh. Over the quadrilateral domain, a network of B-Spline patches is built. In order to reach a user-specified error tolerance, a refinement process takes place. This process adaptively subdivides the quadrilateral domain producing smaller quadrilateral subfaces. A new network of B-Spline patches is necessary to fit the refined surface. Even though this is an effective method for reconstructing B-Spline surfaces, e.g. NURBS, from point clouds, it is computationally expensive in terms of memory and processing needed. Furthermore, only G1 continuity between the patches is guaranteed. Krishnamurthy and Levoy [7] proposed an approach that constructs a polygon mesh which is resampled to produce a regular grid where NURBS surface patches can be fitted. The performance of the approach is poor when it operates on complex surfaces, and it cannot be applied over surfaces with holes. Park, Yun and Lee [4] presented a two-stage algorithm. The initial stage, named model approximation, employs K-means clustering to obtain regions where polygon meshes are constructed, subdivided in triangular meshes, and blended in quadrilateral meshes. The meshes produced by the initial stage are represented by a hierarchical graph structure. The second stage takes the hierarchical graph structure to construct a NURBS patch network. This method, like Eck and Hoppe's method, is computationally expensive and just guarantees G1 continuity. Furthermore, it assumes the point cloud represents a closed surface. Gregorski, Hamann and Joy [5] proposed an approach that subdivides the set of points into a strip tree structure. This structure is used to fit quadratic surfaces which are degree-elevated and blended into B-Spline patches. This approach cannot be applied either on closed surfaces or on surfaces that are occluded with respect to its principal component analysis regression plane (PCAP), like the surface shown in Figure 1. Bertram, Tricoche and Hagen [8], and Yvart, Hahmann, and Bonneau [10] proposed approaches that use triangular B-Spline patches instead of NURBS patches to fit point clouds. Their approaches, like the aforementioned ones, construct polygon meshes and use expensive processes for fitting a network of triangular B-Spline to the point cloud.

Fig. 1. Occluded surface with respect to its PCAP

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3 NURBS NURBS, Non Uniform Rational B-Splines, are parametric tensor product curves or surfaces defined by the following expression

∑ ∑ S (u , v ) = ∑ ∑ n

m

i =0 n

j =0 m

i =0

where

N i , p (u ) N j , q (v) wi , j Pi , j

(1)

N i , p (u ) N j , q (v) wi , j j =0

wi, j are the weights, Pi, j are the control points and N i, p (u), N j,q (v) are

p and q respectively, defined over the non periodic node support S u = {u 0 ,...,u r } and S v = {v0 ,..., v r } [11], which can be the B-Spline basis functions of order

calculated in a recursive way by the Cox and de Boor formula [12] according to (2)

N i , p (u ) =

(u − u i ) N i , p −1 (u ) ui+ p − ui

+

(ui + p +1− u ) N i +1, p −1 (u )

(2)

u i + p +1 − ui +1

A NURBS surface is completely determined by its control points

Pi, j , i.e. the

surface changes in a predictable way according to control points movement. This is known as the local support property and allows the surface to be affected, only locally, by the movement of a control point. The main difficulty when fitting NURBS surfaces is to obtain a suitable parameterization and automatically choose the number of control points and their positions to define the surface topology. The weighting factors wi, j of NURBS surfaces play an important role in the fitting process, since these factors determine how much a control point influences the shape of the surface locally. When the weighting factors of NURBS surfaces are assigned in a homogeneous way and their values are one, the NURBS model is reduced to a particular case known as B-Spline surfaces, which are limited in the representation of free-form and conic surfaces. If we want an approximately close enough set of data that represents free-form surfaces using NURBS, it is necessary to manipulate the NURBS parameters, but as previously mentioned such manipulation implies dealing with non linear problems during the fitting process. Furthermore, negative values or zeros in the weighting factors can degenerate the construction of the surface. Figure 2 illustrates the importance of the weighting factors in the NURBS model. The circles represent control points and the line represents a NURBS curve. It is notable how the weighting factors affect the local geometry of the surface. Figures 2a to 2d show the effect over the NURBS curve of assigning the values 0, 0.5, 1 and 2, respectively, to control point 3.


(a)

(b)

(c)

(d)

165

Fig. 2. Weighting factors effect

When fitting data points using NURBS, we attempt to minimize: n m ⎛ N i , p (u )N j ,q (v )wi , j Pi , j ∑ ⎜ i =0 ∑ j =0 δ = ∑ ⎜ zl − n m l =1 ⎜ ∑i =0 ∑ j =0 N i, p (u )N j ,q (v )wi, j ⎝ np

⎞ ⎟ ⎟⎟ ⎠

2

(3)

where np is the number of control points. If the number of knots and their positions are fixed, the set of weighting factors is known and only the control points

{{P } }

m n i, j i=1 j=1

∈ R are considered during the optimization of (3), we have a linear

problem of least squares. But if the knots or the weights are unknown, it will be necessary to solve a non linear problem. In many applications the knots location is not necessary; therefore, knots values are obtained using some heuristic techniques.

4 Evolutionary Strategies Evolutionary Strategies (ES) were developed in 1964 by Rechenberg and Schwefel at the University of Berlin as an experimental optimization technique [13]. ES try to imitate, in contrast with Genetic Algorithms, the effects of the genetic procedures in the phenotype. ES belong to a kind of probabilistic numerical optimization algorithm, which include Evolutionary Programming (EP), Genetic Algorithms (GA) and Genetic Programming (GP), which are known as Evolutionary Algorithms. The first variant of ES, called (1 + 1) − ES , works based on two only individuals, a parent and a descendant per generation. The descendant is created by applying variations, called mutations, binomially distributed (with mean equal to zero and variance

σ 2 ) to the parent. The descendant can be the parent in the next

166


generation if it is better than the parent;if the opposite is true, the parent will be the survivor for the next generation. (1+ 1) − ES was replaced by ( μ + λ ) − ES and ( μ, λ ) − ES variants, with

μ > 1 parents and λ > 1 descendants per generation. In these new variants, the recombination concept was introduced, in order to create individuals as the cross of the parent attributes. After mutation and the individuals evaluation, the descendants replace the parents if the former are better than the latter. Depending on the selection type, μ new individuals are selected only from the descendant popula-

( μ, λ ) − ES or μ new individuals are selected from the parents and the descendant ( μ + λ ) − ES . Beside mutation and recombination, ( μ + λ ) − ES and ( μ, λ ) − ES control the size of the mutation step by an auto-adaptation process tion

that learns the mutation step size, and optionally the covariance, during the evolutionary searching process [13]. ES use three main operators for changing the population until a stop criterion is reached. Recombination: new individuals are produced by crossing the information contained in the parents. Depending on the individual variable representation some algorithms can be applied for recombination purposes: discrete recombination, local intermediate recombination, global intermediate recombination, point crossover and n-point crossover. The recombination operator allows for the exploration of the searching space. Mutation: After recombination, the descendants are changed with a probability p, by introducing small variations known as mutations. Mutation allows for the introduction of new possible solutions and the exploitation near to a given solution (individual). Mutation follows the scheme given in (4) and (5). Selection: choose the best individuals of the population according to a fitting criterion.

σ i ' = σ i e (τ ⋅N ( 0,1)+τ ⋅N ( 0,1) )

(4)

xi ' = xi + σ i '⋅ N i (0,1)

(5)

0

i

i

N( 0,1 ) is a random variable normally distributed with mean 0 and variance 1, and τ 0 , τ i are constants that control the mutation step. where

5 NURBS Fitting to Unorganized Points Our method for constructing NURBS surfaces from scattered and unorganized point clouds is roughly made up of two stages. The first one regularizes and fits the NURBS surface. The second one optimizes the NURBS surface fitting and trims the surface in order to eliminate points inserted outside the cloud during the regularization process. The following subsections explain each of these stages.


167

5.1 Point Cloud Regularization The regularity analysis of the point cloud takes place in a two-dimensional space. Such space is given by the principal components analysis regression plane (PCAP), which is expanded along the directions of higher dispersion of the cloud. The regularization process consists of the following steps: 1.

Project the points of the cloud onto the PCAP and obtain the extreme projected points.

2.

Construct a grid from the extreme points with density cording to (6)

ρ=

P A

where P is the number of points and mined by the extreme points. 3. 4.

ρ calculated ac(6)

A is the area of the ellipse deter-

For each grid element, do steps 4 to 5. If no point is present, insert one point in its centroid, according to (7) if the inserted point is inner to the cloud (Figure 3) or according to (8) if the inserted point is outer to the cloud (Figure 4).

Pins = AVGNeigh + Pgrid − Pj(AVGNeigh, PlanePCA)

(7)

G (8) Pins = Pext + λ B where AVG Neigh is the neighborhood mean of the point for inserting, Pgrid is the centroid of the empty element analyzed, PlanePCA is the PCAP,

Pj is the projection function, Pext is the point of the cloud closest

to the point for inserting, λ is the distance between them and PCAP basis vector in the inserting direction. 5.

B is the

If there are two or more points in the element, remove the points located farthest from the centroid.

Once the regularization process has been finished, a NURBS surface can be fitted to the entire cloud. In order to reduce the fitting error of the surface, due to the smoothing properties of NURBS, an optimization process is carried out for preserving the sharp features of the original point cloud.

168


Fig. 3. Insertion of a point inside the cloud

Fig. 4. Insertion of a point outside the cloud

5.2 Surface Optimization The surface optimization process is devised to maintain the influence of the original point cloud stronger than the points inserted, so that the topology of the cloud is not affected. Our optimization process is carried out by a ( μ + λ ) − ES . It can be described as follows. Let P = p1 , p 2 ,..., p n be a set of 3D points sampled from a real object,

{

}

e.g. the regularized points, and proximates by (9).

S = {s1 , s 2 ,..., s m } be a NURBS surface that ap-

P . Our problem consists of minimizing the approximation error given


E(S)= d P,S < δ

169

(9)

d P,S is the total distance between P and the NURBS approximation surface S . The parameter δ is a given user error tolerance. The evolutionary strategy obtains the control point weights of S , so that (9) is true. where

The evolutionary strategy will only obtain the weights of the points that belong to the point cloud. The weights of the inserted points will be assigned as follows: i) if the point was inserted inside the cloud, its weight will be the mean of the weights of the neighbor points. ii) If the point was inserted outside the cloud, its weight will be zero. Since the influence of the NURBS surface control points is only local, the sampled points P will be divided into clusters where a local optimization process will be carried out, which reduces the computational cost of the proposed method. The optimization process starts with a clustering of the set of points P, such clustering will be achieved by k-means. The objective of the k-means is to find homogeneous regions where the optimization process can be run without distorting the local shape of the surface. By the last run of the k-means clustering algorithm, it is expected to have found the homogeneous regions where the optimization process was run, as shown in Figure 5. Once P is clustered an evolutionary strategy μ + λ − ES will optimize the local fitting of the NURBS in each cluster. At the boundaries of the clusters, the geometry of the surface is not affected since the evolutionary strategy only modified local weights of the NURBS surface control points, it does not modified de points. The evolutionary strategy configuration is as follows:

(

)

Individuals: the individuals of the strategy are conformed by the weights of the cluster points and the mutation steps σ , as shown in Figure 6 (where wi are the

σ i are the mutation step sizes). Mutation operator: uncorrelated mutation with n mutation step sizes σ is ap-

control point weights and

plied to the individuals, according to (4) and (5).

Fig. 5. Clusters found by k-means

170


w1

w2

…

σ1

wn

σ2

…

σn

Fig. 6. Individual of the evolutionary strategy

Recombination operator: the recombination operator is different for object variables wi and parameters σi . A global intermediary recombination is applied to object variables, according to (10), whereas a local intermediary recombination is applied to mutation step sizes σ i , according to (11). Selection operator: the best individuals according to the aptitude function given in (9). In order to perform a fast computation of the distance between the points P and the NURBS surface S , the points of S are stored in a kd-tree structure, so that the searching process for finding the nearest points between

log(n ) order.

b'i =

1

ρ

P and S is of

ρ

∑b k =1

k ,i

b' i = u i bk1 ,i + (1 − u i )bk 2 ,i

(10)

(11)

i is the allele of the individual, bi is the value of the allele, ρ is the size of the recombination pool and μ is a random number uniformly distributed n [0, 1]. where

Table 1. Optimization process Perform a clustering of P by using k-means For each cluster do Set individual size = cluster size Set population size = μ Initialize the population randomly Evaluate the population in the aptitude function (8) While the stop criterion

δ has not been reached do

λ ⋅ 0.9 do Ind i = mut (Populationrand (1,μ ) )

For i = 1 to End for

λ ⋅ 0.1 do Ind i = rec(Populationrand (1,μ ) )

For i = 1 to End for

Population = select from( μ + λ ) End while End for


171

The algorithm in Table I summarizes the optimization process. After the optimization process, the optimized surface is trimmed to eliminate the points inserted outside the cloud.

6 Experimental Results The proposed method was evaluated in scanned data. Two different models are used to show the method results. The Face model, obtained from Alexander Belyaev's web site from the Heriot-Watt University (www.eps.hw.ac.uk/~belyaev), illustrate the complete NURBS constructing process. The Angel model, obtained from Ohio State University, illustrates the results of the optimization process. The method was implemented in a 1.4GHZ Pentium M with 512MB of RAM. The first stage, point cloud regularization, takes seven seconds to process 15K points. We first project the cloud onto its PCAP, as shown in Figure 7(b); then, the two-dimensional analysis takes place for inserting and removing points where necessary. The two-dimensional regularization is shown in Figure 7(c) (inner and outer inserted points are shown in red and green color respectively), whereas the three-dimensional (point cloud) regularization is shown in Figure 7(d). The NURBS surface fitted to the regularized point cloud is shown in Figures 7(e). The second stage, surface optimization, takes 3 minutes for processing 15K points. After 10 generations average, the evolutionary strategy reached the minimum, i.e. the distance between P and the optimized NURBS surface S reached an average of 14% less than the distance between P and the non optimized NURBS surface. Figure 8 shows the effectiveness of the proposed method application over complex surfaces. The profiles presented in Figures 8(a) – 8(c) show the complexity of the surface. Figure 9 show the improvement of the sharp features obtained with our method. In Figure 9(b) the NURBS surface points follow the control points (red circles) closer than in Figure 9(a), which improves sharp feature representation. To verify that the shape of the original point cloud was not distorted, two metrics were defined. i) The relative error Ebdl between the diagonal length of the bounding box of

P and the diagonal length of the bounding box of S . ii) The

normalized modeling error Eavg , according to (12), is given in [4]. In our tests, the relative error was 0.031% and the modeling error was 0.01,

Davg

∑ =

Eavg =

N −1

i =0

N

Davg L

di (12)

172


Fig. 7. Stages of the proposed method evaluated on scanned data. (a) Original point cloud (b) Point cloud projected onto its PCAP (c) Regularized projection (d) Regularized point cloud (e) NURBS surface constructed from the regularized point cloud (f) Trimmed and optimized NURBS surface

Fig. 8. NURBS fitted over a complex surface applying the proposed method. (a) Right profile of the original dataset (b) Frontal profile of the original dataset (c) Left profile of the original dataset (d) Non optimized NURBS surface (d) Optimized NURBS surface.


173

Fig. 9. Detail of sharp features. (a) Detail of a Non optimized NURBS surface fitted to a set of control points. (b) Detail of an optimized NURBS surface fitted to a set of control points.

d i and N denote the signed distance from the data xi and the number of the total data, respectively. L is the bounding box length. In Table II, the execu-

where

tion time of our method stages are summarized. In Table III, some statistics of the optimization process are presented. Table 2. Execution time of the method stages Stage Regularization Optimization

Execution Time 7 seconds 3 minutes

Table 3. Statistics of the optimization process Processed images Average points per image Average points per cluster Tests per image Time for clustering Time for optimizing Average generations per test

μ λ

Distance reduction Relative error Modeling error

30 15K 854 12 5 seconds 3 minutes 10 5 35 14% 0.031% 0.01

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7 Conclusions and Future Work A new simple method for constructing NURBS surfaces from scattered and unorganized point clouds was presented. Both our method applicability in regular and irregular surfaces and the effectiveness of the method improving NURBS surface sharp feature representation were shown. It was demonstrated that it is possible to fit complete NURBS surfaces to point clouds, without previous construction of either polygon meshes or NURBS patch networks; as a result, the proposed method is computationally efficient. Our method cannot be applied over surfaces occluded with regard to its principal component analysis regression plane, like the surface shown in Figure 1. Occluded surfaces with regard to its principal component analysis regression plane generate collisions in step one of the regularization process and such collisions are eliminated by deleting points, which can eliminate a complete segment of the surface. The optimization method could be used for optimizing approaches that use NURBS patches. In those approaches, the clustering process would not be necessary, since the optimization process would be carried out for each NURBS patch. In order to apply our method over occluded surfaces, some future work can point to that direction. The work of Tenenbaum, De Silva and Langford [14] can be used to analyze the cloud regularity in a parametric space without loosing information, which allows constructing the NURBS surface over closed and occluded surfaces. Some additional future work could focus on i) detecting holes in the cloud before regularizing and then trimming them after the regularization process, and ii) improving the optimization process by establishing automatically the number of clusters where the evolutionary strategy is run

References [1] Myers, A.: Introductory literature review surface reconstruction from three dimensional range data. Technical report, The University of Adelaide, Department of Computer Science (1999) [2] Piegl, L., Tiller, W.: The NURBS Book. Springer, Heidelberg (1995) [3] Ristic, M.: Efficient fitting of Non-Uniform Rational B-Spline surfaces using nonorganized 3D data. In: SPIE’S, vol. 11(1) (December 2000) [4] Park, I., Yun, I., Lee, S.: Constructing NURBS Surface Model from Scattered and Unorganized Range Data. In: Second International Conference on 3-D Digital Imaging and Modeling (1999) [5] Gregorski, B., Hamann, B., Joy, K.: Reconstruction of B-spline Surfaces From Scattered Data Points. In: Proceedings of Computer Graphics International (2000) [6] Eck, M., Hoppe, H.: Automatic Reconstruction of B-Spine Surfaces of Arbitrary Topological Type. In: Proceedings of the 23rd annual conference on Computer Graphics and Interactive Techniques, ACM Press, New York (1996) [7] Krishnamurthy, V., Levoy, M.: Fitting smooth surfaces to dense polygon meshes. In: SIGGRAPH 1996 Conference Proceedings. ACM SIGGRAPH, Addison Wesley, Reading (1996)


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[8] Bertram, M., Tricoche, X., Hagen, H.: Adaptive Smooth Scattered-data Approximation for Large-scale Terrain Visualization. In: EUROGRAPHICS - IEEE TCVG Symposium on Visualization (2003) [9] Shi, X., Wang, T., Wu, P., Liu, F.: Reconstruction of convergent g1 smooth b-spline surfaces. In: Computer Aided Geometric Design, vol. 21 (November 2004) [10] Yvart, A., Hahmann, S., Bonneau, G.: Smooth Adaptive Fitting of 3D models using hierarchical triangular splines. In: Shape Modeling International (2005) [11] Cordero, J., Parejo, J.: Curvas y Superficies para Modelado Geométrico, 1st edn., Alfa Omega, España (2002) [12] De Boor, C.: On Calculating With B-Splines. J. Approximation Theory 6(1), 50–62 (1972) [13] Bäck, T., Schwefel, H.: Genetic Algorithms in Engineering and Computer Science. John Wiley & Sons Ltd., Chichester (1995) [14] Tenenbaum, J., De Silva, V., Langford, D.: A Global Geometric Framework for Nonlinear Dimensionality Reduction. Science 290 (5500), 2319–2323 (2000)


Sealing Faceted Surfaces to Achieve Watertight CAD Models Brandon M. Smith1 , Timothy J. Tautges2 , and Paul P.H. Wilson3 1 2 3

University of Wisconsin-Madison [email protected] Argonne National Laboratory [email protected] University of Wisconsin-Madison [email protected]

Summary. Solid modeling engines are capable of faceting CAD models but may facet each face independent of adjacent faces. Regions of the resulting model have gaps between faces of their boundaries. An algorithm is described to seal faceted CAD models such that the boundary of neighboring faces has the same discretization along their shared edges. The algorithm works by sealing skin edges of geometric face faceting to geometric model edge facets, using vertex-vertex and vertex-edge contraction. Ten intricate CAD models of moderate to high complexity are tested with a range of facet tolerances. The algorithm succeeds in creating watertight models in most cases, with failures only at extreme values of facet tolerance and/or in the presence of geometric features which are outside the normal features encountered in most models. Keywords: faceted model, watertight, seal, tessellation.

1 Introduction Computational simulation relies increasingly on valid, accurate representations of the geometry of objects being simulated. Geometric models are most often constructed in Computer-Aided Design (CAD) systems, then transferred to a discretized representation for use in simulation. This discretized representation is often in the form of a Facet-Based Model (FBM), where geometric Vertices, Edges, and Faces are represented by collections of points, edges, and triangles, respectively. FBMs serve a variety of uses in computational simulation. First, they are often the means of transferring geometric models between applications, often in the form of STereo Lithography (STL) files [1]. FBMs are also sometimes used as the basis for generating the 3d discretization, or mesh, for a given geometric model [2, 3]. In other cases, the

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FBM is used directly, for example in monte carlo radiation transport (based on ray-tracing on FBMs)[4], and clash detection for robotic applications. One of the reasons FBMs are used so widely is that they are provided by virtually all geometric modeling systems. Modeling engines such as ACIS [5] and Open.Cascade [6] all provide API functions for getting the triangular facets for each (geometric) Face and facet edges for each (geometric) Edge. Typically, facet edges and faces (triangles) are guaranteed to be within a facet tolerance of the geometric entities they resolve, and this facet tolerance is input to the modeling engine. However, most FBMs provided by geometric modeling systems suffer from a fundamental flaw which prevents their usage as-is for other applications: the FBMs are not watertight. That is, each Face and Edge in the geometric model is faceted independently, with neighboring Faces not sharing facet points with the Edge where they meet nor with each other. While there may be points in each of those facetings that are coincident along the shared Edge, this is not always the case, and for multi-material models, this is almost never true. For example, Figure 1 shows a model where facetings of two Faces bounding an Edge of a cylinder are far from coincident. This flaw must be fixed before FBMs can be used for the other applications mentioned above. There has been a great deal of previous work on the subject of making faceted models watertight. These efforts fall roughly into two groups: one that views the faceted model similar to a point cloud that is the basis of deriving a closed faceted model, and another group that focuses on fixing localized problems in the model, e.g. by filling and closing holes in a faceted surface. None of the efforts reviewed in this work makes use of the topological information often available from geometric models, and few work on models having non-manifold topological features. Furthermore, these approaches vary greatly in their robustness, and few come with guarantees about what kind of FBMs can be made watertight. Implementations of these methods are also not available as open-source software, for use with other geometric modeling packages. A provably reliable solution for fixing FBMs, in an opensource implementation that could be applied in other modeling environments, is a key capability in developing other applications on FBMs. The contribution of this paper is the demonstration of an automatic algorithm for sealing facet-based models. We demonstrate how using topological information that accompanies a certain class of geometric models simplifies this process, and prove that, under certain weakly-restrictive conditions, the method is guaranteed to succeed. Robustness of our method is demonstrated using non-manifold geometric models from a variety of applications, having thousands of Faces and Regions. The algorithm is freely available as opensource software under the LGPL license. The construction of geometric models targeted in this work is a multi-step process. First, geometric Regions are constructed using primitive (sphere, cylinder, etc.) or free-form (spline surface) shapes and boolean operations between them, or more complex modeling operations like lofting or sweeping.

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Next, Regions are “imprinted” together; in this operation, any Faces and Edges that are partially coincident in space are made fully coincident, by splitting the original Edges into coincident and non-coincident parts. This process can be viewed as the opposite of regularization, a common operation in most CAD systems. Following this, entities coincident in space and with like topology are merged, such that only one entity remains. This merging step modifies the topology of the geometric model, such that the remaining merged entity is adjacent to all entities bounded by the entities that were merged. Faces bounding multiple Regions have a sense (either forward or reverse) with respect to each of those Regions, with the convention that a forward sense implies the Face normal points out of the Region. The result of this process is a model with some Faces shared between neighboring Regions, and Vertices and Edges bounding multiple Regions. The representation of non-manifold solid models, especially those with multiple 3d Regions sharing Faces between them, varies widely across modeling engines, whereas the representation and evaluation of manifold models is quite standardized. For this reason, most applications using this type of non-manifold geometry usually represent their own topology above the underlying solid modeling engine. This approach also simplifies implementation of “Virtual Topology” [7], facet-based geometry [8] and other specialized representations. This type of model is used to generate contiguous meshes for multi-material models by the Cubit [8] mesh generation code. The Common Geometry Module (CGM)[9] also uses this approach, maintaining its own topology graph for these and other purposes. CGM is used for all geometric modeling operations in this paper. The remainder of this paper is structured as follows. The remainder of this Section discusses previous work in fixing FBMs, and the nomenclature and functions we use for accessing FBMs and geometric models. The algorithm for “sealing” FBMs, along with a proof of its reliability, is given in Section 2. Implementation of the algorithm is described in Section 3, with results given in Section 4. Conclusions and future work are described in Section 5. 1.1

Previous Work

Previous work in converting flawed FBMs into watertight faceted models can be grouped in two categories. Region-based approaches may guarantee watertightness of a solid bounded by triangles, but at the cost of reconstructing the entire region using an intermediate representation. Mesh-based approaches fix faceting errors with small perturbations in the vicinity of the error, but offer no guarantee of watertightness for the overall solid [10]. In a recent survey [11], Ju concludes that mesh-based methods have been most successful at repairing CAD models where errors are localized, while region-based methods are most successful at reconstructing polyhedrons from poor quality data, with loss of detail.

