Computational Methods in Engineering & Science

Z. H. Yao M. W. Yuan Computational Methods in Engineering & Science Proceedings of "Enhancement and Promotion of Compu...

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Z. H. Yao M. W. Yuan

Computational Methods in Engineering & Science Proceedings of "Enhancement and Promotion of Computational Methods in Engineering and Science X" Aug. 21-23,2006, Sanya, China

With 241 figures

fglTSINGHUA

\Sg? UNIVERSITY PRESS

£} Springer

EDITORS: Prof. Z. H. Yao Dept. of Engineering Mechanics Tsinghua University 100084, China E-mail: [email protected]

ISBN 10 ISBN 13 ISBN 10 ISBN 13

Prof. M. W. Yuan Academician, Chinese Academy of Sciences Dept. of Mechanics and Engineering Sciences Peking University 100080, China E-mail: [email protected]

7-302-13530-4 Tsinghua University Press, Beijing 978-7-302-13530-2 Tsinghua University Press, Beijing 3-540-48259-8 Springer Berlin Heidelberg New York 978-3-540-48259-8 Springer Berlin Heidelberg New York

Library of Congress Control Number: 2006937887 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-Verlag. Violations are liable to prosecution under the German Copyright Law. © 2007 Tsinghua University Press, Beijing and Springer-Verlag GmbH Berlin Heidelberg Co-published by Tsinghua University Press, Beijing and Springer-Verlag GmbH Berlin Heidelberg Springer is a part of Springer Science+Business Media springer.com 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. Cover design: Frido Steinen-Broo, EStudio Calamar, Spain Printed on acid-free paper

PREFACE The 9th EPMESC was successfully held in Macao in November of 2003. At the end of the conference the Board of the EPMESC series decided that the next conference would be held in a city of the mainland of China. Also I was assigned to be the Chair person of the Conference. No doubt this is a great honor to me and also a challenge because there are so many professional international conferences in computational mechanics happening frequently in the world. After the successful organizing of WCCM6 in Beijing, September of 2004,1 engaged to organize the 10th EPMESC. First of all, I had to choose the venue of the Conference. After some investigation and a lot of negotiation we went to see the site of the venue in Sanya, Hainan Island, the south-most city in China. Finally we made the decision. The most important reason for the choice was the ecological environment of this city. It is beneficial to our health after hard work. We scientists and engineers need a good relaxing place after working hard- a place to enjoy life with friends and family. Sanya is an ideal place and a real green city. Blue sky and white clouds, the peaceful sea and the long beach with white sand, shells and pearls, no pollution and no industry. Everything is so beautiful. After the first call for papers, the response was unexpectedly strong. We got more than 190 abstracts from 23 countries and regions. About half of the participants come from the mainland of China. The rest were from Macao of China, Japan, Portugal, Australia, USA, Germany, Russia, Poland, Singapore, Malaysia, Brazil, UK, Israel, Indonesia, France, Spain, India, Korea, Czech, Chile, Hong Kong, and Taiwan of China etc. We are glad to have so many friends from so many countries and regions getting together to exchange their research results and take part in social activities. We have the honor of inviting many famous experts in computational mechanics to give plenary and semi-plenary lectures in the conference. Serge Cescotto, Genki Yagawa, Zhenhan Yao are the plenary speakers. Win Kam Liu, Fred W. Williams, Roman Lackner, Ioannis Doltsinis, Helder Rodrigues, Nasser Khalili, Nori Miyazaki, Yao Zheng, Yeong-Bin Yang, Ka Veng Yuen, Gui Rong Liu, Chung-Bang Yun and Dajian Han are the semi-plenary speakers. All of them have achieved great success in their own fields of computational mechanics. I appreciate their outstanding contributions to the conference. This is the mark of the scientific level of this conference. One of the highlights, the student paper competition, is a traditional program that has been retained throughout the history of the EPMESC series. This will keep the young students following the latest advances in research in computational mechanics. Also it will encourage them to reach the peaks of science and technology. I'd like to thank my friend, Prof. Kai Meng Mok of the University of Macau, for his assistance in organizing the student paper competition. I'd like to thank my friend, Prof. Zhenhan Yao of Tsinghua University, for his outstanding work on the proceedings. He patiently worked long and hard on the abstracts and full length papers. He carefully read all the papers and abstracts and corrected a lot of mistakes. He made this proceedings a consistent, valuable reference and beautiful looking. I'd also like to thank my colleagues, Dr. Yongqian Chen and Yang Kuei, for their assistance in my work. I'd like to show my appreciation to the China National Science Foundation for their generous support. 21-25 August 2006. Mingwu Yuan Chairman, EPMESC X Professor, Peking University

CONFERENCE BOARD J. Bento

(Instituto Superior Tecnico, Portugal)

S. Cescotto

(University of Liege, Belgium)

Y. K. Cheung

(University of Hong Kong, Hong Kong, China)

C. K. Choi

(Korea Advanced Institute of Sc. & Tech., Korea)

I. Doltsinis

(Stuttgart University, Germany)

D. J. Han

(South China University of Technology, China)

V. P. Iu

(University of Macau, Macao, China)

T. Kawai

(Nippon Marc Co., Japan)

S. P. Lin

(Shanghai Jiao Tong University, China)

W. K. Liu

(Northwestern University, USA)

H. Mang

(Vienna University of Technology, Austria)

E. Arantes e Oliveira

(Instituto Superior Tecnico, Portugal)

B. A. Schrefler

(University of Padova, Italy)

A. I. Tolstykh

(Russian Academy of Sciences, Russia)

S. Valliapan

(University of New South Wales, Australia)

F. W. Williams

(University of Wales, Cardiff University, UK)

G. Yagawa

(University of Tokyo, Japan)

Y. B. Yang

(Taiwan University, Taiwan, China)

M. W. Yuan

(Peking University, China)

W. X. Zhong

(Dalian University of Technology, China)

ORGANIZING COMMITTEE Mingwu Yuan

Chairman

(China)

Dajian Han

Vice-Chairman

(China)

Zhenhan Yao

Secretary -General

(China)

Members: S. Cescotto C. K. Choi I. Doltsinis L. N. Lamas S. Sloan G. R. Liu S. P. Lin W. K. Liu H. Mang E. Pereira B. A. Schrefler A. I. Tolstykh S. Valliappan F. W. Williams G. Yagawa Y. B. Yang

(Universityof Liege, Belgium) (Korea Advanced Institute of Sc. & Tech., Korea) (Stuttgart University, Germany) (National Laboratory of Civil Engineering, Portugal) (University of Newcastle, Australia) (National University of Singapore, Singapore) (Shanghai Jiao Tong University, China) (Northwestern University, USA) (Vienna University of Technology, Austria) (Instituto Superior Tecnico, Portugal) (University of Padova, Italy) (Russian Academy of Sciences, Russia) (University of New South Wales, Australia) (University of Wales, Cardiff University, UK) (University of Tokyo, Japan) (Taiwan University, Taiwan, China)

LOCAL ORGANIZING COMMITTEE Mingwu Yuan

Chairman

(Peking University, Beijing, China)

Zhenhan Yao

Secretary -General

(Tsinghua University, Beijing, China)

Members: Dajian Han,

(South China University of Technology, Guangzhou, China)

Shaopei Lin

(Shanghai Jiaotong University, Shanghai, China)

Kai Meng Mok


Yao Zheng

(Zhejian University, Hangzhou, China)

Jiashou Zhuo

(Hohai University, Nanjing, China)

CONFERENCE SECRETARIAT Yongqiang Chen

Scientific Secretary


Kai Meng Mok

Scientific Secretary


Yazheng Yang

Administration Secretary

(Chinese Society of Theoretical and Applied Mechanics, Beijing, China)

Yanan Tang


(Chinese Society of Theoretical and Applied Mechanics, Beijing, China)

Yang Kuei



CONTENTS Preface

i

Conference Board and Committees

ii * denotes the presenter

in alphabet order of the regular session and first-author's name Plenary Lectures 1

Management of water pollutants based on multi-criteria analysis and fuzzy logics,

1

Cescotto S*, Roubens M, Rigo N, Gao SX et al 2

Enriched element method and its applications to solid mechanics,

15

Yagawa G*, Matsubara H 3

Large-scale boundary element analysis in solid mechanics using fast multipole method,

19

Yao ZH*, Wang PB, Lei T, Wang HT Semi-plenary Lectures 4

Optimization and robustness of deformable systems with randomness,

35

Doltsinis /*, Kang Z 5

Simulation of Stochastic Fluctuating Wind Field Using the Wave Superposition Method with Random Frequencies, Han DJ*, Luo J J

51

6

Monotonic and Cyclic Analysis of Granular Soils,

59 Khalili N*

7

Adaptive meshfree methods using local nodes and radial basis functions,

8

Multiresolution mechanics for material design and manufacturing,

71

Liu GR*, Kee BBT, Zhong ZH, Li GY, Han X 87 Liu WK* 9

Application of computational mechanics to reliability studies of electronic packaging,

88

Miyazaki N*, Ikeda T 10

Topology optimization of structures: applications in the simulation and design of cellular materials, Rodrigues HC*

101

11

Rigid body considerations for geometric nonlinear analysis of structures based on the updated Lagrangian formulation, Yang YB*, Lin SP, Chen CS

113

12

Nano-modeling structure and micromechanical properties of mesoscopic composite systems, Yanovsky YuG*

129

13

An extremely efficient finite-element model updating methodology with applications to damage detection, Yuen KV*

139

14

Dynamic infinite elements for soil-structure interaction analysis in a layered soil medium,

153

Yun CB*, Kim JM List of other semi-plenary lectures

168

ABSTRACTS OF REGULAR SESSIONS Computational Fluid Mechanics 15

Numerical simulation of cavitation generation in tandem cascades,

169

Kang C* Liu D, Yang MG 16

Numerical simulation of atrium fire using two CFD tools,

170

Sin VK, Tarn LM, Lao SK, Choi HF* 17

Investigation of the particulate matters with the aid of CFD, .

18

CFD analysis of fire in a forced ventilated enclosure,

171

Tarn LM, Sin VK, Sun HI*, Wong KI 172

Tarn LM, Sin VK, Lao SK*, Choi HF 19

Development of the diffusion control system using air shutter,

173

20

Asami T*, Nakabayashi Y A high order compact difference scheme for solving the unsteady convection-diffusion equation, Xie ZH* Lin JG, Zhou JT

174

21

Computation of one-dimensional dam-break flow using ENO scheme,

175

Liu YL*, Guo YT, Fan WB 22

Numerical simulation of 2D dam-break flood waves using TVD scheme,

176

Wei WL*, He WY 23

Motion analysis of the elevating ball by the effect of buoyant force,

177

Matsuo Y*, Nakabayashi Y 24

Development of an educational flow simulation system,

178

Nakanishi T*, Shibata H, Sato M 25

Using finite element program generator to solve N-S equation,

179

26

Wan S*, Nielsen MP, Chai G. Experimental and CFD study of the effects of design parameters on Reynolds number in a short duration hypersonic test facility, Al-Ralahi A * YusafT, YusoffMZ

180

27 28 29 30

Investigation of multiphase flows near walls with textures by the lattice Boltzmann method* Chew YT*, Huang JJ, Shu C, Zheng HW Numerical Simulation of the microstructure of magnetorheological fluids in magnetic fields > PengXH*,LiHT Implementation of a 3D multilaminated hydromechanical model for analysis of an unlined high pressure tunnel, Leitao NS, Lamas LN*

181

Prediction of mixing and reacting flow inside a combustor,

184

182 183

De Bortoli AL* 31 32 33

Large Eddy Simulation of Turbulent Reactive Flow, Liu Y*, Chen YQ, Chen JG, Yang R Computational modeling of coal water slurry combustion processes in industrial heating boiler > Zhu U, Gu BQ* Large eddy simulation of unsteady turbulent flow and pressure fluctuation in an axial-flow pump with various SGS models, Cong GH*, Wang FJ, Zhang L

185 186 187

34

Computation of turbulent flows in natural gas pipes with different rectifiers,

188

Li ZL*, Zhang YX 35

Computation of unsteady incompressible turbulent flow by using an implicit SMAC method, Zhang YX*, Li ZL, Zhu BS

189

36

Simulation on vortex effect for superconducting devices,

190 Lei SL, Lao M*, Chan IN

Computational Solid Mechanics 37

Simulation of corrosion rate of carbon steel subjected to elastic/plastic strain,

191

Ridha M*, Aoki S 38

Effect of surface traction on the shakedown limit under moving surface loads,

39

Elasto-plastic finite element analysis of tapered steel silo,

192 Shiau J* 193

Xu CL*, Luo YF, Song HJ 40

Numerical simulation of failure process in heterogeneous elastoplastic materials,

194

Li YZ, Chen YQ* 41

Fatigue damage analysis of reactor vessel model under repeated thermal loading,

195

42

Takagaki M*, Toi Y, Asayama T Interaction of moving interfacial cracks between bonded dissimilar elastic strips under antiplane shear, Qas £*

196

43

Stress intensity factor of a wide range of semi-elliptical partly through-wall crack in a finite-thickness plate, KOU £/>*

197

44

Numerical simulation of the fracture spacing in two-layered material subjected to thermal and mechanical loading, Li LC*f Tang CA, Liang ZZ

198

45

VCCM rule-based meshing algorithm for an automatic 3D analysis of crack propagation of mixed mode, Murotani K*, Yasuyoki K, Fujisawa T, Yagawa G

199

46

The state-of-the-art methodology to compute 3-D stress intensity factors for arbitrary shaped cracks in complex shaped structures,

200

Okada H*, Yagawa G, Kawai H, Shijo K et al 47

3D crack propagation analysis using free mesh method,

201

Osaki H*, Matsubara H, Yagawa G 48

Finite element method for analyzing stress intensity factor of a surface crack in tubular joints, Shao YB*, Du ZF, Hu WD

202

49

The solutions of stress and displacement fields of orthogonal anisotropic plate with edge-crack, Tian ZR*, Sun Z, Li ZY

203

Computational Structural Mechanics 50 51

Nonlinear FE model for RC shear walls based on multi-layer shell element and micro-plane constitutive model, Miao ZW, Lu XZ*, Jiang JJ, Ye LP Numerical modelling and simulation of an internal combustion engine piston with a surface coating, Niezgoda T, Kurowski Z, Malachowski J*

204 205

52

Reliability analysis using saddlepoint approximation,

206 Wang J*, Yuen KV, Au SK

53

Advanced computational method for reliability analysis of concrete-faced rockfill dam,

207

Wu QX*, Zhao KZ, Yang MZ 54

Structural dynamic reliability of solid rocket motor grains,

208 Zhang SJ*, Ren JG

55

Wave propagation in orthotropic elastic shells: theoretical and numerical modeling,

209

Tie B*, Aubry D 56

Study on the criterion of in-plane instability of non-reinforced U-shaped bellows,

210

Chen Y* Gu BQ 57

Coupled thermal-dynamic stability analysis of large-scale space structures by FEM,

211

Li W*, Xiang ZH, Xue MD 58

Numerical modelling of the lateral-torsional buckling of stainless steel I-beams: comparison with Eurocode 3,

212

Lopes N*, Vila Real PMM, Simoes da Silva L, Mirambell E 59

Research on simulation analysis for stability problem of pressure-penstock with imperfection, Meng WY*, Li XQ, Zhuo JS

213

60

Nonlinear finite element buckling analysis of square reinforced concrete long columns confined with carbon fiber reinforced plastic sheets under axial compression,

214

61

Ren QX*, Chen TG, Huang CK, Liu YH A kind of channel-section beam element for transient coupled thermal-structural dynamic analysis, Duan J*, Xue MD, Xiang ZH

215

62

Semi-analytical analysis of super tall building bundled-tube structures,

216 Gong YQ*, Li K

63

Computational design of beam sections under impact loading,

217

Hou SJ*, Li Q, Long SY, Yang XJ 64

Pretension control of the long-span roof structure of South Shanghai Railway Station,

218

Huang Y*, Luo YF, Yu R 65

Damage detection and sensor placement design for two highway bridges,

219

Li YQ, Xiang ZH*, Zhou MS, Swoboda G, Cen ZZ 66

A new iterative method for solution of rectangular elastic structure,

220 Lin FY*

67

Overall stability of the long span steel roof structure of Hangzhou International Conference Center, Luo YF*, Liu X, Wang ZB

221

68

Regionwise modeling approach for the analysis of layered structures,

222

69

Mohite PM, Upadhyay CS* Computational and experimental study of energy absorption metter by composite structures, Niezgoda T*, Barnat W

223

70

The dynamic analysis of main building of Hangzhou International Conference Center,

224

Song HJ*, Luo YF, Xu CL, Yang MW

71

Second-order analysis for steel frame structures with a distributed plasticity numerical model, Wang K*t Tong L w, Li T

225

72

Pretension simulation of the long span truss string supported by the temporary structures,

226

Yu R* Luo YF, Huang Y 73

Accurate form-finding method for cable-dome structures based on catenary element,

227

Zhao XZ*, Tang RW, Shen ZY 74

Higher order modes in thin-walled beam analysis,

228

75

Vieira R*, Virtuoso F, Pereira E Research on rigidity limits of bridge with conventional spans for Chinese high-speed railway, Gao MM*, Pan JY, Yang YQ

229

76

Numerical implementation and calibration of a hysteretic model for cyclic response of end-plate beam-to-column steel joints under arbitrary cyclic loading,

230

Nogueiro P*, Simoes da Silva L, Bento R Finite Element Analysis 77

Finite element analysis of singular inplane stress field around an inclusion's corner tip,

231

Chen MC*, Ping XC 78

Finite element analysis for the metallic gasket effective width,

232

79

Feng X*, Gu BQ, Liu R Finite element analysis of electrochemical-poroelastic behaviors of conducting polymer (PPy) films, Jung WS* Toi Y

233

80

Finite element analyses of multi-material wedges and junctions with singular antiplane stress field, Ping XC*, Chen MC, Xie JL

234

81

Plane strain finite element analysis of a piled bridge abutment on soft ground,

235

Wang HT*, Chen ZP, Xiao U 82

Finite element analysis of a coal liquefaction reactor during lifting,

83

Finite element analysis of welded cruciform joint,

236

Wang ZB* Luo YF, Liu X 237 Wu AH*, Syngellakis S, Mellor BG 84

Static and dynamic testing of the SATUOeiras viaducts,

238

Xu M*, Santos LO, Rodrigues J 85

Finite element approach to resin flow during the resin film infusion process,

86

Numerical simulation of a new complex FRP pipe culvert by FEA,

239

Yang M, Yan SL* 240 Yang MW* Heat Transfer and Temperature Related Problems 87 88

Numerical study of two-dimensional transient heat conduction using finite element method, Choi LY* Keong WS, Woon OH, Kiong SC Chemical reaction, heat and mass transfer on nonlinear MHD boundary layer flow through a vertical porous surface with thermal stratification in the presence of suction, Kandasamy R*, Periasamy K, Sivagnana Prabhu KK

241 242

89 90 91 92

Development of a mathematical model for heat and mass transfer inside a granular medium, Petty VJ*, De Bortoli AL, Khatchatourian O Factor Analysis for convective heat transfer problem by using the ANN method, Tarn HK*, Tarn SC, Ghajar AJ, Tarn LM Thermally induced mechanical changes around a potential nuclear waste repository in china > Liu YM*, Wang J, Cai MF, Wang SR Research on thermo quality transmission problems for large-scale slab with creep,

243 244 245 246

93

Wang JX*, Wang XC Current developments on the coupled thermomechanical computational modeling of metal casting processes, Agelet de Saracibar C* Chiumenti M, Cervera M

247

94

Application of the mushy cell tracking method for Gallium melting,

248

Liang SJ*, Jan YJ, Chung MS High Performance FEM and SBFEM 95 96

New FDM for plane elasticity in polar coordinate, Zhu BQ*, Zhuo JS, Zhou JF Computational strategies for curved-side elements formulated by quadrilateral area coordinates (QAC), Cen S* Song DP, Chen XM, Long YQ

249 250

97

Studies of 4-node membrane element with analytical stiffness-matrix based on the quadrilateral area coordinates (QAC), Du Y*, Cen S Special hybrid multilayer finite elements for 3-D stress analyses around quasi-elliptic hole in laminated composites, Tian ZS*, Yang QP, Zhang XQ

251

98 99

A 3-dimensional assumed stress hybrid element with drilling degrees of freedom,

252 253

Wang AP*, Tian ZS, Zhang XQ 100 Suppression of zero-energy modes in hybrid finite elements via assumed stress fields,

254

Zhang CH*, Wang DD, Zhang JL 101 Analysis of concentrated boundary loads in the scaled boundary finite element method,

255

Vu TH*, Deeks AJ 102 A frequency-domain approach for transient dynamic analysis using scaled boundary finite element method (I): approach and validation, Yang ZJ*, Deeks AJ, Hong H

256

103 A frequency-domain approach for transient dynamic analysis using scaled boundary finite element method (II): application to fracture problems, Yang ZJ, Deeks AJ*, Hong H

