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INVERSE PROBLEMS IN ENGINEERING MECHANICS II International Symposium on Inverse Problems in Engineering Mechanics 2000 (ISIP 2000) Nagano,Japan
Editors
M. Tanaka Department of Mechanical Systenns Engineering Shinshu University, 4-17-1 Wakasato, Nagano 380-8553 Japan
G.S. Dulikravich Department of Mechanical and Aerospace Engineering University of Texas at Arlington Arlington,TX, 76019, USA
2000
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Preface Inverse Problems can be found in many areas of engineering mechanics. There are numerous successful applications in the fields of inverse problems. For example, nondestructive testing and characterization of material properties by ultrasonic or X-ray techniques, thermography, etc. Generally speaking, the inverse problems are concerned with the determination of the input and the characteristics of a system given certain aspects of its output. Mathematically, such problems are ill-posed and have to be overcome through development of new computational schemes, regularization techniques, objective functional, and experimental procedures. Following the first lUTAM Symposium on these topics held in May 1992 in Tokyo, another in November 1994 in Paris, and also the last ISIP'98 in March 1998 in Nagano, we concluded that it would be fruitful to gather regularly with researchers and engineers for an exchange of the newest research ideas. The proceedings of these symposia were published and are recognized as standard references in the field of inverse problems. The most recent Symposium of this series "International Symposium on Inverse Problems in Engineering Mechanics (ISIP2000)" was held in March of 2000 in Nagano, Japan, where recent developments in inverse problems in engineering mechanics and related topics were discussed. The following general areas in inverse problems in engineering mechanics were the subjects of the ISIP2000: mathematical and computational aspects of inverse problems, parameter or system identification, shape determination, sensitivity analysis, optimization, material property characterization, ultrasonic non-destructive testing, elastodynamic inverse problems, thermal inverse problems, and other engineering applications. A number of papers from Asia, Europe, and North America were presented at ISIP2000 in Nagano, Japan. The detailed data of the ISIP2000 is available on the Internet (http://homer.shinshu-u.ac.jp/ISIP20Q0/). The final versions of the manuscripts of sixty-two papers from these presentations are contained in this volume of the ISIP2000 proceedings. These papers can provide a state-of-the-art review of the research on inverse problems in engineering mechanics. As the editors of the topical book, we hope that some breakthrough in the research on inverse problems can be made and that technology transfer will be stimulated and accelerated due to its publication. As the chairpersons of the ISIP2000 Symposium, we wish to express our cordial thanks to all the members of the International Scientific Committee and the Organizing Committee. Financial support from the Japanese Ministry of Education, Science, Sports and Culture (Monbusho) is gratefully acknowledged. Co-organizership by the University of Texas at Arlington, USA and Ecole Polytechnique, France is heartily appreciated. Also, co-sponsorship by the Japanese Society for Computational Methods in Engineering (JASCOME) and helpful support by the staff of Shinshu University in managing the financial support from Monbusho are gratefully acknowledged. June 2000 Masataka TANAKA, Shinshu University, Japan George S. DULIKRAVICH, The University of Texas at Arlington, U.S.A.
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Symposium Chairpersons Prof. Masataka TANAKA Department of Mechanical Systems Engineering Faculty of Engineering Shinshu University 4-17-1 Wakasato, Nagano 380-8553, Japan Fax: +81-26-269-5124, Tel: +81-26-269-5120 E-mail:
[email protected] Prof. George S. Dulikravich Department of Mechanical and Aerospace Engineering The University of Texas at Arlington Arlington, TX 76019, U. S. A. Fax: +1-817-272-5010, Tel: +1-817-272-2603 E-mail:
[email protected] International Scientific Committee Prof. Masa. Tanaka (Chair), Shinshu University (Japan) Prof. G.S. Dulikravich (Co-Chair), The University of Texas at Arlington (USA) Prof. H. Adeli, The Ohio State University (USA) Prof. C.J.S. Alves, Technical University of Lisbon (Portugal) Prof. S. Aoki, Tokyo Institute of Technology (Japan) Prof. M. Bonnet, Ecole Poly technique (France) Prof. H. D. Bui, Ecole Polytechnique & Electricite de France (France) Prof. T. Burczynski, Silesian Technical University of Gliwice (Poland) Prof. G. Chavent, Universite Paris-Dauphine and INRIA (France) Prof. L. Elden, Linkoping University (Sweden) Prof. H.W. Engl, Johannes-Kepler-Universitaet (Austria) Prof. D.B. Ingham, University of Leeds (UK) Prof. A.J. Kassab, University of Central Florida (USA) Prof. M. Kitahara, Tohoku University (Japan) Prof. S. Kubo, Osaka University (Japan) Prof. P. Ladeveze, ENS de Cachan (France) Prof. A.K. Louis, Universitaet des Saarlandes (Germany) Prof. G. Maier, Politecnico di Milano (Italy) Prof. S. Migorski, Jagiellonian University (Poland) Prof. N. Nishimura, Kyoto University (Japan) Prof. K. Onishi, Ibaraki University (Japan) Dr. R. Potthast, University of Goettingen (Germany) Prof. M. Reynier, ENS de Cachan/CNRS/Universite Paris VI (France)
Vlll
Prof. H. Sobieczky, DLR German Aerospace Research Center (Germany) Dr. B.I. Soemarwoto, National Aerospace Laboratory (The Netherlands) Prof. V.V. Toropov, University of Bradford (UK) Prof. N. Tosaka, Nihon University (Japan) Prof. I. Trendafilova, Katholieke Universiteit Leuven (Belgium) Prof. Z. Yao, Tsinghua University (China) Prof. W. Zhong, Huazhong University of Science & Technology (China)
Organizing Committee Prof. Masa. Tanaka (Chair), Shinshu University (Japan) Prof. G.S. Dulikravich (Co-Chair), The University of Texas at Arlington (USA) Prof. T. Matsumoto (Secretary), Shinshu University (Japan) Prof. K. Amaya, Tokyo Institute of Technology (Japan) Prof. S. Aoki, Tokyo Institute of Technology (Japan) Prof. M. Arai, Shinshu University (Japan) Prof. H. Azegami, Toyohasi University of Technology (Japan) Prof. H.D. Bui, Ecole Polytechnique & Electricite de France (France) Prof. T. Fukui, Fukui University (Japan) Prof. K. Hayami, University of Tokyo (Japan) Prof. S. Hirose, Tokyo Institute of Technology (Japan) Prof. Y. Honjo, Gifu University (Japan) Prof. M. Hori, University of Tokyo (Japan) Dr. H. Igarashi, Kagawa University (Japan) Prof. F. Imado, Shinshu University (Japan) Prof. Y. Iso, Kyoto University (Japan) Prof. K. Kagawa, Okayama University (Japan) Prof. J. Kihara, Himeji Institute of Technology (Japan) Prof. K. Kishimoto, Tokyo Institute of Technology (Japan) Prof. E. Kita, Nagoya University (Japan) Prof. M. Kitahara, Tohoku University (Japan) Prof. F. Kojima, Kobe University (Japan) Prof. S. Kubo, Osaka University (Japan) Prof. A. Murakami, Kyoto University (Japan) Prof. M. Nakamura, Shinshu University (Japan) Prof. N. Nishimura, Kyoto University (Japan) Prof. K. Onishi, Ibaraki University (Japan) Prof. N. Tosaka, Nihon University (Japan) Prof. M. Yamamoto, University of Tokyo (Japan)
Participants Abboudi, S.
France
Kishimoto, K.
Japan
Alves, Carlos J.S.
Portugal
Kitahara, M.
Japan
Amaya, K.
Japan
Knudby, C.
Spain
Aoki, S.
Japan
Kobayashi, A. S.
U.S.A.
Aral, M.
Japan
Kobayashi, S.
Japan
Azegami, H.
Japan
Koishi, M.
Japan
Bemtsson, F.
Sweden
Kojima, F.
Japan
Burczynski, T.
Poland
Koyama, Y.
Japan
Chen, W.
Japan
Kubo, S.
Japan
Cheng, J.
Japan
Lacour, F.
France
Chouaki, A.
France
Langenberg, K.J.
Germany
Liu, D.
China
Constantinescu, A. France Contro, R.
Italy
Matsumoto, T.
Japan
Dennis, B. H.
U.S.A.
Matsushima, K.
Japan
Deraemaeker, A.
France
Migorski, S.
Poland
Duhkravich, G. S.
U.S.A.
Murakami, A.
Japan
El-Badia, A.
France
Nakahata, K.
Japan
Elden, L.
Sweden
Nakajima, M.
Japan
Imado, F.
Japan
Nishimura, N.
Japan
Furukawa, T.
Japan
Nishimura, S.
Japan
Ha-Duong, T.
France
Nowak, A.J.
Poland
Hayabusa, K.
Japan
Oida, S.
Japan
Hayami, K.
Japan
Okajima, N.
Japan
Hon, Y.-C.
China
Okayama, S.
Japan
Hori, M.
Japan
Onishi, K.
Japan
Igarashi, H.
Japan
Ooki, R.
Japan
Inoue, H.
Japan
Orlande, H.R.B.
Brasil
Kabanikhin, S. I.
Russia
Park, K. C.
U.S.A.
Kagawa, Y.
Japan
Pires, G. E.
Portugal
Kanevce, G.
Macedonia
Qian, Y.-J.
China
Kanevce, P. L.
Macedonia
Rikards, R.
Latvia
Kanoh, M.
Japan
Sato, K.
Japan
Kassab, A.
U.S.A.
Shigeta, T.
Japan
Katamine, E.
Japan
Shiho, K.
Japan
Kawai, R.
Japan
Singh, K.M.
Japan
Kawai, T.
Japan
Shirota, K.
Japan
Kimura, A.
Japan
Sobieczky, H.
Germany
Soemarwoto, B.I.
The Netherlands
Verchery, G.
Suzuki, M.
Japan
Wang, Q.-C.
France Japan
Tanaka, M.
Japan
Wu, Z.Q.
Japan
Tanaka, T.
Japan
Cao, X.
Japan
Tomokiyo, K.
Japan
Yagola, A.
Russia
Toropov, V.V.
UK
Yamannoto, M.
Japan
Tosaka, N.
Japan
Yao, Z.-H.
China
Toyoda, T.
Japan
Yoshida, F.
Japan
Trendafilova, I.
Belgium
Yoshikawa, H.
Japan
Tsutsumi, K.
Japan
Yoshino. H.
Japan
Vena, P.
Italy
Zhang, W.-Q.