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Fig. 1. A gap exists between faces of the cylinder because each face is faceted independent of bounding edges

Region-based methods [12, 13, 14, 15] reconstruct the model based on point cloud data derived from the faceting. Because they discard the original facets, these methods do not preserve the details of the original geometric topology, at least in cases where those details are not also geometrically distinct. Thus, Faces imprinted on another Face will not be preserved, since they are not geometrically distinct. Although region-based methods are the only option in the case of range-scanned data, they are not applicable to the problems described here, and are not considered further in this work. Bohn and Wozny [16] create a watertight shell by using topology to identify local gaps in the faceting, which they close based in part on heuristics. The algorithm does not move or merge any vertices. Their method begins by identifying free edges, and organizing them into loops, where each loop bounds a region that must be filled in with facets. This filling is done using an “ear clipping” algorithm, where two consecutive edges in the loop are used to form a tringle, with the third edge replacing the two edges in the loop. Edge pairs with the smallest angles are favored in this approach. The algorithm concludes by using edge flips to improve geometric quality of the facets. Barequet and Sharir also approach this problem [17] by idenitfying free edges, but split those edges such that the resulting edges are more consistent in edge length. A distance- and orientation-based heuristic is then used to determine matching groups of edges; gaps between these groups are filled by constructing new facets. In a subsequent effort [18], vertices within a certain distance tolerance are moved to eliminate gaps between triangulated faces. Edges are considered in pairs, rather than by groups forming arcs or loops. Free edges that remain after this process are identified and assembled into loops, and triangulated using [19]. In ambiguous situations, the user is prompted for guidance. Sheng and Meier [20] use a series of increasing distance tolerances, up to a maximum tolerance, to merge vertices then edge pairs whose endpoints have been merged. This ensures that the best candidates will be paired with


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each other. One drawback of this approach is that it requires facet edges of similar lengths to close gaps in the faceting; this is often not the case in typical facetings returned from modeling engines. Geuziec et. al [21] also note that Sheng and Meier’s algorithm may create edges adjacent to more than two triangles, when the same end is merged more than once. They avoid this simply by preventing merges that will create edges adjacent to three triangles. Borodin et al. improves on the merge-only approach by introducing a vertex-edge contraction, or t-joint operator [22, 23], where a free edge is split if a corresponding point is too far from either end but close enough to the edge. Vertex-edge contraction allows model repair to proceed using a smaller tolerance than if implemented with vertex merging alone. Kahlesz et al. [24] use a similar approach. An approach by Busaryev et al. [25] represents face boundaries as strings of intersecting balls. Boundaries of adjacent faces are joined by combining balls using a tolerance. Although the watertight output is produced using Voronoi diagram and Delaunay triangulation techniques, the repair phase of combining balls resembles vertex-vertex contraction. The advances in mesh-based watertightness are well summarized in an algorithm by Chong et al. [26]. First vertex-vertex contraction is performed by proximity. Next vertex-edge contraction is used, then large holes are triangulated. Finally skewed elements are removed by edge swapping or node merging. The last step is important because prior vertex-vertex and vertexedge contraction likely create facets with poor aspect ratios. These mesh-based algorithms attempt to restore missing information, in the form of where facets meet to close gaps, using three methods: 1. Perform vertex-vertex contraction by proximity. 2. Perform vertex-edge contraction by proximity. 3. Triangulate a patch across gaps that minimizes area or another heuristic quantity. However, we observe that much of the information they try to restore based on spatial proximity and searching was known at one point, but was lost or thrown away. For example, vertex-vertex and vertex-edge contractions are usually motivated by sealing neighboring geometric Faces at shared Edges. Knowing these topological relationships greatly narrows the search for matching free facet vertices and edges. In the case where the geometric model is still available, new points in the faceting can even be projected to the geometric entities the faceting is supposed to resolve. We believe that methods for repair of FBMs are greatly improved by taking advantage of geometric model information that previously has not been used. Furthermore, we assert that this information is often available when the original faceted model is created, but has not been used because of limitations in file formats used to transfer FBMs to applications. The sealing algorithm described in this paper uses geometric model information to improve the robustness of obtaining watertight FBMs.

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1.2


Notation and Assumptions

In this work, we consider a geometric model described in the form of a Boundary Representation (BRep). This model consists of geometric Vertices, Edges, Faces, and Regions, indicated with script letters V/E/ F /R, respectively. Note that non-manifold topology is allowed; in particular, many of the models we encounter have multiple Regions, with Faces shared between two of those Regions, and Edges and Vertices shared by multiple Regions. The collection of geometric model entities forms a cell complex; that is, the intersection of two entities of like dimension is either empty or corresponds to one or more entities of lower dimension that are also in the cell complex. Each Vertex, Edge, and Face has a corresponding faceting represented by a set, denoted with V, E, F, and R, respectively, with each of these sets containing points p, and possibly edges and faces e and f , depending on the topological dimension (R is empty, since Regions have no facets, but are still represented in the model, to get adjacency relations with other Faces). We assume that the faceting for each geometric Edge with a single Vertex has at least three facet edges. Edge facetings not satisfying this assumption would not be useful for graphics purposes, and in practice this assumption is satisfied in most observed facetings. Each faceting V, E, and F is itself also a d−dimensional cell complex; however, this cell complex does not share points with the facetings of other model entities, even those bounding the model entity in question. This is also typical of faceted models from most modeling engines. A function d(., .) returns the distance between the indicated entities, and Ω(.) represents the boundary of the indicated entity, which is a collection of (d-1)-dimensional facets. There are two tolerances that are important to the algorithm described in this paper. First, the “merge tolerance” is the distance below which two entities are considered spatially coincident; we denote this as g , recognizing that in effect this also serves as the geometric tolerance for those model entities. This tolerance is used in various places during the merging process described earlier. The facet tolerance f is the maximum distance between a facet edge or face and the geometric Edge or face it resolves. We assert certain things about the faceting, based on guarantees provided by the modeling engines constructing them. First, all facet points are within g of the corresponding model entities. While this is typical of faceted models from most modeling engines, it could also be achieved by projecting facet points to the corresponding model entity, using a function provided by virtually all modeling engines. Second, all facet edges and triangles are within f of the corrresponding model entities. Most modeling engines providing facetings take f as an input parameter, though in practice this input is disregarded if it is much larger or smaller than the default value used by the modeling engine. We also assume that f g ; since both these parameters can be changed through user input, this does limit the variations allowed in one parameter after the other has been chosen. Finally, all points on the boundary of a given faceting, Ω(E) or Ω(F ), are within g of some model


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entity that bounds the corresponding model entity, though which bounding model entity is not known on input. While not stated explicitly by the modeling engines, in practice this has been found to be the case for both ACIS and Open.Cascade. Each faceting E and F is non-degenerate (all points of d−dimensional facets, d > 0, are distinct), and is oriented and non-inverted (tangent or normal of facet is consistent with that of the underlying model entity in the neighborhood of the facet). Although we have not encountered cases where this assumption is invalid, invalidities of this type could be fixed using preprocessing similar to that discussed in Section 2.1. The local feature size (LFS)[27] at point x is the smallest closed disk centered at x that intersects two Vertices, two non-adjacent Edges, or an Edge and a Vertex that is not adjacent to the Edge. We assume that LFS f . We have observed cases where this assumption is not valid; preprocessing is used to fix these cases, as described in Section 2.1.

2 Sealing Algorithm The requirements of the proposed algorithm are: • • • • • • •

Seal faceted Faces along Edges to create a watertight model. To preserve human efficiency, the algorithm must be automatic. New facets must be owned by exactly one Face. Support non-manifold Faces. Fast enough to use as a preprocessing module. Deformation of input model should be minimized. Creation of new triangles should be minimized.

The input to this algorithm is a set of Vertices, Edges, Faces, and Regions with appropriate topological relationships. Faces and Edges are represented by facets and edge elements that resolve their geometric entities to within the facet tolerance. The boundary of each faceted face is not the same as the face’s adjacent faceted Edges. The endpoints of adjacent faceted edges are the same such that facet edges can be joined to form loops. The boundaries of Regions are represented by the facets of their bounding Faces. The facet tolerance is selected based on the feature size and required accuracy of the faceted representation. The input model is faceted, implying that the facet tolerance has previously been selected. 2.1

Preprocessing and Input to Sealing Algorithm

On input, the geometric model may not satisfy the local feature size assumption; that is, the model may contain features that are smaller than the facet tolerance input to the sealing algorithm. For example, the 40◦ section of the ITER model, discussed in Section 4, has multiple Edges and Faces of nearzero length and area, respectively. These features must be removed before the

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sealing process can start. Edges with length less than the facet tolerance are removed by identifying their endpoints together; Edges that average less than the facet tolerance apart from each other are merged together. Faces can be removed if all of their bounding Edges occur in merged pairs. These faces are approximately one-dimensional and are the result of imprinting neighboring regions in the solid model. Regions are removed if all of their child faces have been removed, though in practice this does not occur for any of the models tested for this work. One additional defect was identified for several faceted models used in this work, where skin vertices were distinct but within geometric tolerance (g ) of each other. This results in exactly two extra skin edges per defect. To remedy this, skin vertices within g of each other are merged. The search for these defects is only within individual facets or facet edges, rather than all geometrically proximate facet points, and therefore is relatively inexpensive, scaling linearly with the number of facet edges. After preprocessing is completed, we have the faceted geometric model V /E/F/R (including toplogical relations between them), along with specification of the geometric and faceting tolerances g and f , respectively. 2.2

Sealing

Given the input described in Section 2.1, the goal is to seal facets in the model together, such that the facetings of the Faces, Edges, and Vertices satisfy the requirements of a cell complex. That is, the intersections of points, edges, and triangle facets should correspond to other points and edges in the faceting. In this section, we present the sealing algorithm that is the primary subject of this paper. The algorithm is presented in its entirety here; proof of its success is discussed in the following Section. The general structure of our algorithm is to work Face by Face, sealing the skin of the Face’s faceting to the faceting of bounding Edges. We take advantage of the known topological relations between Faces and their bounding Edges and Vertices, and the correspondence between a Vertex, Edge or Face and its faceting. The only remaining assumption describes the geometric proximity of facet points, edges, and faces, with the geometric entities they are supposed to resolve. We describe the sealing algorithm in two parts. First, the higher-level algorithm for sealing a Face faceting with that of bounding Edges is given in Algorithm 1. This part of the algorithm has three main parts: a) matching points on the skin of the Face faceting to the Vertex points; b) separating the edges on the skin of the Face faceting into arcs Ω j , using the Vertex points as separators; and c) for each arc Ω j , sealing the edges on that arc to those of a corresponding Edge that bounds the Face; this last step requires an algorithm seal, described in a following Algorithm. Part b) uses a function doriented that computes the average distance between two ordered lists of facet edges. This distance function is computed as the summed distance between


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Algorithm 1. Face sealing algorithm. I. ∀F i a. ∀V i ∈a F i 1. f ind e ∈ ΩF i s.t. d(e, V i ) min 2. if d(p ∈adj e, V i ) min, choose p 3. else split e → new p 4. p → V i b. group ΩF i → Ω j F i using p ∈ V k ∈adj F i c. ∀Ω j F i = e 1. f ind E k s.t. doriented (Ω j , E k ) min 2. if E k not sealed yet, E k → Ω j F i 3. else seal(Ω j F i , E k )

parameterized points in both edge sequences normalized for edge length, as the edges are traversed in their respective orders. This traversal directionaware measure will compute a significant distance between two sequences of edges that are exactly spatially coincident but ordered in opposite directions. This algorithm is necessary for distinguishing a number of pathological cases we have observed in the facetings of various real geometric models. In the second part of the algorithm, shown in Algorithm 2, the task is to seal two sequences of facet edges. These sequences are ordered, such that they share the facet point(s) at their start point and, if distinct, their end point. Overall, this algorithm works by considering edges ee and es from the Edge and skin sequences, respectively, that share a point, pc ; and sealing them together, either directly if the edges from either sequence are close enough in length, or by splitting an edge in either sequence then sealing to one of the resulting edges. A graphical depiction of the three cases is shown in Figure 2.

Fig. 2. Three cases for sealing edges from two sequences of facet edges. a) Facet edges are similar in length, and can be sealed directly; b) and c), split skin / edge facets, respectively.

2.3

Proof

Previous methods to obtain watertight faceting have in some cases relied on user guidance. This type of guidance is impractical for the large models

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Algorithm 2. Edge/skin sealing algorithm. otherpoint(e, p) is the point adjacent to e not equal to p; for en , pn = next({e}, e, p), en is the next edge along the ordered sequence of edges {e} that shares p; pn = otherpoint(en , p). Begin: seal arc Ω j = {p}, {e}, E k = {p}, {e} I. Initialize: a. pc = Ωstart (Ω j ) b. es , ps = next(Ω j , −, pc ) c. ee , pe = next(E k , −, pc ) II. while Ω j not sealed a. if pe = ps 1. ee ← es 2. Ω j sealed. b. else if d(ps , pe ) ≤ f 1. pe ← ps 2. ee ← es 3. pc = pe 4. es , ps = next(Ω j , ee , pc ) 5. ee , pe = next(E k , ee , pc ) c. else if d(pe , es ) ≤ f 1. split es with pe , es → e1 (pc , pe ), e2 (pe , ps ) 2. pc = pe 3. ee , pe = next(E k , ee , pc ) 4. es = e2 (ps unchanged) d. else if d(ps , ee ) ≤ f 1. split ee with ps , ee → e1 (pc , ps ), e2 (ps , pe ) 2. pc = ps 3. es , ps = next(Ω j , es , pc ) 4. ee = e2 (pe unchanged)

encountered in our work. Therefore, automation is critical to practical use of this algorithm. Although space does not allow a complete derivation of the proof of reliability of our algorithm, we describe here the general structure of this proof. We develop the proof of reliability of this algorithm in four steps: Can seal Vertices to points on Face skin: By definition, the Vertex facet point p is located on the geometric model, and the facet edges in ΩF i are within f of the geometric Edges on ΩF . Because F i is a cell complex and so is the geometric model itself, each Vertex point p will therefore be within f of ΩF i as well. p will be closest to only one point of ΩF i because LFS f . p can therefore always be sealed to ΩF i , possibly after splitting one of the edges in ΩF i to insert a point closest to p.


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Can separate ΩF i into Ω j F i : ΩF i is a cell complex, with points corresponding to Va F i sealed to ΩF i . Because the geometric model is also a cell complex, these points will separate ΩF i into sequences of facet edges Ω j F i , with each sequence bounded by one or two pV, Va F . j i Ω F corresponds to E k , E k a F i : ΩΩ j = V l , and F i and the geometric model are cell complexes, each Ω j F i will correspond to one E k , and E k are distinct because of the preprocessing done before execution of the sealing algorithm. Can seal Ω j F i with E k : The facet points on Ω j F i and E k are within g of the geometric model; the facet edges on Ω j F i and E k are within f of the geometric model; and Ω j F i and E k are ordered consistently with each other. Therefore, similar to sealing the Vertices to ΩF i , the facetings Ω j F i and E k can be sealed together, possibly requiring splitting some of the edges on Ω j F i or E k . When all Face facetings have been sealed to the facetings of their bounding Edges, the whole model has been sealed.

3 Implementation This algorithm is implemented in C++. Geometric models are initialized by reading their faceting and geometric topology through CGM [9] into MOAB [28]. MOAB represents the geometric model entities as entity sets in the mesh, with the entity sets containing the facets defining the entity (Region sets are empty, since there is no 3d mesh), and topological relations between entities represented using parent/child relations between the sets. The sequences of facet edges used in the algorithm described earlier are represented only as ordered lists of facet points. Facet edges are not represented explicitly, since they are being modified frequently as sealing proceeds. Spatial searches are accelerated using MOAB’s k-d tree decomposition. The final results of sealing are represented in MOAB using the same geometric entity sets used for input. This representation can be output to various mesh formats, or can be left in MOAB for subsequent operations (e.g. facet coarsening and refinement to generate FE meshes).

4 Results Test models displayed in Figure 3 and listed in Table 1 were chosen from previous radiation transport simulations. Models vary in complexity, with the most intricate having more than 29 million triangular facets. Before being sealed, five test models were already watertight by proximity. Although not topologically the same, skin points of adjacent faceted surfaces were within the solid modeling engine’s absolute tolerance of each other. These models were created using the ACIS solid modeling engine. Five test models did not

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(a) ITER Module 13

(b) 40◦ ITER Benchmark

(c) FNG Benchmark

(d) UW Nuclear Reactor

Fig. 3. Detailed CAD models were used to test the sealing algorithm Table 1. Geometric entity count and number of triangular facets [millions] as a function of facet tolerance [cm] Model

Geometric Entity Regions Faces Edges UW Nuclear Reactor 2820 30237 65078 Advanced Test Reactor 2132 11827 22402 40◦ ITER Benchmark 902 9834 20485 ITER Test Blanket Module 71 4870 13625 ITER Module 4 155 4155 10255 ITER Module 13 146 2407 5553 FNG Fusion Benchmark 1162 4291 5134 ARIES First Wall 3 358 743 High Average Power Laser 15 139 272 Z-Pinch Fusion Reactor 24 95 143

Facet Tolerance [cm] 10−1 10−2 10−3 10−4 10−5 2.62 2.62 2.98 8.56 29.1 0.44 0.45 0.84 2.44 7.65 0.32 0.78 2.07 8.76 16.3 0.07 0.08 0.12 0.38 1.57 0.29 0.29 0.34 1.07 2.89 0.28 0.29 0.50 2.54 8.65 0.11 0.11 0.14 0.46 1.14 0.17 0.87 1.21 1.55 2.45 0.15 0.47 0.53 0.61 0.88 0.05 0.29 0.99 1.17 1.53


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Table 2. Number of face sealing failures as a function of facet tolerance [cm] Model

Facet Tolerance [cm] 10−1 10−2 10−3 10−4 10−5

UW Nuclear Reactor Advanced Test Reactor 40◦ ITER Benchmark ITER Test Blanket Module ITER Module 4 ITER Module 13 FNG Fusion Benchmark ARIES First Wall High Average Power Laser Z-Pinch Fusion Reactor

1019 88 18 0 0 2 63 1 0 3

0 0 9 0 0 0 0 0 0 0

0 0 0 0 0 0 0 0 0 0

0 0 18 0 0 0 0 0 0 0

0 0 191 0 0 0 0 0 0 0

originate in ACIS and were not watertight by proximity, due to file format translation, non-manifold surfaces, and imprecise modeling. Table 2 shows the number of Faces that failed to conform to their bounding Edges after sealing, for various models and facet tolerances. Note that for facet tolerances of 10−2 cm to 10−5 cm all but the ITER Benchmark model were watertight. This includes the default facet tolerance of 10−3 cm. The ITER Benchmark failed to completely seal because it contained many Faces with features smaller than the facet tolerance, created by imprinting. Failures occur if the facet tolerance becomes larger than the feature size, as in the 10−1 cm group. 4.1

Triangle Count

The speed of computations performed on the FBM is affected by the number of triangles in the model. The number of triangles increases due to vertexedge contraction (since triangle facets connected to the edge are split with the edge), but decreases due to vertex-vertex contraction of adjacent skin points of the same face. For the default facet tolerance of 10−3 cm, the change in the number of triangles ranged from 0% to 2%. Across the entire test suite of ten models and five facet tolerances, the change in the number of triangles ranged from -36% to 3%. A decrease in the number of triangles was common for the 10−1 cm facet tolerance, since at that tolerance many facet points on the same Face get merged. During sealing, each Face imposes an additional constraint on adjacent Edges. If the faceted Edge itself were to be sealed, it too would impose an additional constraint on the sealing process. By replacing the faceted Edge with a corresponding skin arc as suggested in Algorithm 1, the number of contraints on the edge is reduced by one. This decreases the number of additional triangles due to vertex-edge contractions.

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Table 3. The number of Faces containing inverted facets after sealing as a function of facet tolerance [cm] Model

Facet Tolerance [cm] 10−1 10−2 10−3 10−4 10−5 UW Nuclear Reactor 272 0 1 0 13 Advanced Test Reactor 30 0 0 0 0 40◦ ITER Benchmark 7 7 4 0 10 ITER Test Blanket Module 0 0 0 0 0 ITER Module 4 0 0 0 0 0 ITER Module 13 2 0 0 0 0 FNG Fusion Benchmark 16 0 0 0 0 ARIES First Wall 0 0 0 0 0 High Average Power Laser 0 0 0 0 0 Z-Pinch Fusion Reactor 2 1 0 0 0

4.2

Inverted Facets

When sealing, facets become inverted if a skin point is moved across an edge of the facet. This typically occurs with long, narrow facets. Each inverted facet is removed along with adjacent facets to create a polygon. The polygon is refaceted using the ear clipping algorithm as implemented in [29], with O(n2 ) time complexity, with n the number of points bounding the polygon. It is possible that the new facets are still inverted. If so, the polygon is iteratively expanded and refaceted until the new facets are no longer inverted. The polygon can only be expanded to the skin of the Face, so that the new facets are constrained to the skin. One can think of the Face skin as being owned by Edges instead of the face. Inverted facets that cannot be fixed by this method are a rare occurrence if the input model conforms to the input assumptions, as suggested by Table 3. Although the ear clipping algorithm is O(n2 ), n is typically small. 4.3

Timing

The implementation was executed on an Intel Xeon 3 GHz processor with 64 bit Linux. The sealing algorithm is intended to be used as a preprocessing step for applications that use FBMs. Timing results are shown in Table 4. Results do not include file reading and writing, because as part of an existing application, the use of this algorithm will not incur additional runtime due to file reading and writing. In general, sealing required less than one minute for most models and facet tolerances. 4.4

Application Example: Particle Tracking

This algorithm was developed, in part, to seal faceted CAD models for Monte Carlo radiation transport. One of the causes of lost particles is leakage between Faces. Figure 4 shows the lost particle fraction for each model before


191

Table 4. The time [seconds] to seal each model as a function of facet tolerance [cm] Model

Facet Tolerance [cm] 10−1 10−2 10−3 10−4 10−5 UW Nuclear Reactor 136 65 64 156 587 Advanced Test Reactor 93 16 27 76 235 40◦ ITER Benchmark 6 12 38 71 236 ITER Test Blanket Module 15 9 9 14 30 ITER Module 4 10 8 8 23 67 ITER Module 13 6 5 6 19 67 FNG Fusion Benchmark 7 4 4 9 29 ARIES First Wall 1 3 5 13 36 High Average Power Laser 1 1 2 5 25 Z-Pinch Fusion Reactor 1 1 2 4 12

Fig. 4. Lost particle fraction before and after sealing each model. Error bars indicate one standard deviation.

and after sealing. The default facet tolerance of 10−3 cm was used for all models. The ITER Benchmark, Test Blanket Module, Module 4, and Module 13 models lost significantly fewer particles after sealing. Sealing did not significantly affect the UW Nuclear Reactor, Advanced Test Reactor, FNG Fusion Benchmark, ARIES First Wall, High Average Power Laser, and ZPinch Reactor models. This reflects the input models, of which half were already watertight by node proximity.

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Sealing did not eliminate lost particles. The first three lost particles of each sealed model were investigated. In each case the particles became lost because of a specific defect in the particle tracking algorithm, unrelated to watertightness. Improvement of the particle tracking algorithm is an active research topic.

5 Conclusions A tool was developed to make existing faceted models watertight without human intervention. Faces were sealed to their bounding edges using vertexvertex and vertex-edge contraction. Because sealing progresses by Face, this algorithm naturally supports non-manifold geometry. Ten CAD models were tested over a 4 decade range of facet tolerances. Models were successfully sealed if the input assumptions of the algorithm were met. It is assumed that the facet tolerance is less than the feature size and greater than the merge tolerance. On average, sealed models contained more triangles than unsealed models as a result of vertex-edge contraction. Sealing can create inverted facets, with most occurring when input assumptions are not met. Timing results show that the algorithm is fast enough to be used in preprocessing for applications that use faceted geometry. A particle tracking application was used to test models before and after sealing. One cause of lost particles, leakage between unsealed Faces, was eliminated through application of the sealing algorithm. The cause of remaining lost particles was determined to be roundoff and precision issues in the ray-triangle intersect computation; this is the topic of further research.