257

Inverse Problems 104 A novel fuzzyexpert diagnosis system of inner-faults for three-phase squirrel cage induction motors, Cheang TS*, Chan SL, Sekar BD, Dong MC

258

105 Updating noise parameters of Kalman filter using Bayesian approach,

259

Hoi KI*, Yuen KV, Mok KM 106 Damage detection of vibrating structure from limited natural frequencies,

260

Li XL*, Okuda H, Yagawa G Meshless Methods 107 Error estimations in LBIEM and other meshless methods, Chen HB*, Fu DJ, Guo XF, Zhang PQ

261

108 Parallel computing for enriched free mesh method (EFMM),

262 Kobayashi Y*, Yagawa G

109 3D animation for free mesh method,

263

Nagaoka S*, Inaba M, Yagawa G 110 A stabilized conforming integration procedure for Galerkin meshfree analysis of thin beam and plate, Wang DD *

264

111 Element-free Galerkin method with wavelet basis,

265 Liu YH*, Liu YN, Cen ZZ

Micromechanics and Intelligent Materials 112 The numerical prediction of effective properties of non-continuous carbon nano-reinforced composites by the macro-microscopic homogenization method,

266

Luo DM*, Wang WX, Takao Y, Kamimoto K 113 Molecular dynamics simulation of length size effect on mechanical properties of nano-metal, Huang D * Zhuo JS

267

114 Molecular dynamic simulations of CNT-water nanostructures,

268

Zou J*, Feng XQ, Ji B, Gao H 115 Computer simulation of quantum dot surface under stress,

269 Liu XM, Zhuang Z*, Zhang T

116 3D BEM for piezoelectric solids of general anisotropy,

270

Denda M*, Wang CY 117 Analysis of quantum dots induced strain and electric-field in piezoelectric semiconductor substrate of general anisotropy, Wang CY*, Denda M, Pan E

271

Numerical Algorithms 118 Adaptive under-frequency load shedding scheme by genetic algorithm,

272

Lou CW*, Dong MC, Wong CK 119 An effective computer generation method for the domain with random distribution of large numbers of heterogeneous grains, Yu Y* CuiJZ Han F

273

120 Three-dimensional mesh generation using the crossed circle method,

274

Suzuki H*, Ezawa Y 121 Study on displacement prediction of landslide based on grey system and evolutionary neural network, Qao jy*

275

122 Prediction of ambient PM10 concentration with artificial neural network,

276

Lam LH*, Mok KM 123 A note on the complexity of the PCG algorithm for solving Toeplitz systems with a Fisher-Hartwig singularity, Vong SW*, Wang W, Jin XQ

277

124 One-point integration that handles shear-locking in cubic splines,

278 Wang SM*, Zhang YS

125 An improved ICCG method of large-scale sparse linear equation group, Zhang YJ*, Sun Q

279

126 A parallel computing method of object-oriented FEM based on substructure,

280

Zhao HM*, Zhang K, Dong ZZ 127 Promotion of frontier science research by aid of automatic program generation technology, Wu BX*, Qian HS, Wan S

281

128 Uniformed NURBS surface deformation subject to boundary conditions,

282

Lo KM*, Yang ZX 129 The pseudo-spectral method and Matlab implement,

283 Wang SL, Wu ZR*, Cheng YL

Rock, Soil and Concrete 130 Composite construction in reinforced concrete taking into consideration the non-rigid bond of interfaces in joints, Lindig V* 131 Optimization of observation condition on inverse analysis for identifying corrosion of steel in concrete, Suga K*, Ridha M, Aoki S 132 A study on temperature distribution in a cross section of concrete box girder bridges,

284 285 286

Tan YP*, Han DJ L33 Stress-based effective space anisotropic damage model for concrete,

287

Wu JY* Li J [ 34 Identification of electric conductivity and impedance of reinforced concrete by boundary element inverse analysis, Yoshida M*, Suga K, Ridha M, Aoki S, Amaya K

288

L35 Stresses and cracking caused by hydration heat in massive concrete structures,

289

Zhang ZM*, Song ZT, Zhang Y [ 36 Numerical modeling of consolidation of marine clay under vacuum preloading incorporating prefabricated vertical drains, / / 0 ^M Lok TMH*

290

[37 Drag forces applied on rock matrix by fluid flow through fracture network in rock mass,

291

Chai JR* [38 A 2-D natural element model for jointed rock masses,

292

Yu TT*, Ren QW [39 Numerical implementation of a bounding surface bubble model for structured soils, 293 MaranhaJR*, VieiraA [40 Numerical simulation of nonlinear interaction of soil, superstructure and thick raft with 294 irregular plan, Du YF* Di SK, Li H, Song Y, Dang XH [41 Soil additionally affected by non force loading and its influence on upper structure,

295

Kuklik P*, Broucek M [42 Advances in unsaturated soil mechanics,

296 Mi ZK*, Shen ZJ

[43 Numerical simulations of the behavior of foundations on reinforced soil,

297

Tou CM, Lok TMH* 144 Stress-strain modeling of tire chip-sand mixture,

298 Yu HJ* Lok TMH

Structural Optimization 145 Multi-objective optimization for shape design of arch dams,

299

Sun LS*, Zhang WH, Xie NG 146 Optimal shape control of multilayered piezoelectric smart plate structure,

300

Wang JG*, Ding GF, Qin Y 147 Engineering structural optimization with an improved ant colony algorithm,

301

Gong YB*, Li QY 148 Optimization studies for crashworthiness design using response surface method,

302

Liao XT*, Li Q, Zhang WG 149 Path optimization of large-scale automated three-dimensional garage based on ant colony algorithm, jj*f Yang ZQ, Peng ZR Meng

303

150 A continuous approach to discrete structural optimization,

304

Tan T* Li XS 151 Parametrical analysis and optimization of partial double-layer reticulated shells using uniform design method and second order rotation method, Xiao JC*, Liang T, Liu Y

305

152 Optimum design of spiral grooved mechanical seal based on thermo-hydrodynamics,

306

Zhou JF*, Gu BQ 153 Evolutionary topological design of frame for impact loads,

307

Chen XY*, Li Q, Long SY, Yang XI 154 Topology optimization of space vehicle structures considering attitude control effort,

308

Kang Z*, Zhang C 155 Application and research of structure topology optimization of scraper conveyer with MSC.Nastran.optishape, Sang JB*, Liu B, Xing SF, Yang LC, Qie YH

309

156 Structural topology optimization using level set method,

310

Wang MY* 157 Topological optimization analysis of 3-D continuum structure with stress and displacement constraints, ye HL*, Sui YK

311

Topics of Computer Software Technology 15 8 Cross-level sentence alignment,

312

Ho A*, Oliveira F, Wong F 159 CSAT: a Chinese segmentation and tagging module based on the interpolated probabilistic model > Leong KS*, Wong F, Tang CW, Dong MC 160 Overcoming data sparseness problem in statistical corpus based sense disambiguation,

313 314

Oliveira F * Wong F, Ho AN, Li YP, Dong MC 161 Application of translation corresponding tree (TCT) annotation schema for Chinese to Portuguese machine translations, Tang CW*, Wong F, Leong KS, Dong MC, Li YP

315

162 Development of a knowledge based system for the Portuguese code for building acoustics,

316

Graga JM*, Patricio J, Lopes LS 163 Interfacing vision system with robot for pick and place operation,

317 Lalitha R*

164 A web-based data management and decision support system for slope safety inspection and evaluation, Wang j * Hung

318 MC

Vibration, Impact and Control 165 Simplified doubly asymptotic approximation boundary for foundations dynamic analysis,

319

Lei WJ, Wei DM* 166 Limit state analysis of seismically excited 3d r/c beam bearing structures,

320 Kaufinann N*

167 Earthquake response analysis and energy calculation based on wavelet transform,

321

Wu C* Zhou RZ 168 Elastoplastic impact of the sphere upon the Uflyand-Mindlin plate,

322

Lokteva IA*, Loktev AA 169 Numerical analysis of impact between cue and ball in billiard (effect of tip structure),

323

Shimamura S*, Aoki S 170 A new multi-harmonic method for predicting the forced response of mistuned bladed disks with dry friction damping, He EM> Wang HJ*

324

171 Coupled vibration analysis of multiple launch rocket system by finite element method,

325

Li BS, Xu XQ* 172 Three-dimensional vibration analysis of functionally graded material rectangular plates by Chebyshev polynomials, Li g* Iu VP> Kou KP

326

173 A hybrid elasticity method for bending and free vibration of composite laminates,

327

Lu CF, Chen WQ* 174 Vibration assessment of railway viaducts under real traffic using bridge-track models,

328

Rigueiro C, Rebelo C* Simoes da Silva L 175 In-plane vibration of rectangular plates with rectangular cutouts,

329

Shufrin I, Eisenberger M* 176 Optimal control of temperature gradient in a large size crystal growth by response surface methodology,

magnetic Czochralski silicon

330

Yu HP*, Sui YK, Wang J, Dai XL, An GP 177 Active vibration control analysis of piezoelectric intelligent beam,

331

Wang T, Qin R*, Li GR 178 Interval dynamic analysis using interval factor method,

332 Gao W*

179 Sensibility analysis of violin plates,

333 Razeto M*, Staforelli C, Barrientos G

180 Effect of pier and abutment non-uniform settlement on train running behavior,

334

Xiong JZ*, Yu HB, Gao MM Author Index

335

COMPUTATIONAL METHODS IN ENGINEERING AND SCIENCE EPMESC X, Aug. 21-23, 2006, Sanya, Hainan, China ©2006 Tsinghua University Press & Springer

Enriched Element Method and Its Applications to Solid Mechanics Genki Yagawa **, Hitoshi Matsubara2 1

Center for Computational Mechanics Research, Toyo University, 2-36-5, Hakusan, Bunkyo-ku, Tokyo, 112-8611 Japan 2 Center for Computational Science and Engineering, Japan Atomic Energy Agency, 6-9-3 Higashi-Ueno, Taito-ku, Tokyo, 110-0015 Japan Email: [email protected], [email protected] Abstract In the present paper, we discuss the accuracy improvement for the free mesh method (FMM): a node based finite element technique. We propose here a scheme where the strain field is defined over clustered local elements in addition to the standard finite element method displacement field. In order to determine the unknown parameter, the Hellinger-Reissner Principle is employed. The motivation of the study is to seek the better accuracy in the FMM. Key words: enriched element method, free mesh method, Hellinger-Reissner principle INTRODUCTION As is well known, mesh generation for the finite element analysis[l, 2] becomes very serious and time consuming if the degree of freedom of the analysis model is extremely large and the geometries of the model are complex. In order to overcome the above issue of the FEM, the so called mesh-free methods [3,4] have been studied. The Element-Free Galerkin Method (EFGM) [5,6] is one of them with the use of integration by background-cells instead of by elements, based on the moving least square and diffuses element methods. The Reproducing Kernel Particle Method (RKPM) [7,8] is another mesh-free scheme, which is based on a particle method and wavelets. The general feature of these mesh-free methods is that, contrary to the standard FEM, the connectivity information between nodes and elements is not required explicitly, since the evaluation of the total stiffness matrix is performed generally by the node-wise calculations instead of the element-wise calculations. On the other hand, a virtually mesh-free approach called the free mesh method (hereinafter referred to as "FMM") [9,10] is based on the usual FEM, and has a cluster of local meshes and equations constructed in a node-by-node manner. In other word, the FMM is a node-based FEM, which still keeps the well-known excellent features of the standard FEM. Through the node-wise manner of FMM, a seamless flow in simulation procedures from local mesh generation to visualization of the results without user's consciousness has been realized. The method has been applied to solid/fluid dynamics [11], crack problems [12], concrete problems [13], and so on. In addition, in order to achieve a highly accurate FMM, development of the FMM with vertex rotations was studied [14,15]. BASIC CONCEPT OF FREE MESH METHOD The basic concept of FMM starts with only the nodes distributed in the analysis domain ( Q ) , without the global mesh data, as following equation. />,(*,, v , , 0

V ; .e{l,2,-..,m}

(1)

where m is the number of node, /?,(*;,>>/) are the Cartesian coordinates, and rt is the nodal density information which is used to generate appropriate nodes as illustrated in Fig. 1(a). From above nodal information, a node is selected as a — 15 —

Other nodes Central node Local elements Satellite nodes (a) Domain of analysis

n - in -/* *

y

-'-

Stiffness matrix for ej

-P< &

(b) Clustered local elements

V

Assembling

Stiffness matrix for t'j

p,

S/

S: S* Si Si

S6

Stiffness matrix for />,-

(c) Stiffness matrix for central node Figure 1: A 3D model with spherical particles Concept of Free Mesh Method central node and nodes within a certain distance from the central node are selected as candidate nodes. This distance is usually decided from the prescribed density of the distribution of nodes. Then satellite nodes are selected from the candidate nodes, which generate the local elements around the central node (show in Fig. 1(b)). For each local element, the element stiffness matrix is constructed in the same way as the FEM, however in FMM, only the row vector of stiffness matrix for each local element is necessary. The local stiffness matrix of each temporary element is given by kej=[kPi

ks

ksJ

(2)

where k^ is the row vector of the stiffness matrix for element e, and k^ , k 5 and k 5 are components for node of pi, Sj and Sk (j and k are number of current satellite nodes). Through the above procedures are carried out for all local elements, the stiffness matrix for central node is given by

*„=2X where kp

(3) is the stiffness matrix for central node /?,, and ne the number of local elements. Through the above

procedures for all nodes is carried out, the global stiffness matrix is given by assembling k

which are computed by

node-wise manner. ~km K=

PI

(4)

Brief of the nodal stiffness matrix is shown in Fig. 1(c). After the construction of the global stiffness matrix, a derivation of the solution is processed. The great advantage of FMM is that the global stiffness matrix can be evaluated in parallel with respect to each node through the node-wise manner, and only satellite node information is required with each nodal calculation. Finally, a derivation of the solution is performed as the usual FEM. Thus, the FMM is a node-wise FEM, which still keeps the well-known excellent features of the usual FEM. The features of FMM are summarized as follows, (1) Easy to generate a large-scale mesh automatically (2) Processed without being conscious of mesh generation (3) The result being equivalent to that of the FEM

DERIVATION OF ENRICHED FREE MESH METHOD On the other hand, "Assumed strain on the clustered local elements" is the concept of the EFMM: a revised version of the FMM, as shown in Fig. 2. In the EFMM, the strain field on the clustered local elements and the displacement field — 16 —

^

Mixed

Element-wise displacement

fields

Node-wise strain field

Figure 2: Concept of Enriched Free Mesh Method of each local element are assumed independently. In order to combine these two independent fields, we discuss the approach based on the Hellinger-Reissner principle. In the EFMM based on the Hellinger-Reissner principle [1][17] (hereinafter referred to as "EFMM-HR"), the Hellinger-Reissner (hereinafter referred to as "HR") principle is employed to obtain better accuracy. Let the HR principle of a linear elastic body be defined on the clustered local elements by 1 n(«,«)= [i{€}T[D]{du}dQ--\n{e}T[D]{e}dn-

J[{i/}r{b}dCl-

[g{u}T{i}dS

(5)

where {du} = [B]{u} , {e} = [N']{e}

(6)

with {b} being the applied body force per unit mass, and {t} the applied traction on boundary Sa. {u} is the unknown nodal displacement and {c} the unknown nodal strain. The unknown values ( U , E ) of the HR principle satisfy the following equations in a weak manner, lS{e}T[D]([B]{u}-[N]{e})dil £S{u}T[B]T[D][N]{u}dQ-

=0

(7)

k>0 w < y-\y0x\ is valid. By expanding the fundamental functions into Taylor series, the first integral in Eq. (1) can be evaluated as (2) where Mt is the multipole moment centered at y0, and ft is a regular function related to the vector

y0x.

2. First page Multipole to multipole translation (M2M) The multipole moment can be shifted into a new one centered at y'0 defined as

^ U ) = Z^,(>'o>'o)M'(^)

(3)

i

where y.. is the coefficient of M2M translation. 3. Multipole to local translation (M2L) Assume the inequality \x'0x\ < My'0x\ is valid, the multipole moment can be translated into local moment centered at x'0 defined as

AW)=Z^(>'^)A/'(>'»)

(4)

where u is the coefficient of M2L translation. 4. Local to local translation (L2L) Similar to M2M, the local moment can be shifted from xf0 to x0 as follows:

AW)=Z4(>«)Û)

(5)

where £u is the coefficient of L2L translation. 5. Local expansion The first integral in Eq. (1) can be finally evaluated by

— 20 —

\T:,{x,y)ufi{y)dS{y)= £*,(*o*K(*J

(6)

where g{ is a regular function only related to x0x. The second integral in Eq. (1) can be evaluated in a similar formula. 6. Numerical implementation of fast multipole method The algorithm of fast multipole method consists of 6 steps which in all form the matrix-vector products. 1) Tree Construction: The fast multipole method utilizes the tree structure, which is constructed hierarchically. The root of the tree represents a box that contains all the elements and is at level 0 of the tree. The root is divided into 4 equal-sized child-boxes that are at level 1. Each of these boxes is divided into 4 more child-boxes until every box contains at most some fixed number of elements. The shape of the tree adapts to the distribution of elements because the tree will have more levels in regions having more elements. The resulted tree is called an adaptive quad-tree. An example is shown in Fig. 2, where the tree is constructed on the model of a square plate with 100 randomly distributed inclusions of different sizes. 2) Forming multipole moments of leaves: Leaves are the boxes at the finest levels. For each non-empty leaf, its multipole moments are formed by elements it contains.

oC?°oO 0 ° ° 0 Figure 2: Model of a square matrix with random inclusions (left) and its corresponding tree (right) 3) Upward Stage: For each non-leaf box, the multipole moment is formed by shifting the multipole moment of its children using Eq. (3) and adding up the four shifted multipole moments. This procedure is repeated up to level 2. 4) Downward Stage: For each box starting from level 2, the multipole moment of boxes in the interaction list, as depicted in Fig. 3, is transformed to this box's local moment using Eq. (4). Then the local moment is shifted to its children's one using Eq. (5), and the multipole moment in the interaction list of that children level is transformed using Eq. (4) and added on it. This procedure is repeated down to the finest level.

^

&

&

&

#

&

%

&

&

#

#

e- □

Interaction List of box X at level 3 Interaction List of box X' at level 2

Figure 3: Definition of interaction list (Boxes with dashed lines indicate empty boxes) 5) Evaluation of the integrals in Eq. (1): The integrals in Eq. (1) are evaluated in 2 parts. For each source point, the contributions from elements in the same leaf and neighbor leaves are evaluated directly in the way as conventional BEM. Contributions from others are evaluated using Eq. (6). 6) Update of the iterative vector: The iterative vector is updated and the next iteration is started from Step 2. — 21 —

7. New version of fast multipole method The new version of FMM with exponential expansion is developed in the late 1990s by Greengard and Rokhlin. In procedures of the new version, the dense translation operator from multipole moments to local moments in Eq. (4) is diagonalized by three new steps. 1) Multipole to exponential shift: In order to obtain a new series of exponential moments by multipole moments in the same cube. 2) Exponential to exponential shift: In order to transfer exponential moments from one cube containing field points to another cube containing source points. 3) Exponential to local shift: In order to obtain local moments by exponential moments in the same cube. Since operations needed in FMM to obtain local moments by multipole moments dominate most computer resources for 3D problems, successful simplification of these operations makes the new version FMM much effective than the original one. ACCURACY AND EFFICIENCY OF FMBEM To verify the accuracy and efficiency of the FMBEM, a series of test examples have been computed. 1. Accuracy of FMBEM related to the orders of multipole and local expansion A square sheet with a circular hole in the center under uniform normal displacement along 4 edges as shown in Fig. 4 (plane stress problem) is tested. The side length / = 10cm, the radius of the hole r= 2 cm, prescribed normal displacement un = 0.001 cm. All boundaries are discretized into 4,720 DOF.

■mttmiittmmr \f

4

(3 ; !

zn

^3

fc'sinoo.WAi v-o.t

' ♦ T » » f f * » T » » ' f » » * » l ' t

Figure 4: A square sheet with a circular hole in the center under uniform normal displacement along 4 edges The numerical results of normal displacement at three location of hole boundary, using FMBEM with different orders of multipole and local expansions, are listed in Table 1, in comparison with conventional BEM. The comparison has shown that the FMBEM can obtain as accurate results as the conventional BEM provided enough orders of multipole and local expansions are taken. Table 1 Numerical results of FMBEM in comparison with conventional BEM un mm (0=0°)

relative error

« n mm (0=45°)

relative error

un m m (0= 90°)

relative error

Gauss elimination

-0.009191605

—

-0.009374642

—

-0.009191605

—

FMM (p=20)

-0.009191627

2xl0-6

-0.009374664

2x10~ 6

-0.009191607

2xl0-7

¥MM(p=\5)

-0.009191628

2xl0-6

-0.009374664

2xl0-6

-0.009191620

lxlO-6

¥MM(p=\0)

-0.009187775

4xl0-4

-0.009374608

3xl0"6

-0.009187732

4xl0-4

¥MM(p=5)

-0.009218128

3xl0-3

-0.009401706

2xl0-3

-0.009220656

3xl0-3

2. Accuracy of large-scale computation using FMBEM The plane strain model of a square with a circular inclusion in the center (the left of Fig. 5) is periodically repeated to form larger computational models: 2x2 model with 10,880 DOF (the right of Fig. 5) and 30x30 model with 696,000 DOF. The length / = 10cm, the radius of the hole r= 2 cm. The — 22 —

material properties are: Em= 1000MPa, v m = 0.3 for the matrix, and E\= 5000MPa, vi= 0.3 for the inclusions. The prescribed normal displacements are proportional to the side length of the square: un = 0.002 cm for 2x2 model and wn = 0.03 cm for 30x30 model respectively. In this way the traction and the normal displacement relative to each inclusion center along the inclusion boundary should be the same for different models. The numerical results (Table 2) have shown high accuracy of the large-scale computation using FMBEM.

inclusion

!