Japan
Contents Preface Symposium Chaiq)ersons International Scientific Committee Organizing Committee Participants
v vii vii ^iii ix
Inverse Heat Conduction
A combined use of experimental design and Kalman filter - BEM for identification of unknown boundary shape for axisymmetric bodies under steady-state heat conduction M. Tanaka, T. Matsumoto and T. Yano Estimation of the temperature and the concentration fields in a semitransparent medium: Emphasis on the experimental noise disturbance A. Chouaki, I. Darbord and P. Herve Numerical estimation of the transient heat flux boundary conditions for a flat specimen S. Abboudi and E. Artioukhine Estimation of the thermal state of two bars in dry sliding F. Lacour, Y. Bailly and E. Artioukhine Moisture diffusivity estimation by temperature response of a drying body G.H. Kanevce, L.P. Kanevce and G.S. Dulikravich Parameter estimation in moist capillary porous media by using temperature measurements L.B. Dantas, H.R.B. Orlande, R.M. Cotta and RD.C. Lobo Determination of heat transfer coefficient maps using an inverse BEM algorithm E. Divo, A.J. Kassab, J.S. Kapat and J. Tapley Tracking of phase change front for continuous casting - inverse BEM solution I. Nowak, A.J. Nowak and L.C. Wrobel Application of DRBEM and Iterative Regularization to Inverse Heat Conduction M. Tanaka and K.M. Singh Numerical solution of an inverse steady state heat conduction problem F. Berntsson and L. Elden An inverse heat conduction problem and an application to heat treatment of aluminium F. Berntsson and L. Elden
3 13 23 33 43 53 63 V1 81 91 99
Boundary Data Detection in Solid Mechanics
Inverse analysis to determine contact stresses using photoelasticity H. Inoue, K. Hayabusa, K. Kishimoto and T. Shibuya Identification of tractions based on displacement observations at interior points M. Nakajima, K. Hayami, J. Terao, S. Watanabe and S. Ando Identification of dynamic pressure distribution applied to the elastic thin plate M. Arai, T. Nishida and T. Adachi Determination of unsteady container temperatures during freezing of three-dimensional organs with constrained thermal stresses B.H. Dennis and G.S. Dulikravich Analysis of inverse boundary value problem by the alternating boundary element inversion scheme and its improvement using boundary division S. Kubo and A. Furukawa
109 119 129 139 149
Xll
Material Property Determination
Method for identification of elastic properties of laminates R. Rikards A single integral finite strain characterization of soft connective tissues and parameter identification V. Quaglini, R Vena and R. Contro Identification strategies for recovering material parameters from indentation experiments A. Constantinescu and N. Tardieu Identification of material parameters in constitutive model for shape memory alloy based on isothermal stress-cycle tests F. Yoshida, V.V. Toropov, M. Itoh, H. Kyogoku and T. Sakuma
161 171 181 191
Defect Detection
On the identificafion of a crack in 3D acousdcs H.D. Bui, A. Constantinescu and H. Maigre On the identification of conductive cracks C.J.S. Alves, T. Ha-Duong and F. Penzel Inversion of defects by linearized inverse scattering methods with measured waveforms K. Nakahata and M. Kitahara Fast recovering algorithm for crack shape of steam generator tubes using geometric approach F. Kojima and N. Okajima Structural damage idenfificafion using static test data and changes in frequencies X. Wang, N. Hu and Z.H. Yao Inverse analysis for fracture process zone characterizadon D.K. Tran, M.T. Kokaly and A.S. Kobayashi Idendfication of delamination in bonded dissimilar materials with orthotropic electric conductivity by the electric potendal CT method S. Kubo, T. Sakagami and N. Tanaka Damage detecdon of structure using image processing (1st Report, Detection of joint failure) T. Kawai, N. Ikeda and M. Ito
203 213 219 229 239 249 257 267
Shape Determination
Solution to boundary shape idendficadon problems in eUipdc boundary value problems using shape derivatives H. Azegami Shape optimization of transient response problems Z.Q. Wu, Y Sogabe, Y Arimitsu and H. Azegami Soludon to shape determination problem on unsteady heat-conduction fields E. Katamine, H. Azegami and Y Matsuura
Parameter Identification in Solid Mechanics
277 285 295
Muld-objective parameter idendficadon of unified material models T. Furukawa, S. Yoshimura and G. Yagawa 307 The polar method as a tool for solving inverse problems of the classical laminated plate theory G. Verchery, R Vannucci and V. Person 317 Verification of ultrasonic transducer characteristics determined in an inverse problem based on laser measurements T. Kanbayashi, H. Yoshikawa, N. Nishimura and S. Kobayashi 327
Xlll
Identification of elastodynamics load using DRBEM and dynamic programming filter M. Tanaka and W. Chen A geometrical design technique for impact pistons based on stress waveforms D. Liu, A. Chen and P. Zhu Inversion of stress and constitutive relations using strain data for Japan Islands M. Hori Estimation of parameters in tank model analysis using least squares of residuals with constraints M. Kanoh, T. Hosokawa and T. Kuroki Identification of damped joints parameters using the error in the constitutive relation A. Deraemaeker, P. Ladeveze, E. Collard and P. Leconte Condition monitoring and damage quantification in robot joints using nonlinear dynamics characteristics and inverse classification methods I. Trendafilova and H. Van Brussel
333 343 349 359 367 377
Inverse Problems in Aeronautics and Fluid Dynamics
Waverider design with parametric flow quality control by inverse method of Characteristics Y. Qian, H. Sobieczky and Th. Eggers Improper integrals in the formulation of a supersonic inverse problem K. Matsushima A function estimation approach for the identification of the transient inlet profile in parallel plate channels M J. Cola90 and H.R.B. Orlande Inverse aerodynamic shape design for improved wing buffet-onset performance B.L Soemarwoto, Th.E. Labrujere, M. Laban and H. Yanshah The application of modified output error method and its verification on inverse problems in aeronautics (No.2) Y Koyama and F. Imado
389 399 409 419 429
Inverse Problems in Electromagnetics and Acoustics
Optimization of electroplating on silicon wafer K. Amaya, S. Aoki, H. Takazawa and M. Miyasaka Defect shape recovering for electromagnetic problem using HTS-SQUID gradiometer F. Kojima, R. Kawai, N. Kasai and Y Hatsukade Design optimization of electromagnetic devices with neural network A. Kimura and Y Kagawa On the reconstruction of magnetic source in cylindrical permanent magnets H. Igarashi Extraction of transfer characteristics of vocal tract from speech signals K. Tsutsumi and Y Kagawa
439 449 459 467 477
Uniqueness, lU-posedness, Regularization
Regularizing procedures for solving the general inverse problem of structural chemistry and their applications I.V. Kochikov, G.M. Kuramshina, V.R Spiridonov, Yu.L Tarasov and A.G. Yagola Numerical solution of a Cauchy problem for an elliptic equation J. Cheng, YC. Hon, T. Wei and M. Yamamoto
487 493
XIV
Variational and D-N approaches for a magnetostatic Cauchy problem T. Shigeta
501
Numerical and Computational Algorithms
Identification coefficient problems for elliptic hemivariational inequalities and applications S.Migorski Variational approach for the problem of coefficient identifcation of the wave equation K. Shirota On an inverse phase change problem A. El-Badia On some inverse EEC problems M. Chafik, A El-Badia and T. Ha-Duong Numerical solution of under-determined 2D Laplace equation with internal information Q. Wang, Y. Ohura and K. Onishi
513
529 537 545
Applications of Computational Algorithms
Evolutionary methods in inverse problems of engineering mechanics T. Burczyhski, W. Beluch, A. Dtugosz, P. Orantek and M. Nowakowski Screen detection with near field measurements C.J.S. Alves and G.E. Pires Bayesian estimation for nonlinear inverse problems M. Suzuki and A. Murakami Structural optimization using cellular automata T. Toyoda and E. Kita Author Index
553 563 573 583 591
Inverse Heat Conduction
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INVERSE PROBLEMS IN ENGINEERING MECHANICS II M. Tanaka, G.S. Dulikravich (Eds.) © 2000 Elsevier Science Ltd. All rights reserved.
A COMBINED USE OF EXPERIMENTAL DESIGN AND KALMAN FILTER-BEM FOR IDENTIFICATION OF UNKNOWN BOUNDARY SHAPE FOR AXISYMMETRIC BODIES UNDER STEADY-STATE HEAT CONDUCTION Masa. TANAKA , T. MATSUMOTO and T. YANG Department of Mechanical Systems Engineering, Faculty of Engineering, Shinshu University 4-17-1 Wakasato, Nagano City, 380-8553 Japan E-mail:
[email protected] ABSTRACT This paper presents a combined use of the experimental design and the boundary element method (REM) for the inverse problem of boundary shape determination in rotationally axisymmetric bodies under steady-state heat conduction. The REM andfiltertheory have been so far applied to a number of inverse problems, but their successful applications depend on initial guess of parameters to be identified. This paper aims at constructing a more robust method of inverse analysis in this direction. Inverse analysis is performed by using thefiltertheory-REM after an approximate set of parameters are estimated through the experimental design. Such a two-step method of inverse analysis is proposed in this paper, and its advantages are demonstrated through numerical computation of a few examples. KEYWORDS Inverse Problem, Boundary Shape Determination, Experimental Design, Filter Theory, Boundary Element Method, Rotationally Axisymmetric Rody, Steady-State Heat Conduction INTRODUCTION A two-step method of inverse analysis is proposed in this paper for the inverse problems of boundary shape determination in steady-stae heat conduction fileds. First, a method of experimental design is applied to estimating a set of parameters defining the boundary shape to be identified. An approximate set of parameters thus estimated are then used for the inverse analysis which combinedly uses the boundary element method and the Kalman filter theory [1]. It is assumed that rotationally axisymmetric bodies under consideration are in steady-state heat conduction and that the temperature is measured at a number of points located on the outer boundary. The inverse problem can be stated such that the unknown internal boundary shape should be identified by using the measured temperatures on the boundary. In application of the experimental design [3], wefirstassume that each parameter may have a discrete value belonging to the levels we can assign based on a priori information. According to the number of levels and that of parameters (=factors), we choose an appropriate table of orthogonality [3], and carry out influence analysis of parameters. Analysis of variance is performed for a reduced number of parameter combinations, and then we can know which parameters may
4
M. Tanaka, T. Matsumoto and T. Yano
influence more than the other. The most appropriate parameter combination is thus selected so that a cost function is minimized. Boundary element analysis is carried out for each parameter combination to evaluate the cost function. An application of this method to identification of the blast-furnace refractory was reported in [2]. In this paper, the extended Kalman filter is employed in the second step of inverse analysis, in which the parameter values estimated through the experimental design are used in the second step as their initial guess. In the proposed two-step inverse analysis, the set of parameters estimated by means of the experimental design are used as the initial guess for a second step of inverse analysis, and they are modified through iterative computations as shown in [1]. Some numerical examples are computed by the proposed method of inverse analysis and the results obtained are discussed, whereby the usefulness of the proposed method of two-step inverse analysis are revealed. THEORY Influence Analysis by Experimental Design Experimental design [3] has recently been attracting attention of engineers to select effectively appropriate data sets among a number of given data [4]. Though the experimental design using orthogonal tables we can caluculate effects of factors and find a minimal value of the characteristics. The orthogonal table used in this method is chosen by considering the numbers of factors, levels and interactions. The factors will be assigned in the column, and calculate the characteristic value (in our case the cost function). When there are the interactions between factors, they are assigned based on the linear graph of the table. We shall take the orthogonal table L27(3^^) for example [4], where the superscript 13 denotes the number of rows of the table, the number enclosed with brackets denotes the number of levels of each factor and the maximum number of available factors. Namely, L27(3^^) consists of 13 rows and can treat 13 factors each of which has 3 levels. The number in the table denotes the level of each factor and the characteristic value which is used in the influence analysis is calculated using a set of levels which forms the row of the table. Using the orthogonal table, we can drastically reduce the total number of computations fo the influence analysis. For instance, in a case of three levels and eight factors, only 27 computations of influence analysis is required if the orthogonal table L27(3^^) is employed, whereas 3^ = 6561 computations should be done without use of the orthogonal table. In the variance analysis the influence of a factor on the characteristic value is computed by comparing the ratio of variance (F-ratio). Let us consider the case of a levels. Through r-times computations we can obtain n characteristic values yij. The total sum of square ST has fr degrees of freedom. That is,
1=1 j = i
fT =
(1)
n-l
where y is the totally averaged value of yij. The sum of square .94 and its degrees of freedom are
A combined use of experimental design and Kalman filter
5
given as follows:
tr
\
(2)
J
fA = a-l
where yi. is the averaged value of yij in each level i. The total sum of square ST can be resolved into square sum SA and square sum of errors SE as follows: ST
= SA + SE
(3)
The square sum of errors SE and its degrees of of freedom JE are given as SE
== ST-
IE
= IT-
PSA
PIA
(4)
The variance V is written as a value of each component of square sum divided by corresponding degrees of freedom / , which are written for factors and errors, respectively as follows: VA - SA/JA
(5)
VE = SE/IE
(6)
The F-ratio can be computed using VA and VE as follows: FA = VA/VE
(7)
When F-ratio FA is smaller than 1, we may think that the influence of factor A is small and its effect on the characteristic value is negligible [4]. The F-distribution table is used to examine whether the effect of the factor is significant or not. When the F-ratio is smaller than a critical value given in the F-distribution table, this factor is negligible, and vice versa. For the significant factors the following pure sum of square S'A and effective ratio PA are defined, i.e. S'A
= SA - VE
PA = S'AIST
(8) (9)
The effect of each level Xi can be evaluated as follows: r
x^^Y^mj'
(10)
Boundary Element Analysis The boundary integral equation for the three-dimensional problems of steady-state heat conduction without heat source is given in the Cartesian coordinate system (xi, X2,2:3) by [5] c{yHy) + /J —j^u{x)dT j u'{x,y)q{x)dT =J u*{x, y)qix)dT ^"]^'^^"(^)^r ^
(11)
where u is the temperature at a point on the boundary F, q the outward normal derivative of the temperature, u' the fundamental solution of the Laplace equation, y the source point and c{y) a
6
M Tanaka, T. Matsumoto and T. Yano
constant which depends on the geometry of the boundary at y. The fundamental solution u* is given by u*{x,y) = -—, r = \x-y\ (12) 47rr In this study, we assume rotationally symmetric distributions of variables for rotationally symmetric bodies, and follow the solution by Wrobel [6]. Expressing equation (11) in the cylindrical coordinate (/?, ^, Z), we obtain c{yHy)
+ / / " ^''*^^^^K{x)JdedT
= 1 1 \*{x,y)q{x)JdedT
(13)
where F denotes the meridian of the axisymmetric body and J is the Jacobian which is equal to R in this case. The integrals in equation (13) with respect to 0 can be evaluated analytically. Thus, we obtain c{yHy)
+ JQ* (x, y) u{x)dr = fu* (x, y) q{x)dT
(14)
The details of fundamental solutions U* and Q* can be seen in [1] and [5]. Parameter Identification Using Filter Theory In order to identify the parameters which express the geometry of unknown internal boundary, we combine the boundary element method with the extended Kalman filter [7]. In the following an outline of this procedure [1] will be explained. In the filter theory, we may assume that the measured temperatures compose the observation vector and the unknown parameters defining the geometry of the unknown internal boundary do the state vector. The observation equation which describes the relation between the state vector X and the observation vector y is written as follows: yk = hk{xk) + Vk
(15)
where v is the observation error vector and the subscript A: denotes time. Since there is no temporal change in the temperature, we may regard the subscript k as the iteration number of computation. The state equation describing the transition of state vector about the time can also be assumed as follows: Xfc+i=Ixfc
(16)
where I denotes the unit matrix. Based on a linearized expression of the observation equation (15) and the state equation (16), we can identify the unknown parameters in an iterative manner. The main flow of the iterative inverse analysis can be summarized as follows: 1. Assume the initial state vector Xo/_i and the covariance matrix of estimation errors Po/-i in the extended Kalman filter. 2. Input the measured temperatures y^ and the covariance matrix of observation errors R^.
A combined use of experimental design and Kalman
filter
1
3. Calculate the temperature at observation points using the BEM and the previous estimated state vector i^k/k-i as well as the sensitivity matrix H^, which is defined by
Hfc =
dui dxi duo
^hfc(xfc) dy.k
IXfc=Xfc/fc_i
dui dx2 duo
fi.