Acknowledgement The authors thank Jason Kraftcheck for his advice and assistance using MOAB. This work was supported, in part, by Sandia National Laboratories and the US ITER Project through Sandia contracts #579323 and #866756. This work was also supported by the US Department of Energy Scientific Discovery through Advanced Computing program under Contract DE-AC0206CH11357. Argonne National Laboratory (ANL) is managed by UChicago Argonne LLC under Contract DE-AC02-06CH11357. The submitted manuscript has been created in part by UChicago Argonne, LLC, Operator of Argonne National Laboratory (”Argonne”). Argonne, a U.S. Department of Energy Office of Science laboratory, is operated under Contract No. DE-AC02-06CH11357. The U.S. Government retains for itself, and others acting on its behalf, a paid-up nonexclusive, irrevocable worldwide license in said article to reproduce, prepare derivative works, distribute copies to the public, and perform publicly and display publicly, by or on behalf of the Government.


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A Metric for Automatic Hole Characterization German Sanchez T1, John W. Branch2, and Pedro Atencio3 1

Professor Faculty of Engineering, Systems Engineering Program, Magdalena University, Colombia [email protected] 2 Professor school of system and informatics, faculty of minas. National University of Colombia - Medellin, Colombia [email protected] 3 Student of system and informatics school, faculty of minas. National University of Colombia - Medellin, Colombia [email protected]

Summary. The correct repair of three-dimensional models is still an open research problem, since acquiring processes (methods and technology) still have limitations. Although a wide range of approaches have been proposed, the main limitation is that user intervention is required to decide which regions of the surface should be corrected. We propose an automatic method for hole characterization enabling the classification of real and false anomalies without user intervention by using an irregularity measure based on two geometrical estimations: the torsion contour’s estimation uncertainty, and an approximation of geometrical shape measure surrounding the hole. Keywords: Curve torsion, entropy contour, Integration, surface reconstruction.

1 Introduction The shape reconstruction process requires estimating a mathematical representation of an object’s geometry using a measured data-set from the object [1]. Since there is no an ideal sensor that does not alter the samples obtained, the process should deal with general problems in order to generate models as accurate as possible. In context, there are many measuring drawbacks in the acquisition step: topological characteristic of the objects, sensor structure, physical properties of the object’s material, illumination conditions, among others. These inadequacies represent the main source of anomaly generation, and must be repaired in order to create a valid digital model [2]. The anomalies could be classified into three types: noise, holes artifacts and redundancy. Typically, these anomalies are repaired in a phase called Integration [3]. Whatever the anomaly type is, the process to correct it corresponds to a wide studying area, with many proposed techniques. However, correcting an anomaly is still considered an open problem inside the Computer Vision and Computer

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G. Sanchez, T.J.W. Branch, and P. Atencio

Graphics Community. The difficulty lies, in some cases, in the fact that exact nature of the anomaly’s source is undetermined or unknown, i.e. the noise distribution function [4], or its modeling is complex and does not have an unique solution, i.e. the filling of holes. The classical reconstruction methods need to apply a post-processing procedure after the final stage of surface-fitting. This is mainly due to the difficulty in differentiating the nature of the discontinuity, that is, whether or not it belongs to the actual surface (see Figure 1).

(a)

(b)

(c) Fig. 1. Examples of surface discontinuity, a-b) real discontinuities, c) false discontinuity

One of the main desirable features in surface reconstruction methods is the ability to fill holes or to complete missing regions. Lack of information is caused mainly by the reflective properties of the material, or by occlusion problems in regions inaccessible to the sensor. Some methods make an implicit correction during the fitting phase by means of global reconstruction [1] [5]. This approach has two disadvantages: First, it does not permit to keep or represent false holes, i.e. those belonging to the object, and second, the quality of the portion generated depends on the technique used and not on the analysis of the intrinsic geometry of the object. While taking the surface as continuum and complete, these techniques reproduce visually appropriate solutions. However, the correction of these anomalies is still limited to particular cases when objects are closed. A wide range of works has been proposed, many of them can be classified according to the representation used in the initial data, such as mesh repair techniques and dispersed data. However, other classifications can be made according to the conceptual nature of the used technique: based on geometry, algebraic methods and implicit functions.

A Metric for Automatic Hole Characterization

197

Hole Detection For the process of identifying holes, the most simple and functional procedure, with great ease of implementation consists in the idea that a discontinuity an the surface is easily defined as a succession of boundary edges. A boundary edge is defined as a triangle edge that is not shared by any other triangle. The procedure begins with a general search on the mesh to find an initial boundary triangle [6]. The search continues with the neighboring triangles looking for the next boundary edge which must shared a vertex with the initial edge. The process goes on, until the initial triangle is reached. Finally, a closed path that defines the hole contour is traced. The most important weakness of this simple procedure is that is limited to detecting any discontinuities but does not differentiate between real and false holes, because it assumes a whole closed object surface. Filling Holes After the identification procedure is applied, the hole-filling procedure continues by means of surface segment generation, for which, different techniques have been proposed [7] [8] [9] [10] [11] [12] [13] [14]. There are two general trends in the group of geometry-based techniques: repair based on triangles meshes and repair based on voxels. Liepa [6] describes a method for filling holes in meshes based on Delaunay Triangulations of the hole, after a refinement approach to mesh optimization by minimizing an energy functional related to the neighbor curvature estimation. Branch J, [14] filled the holes basing on the local interpolation of the radial basis function. A new segment is generated over a regular and symmetric mesh taking the original point-set density. This approach correctly reproduces the general geometry of the neighborhood hole contour but it fails when a planar surface contains the hole. Wei [15] proposed an algorithm for filling holes that starts with a hole identification phase and then applies a triangulation of the hole region using the Advancing Front Method. Finally, by solving a Poisson equation, the vertex of the generated triangles is adjusted. Although adequate visual results are obtained with this procedure, it is time costly and depends on the size of the hole. Voxel-based approaches estimate an initial surface volumetric representation by voxel-set. These voxel units are marked with a sign according to their relative position to the surface, that is, inside or outside the surface. Different techniques have emerged to fill the hole in the volumetric space. Curless [16] proposed a method based on volumetric diffusion consisting of a distance function estimation which is used to mark the voxel, and then diffusion is applied through the volume to find the zero-set that define the surface. A similar approach is proposed by Davis [8]. Ju [17] proposed a method of contour surface reconstruction by marking the voxel using an Octree data structure. The procedure is able to fill small gaps, taking into account geometric characteristics. The main limitation is that the hole size must be smaller than the relative surface size. Similarly, Joshua [10] describes an algorithm for hole-filling based on space decomposition by atomic volume regions and defines the model as the union of inner atomic units using a graph cut. User intervention is required to select the way to fill the hole. Chun [18]

198


describes a two-phases approach to 3D model repair. The first phase, a radial basis function interpolation is used to fill the region inside the hole; the second one is a post-processing stage to refine the geometrical detail. In the refinement stage the normals are adjusted to produce adequate visual results. We can conclude that hole-repairing methods are typically part of a pre-process of the surface fitting step, to get an accurate representation of the geometrical characteristics. In order to correctly fill those holes, a manual-assisted correction procedure is needed. Generally, this manual-assisted procedure has been based on both geometric and mathematical concepts. Their most important weakness lies in its application-domain limited nature, mainly due to their narrow flexibility in objects with different topologies. In this paper, we propose a metric for robust hole characterization in 3D models. This metric intends to characterize holes through geometric features measures. Our hypothesis is based on a widely accepted definition [19]: free-form objects are smooth except in regions that represent specific geometric details. Hence, if there are not any problems in the acquisition stage, a false contour anomaly should not have large geometric variations; otherwise, it could be caused by problems in the acquisition stage and constitute an anomaly to be repaired. Thus, if there were any problems during the acquisition process then the data is altered introducing distortion that should not be equal for the segments that define the anomaly. That is, acquisition problems introduce some degree of "contour distortion”. The characterization of each anomaly is based on the quantification of this distortion, which for this particular case is approximated by a quantification of the entropy in the boundary geometry. The paper is organized as follows: section 2 introduces the topic of anomalies characterization; section 3 describes the hole context irregularity measure; section 4 describes the proposed contour’s irregularity; and section 5 presents the experimental design and results.

2 Anomalies Characterization The main problem for the automatic repair of holes is to differentiate whether or not the discontinuity belongs to the object. In this context, the key difference between 3D scattered data and other data types such as images or video, is that 3D scattered data are typically irregularly sampled. The points’ distribution of vertices across the surface is not uniform, so to quantify a measure it must be robust under different distribution of points. In order to get the most accurate estimation of the irregularity of a hole we propose a metric that measures the hole´s most important geometrical features from two points of view: surface irregularity around the hole, and contour curve irregularity measure from the torsion and curvature entropy. A summary of the general diagram is shown in Figure 2.


Ž ŽŶƚĞĞǆƚŝƌƌƌĞŐƵ ƵůĂƌŝƚƚǇŵĞ ĞĂƐƵ ƵƌĞ

Ž ŽŶƚŽƵ ƵƌŝƌƌƌĞŐƵůĂƌŝƚǇǇŵĞĞĂƐƵƌƌĞ

y yes

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High entroppy?

Is ann ano omally

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Iss a faalse hole h

Fig. 2. Diagram of hole characterization

3 Hole’s Context Irregularity Measure Initially, it starts making a cluster environment analysis by regions variability quantification around the hole. For each one of clusters, the centroid is taken to estimate a shape description by geometrical-topological properties. Therefore, this stage implements a general technique of clustering by region growing. The algorithm starts calculating a prior estimation of curvature at each point of surface [6]. It is approximated by principal component analysis obtained resolving the covariance matrix, so that: 1

(1)

1 where is the neighborhood size of , and defined following equation:

is the centroid of each cluster and is

1

(2)

The curvature estimation is approximate by eq. 3, like in [26]: , where

(3)

Note that the eigenvector associated with λ indicates the low variation direction of the data, therefore, aims to approximate the normal vector of the surface in the point , so that indicates quantitatively the variation of the surface tangent plane, and it constitutes a measure of the variation of the surface.

200


Once the approximation of the curve is obtained, the next step is to compute a clustering of surfaces, so that the variance of each cluster does not exceed a fixed threshold β. The general description of this procedure is presented in Algorithm 1 and an example in Figure 3. Algorithm 1: Clustering and center selection of environment. Center selection ( ) Take a Pk random seed point and create a new cluster Ci. while(Nota ll point are clustering) Add a vertex Vk є Nci successively to C, while Vci < β, where Nci is a radial neighborhood of Ci and Vci is the estimation of variance of cluster Ci. Endwhile

Fig. 3. Cluster contour result

In this work we only use the curvature approximation for clustering the point set around the hole-contour. However, the classical surface curvature measures, such as the Gaussian and the mean curvature at a point of a surface, are not very indicative of local shape. Hence, we used a measured independent of the size of the surface so it is locally described as shape of the surface. The shape index [20] is a measure that describes the second order structure of the surface in the neighborhood of any one of its points. This is thus estimated as follow: 2

arctan

(4)

k are the principal curvatures, and s 1, 1 except for the plaWhere k nar region that has an indeterminate shape index. In order to obtain a global holecontour description, an average shape-index of cluster is estimated, due to the fact that shape-index is a point measure. So, the centroid shape-index of each cluster i is thus: 1

(5)


201

Where, is a shape-index of the cluster , and c C is the set of k cluster (see Figure 4). The point-set of size N , such that is the amount of points of the cluster i. In general, this corresponds to a shape-index average, weighted by the amount of points in each cluster.

Fig. 4. Points’ selection for index-shape estimation

4 Measure of Contour’s Irregularity In this step we are interested in measuring the geometrical characteristic of the contour curve. i.e. its means its curvature and its torsion. The aim of these estimations is to quantify its irregularity by means of the uncertainty using the entropy measure. Contours’ Torsion Entropy Measure The two fundamental characteristics of a curve are its curvature and torsion; these allow to measure how a curve bends inside the 3D space, therefore, it constitutes a curve’s particular characteristic. Often, we assumed that discontinuous contour curves in smooth objects without acquisition problem are smooth too. Contour bends give us a measure of irregularity. However, estimating the accurate torsion value of a sample curve, defined by a piecewise linear approximation through an ordered finite collection of points p , is not a trivial task since noise is present. i.e., the points p stay too close to the curve, but not necessarily lie on it. In order to approximate a correct classification of contour curves, we used the τ s N s torsion measure. For spatial curve the torsion is defined by B s where N s r s / r s is the normal vector, s is the arc-length from a specific position r t given by a parametric curve r, to a close position r t and de(see Figure 5). For a non arc-length fined by s t r u rized r t , τ s is thus estimated: (6)

202


Fig. 5. Normal, tangent and osculating plane

To estimate the torsion we adopt the weighted least squares approach and local arc-length approximation [22] [23] [24]. It considers a samples-set p from a spatial curve. The estimation of derivates of r at p is performed with a pointsubset of 2q 1 points such that (see Figure 6): ,

,…,

.

Fig. 6. Weight least square curve

Then a parametric curve x s , y s , z s)) is fitted locally, assuming p r and an arc-length s value associated to the samples p : 1 2 1 2 1 2 ̂ Taking

coordinate, the derivates x , x , x E x ,x ,x

1, s Where w trix inversion:

∑

w x

p

p

x s

,p

1 6 1 6 1 6

(7)

are obtained minimizing [21]: 1 x s 2

1 x 6

s

(8)

. It can be written in terms of ma-


203

(9) A similar approach is used to estimate the

and

derivates getting the vectors:

From the equations system: (10) here, a a a

a a a

The a values and b

b b b

a a a ,

, ,

b b b

are defined thus:

1 6

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,

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,

,

,

1 2

1 4

1 12

1 36

1 2 1 2 1 2

Finally, it defines:

The computation of

,

is straightforward, thus:

,

,

,

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, , ,

b b b

, , ,

204


Due to their nature, hole-characterization problems suggest solutions based on inference, since it needs a process of drawing conclusions from available information that is partial, insufficient and that does not allow to reach an unequivocal, optimal and unique solution. Then we need to make inferences from the available data assuming it is noisy. Specifically, the topic of hole characterization constitutes a highly ambiguous example to take decisions because there are many possible configurations of irregular contours. Both aspects, noise and ambiguity imply taking uncertainty in account. The adequate way to deal with the presence of uncertainty, related to lack of information, is to introduce assumptions about the problem’s domain or a priori knowledge about the data, by means of the notion of degrees of belief. It should be treated using the classical rules of calculus of probabilities. The rules of probability theory allow us to assign probabilities to some "complex" propositions on the basis of the probabilities that have been previously assigned to other, perhaps more "elementary" propositions. However, in order to estimate a measure to characterize contours, we are not interested in just probability estimation about a geometrical characteristic, but also in its variability. High variability could be measured through entropy. Specifically, conditional entropy is used. Given two variables and , the | quantity that measures the amount of uncertainty about one variable x when we have some limited information about another variable is conditional entropy [25]. It is obtained by calculating the entropy of as if the precise value of were known and then taking the expectation over the possible values of y. |

|

|

log

|

In a similar way, log

|

|

(11)

,

forming the contour of a 3D curve definGiven a sequence of points : ing an anomaly, and a set of geometrical characteristic measured associated to each one in . We want to measure the irregularity in from a prior knowledge of some geometrical characteristic measure. It means, the certainty a point is estimated taken a -set : : 1 1 over a sorted sequence of points used to estimate the next value. The certainty or in inverse form, the unpredictably of all is related to entropy.


, Where

:

,

,…,

,

log

|

205

(12)

,, and: .

Contours’ Curvature Measure The principal limitation of a descriptor based on torsion measure is deal with planar curves. Planar curves may appear as a result of occlusions; although these are uncommon the torsion based decision is inappropriate. However, in order to completeness, our metric take into account those cases and use for planar cases a tangent vector variability measure like irregularity. For anomalies in planar cases the tangent variability usually is high, otherwise, real holes show smooth changes between tangents angles (see Figure 7).

a)

b)

c)

d)

Fig. 7. Tangent vector (Red) variability for a) Dragon object and b) eye contour from Mask, c-d) close view of both cases respectively

To estimate this measure, we take the weighted least squares approach and local arc-length approximation made in the section 4. Tangent vector is defines ,, or in derivates terms

. We estimate the entropy

of

angle between successive tangents like the equation 12, replacing the torsions distributions by angle between tangents distribution. And finally quantify the global entropy of the contour by: S

S

S

(13)

206


Finally,

For undefined cases of SI it means for planar case, the S measure is an accurate estimation of irregularity.

5 Experiment Design and Result In order to estimate the and | quantity given a continuous variable of torsion measure, we used a bin size to discrete the domain of an experimental set of hole-contour configurations. The experimental set was obtained from 10 images with real hole anomaly and 10 contours of false hole anomaly, in partial and complete 3d models’ range data. Some images were scales to maintain a fixed scale. It was done by subtracting the mean and dividing by the standard deviation. The parameter r was set to 10% of standard deviation and m set to 2. The irregularity was estimated with equation 13. Figure 8 shows the irregularity estimated for both sets. It shows that the irregularity measure is highly sensitive to the irregularities in the contour. Determining if an anomaly is real or false is straightforward because the values are sufficiently separated. The midpoint of the range of separation is 3.1. The experimental threshold for classification was estimated in this value.

Fig. 8. Irregularity values of false (blues) and real anomalies (red)

The irregularity could increase when increased the separation of the data. The method is highly sensitive to noise; small changes in the regularity of the contour show an equal increase in the estimation of entropy. This method can be used as an initial step in the process of correcting anomalies. We aim to complement the method with the filling process to propose an automatic robust method to correct anomalies.


207

The proposed method allows to estimate a metric for the automatic classification of anomalies in range images. The purpose of the method is to automate a process that has traditionally required user intervention. The method estimates the most relevant geometric characteristics of curves and surfaces to describe them. The anomalies used as the working set were mostly generated by occlusion.

References [1] Curless, B.: New Methods for Surface Reconstruction from Range Images. Stanford University, Stanford (1997) [2] Kumar, A., et al.: A Hole-filling Algorithm Using Non-uniform Rational B-splines, pp. 169–182. Springer, Heidelberg (2007) [3] Turk, G., Levoy, M.: Zippered polygon meshes from range images, pp. 311–318. ACM, New York (1994) [4] Sun, X., et al.: Noise in 3D laser range scanner data. In: IEEE International Conference on Shape Modeling and Applications, SMI 2008, pp. 37–45 (2008) [5] Carr, J.C., et al.: Smooth surface reconstruction from noisy range data, p. 119. ACM Press, New York (2003) [6] Liepa, P.: Filling holes in meshes, pp. 200–205. Eurographics Association (2003) [7] Dorai, C., et al.: Registration and Integration of Multiple Object Views for 3D Model Construction. IEEE Transactions on Pattern Analysis and Machine Intelligence 20, 83–89 (1998) [8] Davis, J., et al.: Filling Holes in Complex Surfaces Using Volumetric Diffusion, pp. 428–438 (2002) [9] Wang, J., Oliveira, M.M.: A hole-filling strategy for reconstruction of smooth surfaces in range images. In: XVI Brazilian Symposium on Computer Graphics and Image Processing. SIBGRAPI 2003, pp. 11–18 (2003) [10] Podolak, J., Rusinkiewicz, S.: Atomic volumes for mesh completion, p. 33. Eurographics Association (2005) [11] Bischoff, S., Pavic, D., Kobbelt, L.: Automatic restoration of polygon models. ACM Trans. Graph. 24, 1332–1352 (2005) [12] Guo, T., et al.: Filling Holes in Meshes and Recovering Sharp Edges. In: IEEE International Conference on Systems, Man and Cybernetics, SMC 2006, vol. 6, pp. 5036– 5040 (2006) [13] Bendels, G.H., Schnabel, R., Klein, R.: Fragment-based Surface Inpainting. In: Desbrun, M., Pottmann, H. (eds.) The Eurographics Association (2005) [14] Branch, J., Prieto, F., Boulanger, P.: Automatic Hole-Filling of Triangular Meshes Using Local Radial Basis Function, pp. 727–734. IEEE Computer Society, Los Alamitos (2006) [15] Zhao, W., Gao, S., Lin, H.: A robust hole-filling algorithm for triangular mesh. Vis. Comput. 23, 987–997 (2007) [16] Curless, B., Levoy, M.: A volumetric method for building complex models from range images, pp. 303–312. ACM Press, New York (1996) [17] Ju, T.: Robust repair of polygonal models, pp. 888–895. ACM, New York (2004) [18] Chen, C.-Y., Cheng, K.-Y.: A Sharpness-Dependent Filter for Recovering Sharp Features in Repaired 3D Mesh Models. IEEE Transactions on Visualization and Computer Graphics 14, 200–212 (2008)

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[19] Besl, P.J.: The free-form surface matching problem, in Machine Vision for ThreeDimensional Scenes. Academic Press, London (1990) [20] Koenderink, J.J., van Doorn, A.J.: Surface shape and curvature scales. Image Vision Comput. 10(8), 557–565 (1992) ISSN:0262-8856 [21] do Carmo, M.: Differential geometry of curves and surfaces. Prentice Hall, Englewood Cliffs (1976) [22] Lewiner, T., Gomes Jr., J.D., Lopes, H., Craizer, M.: Curvature and torsion estimators based on parametric curve fitting. Computers & Graphics 29(5), 641–655 (2005) [23] Lancaster, P., Salkauskas, K.: Surfaces generated by moving least squares methods. Math. Comp. 37(155), 141–158 (1981) [24] Lancaster, P., Salkauskas, K.: Curve and Surface Fitting: An Introduction. Academic Press, London (2002) [25] Ariel, C.: Lectures on Probability, Entropy, and Statistical Physics. Albany: Department of Physics, University at Albany (2008) 0808.0012 [26] Pauly, M., Gross, M., Kobbelt, L.P.: Efficient simplification of point-sampled surfaces, pp. 163–170. IEEE Computer Society, Los Alamitos (2002)

Receding Front Method: A New Approach Applied to Generate Hexahedral Meshes of Outer Domains Xevi Roca, Eloi Ruiz-Gironés, and Josep Sarrate Laboratori de Càlcul Numèric (LaCàN), Departament de Matemàtica Aplicada III, Universitat Politècnica de Catalunya, Jordi Girona 1-3, E–08034 Barcelona, Spain {xevi.roca,eloi.ruiz,jose.sarrate}@upc.edu

Abstract. Two of the most successful methods to generate unstructured hexahedral meshes are the grid-based methods and the advancing front methods. On the one hand, the grid-based methods generate high quality elements in the inner part of the domain using an inside-outside approach. On the other hand, advancing front methods generate high quality hexahedra near the boundary using an outside-inside approach. In this paper we propose the receding front method, an original approach that combines the advantages of both methodologies: we use an inside-outside mesh generation approach by means of a reversed front advance. We apply this approach to mesh outer domains. To reproduce the shape of the boundaries, we first precompute the mesh fronts by combining two solutions of the Eikonal equation on a tetrahedral reference mesh. Then, to generate high quality elements, we expand the quadrilateral surface mesh of the inner body towards the unmeshed external boundary using the pre-computed fronts as a guide. Further research is under way in order to apply the proposed method to more complicated geometries. Keywords: Mesh generation, unstructured hexahedra, eikonal equation.

1 Introduction During the last two decades several general-purpose algorithms for fully automatic hexahedral mesh generation have been proposed, see [1, 2, 3, 4, 5, 6] for a survey. However, none of the existent algorithms is robust, automatic and generates highquality meshes for any initial geometry. There are two families of methods that almost fulfill all these requirements, the grid-based and the advancing front methods. In fact, these approaches are the most successful methodologies to obtain a generalpurpose hex-meshing algorithm. Furthermore, the grid-based and advancing front methods have advantages and disadvantages that complement each other. Thus, we

This work was partially sponsored by the Spanish Ministerio de Ciencia e Innovación under grants DPI2007-62395, BIA2007-66965 and CGL2008-06003-C03-02/CLI and by Universitat Politècnica de Catalunya (UPC).