/■

( ■9 matrix

Figure 5: A square with a circular inclusion in the center (left) and the corresponding 2x2 model Table 2 Numerical results of 2x2 model and 30x30 model FMM (p=20)

un (mm)

/n (MPa)

error

0=0°

2X2

-0.0112337

30X30

-0.0112334

2X2

0. 552577

30X30

0.552595

2xl0-5

2xl0-5

0=45° -0.0152808 -0.0152802 0. 547241 0. 547221

error

error

0=90° -0.0112337

3xl0-5

2xl0-5

-0.0112334 0. 552578

3x10~ 5

2x10~ 5

0. 552595

3. Efficiency and convergency of FMBEM The computation time and memory requirement using FMBEM for different problem scale, are shown in Fig. 6 in comparison with those using conventional BEM. For the FMBEM computation the order of multipole and local expansions is taken as p = 20, and the tolerance for the iteration e- lxlO 5 , and for the conventional BEM computation the package LAPACK is applied for the Gauss elimination. Computation Time (sec)

Memory Requirement (MB)

, 800

Conventional BEM

60O

.,,

,

r.

,

" » Conventional BEM

"

-

400 a 20O

FMBEM

FMBEM 5000

1O000

R - A « f , i A ( ,«£»„„ 5000

15000

Degree of Freedom

■ £

--

At-A

,

A

100O0

,

*

■*

15000

Degree of Freedom

Figure 6: The comparison of computation time and memory requirement between FMBEM and BEM The convergency of the FMBEM is shown in Fig. 7, the order of multipole and local expansion is taken as p=20. The results have shown good efficiency and convergency of FMBEM. — 23 —

Residual _j

j i

01 a

0 01

, , ,

r

C A

DOF=2400 | J)OK=280TW)i

_

A a

^ A A A

1E-3 °

A A

Ifc-b

a

C

1E-6

D

A D A

IE-7 1E-8

•

1

>

1, , , ,

i

1

.n ,

20

,

'A

30

Iteration Number Figure 7: The convergency of the computation using FMBEM APPLICATION OF FMBEM ON SIMULATION OF COMPOSITE MATERIALS The FMBEM has been applied in the 2D and 3D simulation of composite material and the obtained equivalent material properties have shown good agreement with the available results in micromechanics. But the computation using FMBEM can obtain not only the equivalent properties, but also the whole fields of displacements and stresses, which is useful for the further investigation of the failure process of the composite materials. 1. 2D simulation of long-fiber reinforced composite material Long-fiber reinforced composite material can be simulated using 2D model of plane strain. Fig. 8 shows two models of square containing 1600 randomly distributed circular inclusions discretized into 544,000 DOF. Fig. 9 shows the results of equivalent volume modulus obtained using FMBEM in comparison with that obtained using Mori-Tanaka method.

W

M

msmm %s^M*Mk%

Figure 8: Models of square containing 1600 circular inclusions, volume fraction = 0.2 (left), 0.4 (right) 1

1

1

1

1

1

1 -—

i—i

-1—

-T

J

-i

-

|

1

—l

30

"

A

FMBEM ■O

25

.XI'"'" -"'..A--"'

::2V*

-

1.6

1

1

r

"

-

lf>

-

f

-

1

1

1

"

— a — Mori-Tanaka —A— FMBEM

"

CJV

/A

,^'

1.4

/A 20

1

r-

1.5

J ja' . „&' A?"

"

i

' — l —

—O— Mon-Taiiaka

_

J?

1.3

"

..

1

«

1

«

1

«

1

•

1

fiber volume fraction c (%)

Figure 14: The comparison of computation time and memory requirement between FMBEM and BEM As the aspect ratio of fibers increases, for example aspect ratio equals 11 as shown in Fig. 15, the representative volume element will not be big enough. In such case larger-scale computation will be required, and parallel computation is necessary.

lilSfSllI

-"S^MMsi

Figure 15: Models containing 100 longer fibers with different orientation angle: 0° (left), 30° (middle), 90° (right) — 26 —

LARGE-SCALE PARALLEL COMPUTATION OF FMBEM The FMBEM is more difficult to parallelize efficiently than the conventional BEM because of two extra key problems. One is that the elements are non-uniformly distributed and hence the tree is imbalanced. The other is that the tree is a connected data structure that is harder to traverse than simple arrays. Therefore some special approach has to be adopted. 1. Task decomposition by boxes Instead of domain decomposition method (DDM) commonly used in conventional parallel BEM, box decomposition is adopted in the parallel algorithm of FMBEM. The domain is decomposed while splitting the tree by the unit of boxes. The boxes are first sorted in a 2D or 3D space order by a sorting algorithm presented in Ref. [13]. Then each task is assigned with maximal approximately even number of elements by a parameter called decjevel, which stands for the level the tree is decomposed at. The higher the declevel is, the more balanced the decomposition will be. But it cannot be higher than the minimum level of leaves in the tree. Fig. 16 depicts decomposing the model in Fig. 2 into six tasks at dec_level=3. The main advantage of this partition scheme is that the tasks can be decomposed regardless of different phases such as the inclusion and matrix for particle-reinforced composite.

BilU ffi 1

3rfm

If ¥ rCn R-

®rw -fmfOiUF

Wflrefc

Figure 16: Task decomposition boxes Task /

Task /

r >> KJ

Request

^ ^

%

-4

>

r >! r >i r >j L) LJ K ) !

tc ^j M

»

Return

ww w ; j Interaction List of X' box

Figure 17: Remote checks in parallel tree traversing (Empty boxes are not returned) 2. Parallelization of the tree traversing The most time-consuming step in fast multipole method is the downward stage. It is also the most complicated step in parallel formulations since interactions from interaction list and neighbor boxes are needed. And these boxes may be in the same or other tasks, and may be empty anywhere. So a remote-check procedure is used as a simple example shown in Fig. 17 for the case of interaction list only, where box marked with X in task / need the — 27 —

remote boxes in task/ In that case task / sends a checking request toy and a checking result of non-empty boxes are returned. The overhead of this procedure can be neglected since it costs only a few seconds in a model with tens of thousands unknowns and also the checking result does not change from iteration to iteration for elastostatics. 3. Accuracy verification To verify the accuracy of FMBEM, two models with 1 inclusion centered at the origin of the matrix and 343 (7x7x7) inclusions that are periodically distributed are used, as shown in Fig. 18. The edge length of matrix is 20x20x20(cm) and the inclusion volume fraction is 0.1. Each inclusion is discretized into 392 triangle piecewise constant elements and 120,000 elements for the outer boundary of matrix. The multipole/local expansion order is 12 and the residual of GMRES is l.OxlO"7.

it f *

. ''°'s -

* *d

Figure 18: Model with periodically distributed inclusions and local coordinate of an inclusion (right) The relative displacement u.0) - u\ ' and the traction ry are given in Fig. 19. The slight difference of the last 49 inclusions, which are near the front surface (in +JC direction) of the matrix, comes from matrix discretization error. The results have shown satisfactory accuracy of the parallel FMBEM computation.

Figure 19: The results of relative displacement uy —u^' and the traction rx' for point 1 and 0 of each inclusion 4. Performance and Efficiency of Parallel FMBEM The following computations are carried out on a 16-node SMP PC cluster. The nodes are connected via 1,000Mb Ethernet switch hub. Each node owns 2 Pentium IV Xeon processors and 1 GB memory. C++ is used as the programming language and a message passing interface, LAM/MPI, is used for communications. To show the scalability of parallel FMBEM, 6 models with different number of inclusions are used. The edge length of matrix is 20x20x20 (cm) and the radius of inclusion is 0.45 71 (cm). The matrix is applied with an axial traction of lOOMPa. The inclusions are distributed randomly by using Sobol sequence, as shown in Fig. 20. Total 32 processors are used for those calculations. The multipole/local expansion order is 12 and the residual of GMRES is l.OxlO-5. Table 3 demonstrates the performance and memory requirements of the computation. Both time and storage complexity are approximately proportional to the scale of the problem. For the largest model, the running CPU time is about 43 hours and approximately 9GB memory is required. Fig. 21 shows the computing time versus numbers of processors and the speedup for the parallel FMBEM. — 28 —

, "^"-ij 1".' „ ^ > 3 ? ' .'-■".. ■

/ = lOOMPa

/ =100MPa

r = 100MPa 1000 inclusions DOF=l.536,000

500 inclusions DOF=948,000

100 inclusions DOF=477,600

/ =100MPa

r = 100MPa

2000 inclusions DOF=2.71 2.000

4000 inclusions DOF=5,064.000

3000 inclusions DOF=3,888,000

Figure 20: Models with different number of inclusions Table 3 Performance of parallel FMBEM computation Inclusion number

100

500

1000

2000

3000

4000

Volume fraction

0.005

0.025

0.05

0.1

0.15

0.2

Number of DOF

477,600

948,000

1,536,000

2,712,000

3,888,000

5,064,000

Memory requirement (MB)

1,976

3,070

4,067

5,989

7,543

9,182

Computing time (s)

4,660

15,179

32,136

71,636

103,635

161,190

90000

ideal adlkJ speedUfJ ---

\>

X . . - - ■ '

,

70000

1_

60000 1

50000

i

r,x'

../" ,>

,.y

f g.

/-'""

, , ■ ■ • " - "

40000

,

-

-

■

'

30000 20000

\

10000

-.... 4._.._.

-.._ 10

12 14 16 18 20 Number of processors

22

24

26

29

M

32

y

/•"' 18 20 Number of processors

22

24

26

28

30

32

Figure 2 1 : The computing time versus number of processors (left), and the speedup for parallel F M B E M 5. Application on the simulation of fiber reinforced composites Two fiber shapes, bone-shaped short fiber (as shown in Fig. 22) and conventional straight short fiber, are simulated and compared. Fig. 23 shows a typical configuration of RVE model of well-aligned bone-shaped short-fiber reinforced composites. There are 200 fibers in the RVE, the volume fraction is 0.05. It is discretized into 2,596,800 DOF, and the computing time is approximately 38 hours. Fig. 24 shows the comparison of effective tensile modulus for two types of fiber shapes, and a normalized histogram with fitted Weibull probability density functions. — 29 —

Figure 22: Well-aligned bone-shaped Ni-fiber reinforced polyester matrix fiber composites and a fiber model

Figure 23: The BEM model of a RVE of matrix containing 200 randomly dispersed bone-shaped short fibers ;j Two ryp»s of TiBftr tnapes J ! Bone-shaped short fibers I i Conventional straight short fiber*

£

! T A o t> pes of fiber sr»ape% jBorte-*bep*v

;_

"'

■ : ; •; .;

:,';''';

■-; ;>

;

1 v^

„' ' „:. , -j

Figure 33: A sheet containing 4000 microcracks

Dilute SCM DM Feng-Yu GSCM • Numerical

crack density

Figure 34: Effective in-plane bulk modulus versus crack density

Furthermore, the effect of crack non-uniform distribution on effective in-plane bulk modulus is also investigated using the FMDBEM. This work assumes that the microcracked solid contains some local regions having a crack density coL higher or lower than the average crack density co0 and analyze the variation of the effective in-plane modulus with coL when co0 is fixed. Fig. 35 shows a square sheet containing 4 higher crack density region, and Fig. 36 shows the results of Ke /K0 versus coL /co0 for the case of co0 = 0.3 . The results show that the non-uniform distribution of microcracks increases the effective in-plane bulk modulus of the whole microcracked solid. * Numerical restsJte — A « r a g e iame

*

0*

*.s

Figure 35: A square sheet containing 4 local regions

G.O

as

ia

20

26

Figure 36: Ke/K0 versus a>L/CD0 (co0= 0.3)

CONCLUDING REMARKS Based on the progress of FMBEM, the authors' group carried out a series of investigation on the applications of FMBEM in solid mechanics. The investigations on large-scale FMBEM analysis in solid mechanics, including 2D and — 33 —

3D elasticity and 2D fracture problems, have shown its attractive advantages, high accuracy and efficiency. Combining with FMM the boundary element method become suitable to deal with large-scale practical engineering and scientific problems. The first author has been involved in the research on boundary element methods since 1979. The BEM is regarded as an important complement of the widely-applied FEM, but if the BEM is only capable to obtain the same results as obtained by FEM, such complement was not necessary. For the complement it is important to do something, which FEM could not do, or do something significantly better than FEM. The development of FMBEM have shown good prospects at this aspect. FMBEM have been successfully applied in the field of MEMS design and electro-magnetic field analysis. In the field of solid mechanics, the most important thing is to develop practical applications of FMBEM. The further investigation in authors' group is concentrated on several topics, including FMBEM of elasto-plasticity problems, thin structure problems, dynamic and coupling problems. Acknowledgements Financial support for the projects from the National Natural Science Foundation of China, under grant No. 10172053, 10472051 is gratefully acknowledged. REFERENCES 1. Rokhlin V. Rapid solution of integral equations of classical potential theory. J. Comput. Phys., 1985; 60: 187-207. 2. Greengard L, Rokhlin V. A fast algorithm for particle simulations. J. Comput. Phys., 1987; 73: 325-348. 3. Greengard L, Rokhlin V. A new version of the fast multipole method for the Laplace equation in three dimensions. Acta Numerica, 1997; 6: 229-269. 4. Nishimura N. Fast multipole accelerated boundary integral equation methods. Applied Mechanics Review, 2002; 55: 299-324. 5. Peirce AP, Napier JAL. A spectral multipole method for efficient solutions of large scale boundary element models in elastostatics. Int. J. Numer. Meth. Engng., 1995; 38: 4009-4034. 6. Yoshida K, Nishimura N, Kobayashi S. Application of new fast multipole boundary integral equation method to crack problems in 3D. Engrg. Anal. Boundary Elements, 2001; 25: 239-247. 7. Wang HT, Yao ZH. Application of a new fast multipole BEM for simulation of 2D elastic solid with large number of inclusions. Acta Mechanica Sinica, 2004; 20: 613-622. 8. Wang PB, Yao ZH, Wang HT. Fast multipole BEM for simulation of 2-D solids containing large numbers of cracks. Tsinghua Science and Technology, 2005; 10: 76-81. 9. Wang HT, Yao ZH. A new fast multipole boundary element method for large scale analysis of mechanical properties in 3D particle-reinforced composites. Comput. Model, in Engrg & Sciences, 2005; 7: 85-96. 10. Wang HT, Yao ZH, Wang PB. On the preconditioners for fast multipole boundary element methods for 2D multi-domain elastostatics. Engrg. Anal. Boundary Elements, 2005; 29: 673-688. 11. Wang PB, Yao ZH. Fast multipole DBEM analysis of fatigue crack growth. Comput. Mech., in press. 12. Lei T, Yao ZH, Wang HT, Wang PB. A parallel fast multipole BEM and its applications to large-scale analysis of 3-D fiber-reinforced composites. Acta Mechanica Sinica, in press. 13. Warren MS, Salmon JK. A parallel hashed oct-trees N-body algorithm, in Proc. Supercomputing' 93, Portland, Oregon, US, 1993, pp. 12-21. 14. Liu YJ, Nishimura N, Otani Y. Large-scale modeling of carbon-nanotube composites by a fast multipole boundary element method. Comput. Materials Science, 2005; 34: 173-187. 15. Isida M. Effects of width and length on stress intensity factor for the tension of internal cracked plates under various boundary conditions. Int. J. Fract. Mech., 1971; 7: 301-306. 16. Portela A, Aliabadi MH, Rooke DP. Dual boundary element incremental analysis of crack propagation. Comput. Struct., 1993; 46: 237-247. 17. Feng XQ, Yu SW. Estimate of effective elastic moduli with microcrack interaction effects. Theor. Appl. Fract. Mech., 2000; 34: 225-233. — 34 —


Management of Water Pollutants Based on Multi-Criteria Analysis and Fuzzy Logics Serge Cescotto l*, Marc Roubens 2, Nicolas Rigo 3, Shixiang Gao 4, Xiaodong Wang 4, Aiquan Zhang 4, Nelson Lourenco5, Jiti Zhou 4, Xuemin Xiang4, Joao Paulo Lobo Ferreira6 1

Department M&SANAST, University of Liege, Belgium Department of Mathematics, University of Liege, Belgium 3 Department ANAST, University of Liege, Belgium 4 School of the Environment, Nanjing University, Nanjing, China 5 Faculdade de Ciencias Sociais e Humanas da Universidade Nova de Lisboa, Lisboa, Portugal 6 Laboratorio Nacional de Engenharia Civil, Lisboa, Portugal 2

Email: [email protected] Abstract This work has been developed in the frame of the MANPORIVERS research project funded by the European Commission. The goal is to identify effective and sustainable policies for the management of surface and ground water pollutants, taking account of their relationships with food production and human health. A methodology based on the combination of fuzzy logics and multicriteria analysis is proposed as a decision aid tool for the development of such policies. An example of application in the Huai river basin is given. Key words: Rivers, pollutant, methodology, management, policy OBJECTIVES AND ACTIVITIES The goal of the MANPORIVERS project is to identify effective and sustainable policies for the management of surface and ground water pollutants, taking account of their relationships with food production and human health. The aim is the definition of policies with a very broad range of applicability that could be used for many river basins. They can be used interactively for different basins exchanging water. They can also be used at different scales in a recursive manner, from small tributary basins to large basins. Table 1 Description of the tasks 1 Task 1 number WPl.a IWPl.b

IwPl.c WP2.a |WP2.b WPS.a WP3.b WP4.a WP4.b WP4.c WP4.d WP4.e WP4.f WP5 WP6

Description of the tasks The different tasks develop methodologies for : Evaluation of non accidental pollutant input Evaluation of accidental pollutant input Evaluation of input evolution in the future Analysis and selection of surface water pollutant transport models Analysis and selection of groundwater pollutant transport models Identification and analysis of techniques for water cleaning Identification and analysis of accident remediation techniques Assessment of the use of drink water Assessment of the use of irrigation water Assessment of water used in food industries Assessment of water used for fish breeding Assessment of other uses of water Evaluation of water use in the future Analysis of the relationships of pollutants with health Pollution management priority policies by fuzzy logics and multicriteria analysis

| | ] | | | | | j | j j | | | | I |

The activities presented in this paper are summarized in Fig. 1 and Table 1. Although we mainly concentrate here on task WP6, it is worth giving some information on the work achieved in tasks WPl to WP5 as they constitute a support for WP6. METHODOLOGIES FOR THE ANALYSIS OF FACTS WPL WP2, WP3, YVP4, WP5

1

~

METHODOLOGIES FOR THE CHOICE OF PRIORITIES IN THE MANAGEMENT OF POLLUTION IN RIVER BASINS VVP6

Figure 1: General Organization of the activities METHODOLOGIES FOR THE ANALYSIS OF FACTS 1. Methodologies for the evaluation of accidental and non accidental pollutant input (WPl) [1] Pollutant sources are classified into two categories: non accidental pollutant input and accidental pollutant input. The research work completed consists of two parts: (1) Development of evaluation methodologies for the two categories; (2) Application to the Xuyi County (Huai river basin) and to the Xizhijiang River basin (Pearl River delta). 2. Methodologies for the choice of models for the transport of pollutant by surface water and groundwater (WP2) [2-5] The objective is to evaluate existing models for pollutant transport, both for surface waters and groundwater, as tools contributing to the management policies of water pollutants. 1) Transport of pollutant by surface waters The analysis of different softwares has been performed. The following ones were considered and a methodology for their selection and use developed: (a) Mike 11 DHI Danish Hydraulic Institute from Denmark; (b) U.S. Geological Survey (USGS) from USA: a set of 42 sofwares for different purposes; (c) SOBEK Delft Hydraulics from Netherlands; (d) InfoWorks RS; Wallingford from Great Britain; (e) HEC-RAS (Army Corps of Engineer's Hydraulic Engineering Center (HEC)); (f) River Analysis System (RAS) from USA; (f) WOLF software from Belgium. The required parameters, the basic characteristics and use limitations are examined. 2) Transport of pollutant by groundwater The application capabilities of several flow and pollutant transport models available on the Internet are studied, aiming at creating a methodology for their selection and use: (a) FEFLOW (Diersch, 1998); (b) MT3D (McDonald e Harbaugh, 1988); (c) ASMWIN (Chiang et al., 1998); (d) RBCA Tiers Analyser (Roy et al., 2000); (e) AQUA3D (Vatnaskil Consulting Engineers, 1988); (f) FLOWPATHII (Eviksov et al., 1998); (g) WINTRAN (Rumbaugh e Rumbaugh, 1995). Conclusions on the possibilities, data requirements and accuracy corresponding to these softwares are summarized in tables that help decision makers to chose the appropriate model according to the site to be studied. 3. Methodologies for the choice of sanitation and remediation techniques (WP3) [6] The objective is to identify and evaluate existing techniques to decrease pollution levels in waters, due to accidental and non accidental input in order to support management policy decisions by appropriate selection charts and methodology. 1) Non accidental pollutant input Various sanitation techniques are examined according to the specific pollutants. Their working mechanisms and characteristics, the advantages and disadvantages, the suitable application domains, the equipments and technologies used as well as the costs are considered. The emerging innovative techniques are also identified and their potentials are evaluated. The technical specifications and the financial aspects are also taken into consideration so that they could be applied in real industrial cases. 2) Accidental pollutant input This part recapitulates and analyses some of the major accidents in drinking water as well as surface and groundwater pollution in order to gain a better understanding of the causes of accidental pollutant input. It is found that traffic accidents and vehicle overloads are the two main factors causing the release of some toxic chemicals into water body. A variety of remediation technologies are recommended for hazard minimization, among which chemical and biological methods provide successfully techniques. Biological degradation methods are also holding promising perspectives. — 2 —