_
__
OXi
0X2
duN L dxi
,^^ . . .
dui dxn duo
__
(17)
OXn
duA
duM dx2
dXn
k/k-1
First-order derivatives in equation (17) are calculated by a finite difference scheme. 4. Calculate the filter gain Kk and renew the estimated state vector yik+i/k- The filter gain in the extended Kalman filter is computed by Kfc = Pfc/fc-iH^. [H-kPk/k-i^k
+ ^fc]
(18)
5. Check convergence. Unless convergence is realized, go to the step 3 and repeat the computation. NUMERICAL ANALYSIS Analysis Model In this study the parameters in the inverse analysis are used as factors and the cost function as the characteristic value in the terminology of experimental design. The cost function is defined as the square sum of residuals between the temperatures computed by the BEM analysis and the observed temperatures. In actual analysis, the two-step method of inverse analysis is applied: First, the best level in each parameter is selected by influence analysis of experimental design, and then a precise inverse analysis is performed by a combined use of BEM and Kalman filter with the parameter values estimated by the experimental design. Instead of using equation(lO), the best level in each parameter is found as what minimizes the following expression:
W^ = J2 ViJ
(19)
For the parameters which are judged to be not significant, the level selected by the influence analysis is regarded as the estimated value, if the sensitivity for the temperatures are sufficiently small. Figure I shows the analysis model in which the blast furnce hearth is modeled as a rotationally symmetric body about Z axis. The boundary conditions used in this numerical analysis are also shown in the figure. As a parameter we take the distance between a fixed point (0,10) and a point on the internal boundary which lies on the ray issued from the fixed point with a given angle (Figure 1). It is assumed that the ray is placed with an equal angle with the adjacent ones. In addition, we assume that 10 thermocouples which are shown in Figure 1 by o symbol are located, from which we can measure the temperature. The internal surface is produced by interpolating the eight points using cubic spline curve. Three levels are assumed for each parameter based on a priori information, x symbol denotes the first level of each parameter, • the second and • the third.
8
M. Tanaka, T. Matsumoto and T. Yano
In estimation by the extended Kalman filter, it is assumed that the initial values of diagonal arrays of the covariance matrix for estimation errors is assumed to be 1.0 x 10^, and that the corresponding values of the covariance matrix for observation errors is to be 1.0. Two examples are analyzed to check how the assumed levels infulence the estimation results: The first example in which the target geometry lies within range of the assumed levels and the second example in which the target geometry lies outside the range of the assumed levels.
q = 0 [Wm ' ] X •r
/i = 70[Wm ^K •] Ua = 30 PC]
Fig. 1. Inverse analysis model
An example in which the target geometry lies in the assumed levels. Figure 2 shows the estimated geometry through influence analysis of experimental design. Table 1 summarizes Wi for each level and the values denoted by the circle o indicate the selected levels by influence analysis. The solid line in Fig. 2 denotes the estimated geometry of refractory erosion by influence analysis. Figure 3 shows the estimated geometry after two-step inverse analysis in which the result of influence analysis is used as the initial guess of parameters in the combined Kalman filter-boundary elememt method (KF-BEM) of inverse analysis. Only the eighth parameter x^ is not significant and its sensitivity is very small. Therefore, we assume that this value obtained by the influence analysis for the 8th parameter is the final result and no modification is made in the (KF-BEM) inverse analysis. Figure 3 shows an excellent agreement between the target geometry and the estimated geometry by the proposed two-step inverse analysis. An example in which the target geometry lies outside the assumed levels. In this numerical analysis the levels are assumed outside the target geometry. Table 2 summarizes the results obtained by influence analysis, and Fig. 5 shows the estimated geometry through the two-step inverse analysis. A good agreement can also be recognized. Even if the levels are assumed
A combined use of experimental design and Kalman filter
9
outside the target geometry, the second-step of the BEM-F inverse analysis can provide a very good estimation as a converged solution.
10-
ytk^
Estimated geometry
9-
o
Target geometry
8-
xA
76N
o
jl
%1
1
'.•
;
54-
\
X
iI
3^
\ •
X
^^^^^" -* -
•
^
.
-
'-'
\ •
21in
\ )
1
(
2
3 R
5
4
6
•
Fig. 2. Approximate estimation by experimental design (1st step analysis)
Table 1. Effectivity of each level
0-0 I I I I I I M I I I I I n M I M I I M I I I M I I I I 123 1 2 3 1 2 3 1 2 3 1 2 3 123 1 2 3 123 xl
x2
x3
x4
x5
x6
x7
x8
10
M. Tanaka, T. Matsumoto and T. Yano lU
Target geometry
98-
Initial geometry
7-
Estimated geometry
6N
54-
2-
\
\ \ 1
\ \
J \ —-—V \ \
10- ^
^
Fig. 3. Final estimation by Kalman filter-BEM inverse analysis (2nd step analysis)
lU-
Estimated geometry
9-
X
•
«
•
(>
Target geometry
8-
)
76N
X
5-
' 4-
^ •
'.
• 11
1 ' -''
^
\
\ \
2100
1
Fig. 4. Approximate estimation by experimental design (1st step analysis)
A combined use of experimental design and Kalman filter Table 2. Effactivity of each level
0-6 1 I I I I I M I I t I M 1 I I I 1 I I I I I I M M I M I 123 123 123 123 123 123 123 123 xl
x2
x3
10-
x5
x6
x7
x8
Target geometry
9-
Initial geometry
87-
\
Estimated geometry
6N
x4
5-
/
4-
J\
\ \
2-
\ \
()
1
2
3 R
^t
;i
6
Fig. 5. Final estimation by Kalman filter-BEM (2nd step analysis)
11
12
M. Tanaka, T. Matsumoto and T. Yano
CONCLUSION The two-step method of inverse analysis, first by experimental design and then by Kalman filterBEM, has been proposed for estimating the erosion line of a blast-furnace model. A computer code was developed and applied to a few examples, whereby it was revealed that an accurate solution of the inverse problem could be obtained by the proposed two-step method. It can be mentioned that the proposed method of inverse analysis is very effective and robust, because a priori information can be used as much as possible for the first-step analysis by experimental design. As a future work in the direction of the present paper, it can be recommended to apply the proposed method of inverse analysis to predicting the solidification layer which reportedly is produced on the surface of blast-furnace refractory. On the other hand, we may hope that since the proposed method has wide applicability, it could open a door to successful applications to a wide variety of inverse problems.
References (1) M. Tanaka, T. Matsumoto and S. Oida, Identification of unknown boundary shape of rotationally symmetric body in steady heat conduction via BEM and filter theories. In: Inverse Problems in Engineering Mechanics, ed. by M. Tanaka and G.S. Dulikravich, Elsevier Science, Amsterdam, (1998), pp. 121-130. (2) H. Yoshikawa, M. Ichimiya, S. Taguchi, M. Tanaka, Estimation of erosion line of refractory and solidification layer in blast furnace hearth. In: Proc. 4th Conf. on Simulation Technology, Japan Soc. for Simulation Technology, (1984), pp. 75-78. (3) G. Taguchi (1976). Experimental Design Vols. 1 and 2, Maruzen, Tokyo. (4) T. Kashiwamura, M. Shiratori, Q.Yu (1998). Optimization by Experimental Design for NonLinear Problems, Asakura-shoten, Tokyo. (5) M. Tanaka, T. Matsumoto and M. Nakamura (1991). Boundary Element Methods, Computational Mechanics and CAE Series 2, Baifukan, Tokyo. (6) L. C. Wrobel (1981). Potential and Viscous Flow Problems Using the Boundary Element Method, PhD Thesis, Southampton University, UK. (7) T. Katayama (1989). Applied Kalman Filter. Asakura-shoten, Tokyo.
INVERSE PROBLEMS IN ENGINEERING MECHANICS II M. Tanaka, G.S. Dulikravich (Eds.) © 2000 Elsevier Science Ltd. All rights reserved.
13
ESTIMATION OF THE TEMPERATURE AND THE CONCENTRATION FIELDS IN A SEMITRANSPARENT MEDIUM: EMPHASIS ON THE EXPERIMENTAL NOISE DISTURBANCE A. CHOUAKI, I. DARBORD AND R HERVE L3E, Labomtoke d'Energetique et d'Economie d'Energie (Universite Paris X) 1, Chemin Desvallieres 92410 Ville d'Avray FRANCE ABSTRACT This paper deals with the estimation of the steady distributions of the temperature and the concentration fields (T,C) of H2O in a vapor state in a hot gas. Our approach is a non intrusive and a non active method, emission spectroscopy involving an identification algorithm. The (T,C) fields are estimated through the comparison of the analytical intensities with the experimental intensities collected at several locations on the edges of the gas. The analytical spectral intensity transmitted by the gas is given by the equations of radiation transfer for a semitransparent medium. These equations are non-linear Fredholm-type equations. In order to solve them, the (T,C) fields are described using analytical basis functions enabling to recover a wide range of typical profiles encountered in the scope of applications. A special care is given to the sensitivity of the identification procedure to noise influence. First, the sensitivity of eight selected wavelengths with respect to the noise is achieved. Then, the identification procedure described below is applied to a real case study (methane flame). The identification process makes use of the Extended Weighted Least Squares estimator combined to a quasi-newton minimization method. KEYWORDS Semi-transparent medium, emission spectroscopy, identification, temperature, concentration. INTRODUCTION The development of processes of high technology in the scope of hypersonic flows occurs in severe conditions of temperature, pressure (up to 3000 K, and up to Mach 6) and vibration level (more than 180dB). A related issue is the development of hypersonic ramjet launcher. It is performed through the measurement of temperature and concentration of water {H2O) in the exhaust combustion gas. Such measurements could be undertook using intrusive probes. Yet, in the drastic conditions cited above, they are rapidly destroyed. Hence, their success is highly dependent on the development of high technology materials. An alternative to that damagingijetable) technology is optical methods because they operate without any contact with the flow.
14
A. Chouaki, I. Darbord and P. Herve Optical methods may be classified into two categories: passive ones and active ones.
The main active methods are CARS (Coherent Anti Stokes Raman Scattering), DFWM (Degenerate Four Wave Mixing techniques [1], LIIF (Laser Induced Iodine Fluorescence) [2] and Rayleigh Diffusion [3]. Active methods need a light source (more often laser beam) and analyse the incidental light transmission through the medium. The measurements are obtained in one point of the flow so the spatial resolution is very high. Yet, 2D cartography then requires several successive shots. Hence, a though problem encountered is the alignment of the laser beams, especially when the vibration level is high. With CARS or DFWM, local unsteady measurement of temperature and concentration of the main species can be performed. CARS is based on the focus of two different monochromatic laser beams in the medium. The diffused light has properties related to the frequencies of the incidental beams and to the nature of the studied specie. DFWM is founded on the same technique as CARS but mixing four wave laser beams instead of two. LIIF, initially developed to visualize flows is now often used to quantitative metrology. The flow is seeded with a tracer, iodine, and lit by a laser beam. The principle is that the incidental radiation absorption by iodine particles is accompanied by a emission of fluorescence. The concentration of tracer which is activated is function of the flow characteristics (temperature, speed and pressure fields). The absorption lines of iodine are well-known in the visible part of the spectrum. Each line has a more or less large sensitivity to the pressure or the temperature. This method requires an iterative processing of the signal and the knowledge of pressure and temperature in a point of the flow in order to evaluate the constants of the theoretical formula. The use of LIIF for concentration measurement is not possible. In Rayleigh Diffusion, the medium is lit by a monochromatic laser beam. The resulting diffusion is proportional to the density of particules in the gas medium. Assuming the perfect gas equation of state for the medium, the collected signal is then inversely proportional to the temperature. So this method can't perform the measurement of the concentration of a particular specie in the gas. Moreover, to collect the diffused light in only one position, it is necessary to be sure that the laser beam is well focused on a point of the flow. On the contrary, the incident beam is partially absorbed by the studied medium, and the measurements must be carried out on the total length of the beam across the medium. Passive methods don't have need of external sources. Emission Spectrometry (for example Fourier Transform emission spectroscopy) is the method presented in this paper. It consists in the measurement of spectral intensity transmitted by a volume of gas. The measurement is made at the edges of the flame, so it does not change the state of the medium or disrupt the flow. In this paper, our experimental method will be described, before focussing on the inverse problem that is solved as a post processing of the measurement of the intensity, to obtain the (T,C) fields. The paper then recalls the main features of the identification, especially the choice of the experimental wavelengths, the analytical representation of the unknowns and the determination of the cost function. At the end, the results at 8 wavelengths are presented pointing out the sensitivity of the method to experimental noise. This study was carried out on simulated measurements for the
Estimation of the temperature and the concentration fields
15
whole suggested wavelengths. A result starting from a real case of measurement of intensity (5.15mm) on a methane flame is also presented. THE PROPOSED APPROACH The method was first applied in [4] in order to identify the (T,C) fields in a ramjet with hypersonic combustion, then, in [5] in order to determine the pollutant concentration in a turboreactor. Emission spectroscopy consists of the measurement of spectral intensities in different locations at the edge of the gas flow (see Fig. 1). The signal results of the radiative propagative phenomenon emission, absorption and transmission in the semi-transparent medium and which is described by the equation of radiation heat transfer. The equations are written for each measured volume >k
/^
s
1 \ 0
\ Sigr
\\
// 1
\—
\
lAy^
1 1 1 loi 1
(Signal L ( X ) ^
\
>^
Fig. 1. The flame cross section as follows :
j\\X^,T{x,Y^)
i
9r{K.T{xX^C{x,yi))
' L«(A.,T(X.,y)) dr{Xn.nX.y)MX.y)) dy
^^ ^
L.^^^X.^Y,)
^^ ^ L^^^^X^^X.)