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can consider how to obtain a hexahedral meshing approach that presents only the advantages, and avoids the disadvantages, of these two methods. On the one hand, the grid-based methods [7, 8, 9, 10] are the only family of robust and fully automatic hexahedral mesh generation algorithms. In addition, they generate high-quality meshes in the inner part of the mesh. These advantages are possible because the mesh is generated from inside-to-outside. However, the gridbased methods generate low quality hexahedra near the boundary and the final mesh depends on the spatial orientation of the domain. These drawbacks appear because the inner mesh does not have layers of hexahedra that progressively adapt to the boundary shape of the domain. On the other hand, the advancing front methods [11, 12, 13] generate high-quality meshes near the boundary (boundary sensitive) that do not depend on the orientation of the object (orientation insensitive), see details on hex-meshing requirements in [2]. This is possible because the elements are generated layer by layer following the shape of the boundary surface. However, the advancing front methods are less robust and automatic. When the fronts are advanced, from the boundary to the inner part, they collide and can delimit complex voids. Specifically, if the advancing front method starts with a prescribed quadrilateral mesh of the boundary (constrained approach) [11] the resulting void is, in general terms, over-constrained and cannot be meshed. On the contrary, the versions of the advancing front method that start without a prescribed mesh of the boundary (unconstrained approach) [12, 13] can always generate a hexahedral mesh for the void. To this end, usually the unconstrained methods [12, 13, 14, 15] use a tetrahedral mesh to generate a hexahedral one. However, the quality of the mesh of the inner void is not guaranteed because it results from splitting each tetrahedron in four hexahedra. These disadvantages at the inner part are caused because the elements are generated from outside-to-inside. Note that also there are constrained methods that directly transform a tetrahedral mesh into a hex-dominant mesh [16, 17]. Summarizing, by generating elements from inside-to-outside we can avoid the front collisions that lead to unmeshed voids or low-quality inner meshes. Moreover, by generating the elements using fronts (layers of elements) we can obtain meshes that reproduce properly the shape of the domain boundary. In this work we apply these advantages to mesh outer domains. To this end, we propose: • •

To generate layers of elements from inside-to-outside. Hence, we present the first receding front method. That is, a reversed advancing front method. To pre-compute the fronts (or layers of hexahedra) combining two solutions of the Eikonal equation. One solution determines the distance to the inner part and the other the distance to the outer boundary. The level sets of the combination of both solutions determine the fronts.

This work is clearly related to the grid-based and advancing front methods. However, the grid-based methods do not generate layers of hexahedra from insideto-outside that smoothly adapt to the boundary of the domain. In addition, the advancing front methods do not start to generate layers of hexahedra from inside the domain. In this sense, the proposed approach is different to both methodologies.

Receding Front Method: Applied to Outer Domains

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Furthermore, we propose to pre-compute the fronts by solving the Eikonal equation. It is important to point out that there are other mesh generation works that use the Eikonal equation. In his seminar work, Sethian proposes a method to advance structured meshes by solving the Eikonal equation [18]. Another front propagation method based on the Eikonal equation is presented in [19]. In [20, 21], the authors show how to obtain the medial axis transform (MAT) by means of the Eikonal equation. Nevertheless, this is the first work were two solutions of the Eikonal equation are combined to pre-compute the fronts and obtain an unstructured hexahedral mesh. The remainder of the paper is organized as follows. First, we present a 2D motivation example in Section 2. According to this motivation, in Section 3 we detail the receding front method. Specifically, we present how to pre-compute the fronts, generate the meshes between the fronts and refine the resulting hexahedral fronts. Finally, in Section 4 we present several example meshes that demonstrate the capabilities of the first implementation of the receding front method.

2 2D Motivation To illustrate and clarify the basis of the receding front method in 3D, we consider a 2D example. Specifically, we present a smooth domain with a three-branched skeleton to be meshed with quadrilateral (hexahedral) elements. With the help of this domain we first review the main advantages and disadvantages of the grid-based and advancing front methods. Then, we outline the proposed receding front method which combines the advantages of both methods. Given a domain, the grid-based methods first generate a quadrilateral (hexahedral) mesh in the inner part of the domain, Fig. 1(a). Then, the remaining void between the inner mesh and the boundary, Fig. 1(b), has to be meshed. To this end, several new nodes are created on the boundary. These nodes are connected with the quadrilateral elements of the boundary of the inner mesh to form the last layer of hexahedra, Fig. 1(c). Since the boundary of the inner mesh is not adapted to the shape of the domain boundary, the last layer of hexahedra can present low quality elements. We would like to remark that this approach is robust and can be applied to general geometries to obtain meshes with high-quality elements in the inner part.

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Fig. 1. Several steps of a grid-based method: (a) inner mesh; (b) void between boundary and inner mesh; and (c) final mesh

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The advancing front methods generate layers of elements (fronts) that start at the domain boundary and layer by layer reach (advance towards) the inner part of the domain. At the last step, several elements that connect the fronts close the remaining void. There are two families of advancing front methods: the constrained [11] and the unconstrained approaches [12, 13]. The constrained approach generates a first layer of elements, Fig. 2(a), that matches with a prescribed mesh of the boundary. Then, several layers of elements are generated by merging and matching the elements that are in front of the last layer, Fig. 2(b). Since the process starts with a prescribed mesh of the boundary and advance inwards, the last voids in the inner part of the domain can be complex and over-constrained. Specifically, in the 3D case the advance can lead to remaining voids that cannot be meshed, Fig. 2(c). The unconstrained approach relaxes the hex-meshing problem by considering that the domain boundary is not meshed. The meshing process starts at the boundary and provides a decomposition of the domain in several layers, Fig. 3(a). The process stops when the inner void can be discretized with a hex-meshing primitive. Since the inner void results from successive offsets of the boundary, it can be as much difficult to hex-mesh as the initial domain. Thus, there are configurations where the inner void can not be meshed with a high-quality hex-meshing primitive. However, it is always possible to generate a hexahedral mesh. To this end, this approach generates a simplicial mesh of the unrecognized inner void, Fig. 3(b). The simplicial mesh can always be split in quadrilateral (hexahedral) elements, Fig. 3(c). Then, the boundary of this inner mesh is propagated through the layers to the boundary of the domain.

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Fig. 2. Several steps of a constrained advancing front method: (a) first front; (b) last front and contours of the previous fronts; and (c) unmeshed void

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Fig. 3. Several steps of an unconstrained advancing front method: (a) fronts and final void; (b) simplicial mesh of the void; and (c) splitting simplicial mesh


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The quality of the inner elements is not guaranteed because they are originated by a simplicial mesh. Moreover, the boundary of the inner mesh is propagated through the domain to the boundary. Thus, the inner mesh determines the structure and the quality of the inner part of the boundary curves (surfaces). It is important to point out that the element quality close to the boundary features is ensured because it is a front approach, except when the boundary has a very sharp dihedral angle. In addition, this approach is fully automatic and provides high-quality meshes for a wide range of geometries. To combine the advantages of both the grid-based and the advancing front methods, herein we propose the receding front method. To apply this methodology we require an initial mesh (seed) of the inner part of the domain, Fig. 4(a). Note that for the specific case of meshing the outer domain of a given body the initial mesh is a quadrilateral mesh of the body surface. This way, we can decouple the problem of generating the inner seed from the front generation process. The fronts that determine the layers of elements can be pre-computed. Specifically, we generate an offset of the shape of the inner seed towards the boundary, Fig. 4(b), and an offset of the boundary shape inwards, Fig. 4(c). These offsets are obtained as the level sets of two solutions of the Eikonal equation, Section 3.1. One solution is related to the boundary of the inner part and the other one to the outer boundary. To compute these solutions we use an edge-based solver on a triangular (tetrahedral) mesh [22]. Then, we combine both solutions to obtain a function of the domain that reproduces the inner shape in the inner part and the boundary close to the outer part, Fig. 4(d). Then, the fronts are obtained as the level sets of this function. Finally, these fronts can be used as a guide to generate layers of elements starting from the inner seed and finishing in the outer boundary, Fig. 5. To this end, we need to consider a set of templates that determine how to offset the previous layer of elements to

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Fig. 4. Pre-computing the fronts: (a) outer boundary and inner seed; (b) level sets from inside to outside; (c) level sets from outside to inside; and (d) combining inside-to-outside with outside-to-inside level sets

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Fig. 5. Layers of elements for the receding front method: (a) first layer; (b) second layer; (c) third layer; and (d) final layer

the new front, Section 3.2. Moreover, we have to consider a set of refinement rules that ensure that the element size is not surpassed. The resulting procedure generates layers of elements that progressively morph from the shape of the inner seed to the shape of the outer boundary. Moreover, starting from the inner part we can avoid over-constrained or complex inner voids.

3 The Receding Front Method The receding front method is decomposed into two steps. First, we pre-compute a set of fronts between the inner and the outer boundaries. Second, we expand the quadrilateral mesh of the inner boundary towards the unmeshed outer boundary according to the pre-computed fronts. 3.1

Pre-computing the Fronts

Given a domain Ω ⊂ Rn , the Eikonal equation is the following non-linear partial differential equation

∇d = f in Ω (1) d|U⊂{Ω∪∂Ω} = 0, where f is a known function and · is the euclidean norm. To solve the Eikonal equation, we first generate a tetrahedral mesh of the domain. Then, we obtain the solution for that discretization by means of an edge-based solver [22]. 1 For f = 1 the solution d is the distance from ∂Ω. For f = h(x) the level sets of the solution d follow the size field h(x) defined for each x ∈ ∂Ω. In this work we consider f = 1.


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In our applications we consider a domain bounded by an inner object (which defines the inner boundary of the domain, ∂Ωin ) and a smooth outer boundary denoted by ∂Ωout . In order to find a distance field that takes into account the distance from each inner point to both boundaries we first consider the following problem, see Figure 6(a):

∇dout = 1 in Ω (2) dout |∂Ωout = 0, where dout 0. The solution of this problem provides the distance to the outer boundary. Then, we consider the problem, see Figure 6(b):

∇din = 1 in Ω (3) din |∂Ωin = 0, where din 0. The solution of this problem provides the distance to the inner boundary. Note that we use the same tetrahedral mesh to solve Equations (2) and (3). The combined distance field, u, is defined as: u :=

dout . dout + din

(4)

Note that the combined distance field verifies 0 u 1, and at the boundaries of the domain it also verifes that u|∂Ωout = 0 and u|∂Ωin = 1. Moreover, the contours of u close ∂Ωout are similar to dout whereas the contours of u close ∂Ωin are similar to din . That is, the combined distance field reproduces the shapes of the inner and outer boundaries close to them. Finally, we extract m level sets of the combined distance field u. These level sets determine the fronts used to advance the mesh from the meshed inner boundary towards the meshed outer boundary. Algorithm 1 details the proposed procedure to generate the level sets of the distance field.

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Fig. 6. Solution of the Eikonal equation starting from: (a) the outer boundary (∂Ωout ) and, (b) the inner boundary (∂Ωin )

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Algorithm 1: Generate level sets ordered from inside to outside

2

Input: ∂Ωin , inner boundary; ∂Ωout , outer boundary; m, number of level sets Output: L, level sets ordered from inside to outside Scalar field din ← solve Eikonal equation in Ω starting at ∂Ωin Scalar field dout ← solve Eikonal equation in Ω starting at ∂Ωout

3

Scalar field u ←

4

Level sets L ← extract m iso-surfaces of u from 1 to 0

1

dout dout +din

Algorithm 2: Generate hexahedra between level sets

1 2 3 4 5 6 7 8

Input: L, ordered level sets; Q0 , quad surface mesh of the inner object Output: H, final hexahedral mesh Hex-mesh H ← {∅} foreach level set in L do Quad-mesh Q ← get the quadrilateral mesh on detect geometric features of Q Level set next ← get next level set to Hex-mesh H ← offset Q from to next H ← H ∪ H refine H according to the prescribed element size

3.2

Meshing the Fronts

From a quadrilateral surface mesh of the inner boundary we want to generate an unstructured hexahedral mesh of the domain without prescribing a quadrilateral surface mesh of the outer boundary. To this end, we detail in Algorithm 2 the proposed procedure. Each level-set will delimit a layer of hexahedral elements. Therefore, we have to describe the procedure to expand a quadrilateral mesh on level set to level set + 1. We first detect the geometric features of Q and classify the edges of the quad mesh as corner, reversal and end according to the angles defined by the adjacent faces, φ. An edge is classified as corner if π/4 ≤ φ < 3π/4; as reversal if 3π/4 ≤ φ < 5π/4; and as end if 5π/4 ≤ φ < 7π/4. Otherwise the edge is not a feature and it is classified as side. Once the edges are classified, we proceed to classify the nodes of the quadrilateral mesh. A node is classified as feature node if it belongs to three o more feature edges. Otherwise, the node is classified as nonfeature node. Note that these two classifications will determine the topology of the hexahedral mesh for the considered level. Once the edges and the nodes of the quadrilateral surface mesh are classified, we expand the quadrilateral mesh to the next level set. The expansion process is performed taking into account the topological entities of the quadrilateral mesh. First, we consider the faces of the quadrilateral mesh. Figure 7 illustrates the expansion of a quadrilateral face. Second, we expand the feature edges of the quadrilateral


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mesh according to their classification. Figure 8 illustrates the templates used to expand feature edges of the quadrilateral mesh. Third, we expand the feature nodes of the quadrilateral mesh according to the classification of adjacent feature edges. Figure 9 illustrates the templates used to expand feature nodes of the quadrilateral mesh. The size of the new hexahedra may differ from the prescribed element size. For this reason, a local refinement process is performed at each level. The edges of the

Fig. 7. Face template

Fig. 8. Edge templates

Fig. 9. Vertex templates

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new boundary that are longer than the prescribed element size are marked to be refined. Then, each hexahedron is replaced by a given template of hexahedra depending on the marked edges to be refined. To this end, we use the method proposed in [23] where the templates introduced by [9, 10] are adapted to marked edges. At this point, the algorithm of front meshing is iterated until all the fronts are discretized.

4 Examples In this section we present four meshes of the outer domain of a given object created with the receding front method. In all the cases the starting seed is a quadrilateral mesh of the inner surface. The user input is the element size of the quadrilateral mesh and the number of levels of the mesh. 4.1

Long Box

The first example presents a mesh generated on the exterior domain of a long box. The box is located inside a smooth domain. Note that the inner boundary only contains feature edges classified as corner, see Section 3.1. Figure 10(a) presents the tetrahedral mesh used to compute the solution of both Eikonal equations. Figure 10(b) presents the pre-computed fronts as detailed in Section 3.1. Note that we have prescribed four levels in order to generate the mesh. Figure 10(c) shows a general view of the hexahedral mesh while Figure 10(d) illustrates a longitudinal cut of the mesh. Although the quadrilateral surface mesh of the inner box is structured, the final mesh contains unstructured nodes both in the interior and on the boundary of the mesh. For instance, in Figure 10(c) we highlight a node with three adjacent hexahedra and in Figure 10(d) we mark an inner node with six adjacent hexahedra. 4.2

Pentagonal Star

The second example presents the generated mesh for the domain delimited by a star placed inside a sphere. In this case the definition of the domain contains feature edges classified as corner and end. The final mesh is composed by eight levels of hexahedral elements. Figure 11(b) shows a cut of the mesh and Figure 11(c) presents a detail of the unstructured mesh. Note that the expansion of the seed surface mesh generates unstructured elements in order to reach properly the outer boundary. 4.3

Smooth Object with a Reversal Feature

The objective of the third example is to show that using a refinement procedure we can respect the prescribed element size in the final mesh. To this end, we discretize a domain delimited by a flat object inside an ellipsoid. This geometry only contains feature edges classified as reversal. First, we generate a hexahedral mesh without using the local refinement process described in Section 3.1. Figure 12 shows the final mesh. Note that the obtained element size near the outer boundary is greater


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Fig. 10. Hexahedral mesh for the exterior domain of the long box. (a) Tetrahedral mesh used to solve the Eikonal equation. (b) Level sets of the combined distance field. (c) General view of the hexahedral mesh. (d) Longitudinal cut of the hexahedral mesh.

than the obtained element size near the inner boundary. In order to preserve the prescribed element size, in each level we perform a local refinement. Figure 13 illustrates that the final mesh reproduces with more fidelity the prescribed element size. Note that in both cases an unstructured mesh is obtained. 4.4

Space Capsule

One of the advantages of the proposed approach is that it is straightforward to stretch the elements in the normal direction of the fronts. To this end, we use a blending function [24] that modifies the combined distance field u introduced in equation (4): u 0=

eαu−1 , eα − 1

(5)

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Fig. 11. Hexahedral mesh for the exterior domain of the pentagonal star. (a) General view. (b) Vertical cut. (c) Detail of an unstructured region.

where α ∈ R. If α < 0, the levels are concentrated towards the outer boundary. If α > 0, the levels are concentrated towards the inner boundary. To illustrate the behavior of the blending function (5), we present in Figure 14(a) a uniform level set distribution defined on a simple geometry. Figure 14(b) presents the new the level set distribution when equation (5) is applied with α = 5. Note that the level sets are concentrated towards the inner boundary. Figure 15 presents the mesh generated on the exterior domain of a space capsule. In this mesh, we apply a boundary layer by using the blending function (5) with α = 7. The mesh is generated using 28 levels. Figure 15(b) shows a general view of the final mesh while Figure 15(b) shows a detail of the final mesh inner boundary.


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Fig. 12. Hexahedral mesh without local refinement for the exterior domain of the smooth object with a reversal feature. (a) General view of the outer boundary mesh. (b) Longitudinal cut.

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Fig. 13. Hexahedral mesh with local refinement for the exterior domain of the smooth object with a reversal feature. (a) General view of the outer boundary mesh. (b) Longitudinal cut. (c) Detail of the inner levels.

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Fig. 14. Distribution of the level sets: (a) uniform; and (b) concentrating towards the inner boundary (α = 5)

5 Concluding Remarks and Future Work In this work we have proposed the receding front method, a new approach for generating unstructured hexahedral meshes applied to generate hexahedral meshes of outer domains. Specifically, the two main contributions of this work are to precompute the meshing fronts by combining two solutions of the Eikonal equation, and to advance unstructured hexahedral elements from inside-to-outside (recede) guided by the pre-computed fronts. The former allows us to obtain meshes that reproduce the domain shape close to the outer boundary. The latter allows us to avoid the collision of constrained meshing fronts. We have implemented the proposed method in the ez4u meshing environment [25]. The first results show the possibilities of the receding front method applied to the unstructured hexahedral mesh generation of exterior domains. Moreover, we show that it is straightforward to obtain stretched meshes along the normal direction of the domain boundaries. Our long-term goal is to obtain a general-purpose unstructured hexahedral mesh generator based on the receding front method. In this sense, the first implementation of the method presents several issues that should be investigated and solved in the near future. First, we are currently including additional advancing and refinement templates. These templates allow us to improve the quality of the meshes obtained by advancing the elements from one layer to the following one. Second, we want to extend the presented approach to mesh the exterior domain of several objects and objects with holes, for instance a torus inside a sphere. Third, we want to apply the exterior domain meshing tool to outer boundaries with feature curves and vertices. To this end, we need to develop an imprinting technique that allows to propagate through the fronts the features of the outer boundary towards the inner boundary. These imprints would determine a decomposition of the domain in sub-volumes that connect the outer boundary with the inner boundary. Then, we can restrict the receding front method to each one of the sub-volumes to advance layerby-layer unstructured hexahedra from the inner mesh to the outer boundary. The


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Fig. 15. Hexahedral mesh for the exterior domain of the space capsule. (a) General view of the outer boundary mesh. (b) Longitudinal cut. (c) Detail of the inner levels.

resulting hex-meshing primitive would respect the boundary features and would be equivalent to a fully unstructured sweeping (regular sweeping is semi-structured). Fourth, we will analyze how to deal with narrow regions where the thickness of the part is significantly smaller (for instance one order of magnitude) than the surrounding volume. Since our approach generates the same number of levels in the whole domain, the distance between two consecutive level sets is variable. Therefore, it could be interesting to generate different number of hexahedral layers in different regions bounded by two consecutive level sets. To this end, we will investigate how to discontinue a layer and connect it to the boundary in one part of the model, but continue advancing the fronts in other parts. Fifth, we have to investigate how to automatically generate an inner hexahedral mesh that approximately reproduces the

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skeleton of the domain. To this end, we have considered to use a similar technique to the one proposed in [20, 21]. Then, we can obtain an automatic unstructured hexahedral mesh generator by means of advancing the fronts from inside-to-outside with the receding front method. Finally, we have to analyze how the accuracy of the Eikonal equation solution influences in the resulting hexahedral mesh.

References 1. Owen, S.J.: A survey for unstructured mesh generation technology. In: 7th International Meshing Roundtable, pp. 239–267 (1998) 2. Blacker, T.D.: Automated conformal hexahedral meshing constraints, challenges and opportunities. Engineering with Computers 17(3), 201–210 (2001) 3. Tautges, T.J.: The generation of hexahedral meshes for assembly geometry: survey and progress. International Journal for Numerical Methods in Engineering 50(12), 2617–2642 (2001) 4. Baker, T.J.: Mesh generation: Art or science? Progress in Aerospace Sciences 41(1), 29–63 (2005) 5. Shepherd, F.J.: Topologic and geometric constraint-based hexahedral mesh generation. PhD thesis, The University of Utah (2007) 6. Roca, X.: Paving the path towards automatic hexahedral mesh generation. PhD thesis, Universitat Politècnica de Catalunya (2009) 7. Schneiders, R., Bünten, R.: Automatic generation of hexahedral finite element meshes. Computer Aided Geometric Design 12(7), 693–707 (1995) 8. Schneiders, R.: A grid-based algorithm for the generation of hexahedral element meshes. Engineering with Computers 12(3), 168–177 (1996) 9. Zhang, Y., Bajaj, C., Sohn, B.S.: 3D finite element meshing from imaging data. Computer Methods in Applied Mechanics and Engineering 194(48-49), 5083–5106 (2005) 10. Zhang, Y., Bajaj, C.: Adaptive and quality quadrilateral/hexahedral meshing from volumetric data. Computer Methods in Applied Mechanics and Engineering 195(9-12), 942–960 (2006) 11. Blacker, T.D., Meyers, R.J.: Seams and wedges in Plastering: a 3-D hexahedral mesh generation algorithm. Engineering with computers 9(2), 83–93 (1993) 12. Staten, M.L., Owen, S.J., Blacker, T.D.: Unconstrained paving and plastering: A new idea for all hexahedral mesh generation. In: 14th International Meshing Roundtable (2005) 13. Staten, M.L., Kerr, R.A., Owen, S.J., Blacker, T.D., Stupazzini, M., Shimada, K.: Unconstrained plastering-hexahedral mesh generation via advancing-front geometry decomposition. International Journal for Numerical Methods in Engineering 81(2), 135–171 (2009) 14. Kowalski, N., Ledoux, F., Staten, M.L., Owen, S.J.: Fun sheet matching - automatic generation of block-structured hexahedral mesh using fundamental sheets. In: 10th USNCCM (2009) 15. Roca, X., Sarrate, J.: Local dual contributions: Representing dual surfaces for block meshing. International Journal for Numerical Methods in Engineering 83(6), 709–740 (2010) 16. Meshkat, S., Talmor, D.: Generating a mixed mesh of hexahedra, pentahedra and tetrahedra from an underlying tetrahedral mesh. International Journal for Numerical Methods in Engineering 49(1-2), 17–30 (2000)


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17. Owen, S.J., Saigal, S.: H-Morph: an indirect approach to advancing front hex meshing. International Journal for Numerical Methods in Engineering 49(1-2), 289–312 (2000) 18. Sethian, J.A.: Curvature flow and entropy conditions applied to grid generation. J. Comp. Phys (1994) 19. Wang, Y., Guibault, F., Camarero, R.: Eikonal equation-based front propagation for arbitrary complex configurations. International Journal for Numerical Methods in Engineering 73(2), 226–247 (2007) 20. Xia, H., Tucker, P.G.: Finite volume distance field and its application to medial axis transforms. International Journal for Numerical Methods in Engineering 82(1), 114–134 (2009) 21. Xia, H., Tucker, P.G.: Distance solutions for medial axis transform. In: Proceedings of the 18th International Meshing Roundtable, pp. 247–265 (2009) 22. Sethian, J.A.: Level set methods and fast marching methods. Cambridge University Press, Cambridge (1999) 23. Carreras, J.: Refinament conforme per malles de quadrilàters i hexàedres. Master’s thesis, Facultat de Matemàtiques i Estad´ıstica. Universitat Politècnica de Catalunya (2008) 24. Thompson, J.F.: Handbook of Grid Generation. CRC Press, Boca Raton (1999) 25. Roca, X., Sarrate, J., Ruiz-Gironés, E.: A graphical modeling and mesh generation environment for simulations based on boundary representation data. In: Congresso de Métodos Numéricos em Engenharia (2007)


EBMesh: An Embedded Boundary Meshing Tool Hong-Jun Kim1 and Timothy J. Tautges2 1

Argonne National Laboratory, Argonne, IL, U.S.A [email protected], [email protected]

Summary. This paper describes a method for constructing Cartesian meshes for embedded boundary algorithms by using a ray-tracing technique. In this approach, each mesh cell is distinguished as being inside, outside, or on the boundary of the input geometry, which is determined by firing rays parallel to x/y/z coordinates. The most expensive process of the embedded boundary mesh generation, an edgegeometry intersection test, is performed for the group of edges on a fired ray line together, which decreases the computational complexity of the whole method significantly. Produced boundary cells also have edge-cut fraction information and volume cut fraction information for each material. This work is implemented to be enable to directly import various CAD-based solid model formats and as an opensource code to be used easily in many engineering simulation fields. Keywords: Embedded boundary mesh, cut-cell, ray-tracing.