4. Methodologies for the evaluation of water use (WP4) [7] In order to develop and apply an effective and sustainable policy for the management of water use, it is necessary to have clear and quantitative responses to the following questions: (a) What is this water used for? (b) Where does this water come from? (c) What are the quantities used for different purposes? (d) How do these quantities evolve in a year time? (e) What is the foreseen evolution in the future? The objective of this work package is to develop methodologies to answer these questions and demonstrate their applicability in the chosen case areas. The following methodologies have been developed: (a) Methodology for evaluating the present water use; (b) Methodology for evaluating water demand in future; (c) Methodology for evaluation of the rationality of water use structure. 5. Methodologies for the evaluation of relationships of pollutants with human health (WP5) [8] The objective is to build a quantitative and qualitative database on the influence of various pollutants in river basins on human health. To reach this goal different complementary approaches have been used. (1) Various human health risk assessment methodologies have been evaluated. The data and information on human exposure to pollutants and their potential health effects have been catalogued. The realistic health risks that environmental pollutants impose on human have been evaluated by comparing their exposure concentration and their toxic potency, leading to a priority list of main pollutants in river basins. (2) Considering that almost all data on health effect and toxicological information are based on animal studies and that there is great difference between animal and human beings, a short-term in vitro Micronuclei Assay based on human peripheral blood lymphocytes micronuclei has been developed and applied to directly evaluate the human health effect of pollutants. Toxic potencies of some aromatic compounds have been tested by this assay. (3) Important efforts have been devoted to develop the Quantitative Structure Activity Relationships (QSAR) models. These are mathematical models that relate the biological activity (e.g. toxicity) of molecules to their chemical structures and to the corresponding chemical and physicochemical properties. This development aims at filling in the data gap for some pollutants. They can be used as indicators of the human health effect of pollutants and also for the development of to predict health effect of other pollutants. METHODOLOGIES FOR THE CHOICE OF PRIORITIES IN THE MANAGEMENT OF POLLUTION IN RIVER BASINS BASED ON MULTICRITERIA ANALYSIS AND FUZZY LOGICS (WP6) 1. The MANPORIVERS methodology [9-11] Based on the combined use of multicriteria analysis and of fuzzy logics, a general methodology has been developed to help decision makers establish sustainable management policies for priority water pollutants and their effects on foods and human health. It is called the MANPORIVERS methodology (Fig. 2). Strategic socio-economic development in a given river basin Action i (i= 1 ,n) Sink j (j=l ,m)

I

Impacts

I

Methodologies for the analysis of facts

Interface

Action data

Basin data

Evaluation tools

Ranking

Figure 2: General Scheme of the MANPORIVERS Methodology This methodology is a major original contribution of this research project and constitutes an efficient tool to help decision makers to undertake environmentally, socially and economically sustainable actions in a river basin. It is applicable to any river basin and takes account not only of the environmental impacts (typically the concentrations of pollutants in water) but also of the social and economical aspects of the problem. The MANPORIVERS methodology is a tool that is able to rank different Actions or Scenarii (i.e. combinations of Actions) in order to maximize their positive effects and minimize their negative effects. — 3 —

The terminology used in Fig. 2 is detailed below. Actions: any type of industrial, economical, political... action or event. Sinks: different entities possibly affected by the impacts of the considered strategic socio-economic development in the region. Action data: specific to a given action (economic, social, pollution data). Regional data: specific to a region, we consider variable regional data (population, migration...) and permanent regional data (geology, rainfall, topography...). Impacts: the different effects of actions, e.g. investment, annual cost, modification of pollutant levels, migration of population, aspect of landscape, ...; each impact must be expressed by a quantifier which can be quantitative (e.g. amount of money invested, pollutant concentration or mass, ...) or qualitative (e.g. good, neutral or bad effect on landscape). Methodologies: all the means (software, data bases, selection charts, recommendations ...) by which the impacts quantifiers of different actions can be obtained. Interface: integration of different impacts in three stages: a geographic integration, an integration within the set of all major pollutants and the integration within the set of sinks. Scenarii: different combinations of Actions and different weights for the social, economical and pollution impacts. Evaluation tools: softwares based on multicriteria analysis and fuzzy logics taking account of the impacts and their respective weights to give a classification of different actions. Ranking: ordering from the best to the worst action. Of course, this global methodology makes use of the specific methodologies for the analysis of facts presented in section 3 above. In must be noted that the MANPORIVERS methodology is capable to take account of: (a) the different uses of water; (b) the effects of pollutants on human health; (c) the direct and indirect costs of Actions (investments, maintenance, functioning,..); (d) the effects of Actions on population (creation or loss of jobs, change of water price, change of water consumption,...; (e) the effects of Actions on landscape, on the quality of life, ...; (f) the coherence and feasibility of Actions. It is able to consider not only deterministic quantitative criteria but also fuzzy criteria as well as qualitative criteria (such as linguistic statements). The MANPORIVERS methodology is also characterized by the fact that it allows decision makers to estimate the robustness of their decision: by modify the weights given to the different criteria, they can see if their decision is modified by slight weight changes (no robust decision) or if, on the contrary, the decision is not affected by reasonable weight changes (robust decision). To use this methodology, it is necessary to be able to evaluate the different scenarii on the basis of different criteria. In others words, it is necessary to quantify or qualify the impacts of the actions on a set of comparison indicators that must be defined by the decision makers. After having quantified/qualified these different impacts (using the specific methodologies presented in the last section), it is possible to use a decision aiding software based on multicriteria and fuzzy logics to select the best scenario according to the weight given to the different comparison criteria. For complex management projects, it is necessary to have a global view, i.e. it is important to consider social, economic and environmental aspects but also, the concept of the coherence and the feasibility of the measures. The selection of the best scenario must be made according to these different aspects. We have to evaluate the different impacts of the measures on the criteria. However, the evaluation of some impacts is quite difficult because of the lack of data or the complexity of the reality. Then, the use of quantitative model is impossible and we have to use qualitative models. This is where fuzzy logics enter into the picture: in classical multicriteria analysis, impacts are normally expressed by figures, usually some amount of money. In the methodology, fuzzy logics has been introduced: it is possible to consider criteria expressed by fuzzy expressions and even to use linguistic scales (see Fig. 3). Furthermore, the importance of the different comparison criteria is also essential in the selection. The decision makers will choose according to their convictions, their points of view. So, the methodology allows them to weigh the different criteria according to their convictions. Then, it is clear that the final choice will be different according to the specific weights given to each comparison criterion. In fact, we can say that the weighting is the mathematical translation of the convictions of the decision makers. — 4 —

I 1

Types of measurement scales or « quantifiers » Quantitative

«■+>

An integer or a decimal number e.g.: 1250 persons, 10,25 kWh Ordinal e.g.: good/neutraMad Nominal e.g.: yes/no; green/red A statement involving uncertainties e.g.: probably between 15 and 20 but certainly not below 12 and not above 24

1

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Figure 3: Examples of Measurement Scales or Quantifiers Among the comparison criteria, the different pollutants of water constitute very important indicators for the final selection of a management strategy for the basin. However, there are many pollutants that are important in the analysis. The large number of pollutants could be an obstacle to the clarity of the reasoning. That is why the MANPORIVERS methodology contains a functionality allowing to "integrate" the effects of the different pollutants. This functionality involves 3 levels of integration: (1) Integration of the impact of a pollutant over the entire basin: this takes account of their potential danger of the pollutant over the considered period of time and of the population potentially affected. (2) Integration on the pollutants : this takes account of the relative toxicities of the different pollutants present in the basin. (3) Integration on the sinks: the sinks correspond to the different fields of water use: consumption, irrigation, fishing, food and domestic use. The methodology proposes a linguistic scale to give a weight to each of them. Thanks to this functionality, it is possible to characterize the pollution of the considered basin by a simple result integrating not only the concentrations of the different pollutants but also their toxicity and the populations potentially affected. For the economic criteria (direct cost, investments, maintenance costs, interest rates,...) classical methods of economy can be used to integrate them in order to get the global cost of an Action. So, this point is considered as classical and well known. For the social criteria and for the criteria of coherence and feasibility, no attempt is made to try to "integrate" them. They are considered one by one. Indeed, the "aggregation" of criteria like number of jobs created or lost, modification of water consumption, change of water price, change of land use,... .does not seem reasonable and is not easily feasible in practice since linguistic scales are often used for some of these criteria. Once all the criteria defined, evaluated on the basis of the methodologies for the analysis of facts and integrated if necessary, a multicriteria table is obtained (see an example in Table 8). Then, a multicriteria decision aiding software can be used. In this research, we use the software Decision Lab using the PROMETHEE methodology. It is a multicriteria decision aiding software including fuzzy logic aspects. It allows the ranking of different Actions or Scenarii taking account of the different criteria and of their respective weights. 2. Integration of pollutants: the global Environment Pollution Risk Criterion (EPRC) Among the comparison criteria, the different pollutants of water constitute very important indicators for the final selection of a management strategy for the considered river basin. However, there are many pollutants that are important in the analysis. The large number of pollutants could be an obstacle to the clarity of the reasoning. That is why the MANPORIVERS methodology contains a functionality allowing to "integrate" the effects of the different pollutants. This functionality is the global Environment Pollution Risk Criterion (EPRC). — 5 —

It involves 3 levels of integration: (1) Integration of the impact of a pollutant over the entire basin: this takes account of their potential danger of the pollutant over the considered period of time and of the population potentially affected. (2) Integration on the pollutants: this takes account of the relative toxicities of the different pollutants present in the basin. (3) Integration on the sinks: the sinks correspond to the different fields of water use: drinking water, irrigation, fish breeding, water for food industries and water for non food industries. The methodology proposes a linguistic scale to give a weight to each of them. Thanks to this EPRC, it is possible to characterize the pollution of the considered basin by a simple result integrating not only the concentrations of the different pollutants but also their toxicity and the populations potentially affected. 1) Geographic integration The goal of this first step is to evaluate the effect of each pollutant on the considered basin. It is suggested to divide the basin in different zones in order to determine the impact of a pollutant on the basin with precision. Actually, it would be dangerous to consider the basin as a whole because, in that case, the calculated impact would show an unrealistic homogeneity. We have to take the diversification of the area into account, particularly for the density of population and for the pollutants concentration. The proposed methodology is based on these basic considerations. The ideal would be to have a division highlighting zones with various pollutants concentrations and densities of population in order to have an optimal representation of reality. Concerning the pollutants concentration distribution and the evolution of these concentrations with time, they can be obtained by the complementary use of pollutant transport models and in field data analysis. Given the pollutant input data, they enable to predict the concentration c(x, y, t) of a given pollutant at point (x,y) of the considered basin at time t. These models can be calibrated with the help of in situ measurements. Geographic integration formula In order to aggregate the impact of a pollutant over the entire basin, we propose the following expression in which the notation « year » indicates the considered year (we cannot forget the evolution with time; the "sustainable development" concept is very important). Iijk (year) = J Dijk (x, y, year) x F™ (x, y, year)dA A

Let us clarify the content of this expression: (1) The subscripts i,j and k represent the action or scenario /, the sinky, and the considered pollutant k. (2) A is the area of the considered basin entire basin. (3) Dijk(x, y, year) defines the « potential danger » of pollutant k for one year. It is based on the concentration of the considered pollutant c,y* (x, y, t) and is expressed in jug 11; it is defined below; (4) Ffnp(xy ,year) is a function of the distribution of population density in the considered year expressed in pers./km2; it is also defined below; (5) The integrated impact is expressed in: pers x fig jl Definition of function Dyk(x, y, year) The definition of the potential danger of a pollutant must be considered with care. A first idea is be to integrate the concentration over 1 year: this gives an idea of the average amount of pollutant to which the population is exposed during the year under consideration. However, this approach erases potential concentration peaks presenting severe danger for human health. That is why we propose to consider another element: the maximum of the pollutant concentration. Then, the questions to answer are: "When should we use the maximum?" "When should we use the average?" Usually, each pollutant is characterized by a critical threshold that cannot be exceeded: over this critical threshold, the consequences for human health could be very serious. This threshold is used as discriminant. Let us consider the two following cases for a given pollutant. First, we consider the case where the yearly trend of the pollutant concentration is quite stable (Fig. 4). In this case, there is no reason to include the maximum of the concentration because it is lower than the critical concentration, and therefore, the average represents fairly enough the reality. — 6 —

Then, we consider the following case (Fig. 5). The maximum concentration of pollutant k is higher than the critical level. By itself, the average does not represent correctly the reality because it doesn't reflect this peak. Then, the use of the maximum of the concentration would be favourable. That is why; we propose the following expression to calculate the potential danger of pollutant k: c(t) ^critical

average

Figure 4: Stable evolution of pollutant concentration

c(t)

average

Figure 5: Evolution of pollutant concentration with peak DiJtk (x, y, year) = a — f cijk (x, yy t)dt 4- 0 max cijk (x, y, t) A^ = lyear

{ccrilical

[(1,0) if

\/t:ciJk(x,y,t) -0.101

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Figure 6: The final ranking of the scenario for the horizon 2005 with all criteria in same weight We can continue the analysis and compare the ranking of the scenarii if we change the weights of the criteria. For example, if we put emphasis on the social point of view, we give more importance to the social criteria. It is just an example to illustrate the possible different final results according to the different points of view of the decision makers. In fact, we keep the weight 1 for all the criteria except for the three social criteria: we give the weight 3 to the employment and the increased demand of water, and 5 to the public health risks. The result for the horizon 2005 is shown in Fig. 7. — 12 —

|

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PROMETHEE 2 Complete Ranking PROMETHEE 1 Partial Ranking j

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Figure 7: The final ranking of the scenario for the horizon 2005 with all criteria in different weight In the present case, the best scenario is the scenario 3 while scenario 4 is placed at the second rank. The result is different compared to a homogeneous distribution of weights. This demonstrates that the MANPORIVERS methodology makes it possible to analyze the robustness of the best scenario. To summarize, we can say that the MANPORIVERS methodology makes possible to construct pollution management scenarii, to compare them according to criteria defined previously and to propose a final ranking indicating which scenario is the best. CONCLUSION The MANPORIVERS methodology is based on the combined use of multicriteria analysis and of fuzzy logics. It constitutes a decision aid tool to help decision makers establish environmentally, socially and economically sustainable management policies for priority water pollutants taking account of their effects on foods and human health. It is applicable to any river basin and takes account not only of the environmental impacts (typically the concentrations of pollutants in water) but also of the social and economical aspects of the problem. This methodology can be used interactively for different basins exchanging water. It can also be used at different scales in a recursive manner, from small tributary basins to large basins. These properties constitute a strong incentive to harmonize and coordinate the policies of different basins but these policies can be applied progressively according to the circumstances and available budgets, starting from small basins, since the results of small basins can be directly utilized for larger ones. Acknowledgements This reseach was performed under the contract number: ICA4-CT-2001-10039 MANPORIVERS: 01/01/2002 to 31/12/2005 of the European Community. The authors acknowledge the support of the European Community. REFERENCES 1. Gao Shixiang, Zhang Aiqian, Wang Xiaodong, Han Shuokui, Wang Liansheng, Wang Xiangde, Zhu Bin, Zhang Xiujuan, Ndiaye Alassane, MarchaL Jean. Methodology for Pollutant Input Evaluation. Final Report on WP1: Deliverable D15, Annex 1 to the Second Annual Report of the MANPORIVERS research project, EC Contract ICA4-CT-2001 -10039. 2. Lejeune Andre, Pirotton Michel, Novo ME, Ikavalko VM, Diamantino C, Lobo-Ferreira JP, Brouyere S, Orban Ph, Dassargues A. Pollutant Transport Models. Final Report on WP2, part 1: Deliverable D16, Annex 2 to the Second Annual Report of the MANPORIVERS research project, EC Contract ICA4-CT-2001-10039. 3. Lejeune Andre, Pirotton Michel. Transport of Pollutants by Surface Waters. Final Report on WP2, part 1: Deliverable D16, Annex 2 to the Second Annual Report of the MANPORIVERS research project, EC Contract ICA4-CT-2001-10039. 4. Brouyere S, Orban Ph, Dassargues A. Applicability of Groundwater Pollutant Transport Models. Final Report on WP2, part 2: Deliverable D16, Annex 2 to the Second Annual Report of the MANPORIVERS research project, EC Contract ICA4-CT-2001-10039.

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5. Novo ME, Ikavalko VM, Diamantino C, Lobo-Ferreira JP. Generic Characterization of Mathematical Models to Simulate the Pollutant Transport in Groundwater. Final Report on WP2, part 3: Deliverable D16, Annex 2 to the Second Annual Report of the MANPORIVERS research project, EC Contract ICA4-CT-2001-10039. 6. Zhang Ying, Zhou Jiti, Xiang Xuemin, Xu Yanli, Wang Xiangde, Liang Lianluan, Zhang Xiujuan. Selection Methodology for Sanitation and Remediation Techniques. Final Report on WP3: Deliverable D17, Annex 3 to the Second Annual Report of the MANPORIVERS research project, EC Contract ICA4-CT-2001 -10039. 7. Wang Xiangde, Li Chuanhong, Wang Min, Huang Xihua, Zhang Xiujuan, Xiang Xuemin, Zhou Jiti, Xu Yanli. Methodology for the evaluation of water use. Final Report on WP4: Deliverable D18, Annex 4 to the Second Annual Report of the MANPORIVERS research project, EC Contract ICA4-CT-2001-10039. 8. Wang Xiaodong, Gao Shixiang, Zhang Aiqian, Sun Cheng, Han Shuokui, Wang Liansheng. Relationships between Pollutants and Human Health. Final Report on WP5: Deliverable D19, Annex 5 to the Second Annual Report of the MANPORIVERS research project, EC Contract ICA4-CT-2001-10039. 9. Roubens Marc, N'Diaye Alassane, Rigo Nicolas, Cescotto Serge, Lourenco Nelson, Zhang Aiqian, Xiang Xuemin, Gao Shixiang. Development of the Manporivers Methodology. Final Report on WP6, part 1: Deliverable D23, Annex 1 to the Third Annual Report of the MANPORIVERS research project, EC Contract ICA4-CT-2001-10039. 10. Roubens Marc, N'Diaye Alassane, Rigo Nicolas, Cescotto Serge, Lourenco Nelson, Zhang Aiqian, Xiang Xuemin, Gao Shixiang. Application of the Manporivers Methodology in the Huai River Basin. Final Report on WP6, part 2: Deliverable D23, Annex 1 to the Third Annual Report of the MANPORIVERS research project, EC Contract ICA4-CT-2001-10039. 11. Erpicum Sebastien, Pirotton Michel, Lejeune Andre, Mao Yuanyuan, Wang Xiangde, Li Chuanhong, Wang Ming, Huang Xihua, Zhang Xiujuan. Application of the Manporivers Methodology in the Pearl River Delta. Final Report on WP6, part 3: Deliverable D23, Annex 1 to the Third Annual Report of the MANPORIVERS research project, EC Contract ICA4-CT-2001-10039.