(1)
(2)
where L^{Xn^T{x^y)) is the. Planck function, r(An, r ( x , y), (7(a:, y)) is the transmissivity, Lexvi^n-, Yi) is the experimental spectral intensity on the edge x at the Yi location and Le^j,{\n^Xi) is the experimental spectral intensity on the edge y at the Xi location. The whole quantities are given at the wavelength A^. Seeing the form of the equation, a post processing based on the solution of an inverse problem or identification is required to determine simultaneous (T,C) fields starting from measurements. The identification method requires the knowledge of the emission spectra of the studied species, giving the evolution of emission lines with respect to temperature and pressure. The emission spectra is obtained either with the help of a database or with a molecular model [6] and enables to select the most interesting wavelength. Hence, two difficulties appear : As the measurement is carried out on a volume of gas, emission spectroscopy's spatial resolution is not so accurate than with active methods. Moreover, emission spectroscopy can be used to detect gases present in sufficient quantity in the medium or gases at a sufficiently large temperature so as to collect a significant signal. Yet, the performance of actual detectors is increasing and enables to apply the method even for unsteady flows.
16
A. Chouaki, I. Darbord and P. Herve
Emission spectroscopy also answers to several constraints of the problem: the experimental device is reliable, simple and non intrusive. It can also be noted that compared to active methods, methods based on emission are not so sensitive to particles diffusion coming from erosion of the combustion chamber. THE IDENTIFICATION PROCESS In the direct problem, the simulation of the (T,C) fields remains a difficult task, even knowing the combustion and the geometrical parameters. A turbulent flow and the air induction at the interface of the flame are not always well modeled. In order to obtain a better representation of the temperature and the concentration fields, they are derived directly from the spectral intensities measurements. Then, these (T,C) fields could be used in order to improve the analytical parameters involved in the direct problem. The achievement of the identification process is performed around three key-points: • the choice of the experimental measurements, • the analytical representation, • the characterization of the test/analysis distance. The choice of the experimental measurements A first choice consists of defining efficient sensors' locations with respect to the model to identify. In our case, as it will be seen further (), the sensors' locations are not critical to determine. A more difficult task is the choice of the experimental wavelengths. Actually, the experimental tests are achieved at several wavelengths. Generally, one tries to choose the wavelengths which produce high temperatures and concentrations. Nevertheless, the sensitivity of the chosen wavelengths with respect to the temperature and the concentration fields is rarely taken into account. One originality of this work is that the experimental wavelengths are selected considering the sensitivity of the spectral intensities to the (T,C) fields. Below is shown a typical curve of the spectral intensities with respect to the temperature at several wavelengths. Hence, the wavelengths 5.15 fim and 1.82 fxm are chosen because the collected signal is high. Though the signal is low, the wavelength 0.72 fim is also selected because of the high evolution of the spectral intensity with respect to the temperature. The analytical representation The temperature and the concentration fields have to be projected over a function basis. This basis should be able to represent most of the usual profiles encountered in the field of interest. In our case, it means that the basis allows us to represent most of the profiles encountered for example in the field of propane or hydrogen-air gas jet, waste incineration and charcoal power station fires. Hence, the optimization parameters p are the ones involved in the basis' functions and the contributions of these functions to the corresponding field: nt
T{x, y,z) = Y^ Pi^i{x, y, z)
(3)
1=1
nc
C{x,y,z) = Y,Pj'^ji^^y^^) with:
(4)
Estimation of the temperature and the concentration fields • ?li=5.15|a,m A.2=1.82nm X3=0.72 ^im
Spectral intensities (W.m-2.Sri) 40-^
Spectral intensity ^Qo (W.m-2.Sri)
.^'
6.10*
^f 1000
17
4.10* 2.10-"
1500
2000
2500
Temperature (K)
Fig. 2. A typical evolution of the spectral intensities with respect to the temperature at several wavelengths T C nt nc Pi Pj ^i
the temperature field the concentration field the number of the temperature basis functions the number of the concentration basis functions the temperature optimization parameters the concentration optimization parameters the temperature basis' functions the concentration basis' functions
It should be noted that other optimization parameters could be added inside the basis' functions (for example: a parametrized sine or a parametrized normal law). The characterization of the test/analysis distance The identification process seeks to fit the experimental results to the analytical ones through the optimization of the parameters. Nevertheless, the experimental data are incomplete and corrupted by the experimental noise. In the control theory for system identification, the influence of the noise has been investigated by many authors such as [7], [8] and [9]. Hence, the identification process becomes an estimation process where "estimators" are used depending on the a priori known information (for example the density probability function of the the noise, the density probability function of the parameters, the density probability function of the measurements ... ). The estimators can be classified from the one requiring the most amount of a priori information to the one requiring less, as follows: 1. the bayesian estimator, 2. the maximum likehood estimator, 3. the markov estimator and 4. the ordinary least squares estimators.
A. Chouaki, I. Darhord and P. Herve Moreover, the estimation problems can be written in terms of a minimization of a cost function. In the case where the measurements and the parameters are supposed to follow normal laws, the bayesian estimator leads to the Extended weighted Least Squares (E.W.L.S.) estimator characterized by the following cost function ([10]): J{P) = r{pr[Gr]r{p) with:
p. . \Gr]
[GA a
if
+ a{p-
^i—
p,J*[G,](p
^—xni ^
-
p j
(5)
residual vector depending on p optimization parameters vector initial values of the parameters to estimate the inverse of the covariance matrix of r the inverse of the covariance matrix oip a ponderating scalar Hermitian quantity
The first term of J(p) is called the penalty term. It controls the estimator and makes the minimisation problem convex. The second term of J(p) is called the regularization term (in the Thikhonov sense). The scalar a is a ponderating factor which determines the weight of the terms involved in J(p). The values of a enables to control the confidence that has the user in the measurement. For example, in the case where the experimental noise is high, a will have a high value. In the case where the initial parameters are far from the realistic ones, the minimization process becomes very critical. Hence, an alternative to this problem is to decrease the weight of the penalty term at the beginning of the minimization by increasing the value of a. In tough situations, a is determined using a cross validation check [10]. It should be noted that the E.W.L.S. estimator is known to be unbiased and with minimal variance which are very convenient properties. In our case, the residual vector r(p) is defined as follows: r^-^(p)
i^-^(p)
exp
(6)
J-'exp
where L^''^{p) is the analytical spectral intensity at a given wavelength A^ at a given location j , and Lg^'^ is the corresponding experimental spectral density. Hence, the cost function J{p) becomes: n\
j(p) = — nX nl
nl
EEE'"(P)1^^]^'"(£) ^-{P-2.J*\G,]{p-p_J
Lt=i
il)
i=i
where n\ is the number of the involved wavelengths, nl the locations' number and np is the number of the parameters. From the definition of J(p), knowing that the parameters p are initialized to 1, it is obvious that for a = 1 the penalty term has the same weight as the regularization term. Hence, the determination of the value of a is not critical. EXAMPLES As mentioned above, the choice of the wavelengths is of a paramount importance. After choosing the wavelengths with respect to their sensitivity to the temperature and the concentration, it
Estimation of the temperature and the concentration fields
19
remains to select those of them that are less sensitive to the experimental noise. Sensitivity study This study deals with the sensitivity of the identification process with respect to the experimental noise, in the case the 7^20 molecule in methane combustion. After first tests, the following wavelengths: [0.0072 0.0113 0.0117 0,0468 0.0515 0.0546 0.1041 0.1048] lO^^m have been selected. First, the relation between the black body spectral intensity and the temperature has been established using the Hitemp software database database. Then, six analytical basis functions have been chosen (three for each field). Next, a set of "experimental" parameters (p ) is selected in order to simulate spectral intensities. Finally, a noise (with a zero mean, following a uniform law) is added to the simulated spectral intensities. The covariance matrices are taken to be equal to the identity matrix. Two ways have been investigated in order to check the sensitivity of the identification process with respect to the noise: 1. the value of the cost function: the analytical spectral intensities are computed using (p ). Since the experimental spectral intensities are corrupted by the added noise, the value of the cost function is not equal to zero. Hence, Table 1 has been obtained. Each calculation presented in Table 1 has been Noise levels (%) 0.72 1 0.0088 5 0.020 10 0.083 15 0.19 20 0.34 30 0.72
Table 1 The values of the cost function. Wavelengths (^m) 1.13 1.17 4.68 5.15 5.46 10.41 10.48 0.0083 0.0080 0.0093 0.0083 0.0080 0.0084 0.0081 0.021 0.021 0.023 0.022 0.021 0.021 0.021 0.078 0.090 0.088 0.085 0.086 0.083 0.082 0.20 0.20 0.19 0.19 0.21 0.19 0.19 0.35 0.36 0.33 0.31 0.35 0.34 0.35 0.85 0.76 0.81 0.75 0.85 0.76 0.82
obtained as a mean of 40 runs. From this first study, it possible to check the sensitivity of the wavelengths with respect to the noise. The Table 1 shows that the sensitivities are very close, hence, no wavelength will be privileged in the identification process. 2. The values of the identified temperature and concentration fields: Here, the values of the identified temperature and concentration fields are studied in terms of the maximal values and the mean values. For each noise's level, the (T,C) fields have been identified after the minimization of J{p) using the Broyden Flecher Goldfarb Shano (B.F.G.S.) method. The derived (T,C) fields using p have been compared with those obtained using the identified parameters. It leads to Table 2 where: D
D
\J-ana~ -texpl __
I J- exp I
\Cana-Cexp\ \Cexp\
max{) : the maximum value. mean{) : the mean value. From Table 2, one can conclude that the selected wavelengths are less sensitive to the concentrations than to the temperatures.
20
A. Chouaki, I. Darbord and P. Herve Table 2. Comparison of the values of the (T,C) fields. Noise (%) max{RT) (%) mean{RT) (%) max (Re) (%) mean (Re) (%) 0 0.3 0.0 1.3 0.4 1 5.4 1.4 19.3 4.9 5 34.7 9.5 71.6 15.8 10 49.8 12.5 113 26.3 20 85.0 22.0 271 74.0
A real case study The experimental setup The objective is to measure the spectral intensity emitted by a flame simultaneously at 2 wavelengths. The whole optical bench is mobile in one direction in front of the flame, to get the intensity from several locations. Here, the (T,C) distributions are assumed to be axisymmetrical. A diaphragm selects a small volume of gas. A separating blade and two detectors (photomultiplier and INSB) are used to collect simultaneously the signal of the same volume of gas at 0.72fim and 5.15//m. The experimental noise is limited by the use of a modulator and a synchrodectector. 40 measurements are made with an interval of 1mm. Calibrated hole
i
0
^
i
Modulator
c
0
.4!^
i
i
3 0
M^, a oscillations begins to appear and amplifies as M increases., The figures 3c shows, for M=201, well the existence of these oscillations linked to the influence of local systematic errors. The difference between exact and estimated solutions shows that the average fluctuation is very zero neighbor. 2500
:
:
^2000 E1500
;
(3)
:/ ; \
t
;
;
; ivi=2i
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7 ; \ '/ ^ E s \
x1000
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xiooo
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0.8
/I 0.9
Fig. 3a, b and c. Comparison between exact (Ex) and estimated (Es) heat flux density For the three values, M=21, M=ll and M=201, of the number of approximation parameters, we present respectively, in figure 4 a, b and c, at the locations X^ and X^, the temperatures histories calculated from the estimated heat flux density and in figures 5a, b and c, the difference between simulated and estimated temperatures histories at the same locations. These curves explain remarks describe above mentioned and confirm numerically the high efficiency of the methods of parametric regularization developed in this work for to estimate the heat flux density. 1000 800 O
600
f
400 200
;
;
;
:
i^^)
-Tcl.J^
;{b)
IVI=21
=11
;
1 ! 1 \ 1
,Tc2
...;.-.
0 Lr-rT+rrTiT.... 1,,., 1,,., , , , , i , . , , i , . . . 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 time (s)
;
itz
(.,^-771',,,, 1,,,, 1,,,, 1,,,, , , , , t , , , , l , , , .