1 Introduction The Finite Element (FE), Finite Difference (FD), and Finite Volume (FV) methods solve Partial Differential Equations (PDEs) by using different types of discretization for the spatial domain. FE and FV methods are popular because they can resolve complicated geometric domains through the use of body-fitted grids. FD methods are straightforward to implement, and can achieve higher-order approximations of derivatives on structured grids, at the cost of those grids being difficult to generate for complex domains. Another class of simulations, based on the Embedded Boundary method, splits the difference between these by using structured grids on domain interior, and unstructured polyhedral cells or volume fraction approximation for cells intersecting the domain boundary [1] [2] [3] [4]. For this method, the mesh generation challenge is in generating such “Embedded Boundary” grids. Previous methods for generating EB grids rely on computing the intersection of each cell from a structured Cartesian grid with geometric model boundaries. Various approaches are used, depending on the representation of the geometric model boundaries. In the method by Aftosmis et al. [1], surface triangles of componentbased geometries are preprocessed to remove interior triangle pairs and break

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intersected triangles to smaller ones. Remaining boundary triangles are stored in a special tree structure to reduce the time to compute cell-boundary intersections. Cartesian cells found to intersect this boundary are converted to polygons using intersection points along cell edges. With no optimization, the cost for whole triangle-cell intersection test, which is usually most expensive task, will be approximately ANtN3, where Nt is the number of triangles on the boundary, N is the number of divisions on each side of the entire Cartesian mesh block. Aftosmis et al. uses optimization to reduce this time; Alternating Digital Tree (ADT) [5] gives a list of intersection candidate triangles for each cell in logNt time and the intersection test is performed for the candidate triangles. However, since the tree search is performed for all N3 cells, the approximate scaling behavior of intersection test for the method will be AN3logNt. Colella et al. [2] generate EB meshes by using implicit functions to represent geometric boundaries of a solid. Interior and exterior regions are indicated by negative and positive values of this implicit function φ, respectively; the function value is zero on the boundary. Colella et al. incorporate the definition of this implicit surface into PDE terms that evaluate fluxes over cell surfaces (or over the boundary, for cells intersecting the boundary). Although not explicitly stated, they use recursive subdivision of the Cartesian space to determine which cell is boundary out of N3 cells and the implicit surface evaluations are performed to them. Therefore, it appears that the scaling for the whole intersection test is AN3logN where each search is performed in logN time. In summary, both approaches reviewed here are bounded below by AN3, with the term either logN or logNt. Aftosmis et al.’s implementation is available under license from NASA, but it is not free for non-government use. Colella et al.’s method is available as part of the Chombo package [6]; however, the user is responsible for providing an implicit surface definition based on their definition of the surface boundary. In addition, it appears that neither method is able to handle domains with multiple materials, though this feature may be straightforward to implement with either method. In this paper, we propose a new EB mesh generation method, with contributions in several aspects. First, our algorithm achieves AN2logNt scaling performance of cell-triangle intersection test; this reduces the constant factor from previous results. Second, our method handles multi-volume assembly models without modification, without repeating cell-surface intersections for surfaces shared by multiple volumes. Third, our method supports EB mesh generation not only for faceted or triangle-based surfaces, but also directly from CAD geometries. We demonstrate the robustness of the method using a variety of geometric models from several simulation applications. The remainder of this paper is organized as follows. Section 2 describes our raytracing technique and how it is accelerated to get better performance. Section 3 describes the overall procedures for constructing EB meshes using our ray tracing method. Section 4 shows performance results of the proposed method with several examples. Section 5 concludes this paper and suggests further future research.

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2 Ray-Tracing Using Hierarchical OBB Trees Triangle-based ray tracing is widely used in graphics applications, since most surfaces are visualized as sets of triangular facets. Recently, we have also used triangle-based ray tracing to support Monte Carlo radiation transport [7], where radiation is tracked as individual particles propagating through absorbing media. Our EB meshing procedure is based on the same ray tracing procedure described in Ref. [7]. This procedure is summarized below. The input to our method is a non-manifold geometric model. The model is specified in the form of a Boundary Representation (Brep), as a collection of Vertices, Faces, Edges, and Regions, with topological relations between those entities. Models are accessed through the Common Geometry Module (CGM) [8], which is the same modeling interface used by the CUBIT mesh generation toolkit [9]. We read the facet-based description of the geometric model, including the topological relationships between the original geometric model entities (represented as groups of facets), into the MOAB mesh database [10]. Ray tracing is implemented in the DagMC library provided with MOAB. Since ray tracing accounts for the overwhelming majority of execution time in Monte Carlo radiation transport, a great deal of effort has been made to optimize that computation. This work has been reported in [7], and is summarized below. Construction of recursive space subdivision structure could be used to optimize ray-triangle intersections. However, axis-aligned boxes would have large volumes for any collection of facets not aligned with coordinate axes, increasing the likelihood of rays intersecting those boxes. In contrast, boxes oriented to minimize the extent in at least one box dimension are much less likely to be hit by a given ray, at the cost of having to transform the ray into the oriented box coordinate system. We have found that in practice the savings in ray-box intersections is well worth the cost of ray transformations. The advantage of the oriented bounding box for ray-tracing is amplified when it is combined with hierarchical tree structure, referred to as an OBB tree. An OBB tree is a binary tree of oriented bounding boxes of polygons. OBB trees were originally developed to accelerate collision detection problem in robot motion planning [11]. Tree construction starts from the root bounding box node encompassing a collection of polygons. The set of polygons is divided geometrically into two subsets, and a bounding box is formed for each subset of polygons. The process is repeated until each leaf box encloses a specified number of polygons. Figure 1 shows an example OBB tree. In MOAB's OBBTree representation, the collection of facets for each original geometric surface forms the starting point of the OBBTree subdivision. Once the OBBTree has been formed for the facets from each geometric surface, the root nodes for all surfaces bounding each geometric region are combined, leaving one OBBTree for each Region in the original geometric model. Ray tracing on a Region is performed by firing the ray first at the root box in the tree, first transforming the ray into the coordinate system of that node. If the ray hits a given node, the procedure is repeated down the tree, transforming the ray in the process. If any leaf nodes are intersected, the ray is tested against all

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Fig. 1. OBB tree construction

triangles in that node. Valid ray-triangle intersections are returned in order of increasing distance from the ray starting point.

3 Embedded Boundary Mesh by Ray-Tracing Given a facet-based geometric model, and the ability to compute ray-tracing on that model, it is straightforward to construct an EB mesh generation algorithm. This algorithm consists of the following steps: • Initialize the geometric model and hierarchical OBBTree for ray-tracing • Find the Cartesian box that surrounds all Regions of the model, and its subdivisions in cells • Use ray-tracing along cell edges to find cell intersections with geometric boundaries • Store the EB mesh on Cartesian cells These steps are described in the following sub-sections. 3.1 Initializing Geometric Model and Hierarchical OBBTree Geometric models are imported by using the Common Geometry Module (CGM) [8]. CGM provides a common interface for interacting with models stored in a variety of underlying solid modeling engines, including ACIS [12] and OpenCASCADE [13]. Both these engines support import and cleanup of IGES and STEP model exchange formats. CGM provides facet-based representations for each Edge and Face in the geometric model, and functions for evaluating overall model topology. The MOAB mesh library [10] uses a data model consisting of mesh entities (vertices, triangles, etc.), entity sets (arbitrary collections of entities and other sets), the databased instance, and tags (a named datum assigned to the previous


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three data types). Entity sets can have parent/child relations with other entity sets; these are distinct from the “contains” relations described earlier. We import the facet-based geometric model from CGM into MOAB, representing each geometric entity using an entity set, and topological relations between geometric entities using parent/child relations between the corresponding entity sets. This representation provides the basis for ray-tracing on the facet-based model. A hierarchical OBB tree is constructed on the facet-based model in MOAB, with one tree for each geometric Region set. For simplicity, a single tree node is inserted at the top, and linked to the root node for all Regions. The OBB tree provides ray-tracing functions and the coordinates of a box surrounding the entire OBB tree. 3.2 Building a Cartesian Mesh After constructing the OBB tree, the coordinates of the top-level box indicate the geometric extents of the model. These coordinates are used to create a Cartesian mesh encompassing the whole model. The number of divisions of that mesh in each coordinate direction is computed by using the number of facets and the box size, or from user input 1 . The Cartesian mesh is stored in MOAB’s structured mesh representation [10]; this representation requires only about 24MB per million elements in the box, based on double-precision storage of vertex coordinates. 3.3 Firing Rays In order to find intersections between cells and the model Faces, rays are fired along edges of the cells. Since we use an axis-aligned Cartesian mesh, edges in a column of cells line up into straight lines. A ray trace is performed along each of these lines, with the starting position of the ray at the surface of the Cartesian mesh; intersections with the model are returned as distances along that line and surfaces in the model that are intersected. Since we use equal spacing of elements in the Cartesian grid, these distances can be converted to fractional distances along edges in the grid. The surfaces intersected are used to assign the model Regions to sections of edges between intersections, starting with a designation of “outside”, since the ray begins on the boundary of the Cartesian mesh outside any model Regions. Rays are traced in the three coordinate directions, at each interior node on the corresponding coordinate plane. This process results in ray traces along all interior edges of cells in the mesh. If the number of divisions on each side of the box is N, then only a total of 3(N-1)2 ray tracing operations is performed, each operation returning all Face intersections along the ray (Fig. 2). The second figure in Fig 2 shows ray-tracing covers N edge intersection test with a ray.

1

Scaling numbers discussed in this paper assume a single number of divisions on all sides for simplicity only.

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Fig. 2. Rays fired in three directions along cell edges, A ray covers N edges

If the intersection test is performed edge (cell) by edge (cell) like in other methods, abundant tree searching has to be performed to the non-boundary edges as in Fig 3. The search in the Fig 3 is not stopped until SS (Spatial Search) 4 block dose not have any facet triangles inside and conform that the edge is not on the boundary.

Fig. 3. Spatial search is performed for non-boundary edge

Ray-firing function of our implementation returns intersected triangles and distances from ray starting point. Since OBB tree is constructed for geometry surfaces, it also returns which surface is hit. With these information and previous edge status, each edge and element is easily determined if it is inside, outside or boundary to geometry and edge cut fraction information can be obtained for boundary elements. To avoid cell by cell job such as setting all element status, element status default value is set and only the elements near intersections are considered to the status determination and have real values. In rare cases, ray tracing fails to find correct intersections with surfaces that they should. This failure is due to the use of non-watertight faceting returned by CGM and precision issues detecting ray-triangle intersections. Modifying the faceting of a non-watertight geometric model is outside the scope of this paper, but is reported elsewhere [14] [15]. If the precision issues are detected such that


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ray intersect triangle edges or vertices, the ray is displaced slightly and the ray trace is repeated. This strategy fixes all examples we have seen of this problem. The intersection along this modified ray is moved to the original edge at the appropriate position, with that offset very small relative to the cell edge length. 3.4 Storing Information MOAB allows applications to assign tag values to the entities in a mesh. Tags are uniquely identified by a name, and are referenced by a tag handle for efficiency. MOAB provides tags of three storage types: “sparse” tags, stored in (entity handle, tag value) tuples; “dense” tags, stored as an array of tag values for contiguous sequences of entity handles; and “bit” tags, in which each tag value is a user-defined number of bits. Dense tags are more memory-efficient when assigning a tag to many entities in a mesh. Tags of all three storage types can be created with default values; for cases where the majority of entities will be assigned the same value for a given tag (e.g. all cells on the interior of a solid), a default value eliminates the storage for this value on those entities. The EB mesh generation method described here stores two types of information. First, each cell in the Cartesian mesh is marked as inside (0), outside (1), or boundary (2), using the 2-bit tag ELEM_STATUS_TAG. Second, for cells marked as boundary, the position of intersection along each edge of the cell must be marked. Although each cell is bounded by 12 edges, only three edges for each cell must be marked, corresponding to the left, bottom, and front edges; the other edges are marked by neighboring cells. Edge fractions are measured from the left, front and bottom corner of boundary cell to x/y/z directions as in Fig. 4. If a ray hits the boundary from inside to outside, the fraction value is stored with a negative sign. In contrast, for the case of a ray intersection from outside to

Fig. 4. An example of edge cut fraction tag information on boundary cell

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inside, the value is marked as positive. Since multiple intersections can exist in each direction, the EDGE_CUT_FRAC_LENGTH_TAG is created to store the numbers of edge fraction values on the x, y, and z edges connected to the bottom/left/front corner of each cell. The EDGE_CUT_FRAC_TAG stores an arbitrary number of double precision values, listing all the intersection points along the x, then y, then z edges of a cell. Edges that are completely outside (inside) the Region are designated with zero (one) for edge intersection fractions. All tags except ELEM_STATUS_TAG are sparse tags, whose default values are chosen to indicate an interior cell. This tag is assigned explicitly to boundary and exterior cells. This approach saves substantial amounts of memory when there is a majority of interior cells, which is usually the case (since the Cartesian box is chosen to fit the geometric model tightly). 3.5 Calculating the Volume Fraction Multiple material compositions are required in each mesh cell for many simulation analyses, such as deterministic radiation transport. As an approximation to volume fractions within a cell, we use a method similar to that of Riper [16] [17], where ray tracing is performed on a subdivision grid over the cell. The volume fraction for a given Region intersected by the cell is equal to the total ray traversal length in that Region, summed over all rays fired over the subdivided cell, normalized by the total ray traversal length over the cell. The volume fraction calculation is performed by firing sub-rays to 3 directions parallel to x/y/z coordinates in boundary cell. All ray lengths inside each material geometry are summed and divided by all fired ray length sum as in equation (1) [17].

FM = ( Mj) / (DJ) FM : fraction of material M in a cell LMj : ray path length in material M for ray j J : total number of rays D : sum of all ray lengths (1) The number of divisions in the subdivision of each boundary cell is equal in our implementation and is assigned by the user. To store these information for boundary cells, MAT_FRAC_ID_TAG and VOL_FRAC_TAG are created as sparse tags for each material. Each tag has an arbitrary number of material ids and volume fractions. 3.6 Export Meshes or Query Function as Library The Cartesian grid, edge intersection data, and (optionally) volume fractions stored on that grid, can be exported from MOAB in a variety of formats. Alternatively, the data can be queried directly from MOAB by using existing functions for accessing mesh and tag information. In addition, in order to avoid the overhead of interacting through files, the EB meshing method can be linked as a library directly to applications. Along with


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functions for specifying the Cartesian box divisions and for requesting the generation of the EB mesh, there are several query functions for retrieving cell inside/outside status, edge-cut fractions and volume fractions of boundary cells. Currently, these functions are planned to be used for electromagnetic analysis [18] and radiation transport simulation [19]. The API specification for these functions appears in the Appendix of this paper.

4 Results The algorithm described in Section 3 has been tested on several example models of varying complexity. Performance data is measured by using a Linux workstation with Xeon 3GHz CPU and 16GB of memory.

Fig. 5. Example model and Cartesian meshes created from many primitives

An example model used to test the method for a model of moderate complexity was created by combining many primitive geometries as in Fig. 5. Cartesian mesh elements, distinguished as different colors by attached element status tags, are also shown in the Fig. 5.

Fig. 6. Timing and memory usage results for multi-primitive model

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Cartesian meshes were produced with different mesh sizes; their computation times and maximum memory usages are plotted in Fig. 6. Mesh generation took 4.64 seconds including geometry importing time to produce 10.5 million elements with a maximum memory of 432 MB. The next examples are standard STL format files produced by 3D scanning, which have complex boundary representations as in Fig. 7 [20].

Fig. 7. STL 3D “statue” and “horse” models [20]


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Figure 8. shows their Cartesian meshes produced by EBMesh.

Fig. 8. Cartesian mesh elements of 3D STL models

Some performance results are also plotted with different mesh sizes in Fig. 9. It takes 6 seconds to produce 11.8 million elements with a maximum memory of 480 MB for the first statue model and 2.3 seconds for 8.8 million elements with a maximum memory of 360 MB for the second horse model.

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Fig. 9. Timing and memory usage results for “statue” and “horse” STL models

The last example is an accelerator cavity model for the International Linear Collider (ILC), used to optimize electromagnetic performance of the cavities by adjustment of geometric design parameters [21]. In Fig. 10, the model geometries are shown, including the complex higher-order mode (HOM) coupler region.

Fig. 10. Accelerator cavity models


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Their meshed results are shown with different element status in Fig. 11.

Fig. 11. Cavity model mesh results

In Fig. 12, timing results and maximum memory usages are plotted with different mesh sizes. Mesh generation time is 68.8 seconds for this model, including geometry loading time for 10.7 million elements with a maximum memory of 582 MB. Since the cavity model is very complex, the time for importing, faceting and tree construction takes large portions of the overall computation time.

Fig. 12. Timing and memory usage results for ILC-coupler cavity model

The complexity of our algorithm is estimated to be scaled by N2, as discussed in Section 3. That is, total computation time can be expressed as T = A1N2logNt + A2Nt + A3. The first term corresponds to the EB meshing time. The second term is for triangle faceting and tree construction, and depends mostly on geometric model complexity. The third term is for some constant-time jobs involved. When the same input geometry is used for timing, the second and third terms are

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constant, with the remaining time proportional to N2. Therefore, a log-log plot of N and meshing time TM is expected to have slope 2. In Fig. 12, the meshing time for the ILC-coupler model is plotted when N is increased; it has a slope of approximately 2, as expected.

Fig. 13. Log-log plot of N and meshing time has a slope of 2

5 Conclusions and Future Work We have presented an embedded boundary Cartesian mesh generation algorithm that produces better performance than traditional EB approaches for complex geometries. It reduces mesh computation time as much as of 1/N compared with the cell-by-cell method by checking intersections edge line by line. As a result, all mesh elements are distinguished as being inside, outside, or on the boundary of the geometry model. Boundary cells also have edge-cut information stored as a tag. Optionally, volume fraction information about each material for boundary cells can be calculated. The mesh and related information can be written to a variety of mesh file formats or used directly by an analysis program. This work is implemented as an open-source code in Ref [22]. We envisioned several avenues of future work. For example, it may be useful to perform adaptive mesh refinement in the region of rapidly changing curvature geometry. The current volume fraction calculation function is similar enough to the refinement that we can approach it as a start. With curvature information directly obtained from geometry, we will produce better refinement meshes. Another future activity is generating a hex-dominant mesh by making polyhedral meshes instead of boundary hexes. Combined with OBB, which makes meshes oriented to natural geometry direction, it may produce good-quality meshes independent of geometry transformation. Lastly, parallel implementation of EBMesh and actual CAD surface intersection test will be considered for faster and more accurate results.


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Acknowledgements This work was supported by the US Department of Energy’s Scientific Discovery through Advanced Computing program under Contract DE-AC02-06CH11357.

References 1. Aftosmis, M.J., Berger, M.J., Melton, J.E.: Robust and efficient Cartesian mesh generation for component-based geometry. AIAA Journal 36(6), 952–960 (1998) 2. Colella, P., Graves, D., Ligocki, T., Trebotich, D., Straalen, B.V.: Embedded boundary algorithms and software for partial differential equations, SciDAC 2008. Journal of Physics, Conference Series 125 (2008) 3. Nieter, C., Cary, J.R., Werner, G.R., Smithe, D.N., Stolz, P.H.: Application of DeyMittra conformal boundary algorithm to 3D electromagnetic modeling. Journal of Computational Physics 228(21), 7902–7916 (2009) 4. Pattinson, J.: A cut-cell, agglomerated-multigrid accelerated, Car-tesian mesh method for compressible and incompressible flow, Ph.D. thesis, University of Pretoria (2006) 5. Bonet, J., Peraire, J.: An Alternating Digital Tree (ADT) Algorithm for Geometric Searching and Intersection Problems. Int. Journal of Numerical Method and Engineering 31, 1–17 (1991) 6. Colella, P., et al.: Chombo Software Package for AMR Applications Design Document, Applied Numerical Algorithm Group. Lawrence Berkeley National Laboratory, Berkeley (April 2009) 7. Tautges, T.J., Wilson, P.P.H., Kraftcheck, J.A., Simth, B.M., Henderson, D.L.: Acceleration Techniques for Direct Use of CAD-Based Geometries in Monte Carlo Radiation Transport. In: Proceedings of International Conference on Mathematics, Computational Methods & Reactor Physics, NY, May 3-7, Saratoga Springs (2009) 8. Tautges, T.J.: CGM: A geometry interface for mesh generation, analysis and other applications. Engineering with Computers 17, 486–490 (2005) 9. Sjaardema, G.D., Tautges, T.J., Wilson, T.J., Owen, S.J., Blacker, T.D., Bohnhoff, W.J., Edwards, T.L., Hipp, J.R., Lober, R.R., Mitchell, S.A.: CUBIT mesh generation environment. Users manual, vol. 1. Sandia National Laboratories (May 1994) 10. Tautges, T.J., Meyers, R., Merkley, K., Stimpson, C., Ernst, C.: MOAB: A MeshOriented Database, Sandia National Laboratories (2004) 11. Gottschalk, S., Lin, M.C., Manocah, D.: OBBTree: a hierarchical structure of rapid interference detection. In: Proceedings of the 23rd annual conference on Computer graphics and interactive techniques, pp. 171–180 (1996) 12. ACIS 3D Modeling, Spatial Technology Inc. (2010), http://www.spatial.com/products/3d-acis-modeling 13. OpenCASCADE Technology (2000-2010), http://www.opencascade.org/ 14. He, X.J., Chen, Y.H.: A Haptics-guided Hole-filling System Based on Triangular Mesh. Computer-Aided Design & Applications 3(6), 711–718 (2006) 15. Boonma, A.: Haptic-Based Sharp Edge Retaining and Gap Bridging Algorithms for Computer Aided Design (CAD) and Reverse Enineering (RE), Master Thesis, North Carolina State University (2006)

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16. Riper, K.A.V.: Mesh and Volume Fraction Capabilities in Moritz, KAV. In: Workshop on Common Tools and Interfaces for Deterministic Radiation Transport, for Monte Carlo, and Hybrid Codes (3D-TRANS-2003), France, pp. 25–26. Issyles Moulineaux (September 2003) 17. Riper, K.A.V.: Moritz Geometry Tool (2006), http://www.whiterockscience.com/moritz.html 18. Stoltz, P., Veitzer, S., Niether, C., Messmer, P., Amys, K., Cary, J., Lebrun, P., Amundson, J., Spentzouris, P., Kim, H.J., Tautges, T.J. (2010) Recent Progress in Accelerator Physics Simulations with the VORPAL code. In: Sci-DAC (2010) (in preparation) 19. Smith, B., Wilson, P.P.H., Sawan, M.E.: Three dimensional neutronics analysis of the ITER first wall/shield module 13. In: 22nd IEEE/NPSS Symposium on Fusion EngineeringSOFE 2007, Institute of Electrical and Electronics Engineers Inc., Piscataway (2007) 20. STL files, Jelsoft Enterprises Ltd. (2000-2010), http://forum.carvewright.com/showthread.php?t=12452&page=2 21. Ko, K., et al.: Impact of SciDAC on Office of Science Accelerators through Electromagnetic Modeling. In: SciDAC 2005 (June 2005) 22. EBMesh Tool, http://trac.mcs.anl.gov/projects/fathom/wiki/EBMesh

Appendix A Presented here is the API specification for the query functions. • bool get_grid_and_edges ( @return double boxMin[3] : entire grid box minimum @ return double boxMax[3] : entire grid box x maximum @ return int nDiv[3] : number of divisions @ return vector cutCellIndices: boundary cell index vector @ return vector cutFracLength : number of edge cut fractions for boundary cells @ return vector<double> cutFraction : edge cut fraction vector ); • bool get_volume_fractions ( @ return vector materialID : material id vector @ return vector boundaryCell : boundary cell vector @ return vector<double> volumeFrac : volume fraction double vector ); The submitted manuscript has been created by UChicago Argonne, LLC, Operator of Argonne National Laboratory ("Argonne"). Argonne, a U.S. Department of Energy Office of Science laboratory, is operated under Contract No. DE-AC0206CH11357. The U.S. Government retains for itself, and others acting on its behalf, a paid-up nonexclusive, irrevocable worldwide license in said article to reproduce, prepare derivative works, distribute copies to the public, and perform publicly and display publicly, by or on behalf of the Government.