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COMPUTATIONAL METHODS IN ENGINEERING AND SCIENCE EPMESC X, Aug. 21-23, 2006, Sanyo, Hainan, China ©2006 Tsinghua University Press & Springer

Adaptive Meshfree Methods Using Local Nodes and Radial Basis Functions G R. Liu**, Bernard B. T. Kee \ Z. H. Zhong 2 , G Y. Li 2 , X. Han 2 1 2

Centre for Advanced computations in Engineering Science (ACES), 10 Kent Ridge Crescent, National University of Singapore, Department of Mechanical Engineering, Singapore 11920 State Key Laboratory of Advanced Technology for Vehicle & Manufacture, M.O.E., Hunan University, Changsha, 410082 China

Email: [email protected], [email protected] Abstract In this paper, adaptive meshfree methods using local nodes and radial basis functions (RBFs) which based on strong-form formulation is presented. In this present formulation, radial basis functions are used in the function approximation for the discretization of the governing system equations. Regularization techniques are suggested and examined to stabilize the solutions in order to obtain stable and accurate results. Different schemes for constructing regularization matrix are compared and discussed. As stability is restored, meshfree strong-form method using local nodes and RBFs can facilitate an easier implementation for adaptive analysis to achieve desired accuracy. Residual based error indicator is devised in the adaptive scheme. A simple and practical node refinement procedure is presented for node insertion at each adaptive step. Key words: adaptive meshfree methods, radial basis functions, strong-form formulation, regularization technique, error indicator INTRODUCTION Since Kansa's work [12] in 1990, radial basis functions (RBFs) have been extensively used for solving partial differential equations (PDEs) numerically [11,17,21]. In these methods, full and ill-conditioning coefficient matrix is always obtained as RBFs is used globally with all the nodes in the problem domain. Recently, methods that use local nodes and RBFs for solving PDEs has been proposed [4-9], which resulted in banded coefficient matrix. It is found that the stability of the solution becomes, however, an important issue [2,6,7], Radial point collocation method (RPCM) is a meshfree strong-form method using radial basis functions with irregularly distributed local nodes [4-7]. It is regarded as a truly meshfree method as mesh is required neither in the formulation procedure nor function approximation. Since no mesh is used, mesh-related problems can therefore be avoided. The strong formulation is simple and straightforward [1,2], which can facilitate an easier implementation of adaptive analysis if effective stabilization techniques can be developed. In this paper, regularization techniques, which are often been used for stabilizing the solution of ill-posed inverse problem [3], are suggested to restore the stability of the RPCM solution. Four different regularization schemes are provided for constructing the regularization matrix that modifies the system coefficient matrix and hence provides stability in the solution. FUNCTION APPROXIMATION A local radial point interpolation scheme is used to approximate unknown field function using radial basis functions (RBFs) locally. Consider a smooth field function u can be approximated at interest point x in the problem domain as

«*W = I«,l(|x-x i |) + £Vy i=l

(D

y=l

where n is the number of supporting nodes in the local support domain, m is the number of monomials in polynomial — 71 —

function, rt (|| • ||) is radial basis function and pj is the monomial of polynomial function for augmentation. ai and bj are the coefficients of radial basis function and monomial of polynomial function. By enforcing the interpolation passing through the nodal values, the following expression can be obtained,

idlxi-xill) ^lxi-x2ll) ••• i(h- x »I) fidk-xj) U^-xJl)

r

p

^

2(j| x 2- x 2 |) *•• ^ | x 2 - x „ | | ) px{x2)

rmfam-x2\)

-

r.Jx.-x.ll)

A

(*J

"'

Pm

^

•••

pm(x2)

••

^(xjj

or

U = [R P]

(2)

where U is the vector of unknown nodal values, a and b are the vector of coefficients of radial basis functions and monomials of polynomial function respectively. With orthogonal condition [13,14], Pra = 0

(3)

The vectors of coefficients can be obtained as R

-i-i 1111

Pr

= G'

(4)

Therefore, the approximated field function u at interest point x can be expressed as

«*(»)=foto M

-

* *

*

V

Neumann boundary I~"t

*

*

Field nodes Regularization points*

*

*■ a

*

*

•

\

*

/

Dinchlel boundaiy Tû

Figure 5: Regularization points of Scheme IV The regularization equations can be collocated at regularization points respectively and assembled in the following matrix form as Kr4U = Fr4,

(28)

where K r 4 is the regularization matrix and F r4 denotes as the regularization vector of scheme IV. DETERMINATION OF REGULARIZATION FACTOR The degree of regularization depends on the regularization factor a . If a = 0, the regularized least-squares radial point collocation method (RLS-RPCM) will be reduced to the original radial point collocation method. Therefore, it is very crucial to determine an appropriate regularization factor in the regularization procedure. In this work, the L-curve approach [15] is adopted to determine the a . The idea of L-curve approach is to achieve an optimal regularization factor such that the residual of the both terms of the functional 77 in Eq. (15) are minimal. The general guideline for determining an appropriate regularization factor is that a should be positive and just small enough to prevent the ill-condition. From our observation, the optimal regularization factors for scheme Ij scheme II and scheme HI are is remained unchanged throughout the adaptive analysis. Hence, regularization factor is only required to determine at the initial step of the adaptive analysis. Scheme IV is a very special case, the vector of unknown nodal values can be solved either by discretized governing system equations, Eq. (13), or discretized regularization equations, Eq. (24). However, none of the solutions is stable. By solving the same status of problem with two differences sets of discretized system equations, the regularization procedure provides a compromise solution between two set of discretized system equations. Therefore, in this scheme, a = 1 is a logical choice to be used. ADAPTIVE STRATEGY In this work, a residual based error indicator is used. The problem domain can be discretized using Delaunay diagram. The governing equation is satisfied solely at the field nodes but elsewhere. The local error indicator can be evaluated at the mid point of the Delaunay cell as

where Aj is the area of they* Delaunay cell and \LUj - f \

is the L2 norm of the residual of the governing equation

at the mid point of the cell. The estimated global residual norm can therefore be obtained by integrating Eq. (29) over the entire problem domain as

^ = V«=z{4lK-/ J IL-

(3°)

where nc is the total number of Delaunay cells in the problem domain. Additional node will be inserted in the middle of they* Delaunay cell if the local error indicator for they* cell meets the refinement criteria, — 76 —

Tlj>K,11m,

(31)

where K, is a predefined coefficient for local refinement and 77M=rnax(/7,.).

(32)

Adaptive process will progress until the stopping criteria based on the global error indicator is met as 7,^*,^,

(33)

where Kg is the global residual tolerant and rjmg is the maximum ofthe estimated global residual norm throughout the adaptive process. NUMERICAL EXAMPLE In this numerical example, adaptive meshfree method using radial basis functions and local nodes is demonstrated. Multi-quadratic (MQ) RBFs augmented with completed second order polynomial function is used for the function approximation. The shape parameters of MQ-RBFs with arbitrary real parameters are adopted, and the recommended parameters by Liu et al. [20] are used: q = 1.03, ac = 3 . 0 . The coefficient for local refinement and global residual tolerant are predefined as Kl = 0.1 and Kg = 0 . 2 for all the cases studied in this work. The following norm is defined to indicate the true error ofthe approximated solution:

i z(s-r

(}

■

where s stands for the solution that can be displacements or stresses. In this work, a benchmark plane stress problem of solid mechanics is studied. A cantilever beam subjected to a parabolic shear stress at right end is considered (see Fig. 6). The geometries and material properties are given as: Young's modulus £ = 3xl0 7 , Poisson's ratio v = 0.3 , length of cantilever is L = 48.0m and the height is H = 12.0/w . The parabolic shear stress at the free end is given as r^ =

p

(H2 / 4 - V 2 \, where I is the moment of

inertial ofthe cross section of cantilever beam and P = 1000N.

Figure 6: Cantilever beam subjected to a parabolic shear stress at the right end The governing equations ofthe solid mechanics problem are well known as: 0-..+6,. = 0 i n / 2 ,

(35)

with Neumann boundary condition aijnj = /,. on rt,

(36)

and Dirichlet boundary condition K,=5i<m/;.

(37)

In this example, the exact displacements are imposed along the left edge and Neumann boundary conditions are imposed along the rest ofthe edges. The analytical solution of this problem can be found in [19]. The results of adaptive analysis using radial basis functions associated with different regularization schemes are shown as follows. — 77 —

1. Scheme I In this scheme, a = 0.01 is used as a regularization factor. The adaptive analysis started with 55 regular distributed nodes and ended with 1258 nodes randomly distributed in the problem domain at 4th step as shown in Fig. 7. The estimated global residual norms are plotted in Fig. 8, and the error norms of displacements are shown in Fig. 9.

Total fl*MMM(««-1CS

Total N M r w f b v t t

ife"VVW:?J!"VT'?&"?!'^?'V,"''r!']

Figure 7: Nodal distribution of Scheme I in adaptive analysis Estimated global residual norm

Total no. of nodas

Figure 8: Estimated global residual norms of Scheme I in adaptive analysis The error norms of the displacements have been drastically reduced from 9.38% and 10.39% to 0.69% and 0.58% in the x, y directions respectively. The error norms of o^ and r^ are plotted in Fig. 10. The error norms of the stresses reach 0.72% and 1.37% respectively at the final step of the adaptive analysis.

Total no. of nodaa

Figure 9: Error norms of displacements of Scheme I in adaptive analysis — 78 —

Totelne. ofnoifea

Figure 10: Error norms of stresses of Scheme I in adaptive analysis 2. Scheme II In scheme II, a = 0.1 is used as a regularization factor Similar to scheme I, the adaptive analysis takes four steps to complete in this case. The nodes has been increased from 55 regularly distributed nodes to 1185 irregularly distributed nodes in the problem domain at the final step as shown in Fig. 11. The estimated global residual norms are plotted in Fig. 12, and the error norms of displacements are shown in Fig. 13.

m ToM«*ttn«4*i»lia6

Figure 11: Nodal distribution of Scheme II in adaptive analysis Estimated global residual norm

Figure 12: Estimated global residual norms of Scheme II in adaptive analysis From Fig. 13, the error norms of the displacements are shown tremendously reduced from 2.72% and 1.05% to 0.05% and 0.06 % in the x and j> direction respectively. The error norms of the a^ and r ^ have been reduced from 4.55% and 21.73% to 0.18% and 0.65% respectively, as shown in Fig. 14. — 79 —

Displacements error norm

Total no. of nodes

Figure 13: Error norms of displacements of Scheme II in adaptive analysis Stresses error norm

Total no. of nodes

Figure 14: Error norms of stresses of Scheme II in adaptive analysis 3. Scheme III In this scheme, a = 1 x 10~5 is used as a regularization factor. The adaptive analysis takes five steps to complete in this case. Nodal distributions at 1st, 3 rd , 4th and 5th step are plotted in Fig. 15. The adaptive analysis ended with 1533 nodes in the problem domain. The estimated residual norms are shown in Fig. 16 and the error norms of the displacements are plotted in Fig. 17. Step=3

Step=1 20

20

10

10

* * * * * * * * *

0

0

-10 -20

-10

()

20 40 Total field nodes-55

60

-20

(

Step=4

Step=5

20

20

10

10

0

o

-10

-10

-20

()

20 40 Total field nodes=56 0

60

-20

20 40 Total field nodes=354

0

20 40 Total field nodes=1533

60

Figure 15: Nodal distribution of Scheme III in adaptive analysis — 80 —

Estimated global residual norm

Total no. of nodes

Figure 16: Estimated global residual norms of Scheme III in adaptive analysis As shown in Fig. 17, the error norms of the displacements are reduced significantly from 135% and 134% to 5.10% and 5.15% in the x and>> direction. The error norms of the a^ and r have achieved 5.29% and 5.93% from 148% and 244% respectively as shown in Fig. 18. Displacements error norm

10

10* Total no. of nodes

Figure 17: Error norms of displacements of Scheme III in adaptive analysis Stresses error norm

Figure 18:. Error norms of stresses of Scheme III in adaptive analysis 4. Scheme IV The adaptive analysis of scheme IV began with 55 evenly distributed nodes and ended with 1277 nodes at fourth step as shown in Fig. 19. The estimated global residual norms are plotted in Fig. 20, and the error norms of displacements are plotted in Fig. 21. 81

Stcp*2

Total Mdnottoa-IGO

Total M4n»1

Step 2

Step 4

Step 3

Step 5

Scheme

u

V

u

V

u

V

u

V

u

I

9.38%

10.39%

2.69%

2.73%

0..66%

0.31%

0.69%

0.58%

-

II

2.72%

1.05%

1.85%

1.38%

2.05%

1.26%

0.05%

0.06%

-

v

III

135%

134%

7.42%

6.76%

19.08%

17.53%

5.06%

5.26%

5.10%

IV

3.16%

3.97%

0.81%

0.76%

1.21%

0.87%

0.20%

0.10%

-

RPCM

175%

179%

205%

201%

47.66%

51.15%

56.58%

53.79%

5.76%

5.15% 4.63%

*In the case of using RPCM without stabilization technique is used, twenty of the local nodes is used to construct the shape functions.The number of nodes in the problem domain at each step is: 55,112,221,375,593. Table 2 Error norm of stresses at each adaptive step Stresses error norm Scheme

Step 1 ^xx

Step 2 T

v

°xx

Step>4

Step 3 T

*y

(j

T

XX

*y

, and (j), are the measured natural frequency and mode shape of the ith mode; co, and (|)t are the model natural frequency and mode shape of the ith mode; and cc, and p, are the weighting, where i = 1,2,..., m. Note that one of the major difficulties is on the mode matching, that is, it is necessary to determine which model mode matches which measured mode. In the case when only measurements of partial mode shapes are available, this is not an easy task. Another major difficulty is that the m measured modes might not necessary be the lowest m modes in practice. Furthermore, in the case when there is possible damage in the structure, the order of the modes might switch, which increases the difficulty of this problem. The first attempt for solving this model updating problem without mode matching was made in [10] and [12] with the Bayesian probabilistic framework. This paper presents a very efficient finite-element model updating method using incomplete measured frequencies and mode shapes with applications to damage detection. The frequencies and mode shapes are measured. The proposed method does not require any mode matching between the model modes and measured modes. Furthermore, the pro posed method operates in an efficient iterative procedure instead of nonlinear optimization/searching. First, let's introduce the key idea of the proposed approach with a simple ideal case. Consider the eigenvalue problem of a structure with d degrees-of-freedom (DOFs): K^

= CG^MC^

(2)

— 139 —

where M and K are the d x d mass and stiffness matrix of the structure, respectively; coi and (^ are the natural frequency and mode shape of the first mode. The stiffness matrix K can be parameterized as follows: K = K0+Xe,K;

(3)

7=1

where 0 = [0i, 02,..., 0 n ] r is the uncertain/unknown parameter vector that governs the stiffness matrix and the matrices Ky, j = 0 , 1 , . . . , n, are known. Then, the eigenvalue problem becomes: [K1^1,...,K^1]0-(co?M-KoH1

(4)

Therefore, the stiffness parameter vector 0 is readily obtained: 0 = [K 1 ^ 1 ,...,K^ 1 ]- 1 (cofM-K 0 )(t) 1

(5)

However, complete mode shape measurements are not available in practice. Furthermore, the measurements are con taminated by measurement noise. In the next section, the formulation of the problem is described. Then, an expanded mode shape technique is introduced following with a method for fine tuning the noisy measurement of the natural fre quencies. Then, the stiffness parameter updating methodology will be introduced. Finally, the proposed approach will be presented which is an iterative process, that updates the mode shapes, natural frequencies and stiffness parameters alternately. Examples with a ten-story building and a three-dimensional braced frame will be used to demonstrate the proposed method with applications to damage detection. PROBLEM DESCRIPTION Assume that m modes are measured. Note that they are not necessary the first m modes. Consider the eigen-equation of the ith measured mode of a structure with d DOFs: ftfc = fltfMfc, i = l,2,...,m

(6)

where M and K are the d x d mass and stiffness matrix of the structure; and CD; and ()), are the natural frequency and mode shape of the ith measured mode. Use d>, and (j), to denote the measurements of the natural frequency and mode shape for the ith measured mode, respec tively. These measurements are contaminated by measurement noise, that is, there is a difference between the measured and the actual frequency and mode shape. Let 8 denote the difference operator. For example, 8co? is the difference, or measurement noise, between the actual and measured natural frequency of the ith measured mode:

5co? = ($-&}

(7) th

Similarly, the difference between the actual and measured mode shape of the i measured mode is given by Sfc-= ♦,-♦,■

(8)

In the case when a component of the mode shape is unmeasured, one can simply assign zero to its measured value. Then, all these random differences are grouped into an error vector: Z={h

^C02,((!

(10)

^

which can be obtained by Bayesian modal identification methods [13-16]. Note that the error of the modal frequen cies/mode shapes of different modes might be correlated. For the unobserved components of the mode shapes, one can simply assign a large value for their variance without correlation with other modal quantities. — 140 —

By using a substructuring approach, the stiffness matrix K can be parameterized as follows: (11) where 6 = [61,62,.. , 6 n ] r is the unknown parameter vector that governs the stiffness matrix and the substructural stiffness matrices K,, j = 0,1,...,«, are known up to a scaling 6 7 . Then, in the similar fashion as in Eq. (7) and (8), the difference between the identified values and the nominal values 6 for the stiffness parameter vector is denoted as 1 = 6-9

(12)

Therefore, the difference between the actual and identified stiffness matrix depends on the difference between the actual and identified stiffness parameters 86 as follows: 5K = K - K = £ (6, - Qj)Kj = £ 86,K,

(13)

MODE SHAPES EXPANSION If sensors are available at s DOFs, the mode shape components can be measured at these DOFs only. In order to re cover the unmeasured components, many existing mode shape expansion techniques were proposed. A comprehensive summary and comparison among them was reported in [17]. Note that in the existing methods, the measured mode shape components are fixed at their observed values. Therefore, the expanded unobserved mode shape components are sensitive to the measurement noise for the observed components. In the following, a 'flexible' and computationally efficient method is presented. It does not completely constrain the observed values. Instead, the measured components will be probabilistically constrained by the measurements. In other words, the measured components can be different from their observed values and the allowable difference is controlled by the covariance of the measurement uncertainty. This is expected to improve the accuracy of the expanded mode shapes. Now, the mode shape mJ

fc-$2

"♦i-§i :

V

2_^2

(19)

An-ftd

. ♦ m - m_ th

where <j>,-, i — l,...,m is the mode shape measurement of the i measured mode. For those unmeasured components, zero value can be assigned. 1^ is the covariance matrix of all the mode shapes. Minimizing this objective function is equivalent to minimizing the equation error with probabilistic constraint for the measured components of the mode shape. It allows the measured components to vary from the measured values but the variation will be constrained by a scale of its standard deviation of the error in the measurement. It can be easily shown that the updated mode shapes (recovering the unmeasured components and fine tuning the measured components) is given by

$2

-(F^:/3^^-1)-1!,-1

$2

(20)

where the (symmetric) matrix F^3^ is given by I©JM - K 3

F< > = 0

0

a> 2 M-K

(21)

fi&M

- K

dmxdm

FINE TUNING OF THE MEASURED NATURAL FREQUENCIES Note that the frequencies are measured with noises so it is possible to further improve the prediction by the following procedure. Now, the squared natural frequency co2 is updated using Eq. (6) but the stiffness parameters 0 (and so the stiffness matrix K) and the mode shape §t are uncertain. By using Eq. (8) and (13), Eq. (6) can be rewritten by neglecting the second order error: (K - co2M)$( = (co2M - &)&|>, - X M y M ,

' (22)

where £"_, 807K/<j), can be written in terms of 88: 2 8 e ; K A = [K l $,,...,K n t.]8e

(23)

Define the equation error zeq ^ = (u)2M — K)8,- — [Ki <j>;,..., K„^,]80, which is the right hand side of Eq. (22). There fore, the covariance matrix of the equation error~Leqlsf.is given by 2 ^

= GOS (o2 G(» r + G(2)ZeG(2)r

(24)

where the matrices G ^ and G^2^ are given by co2M-K G =

co^M-K

(25) 2

co M-K

dmxdm

142 —

G =

-Ki$! —Ki<j>2

-K2$! — K2$2

-K,$m

-K2$m

Knf, Kn$2

(26) dmxn

Then, the natural frequencies can be updated by minimizing the following objective function: '(K-0)fM)$ 1 " , r (K-cofM)^

Jtf =

■(R-cofM)$," (K-coJM)^

y-l

e^co2

Cof-Cbj" 0^-0)2

+

r

*v 9

(K-co^M)^

^9

"cof-fi)?" ©1-^

(27)

«m-«>mj

where d),, i = l,,..,m is the measurement of the natural frequency of the ith measured mode. Z^ is the covariance matrix of all the measured frequencies. Minimizing this objective function is equivalent to minimizing the equation error with probabilistic constraint for the measured frequencies. It allows them to be different from the measured values but the variation will be constrained by a scale of the standard deviation of the error in the measurement. By minimizing J^p. with respect the co2, the natural frequencies can be fine tuned as follows:

rv[V

3

= (G( >+Z- 2 ')-

2

( \

"Kr 2"

1

K$ 2

2

+ *3

eq,m.

(28) ?

H0) =

(36)

mm cofM-K H (2)

(KM

_

-

K

(37) dm -dm

Therefore, the covariance matrix of the equation error is given by I s / , = [H ( ".H< 2, ]I fc .[H (1 ».H ,2, ] r

(38)

where the matrix I,, is the covariance matrix of e given in Eq. (10). It is shown in the Appendix that the optimal estimate of the stiffness parameter vector 0 is given by e=(A r X 6 / ,'A)

'A r I W ) 'b

(39)

where the matrix Z§£ is the same as I5/,, except that it replaces the zero eigenvalues of Its/?by a small value. The matrix A X&, A is invertible if enough constraints (number of sensors and measured modes) are available. Otherwise, more sensors or measured modes are needed. Note that the least squares solution is a special case of Eq. (39) under the condition when X^ = a 2 1: e=(A r A)~ 1 A 7 "b

(40)

Uncertainty Estimation The uncertainty of the estimation can be quantified by its covariance matrix which is given by U =

(xTiSblAyl

(41)

The diagonal elements give an estimation of the variance for each stiffness parameter.