0
0.1
0.2
0.3
0.4
0.5 0.6 time (s)
0.7
0.8
0.9
1
30
S. Abboudi and E. Artioukhine ;(c)
;
;
;
' ' ^ Z " ^ " ' "ITCIT;
Mf201 —
L/i.-iV. Tc2 ;
[/ 0
1 ^j 0.1
0.2
"~'~C .. L... L.. J ^ ^ ^ . . . . 1 , . , . 0.3
0.4
0.5 0.6 time (s)
0.7
0.8
0.9
1
Fig. 4 a, b and c. Evolution of the estimated (or calculated T^) temperatures at locations X^ and X^
{
(b)
'\
Tc1-Tm1
X^ 1
IAA imm V ¥f
o -20 -40
M=11
\
Tc2-Tm2
-80 0
0.1
0.2
0.3
0.4 0.5 0.6 time (s)
0.7
0.8
0.9
-100
1
0
(C)
80
0.1
0.2
0.3
0.4 0.5 0.6 time (s)
0.7
0.8
0.9
1
M=201
60 40 S
20
•7
0
J -20
?^\ii^'^l\tf*A*Atf#/^^^
-40 -60 -80 0
0.1
0.2
0.3
0.4
0.5 0.6 time (s)
0.7
0.8
0.9
1
Fig. 5a, b, and c. Evolution of the difference between estimated and simulated temperatures
CONCLUSIONS To estimate the heat flux evolution from measured internal temperature histories, a method and iterative algorithms are proposed. The method is based on the residual functional minimization in the L2 space of parameterized functions. The number of approximation coefficients is the regularization parameter in the method. The residual criterion in the form (28) is used to choose the regularization parameter. An example of computational experiments was presented for estimating the evolution of heat flux density. The comparison between results, obtained for different values of the number of approximation parameters, show a high efficiency of the iterative algorithm for the optimal value
Numerical estimation of the transient heat flux boundary conditions M^ = 21 for the example studied. For M>M^, instability of the solution appears as M increases and for M <M^, the solution begins to deform as M decreases. REFERENCES 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12.
Tikhonov, A., Arsenin V., 1977, Solution of Ill-posed Problems, Wiley, New York. Ahfanov, O. M., Artyukhin, E. A., and Nenarokomov, A. V., 1987, Spline-Approximation of the Solution of the Inverse Heat Conduction Problem, Taking Account of the Smoothness of the Desired Function, High Temperature, Vol. 25, No.5, pp. 520-526. Ahfanov, O. M, Artyukhin, E. A and Rumyantsev, S. V., 1995, Extreme Methods for Solving III Posed Problems with Applications to Inverse Heat Transfer Problems, Begell House, Inc., New York. Beck, J. v., Blackwell, B., St. Clair, C. R., 1985, Inverse Heat Condition : Ill-Posed Problems, Wiley Intersc, New York. Beck, J. v., 1993, Comparison of the Iterative Regularization and Function Specification Algorithms for the Inverse Heat Conduction Problem, in Zabaras et al., 1993. Beck, J. v., Blackwell, B., and Hajisheikh, A., 1996, Comparison of Some Inverse Heat Conduction Methods Using Experimental Data, International Journal of Heat and Mass Transfer, Vol. 39, No. 17, pp. 3649-3657. Hensel, E., 1991, Inverse Theory and Applications for Engineers, Prentice Hall, Englewood Cliffs, New Jersey. Murio, D., 1993, The Mollification Method and the Numerical Solution of Ill-Posed Problems, John Wiley and Sons, Inc., New York. Zabaras, N., Woodbury, K., and Raynaud, M., (Eds), 1993, Proceedings of the First International Conference on Inverse Problems in Engineering: Theory and Practice, Palm Coast, Florida, USA, June 13-18 1993, ASME. Delaunay, D., Jamy,Y., and Woodbury, K., (Eds), 1996, Proceedings of the 2nd International Conference on Inverse Problems in Engineering: Theory and Practice, Le Croisic, France, June 9-14 1996, ASME. Artyukhin, E. A, Iterative methods for estimating temperature-dependent thermophysical characteristics. High Temperature- High Pressures, 1997, Vol. 29, pp. 533-539. Artyukhin, E. A. and B. Gamier, Estimating of the thermophysical characteristics of a composite material. High Temperature- High Pressures, 1997, Vol. 29, pp. 541-547.
31
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33
ESTIMATION OF THE THERMAL STATE OF TWO BARS IN DRY SLIDING
F. LACOUR, Y. BAILLY, E. ARTIOUKHINE Institut de Genie Energetique - Universite de Franche-Comte 2, Avenue Jean MOULIN 90 000 BELFORT - FRANCE Tel: (33) 03 84 57 82 00 - Fax : (33) 03 84 57 00 32 - Email:
[email protected] ABSTRACT In this communication, a thermal study of two bars in dry sliding is presented. The experimental device has been designed by carrying out tri-dimensional computations with Fluent software in order to apply a one-dimensional thermal model. The heat transfer within materials and the thermal conditions at the sliding interface are obtained by solving the inverse heat conduction problem with the use of the iterative regularization method. KEYWORDS Dry sliding , heat generation, inverse problem, measurements, estimation, numerical solution.
INTRODUCTION The problem of analyzmg the thermal state of two solids in dry sliding is of a grate interest during about three part decades. This interest is connected with different applications such as the thermal analysis of brake systems, clutch systems, bearing and others. The physical basis of the thermal conditions during dry sliding is given in [1,2]. It is shown that the main difficulty of the thermal analysis is that two thermal phenomena have to be taken into account: the heat generation at the friction surface and the thermal resistance of sliding contact. This study deals with the problem of estimating the thermal conditions between two solids which are in dry sUding. This work is a part of a thermomechanical characterization of devices used m brake systems. Because in practice the braking operation is generally short, the thermal study is essentially a transient phenomenon. That is why the goal is to estimate the evolution of thermal conditions at the contact surface by measuring temperature histories inside sliding bodies. The main peculiarity of the system under analysis is that no sensor can be placed at the sliding interface during the braking. The developed method consists in using
34
F. Lacour, Y. Bailly andE. Artioukhine
computational and experimental approach based on solving the inverse heat transfer problem. Experimental data are obtained by the use of thermal micro-sensors located in each solid and provided the data which are necessary for solving the inverse problem. EXPERIMENTAL SETUP The materials used for the dry sliding tests are copper and ferrodo. The purpose of using of copper is to have good homogenization of the heat flux near the sliding interface because of its high thermal conductivity. In brake systems, the observed temperature region is quite large, that is why the temperature dependence of the thermophysical characteristics of materials has to be taken into account. The characteristics of copper are well known and largely described in the literature. To obtain the temperature-dependent thermophysical characteristics of ferrodo, preliminary experiments have been carried out with the use of an experimental device similar to that described in [3]. Experimental data processing was realized by solving the non-linear inverse heat conduction problem of estimating simultaneously two temperature functions.
Thermal considerations for the experimental setup To simplify the interpretation of experimental results and numerical calculations, the first step was to define the geometrical parameters of the specimen required to use a one-dimensional model. Many numerical calculations of non-stationary heat transfer were performed by using the Fluent software. The main difficulty consists here in obtaining one-dimensional configuration during long enough time period (a few minutes after the contact) while conserving a possibility to make the specimen mechanically. Fig. 1 gives a typical sample of the geometry simulated with Fluent. A bar made of copper and another one made of ferrodo are in sliding contact. Their lateral surfaces are insulated with the first layer of a special "high temperature" insulator (up to 1700°C) which is based on aluminum oxide. The second insulation layer made of a ceramic which has the thermal conductivity about 0.3 W/m°C near 1000°C (and < 0.6 W/m°C near 250°C) completes the insulation. The fig. 2 shows typical results provided by Fluent with bars of diameter 10 mm. Of course, the temperature distributions are different for two deferent materials under consideration but inside the bars one can see that the cross-sections are isothermal. This actually a characteristic of a onedimensional thermal problem. Fig. 3 shows the temperature profiles calculated along the bars for various radial positions. All curves are superposed. This fact proves that the specimen can be conceded as one-dimensional. Experimental device The experimental apparatus is based on a machine tool used for industrial activities. This system has the property to ensure a nearly perfect alignment of the bars. The copper bar is in rotation while that of ferrodo is fixed. An additional mechanics is used to apply an axial pressure between the bars to generate the friction. The temperature measurements are carried out by means of a set of K type micro-thermocouples (50 jim) [4]. To limit measurement uncertainties related to the use of a turning collector, a specific system of rolling up the wire has been used. This device allows to perform experiments during 90 about seconds at a rotational speed of approximately 2500 rpm. The data acquisition is carried out by using a set
35
Estimation of the thermal state of two bars in dry sliding of amplifiers with cold welding compensation. A PC with the LabVIEW software permits to drive the experiments and the temperature histories are recorded with a samphng rate of 1 kHz for each channel. The sensors are located both at the lateral surface of the bars and inside of them (fig. 5) to verify if the heat transfer process is actually one-dimensional or not. In order to avoid problems of electrical insulation of the thermocouple wires and to obtain good enough precision of determining the sensor locations inside the bars, the wires are inserted through a rigid ceramic sheath. The diameter of the sheath is 800 \xxrL (fig. 4).
INVERSE PROBLEM To simplify the use of the measured temperature histories in computations, the direct problem is formulated by considering the bars as a multi-layer system. The interfaces of this system coincide with the thermocouple locations. The contact between the successive layers is rigorously perfect (fig. 6). At the interface between copper and ferrodo two thermal phenomena are taken into account: the presence of the thermal contact resistance of sliding and the heat source generated by the friction. The mathematical model of the process under analysis is based on the nonlinear heat conduction equation. This model is formed by the following system of equations :
^*^^*^^ = £ ( ^ * ^ ' ^ ^ ] - 4-1 < ^ < 4 . ^ = 1.2,-,^ .t,0
+ (l-s)KoLuBi„[l-^{l,r)j
+ Pn^jM
dx
dX
atX=7,forr>0
+ BiJ(l-{l.r)] = 0,
"
dx
^^ '^
55
(lc,d) (le,f) (ig) (Ih)
where the following dimensionless groups were defined 4x,r)=^^^^, u-u
d^X.T)J^^^^^. s-Tn
BiA
Pn=s'^. u^-u
Bi^M, lu = '^, . 4 ' ,
Ko='-'^^^^. cT,
X='-. Q^-^—.
(2a-d)
(2e-j)
The properties of the porous medium appearing above include the thermal diffusivity {a), the moisture diffusivity {am), the thermal conductivity (A:), the moisture conductivity {kn^ and the specific heat (c). Other physical quantities appearing in the dimensionless groups of eqs. (2) are the heat transfer coefficient Qi), the mass transfer coefficient (hm), the thickness of porous medium (/), the prescribed heat flux (g), the latent heat of evaporation of water (r), the temperature of the surrounding air (Ts), the uniform initial temperature in the medium {To), the moisture content of the surrounding air {u*), the uniform initial moisture content in the medium {Uo), the thermogradient coefficient {5) and the phase conversion factor {s), Lu, Pn and Ko denote the Luikov, Possnov and Kossovitch numbers, respectively. Flow of dry air ^ ^ Heat and moisture transport Moist porous sheet Hot plate
Heat supply rate: q
T
Fig. 1. Geometry for the drying of a moist porous medium Problem (1) is referred to as a Direct Problem when initial and boundary conditions, as well as all parameters appearing in the formulation are known. The objective of the direct problem is to determine the dimensionless temperature and moisture content fields, %{X,f) and ^{X,f), respectively, in the capillary porous media. INVERSE PROBLEM For the inverse problem of interest here, the parameters Lu, Pn, Ko, e. Big and Bim are regarded as unknown quantities. For the estimation of such parameters, we consider available the transient temperature measurements Ytm taken at the locations Xm, m=l,...,M The subscript /
56
L.B. Dantas et al
above refers to the time at which the measurements are taken, that is, r^ for z-1,...,/. We note that the temperature measurements may contain random errors, but all the other quantities appearing in the formulation of the direct problem are supposed to be known exactly. Inverse problems are ill-posed [12-15]. Several methods of solution of inverse problems, such as the one used here, involve their reformulation in terms of well-posed minimization problems. By assuming additive, uncorrected and normally distributed random errors, with constant standard deviation and zero mean, the solution of the present parameter estimation problem can be obtained through the minimization of the ordinary least-squares norm. Such a norm can be written as 5(PHY-e(P)]^[Y-0(P)]
(3)
where P=[Iw, Pn, Ko, e, Big, Bim] denotes the vector of unknown parameters. The superscript T above denotes transpose and [Y - 0(P)]^ is given by [Y-e(P)f-[(f, -0,),{Y, -e,),...
,(f, -0,)]
(4a)
where (}^ - 6^ j is a row vector containing the differences between the measured and estimated temperatures at the measurement positions JC;,, 7W=1,...,M, at time /,, that is,
^-9)AYa-0.vY„-e,„...,Y.^
-9,^)]
(4b)
The estimated temperatures ^tm are obtained from the solution of the direct problem, Eqs. (1), at the measurement location Xm and at time ti. The present inverse problem of parameter estimation is solved with the Levenberg-Marquardt method of minimization of the least-squares norm [12,13,17,18]. The iterative procedure of such method is given by p^+l = p^ + [(j^)^j^+^^Q^]-l(j^)^[Y-e(P^)]
(5)
where J*^ is the sensitivity matrix, p. is a positive scalar named damping parameter, Q is a diagonal matrix and the superscript k denotes the iteration number. k
k
The purpose of the matrix term // Q in equation (5) is to damp oscillations and instabilities due to the ill-conditioned character of the problem, by making its components large as compared to those of J^J, if necessary. The damping parameter is made large in the beginning of the iterations. With such an approach, the matrix J\F is not required to be non-singular in the beginning of iterations and the Levenberg-Marquardt Method tends to the Steepest Descent Method, that is, a very small step is taken in the negative gradient direction. The parameter // is then gradually reduced as the iteration procedure advances to the solution of the parameter estimation problem and then the Levenberg-Marquardt Method tends to the Gauss Method [12]. However, if the errors inherent to the measured data are amplified generating instabilities on the solution, as a result of the ill-conditioned character of the problem, the damping parameter is automatically increased. Such an automatic control of the damping parameter makes the Levenberg-Marquardt method a quite robust and stable estimation procedure, so that it does not require the use of the Discrepancy Principle in the stopping criterion to become stable, like the conjugate gradient method [13,14].