Sharp Feature Preservation in Octree-Based Hexahedral Mesh Generation for CAD Assembly Models Jin Qian and Yongjie Zhang Department of Mechanical Engineering, Carnegie Mellon University, Pittsburgh, PA 15213, USA Tel.: (412) 268-5332; Fax: (412) 268-3348 [email protected]

Abstract. This paper describes an automatic and robust approach to convert 3D CAD assembly models into unstructured hexahedral meshes that are conformal to the given B-Reps (boundary-representations) with sharp feature preservation. In previous works, we developed an octree-based isocontouring method to construct unstructured hexahedral meshes for manifold and non-manifold domains. However, sharp feature preservation still remains a challenge. In the current algorithm, boundary features such as sharp curves and NURBS (non-uniform rational B-Splines) surface patches are first extracted from the given B-Reps. An octree-based isocontouring algorithm is then carried out to create unstructured hexahedral meshes, detecting and preserving all the sharp features. After this stage, a two-step pillowing technique is adopted to eliminate triangle-shaped quadrilateral elements along the sharp curves and “doublets”. To further improve the mesh quality, a combination of smoothing and optimization is used. The developed algorithm is robust, and it works for single-component manifold domain, as well as multiple component non-manifold domains. We have applied our algorithm to several complicated CAD models and assemblies. Valid meshes are generated and quality statistics are given. Keywords: Hexahedral Mesh, Sharp Feature, Pillowing, Quality Improvement, CAD Assembly, Manifold and Non-Manifold Domain.

1 Introduction Mesh generation for arbitrary 3D CAD models has played an important role in engineering numerical simulations. These complicated 3D domains like


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Fig. 1. The resulting hex mesh of a four-component gear assembly. (a) The assembly; (b-c) Zoom-in details of two corners; and (d-f) Three single components.

single-component CAD models or multiple-component assemblies (Fig. 1) need to be discretized into simple tetrahedra or hexahedra , while hexahedral (hex) meshes are more preferred. The 3D CAD models contain sharp features consisting of sharp curves and surface patches. Extracted from BReps (boundary-representations), these features need to be incorporated into hex mesh generation so that the meshes conform to the given CAD model. Traditional methods such as sweeping [4], plastering [18] and whisker weaving [2] attain some success, while fully automatic unstructured hex mesh generation with sharp feature preservation still remains a challenging problem. Among the recent advances, some grid methods prove to be robust and effective. The octree-based isocontouring method analyzes each interior grid, and generates a dual mesh of the grids [21, 22, 23]; The feature embedding method finds an embedding of the geometric topology into a base mesh and then maps the base mesh to the given geometry [9, 16, 6]. While these methods provide important insights in single-component domain meshing, mesh generation for domains with multiple components is still a challenge. In this paper, we develop an automatic and robust method to generate conformal hex meshes with sharp feature preservation for single-component models, as well as multiple-component assemblies. The CAD model is first

Sharp Feature Preservation of CAD Assemblies

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discretized into a triangle mesh, and then a binary grid data is calculated by distinguishing interior and exterior regions. For domains with multiple components, instead of using a binary data we attach a component index to each grid point. From this data, we use an octree-based isocontouring algorithm to construct all-hex meshes [21, 22, 23]. Based on the generated mesh, sharp features are detected and preserved. Critical points are identified first, considering the curve topology in the given B-Reps. Starting from one ending of each sharp curve, we track the complete path of the curve until the other ending is reached. Then we analyze each boundary quadrilateral (quad) element and decide which NURBS surface patch it belongs to. Having already detected all the boundary features, we classify all the vertices into four groups: critical points, curve vertices, surface vertices and interior vertices, and relocate them to appropriate positions via smoothing. The smoothing process is combined with a feature parametrization so that the features are preserved during vertex relocation. Then the mesh conforms to the B-Reps, however, there are triangle-shaped quads along the sharp curves and “doublets”. To eliminate them, a two-step pillowing technique is developed. Since the mesh still contains a few elements with bad aspect ratios, a modified Laplace smoothing and Jacobian optimization are used to further improve the mesh quality. We have applied our algorithms to several complicated CAD models including two manifold domains with one single component and two non-manifold assemblies with multiple components. The generated meshes are all valid, and the mesh quality statistics are given. The remainder of this paper is organized as follows: Section 2 summarizes related previous work. Section 3 describes the overview of our algorithms. Section 4 talks about sharp feature detection and preservation. Section 5 explains the detailed pillowing algorithm. Section 6 talks about quality improvement based on the constructed mesh. Section 7 presents the results and discussion. Finally, Section 8 draws conclusions and outlines future work.

2 Previous Work Unstructured Hex Mesh Generation: Generally there are two kinds of unstructured all-hex meshing methods: indirect and direct [8]. The indirect methods generate a tetrahedral (tet) mesh first and then convert it into a hex mesh. Compared to indirect methods, the direct method generates hex elements directly. It includes five distinct methods: grid-based, medial axis, plastering, whisker weaving and iso-surface extraction. The grid-based method generates a fitted 3D grid of hexes in the volume, and then adds hex elements at the boundaries to fill gaps [13, 14, 15]. The medial axis algorithms decompose the volume into simpler primitives that can be meshed using a structured algorithm [10, 11]. The plastering algorithm starts with a quad boundary mesh and places hexes onto the quads using the advancing front methods [1, 18]. The whisker weaving method first constructs the dual of a hex mesh then fits the hex elements into the volume [20]. The isosurface

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extraction method extracts the boundary surface and constructs uniform and adaptive hex meshes from volumetric imaging data [22, 21]. Furthermore, this method has been extended to meshing domains with multiple materials [23, 12]. Pillowing: Pillowing is a sheet insertion method that refines the mesh boundary. This method eliminates the situations where two neighboring hexes share two faces, called a “doublet” [7]. The doublet needs to be eliminated because the dihedral angle between these two faces are usually large and it becomes impossible to obtain all positive Jacobians. The pillowing method inserts a new layer of elements over the selected region called “shrink set”, and splits the doublet hexahedron into several. It looks as if the selected quads are inflated to become a new hex. This method is not only useful in removing doublets but also proves to be one powerful tool to insert element layers in existing meshes [16, 17, 19]. Incorporating Sharp Features: The plastering [1], whisker weaving [20, 2] and unconstrained plastering [18] algorithms begin with a well-defined boundary and then progressively propagate inside. Advancing front algorithms are implemented to fill the interior void. Differently, the grid-overlay and octree methods begin with a base mesh and then embed the geometric features into the base mesh. After building the base mesh, vertices are mapped to their appropriate positions according to boundary features. In the octree methods the geometric features are captured using a template-based refinement [14, 3, 21]. The octree-based isocontouring method detects and preserves features during mesh generation, which is especially suitable in meshing volumetric datasets [21, 22, 23]. These methods are effective mostly for biological models where sharp features are not dominant. For models involving critical sharp features, features can be embedded into a non-conforming base mesh and then be preserved through feature embedding [16, 9, 6]. The above mesh generation and feature preservation methods were mainly designed for manifold single-component domains. In this paper, we will talk about an automatic and robust approach for hex mesh generation with sharp feature preservation for both single-component CAD models and multiplecomponent assemblies.

3 Overview Fig. 2 shows the pipeline of our hex mesh generation algorithm with sharp feature preservation. First we extract NURBS curves and surfaces from the given CAD model, and triangulate the model using NURBS sampling. Then a binary grid data is created to distinguish the interior and exterior regions for each component. For domains with multiple components, the binary sign for each grid point is replaced by a component index. Based on the constructed grid data, we use an octree-based method to construct hex meshes


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[21, 22, 23]. One minimizer point is calculated for each' cell to minimize a predefined Quadratic Error Function (QEF): QEF (x) = (ni · (x − pi ))2 , where pi and ni are the position and normal vectors at the intersection points. We analyze each interior grid point for all-hex mesh generation. In this octree data structure, each grid point is shared by eight cells, and we use the calculated eight minimizers to construct a hexahedron. Sharp curves and surface patches are detected and mesh vertices are classified into four groups: critical points, curve vertices, surface vertices and interior vertices. If the domain consists of multiple components, the common curves and patches shared by multiple components need to be identified as well. Then we relocate each group of vertices to appropriate positions while preserving all the sharp features. Due to the triangle-shaped boundary quads along the sharp curves and “doublet” elements, each surface patch is pillowed individually first, and then the whole boundary is pillowed. Till now the mesh still has some elements with bad aspect ratios, therefore a modified Laplace smoothing and Jacobian optimization are used to make the final mesh valid. In the following, we will talk about sharp feature detection and preservation, a two-step pillowing technique, as well as quality improvement in detail.

Fig. 2. Meshing pipeline. The modules in orange are critical steps involving sharp feature detection and preservation.

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4 Sharp Feature Detection and Preservation Starting from the constructed binary data, we first use the octree-based isocontouring method to generate a base mesh, and then detect sharp features and embed them into the base mesh. Two kinds of features are preserved: sharp curves and surface patches. For multiple-component assemblies, the boundary features may be shared by two or more components. To preserve features, we find the appropriate position of each boundary vertex according to the given B-Reps. 4.1

Sharp Curve Detection

We begin from detecting the endings of each curve and then track the mesh edges to locate the complete curve trajectory. If the two endings of one single curve happen to be the same vertex, this is a closed curve, like a circle. Starting from one ending of the curve, we track its neighboring boundary edges to find the next curve point. To decide which vertex should be chosen, two criteria are considered: how small the distance from the vertex to the curve and how far the vertex “advances” along the curve. Generally the vertex closest to the curve is chosen as the next curve point. However, with a similar distance, the vertex having a greater “advance” is selected. In Fig.3, V0 is one ending point of the red curve, V1 has the minimum distance from the curve, but V2 has greater “advance” along the curve. In this case, V2 is selected as the next curve point. We keep tracking the boundary edges until the other ending point is reached, and these continuous edges form a complete mesh curve. To preserve the correct curve topology, we restrict any two curves can not share any edge segments. There are two special cases violating this restriction. We can change either the non-manifold point or the curve paths to overcome this problem. When three curves share one ending point in the B-Reps topology, it is possible to form the red curves in the constructed hex mesh as shown in Fig. 4(a). Here V0 is the detected non-manifold point, Curve 1 contains V0 − V1 − V2 , Curve 2 contains V0 − V1 − V3 , and Curve 3 contains V0 − V4 . Curve 1 and Curve 2 share one edge V0 − V1 . To preserve the correct topology, V1 is set as the new non-manifold point. Now Curve 1 becomes V1 − V2 , Curve 2 becomes V1 − V3 , and Curve 3 becomes V1 − V0 − V4 , as

Fig. 3. V2 is chosen as the next curve vertex due to its greater advance along the curve. The distances from V1 , V2 to the curve are similar.


(a)

(b)

(c)

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(d)

Fig. 4. Correct curve topology preservation. Red lines are curve segments, and red dots are non-manifold points. Different colors represent various surface patches.

shown in Fig. 4(b). When four or more curves share one ending point, we may have another special case as shown in Fig. 4(c). Here V0 is the detected nonmanifold point, Curve 1 contains V0 − V1 − V2 , Curve 2 contains V0 − V1 − V3 , Curve 3 contains V0 − V4 , and Curve 4 contains V0 − V5 . Curve 1 and Curve 2 share the common edge V0 − V1 . In this situation, we can not choose V1 as the non-manifold point as what we did for the first special case, because if we do that Curve 3 and Curve 4 will share the edge V0 − V1 . Therefore, we still choose V0 as the non-manifold point, but we change the path of Curve 2. In this way as shown in Fig. 4(d), Curve 2 becomes V0 − V6 − V3 , while Curves 1, 3 and 4 stay the same. After handling these two special cases, the complete path of each sharp curve is identified with the correct topology. 4.2

Surface Patch Detection

In addition to sharp curves, we need to preserve NURBS surface patches as well. Similarly, we need to find out the set of quads corresponding to each NURBS patch. To obtain these sets, we check each boundary quad to see which patch it belongs to. The four vertices of each quad are compared with the triangle mesh generated from the NURBS patch. In order to save the computation time, we set a small visual box around the targeting quad, and only check the quad vertices in that box. We check which triangle the vertex is closest to, and then decide which patch it belongs to. Note that vertices on sharp curves are shared by multiple patches and thus should not be counted. If all the vertices (except the ones on the curve) of one quad belong to one patch, then this quad belongs to that patch. When the four vertices of one quad all lie on sharp curves, we use the quad geometric center to decide which surface patch it belongs to. Following these guidelines we can find the set of quads belonging to each surface patch. These sets can also be identified using a propagation algorithm. From the B-Reps we know all the curves surrounding each patch. Beginning from the quads along these curve vertices we propagate to other elements until the whole set of quads are found. Sometimes the surface patches need to be modified. For example in Fig. 5, the assembly consists of two components: an inner cylinder and an outer ring.

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(a)

(b)

(c)

(d)

Fig. 5. One CAD assembly with an inner cylinder and an outer ring. (a) The two-component assembly; (b) The inner cylinder; (c) The outer ring; and (d) The modified cylinder patch. The pink surface in (b) is divided into three.

The cylinder has a big pink patch. However, the shared surface is only part of it. Therefore the pink surface patch needs to be split into three smaller ones (green, pink and blue) to match the two components. 4.3

Sharp Feature Parametrization and Preservation

Having detected all the vertices on sharp curves and surface patches, we classify them into four categories: critical points, curve vertices, surface vertices and interior vertices. In order to preserve all the sharp features, we relocate each vertex to its appropriate location via smoothing. Critical points are fixed. Curve and surface vertices are smoothed and projected to the given NURBS curves and surface patches. Finally, the interior vertices are smoothed to their volume center via a weighted-averaging method. The critical points are defined as the points where multiple curves meet. Note that open curves are always split if they meet with other curves in the middle, while close curves are not. Therefore all the ending points of open curves are critical points. Following this rule the critical points can be among the following three situations: the shared ending of two open curves (Fig. 6(a)), the intersection of one open curve and one closed curve (Fig. 6(b)), or the joint point of three or more curves (Fig. 6(c)). All critical points are fixed during smoothing. For other curve vertices, they are “interior” in the curve compared to the critical points. During smoothing, we need to calculate a new position for

(a)

(b)

(c)

Fig. 6. Critical points (the red points). (a) The shared ending point of two open curves; (b) The intersection of one open curve and one closed curve; and (c) The joint point of three curves.


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each curve vertex. Here, we first find its two neighboring vertices on the curve, calculate their middle point, and then project the middle point to the curve via a NURBS curve parametrization. The NURBS curve in the given B-Reps is represented as: n

C(u) =

ωi Pi Ni,d (u)

i=0 n

,

(1)

ωi Ni,d (u)

i=0

where Pi are the control points, ωi are the weights, Ni,d (u) are B-spline basis functions of degree d, and u is the parameter ranging from 0 and 1. In order to find the projection of one curve vertex on the NURBS curve, we need to first find its parameter u on the NURBS curve, and then compute its exact position according to Equation (1). The NURBS curve is first discretized into a series of line segments using equally-spaced parameters u, and then we find the segment closest to the targeting curve vertex. Based on the u values of the two ending points of that segment, we compute the parameter u value for the targeting vertex using a linear interpolation. Then Equation (1) is used to compute its exact position. Finally the curve vertex is moved toward this position iteratively during smoothing. The new position for one surface vertex is calculated as follows: we find the area center of its neighboring boundary quads, and project the area center to the detected NURBS surface patch via a NURBS surface parametrization. The NURBS patch can be represented as: n m

S(u, v) =

ωij Pij Ni,d (u)Nj,d (v)

i=0 j=0 n m

,

(2)

ωij Ni,d (u)Nj,d (v)

i=0 j=0

where Pij are the control points, ωij are the weights, and Ni,d (u), Nj,d (v) are B-spline basis functions of degree d, and (u, v) are the parameters ranging from 0 to 1. To find the projection of one mesh vertex on the NURBS patch, we discretize the NURBS patch into triangles, compute the vertex parameter values (u, v) using a barycentric interpolation, and obtain the exact position using Equation (2). In concave corners, the smoothing algorithm may produce bad aspect ratios. As shown in Fig. 7(a), the quads tend to have large angles, therefore we project them to the corner angular bisector like Fig. 7(b). In addition, some other vertices tend to be very close to the concave corners, see Fig. 7(c), therefore we move them away from the concave corner as in Fig. 7(d). After smoothing and feature parametrization, the mesh is smoothed and all the sharp features are preserved. Fig. 8 shows one example.

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(a)

(b)

(c)

(d)

Fig. 7. Mesh correction near concave corners. (a-b) Nearby quads have large angles, therefore some vertices are projected to the corner angular bisector; and (c-d) Nearby vertices are very close to the corner, therefore they are moved to the middle points of the corner and themselves.

(a)

(b)

Fig. 8. One sharp curve before (a) and after (b) NURBS curve tracking, smoothing and projection for hook2 model. The boundary surface is also smoothed and projected to the given NURBS surface patches.

5 Pillowing to Create Boundary Layers After sharp feature preservation, the constructed hex mesh conforms to the given CAD model. However, some boundary quads may have two edges lying on the same curve, which leads to triangle-shaped quads along the curves. In addition, some hex elements may have two or three faces lying on the same surface patch, which leads to big dihedral angels. These two situations decrease the mesh quality and make it impossible to obtain all positive Jacobians via smoothing and optimization. On the contrary, the desired meshes should have two attributes [16, 23]: each hex element has at most one quad on the boundary, and each boundary quad has at most one edge on the sharp curves. To make the mesh fundamental, we use a two-step pillowing technique


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to improve the mesh: (1) pillow each individual surface patch, and (2) pillow the overall boundary surface. 5.1

Manifold Domain with Single Component

For single-component domains, the first pillowing step eliminates the triangleshaped boundary quads along sharp curves. As shown in Fig. 9(a-c), the shrink set is defined as the set of quads corresponding to each surface patch. These sets of quads are already identified in the previous steps. Then we duplicate the nodes of each quad belonging to the shrink set, and connect them to the original nodes to form a new layer of hexes. In this procedure, these quads seem to be inflated to form new hex elements. Finally we project these new vertices to the original surface patch. In this case, the original mesh boundary remains unchanged, however, the inserted hex layer now becomes “flat”. This creates elements with multiple faces on the same boundary patch, therefore another layer insertion is necessary. The second pillowing step eliminates the hex elements with multiple quads on the same surface patch. We choose the whole boundary as the shrink set, and then duplicate all the boundary vertices at the original positions. Finally we connect the new vertices to the original ones. Since we keep the new vertices at the original positions while move the old vertices inside, the original mesh boundaries are kept. The effect of these two layer insertions are shown in Fig. 9(d-f). The yellow elements are the original ones, the blue ones are the first pillowed layer and the pink ones are the second pillowed layer. From Fig. 9(f), we can observe that the corner angle is divided into two. If the corner angle is small, the divided angles are smaller and thus decrease the mesh quality. Therefore, we choose not to pillow these small angles to attain better mesh quality. This can be done by modifying the shrink set in the first pillowing step, as shown in Fig. 10. When we define the shrink

(a)

(b)

(c)

(d)

(e)

(f)

Fig. 9. two-step pillowing. (a-c): Inserting one boundary layer for the top surface. The shrink set (yellow) is defined in (a), pillowed in (b) and projected to the top surface patch in (c). (d-f): Inserting two boundary layers. (d) and (e) are the original mesh, and (f) is the resulting mesh. In (f), the yellow elements are the original mesh, the blue and pink ones are the first and second pillowed layers, respectively.

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Fig. 10. Small sharp angle preservation. (a): A meshing result before pillowing; (b): Each surface patch is pillowed and all the sharp angles are split; (c): Sharp angles in (d-f) are preserved by modifying the shrink set, while the sharp angle in (g-i) has to be split because all the four vertices of that quad are on the sharp curves; (d-i): Zoom-in details of two corners.

set, if one quad contains a small angle formed by two sharp curves, the quad is excluded from the shrink set. In this manner, that quad is not pillowed, and the small sharp angle is preserved as shown in (d-f). Note that if all the four vertices of a quad are on the curves, see the sharp corner in (g-i), we cannot exclude that quad and the sharp angle has to be split. Otherwise, the triangle-shaped quad cannot be removed. After two boundary layers are inserted, each boundary quad has at most one edge on the sharp curves while each hex element has at most one quad on the boundary. Fig. 11 shows the pillowing result of the multiaxis model. 5.2

Non-manifold Domain with Multiple Components

For domains with multiple components, patch matching is essential especially for patches shared by multiple components, here we use different matching strategies for the first and second pillowing steps. In multiple-component models there are two kinds of surface patches: patches contained in one component and the common patches shared by two components. During the first pillowing step, we first mark the sets of common patches. Then we pillow the faces that belong to only one component in the same way as the single-component domain, and one layer of elements are inserted at one time. Whenever we come across a common patch, two instead of one hex layers are inserted. If the original nodes on the common patch is Set A (yellow in Fig. 12(a)), we duplicate two different layers, Sets


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Fig. 11. Meshing results before (a) and after (b) inserting two boundary layers. (c) and (d) show the details.

(a)

(b)

(c)

Fig. 12. The first pillowing step for a common patch. (a) The original two components share one common patch A; (b) Sets B and C are created, then Sets A and B, A and C are connected; and (c) A is projected to B and C.

B (blue) and C (green). Then, we connect Sets A and B, A and C to form two hex layers. Note that Sets B and C have the same vertices on the red boundary curve, so all the curves are automatically conformal to each other. Finally Set A is projected to B and C. For the second pillowing step, patch matching is rather straight-forward. Like what we did for one single component, when pillowing one surface we shrink the original nodes inside while keep the new nodes at the boundary. Therefore, the nodes in other components connecting to the original boundary should be connected to the new boundary. Therefore, the resulting meshes automatically match at the common patch shared by multiple components. Fig. 13 shows an evolution of the two-cylinder assembly. After inserting two pillowed layers, the resulting meshes for the two components automatically match with each other. Fig. 14 shows meshes of the cylinder-ring assembly

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(a)

(b)

(c)

(d)

(e)

Fig. 13. (a-b): A two-cylinder assembly and the resulting mesh; and (c-e): one cross section of the original mesh, the mesh with the first pillowed layer, and the mesh with two pillowed layers, respectively.

(a)

(b)

(c)

(d)

Fig. 14. The resulting meshes of a cylinder-ring assembly. (a) The assembly; (b) The inner cylinder; (c) The outer ring; and (d) One cross section of the assembly.

in Fig. 5. Note that in (b), the mesh is denser around the top and bottom of the cylinder. This is because the cylinder face is divided into three small patches as discussed earlier, and all these three patches are pillowed separately resulting in denser meshes over there.

6 Quality Improvement After preserving sharp features and inserting two boundary layers, the resulting hex mesh conforms to the B-Reps and is free of triangle-shaped quads or doublets. However, the mesh may still have bad aspect ratios along the boundary and negative Jacobians may exist. Therefore a modified Laplace smoothing is carried out to improve the overall quality followed by a Jacobian optimization. We use the same smoothing techniques as developed in Section 4.3 except that the Jacobian is integrated to decide the movement for each interior vertex. The interior vertex is relocated to the weighted volume center of its neighboring elements only when the movement succeeds in improving the local Jacobian. The optimization method starts with looping all the vertices to compute their Jacobians, and then the very vertex with the worst Jacobian is found and improved using a conjugate gradient method, in which


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the objective function is the Jacobian of that vertex. Then a new loop begins and the new worst region in the improved mesh is found and optimized in the same manner. We keep improving the worst Jacobian until it is greater than a pre-defined threshold. We choose the traditional definition of the Jacobian matrix using the finite element basis functions [23]. Generally if the eight corners of one hex all have positive Jacobians, then the Jacobian inside the hex is usually positive. In order to guarantee that some numerical analysis works properly, we can check the Jacobian at each Gaussian integration point and include it in the loop.

7 Results and Discussion We have applied our algorithms to two single-component CAD models (Figs. 15, 16) and two multiple-component assembly models (Figs. 1, 17). Our techniques generate quality meshes with all sharp features preserved. All the computations are performed on a PC equipped with an Intel Q9600 CPU and 4GB DDR-II memories. The resulting meshes are valid with conformal boundaries and sharp feature preservation. Table 1 shows the statistics before and after quality improvement of the meshes. Note that during optimization, we use the Jacobian defined in the finite element method as the objective function; while in the statistics, we measure the mesh quality with the Jacobian defined by three edge vectors

Fig. 15. The resulting mesh for the hook model. (a-b) Snapshots from two view angles; and (c-d) Zoom-in details of the boundary layer and one corner.