PROPOSED ITERATIVE APPROACH The proposed methodology updates the natural frequencies, mode shapes and stiffness parameters in an alternating manner with the aforementioned methods. It is recommended to operate in the following order: 1) Assume an initial trial vector for 9 and calculate the stiffness matrix K with 8. 2) Expand and fine tune the mode shapes ,, / = 1, 2, . . . , w, using Eq. (20). Note that this mode shape expansion technique can be used for reducing measurement noise even when complete measurements of mode shapes are available. 3) Fine tune the squared natural frequencies ci)*-, / = 1, 2 , . . . , m, using Eq. (28). 4) Update the estimation of the stiffness parameter vector 9 by using Eq. (39). In the case if a component becomes negative, replace it by the value in the last iteration. — 144 —

5) Iterate the previous steps 2-4 until the stiffness parameter vector 9 converges. 6) Calculate the covariance matrix of the estimates of the stiffness parameters using Eq. (41). It is recommended to skip step 3 in the first few, say 10, iterations in order to have faster convergence. ILLUSTRATIVE EXAMPLES 1. Example 1: Ten-story Building In this example, a ten-story building is considered. It is assumed that this building has uniformly distributed floor mass and stiffness across all floors. The mass per floor is taken to be 100 tons, while the interstory stiffness is chosen to be kQ= 176.729 MN/m so that the first five modal frequencies are 1.00,2.98,4.89,6.69 and 8.34 Hz. The coefficient of variation of the measurement error of the squared natural frequencies and mode shapes are taken to be 1.0% for all modes. The initial trials of the stiffness parameters are taken to be uniformly distributed from 2#o to 3#o, where k0 is the actual interstory stiffness. In this case, the initial trial is significantly overestimated and the variation between different interstory stiffness is substantial. Table 1 shows the initial trial values and identified values of the stiffness parameters using different number of measured modes, where the actual value is 176.729 MN/m for all these parameters. The estimated standard deviations, calculated using Eq. (41), are shown in parenthesis. In these cases, complete measurements at all DOFs are utilized. It is not surprising that the uncertainty reduces when the number of measured modes increases. Table 1 Identification results using different number of measured modes (Example 1) Parameter 9l

°2

1 e3 e4 e. e6 e7 e8 e9

Oio

Initial trial 359.29 435.09 489.67 400.89 456.37 518.83 425.93 415.03 379.62 407.28

2 modes 177.01 (2.95) 175.59(4.73) 178.42(6.69) 187.88(11.5) 182.44(7.28) 173.69(4.70) 179.92(4.22) 171.88(3.85) 174.06(4.76) 183.02(6.71)

3 modes 173.96(1.81) 181.11 (3.56) 179.42(4.54) 177.44(2.83) 178.77(2.52) 177.22(2.63) 176.81 (3.45) 173.81 (2.17) 179.14(2.14) 174.05(2.09)

4 modes 174.05(1.45) 182.25(3.36) 175.66(1.83) 175.81 (1.62) 177.37(2.33) 176.66(1.49) 175.62(1.62) 175.74(1.81) 175.86(1.27) 174.93(1.21)

5 modes 174.53(1.27) 178.99(2.08) 176.06(1.25) 176.28(1.51) 177.19(1.32) 176.79(1.16) 175.42(1.17) 176.03(1.12) 176.16(1.04)

175.93(0.95) 1

Incomplete mode shape measurements: Consider only five sensors on the first, fourth, fifth, seventh and top floor. Table 2 shows the initial trial values, identified values, standard deviation and the coefficient of variation (C.O.V.) of the stiffness parameters. Fig. 1 shows the iterative history of the identification process. It converges with 50 iterations. The difference in 8, is smaller than l.OxlO-6 compared to those estimated after 1000 iterations. The CPU time is about one second with a normal personal computer with 2.4 GHz under the MATLAB environment [18]. It is seen that the iteration virtually converges after ten iterations. Table 2 Identification results with measurements of five sensors and five modes (Example 1) Parameter 01

02

e3 e4 85

e6 e7 e8 e9

Oio

Initial trial 359.29 435.09 489.67 400.89 456.37 518.83 425.93 415.03 379.62 407.28

Identified values 0 175.54 178.52 174.04 173.92 178.62 178.17 175.97 176.19 175.33 176.41

— 145 —

Standard deviation Ge 1.84 2.61 2.09 2.09 2.03 1.47 1.60 1.68 1.56 1.26

C.O.V. 0.010 0.015 0.012 0.012 0.011 0.008 0.009 0.009 0.009 0.007

600

|

l! !

l

fi \ I! I

i

jl ■

i

I

I

I

1

I

1

1

j| | ; ; j

550^ r

M■ M I

450 ■ - ■ ■ ■ ,

400

fi

i

;

I!

~

3501 300

~

250

\ i/\l

I'M-

-

200 L i / .

;' i , ,f

s^f-J™**"'*-"■-■"

J

—

■ ■ ■ - ' - - • ' ' - ■ ' ■ - " '•■•A,...™

....-,....

..-...■■.■...■...»..

150

1A ;W'' ;/ 100

It/

lV

v

' 1

1

10

1

15

1

I '

1

20

25

30

35

1

40

1

45

50

Number of iterations

Figure 1: Iteration history for the stiffness parameters with incomplete measurements of mode shapes (Example 1) 2. Example 2: Three-dimensional braced frame The proposed method is applied to update the finite-element model of a three-dimensional five-story braced frame. It has a square section with width a = 4 m. There are four columns for each floor, one at each corner. Each of them have interstory stiffness 10 MN/m and 15 MN/m in the x andy direction, respectively. Furthermore, each face in each floor is stiffened by a brace and its stiffness is taken to be 20 MN/m. As a result, the interstory stiffness is 80 MN/m and 100 MN/m in the x andy direction, respectively. The floor mass is taken to be 10 tons for each floor. As a result, the first five natural frequencies of the structure are 4.05,4.53, 7.44, 11.83 and 13.22 Hz.

(+y) (1)

L±ii_

h+*

Û

(-x)

i (2)

o* - p ^

^r>

Oi

(+x)

**.-» (3)

(-y) Figure 2: Floor plan for the 12-DOF model for identification (Example 2)

— 146 —

In order to locate the face(s) that sustain damage, four stiffness parameters are used for each story to give twenty stiffness parameters, ©4(/-i)+i = ^/.+.v, &4(i~i)+2 ~ ^/,+V' ^4(/--i)+3 ~ ty.~x a n d ®4(/-i)+4 — £/,->■•> / = 1, 2, . . . , 5, where the index / represents the story number and '+JC', '+y\ 6-x9 and '->>' represent the direction of the outward normal of a face. The actual values of these stiffness are A/.+.v = A/.-* = 50 MN/m and A/.+v = A7. _ v - 40 MN/m, / = 1, 2 , . . . , 5. In other words, 61 = 63 = • • • = 819 — 50 MN/m and 62 = O4 = • • • — 02o = 40 MN/m. The floor plan is shown in Fig. 2. The point Oj (xj, yj) is the stiffness center of story /, where xj and >7 , / = 1,2,. . . , 5, are given by a(ki+x-ki,

-x)

2(kx+x + klf.xy

yi ■

'"

«(A7, + v-/v7,-v)

(42)

2(*,. + r + */._,..)

The stiffness matrix for story / with respect to the O'j coordinates is iL+Y + ki,~Y 0 0

K; =

£/,+v - ^/.-i 0 0

0 ^/.+.r + ^7,-.r 0 0 -A7. + . v - kj^x 0

0 0 kit 0 0

-kn

-A7.+V - A7,-.i 0 0 A7i+v-hA7.-v 0 0

xifk,.+x + ( - -yi)2k,.+y + ( - +x,)2ki.~x + ( -

kit = (--

0 —ki,+x-ki.„x 0 0

fc/,+.r + fy-.r 0

+y,)2k,...y

0 0 -kn 0 0

(43)

ku (44)

Here, the first three DOFs and the last three DOFs correspond to the lower floor and the upper floor of the story, respectively. Each of these sets of three DOFs correspond to the x-translational, y-translational and torsional motion. The stiffness matrix for story / with respect to the (1), (2) snd (3) DOFs in Fig. 2 of the upper and lower floor is given by K/ = T r KjT

(45)

where T is given by T =

1

T

0

0 T

1

1 0 - S + y/ T = 0 1 -$-xi 1 0 2 + v/

(46)

The stiffness matrix for the 12-DOF structural model is assembled from those of the floors. The DOFs for this stiffness matrix are the (1), (2) and (3) shown in Fig. 2 for each floor. However, this stiffness matrix is not linear to the stiffness parameters 9 ; . In this case, one can linearize the relationship between the stiffness matrix and the stiffness parameter: 20

K=Ko+XKA-

(47)

7=1

where

.20

(48)

and K0 = K - 2^ K7-0y

(49)

where the matrix Ko will need to be updated in every iteration. To increase the difficulty, it is assumed that only first three x-directional and ^-directional modes are measured but not any of the torsional modes. This is done deliberately to simulate the reality that some of the modes might not be excited so they are not able to be measured. In the identification, it is assumed that we do not know there are missing modes. These measured modes correspond to the 1 st (4.05 Hz), 2nd (4.53 Hz), 4th (11.83 Hz), 5th (13.22 Hz), 7th (20.84 Hz) and 9th (23.95 Hz) mode. Sensors are placed at the +y, -x and -y face of the 1st, 3rd and 4th floor to measure the — 147 —

natural frequencies and mode shapes. The coefficient of variation of the modal data is taken to be 0.5%. Initial trials are taken to be 100 MN/m, which overestimates by 100% and 150% for the ±x and ±y faces, respectively. The iteration history is shown in Fig. 3 and it virtually converges after 20 iterations. It took about 3.1 seconds after 50 iterations. These stiffness parameters converges into two branches, one for the ±x faces (approximately 50 MN/m) and the other for the ±y faces (approximately 40 MN/m). Table 3 shows the actual values, identified values, standard deviations and coefficients of variation (C.O.V.) of the stiffness parameters. The standard deviations are computed using Eq. (41) and they are up to about 1%. The estimates are all close to the actual values. The difference between the actual and identified values are of similar order to the corresponding calculated standard deviations. 100

20

25

30

50


Figure 3: Iteration history for the stiffness parameters of the undamaged structure (Example 2) Application to damage detection: The structure is assumed to be damaged on the +y face of the first story and the +x face of the third story. One third and a quarter of the stiffness reduction of the corresponding brace is imposed to these two faces, respectively. It corresponds to 16.67% and 10% stiffness reduction of these faces, respectively. It is noted that there is no sensor at the +JC face. The first five natural frequencies of the damaged structures are 3.98, 4.50, 7.37, 11.67 and 13.19 Hz. Note that these damages alter the order of the modes. Furthermore, the translational and torsional modes are mixed, especially for the high modes. In this case, the first six translational modes become the 1st (3.98 Hz), 2nd(4.50 Hz), 4th(l 1.67 Hz), 5th(13.19 Hz), 6th(18.47 Hz), and 7th(20.67 Hz) mode. Initial trials are again taken to be 100 MN/m. Independent modal data is used to identify this damaged structure. The iteration history is shown in Fig. 4. It took about 3.3 seconds for 50 iterations. These stiffness parameters converges into four branches, one for the +y face of the first story, one for the other ±y faces, one for the +JC face of the third story and the last one for the other ±JC faces. Identification results are shown in Table 4, which is similar to the one in Table 3. It is clearly seen that the +y face of the first story and the +x face of the third story has substantial stiffness reduction compared to the undamaged structure. In order to visualize the damages, the means and standard deviations for the stiffness parameters are used to find the probability that a given stiffness parameter 9/has been reduced by certain fraction d, compared to the undamaged state of the structure. An asymptotic Gaussian approximation [19] is used for the integrals involved to give:

,

p}™(d) = P{&f < (1 -d)Vf)^ 4> ^'

(l-d)ef-ef l

J

-

^ ( l - d ) 2 ( < ) 2 + ( ,+.v 40 39.805 0.17 05.4-y 50 49.839 0.14 85. - r 40 39.759 0.18 ! e*-, 1

| C.O.V. 0.0069 0.0050 0.0052 0.0051 ! 0.0080 | 0.0055 0.0072 0.0055 0.0068 0.0108 0.0053 0.0109 0.0067 0.0060 0.0058 0.0058 0.0033 0.0044 0.0029 0.0046 1

100

90 I

1

0

5

10

15

20

25

30

35

40

45

50


Figure 4: Iteration history for the stiffness parameters of the damaged structure (Example 2) where 0( •) is the standard Gaussian cumulative distribution function; B'f and &li denote the most probable values of the stiffness parameters for the undamaged and (possibly) damaged structure, respectively; and &f and tf are the corresponding standard deviations of the stiffness parameters. — 149 —

Table 4 Identification results of the damaged structure (Example 2) Parameter 6i. + . v ©l.+v 6i._, 6,,_ v

e2.+,. 62,+v

e2._.v e2._,, 63.+,-

e3.+v

63._,. 63.-.V ^4,+.Y

e 4 . +v ®4,-x

e 4 .- v e5.+.v ^5,+v

e5.-v 05.-,

Actual values 0 50 33.33 50 40 50 40 50 40 45 40 50 40 50 40 50 40 50 40 50 40

Identified values 0 50.246 33.465 50.030 40.400 50.415 38.826 50.486 40.160 45.258 39.946 50.379 39.423 49.861 40.147 50.179 39.669 50.334 39.686 50.254 40.106

Standard deviation Ge 0.33 0.19 0.26 0.21 0.45 0.36 0.46 0.39 0.35 0.27 0.26 0.25 0.34 0.24 0.28 0.24 0.19 0.17 0.17 0.17

C.O.V. 0.0066 0.0057 0.0052 0.0053 0.0090 0.0090 0.0092 0.0099 0.0079 0.0069 0.0053 0.0064 0.0068 0.0060 0.0057 0.0060 0.0038 0.0043 0.0035 0.0044

M,+y)

:3,+x)

-0.05

0.1

0.15

0.2

0.25

Damage extent d

Figure 5: Probability of damage (Example 2) The probabilities of damage for the twenty 07 are shown in Fig. 5. It can be clearly seen that the +y face of the first story and the +x face of the third story certainly have damage with probability almost unity. The mean of the damages

— 150 —

are 16.1% and 9.5% and their target values are 16.61% and 10%. Furthermore, uncertainty of these estimates are 0.6% and 0.9%. The figure can be interpreted as follows. Consider the probability of damage curve of the +y face of the first story. Damage is 15% damage or more with probability 0.95. All the other eighteen faces are undamaged. The stiffness parameters difference between the undamaged and damaged structure is within the uncertainty level. This probability of damage can be used directly to set up the confidence interval for damage with a given percentage of confidence. CONCLUDING REMARKS A finite-element model updating methodology is presented with applications to structural damage detection. The proposed method does not require any matching between the measured and calculated modes from the finite-element model as it does not require any calculation of the natural frequencies and mode shapes of the finite-element model. Furthermore, it operates in a computationally efficient iterative manner without nonlinear optimization programming. The illustrative examples confirm the efficiency and effectiveness of the proposed approach. Acknowledgements This work was supported by University of Macau under research grants RG097/03-04S/YKV/FST and RG068/0405S/YKV/FST. These grants are gratefully acknowledged. APPENDIX Consider a set of linear algebraic equations: A9 = b + 5b

(51)

where 5b is a random vector that describes that the equation error with zero mean and covariance matrix !&,. Since X§& is symmetric, it is diagonalizable: X^VDV"1

(52)

where the diagonal matrix D contains the eigenvalues of I&, and the matrix V contains the corresponding eigenvectors. With a proper choice of the scaling factors of the eigenvectors, a matrix that satisfies V r = V" 1 can be chosen since Ega is symmetric. By pre-multiplying Eq. (51) with \ T , one obtains: V r A9 = V r b + 5b'

(53)

where 5b' = V r 5b has zero mean and covariance matrix V r I^,V = D, which is diagonal. This implies that the random components in 5b' are uncorrected. Note that the covariance matrix X&, is in general singular. Therefore, some of the diagonal elements of D are zero. The equations associated with these zero eigenvalues will have zero error, that is, they give exact constraints to the unknown parameters. One possible solution is to consider those constraints to eliminate some unknown parameters. Then, one can solve the remaining unknown parameters by considering the remaining equations weighted by the non-zero entities in the diagonal of the matrix D. However, it is suggested to solve this problem with the following approximated procedure. The values on the diagonal elements of D indicate the variance of the error of each equation. A larger value implies larger uncertainty. For those diagonal elements that are zero or close to zero, one can replace them by a small number, say 10 6 / V/v? , where Kn-g denotes the average of the eigenvalues of X&,. This means that the weighting for those equations are still large compared to the others. There are two reasons for this procedure: 1) The matrix D becomes invertible; 2) Instead of completely rely on those equations with zero eigenvalues, it just gives large weighting for them because the covariance matrix £&> might not be exact, e.g., due to the fact that the second order errors are neglected. Use D to denote that the diagonal matrix that replaces the zero diagonal elements by small values in D. Then, by pre-multiplying D 7, one obtains I ) H T 7 A e - D"i V r b + 5b"

(54)

where 5b" — D ^Sb' has zero mean and unity covariance matrix I. Then, all the equations have independent errors with the same variance. Therefore, the optimal solution of the unknown parameter vector 0 is the least squares solution: — 151 —

0 = [(i)-lvTA)T{D-iYTA)]-\D-?VTA)Tii-iVTb

(55)

Then, the unknown parameter vector 0 can be estimated as follows: e=(A%1Ar1ArS»b

(56)

where the matrix I&, is the same as Z&,, except that the zero eigenvalues are replaced by a small value. REFERENCES 1. Natke MA, Yao JTP eds. Proceedings of the Workshop on Structural Safety Evaluation Based on System Identification Approaches. Vieweg and Sons, Wiesbaden, Germany, 1988. 2. Agbabian MS, Masri SF eds. Proceedings of the International Workshop on Nondestructive Evaluation for Performance of Civil Structures. Department of Civil Engineering, University of Southern California, Los Angeles, CA, USA, 1988. 3. Chang FK ed. Proceedings of the 4th International Workshop on Structural Health Monitoring. Stanford University, USA, 2003. 4. Mazurek DF, De Wolf JT. Experimental study of bridge monitoring technique. ASCE J. Structural Engineering, 1990; 116(9): 2532-2549. 5. Hearn G, Testa RB. Modal analysis for damage detection in structures. ASCE J. Structural Engineering, 1991; 117(10): 3042-3063. 6. Lam HF, Ko JM, Wong CW. Localization of damaged structural connections based on experimental modal and sensitivity analysis. J. Sound and Vibration, 1998; 210(1): 91-115. 7. Hemez FM, Farhat C. Structural damage detection via a finite element model updating methodology. International J. Analytical and Experimental Modal Analysis, 1995; 10(3): 152-166. 8. Pandey AK, Biswas M. Damage detection in structures using changes in flexibility. J. Sound and Vibration, 1994; 169:3-17. 9. Beck JL, Katafygiotis LS. Updating models and their uncertainties. I: Bayesian statistical framework. ASCE J. Engineering Mechanics, 1998; 124(4): 455-461. 10. Beck JL, Au SK, Vanik MW. Monitoring structural health using a probabilistic measure. Computer-Aided Civil and Infrastructure Engineering, 2001; 16(1): 1-11. 11. Ching J, Beck JL. New Bayesian model updating algorithm applied to a structural health monitoring benchmark. Structural Health Monitoring, 2004; 3: 313-332. 12. Yuen KV, Beck JL, Katafygiotis LS. Efficient model updating and monitoring methodology using incomplete modal data without mode matching. Special issue in memory of Prof. T.K. Caughey, J. Structural Control and Monitoring, 2006; 13(1): 91-107. 13. Yuen KV, Katafygiotis LS. Bayesian time-domain approach for modal updating using ambient data. Probabilistic Engineering Mechanics, 2001; 16(3): 219-231. 14. Katafygiotis LS, Yuen KV. Bayesian spectral density approach for modal updating using ambient data. Earthquake Engineering and Structural Dynamics, 2001; 30(8): 1103-1123. 15. Yuen KV, Katafygiotis LS. Bayesian modal updating using complete input and incomplete response noisy measurements. ASCE J. Engineering Mechanics, 2002; 128(3): 340-350. 16. Yuen KV, Beck JL, Katafygiotis LS. Probabilistic approach for modal identification using non-stationary noisy response measurements only. Earthquake Engineering and Structural Dynamics, 2002; 31(4): 1007-1023. 17. Levine-West M, Milman M, Kissil A. Mode shape expansion techniques for prediction: experimental evaluation. AIAA J., 1996; 34(4): 821-829. 18. MATLAB. Matlab User's Guide. The MathWorks, Inc., Natick, MA, USA, 1994. 19. Papadimitriou C, Beck JL, Katafygiotis LS. Asymptotic expansions for reliability and moments of uncertain systems. ASCE J. Engineering Mechanics, 1997; 123(12): 1219-1229.