Parameter estimation in moist capillary porous media
57
The sensitivity matrix is defined as
dO[_
dol_
del dOl del_
del_
de^ j(p)-
9e^(P) ap
dP\
36^ dP2
dP,
dP2
dej
deJ deJ
dPi
dP2
dPN
dp^
dP^
(6a)
dOj dPM
The elements of the sensitivity matrix are the sensitivity coefficients. They are defined as the first derivative of the estimated temperatures with respect to each of the unknown parameters Pj, j=\,...,N. The sensitivity coefficients are required to be large in magnitude, so that the estimated parameters are not very sensitive to the measurement errors. Also, the columns of the sensitivity matrix are required to be linearly independent, in order to have the matrix J J invertible, that is, the determinant of J^J cannot be zero or even very small. Such a requirement over the determinant of J^J is better understood by taking into account a statistical analysis, as described below. STATISTICAL ANALYSIS AND EXPERIMENTAL DESIGN After the minimization of the least squares norm given by Eq. (3), a statistical analysis can be performed in order to obtain confidence intervals and a confidence region for the estimated parameters [12]. Confidence intervals at the 99% confidence level are obtained as: Pj-2.576a p
^^ are explicitly set to zero. We use the MATLAB code ode45 for solving the system of ordinary differential equations. Finally, the nodes at the inner radius of the annulus are mapped onto a set of non-uniform grid points on V2, using the conformal mapping. The solution T|r2 is then obtained by cubic spline interpolation. For this scheme to work we need a method for computing numerically the conformal mapping n' = 0(0). To find such a mapping is in general a difficult problem. For simply connected
96
F. Berntsson and L. Elden
-1
-0.5
0
0.5
1
0
0.2
0.4
0.6
0.8
1
Figure 4: An orthogonal grid on an annulus is mapped conformally onto il. polygonal regions this problem is solved by the Schwarz-Christoffel mapping function. A similar mapping function exist for doubly connected domains [5, 6]. A software package for this case exists, and is freely available from Netlib^ The mapping displayed in Figure 4 was computed using this code.
NUMERICAL EXPERIMENTS In this section we give numerical results intended to demonstrate that the proposed methods work well. The test problems were created in the following way: First we selected the function T, which is harmonic in fi, and computed the corresponding temperature and Jieat-flux data, o and /i, on Fi. Normally distributed noise was added giving vectors g^, and h^. By solving numerically the Cauchy problem (1) we obtained approximations of the steady state temperature on r2. We conducted two separate tests, using different functions T as exact solutions. In Figure 5 we display the results obtained using Tikhonov regularization on the finite difference matrix, i.e. by solving (4), where iy is a discretization of a second derivative. In this experiment we used a grid with 504 nodes on Pi and 304 nodes on r2. Thus we solved a least squares problem with a matrix of size 16633 x 16129, and with 57541 non-zero elements. In Figure 6 the results obtained by solving an equivalent problem on the annulus are displayed. This method requires that the heat-flux h is zero. Thus, as a preprocessing step, we solved a well-posed problem boundary value problem. When solving the Cauchy problem on the annulus we needed to compute the F F J of vectors of size 4096. From these tests we conclude that the proposed methods work well. In all cases we managed to solve the inverse problem with accaptable accuracy.
^http://www.netlib.org/toms/785
Numerical solution of an inverse steady state heat conduction problem
0
0
0
0.5
97
1
0
Figure 5: Results obtained using the finite difference method, and Tikhonov regularization. We display the numerical solution in H (left), and at r2 (right,solid). Also we display the exact solution at r2 (right,dashed). The vertical lines mark the locations of the comers. We solved two different problems, in both cases the noise were of variance 10"^ and we used the regularization parameter A = 10"^.
98
F. Berntsson andL. Elden
I
'
I
>
I
I
Figure 6: Numerical results using conformal mapping, and solving an equivalent problem on the annulus, using the spectral method. Two different problems were solved, and we display the approximate (solid) and the exact solution (dashed). The vertical lines mark the locations of the comers. Two different problems were solved, in both cases the noise were of variance 10"^ and we used the "cut off" frequency ^c = 8.
ACKNOWLEDGEMENT The work of Fredrik Berntsson is supported by a grant from the National Graduate School for Scientific Computing.
REFERENCES 1. Dietrich Braess. (1997). Finite Elements. Cambridge University Press, Cambridge. 2. B. H. Dennis and G. S. Dulikravich. (1998). In: Inverse Problems in Engineering Mechanics. International Symposium on Inverse Problems in Engineering Mechanics 1998 (ISIP 98), Nagano, Japan, M. Tanaka and G.S. Dulikravich (Eds.), Elsevier, Oxford, pp. 61-72. 3. L. Elden, F. Berntsson, and T. Reginska. (2000). SIAMJ. Scient. Comput., to appear. Also: Technical Report LiTH-MAT-R-97-22, Department of Mathematics, Linkoping University, Sweden. 4. R C. Hansen. (1997). Rank-Deficient and Discrete Ill-Posed Problems. Numerical Aspects ofLinear Inversion. Society for Industrial and Applied Mathematics, Philadelphia. 5. P. Henrici. (1993). Applied and Computational Complex Analysis, volume 3. Wiley. 6. ChenglieHu. (199S). ACM Transactions on Mathematical Software, 24(3):317-333.
INVERSE PROBLEMS IN ENGINEERING MECHANICS II M. Tanaka, G.S. Dulikravich (Eds.) © 2000 Elsevier Science Ltd. All rights reserved.
99
AN INVERSE HEAT CONDUCTION PROBLEM AND AN APPLICATION TO HEAT TREATMENT OF ALUMINIUM
FREDRIK BERNTSSON and LARS ELDEN Department ofMathematics, Linkoping University S'581 83 Linkoping, Sweden ABSTRACT We consider an inverse heat conduction problem, the sideways heat equation, which is a model of a problem where one wants to determine the temperature on the surface of a body using internal measurements. The problem is ill-posed in the sense that the solution does not depend continuously on the data. We discuss the nature of the ill-posedness as well as methods for restoring stability with respect to measurement errors. Successful heat treatment requires good control of the temperature and cooling rates during the process. In an experiment a aluminium block, of the alloy AA7010, was cooled rapidly by spraying water on one surface. Thermocouples inside the block recorded the temperature, and we demonstrate that it is possible to find the temperature distribution in the region between the thermocouple and the surface, by solving numerically the sideways heat equation. KEYWORDS ill-posed, inverse heat conduction problem, heat treatment INTRODUCTION In several industrial applications it is desirable to determine the temperature on the surface of a body, where the surface itself is inaccessible for measurements. Another reason is that locating a measurement device on the surface would disturb the measurements and an incorrect temperature is recorded. In such cases one is restricted to internal measurements, and from these one wants to determine the surface temperature. The sideways heat equation is a model of this situation. Find the unknown temperature T that satisfies, ( {kTx)x = pCpTt, —CO < t < oo, { T{L,') =g, -oo-
-OOCXXH
ooooo Applied load (a) Variation of light intensity.
Applied load (b) Variation of phase.
coooo
Applied load (c) Increment of relative phase.
Fig. 1. Determination of the difference of the principal stresses. //
=
Acos^O^B,
0 = Tratad^
(2) (3)
where A is a constant of intensity modulation, B is the background light intensity, a is the photoelastic sensitivity, t is the thickness of the specimen, ad is the difference of the principal stresses, 0 is the retardation phase, and Id and // are hght intensities of dark and hght fields, respectively. The normalized light intensity of dark field /^ is represented by T*
h — ho Iio — ho
sin' 0,
(4)
where ho and //o are light intensities of dark and hght fields at no loading, respectively.
DETERMINATION OF THE DIFFERENCE OF THE PRINCIPAL STRESSES The technique developed in Hayabusa et al. [4] is employed for determining the difference of the principal stresses from a series of isochromatic fringe images obtained by changing the load incrementally. As shown in Fig. 1(a), the normahzed hght intensity at a point of the specimen varies sinusoidally with the load according to Eq. (4). The inverse relation to Eq. (4) is formally expressed as Orel = Sin-^ y/lj. (5) Substituting the hght intensity at each loading level into Eq. (5), the "relative phase" is obtained as shown in Fig. 1(b). However, in order to determine the difference of the principal stresses, the absolute retardation phase shown in Fig. 1(b) is required. If the increment of the relative phase at each load increment is calculated, it is understood that the variation of the phase increment is stepwise and that the absolute value of the phase increment is constant for all load increments as shown in Fig. 1(c). Therefore, the "absolute phase" may be obtained by taking a cumulative sum of the absolute value of the phase increment as shown in Fig. 1(b). Once the absolute retardation phase is obtained, the difference of the principal stresses is readily determined by Eq. (3).
ESTIMATION OF THE CONTACT STRESS Suppose that the difference of the principal stresses is given over a region scanned with a camera (referred to as measured region). An analysis region 0 is defined inside the
112
H. Inoue et al
Load Contact region To Analysis region
Measured region Fig. 2. Description of contact problem. measured region and its boundary is referred to as F = To U Fi as shown in Fig. 2. It is assumed that the boundary condition is known on boundary Fi (ex. free surface) while it is unknown on boundary FQ (ex. contact region or virtual boundary). From Somigliana's equation, the stress component crij at any point p inside 0 is given by a.,(p) = j ^ D.,k{p, Q)h(Q)dT
- ^ 5.,,(p, Q)uk{Q)dT
(6)
where ^A:(Q) and Uk{Q) are traction and displacement components at a point Q on F, respectively, while AjA;(p,Q) and Sijk{p, Q) are derived from the fundamental solutions of two-dimensional elasticity. Discretizing the boundary into elements in a similar manner as in the BEM, the following equation is obtained: (Tij = Gc^ — HcU
(7)
where t and u are vectors composed of tractions and displacements at all nodal points on the boundary F, respectively, and Gc and He are known matrices (row vectors) derived from the fundamental solutions. The boundary integral equation is used as an additional requirement. It is given as
c.,(P)«,(P) = J^ t/.,(p, QMQ)dr - J^ r,,(p, Q)u,{Q)dr,
(8)
where both P and Q are points on the boundary F and Cij is a constant determined by the geometry of the boundary. Discretizing Eq. (8) in the same manner as in Eq. (7), another system of equations with respect to t and u is obtained:
Gt-Hu
=0
(9)
where G and H are matrices derived from the fundamental solutions. Substituting Eq. (9) into Eq. (7) and applying boundary conditions on Fj, the following equation is obtained: Cto
(10)
where ^o is a vector composed of unknown traction components on the boundary FQ and C is a known matrix (row vector) derived from known boundary conditions and the fundamental solutions.
Inverse analysis to determine contact stresses usingphotoelasticity
113
Light Source ^Quarter wave plate ] Quarter wave plate
Polarizer CCD camera Specimen
Fig. 3. Experimental setup for two-dimensional photoelasticity. The difference of the principal stresses cid is expressed in terms of stress components as CTi = ^(^11 - a22f + 4^ direction
-5
10
Fig. 9. Contact stresses on FQ estimated from Fig. 7. (A: x-direction, Q - y-direction)
Inverse analysis to determine contact stresses using photoelasticity Estimation
111
of the Contact Stress
Inverse analysis based on the difference of the principal stresses shown in Fig. 7 was conducted. The tractions on Boundary AB, that is the contact stresses were estimated as shown in Fig. 9. The curves with plots represent estimated tractions while the curves without plots represent the exact solutions given by Eqs. (14) and (15). Although the estimated results are somewhat oscillatory, they agree fairly well with the exact solutions for the elastic half plane indented by perfectly bonded rigid flat stamp. It can be considered that the friction between the steel stamp and the epoxy plate was relatively high.
CONCLUSIONS In this paper, a new technique has been developed for measuring contact stresses using photoelastic images of one of the contacting bodies. The contact stress is obtained by determining the distribution of difference of the principal stresses from a series of isochromatic fringe images and then by estimating the unknown tractions. The inverse problem for estimating the tractions are efficiently formulated and successfully solved by BEM. Although the estimated contact stresses were somewhat oscillatory, it may be improved to some extent by applying some regularization technique.
REFERENCES 1. Oda, J. and Moto, S. (1990). Trans. JSME 56A, pp.1479-1484 (in Japanese). 2. Oda, J. and Hattori, M. (1992). Trans. JSME ^8A, pp.106-111 (in Japanese). 3. Hayabusa, K., Inoue, H., Kishimoto, K. and Shibuya, T. (1998). In: Inverse Problems in Engineering Mechanics, M. Tanaka and G.S. Duhkravich (Eds). Elsevier Science, Amsterdam, pp.603-612. 4. Hayabusa, K., Inoue, H., Tanaka, S., Kishimoto, K. and Shibuya, T. (2000). Trans. JSME^ accepted for pubhcation (in Japanese). 5. Gladwell, G.M.L. (1980). Contact Problems in the Classical Theory of Elasticity. Sijthoff & Noordhoff, Alphen aan den Rijn.
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INVERSE PROBLEMS IN ENGINEERING MECHANICS II M. Tanaka, G.S. Dulikravich (Eds.) © 2000 Elsevier Science Ltd. All rights reserved.