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Fig. 16. The resulting mesh for the varco3 model. (a-b) Snapshots from two view angles; and (c-e) Zoom-in details of three corners. Table 1. Statistics of meshes before and after quality improvement Mesh Size Jacobian Condition Inverted Time (Vert , Elem ) (Worst, Best) (Worst, Best) Elem (s)

Model

Mesh

Multiaxis

Original Improved

(1755, 1187) (4505, 3707)

Hook

Original Improved

(2556, 1663) (6327, 5121)

0 0

29

(-0.0003, 1.0) (20553.31, 1.0) (0.029, 1.0) (34.48, 1.0)

3 0

49

Original (6130, 4658) (-0.0004, 1.0) (25659.53, 1.0) Improved (13627, 11448) (0.016, 1.0) (61.23, 1.0)

21 0

84

Bearing Original (4252, 2979) (-0.0005, 1.0) (8509.72, 1.0) Assembly Improved (12208, 10231) (0.016, 1.0) (63.34, 1.0)

21 0

92

Gear Original (7434, 4938) (0.0003, 1.0) (3084.63, 1.0) Assembly Improved (23671, 19454) (0.021, 1.0) (46.18, 1.0)

0 0

152

Varco3

(0.014, 1.0) (0.24, 1.0)

(73.05, 1.0) (4.10, 1.0)


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Fig. 17. The resulting hex mesh of a three-component bearing assembly. (a) The bearing assembly; (b-c) Zoom-in details of two corners; and (d-f) Three single components

[5, 21, 23]. After quality improvement, all the negative Jacobians are removed, and the worst Jacobian as well as the worst condition number are improved significantly. The success of sharp feature detection relies on a good base mesh. Fortunately the octree-based isocontouring method provides us such meshes. Even though the base mesh has some bad elements on the boundary, it is improved through the two-step pillowing. In the pillowing, the number of elements increases sharply, which leads to large memory and CPU requirements. The mesh generation computing time is optimized so that generating 10,000 elements (after pillowing) takes about 90 seconds. In the pipeline, sharp feature detection and pillowing are fast, the most time-consuming part is the overall smoothing, which is linear in complexity. From Table 1 we can also tell that the computational time is nearly proportional to the element number (after pillowing).

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8 Conclusion and Future Work We have developed an automatic and robust approach to generate unstructured hex meshes with sharp feature preservation for single-component CAD models and multiple-component assemblies. The base mesh is generated using the octree-based isocontouring method. Features such as sharp curves and surface patches are detected first, and then the mesh is smoothed while incorporating these features. After this stage, a two-step pillowing technique is used to eliminate triangle-shaped quads along the sharp curves and doublets. Then a modified Laplace smoothing and Jacobian optimization are used to improve the mesh quality. Several CAD models are tested and statistics of the resulting meshes are given. For large datasets, the required memory and CPU time increase significantly. In the future, we will optimize our data structure and develop parallel meshing algorithms to construct non-manifold hex mesh efficiently for very large datasets. To reduce the rapid increase in the element number, we will implement local refinement algorithms instead of pillowing two whole boundary layers. In the future, we will also apply the algorithm to more applications, and include material properties into our mesh generation and quality improvement procedure.

Acknowledgement We would like to thank Wenyan Wang for useful discussions on NURBS curve and surface path parametrization, and Matt Staten for providing the multiaxis, hook2, hook and varco3 models. The research was supported in part by Y. Zhang’s ONR-YIP award, an ONR grant N00014-08-1-0653 and a NRL contract N00173-08-C-6011 as part of the HPCMP CREATE program.

References 1. Blacker, T., Myers, R.: Seams and Wedges in Plastering: A 3D Hexahedral Mesh Generation Algorithm. Engineering With Computers 2(9), 83–93 (1993) 2. Folwell, N., Mitchell, S.: Reliable Whisker Weaving via Curve Contraction. Engineering With Computers 15(3), 292–302 (1999) 3. Ito, Y., Shih, A., Soni, B.: Octree-based Reasonable-quality Hexahedral Mesh Generation using a New Set of Refinement Templates. International Journal for Numerical Methods in Enginnering 77(13), 1809–1833 (2009) 4. Knupp, P.: Next-generation Sweep tool: A Method for Generating All-hex Meshes on Two-and-one-half Dimensional Geometries. In: Proceedings of 7th International Meshing Roundtable, pp. 505–513 (1998)


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5. Knupp, P.: Achieving Finite Element Mesh Quality via Optimization of the Jacobian Matrix Norm and Associated Quantities. Part II–A Framework for Volume Mesh Optimization and the Condition Number of the Jacobian Matrix. International Journal for Numerical Methods in Engineering 48(8), 1165–1185 (2000) 6. Marcha, L.: Advances in Octree-Based All-Hexahedral Mesh Generation: Handling Sharp Features. In: Proceedings of 18th International Meshing Roundtable, pp. 65–84 (2009) 7. Mitchell, S., Tautges, T.: Pillowing Doublets: Refining a Mesh to Ensure That Faces Share at Most One Edge. In: Proceedings of 4th International Meshing Roundtable, pp. 231–240 (1995) 8. Owen, S.: A Survey of Unstructured Mesh Generation Technology. In: Proceedings of 7th International Meshing Roundtable, pp. 26–28 (1998) 9. Owen, S., Shepherd, J.: Embedding Features in a Cartesian Grid. In: Proceedings of 18th International Meshing Roundtable, pp. 117–138 (2009) 10. Price, M., Armstrong, C.: Hexahedral Mesh Generation by Medial Surface Subdivision: Part I. International Journal for Numerical Methods in Engineering 38(19), 3335–3359 (1995) 11. Price, M., Armstrong, C.: Hexahedral Mesh Generation by Medial Surface Subdivision: Part II. International Journal for Numerical Methods in Engineering 40(1), 111–136 (1997) 12. Qian, J., Zhang, Y., Wang, W., Lewis, A., Qidwai, M., Geltmacher, A.: Quality Improvement of Non-manifold Hexahedral Meshes for Critical Feature Determination of Microstructure Materials. International Journal for Numerical Methods in Enginnering 82(11), 1406–1423 (2010) 13. Schneiders, R.: A Grid-based Algorithm for the Generation of Hexahedral Element Meshes. Engineering with Computers 12(3-4), 168–177 (1996) 14. Schneiders, R., Weiler, F.: Octree-Based Generation of Hexahedral Element Meshes. In: Proceedings of 7th International Meshing Roundtable, pp. 205–216 (1996) 15. Shephard, M., Georges, M.: Three-Dimensional Mesh Generation by Finite Octree Technique. International Journal for Numerical Methods in Engineering 32(4), 709–749 (1991) 16. Shepherd, J.: Conforming Hexahedral Mesh Generation via Geometric Capture Methods. In: Proceedings of 18th International Meshing Roundtable, pp. 85–102 (2009) 17. Shepherd, J., Zhang, Y., Tuttle, C., Silva, C.: Quality Improvement and Boolean-like Cutting Operations in Hexahedral Meshes. In: The 10th ISGG Conference on Numerical Grid Generation, FORTH, Crete, Greece, September 16-20 (2007) 18. Staten, M., Kerr, R., Owen, S., Blacker, T.: Unconstrained Paving and Plastering: Progress Update. In: Proceedings of 15th International Meshing Roundtable, pp. 469–486 (2006) 19. Staten, M., Shepherd, J., Ledoux, F., Shimada, K.: Hexahedral Mesh Matching: Converting non-conforming hexahedral-to-hexahedral interfaces into conforming interfaces. International Journal for Numerical Methods in Enginnering 82(12), 1475–1509 (2009)

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20. Tautges, T., Blacker, T., Mitchell, S.: The Whisker-Weaving Algorithm: a Connectivity Based Method for Constructing All-hexahedral Finite Element Meshes. International Journal for Numerical Methods in Engineering 39(19), 3327–3349 (1996) 21. Zhang, Y., Bajaj, C.: Adaptive and Quality Quadrilateral/Hexahedral Meshing from Volumetric Data. Computer Methods in Applied Mechanics and Engineering 195(9-12), 942–960 (2006) 22. Zhang, Y., Bajaj, C., Sohn, B.: 3D Finite Element Meshing from Imaging Data. The special issue of Computer Methods in Applied Mechanics and Engineering on Unstructured Mesh Generation 194(48-49), 5083–5106 (2005) 23. Zhang, Y., Hughes, T., Bajaj, C.: An Automatic 3D Mesh Generation Method for Domains with Multiple Materials. Computer Methods in Applied Mechanics and Engineering 199(5-8), 405–415 (2010)

Pen-Based User Interface for Geometric Decomposition for Hexahedral Mesh Generation Jean Hsiang-Chun Lu1 , Inho Song1 , William Roshan Quadros2, and Kenji Shimada1 1 2

Carnegie Mellon University, Pittsburgh, PA [email protected], {songphd,shimada}@cmu.edu Sandia National Laboratories , Albuquerque, NM [email protected]

Summary. In this work, we present a pen-based user interface (UI) that makes manual geometry decomposition easier and helps reduce user time spent on geometry decomposition for hexahedral meshing. This paper presents the first attempt to apply a pen-based UI for geometry preparation for mesh generation. The proposed UI is more natural, intuitive, and easier to use than existing interfaces such as Window-Icon-Mouse-Pointer and commandline interfaces. The pen-based UI contains the following features: (1) drawing freeform strokes, (2) fitting and aligning strokes to CAD geometry automatically, (3) creating cutting surfaces, and (4) performing various tasks including webcutting through gestures. The proposed pen-based UI has been tested on a few models to demonstrate its effectiveness in decomposition, defeaturing and controlling mesh orientation. Keywords: pen-based interface, decomposition, hexahedral meshing.

1 Introduction Manual geometry decomposition is one of the most time-consuming steps in the mesh generation process. As shown in Fig. 1, geometry decomposition dominates time spent during hex-meshing tasks at Sandia National Lab. In order to perform geometry decomposition in existing preprocessing / meshgeneration tools (e.g. CUBIT, TurboMesh), users have to either select an appropriate command via a Window-Icon-Mouse-Pointer (WIMP) Graphical User Interface (GUI), or type the command in the command-line interface

Sandia is a multiprogram laboratory operated by Sandia Corporation, a Lockheed Martin Company for the United States Department of Energy’s National Nuclear Security Administration under contract DE-AC04-94AL85000.

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Fig. 1. The user time spent for various tasks in mesh generation process at Sandia National Laboratories [1]

(CLI). Though these traditional interfaces are powerful, they are cumbersome to a novice user. The decomposition of models into meshable geometries has received a great deal of attention due to the limitations of current hexahedral meshing algorithms. The practical approach is to manually decompose the complex geometry into meshable regions. These meshable regions are then imprinted and merged before performing surface and volume meshing. The surfaces of these regions are meshed using existing algorithms such as paving, mapping, and sub-map [2]. Then volume meshing schemes such as sweeping (which requires specifying source and target surfaces) and mapping are used to mesh the sub-domains. The pen-based UI proposed in this paper improves the interface of decomposition operation in order to make the manual decomposition process more efficient. A recent trend in computer-aided engineering software is moving toward more natural and accessible interfaces, of which pen-based interfaces are a prime example. In the early design stage, pen-based interfaces that use a stylus pen and tablet have already been playing an important role and have been used in freeform styling design. However, deposit its potential, pen-based UI has not been commonly utilized in the engineering analysis. In this paper, the benefits of pen-based interfaces are brought to engineering analysis tasks by developing a new UI for easier and faster manual geometry decomposition for hexahedral meshing as shown in Fig. 2. This work is the first attempt to apply pen-based interface in mesh generation and is in its early development stage. More pen-based functions will be implemented in the future. The overview of the proposed UI is as follows. The program first takes freehand input as sequences of screen coordinates. The freehand inputs are then re-sampled and smoothed. The processed input point set are then identified as one of the three types: lines, circles, or splines.

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Fig. 2. Decomposing a model with the proposed pen-based UI

The program then fits appropriate 1D geometry entity to each input according to the type. Cutting surfaces are created by sweeping the 1D geometry entity in a given sweeping direction. The solid model is then decomposed using the cutting surfaces. The rest of the paper is organized as follows: Section 2 discusses the previous related works. Section 3 covers the beautification and snapping algorithms to process the freehand inputs. In Section 4, we describe how to create cutting surfaces for webcutting [3]. Section 5 includes results that illustrate decomposition targeting better mesh quality, controlling orientation of mesh and defeaturing for hex meshing.

2 Related Work 2.1

MARK-IT

The approach of Yang et al. to improve user interface (UI) with MARK-IT [4] successfully simplifies the WIMP GUI as shown in Fig. 3 and saves user time by avoiding switching among different panels in a conventional GUI interface. Decomposition commands are executed by selecting entities and drawing a corresponding mark with the mouse. MARK-IT covers over 50 CUBIT operations with different marks. To help users in drawing a correct mark, a label appears by the cursor to display the associated command. When

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Fig. 3. Commands shown in MARK-IT interface [4]

a user needs to input a value, MARK-IT provides a “virtual slider” that can be dragged to adjust the value. MARK-IT shows the potential of a marking interface that is integrated with a complex meshing system such as CUBIT. MARK-IT avoids unnecessary movement of the mouse by improving access to commands and operations. However, it does not reduce the time spent in the manual decomposition process. 2.2

ITEM

CUBIT offers the Immersive Topology Environment for Meshing (ITEM) [5], a wizard-like environment that uses the approach used by [6] to determine the possible cutting surfaces. First, all the curves that form a dihedral angle less than an input value (default is 135 degrees) were collected. Second, a graph-based algorithm was used to determine all the closed loops or the open loops that can be closed using a boundary edge. Third, all the surfaces that bound the loops were collected. Fourth, extensions of the collected surfaces were used in defining the cutting surfaces. The extensions of the surfaces are presented as suggest cutting surfaces to users. Fig. 4 shows some of the cutting surfaces suggested by ITEM. However, those suggested cutting surfaces might not result in a meshable model. The user has to search through the available options and make meaningful selections if exists. If there are no meaningful cutting surfaces, then the user has to use CLI or GUI to manually create the desired cutting surfaces. Thus, time-consuming manual decomposition operations are still required. 2.3

Pen-Based Interface

Numerous researches have focused on using pen-based interface to create 3D models. Free-form surface generation or model creation systems such as [7, 8, 9] reconstruct a 3D model or surface from 2D sketch inputs. The template-based system proposed by Yang et al. [10] applies sketch recognition algorithms to match points and curves to the given 2D templates. Kara


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Fig. 4. Suggested cutting locations provided by ITEM

and Shimada [11] use computer vision techniques to align a template with an input sketch for users to deform a model. By tracing over the sketch, user can simply draw the new shapes on the model for deformation. Kho et al. [12] use strokes to define a free-form skeleton and the region of the model to be deformed. Nealen et al. [13] first detect an object’s silhouette, and then let the user create new features or modify the existing one by sketching an approximate shape onto the model. Cheutet et al. [14] allow the user to directly operate on an object by modifying strokes with a stylus pen. The gesture based interface used by Teddy [7] and [15] uses user strokes for operations such as extension, bending, and other modification functions. An optimization-based algorithm is used to determine the depth of 2D sketch vertices in the work presented by Masry et al. [16]. It constructs a 3D wireframe shape from a 2D freehand input for a physical simulation. Though pen-based interfaces have many contributions in the CAD modeling field, their advantages have not been applied in geometry preprocessing for hex meshing.

3 Drawing, Beautification and Snapping of Freehand Inputs The proposed UI interprets the user’s freehand input as a “hint” in order to guess the intention and suggests the best matching result. It solves the problems of inaccuracy and repeatability of freehand input strokes by beautifying them with re-sampling, smoothing filters and geometric fitting to 1D entities, and then snapping the 1D entities by offsetting and overlapping algorithms. The user draws a freehand input P , then the input is re-sampled and smoothed to P˜ . P˜ is represented as a 1D geometric entity C(t) such as a line, a circle, or a spline. In the future implementation, the system will search on the object surface for edges of the same geometry type and calculate the

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distance between the matching edge and C(t). C(t) will then be snapped as an offset or overlap of the closest edge. Another feature of the proposed UI is using gestures for easily access commands. Currently, ten gestures can be recognized by the systems. The user can draw gestures to execute the linked commands. 3.1

Drawing on Object Surfaces

The users intentions are either to create 1) a geometric entity, or 2) to specify an operator that works on the geometry entity. In the case of geometric entities, the intensions are to create 1D entity of one degree (e.g. line, polyline), two degrees(e.g. arc, circle, ellipse, parabola), or higher degrees(e.g. cubic splines, Bezier, NURBS). However, currently only lines, arcs, circles, and splines can be identified. The intended operators generally work on the 1D entities or can operate on 2D entities (e.g. cutting surface). The operators that work on 1D entities include alignment, trimming, offsetting, extending, sweeping. The operators that work on 2D entities include webcutting using a cutting surface. These operators are specified via gestures as shown in Fig. 5 (the dots on the gestures indicate the starting points) or key shortcuts. As shown in Fig. 6, the gesture can still be recognized even the directions and sizes are different from the template. Currently, a subset of operators are implemented and linked with the gestures, and rest of the gestures will be linked in the future with other advanced operators. 3.2

Beautification: Resampling, Smoothing and Geometric Fitting

The proposed interface uses three steps to beautify freehand inputs: 1) resampling, 2) smoothing, and 3) geometric fitting. Resampling will uniformly distribute the points carried by the freehand input. Smoothing removes noise while geometric fitting matches the freehand input to an intended shape such as a line, circle, and spline.

Fig. 5. Gesture set


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Fig. 6. An example of alpha gesture recognition. The program is able to recognize the incurrate alpha gestures correctly though they are in different sizes and drawing directions.

A freehand input is noisy due to the hand control of a user as well as sampling errors caused by stylus and tablet. The number of samples in a given input stroke depends on the pen speed (i.e. point density is inversely proportional to the users pen speed) and varies among devices. The system needs to re-sample the input to obtain data points uniformly spaced along the stroke before Laplace smoothing [17] can be applied to remove noise from the inputs. The noisy freehand input is replaced with a re-sampled and smoothed stroke, shown in Fig.7(a) and (b), respectively. The third step is to fit the smoothed inputs to common predefined shapes, such as line segments (Fig. 8(a)), circles (Fig. 8(b)), and splines (Fig. 8(c)), if they match. We define a set of tolerances to determine how a stroke should fit to each predefined shape. This tolerance is defined as the ratio of the length between the endpoints of the stroke, Llength and the actual curve length of the stroke Lo . If the tolerance is greater than 0.99, then the stroke is fit to a line segment. If the tolerance is less than 0.06, then the stroke is fit to a closed circle. We assume that when the user would like the freehand input be fit to a circle with radius r, the length of the input Lo is close to 2πr and Llength is within 0 to r/3 as shown in Fig. 9. Otherwise, the stroke is returned as a spline as shown in Fig. 8(c). 3.3

Identifying Candidate Edges and Snapping

In this section, we describe how to snap 1D input entities to fit the object edges of the same type. It is important to identify the 1D entities types before snapping to reduce the search space. In order to snap the 1D entities correctly to the closet edges on the object surfaces, we first identify the 1D entities as different types: lines, circles and splines, then the system can search

(a) Before smoothing.

(b) After smoothing.

Fig. 7. Smoothing the freehand input

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(a)

(b)

(c)

(d)

Fig. 8. Freehand inputs are fitted to (a) a line segment; (b) a circle; (c) an arc; (d) a spline

Fig. 9. Snap a freehand input to a circle. Thick solid line represents a freehand input, which will be fitted to a circle shown with a dash line.

Fig. 10. The algorithm to identify candidate edges

on the object surfaces for candidate edges which have the same type. Next the system compares the average distances between different candidate edges and the 1D entity. If the average is within the tolerance TL , the edge will be selected as a candidate. The algorithm for identification is shown in Fig. 10. To snap a 1D entities to edges on object surfaces, we apply the method proposed by Song and Shimada [18] as shown in Fig. 11. After the system


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Fig. 11. The snapping algorithm if candidate edges exist

identify the candidate edges of the same type as the 1D entity, it calculates the average distance davg shown in Fig. 12(a), between the 1D entities and those edges. The davg is then compared with the overlapping tolerance To and parallel tolerance Tp . When davg is less than the overlap tolerance To , the 1D entity will be trimmed to its closest edge length. When davg is within To and Tp , the 1D entity will be offset davg from the edge displayed in Fig. 12(b) as the result shown in Fig. 12(c). The offsetting result to the circle edge is shown in Fig. 13. If davg is greater than Tp , the 1D entities will not be offset to the edge.

4 Creating Cutting Surfaces and Decomposition The user can sweep a beautified and snapped 1D entity to create a cutting surface. The system first picks a surface of the given model by the midpoint of the entity then projects the stroke on the surface. The entity will be swept to create a cutting surface in one of the following directions: 1) the object’s surface normal direction; or 2) the current viewing direction. More sweeping directions will be implemented in the future. Next, the system determines the target volume with the midpoint of the stroke. Then the stroke is swept along the sweeping direction until it reaches

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(a) The original input stroke

(b) The snapped stroke

(c) The distances from strokes (1) and (2) to the upper edge are smaller than TP so they are offset to the upper edge. Stroke (3) is not offset to any edges. The distance between (4) and the lower edge is greater than TP thus (5) is not offsett to the lower edge. (6) is offsett to the lower edge.

Fig. 12. Offset snapping illustration

Fig. 13. Concentric circle snapping

(a) Sweeping along normal direction

(b) Sweeping along current viewing direction

Fig. 14. Creating cutting surfaces for webcutting


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the bounding box size of the target volume. Fig. 14(a) and (b) show the cutting surface and the decomposition along the surface normal direction and the viewing direction, respectively.

5 Results and Discussions The first example demonstrates the geometry decomposition using freeform strokes to obtain better mesh quality. The second example demonstrates the use of the proposed UI on defeaturing to obtain a meshable model. In the last example we show the potential of using the proposed interface to control the orientation and the size of the mesh. 5.1

Decomposition with Pen-Based Interface for Hexahedral Meshing

The hexahedral mesh generation is a hard problem even for simpler geometries and generally requires geometry decomposition as a pre-process. For example, hex meshing algorithms such as sweeping and mapping fail to mesh the model shown in Fig. 16(a) since the model contains two holes in orthogonal direction. The model can be meshed via a sweeping algorithm if it can be decomposed so that the two holes are in two different sub-domains. As shown in Fig. 15, the ITEM does not provide any cutting surfaces in ideal region between the two holes as there are no geometric features in this ideal region. Using the CLI user will be able to webcut the volume using a planar surface; however, the mesh quality will not be good because of the bad angles. The cutting surface in the ideal region should be orthogonal to the sweeping source and target surfaces (see Fig. 16(b)). Using the pen-based interface the user can draw a stroke in the desired direction orthogonal to the source and the target surface in the ideal region as shown in Fig. 16(b). Fig. 16(c)

Fig. 15. None of the suggested cutting surfaces by ITEM locate in the ideal region

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(a) The original model

(b) Create the desired cutting surface with a freeform stroke in the ideal region

(c) The decomposition result via pro-(d) The hex mesh quality result uses the Scaled posed pen-based interface Jacobian matrix

Fig. 16. Decomposition with freefrom stroke via pen-based interface

shows the meshed model and the mesh quality is shown in Fig. 16(d). The decomposition via the proposed interface has given a min Scaled Jacobian of 0.7255. While cutting with a planer surface using the command line the min scaled Jacobian was 0.6513. 5.2

Defeaturing via Pen-Based Interface

In the last couple of decades the CAD technology has evolved significantly and has enabled the users in creating detailed CAD models; however, this has brought new challenges to the mesh generation phase. In order to obtain a conformal high-quality hex mesh with a small number of mesh element, irrelevant detailed features should be removed. Though much focus on automatic defeaturing has been given [19], user interactions are still required in many cases, especially during hexahedral mesh generation. In this example, we demonstrate an application of the proposed pen-based UI in the area of defeaturing. Fig. 17(a) shows a typical industrial model containing two small features: threads and slots. Fig. 17(a) also shows the two input strokes marked on the model surface to indicate the regions that need to be defeatured. Fig. 17(b) shows that the small threads and slots are decomposed from the main solid and in Fig. 17(c) those features are removed.


(a) The original model: unmeshable due to small features, the slot and threads

(b) The defeatured model

(c) The decomposed and defeatured model

(d) All hex mesh output Fig. 17. Defeaturing to achieve a meshable model

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Fig. 17(c) and Fig. 17(d) show the meshable sub-domains and the final mesh obtained by combining the proposed UI with other existing tools in CUBIT. 5.3

Controlling Mesh Orientation and Size via Freeform Stroke

In this example, we show the potential of our pen-based interface in controlling mesh orientation and size. Currently, there are no general automatic hex meshing algorithms that can utilize tensor fields [20] based on physical parameters such as temperature distribution, stress contours, or streamlines in flow problems. With the proposed UI, the user can draw freeform strokes along the contours based on domain knowledge. Fig. 18(a) shows a typical streamline pattern in the flow around an airfoil. To generate a hex mesh that aligns with the flow pattern, the user can draw strokes along typical streamlines. The

(a) A flow pattern around an airfoil [21]

(b) Drawing strokes along the streamlines to decompose the model. The strokes are marked as orange

(c) The oriented mesh along the streamlines

(d) The oriented hex mesh along the streamlines

Fig. 18. Controlling mesh orientation


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model is then decomposed along the strokes as shown in Fig. 18(b). Fig. 18(c) shows that mesh is oriented along the streamlines and the mesh size at the boundary layer at the bottom of the airfoil can be controlled by maintaining a required spacing between the freeform strokes. A smaller spacing produces finer elements at boundary layers and a larger spacing produces coarser elements.