— 152 —


Application of Computational Mechanics to Reliability Studies of Electronic Packaging N. Miyazaki *, T. Ikeda Department of Mechanical Engineering and Science, Kyoto University, Kyoto, 606-8501 Japan Email: [email protected] Abstract Computational mechanics approaches, especially computational fracture mechanics, have been utilized as a powerful tool for reliability studies of large-sized structures such as air crafts, pressure vessels, piping and so on. Electronic packages are considered as small-sized structures to protect electrical integrity of electronic devices. So the computational mechanics can be successfully applied to the reliability studies of electronic packaging. In the electronic packages, there exist a lot of interfaces between different materials, so that the interface fracture mechanics is very useful for such studies. In the first part of the present paper, we summarize the interface fracture mechanics, and then we provide two examples of the application of computational mechanics to the reliability studies of electronic packaging. One example is the strength evaluation of a plastic package during solder reflow process, and another is the delamination evaluation of anisotropic conductive adhesive films under moisture/reflow sensitivity tests. Key words: interface fracture mechanics, stress intensity factors, electronic package, plastic package, anisotropic conductive films, moisture absorption

INTRODUCTION Computational mechanics approaches, especially computational fracture mechanics, have been utilized as a powerful tool for reliability studies of structural components. The fracture mechanics parameters such as the stress intensity factor and ./-integral calculated using the computational mechanics have been successfully utilized in assessing the structural integrity of large-sized structures such as air crafts, pressure vessels, piping and so on. Electronic devices are incorporated not only into electric appliances but also into various kinds of mechanical systems. For example, they are used to control the ignition of the engines in automobiles. In such a case, the failure of the electronic devices may induce the safety problems of the automobile. So the reliability of electronic devices is one of key issues of the safety of automobiles. The protection of electronic devices from the external environment and the electric connection between the electronic devices and substrate are generally called as electronic packaging. Electronic packages are considered as small-sized structures to protect electrical integrity of electronic devices. So the computational fracture mechanics can be successfully applied to the reliability studies of electronic packaging. In the electronic packages, there exist a lot of interfaces between different materials, so that the interface fracture mechanics is very useful for such studies. In the first part of the present paper, we summarize the interface fracture mechanics. Then we provide two examples of the application of computational mechanics to the reliability studies of electronic packaging to show the effectiveness of the fracture mechanics to the reliability studies of electric packaging. One is the strength evaluation of plastic packages during solder reflow process. Another is the delamination evaluation of anisotropic conductive adhesive films under moisture/reflow sensitivity tests.

INTERFACE FRACTURE MECHANICS Here we summarize the interface fracture mechanics useful to the reliability studies of electronic packaging. 1. Asymptotic solution in the vicinity of a corner of jointed dissimilar materials Chen and Nishitani [1] derived the asymptotic solution in the vicinity of a corner of jointed dissimilar materials whose geometry is shown in Fig.l. — 88 —

IMM ■ji—\

Mode II

tHt* Mode I

Figure 1: A coordinate system around a corner of jointed dissimilar materials The orders of stress singularity for the mode I and mode II deformations 1 - \ and 1 - A^ are obtained from the following characteristic equations: Dl(a,$,y,X) = (a - P)2A,2(1 - c o s 2y) + 2 A ( a - p)siny{sin Xy + sin M2n - y ) } + 2A,(a- P)psiny{sin A.(a -p)psiny{sin X(2n - y) - sinty } + (1 - a 2 ) - (1 - p2)cos 2Xn

(2)

+ (a2-p2)cos{2A0

(5)

a unique real solution for \ also can be obtained from Eq. (1). The stress field in the vicinity of the corner of jointed dissimilar materials selected as in Eq.(5) is shown as (6) where au t(kl= xx,yy,xy) represent the stresses in the region of material i, K, ^ and Kn ^ represent the stress intensity factors of the corner for Mode I and Mode II deformations, respectively, and fluid) and f"Li{9) are the coefficient functions of 9. The displacements around the corner are described by the stress intensity factors K, ^ and K

u,x> a s r*K„

«ZW>

IÎKG,

2-JlxG,

(7)

where ut and vt are components of the displacement in the x and >> directions within the region of the materials /, and #(/(#)> g"i(&)> giAfy

and

g2,i(0)

are the

coefficient functions of 9. — 89 —

The stress intensity factors Kj ^ and Kn ^ are calculated from Eq. (7) using the displacement extrapolation method. 2. Stress intensity factors of an interface crack between dissimilar materials If a coordinate system around an interface crack is defined as shown in Fig. 2, the asymptotic solution of the stress components along the jc-axis in the vicinity of an interface crack tip is expressed as [2]

fc+^L

ie _K1 + iKIL( \ r V2/zr

(8)

where Kt and Kn represent the stress intensity factors of an interface crack for the mode I and mode II deformations respectively, lk is the characteristic length, which can be taken as an arbitrary constant, and £ is the bimaterial constant defined as

/

1 "\ If

Vî

T

8=—ln| 2TT

iJ

V (9)

\KJ2

Material 1

Figure 2: A coordinate system around an interface crack The value of characteristic length / should be fixed at a certain value. It is difficult to decide the most suitable value of Z^, but it is expected to be close to the size of a fracture process zone [3]. For convenience, we use 4=10 //m in this study. When the value of lk is changed to tk, Kl and Kn are transformed into K) and Kin as follows [4]:

K

\ uJ

cosQ

-sinQ

_sinQ

cosQ

(10)

Q = s\x\>>0 andy x ) = S ^ u f

(©) - A t f - ( © )

(16)

where A is a constant transformation matrix. When a nonlinear analysis is carried out by changing its shear modulus and damping ratio which are equivalent to a reference strain in the region, the dynamic stiffness matrix for the near-field soil shall be modified at each iteration step. However, the earthquake force vector can be assumed unchanged during the iteration process, if the far-field region was selected so far from the structure that nonlinear behavior of soil may be negligible in the region. For the purpose of time domain earthquake response analysis, the equivalent earthquake input force, rbee (co), is firstly computed along the interface in the frequency domain. Then, the earthquake forces on the interface in the time domain, fb (t) = xbee (t), are computed through the discrete Fourier transform technique. Finally, the earthquake responses are computed by solving Eq. (15), in which force vector fs(t) is equal to zero [41]. VERIFICATION EXAMPLES 1. A Rigid Strip Foundation on an Elastic Layer with Rigid Bedrock (2D IE) A rigid strip foundation on an elastic soil with rigid bedrock is modeled as in Fig. 6. Two Rayleigh waves and two body waves are used in the infinite element formulation. The vertical compliance of the foundation was obtained using the numerical and closed-form frequency-dependent infinite elements, and compared with each other along with reference solution by Tassoulas [48] in Fig. 6. Very good agreements have been found between them. 2. A Rigid Circular Disk on Layered Hals-Space Medium (Axisymmetric IE) Horizontal impedance has been obtained for the case that the depth of the upper horizontal layer (h) is RQ . Both the horizontal layer and the halfspace are assumed to be elastic, homogeneous and isotropic but having different soil properties; i.e., shear wave velocities (C 5l / Cs2 = 0.8), densities (p 1 / p 2 = 0.85) and Poisson's ratio (v t = v2 = 0.25). Two body waves and three Rayleigh waves have been also used for infinite elements. The effect of the horizontal dimension of the nearfield has been also investigated using three different meshes shown in Fig. 6; i.e., r0 = 12RQ , \5RQ and 2RQ . The results shown in Fig. 7 indicate that reasonable solution can be obtained if the horizontal dimension of the near-field is taken to be larger than 15RQ . The results are also found to be in good agreement with the analytical solutions obtained by Luco [49]. — 159 —

Tassoulas o

Numerical E

*

Analytical E

H=2b\

a0

(a) FE & IE mesh Figure 6: r*

-cobIcs

(b) Compliance of vertical motion

Vertical compliance of a rigid strip foundation on a layer with fixed bedrock (2D analysis) **--

■■-■■H

1 r™

V f h h TTT\y

-*M)

h«-

-■•»-j

^ 1 11 r' \1

y |7\r\

^X&\ P\n WE

^V)

■-H

^ 11 1 1 / \\\

\1 /

Ar\\

f 1.4 | Luco

r0 / /2o = 15 r 0 //?o = 2.0

r

stu(

jy

J

=

Figure 7:

w^

Horizontal impedances of a rigid disk on a layered half-space using three different finite element meshes for the near-field (h IR^ = 1.0) (Axisymmetric analysis) Chow(1987) This study

*\

Ox / /xO

(a) Vertical impedance (h/B=4)

\

/ 0

(b) Vertical impedance (h/B=6)

Figure 8: Vertical impedance of rectangle footing on a soil layer with bedrock (L/B=2, v=0.33, \ =0.05) (3D analysis)

— 160 —

3. A Rigid Rectangular Disk on Layered Half-Space (3D IE) A rigid rectangular footing on a horizontal soil layer with an underlying rigid rock is analyzed. The impedance function of the rectangular footing is expressed as kz(a0) = ksz'a(k+ia0c)(l+2iZsd)^ w h e r e ks2'a ig t h e v e r t i c a l s t a t i c s t if m e ss. HIE and HCIE are used for the horizontal exterior region of the soil layer, while the boundary condition along the interface with the bed rock is taken as fixed. Two cases with different depth ratios (h = 4B and 6B ) are considered. Fig. 8 shows the results for the rectangular footing along with the results by Chow [50]. The present results are found to be in good agreement with the reference values. 4. A Strip Foundation on a Two-Phase Medium (2D Body) As a demonstrative extension of the present methodology, we present results of forced vibration of a strip foundation on a two-phase soil layer with fixed bedrock, as in Fig. 9. In the two-phase layers, the wave propagation mechanism is much more complicated than that in single-phase soil, e.g., three body wave components exist including P1-, P2- and S-waves. However, the infinite element formulation essentially remains the same and straightforward [37]. Fig. 9 shows the accuracy of the present method, and indicates that multiple wave components shall be considered in the analysis even though other researchers [35, 36, 38] obtained reasonable results by considering one or two wave components only. 20MPa —}

tyKMitt

iiimm HIE HIE

5m

m

HIE HIE HIE

\—1

= 20 rad/sec)

Xo/B = 11

X-coordinate (m)

Frequency (rad/sec)

Figure 9: Frequency response of a 2D strip foundation on a two-phase soil layer upon fixed bedrock i(Exact:

.) (FE only: ~^~)

(FE-IE : ♦ = including 1 wave, ■ = 2 waves, • = 3 waves)

5. A Rigid Circular Disk on Flexible Hollow Caisson (Axisymmetric IE and 3D IE) A rigid circular footing on a RC caisson shown in Fig. 10 is analyzed using the FE-IE approach. The axisymmetric and 3D infinite elements are employed for modeling the far-field of soil. The material properties are given in the figure. The impedance functions are normalized by GSR0 for the horizontal and the vertical components, while they are normalized by GSR03 for the rocking and the torsional components. The results are shown in Fig. 10, and compared with those based on the indirect boundary element method by Chen & Penzien [51]. Excellent agreements have been achieved using both types of axisymmetric and 3D infinite elements. APPLICATION EXAMPLES 1. Simulation of Forced Vibration Test & Earthquake Responses (Axisymmetric Body) This example presents the result of an international cooperative research on the post-correlation analysis of forced vibration tests and the prediction of earthquake responses of a large-scale seismic test (LSST) structure. Through the system identification technique, the properties of the soil layers are revised so that the best correlation in the responses is obtained as shown Fig. 11. The revised values are shown in Table 2 [41]. Utilizing the revised soil properties as the initial linear value, the seismic responses are predicted for an earthquake using the equivalent linearlization technique. It has been found that the predicted responses by the equivalent nonlinear procedure are in excellent agreement with the observed responses as shown in Fig. 12. — 161 —

Ve"»'

,m

| Ec/&=500 1 R;/ft=1.5 \ v c =v,=025 \ ^=0.02

^=oo

\

\\

1 1

* 30

I 0.6 m ME ME

12m

,.

-

0)

impec

mmm

£

40

Cats*n

"3

ME

N

ME

0

2°

Chen and Penzien o

V

Presentstudy

CHH

-

~-^r—1>^::8

(

^HH

ME

Pi"

ME

Ro=4m

a i

ME

«-

_

vm

VIE VIE

CE

in

0.0

0.5

1.0

i

i

i

1.5

i

i

i

i

2.0

, (oR>

(a) Using axisymmetric FE-IE Chan & Penzlen(19B6)

I

This study

120

i

k„M

•• •

(b) Using 3D FE-IE Figure 10: Impedances of a rigid disk on a hollow caisson embedded in homogeneous half-space 8.0 6.0

o Test (D^

H15

—Calculated

4.0"2.0 0.0-

n 5

1 ■ r10 15 Frequency (Hz) ■

Figure 11: Simulated and measured forced vibration test (Di-direction) — 162 —

20

1 10 Frequency (Hz)

1 10 Frequency (Hz)

Figure 12: Response spectra of simulated and measured accelerations in structure (Di-direction) Table 2

Identified values of the parameters for from the Hualien LSST structure using FVT data, and equivalent linear properties (Note: Values in parentheses are hysteretic damping ratios in percent.) FVT simulation & Linear Earthquake Response Analysis

Parameters (V s in m/sec, E in GPa)

133(2.0),

Sand-l,Sand-2(V s )

Nonlinear Earthquake Response Analysis (May 1, 1995)

231(2.0)

113(4.7),

199(4.5)

Backfill-1, SB-1 (Vs)

270 (2.0)

244 (2.4),

207 (5.3)

Backfill-2, SB-2 (Vs)

325 (2.0)

257 (4.8),

234 (6.4)

Gravel-1, SG-1 (Vs)

308 (2.0)

262 (5.0),

248 (6.0)

Gravel-2, SG-2 (Vs)

281 (2.0)

214(7.0),

203(7.9)

211(7.2),

340(4.4)

Gravel-3, Gravel-4 (Vs)

333 (2.0),

388 (2.0)

Roof & Base (E)

28.2 (2.0)

28.2 (2.0)

Shell (E): Upper, Middle, Lower

19.7 (2.0), 21.3 (2.0), 21.8 (2.0)

19.7 (2.0), 21.3 (2.0), 21.8 (2.0)

|

2. Time Domain SSI Analysis For verification of the proposed time-domain SSI analysis procedure, earthquake response analysis of a multi-layered free field half-space shown in Fig. 13 is carried out. The near-field soil region is discretized with plane strain finite elements and the remaining far-field soil region is modeled by the analytical frequency-dependent infinite elements [39]. The properties of the soil layers are shown in Table 2. A horizontal acceleration record is used as the input control motion on the ground surface, which is the NS-component of an earthquake measured at Hualien, Taiwan on January 20, 1994. The peak ground acceleration is 0.0318g, and the time history is shown in Fig. 13. The acceleration histories are compared with those of the free field analysis which are obtained based on the frequency domain method. The results in Fig. 13 show excellent agreements. (a) Control motion (Free surface)

9-node FE AI Sand 1

* * * *

2.0 m

V

> \

3.15 m

Sand 2

I

Graved

i

7.0 m

J

' Half-space Boundary for

|

|

|

|

|

|

|

êarthquake force input

3-nodevlE

C 0.00 < -0.04

Vertically Incident P-wave and SV-wave

(a) Free field analysis problem

Time (sec)

(b) FE-IE mesh

(c) Time history responses

Figure 13: Earthquake response analysis of a layered soil medium — 163 —

3. Flexible RC tank on a layered half-space (Axisymmetric Body) In order to demonstrate the soil-structure interaction effect on the member forces of a liquid storage tank, a stress analysis is carried out for a structure depicted in Fig. 14 under various soil conditions. The structure is supported by a horizontal layer with the underlying bedrock. Three values of the shear wave velocity for the horizontal soil layer, i.e., 500m/s, 800m/s, and 5000m/s, are considered in this investigation. The other material properties for the structure and soil regions are given in Fig. 14. The frequency-dependent mass matrix associated with the DOF along wetted interface is included in the equation of motion in the frequency domain, in which the fluid motion is represented by Bessel functions [46]. An acceleration time history with PGA of 0.14g, which is compatible with a design response spectrum for a rock site is simulated for the earthquake input as in Fig. 12. In this analysis, the control acceleration is assigned at the top of bedrock as a horizontal outcrop motion. Thus, seismic motion can be amplified at the ground surface depending on the properties of the horizontal soil layer. Member forces are calculated on the vertical shell for three different soil conditions including both the fluid-structure interaction and soil-structure interaction, and their maximum values are plotted along the height of the structure in Fig. 14. For the purpose of comparison, the maximum member forces are also computed using ANSYS program for the same structure but on a rigid ground. A fully coupled fluid-strucrure-soil interaction analysis cannot be carried out by ANSYS program. In ANSYS analysis, the input ground acceleration at the fixed base is prepared for each soil condition by carrying out the free-field analysis using SHAKE91 program. Thus, the solution by ANSYS can be considered as the response for the same input motion but excluding the soilstructure interaction effect. Two sets of the results for a rigid soil condition by the present and ANSYS analysis (in Fig. 14) are found in good agreements, which confirms the accuracy of the present analysis. The results for the softer soil conditions indicate that the member forces on the shell reduce considerably as the soil stiffness decreases. This result re-confirms that accurate dynamic analysis of a large liquid storage tank considering the soil-structure interaction may yield cost-effective cross-section for the structure. C.L.

I H=40m

50m 3m Compliant Soil

/?=30m

! water pw=1.0Mg/m3

RC Shell pc=2.6Mg/m3, Es=30MPa, vc=0.2, h=2%

"f" 20m

(p s =2.5Mg/m\ v s =0.2, h=2%) Bedrock (ps=2.5Mg/m\ Vs=5000m/sec vs=0.2, h=2%)

104m

^ 1 SSI excluded ^ z | (ANSYS) SSI included (Present Study)

2000 4000 6000 Member force (KN/ m)

(a) Ks=5000m/s

2000 4000 6000 Member force (KN/ m)

(b) Ks=800m/s

2000 4000 6000 Member force (KN/m)

(c) Ks=500m/s

Figure 14: RC liquid storage tank and maximum member force profiles for a RC liquid storage tank (N t at 0 = 0°, N z at 0 = 0°, and Nte at 0 = 90°) — 164 —

CONCLUDING REMARKS This paper presented infinite element formulations for the dynamic SSI analysis of an axisymmetric body subject to 3-D loads, a 2D problem, and a 3D analysis both in frequency and in time domains. The displacement shape functions of the infinite elements were constructed using approximate expressions of analytical solutions in frequency domain to represent the characteristics of multiple waves propagating into the infinite multi-layered soil medium. For the sake of time domain analysis, the dispersive wavenumbers included in shape functions were linearized to obtain closed-form frequency-dependent element matrices, which can analytically be transformed into the time domain terms by a continuous Fourier transform. The proposed elements were verified using several benchmark examples. Comparisons with the results by other studies showed that the present formulations are very effective for the SSI analysis either in frequency or in time domain. Example applications to actual SSI problems were also given to demonstrate the versatility and effectiveness of the present methodology. Acknowledgement This study has been supported by SISTEC (KOSEF/MOST: Grant No. Rl 1-2002-101-03001-0). Their financial supports are greatly acknowledged. Also the second author would like to thank KOSEF (Grant No. R01-2003-00010635-0) for the financial support to this study. REFERENCES 1. Wolf JP. Dynamic Soil-Structure Interaction, Prentice-Hall, 1985. 2. Ungless RF. An infinite finite element. MS Thesis, University of British Columbia, Vancouver, Canada, 1973. 3. Bettess P. Infinite elements. Int. J. Num. Meth. Eng., 1977; 11: 54-64. 4. Astley RJ. Wave envelope and infinite elements for acoustic radiation. Int. J. Num. Meth. Fluids, 1983; 3: 507-526. 5. Zienkiewicz OC, Emson C, Bettess P. A novel boundary infinite element. Int. J. Num. Meth. Eng., 1983; 19: 393-404. 6. Chow YK, Smith IM. Static and periodic infinite solid elements. Int. J. Num. Meth. Eng., 1981; 17: 503-526. 7. Beer G, Meek JL. Infinite domain elements. Int. J. Num. Meth. Eng., 1981; 17: 43-52. 8. Lynn PP, Hadid HA. Infinite elements with 1/r" type decay. Int. J. Num. Meth. Eng., 1981; 17: 347-355. 9. Rajapakse RKND, Karasudhi P. Elastostatic infinite elements for layered half space. J. Eng. Mech., ASCE, 1985; 111: 1144-1158. 10. Koh KH, Lee SR. p-Version static infinite element for representing 1/r" type decay problems in unbounded media. Computers and Geomechanics, 1998; 22: 73-89. 11. Schrefler BA, Simoni L. Non-isothermal consolidation of unbounded porous media using mapped infinite elements. Comm. Num. Meth. Eng., 1987; 3: 445-452. 12. Karpurapu GR. Composite infinite element analysis of unbounded two-phase media. Adv. Eng. Software, 1988;10:202-209. 13. Honjo Y, Pokharel G. Parametric infinite elements for seepage analysis. Int. J. Num. Anal. Meth. Geomechanics, 1993; 17: 45-66. 14. Zhao C, Valliappan S. Transient infinite elements for seepage problems in infinite media. Int. J. Num. Meth. Eng., 1993;17:323-341. 15. Bettess P, Zienkiewicz OC. Diffraction and refraction of surface waves using finite and infinite elements. Int. J. Num. Meth. Eng., 1977; 11: 1271-1290. 16. Lau SL, Ji Z. An efficient 3-D infinite element for water wave diffraction problems. Int. J. Num. Meth. Eng., 1989; 28: 1371-1387. 17. Park WS, Yun CB, Pyun CK. Infinite elements for evaluation of hydrodynamic forces on offshore structures. Computers & Structures, 1991; 40: 837-847. 18. Park WS, Yun CB, Pyun CK. Infinite elements for 3-dimensional wave-structure interaction problems. Eng. Structures, 1992; 14: 335-346. — 165 —