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I D E N T I F I C A T I O N OF T R A C T I O N S B A S E D O N D I S P L A C E M E N T O B S E R V A T I O N S AT I N T E R I O R P O I N T S Masayuki NAKAJIMA, Ken HAYAMI, Jiro TERAO, Seigo WATANABE* and Shigeru ANDO Department of Mathematical Engineering and Information Physics Graduate School of Engineering, University of Tokyo 7-3-1, Hongo, Bunkyo-ku, Tokyo, 113-8656 Japan e-mail:
[email protected] * Nissan Corporation, Yokosuka, Japan ABSTRACT This paper discusses the inverse problem of estimating the tractions on the boundary of an elastic body from displacement information observed at interior points. The problem arises, for instance, when estimating the tractions and the position where they are applied on the surface of a tactile sensor made of elastic body, from displacement observations obtained by ultrasonic cells inside the sensor body [1, 2]. We will formulate this inverse problem using the three-dimensional Boundary Element Method (BEM) and propose an algorithm to solve it. Finally, we will demonstrate the effectiveness of the algorithm through some numerical experiments. KEYWORDS Inverse Problem, Elastostatics, Boundary Element Method, Tactile Sensor, Generalized Inverse.
INTRODUCTION Previous research on inverse analysis in elastostatics have mainly dealt with estimating unknown boundary quantities (displacement and/or traction) from over-specified boundary conditions for two-dimensional elasticity such as in [3] or [4]. In this paper we will deal with the inverse problem of identifying boundary tractions from displacement observations at internal points. Related work has been done in [5], where displacements on the boundary are estimated using displacement information at interior points using BEM for plane elasticity. The present work [6, 7] identifies concentrated tractions and the positions where they are apphed on the boundary of a three-dimensional elastostatic body with the aim of applying it to tactile sensors [1, 2].
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T H E FORWARD PROBLEM Let r = r ( ^ ) U r(^) be the boundary of the region (elastic body) 0 C R^, where traction is given on F^^^ and displacement is given on V^^\ Assume that F is discretized into boundary elements 7ri{l V%*(p, Q) + phK'iv*{p, Q) = S{r), r = |pQ|,
(7)
where S{r) is a Dirac's delta function. Namely, w*{p^ Q) is a deflection on the source point p when a unit concentrate and impulsive load 6{i) is applied on the point Q. The concrete form of w* is expressed by Kelvin's function:
Identification of dynamic pressure distribution
**(p,Q) = - ^ ^ K e i o ( Z ) ,
131
(8)
where 77 = KJph/D and Z = yjr\r. Multiplying the fundamental solution 'ii)*(p, Q) with Eq.(6) and expanding it according to Gauss's divergence theorem, we have the following boundary relation between physical quantities on F and the internal deflection it'(p) in Q.
^(p) - /^{^*(P, Q)K(Q) - f:(p, Q)M.(Q) +
M:(p,Q)fn(Q) - K * ( P , Q ) ^ ( Q ) } ^ r ( Q ) + |^7i;*(p,R)/(R)dO(R),
(9)
where Q is a integral point defined on the boundary F. T^, M* and V* are fundamental solutions derived by substituting Eq.(8) into Eq.(2),(3) and (5), respectively. In the present study, the total boundary F is treated as smooth without corner point, therefore we may neglect a discontinuous term about twisting moment in Eq.(9). Moving the source point p to the point P on the boundary, the following boundary integral equation can be derived: i ^ ( P ) = / ^ { ^ * ( P , Q)V;(Q) - f : ( P , Q ) M . ( Q ) 2
+
M : ( P , Q ) f . ( Q ) - F ; ( P , Q)^(Q)}tiF(Q) + \^ ^ * ( P , R)/(R)ci^(R),
(10)
Then, let us approximate the density functions vf^^.. .y* and transverse pressure /(-R) in Eq,(10) by a suitable boundary element or regional element with a polynomial function, respectively. According to the conventional BEM formulation for the plate bending problems, we have the matrix relation as the form: A{K)X^K)
= y(Ac) + B{K)f{K),
(11)
where ^ is a unknown vector of the density functions, y is a known vector derived by the boundary conditions, A and B are coefficient matrix, respectively. Note that the all vector and matrix in the above equation defined on the Laplacd-transformed domain and pressure / is still unknown in the present inverse analysis. Here we let employ the bending strain as the supplementary infromation for the inverse analysis. Differentiating the Eq.(9) according to the definition of bending strain, we have the following boundary integral representation:
e-(p) = /^{^'(P,Q)K(Q) - fI(p,Q)M„(Q) +
Mlip, Q)f„(Q) - flip,
Q)w(Q)}dr(Q) + / r ( p , R ) / ( R ) d n ( R ) ,
(12)
where notation "^' indicates differentiation according to the definition of strain. The above equation can be rewritten as following matrix form: 5(1^) =C{I
1
0.40
'B
o ^ 0.20 0.00 1.00
1.04
1.08
1.12
1.12
Fig. 1. [Left]: A typical stress-strain curve from a tensile test on periprosthetic capsular tissue. The nominal stress in the loading direction is plotted against the stretch ratio A; [Right]: Uniaxial stress-stretch curves are insensitive to the strain rate. 1.00-t +
100
Stretch=1.04
O
stretch=1.07
^
stretch=l.l
200 time (sec)
300
400
Fig. 2 Stress relaxation curves at different stretch levels can be reduced to a unique normalized stress relaxation function, independent from the applied stretch A. The behavior of a nonlinear viscoelastic material undergoing finite strains is fully described by the sum of the series of integrals
^(o=t^[,--iy,,^,^..^^^^^^^^
(1)
where S and E are the stress and strain tensor respectively, R^^"^ is the stress relaxation response from a single-step relaxation test and R^""* is the difference between the actual relaxation function from a n-steps test and the prediction based on (n-l)-steps data [8]. For many soft collagenous tissues the time dependence of the response is adequately well defined by a single-step relaxation test, and, as a consequence, the stress can be expressed as a function of one time variable only r^ = T . Then the first term of the integral series (1) yields an adequate approximation of the material behavior in other than single step histories [1,6].
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V. Quaglini, P. Vena and R. Contro
A Single Integral model is therefore proposed for the behavior of periprosthetic capsular tissue. The model is expressed by the relation: S0)= r
rf,R[E(r);r-r]
J-QO
collagenous
(2)
where R is the response to a step strain history in which the strain tensor is changed from 0 to E at time t = 0\ it is called the stress relaxation tensor and it is in general a nonlinear function of E, with R ( 0 , r ) = 0 . At times r for which E(r) is differentiable, d^K can be written as (dot denotes time derivative) ..R[E(r),.-r]=^Bfe^E(r>/..
(3)
At jump discontinuities of E(r) the expression J E R is defined through the integral property J'_JER[E(r),r-r]-R[E(5"),/-5"]-R[E(5-)^-5-] .
(4)
For a smooth strain history the Single Integral model takes the form
S(0= f d^m^-rh
r
'^^^tiryir
(5)
or, by integration by parts
S(0=R[E(4o].j:^5^g^^r.
(6)
It should be noted that, as the model employs objective stress and strain measures, equations (5) and (6) are objective too. Constitutive equations are defined on the basis of the experimental results from the mechanical tests on the tissue. As in stress relaxation tests at different strain levels the ratio between the stress at time t and the stress at time zero is independent from the magnitude of the applied strain, R can be split into the product of two functions, one of strain only and the second one of time only: R(E,/) = S^(E).G(0 .
(7)
S^(E) can be defined as the response to an instantaneous strain history: it is uniquely related to the strain tensor and it is called the "elastic response function". The temporal response is carried by the "reduced relaxation function" G{t): its components Gij(0 are dimensionless and subjected to the condition Gjj(0 = 0. Elastic response function. As it appears from constant strain rate tensile tests along different directions, the capsular tissue is isotropic in the plane of the samples. The existence of a potential W^ for the elastic response S^ is assumed of the form
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A single integral finite strain characterization
(8) / i , h, h are the invariants of the metric tensor C. The typical shape of the stress-strain curves (Fig. 1) suggests the following choice of the potential W^ =a{exp[^(/, - 3 ) ] - l } - a M / 2 - 3 ) + / f e ) + g f e + 4
+^1,).
(9)
which corresponds to an exponential form for the elastic function S^. In the above expression, the assumption of incompressibility I3 = 1, which is very common for soft living tissues [1] is made. The principal stresses can be obtained by differentiation of eq. (9):
SI, =«^(exp[/?(£„ + £ , , + £ 3 3 ) ] - £ „
81.=-^.-^ a^33
5^33
-EJ-
2^22 + 1
(10,a-c)
2^33 + 1
In uniaxial tensile tests, S^^'is drawn from the applied force and the stretch ratio /li [7], while 1^22 = 0 . From eq. (10,b) the hydrostatic pressure;? is obtained and after substitution in eq. (10,A) it yields SI, = a ^ : ^ ^ _ ^ { - l - 2 £ 3 3 +2exp[y9(£„ 2^jj +1
+E,,+EJI
(11)
Due to the low thickness of the tissue, the hypothesis of plane state of stress 1^33 = 0 is made. Functions/and g must be chosen in order that eqation (10,c) is identically satisfied. Reduced relaxation function. Stress relaxation is found to be isotropic in the plane of the sample; it is assumed the isotropy of G(t) along the direction of the thickness too, and thus the tensorial function G(t) reduces to the scalar-valued function G(t). As a consequence of the scarce sensitivity of the tissue to the strain rate, the following form of reduced relaxation function, which corresponds to a uniform relaxation spectrum, is chosen [2] \-\-K{E^
^1^ -E,
^1 v^iy
G{t): 1 + ^ln with E^{y) =
/•oo
Jy
^
7
dz .
' t ^
(12)
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V. Quaglini, P. Vena and R. Contro
PARAMETER IDENTIFICATION The proposed Single Integral viscoelastic model is conceived as a multiplicative decomposition of the elastic response and the time-dependent response. This feature allows to split the whole set of parameters into two distinct sets. The first set contains the parameters a, P of the elastic function S^ only, the second set contains the parameters AT, 6\, Bi of the reduced relaxation function G{t). The identification procedure relies in the minimization of an error function which is a measure of the difference between the measured response yk, on laboratory tests, and the model prediction hk(x) calculated by using equations (11) or (12). The elastic function parameters can be identified through eq. (11) fi*om the loading branch of a tensile test at infinite strain rate. Actually, tensile tests at the highest strain rate (300%/min) were used to approximate the infinite strain rate test. The reduced relaxation function parameters of equation (12) were identified on the basis of normalized stress relaxation data. The two sets of parameters were estimated for each patient. In order to take into account data from different specimens from the same capsule, a two steps procedure had been adopted. An estimation of the parameters is firstly obtained by using data from the test on one specimen. The estimation was performed by minimizing the following function: Min % = (y, - h ( x ) ) " w ( y , - h ( x ) ) S.T.
^^^^
g(x), V. V. TORGPOV^), M. ITOff>, H. KYOGGKU^^ and T. SAKUMA^^ 1) Department of Mechanical Engineering, Hiroshima University, Japan 2) Department of Civil and Environmental Engineering, University of Bradford, UK 3) Department of Mechanical Engineering, Chiba University, Japan 4) Department of Mechanical System Engineering, Kinki University, Japan 5) Central Research Institute for Electric Power Industries, Japan ABSTRACT A set of material parameters in a thermo-mechanical constitutive model for a shape memory alloy (SMA), that can describe the pseudoelasticity and shape memory effect, were identified using experimental data obtained from isothermal stress-cycle tests conducted at various temperatures for Ti-Ni-Cu SMA. The identification was performed by minimizing the difference between the test data and the results of the corresponding numerical simulation using an advanced optimization technique based on iterative multipoint approximation concept. This approach allows to determine the phase transformation temperatures directly from the identification, while they are usually obtained only by some special experimental techniques such as DSC (differential scanning calorimetry). KEYWORDS Shape memory alloy, Ti-Ni-Cu alloy, constitutive model, material parameters identification, isothermal stress-cycle, phase-transformation temperature INTRODUCTION When a shape memory alloy (abbreviated to SMA) is isothermally loaded, it exhibits pseudoelasticity at high temperature and reversible shape memory effect at low temperature [14]. Such thermo-mechanical behaviour, which is due to the martensitic transformation (austenite [=parenf (p)] —*'martensite[m]) and its reverse transformation {m-^p), is described by a thermomechanical constitutive model. In general, all constitutive models incorporate some material parameters to be determined from experiments. However, especially for SMAs, the identification of material parameters in a constitutive model by conventional trial-and-error (curve fitting) approach is almost impossible because the constitutive equations are higly nonlinear, and also the number of material parameters in the model is quite large. Therefore, the identification problem should be treated as an optimization problem.