6 Conclusions This paper proposes a new application of pen-based UI in the area of geometric processing for mesh generation. The proposed UI makes the manual geometry decomposition for hexahedral meshing very intuitive, easier, and faster. The proposed approach beautifies and snaps the freehand input strokes (drawn directly on the CAD model) and uses gestures to create cutting surfaces to decompose the CAD model into meshable sub-domains. The pen-based UI can also be used in controlling the size and orientation of the hex mesh and in defeaturing irrelevant small features. Work is underway to add more features to the UI and to apply it to related areas in pre-processing stage.

Acknowledgement We would like to thank Dr. Soji Yamakawa, Mr. Ved Vyas and Mr. Erick Johnson for their valuable advises. We would also like to thank Cubit users, in particular Kristin Dion and Ricardo Garcia, for their valuable feedback.

References 1. Hardwick, M.: DART System Analysis Presented to Simulation Sciences Seminar (2005) 2. Owen, S.J.: A survey of unstructured mesh generation technology. In: Proceeding of the 7th International Meshing Roundtable, Sandia National Laboratories, pp. 239–267 (1998) 3. Sandia National Laboratories, Cubit 12.1 on-line user’s manual: Web cutting, http://cubit.sandia.gov/help-version12.1/cubithelp.htm 4. Yang, N., Forsberg, A., Shepherd, J., Garcia, R., Merkley, K.: Mark-it: A marking user interface for cutting decomposition time. In: Proceeding of the 13th International Meshing Roundtable, Sandia National Laboratories, pp. 231–242 (2004) 5. Owen, S.J., Clark, B.W., Melander, D.J., Brewer, M., Shepherd, J.F., Merkley, K., Ernst, C., Morris, R.: An immersive topology environment for meshing. In: Proceeding of the 16th International Meshing Roundtable, Sandia National Laboratories, pp. 553–578 (2007)

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6. Lu, Y., Gadh, R., Tautges, T.: Volume decomposition and feature recognition for hexahedral mesh generation. In: Proceeding of the 8th International Meshing Roundtable, Sandia National Laboratories, pp. 269–280 (1999) 7. Igarashi, T., Matsuoka, S., Tanaka, H.: Teddy: A sketching interface for 3D freeform design. In: Proceeding of the 26th annual conference on Computer graphics and interactive, pp. 409–416 (1999) 8. Tsang, S., Balakrishnan, R., Singh, K., Ranjan, A.: A suggestive interface for image guided 3D sketching. In: Proceeding of the SIGCHI conference on Human factors in computing systems, pp. 591–598 (2004) 9. Bae, S., Balakrishnan, R., Singh, K.: Ilovesketch: As-natural-as-possible sketching system for creating 3D curve models. In: Proceeding of the 21st annual ACM symposium on User interface software and technology, pp. 151–160 (2008) 10. Yang, C., Sharon, D., van de Panne, M.: Sketch-based modeling of parameterized objects. In: Proceeding of the SIGGRAPH 2005: ACM SIGGRAPH 2005 Sketches, p. 89 (2005) 11. Kara, L., Shimada, K.: Construction and modification of 3D geometry using a sketch-based interface. In: Proceeding of the EUROGRAPHICS Workshop on Sketch-Based Interfaces and Modeling, pp. 59–66 (2006) 12. Kho, Y., Garland, M.: Sketching mesh deformations. In: Proceeding of the Symposium on Interactive 3D Graphics and Games, pp. 147–154 (2005) 13. Nealen, A., Sorkine, O., Alexa, M., Cohen-Or, D.: A sketch-based interface for detail-preserving mesh editing. In: Proceeding of the ACM SIGGRAPH 2005 (2005) 14. Cheutet, V., Catalano, C., Pernot, J., Falcidieno, B., Giannini, F., Leon, J.: 3D sketching for aesthetic design using fully free-form deformation features. Computer Graphics 29(6), 916–930 (2005) 15. Eggli, L., Br¨ uderlin, B.D., Elber, G.: Sketching as a solid modeling tool. In: Proceeding of the 3th ACM symposium on Solid modeling and applications, pp. 313–322 (1995) 16. Masry, M., Kang, D., Lipson, H.: A freehand sketching interface for progressive construction of 3D objects. Computers and Graphics 29(4), 563–575 (2005) 17. Taubin, G.: Curve and surface smoothing without shrinkage. Computer Vision, 852–857 (1995) 18. Song, I., Shimada, K.: Sketch-based computer-aided design tool for configuration design of automobile instrument panel. Computer Aided Design and Application 6(5), 585–594 (2009) 19. Quadros, W.R., Owen, S.J.: Defeaturing CAD models using a geometry-based size field and facet-based reduction operators. In: Proceeding of the 18th International Meshing Roundtable, Sandia National Laboratories, pp. 301–318 (2009) 20. Vyas, V., Shimada, K.: Tensor-guided hex-dominant mesh generation with targeted all-hex regions. In: Proceeding of the 18th International Meshing Roundtable, Sandia National Laboratories, pp. 377–396 (2009) 21. Massachusetts Institute of Technology, Marine hydrodynamics, http://web.mit.edu/13.021/13021003/Lifting20surfaces/lectureC.htm

Particle Systems for Adaptive, Isotropic Meshing of CAD Models Jonathan R. Bronson, Joshua A. Levine, and Ross T. Whitaker Scientific Computing and Imaging Institute, Salt Lake City, UT, U.S.A {bronson,jlevine,whitaker}@sci.utah.edu

Summary. We present a particle-based approach for generating adaptive triangular surface and tetrahedral volume meshes from CAD models. Input shapes are treated as a collection of smooth, parametric surface patches that can meet non-smoothly on boundaries. Our approach uses a hierarchical sampling scheme that places particles on features in order of increasing dimensionality. These particles reach a good distribution by minimizing an energy computed in 3D world space, with movements occurring in the parametric space of each surface patch. Rather than using a pre-computed measure of feature size, our system automatically adapts to both curvature as well as a notion of topological separation. It also enforces a measure of smoothness on these constraints to construct a sizing field that acts as a proxy to piecewise-smooth feature size. We evaluate our technique with comparisons against other popular triangular meshing techniques for this domain. Keywords: Adaptive meshing, particle systems, tetrahedral meshing, CAD.

1 Introduction Tetrahedral mesh generation is a key tool in the computer-aided design (CAD) pipeline. In particular, the conversion of shapes presented by the output CAD systems and solid modeling geometry kernels is necessary to provide input meshes for structural analysis, CFD and other CAE applications. The data from these systems is usually in the form of a boundary representation (B-Rep) made up of hierarchical connectivity (topology) and associated geometric entities. When the B-Rep is manifold and topologically closed, shapes of arbitrary geometric complexity can be produced. Care must be taken to provide accurate representations of these inherently piecewisesmooth solids while robustly preserving the curved features defined by the input topological description [18]. While many types of meshes are suitable for visualization and simulation, simplicial meshes have emerged as one of the dominant forms. Their popularity

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J.R. Bronson, J.A. Levine, and R.T. Whitaker

can be attributed to both the ease at which simplicial meshing algorithms can be implemented as well as the guarantees of termination and quality that can often be shown. Technologies to construct simplicial meshes vary greatly. Some of the most important paradigms include advancing-front [19, 20, 26], Delaunay refinement [10, 25, 29], and particle systems [22, 33]. However, to build meshes that are adaptive, many variants of these techniques require an input oracle that evaluates a sizing field over the domain [4, 8, 23, 27, 32]. An early exception is the approach of Dey et al. [12] that uses Delaunay refinement for meshing smooth domains. Using the dual Voronoi diagram and the concept of poles [2], this algorithm automatically refines based on a simultaneously computed approximation of the local feature size (distance to the medial axis) of the shape whose accuracy increases as mesh density increases. Local feature size of smooth shapes is a natural choice to use as a field to adapt to; however, most CAD models are inherently non-smooth. A notion of local feature size for piecewise-smooth shapes has been defined [8] by coupling local feature size for the smooth regions with a topological condition called gap size [9]. Computing this measure robustly is a significant challenge. The approach in this work aims to automatically infer a global sizing field of equivalent expressivity to [8] while using only locally available information as done by [12]. Such a technique must force a compromise, ours is to construct a proxy for feature size that is Lipschitz continuous by coupling curvature adaptivity with a topological separation condition. Particle systems are an ideal framework for sampling parametric domains since they only require local calculations. We can minimize energy by movements solely within the parameter space of each surface patch while knowing each particle stays within the confines of the surface in world space. Computing good quality triangulations from these samples can leverage the same benefits. We directly build triangulations on the particles in parameter space using 2D Delaunay triangulations (implemented by Triangle [28]). As 3D Delaunay triangulations can be expensive to compute, this provides a significant savings when only surface meshes are required. While this approach cannot immediately build triangles that are 3D Delaunay, we can improve their quality significantly by local modifications (e.g. edge flipping) that consider the world space positions of vertices. The resulting surfaces meshes make ideal inputs for volumetric meshing approaches, such as TetGen [30]. 1.1

Contributions

This work focuses on automatic techniques for building triangular meshes of the boundary surface, and ultimately tetrahedral representations of the interior solid. We also improve the state-of-the-art for particle system-based techniques; our contributions can be summarized as the following:

Particle Systems for Adaptive, Isotropic Meshing of CAD Models

• • • • •

281

An automatic technique for constructing isotropic surface meshes by minimizing a world-space energy through parameter-space movements. Hierarchical sampling of features in increasing dimension, inspired by weighted Delaunay-based approaches [8, 11]. Algorithmic control for both uniform and adaptive sampling, without requiring a pre-computation of global feature size needed by similar particlebased approaches [23]. Fast mesh generation of these samples through the use of the 2D Delaunay triangulation in parameter space and 3D Delaunay edge flipping [6]. Experimental evaluation that compares our approach to existing techniques [7, 15] for mesh generation of CAD domains.

2 Related Work and Background While the history of tetrahedral mesh generation began much earlier, a shift in the emphasis of techniques has become popular within the past decade. In particular, variational approaches, i.e. based on energy minimization, have become one of the most important tools for mesh generation. Alliez et al. [1] describe a variational technique for mesh generation that couples Delaunay refinement with a relaxation process for vertex locations. This algorithm and later variants [31, 32, 34, 37] base their energy minimization on a sizing field for particle density coupled with an energy minimization grounded in the notion of a centroidal Voronoi diagram [14] and its dual, the optimal Delaunay triangulation [5]. Consequently, these meshing algorithms can generate nearly isotropic elements, as a byproduct of the centroidal Voronoi condition, as well as leveraging many of the benefits of Delaunay refinement techniques. However, one deficiency is the need for knowledge of an element sizing field a priori. Computing a sizing field is considered expensive. Often, approaches for computing sizing fields are based on the medial axis [13] or quadratures of mesh elements [3], and thus can require O(n2 ) computations of dense point clouds to build accurate results. One recent solution of Tournois et al. [32] solves this problem by alternating a variational phase with a refinement phase. After each level of refinement, the sizing function is updated before switching back to variational phase. This interleaving allows the available information to drive the computation of a sizing field instead of necessitating a preprocessed computation. We aim to improve upon this theme by allowing an energy minimization based on particle systems to automatically improve its approximation of the sizing field. A second thrust of recent algorithms is to provide provably algorithms for meshing piecewise-smooth shapes. This general class describes shapes with a topological description in the form of a piecewise-smooth complex of kcells that are compact subsets of k-manifolds. We use the same definition as Cheng et al. [8]. In summary, surface patches (2-cells) can meet non-smoothly at curves (1-cells) bounded by points (0-cells). Two k-cells are adjacent if one is on the boundary of the other.

282


Similar to the B-Rep definition, each k-cell has an associated geometric description. Recent Delauany-based approaches [8, 24] for meshing this domain have been able to provide topological correctness guarantees using either weighted Delaunay triangulations [8] or bounding the angle deviations between smooth patches [24]. A missing piece to the implementations of these algorithms is the ability to adapt to a sizing field, primarily because there is no consensus on what is the correct sizing field for non-smooth shapes and how best to compute it. However, they do show that a careful sampling of points on sharp creases can preserve the features of a shape. Our approach is a natural extension of this work, but instead of requiring an accurate sizing field to guarantee topological correctness, our scheme will build watertight meshes provided a few easily satisfied conditions are met by the particle system (described in Section 4.3). 2.1

Particle Systems

At the core of our meshing scheme is a paradigm for sampling shapes using particles. The idea of using repulsive point clouds to (re-)sample a mesh was first introduced by Turk in the context of polygonal remeshing [33]. The first full particle system for meshing was later developed by Witkin and Heckbert [35]. Their technique was primarily used as a mechanism to sample and control implicit surfaces, which was notoriously difficult under other schemes at the time. The key idea behind their work was the introduction of a Gaussian energy function to control the interaction between particles. Improvements to their scheme were made by Hart et al. [17]. Yamakawa and Shimada proposed a meshing scheme similar to particles by using packings of ellipsoidal bubbles [36]. Meyer et al. [21] formulated a more robust and stable solution for evolving particle systems. The new energy kernel was a modified cotangent function, with finite support. By falling off to a finite range, the resulting particle systems were more stable and more quickly lead to ideal packings. Additionally, this kernel was nearly scale invariant. Meyer et al. [23] later introduced a hierarchical scheme for particle-based sampling multi-material surfaces. For such datasets, the boundaries between the different materials can be represented as a piecewise-smooth complex. While without the formal guarantees of [8], they use a similar strategy of hierarchically sampling topologically features in increasing dimension to build consistent, watertight meshes.

3 Particle System Formulation In this section we provide the mathematical formulation behind our particle system. We define the total energy in the system as the sum of each energy Ei calculated with respect to particle pi . Each particle pi has a corresponding σi value representing the radius of its ball of influence Bi centered at pi . It is the varying of σi that provides adaptivity. Each energy Ei is the sum of the


283

energies between particle pi and all neighboring particles pj . Particles pi and pj are considered neighbors if either pj falls within Bi or if pi falls within Bj . We use a variation of the modified cotangent for the energy (1) between any two particles, Eij . By varying σi , the potential function must be scaled to account for this new lopsided interaction between particles. Thus, we scale both the modified cotangent function and its derivative (2) by σi . Eij = σij cot(

|rij | π |rij | π π )+ − σij 2 σij 2 2

dEij π |rij | π = 1 − sin−2 rij 2 σij 2

(1)

(2)

In this form, |rij | is the distance between particles pi and pj and the value σij is taken to be the max of σi and σj . The hexagonal packings that result from this and related particle systems requires the particles to reach a critical density on the surface being sampled. For any surface and any set of σ values, there will always be an ideal number of particles, but calculating this number is not tractable. Like previous systems, we use splitting and deleting to control energy densities. Particles follow the rules: Ei∗ = Ei (1 + ) if if

Ei∗ Ei∗

< 0.35Eiideal > 1.75Eiideal

Split Delete

(3) (4) (5)

Using a hexagonal packing as our notion of an ideal distribution, the ideal energy Eiideal for a particle pi is six times the energy felt between pi and pj at the characteristic distance of approximately 0.58 [21]. Given that a tworing particle pj is at distance 1.0, Equation (6) describes this relationship. Additionally, we scale this value by σi to match the scaling of actual energies. Eiideal = σi 6E(β), with

|rij | 0.5 =β= ≈ 0.58 σij cos(π/6)

(6)

Since one cannot predict what an ideal neighborhood will look like in the adaptive case, the ideal energy is less precise than in the constant case. This leads to more frequent splits and deletes for higher local variation, but ultimately provides much better packings than if the original energy was not scaled proportional to σ. An alternative to this approach would be to use a notion of scaled distance d = σd , and forego the σi scaling. Then, to still achieve the high quality packings, a different scheme for deletion of poorly configured particles would need to be devised. To allow the system to adapt to splits and deletes, Ei is biased by a small random number, 0 ≤ ≤ 1, in Equation (3). This makes the discrete energy jumps have less of an impact on the speed at which the system stabilizes, by allowing time for the system to adapt between jumps. Additionally, this

284


can help resolve any regions which are stuck in bad configurations. As the solution to the system converges, this bias can be adjusted to stop splits and deletes all together, ensuring termination. To find the ideal packing of particles, we use a Newton-Raphson scheme, updating particle information after each movement (Equations (7), (8), and (9)). Each particle maintains its position in both worldspace (xxyz ) and pai rameter space (xuv ). Particles move with a velocity v generated by interi i particle forces between neighbors. Though the energies between particles are computed in 3D world space, particles move strictly in parametric space (9), avoiding the error-prone projection onto the surface that results from 3D movements. Taking these steps in parametric space only requires a change of coordinates, using the inverse Jacobian, J −1 . rij vxyz = dEij · (7) i |rij | vuv i xuv i

= =

j∈N vxyz · J −1 i uv xuv i + vi

(8) (9)

As mentioned earlier, we use a hierarchical sampling scheme, which works well for parametric models. First, we place particles on the 0-cells, the intersection of edges on the models. Next, particles are placed on the 1-cells and are optimized. Finally, particles are placed on the surface patch interiors and the final optimization proceeds. At each phase, the new optimization uses the fixed positions from the previous phase, ensuring consistency across surface patch boundaries. 3.1

Inferred Sizing Field

We recognize that there are several factors that often determine good sizing fields: local curvature, some notion of feature size, and a desired level of adaptivity. Additionally, users may have desires for mesh resolution limits, both minimum and maximum triangle or edge size. Other domain-specific factors also often come into play. In this section, we illustrate the constraints we would like to place on a sizing field. We show that these constraints can be inferred in a reliable way and used to form a smooth sizing field during energy minimization. We aim for meshes that provide arbitrary levels of geometric accuracy and adaptivity, using high quality isotropic elements. In order to provide high quality elements, particle systems require enough spatial freedom to be able to move to lower energy states. Thus, the distance between nearby k-cells imposes its own sizing constraint on the particles. Thus, in order to determine the sizing field value σi at a particular point pi on a model, we must consider the constraints placed on this location by curvature, topological distance, and desired level of adaptive continuity. We refer to these constraints as σκ , στ , and σL , respectively. The actual sizing field value at a particle location is


285

resolved by finding the σi that respects all constraints. This can be expressed compactly as: σi = max {σmin , min {σmax , σκ , στ , σL }}

(10)

Curvature Since the curvature at a point is defined as the inverse of the radius of the osculating circle at that point, a reasonable default sizing field value is the radius of that circle itself. Thus, we use σκ = κ1 , which can be easily computable for parametric surfaces, or queried by middleware packages. To increase or decrease the field relative to this radius, a scaling factor sκ is exposed as a user parameter. Given a unit system, this value can be used to provide constraints to re spect geometry to arbitrary levels of accuracy. Finally, κmin and κmax values are user parameters used to handle straight edges and arbitrar ily high curvature, respectively. These form the total bounds for the sizing field as: σmin = 1/ (sκ κmax )

(11)

σmax = 1/ (sκ κmin )

(12)

Fig. 1. Default curvature constraint on sizing field

For 2-cells, we use the maximum principal curvature, since this size will dominate an isotropic sampling. For 1-cells, using the curvature of the edge itself is insufficient. The maximum principal curvature on both intersecting surfaces must also be considered, since the curve may either be a trim or a boundary curve, and there is no way of knowing which curvature will dominate. Last, 0-cells use the maximum curvature of all 1-cells terminating at its point. Gap Size If available, using the distance to the model’s medial axis would provide a sizing field constraint that generates good samplings in a particle system. However, computing the medial axis on parametric models is a difficult task and still an active area of research. Instead, we use the notion of gap size, introduced by Chang & Poon [9] in the context of piecewise linear mesh generation. For a point p on a k-cell c, its gap size is the distance to the nearest non-adjacent (i.e. not on the boundary of c) cell. This measure also preserves topological features inherent to the model’s decomposition into parametric patches. Depending on the model and the way it was created, this measure may sometimes be equivalent to definitions of local feature size. Figure 2 shows an example where the two are equivalent by a factor of one half. We make the assumption that the topological representation provided as input for the CAD model should be respected in an output mesh. A byproduct of this approach is that some models have adaptivity in regions that

286


are of little benefit to representing the geometry of the model. One could remove adaptivity in regions that do not actually need it by taking a pass over the model and detecting topological junctions that are G1 continuous, and flagging them to be ignored. The remaining geometrically discontinuous junctions could then be preserved using our sampling scheme. Gap size is approximated directly from inter-particle relationships. Particles store which k-cell they lie on, and each k-cell stores which particles lie on it. We define the topological constraint στ to be the shortest distance from particle pi to another particle pj lying on a non-adjacent feature. That is, a 0cell particle interacting with another 0cell particle, a 1-cell particle interacting Fig. 2. Gap size constraint on siz- with another 1-cell particle, or a 0-cell ing field. In this case, the gap size is particle interacting with a 1-cell particle equivalent to the distance to the me- that does not terminate at that 0-cell. dial axis by a factor of two. This notion can be extended to 2-cells as well. We further provide a scaling factor sτ as a user parameter to allow for higher densities of particles within these regions. This proves useful when sampling highly elongated surfaces, with parallel k-cells. Scaling the distance στ allows more rows of particles, allowing for better energy minimization, and ultimately better triangulations. Lipschitz Continuity In order to provide finer control over the adaptivity of the particle samples, the system adheres to a Lipschitz constraint σL that enforces the Lipschitz continuity L on the sizing field. The Lipschitz condition can be expressed in terms of our formulation as: |xi − xj | ≤ L |σi − σj |

(13)

The σL induced by this constraint is simply the minimum allowable value that satisfies this condition: σL = min {|rij | L + σj } j∈N

(14)

Respecting this Lipschitz continuity provides more gradual adaptivity between areas of high and low particle densities. Lower values of L produce samplings that result in more isotropic triangles, while large values provide increased levels of adaptivity, at the cost of isotropy. In the limit when L goes to zero, a uniform sizing field is produced, fitting the smallest constraint on the model. We found a default value of 0.3 provides a good trade-off between triangle quality and adaptivity.


287

It is worth noting that the Lipschitz continuity is not satisfiable for arbitrary surfaces. Since we place samples hierarchically, it is possible the sizing field may need to adapt more quickly on the interior of the surface than it does on the edges. In these situations, the Lipschitz constraint needs to be relaxed to allow the sizing field to adjust.

4 Algorithm Our implementation takes as input a parametric model and outputs a triangular mesh. We use the middleware package CAPRI [16] to provide us direct geometry access to shapes generated by CAD software. It also gives access to the topology of the model, including access to the 0-cells, 1-cells, and 2-cells, and their boundary adjacencies. In this section, we elaborate only on the parts of the update algorithm that are independent from the middleware. 4.1

Particle Optimization

The sampling algorithm consists of three phases: Phase 1 optimizes 0-cell and 1-cell samples based strictly on the curvature and the Lipschitz constraints, σκ and σL . Phase 2 continues the 0-cell/1-cell optimization, but includes the topological constraint στ . Finally, Phase 3 optimizes samples with surface patches. A phase is considered complete when the change from one iteration to the next drops below some threshold. We initialize Phase 1 by placing one particle on each 0-cell, and one particle on the midpoint of each 1-cell. Along the 1-cells, splitting happens to increase particle density as the sizing field is inferred. Similarly, if user parameters make any 1-cell particle unnecessary, it will be deleted. Phase 3 is initialized by placing k random samples in the parameter domain of the surface. Each iteration of the optimization, a particle updates both its position as well as its sizing field value σi . A scaling factor λi is used to increase stability. Pseudocode for the updates of particle positions is shown in Algorithm 1. 4.2

Corner Cases

The motivation for splitting the optimization of 0-cells and 1-cells into two phases is illustrated in Figure 3. When it comes to enforcing the topological condition, just as feature size goes to zero in discontinuous corners, so does our notion of topological feature size. Left unchecked, particles in corners would continually shrink their στ , split and move in closer to the corner. To curtail this response, we detect and label corners in the first phase. Figure 3(a) shows what one corner might look like after Phase 1 has completed. Notice only the curvature and Lipschitz constraints have been met. The σi value of the particle on the 0-cell is saved as the size of the 0-cell’s corner ball. This is similar to the protecting ball idea in Delaunay meshing [9]. Figure 3(b)

288


Algorithm 1. Position Update 1: for all particles do 2: Compute energies Ei , dEi (Equations 1,2) 3: Compute velocity vxyz (Equation 7) i 4: Transform to parameter space, obtain vi∗ (Equation 8) 5: Compute scaling vi∗new = λi vi∗ 6: Compute new particle position unew (Equation 9) i 7: Transform to world space xnew i 8: Compute the new energy value, Einew 9: if Einew >= Ei then 10: if λi

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