19. Astley RJ, Macaulay GJ, Coyette JP. Mapped wave envelope elements for acoustic radiation and scattering. J. Sound and Vibration, 1994; 170: 97-118. 20. Bettess P, Abram RA. Finite and infinite elements for a simple problem in quantum mechanics. Comm. Num. Meth. Eng., 2002; 18: 325-334. 21. McDougall MJ, Webb JP. Infinite element for the analysis of open dielectric waveguide. IEEE Trans. Microwave Theory and Techniques, 1999; 37: 1724-1731. 22. Bettess P. Infinite elements. Penshaw Press, UK, 1992. 23. Astley, RJ. Infinite elements for wave problems: a review of current formulation and an assessment of accuracy. Int. J. Num. Meth. Eng., 2000; 49: 951-976. 24. Medina F, Penzien J. Infinite elements for elastodynamics. Earthq. Eng. & Str. Dyn., 1982; 10: 699-709. 25. Medina F, Taylor RL. Finite element techniques for problems of unbounded domains. Int. J. Num. Meth. Eng., 1983; 19: 1209-1226. 26. Rajapakse RKND, Karasudhi P. An efficient elastodynamic infinite element. Int. J. Solids & Structures, 1986; 22: 643-657. 27. Yang SC, Yun CB. Axisymmetric infinite elements for soil-structure interaction analysis. Eng. Structures, 1992; 14: 361-370. 28. Karasudhi P, Liu YC. Vibration of elastic and viscoelastic multi-layered spaces. Str. Eng. & Mech., 1993; 1: 103-118. 29. Zhang C, Zhao C. Coupling method of finite and infinite elements for strip foundation wave problems. Earthq. Eng. & Str. Dyn., 1987; 15: 839-851. 30. Yun CB, Kim JM, Hyun CH. Axisymmetric elastodynamic infinite elements for multi-layered half-space. Int. J. Numer. Meth. Eng., 1995; 38: 3723-3743. 31. Yang YB, Hung HH. A 2.5D finite/infinite element approach for modelling visco-elastic bodies subjected to moving loads. Int. J. Num. Meth. Eng., 2001; 51: 1317-1336. 32. Zhao C, Valliappan S. A dynamic infinite element for three-dimensional infinite-domain wave problems. Int. J. Numer. Meth. Eng., 1993; 36: 2567-2580. 33. Park KR. Development of three-dimensional frequency-dependent dynamic infinite elements for multi-layered elastic media. PhD Thesis, Kyoto University, Japan, 2004. 34. Seo CK, Yun CB, Kim JM. Three-dimensional cuboidal infinite elements for soil-structure interaction analysis in multi-layered half-space, (in preparation). 35. Khalili N, Valliappan S, Yazdi JT, Yazdchi M. ID infinite element for dynamic problems in saturated porous media. Comm. Num. Meth. Eng., 1997; 13: 727-738. 36. Khalili N, Yazdchi M, Valliappan S. Wave propagation of two-phase saturated porous media using coupled finite-infinite element method. Soil Dyn. & Earthq. Eng., 1999; 18: 533-553. 37. Kim JM. Two-dimensional infinite element for dynamic analysis of saturated two-phase soil. J. Earthq. Eng. Soc. Korea, 2005; 9: 67-74. 38. Wang G, Chen L, Song C. Finite-infinite element for dynamic analysis of axisymmetrically saturated composite foundations. Int. J. Num. Meth. Eng., 2006 (in press). 39. Yun CB, Kim DK, Kim JM. Analytical frequency-dependent infinite elements for soil-structure interaction analysis in two-dimensional medium. Eng. Structures, 2000; 22: 258-271. 40. Kim DK, Yun CB. Time domain soil-structure interaction in two-dimensional medium based on analytical frequency-dependent infinite elements. Int. J. Num. Meth. Eng., 2000; 47: 1241-1261. 41. Kim DK, Yun CB. Earthquake response analysis in time domain for 2-D soil-structure systems using analytical frequency-dependent infinite elements. Str. Eng. Mechanics, 2003; 15: 717-733. 42. Yun CB, Choi JS, Kim JM. Identification of the Hualien soil-structure interaction system. Soil Dyn. & Earthq. Eng., 1999; 18: 395-408. 43. Choi JS, Yun CB, Kim JM. Earthquake response analysis of the Hualien large scale seismic test structure using updated soil-structure properties based on vibration test data. Earthq. Eng. & Str. Dyn., 2001; 30: 1-26. — 166 —

44. Choi JS, Lee JS, Yun CB. Input and system identification of the Hualien soil-structure interaction system using earthquake response data. Earthq. Eng. & Str. Dyn., 2003; 32: 1955-1975. 45. Choi JS, Lee JS, Yun CB. Identification of the soil-structure interaction system using earthquake response data. J. Engrg. Mech., ASCE, 2004; 130: 753-761. 46. Kim JM, Jang SH, Yun CB. Fluid-structure-soil interaction analysis of cylindrical liquid storage tanks subjected to horizontal earthquake loading. Str. Eng. Mechanics, 2002; 13: 615-638. 47. Yun CB, Kim JM. KIESSI - A computer program for soil-structure interaction analysis using finite and infinite element techniques. Res. Rep., Dept. of Civil Eng., KAIST, Korea, 1996. 48. Tassoulas JL. Elements for the numerical analysis of wave motion in layered media. PhD Thesis, MIT, Cambridge, Mass., USA, 1981. 49. Luco JE. Impedance functions for a rigid foundation on a layered medium. Nucl. Eng. Des., 1974; 31: 204217. 50. Chow YK. Vertical vibration of three-dimensional rigid foundation on layered media. Earthq. Eng. & Str. Dyn., 1987; 15: 585-594. 51. Chen CH, Penzien J. Dynamic modeling of axisymmetric foundation. Earthq. Eng. and Str. Dyn., 1986; 14: 823-840. 52. ANSYS Inc., ANSYS 5.5, User's Reference Manual, 1999. 53. Schnabel PB, Lysmer J, Seed HB. SHAKE91 —A computer program for earthquake response analysis of horizontally layered sites. Earthquake Engineering Research Center, UCB, USA, 1991.

— 167 —

COMPUTATIONAL METHODS IN ENGINEERING AND SCIENCE EPMESC X, Aug 21-23, 2006, Sanya, Hainan, China ©2006 Tsinghua University Press & Springer

Monotonic and Cyclic Analysis of Granular Soils N. Khalili *, M. A. Habte, S. Valliappan School of Civil and Environmental Engineering, The University of New South Wales, Sydney, NSW, 2052, Australia Email: [email protected] Abstract A simple yet rigorous bounding surface plasticity model is developed to describe the stress-strain behaviour of variably saturated granular soils subjected to monotonic and cyclic loading. The model is formulated incrementally within the critical state framework using the effective stress approach. Both plastic volumetric strain and matric suction are introduced as hardening parameters. Cyclic behaviour is captured through a radial mapping rule in which the point of stress reversal is taken as the centre of projection. The effect of particle crushing at high stresses is considered through a three-segmented critical state and isotropic compression lines. A non-associative flow rule is employed to generalise application of the model to all soil types. The model is validated using experimental data from the literature. Key words: bounding surface plasticity, effective stress, granular soils, cyclic loading, suction hardening INTRODUCTION Response of granular materials to monotonic and cyclic loadings is complex due to the pressure and specific volume dependency of the stress-stain relationship and the highly nonlinear behaviour of the soil matrix. This is particularly the case under undrained conditions in which repeated loading and unloading can lead to a substantial rise in pore water pressure and a sudden loss in the shear strength and the stiffness of the soil. In fact, the study of loading and unloading response in soils and development of relationships for its prediction in natural formations and engineered materials has been a major area of research in modern geomechanics. Concerted effort has been made to develop predictive capabilities associated with such topics as earthquake engineering, soil-structure interaction, soil liquefaction, off-shore engineering, etc. The objective of this paper is to present a comprehensive and unified elasto-plastic constitutive model for monotonic and cyclic loading of granular soils. The model is formulated using the bounding surface plasticity theory within a critical state framework. A new bounding surface along with a radial mapping rule is introduced to obtain a more realistic response under monotonic and cyclic loading. Other crucial aspects taken into account are the nonassociativity of the flow rule, and the crushing of the soil particles at high stresses. Crushing of the particles is captured through the introduction of a three-segmented critical state and limiting isotropic compression lines. The model is developed within the context of both fully and partially saturated soils. Unsaturated behaviour is taken into account using the effective stress principle, and introducing suction as a hardening parameter. PRELIMINARIES 1. Notation The material behaviour is assumed isotropic and rate independent in both elastic and elastic-plastic responses. For simplicity, triaxial stress notation q~ p is adopted throughout; p' = ^{cr[ + 2<j'7)) is the mean effective stress and q = (&[-&[)

is the deviator stress, where a[ and G[ are the axial and radial stresses

respectively. The corresponding work conjugate strain variables are the soil skeleton volumetric strain sp = s{ + 2s3 and shear strain sq = 2/3 (î - £3). The pairs of stresses and strains are abbreviated in the vector form as a' = [p', qf and € = [ep, sq ] . Following soil mechanics convention, compression is considered positive and tension is negative. 2. The Effective Stress In its incremental form, the effective mean stress is expressed as [1] — 59 —

P'' = P - Pa+V(Pa-

PW)

= Pnet+VS

0 ) f

Where, a superimposed dot indicates the rate of change, p is the effective mean stress, p is the total mean stress, pw is the pore water pressure, pa is the pore air pressure, pnet = p- pa is the net mean stress, s = pa— pw is the matric suction, and y/ is the incremental effective stress parameter, attaining a value of one for saturated soils and zero for dry soils. The effective stress parameter describes the contribution of suction to the effective stress. It is a scaling factor that averages matric suction from the pore-scale level to a macroscopic level over a representative elementary volume. In this formulation, the effective stress parameter is defined using an experimentally obtained correlation between soil suction and the effective stress [1-3],

for

1

s <se

-0.55

¥ =

.45U 0

|

far se <s 25se

(2)

where se is the suction value separating saturated from unsaturated states. For wetting processes, se = s^, and for drying processes se = sae, in which sa is the air expulsion value and sae is the air entry value. For fully saturated soils, y/ — 1 and equation (1) reduces to p' = p — pw. 3. The Critical State The critical state (CS) is an ultimate condition towards which all states approach with increasing deviatoric shear strain. To take into account crushing of soil particles at high stresses [4-6], the critical state line in the v ~ In/?' plane is assumed to take the form of three linear segments [4].

Figure 1: Illustration of critical state line (CSL), limiting isotropic compression line (LICL), state parameter © and elastic unload/reload line on v ~ In/? 'plane As shown in Figure 1, six parameters are used to define the critical state line (CSL) in the u ~ In/?' plane: Ao, r0, ucn Acr, Uf and Af. The parameters Ao and r0 are the slope of the initial portion of the CSL and its specific volume at p' = IkPa, respectively; vcr is the specific volume at the onset of particle crushing; Acr is the slope during the particle crushing stage; and Vf and Af are the specific volume at the end of crushing and the slope of the CSL at extremely high stresses, respectively. The CSL in the q~ p' plane is defined using a straight line passing through the origin. The slope of the CSL, denoted by Mcs, is linked to the critical state friction angle ((j)'cs) through, A/ =

6sin^ 3r-sin^

(3)

with t = +1 for compression and t = -1 for extension. In addition, certain features of the model are linked to the state parameter (%), which is a dimensionless parameter defined as the vertical distance between the current state and the CSL in the v~\np 'plane (Fig. 1).

— 60 —

4. Limiting Isotropic Compression Line Similar to the critical state line, the limiting isotropic compression line (LICL) is a reference line in the v~\np' plane where the stress state approaches with increasing isotropic compression. For granular soils, the limiting isotropic compression line can also be regarded as the locus of the loosest possible state that a soil can achieve for a given effective mean stress. In the present investigation, the LICL is taken as parallel to CSL at a constant shift along the K line in the v ~ In pf plane (Figure 1). For partially saturated soils, the stiffening effect of suction on the soil response is manifested through its effect on the position of the LICL. In general, a suction increase has a dual effect of increasing the slope and vertical positioning of the LICL [12]. STRESS-STRAIN RELATIONSHIP The total strain rate is decomposed into elastic and plastic parts according to, £ = ee+sp

(4)

where superscripts e and p denote the elastic and plastic components, respectively. Incremental elastic strains are linked to the incremental stresses through, d = Deee

(5) e

where D is the elastic property matrix. The incremental plastic strain-stress relationship is written as, e p =-mnPo) = t\ q-

AMcsp'

p'

A-\

(14)

A-l{Po

Typical shapes of the plastic potential are shown in Figure 5. Notice that two families of curves are identified: M*s for compressive loading (q > 0) and M~s for extensive loading (q C3-H > C4-H. The least energetically favorable contact is thus observed for the system "polymer - H-terminated graphite surface", while the most energetically favorable contact for the case of polymer adsorption on the hydrogen-terminated surface containing defects with sp2/sp3-hybridized carbon atoms and having hole-shaped and five-membered defects C4-H. Diagrams of forces of friction for the studied surfaces also substantiate the obtained conclusions. Really, force of shear microscopic friction is minimum for the case of "molecular friction" of a polymer chain segment along a strictly periodic graphite surface (Cl-H). The same result has been obtained for models both nonterminated and terminated by hydrogen. The energy of shear strain for a polymer segment along the hydrogen-terminated surface is also maximal for the case of the most defect surface (C4-H). Comparison of results obtained for nonterminated and terminated surfaces shows that force of microscopic friction along the terminated surface is about 0.9-0.7 of microscopic friction along the nonterminated surface. 3. Structure and adsorption properties of the complex "polymer - 3D carbon particle" Let us now turn to the structure and surface adsorption characteristics of 3D carbon particles imbedded into a medium of polymer molecules of different chemical structure. Commercial carbon we simulated by two 3D clusters with 170 and 670 atoms. The first particle was ~ 15 A in diameter and the second one ~ 25 A. The core of the particles had the structure of carbon in sp3-configuration, while their surface had both C-sp2 and C-sp3 structure, i.e. the surface contained both purely graphite and diamond-like structures arranged without short-range order and presented amorphous carbon. Consider results for adsorption complexes with the basic model of an amorphous carbon-black particle in the form of a carbon cluster ~ 15 A in diameter with 170 atoms. Adsorption of polymer molecules of different chemical nature was investigated, namely, polyethylene, polypropylene, polyacrylic acid, polyurethane, and polyvinyl acetate (oligomers CH3-(R)„-CH3, where n = 5). Adsorption of these molecules on the surface of model carbon-black particles with hydrogen-nonterminated and terminated surface was calculated; corresponding structures of complexes are given in Table 1. Table 1 Optimized models of adsorption complexes Polymer fragment

Adsoiption on nonterminared carbon black surface

Adsoiption on H-terminated carbon black surface

Polyethylene (PE)

Polypropylene (PP)

Polyacrylic acid (PAA)

Polyurethane (PU)

Polyvinyl acetate ' (PVA)

For analyzed adsorption complexes we calculated binding enthalpy and energy of shear strain during motion of a polymer fragment by 1 A along the carbon-black surface, which are referred to one monomelic unit (Figs. 2 and 3). — 132 —

Based on the above-given data, the following conclusions can be made. Adsorption energy and force of a polymer fragment on the amorphous carbon-black surface depends on the hydrophobic contact area (groups CH2 and CH3) as well as on the presence of electronegative atoms (O, N) in the polymer structure. Besides, conformation of a polymer fragment, namely, complementary character of its geometry on the carbon-black surface, also influences its adsorption. For example, adsorption enthalpy and energy of shear strain of the polypropylene monomer is ~ 1.5 times higher than that of polyethylene, which is due to a larger hydrophobic contact area of polypropylene. Adsorption enthalpy and energy of shear strain of the polyurethane monomer is ~ 1.7 times higher than that of polyethylene. At a relatively similar contact area, electronegative atoms in the polyurethane structure play a crucial role in this case. Polyvinyl acetate has the most favorable contact of a polymer fragment (~ 2.0 times higher than for polyethylene), which can be due to a simultaneous influence of two factors such as hydrophobic contact area and electronegative oxygen atoms.

4U

■£ r. o aM

-3

A

-2 3

t

PE ▲

t

-3 2 PP

-4

-2 0 PAA

A

t

-3 6 PV

A

f_

W

-4 1 PVA

35

& 30 %»< 0» -^

.=J c:

re l>

20 M

15 Oligomers of different chemical nature (see Table 1)

Figure 3: Adsorption enthalpy (a) and force of microscopic molecular friction (b) for an organic polymer segment CH3-(R)„-CH3, n = 5, calculated for one monomelic unit -R- for the model of a carbon black particle with hydrogen-terminated surface — 133 —

For polyacrylic acid having electronegative atoms, conformation of a polymer chain fragment adsorbed on the carbon-black surface is not optimal for contact interaction. This is demonstrated by a relatively low adsorption enthalpy and energy of shear strain of its monomer on the carbon-black surface. For a hydrogen-terminated surface, all regularities revealed above are qualitatively similar. However, adsorption enthalpy and energy of shear strain on such a surface are ~ 0.7 times lower. 4. Strain characteristics of the adsorption complex "polymer - 3D carbon particle". Uniaxial deformation and molecular friction Let us now consider strain characteristics of adsorption complexes that contain a cluster model of an amorphous carbon-black particle with a hydrogen-terminated surface ~ 15 A in diameter with 170 atoms and polymer molecules, such as polyethylene, polybutadiene and polyisoprene. Model oligomers were individual molecules CH3-(R)„-CH3, where n = 10. In the quantum-mechanical approximation we calculated energetically optimal structures of the mentioned adsorption complexes and their geometrical, energy and strain characteristics (Table 2). Table 2 also gives minimum distances between the polymer molecule and carbon black particle R, binding enthalpy AHbind calculated for one monomeric unit of polymer, maximum uniaxial tensile force (critical force of rupture) Fdeform,max of a polymer chain, and maximum microscopic friction force FShift,maxFor each model adsorption complex two microscopic strain characteristics were calculated: (1) the dependence energy of strain of a polymer chain contacting with a carbon-black particle on the elongation value (uniaxial tension of the polymer chain up to molecular bond rupture) and (2) force of microscopic friction (change in the adhesive force of the polymer chain and particle surface when a polymer molecule moves along its surface). Corresponding force curves are given in Fig. 4. One can see from the obtained data that stress-strain curves for polyethylene, polybutadiene and isoprene molecules have a common regularity from the qualitative standpoint. In the first curve portion the force of strain is almost constant and corresponds to low tensions; this is an entropy region of torsional and conformational changes of the polymer chain. In the second portion the curve changes abruptly; this is Hook's or enthalpy region where deformation (elongation) of valence bonds takes place. The maximum in the curve corresponds to the critical rupture force of the polymer chain, after which the analyzed curve drops sharply. From the qualitative point of view, it can be stated that in the series from polyethylene to polybutadiene and isoprene the entropy region of uniaxial deformation increases and the critical rupture force of the chain decreases. This bears witness to the fact that at the microscopic level elastic properties of polymer molecules improve in the considered series, and they are most pronounced for polyisoprene. This is known to correspond to experimentally observed facts. Table 2 Model adsorption complexes of polymer molecules with a carbon black particle and their geometrical, energy and strain characteristics calculated in quantum-mechanical simulation Optimized structure of the complex

us

Complex Polyethylene (PE) - carbon black particle

* Polybutadiene (PB) - carbon black particle

Polyisoprene (PI) - carbon black particle

&

-Fdefomuiwx^

kcal mole

kcal, (mole •

A)

■''shift.max-

kcal(mole • A) 8.3

3.2-3.4

-3.9

120.3

3.0-2.9

-4.2

114.8

16.1

2.6-2.8

-5.5

103.1

23.9

&

Ot

lH*

— 134 —

(a)

^ 120 r

)

•< 120

•

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