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RYoshidaetaL
The authors [5, 6] recently proposed a method of the identification of a set of material parameters in a constitutive model of cyclic plasticity for monolithic, as well as bimetallic, sheet metals from cyclic bending tests. In these research works, material parameters were successfully identified by minimizing the difference between the test results and the results of the corresponding numerical simulation using an advanced optimization technique based on iterative multipoint approximation concept presented by Toropov et al, [7-9]. Furthermore, the authors [10] applied this optimization technique to the identification of material parameters in a thermo-mechanical constitutive model for SMA [11-12], and six material parameters were determined using stressstrain data obtained from an individual isothermal stress-cycle test. As an extension of the above-mentioned study, in the present work, a set of material parameters in a modified constitutive model for Ti-Ni-Cu SMA were identified. For the identification, the whole experimental data obtained from isothermal stress-cycle tests conducted at various temperatures were employed simultaneously. It is worth notinig that this approach allows to determine the phase transformation temperatures directly from the identification as two additional material parameters, while they are usually obtained only by some special experimental techniques such as DSC (differential scanning calorimetry). Furthermore, thermo-mechanical behaviour of the SMA in another type of experiment, so-called stress-recovery test, is discussed by comparing the experimental observation with the prediction by the constitutive model incorporating thus identified material parameters. THERMO-MECHANICAL BEHAVIOR OF SMA AND ITS CONSTITUTIVE MODEL Shape Memory Effect and Pseudoelasticity Uniaxial tension-unloading tests for a wire of 3-mm diameter of Ti-Ni-Cu SMA (Ti-49.8Ni41.7Cu-8.5at%) were performed in a water pool for several temperatures ranging from 20 to 80 °C. Figure 1 shows the experimental results of stress-strain responses in uniaxial tension and the subsequent unloading under various isothermal conditions. In this figure, two typical stressstrain responses are observed. At low temperatures (20 and 60 °C in Fig. 1), a certain amount of residual strain remains after unloading. After that, when the specimen is heated up above a certain temperature (e.g., 80°C), the strain completely recovers, as illustrated with the dotted line in Fig. 1. The phenomenon of the recovery of residual strain is called shape memory effect. At high temperature (70 and 80 °C in Fig. 1), the stress-strain response during loading process looks like elastic-plastic behaviour, however, the tension-induced large strain recovers in the subsequent unloading process. This behaviour is cdlXtd pseudoelasticity. The above-mentioned thermo-mechanical behaviour of SMAs results from the martensitic and reverse transformations. The temperature- and stress-dependent phase transformation of a SMA is schematically illustrated in Fig. 2. At high temperature, the SMA has fully austenitic phase (sometimes we call this austenite phase 'parent'). When it is cooled down, the martensitic transformation starts at temperature M^ and becomes fully martensitic phase at M^ (the transformasion can be indicated as /? -*^ m). Starting with the full martensite, the austenitic transformation (reverse transformation of m-^p) starts at A^ when it is heated up and finishes at A^ In our SMA, the orthorhombic martensite transformation and its reverse transformation takes place. The transformation temperatures under stress-free condition (hereafter they are denoted by M^^, M^^, A^^ and A^^) are 46, 51, 59 and 62 °C, respectively. It should be noted that the inelastic deformation of martensite is due to the reversible movement of the twin or intervariant
Identification of material parameters in constitutive model
193
350
CO Q.
CO
en 0
L_ +-•
0.01
Strain
0.015
0.025
Fig. 1 Stress-strain responses of Ti-Ni-Cu SMA under isothermal uniaxial tension and the subsequent unloading at various temperatures.
CO CO
CO
(p^m)
Temperature
Fig. 2 Schematic illustration of temperature- and stress-dependent phase transformations of SMA.
boundaries under applied stress (m^^m), The shape memory effect results from the (m -^ m) deformation (from A-^B-^A at T^ in Fig. 2) and the subsequent {m-^p{-^m)) transformations during heating up followed by cooling down (A - • C at T^ (-^ A)). The phase transformations are strongly stress-dependent, as schematically shown in Fig. 2. For example, let us consider the phase transformations in isothermal loading cycle starting from a stress-free point at a high temperature above A (point D at T^in Fig. 2). In this case, the stressinduced (p—*'m (—*^m)) transformation occurs during the loading process (E—»'F in Fig. 2), and
194
F.Yoshidaetal
(m -*p) reverse transformation during the unloading process (G -* H in Fig. 2). The above stress-induced transformations accompanied by (m —^ m) deformation are the cause of pseudoelasticity. Constitutive Model In order to describe the thermo-mechanical behaviour of the SMA, a constitutive model proposed by Tanaka, etal.[\\-\2] was used. In this model the uniaxial stress-strain relation in rate-type is given by the equation: (1)
G = De-\-pDt + Q,^
where \x)
L ^f-w'*''"
(10)
dS'u
After a limiting process x -> F the last equation becomes:
cj>\z) = 0W + ( H
/,
^(y)-^G*(z,y) ony
dS'u
(11)
(pv) means that the integral should be take in the sense of Cauchy principal value. This equation is usually solved using the Born approximation. The Bom approximation is founded on two simplifying assumptions:
206
H.D. Bui, A. Constantinescu and H. Maigre
• the diffracted wave is created just by the lightened part of the crack F+ • the integral term in the right hand side of (11) is negligible compared with the first one. This conducts to the final equation: ^"(z)
=
2(j>v) /
| r ( y ) - ^ G * ( z , y ) dSy dny
(12)
which will be used afterwards in the identification problems. PERTURBATION OF THE TRANSIENT EQUATION In order to obtain general identifiability results and to eliminate the limitations created on the one hand by the treatment in the frequency domain due to the Helmholtz equation and on the other hand by the Bom approximation let us consider the complete transient problem. This will also enable a the construction of an exact solution in the case of a bounded domain. The reasoning presented in the sequel is constructed using a perturbed acoustic problem. The perturbation, a small viscosity term, does vanish for the asymptotic solution of the physical problem and is used only as a mathematical artifact. The perturbed equations of the direct problem have now the following form: Ce{w) - dtdtu -Au
+ edtu = 0
(13)
with initial conditions: u{x, t /?: Vix',t)
= [uix',t)]
(34)
We shall now analyze the temporal Fourier transform [14] associated with D: /•OO
K[x',q)=
I
V(x',t)cxp{iqt)dt
(35)
The inverse Fourier transform writes : 1 f^ V(x', t) = — exp{-iqt)}C{x', q) dq (36) 27ry_oo On can remark that the spatial support of V is independent of the time t and bounded in P by the extension of the crack surface ( suppX>(x', t) C F ). This implies that the spatial supports of /C(x', q) and P(x', t) are exactly the same. The identification problem of the geometry of the crack is now equivalent with the identification problem of the supports of K, and V. In order to determine the support of /C, which characterizes also the geometry of the crack, its sufficient to study the behaviour of )C{x', q) with ^ > 0 arbitrary fixed. As: dnWq^'\x^=Q = 5x3^9 e'1x3=0 = ( I ^ T " 9^ " ^^?) ^
(37)
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On the identification of a crack in 3D acoustics
The reciprocity gap can be rewritten an allows the determination of /C(x', q) from the following equation:
/.
/C(aj', q) exp(-z i' • x') dx' - nB{i', q)(\if
- q^ - leq)"^ : - Q{i')
(38)
The right hand side of the preceding equation denoted by 6(^') is a C°° function. This is assured by the term eq> ^. Let us continue analytically all the functions depending on (J in a neighborhood of the real axes in a complex space, i.e. s' — ^' + ir)', r}' e B?. Equation (38) becomes: /
/C(x',^)exp(-zs'. x')dx' - 7^i3(s^^)(|sf - q^ - leq)''^
(39)
As the body has been supposed to be bounded in space, Q C [-a, a] , one can obtain for sufficiently large \s'\ the following inequality: \G[s')\ < Cexp(a(|5i| + |s2|))exp (a (l^il^ + \s2\^)'^) < Cexp(2a(|5i| + \s2\))
(40)
with C > 0 is a real constant. We recall the inequality (40) assures that a distribution is of exponential type < 2a. Therefore G{s^) is a distribution of an exponential type and applying the Paley-Wiener theorem ([14], p. 271) implies that Q{s') is the Fourier transform of a tempered distribution with a compact support. Passing to the limit for e ^ 0+, with a fixed q > 0 gives the following expression of the function /C : ^x',
9) = ^
/ ^ 7lB(r, 9)exp(i x' • ^'Mf
- q'-
*0+)-5 d^'
(41)
The explicit computation of this integral gives the support of /C, consequently the support of V = [w] and therefore the geometry of the crack. CONCLUSION This work proposes a reconstruction method for the position and shape of a planar crack from transient acoustic measurements in a bounded body. The work has been based on the reciprocity principle and can be extended to elastodynamics. ACKNOWLEDGEMENT This paper was partially supported by EC Contract no. ERBIC15 CT97 0706. REFERENCES [1] ACHENBACH J.D. 1980 Wave Propagation in elastic solids North Holland Publications, Amsterdam
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[2] ALESSANDRINI G . et DIAZ VALENZUELA A., 1994. Unique determination of multiple
cracks by two measurements. Quad. Mat. Univ. Trieste.
[3] ALVES C . J . S . et HA-DUONG T., 1997. On the far field amplitude for elastic waves. Modern Mathematical Methods in Diffraction Theory and its Applications in Engineering, Ed. E. Meister (Frankfurt: P. Lang) [4] ALVES C.J.S. et HA-DUONG T., 1997. On inverse scattering by screens. Inverse Problems, 13, p. 1161-1176. [5] ALVES C.J.S. et HA-DUONG T., 1999. Inverse Scattering for elastic plane waves. Inverse Problems, 15, p. 91-91. [6] ANDRIEUX S., BEN ABDA A. et Bui H.D., 1997. Sur identification de fissures planes via le concept d'cart la rciprocit en lasticit, C. R. Acad. Sci. Paris, 324, Srie I, p. 1431-1438. [7] ANDRIEUX S., BEN ABDA A. et Bui H.D., 1999. Reciprocity principle and crack identification. Inverse Problems, 15, p. 59-65. [8] Bui H.D., CONSTANTINESCU A., MAIGRE H., 1999. Diffraction acoustique inverse de fissure plane: solution explicite pour un solide borne C. R. Acad. Sci. Paris, 327, Srie lib, p.971-976. [9] CALDERON, A . , 1980. On an inverse boundary problem, Seminar on Numerical Analysis and Application to Continuum Physics, Rio de Janeiro, p.65-73. [10] COLTON D. AND KRESS A., 1992. Inverse Acoustic and Electromagnetic Scattering Theory, (Applied Mathematical Sciences 93), Springer, Berlin [11] HA-DUONG T., 1997. On boundary integral equations associated to scattering problems of transient waves. ZAMM, 6, p. 261-264. [12] FRIEDMAN A. et VOGELIUS M . , 1989. Determining cracks by boundary measurements, Indiana Univ. Math. J., 38, 3, p. 527-556. [13] LIONS J. L., 1986 Controlabilite exacte des systemes distribues, C. R. Acad.Sci. Paris, 302, seriell, 1986, p.471-475 [14] SCHWARTZ L., 1978. Thorie des distributions, Hermann, Paris.
INVERSE PROBLEMS IN ENGINEERING MECHANICS II M. Tanaka, G.S. Dulikravich (Eds.) © 2000 Elsevier Science Ltd. All rights reserved.
213
ON THE IDENTIFICATION OF CONDUCTIVE CRACKS
C. J. S. ALVES ^^\ T. HA DUONG ^2) and F. PENZEL ^^^ ^^Wentro de Matemdtica Aplicada, Instituto Superior Tecnico Av.Rovisco Pais 1, 1049-001 Lisboa, Portugal Email: calves@math. ist. utl.pt ^^^ Universite de Technologie de Compiegne, BP 20529, 60205 Compiegne, France Email: e-mail:
[email protected] ^^^ Technische Universitdt Darmstadt, Schlossgartenstrasse 7, Fachbereich Mathematik, Arbeitsgruppe 12, 64289 Darmstadt, Germany, Email: e-mail:
[email protected] ABSTRACT We consider the problem of identification of a connected crack in a bounded domain. Conditions on the boundary data are presented such that the crack can be identified by the corresponding measurement. An admissible crack (or screen) is considered to be a part of a boundary of an open set with Lipschitz regularity. We show that in the case of admissible connected shapes, a single measurement is enough to determine the position and the shape of a conductive crack, or an acoustic screen. KEYWORDS Inverse problems, uniqueness, cracks, screens. INTRODUCTION Recently, results have been obtained on the identification of cracks in unbounded and bounded domains (an account of the state of the art can be found in [1]). The case of cracks in an infinite domain was first considered in [2] and in [3]. In [3] (Remark 5, Proposition 4) it was proved that the knowledge of three far field patterns generated by incident plane waves are necessary and sufficient to determine the location of a plane crack in R^. In this paper we shall study the problem of identification of a single crack in a bounded domain (see the references [4], [5] and [6] for recent developments). If one prescribes an infinite set of boundary data, the crack can be uniquely determined, see the citations no. 6-11 in [7]. In the paper by Friedman and Vogelius ([7]) it was proved that two measurements are necessary and sufl&cient to determine an insulating crack in a two-dimensional domain. More recently, new results were obtained by Alessandrini and
214
CJ.S. Alves, T. Ha-Duong and F. Penzel
Diaz Valenzuela [8], and Alessandrini and DiBenedetto [9], proving that in the case of conductive cracks two measurements are also sufficient to establish uniqueness in the three dimensional case. However like in [7], the authors obtain these results using input data which are differences of two Dirac masses on the boundary, therefore excluding more regular data such as continuous or integrable functions (see also [10] where other measurements are proposed). In this text we prove that for the case of a connected conductive crack (or acoustic screen) one measurement is enough to determine the crack if we prescribe any data that is continuous and non vanishing on the boundary-. CONDUCTIVE CRACK PROBLEMS Throughout the paper we denote by Q an open, bounded, simply connected set in R**, (usually d = 2, or d = 3) and by F its boundary, which we assume to be Lipschitz. For the definition of an open set with Lipschitz boundary we refer to [11]. Let cj"^ C n be a connected, open set with a Lipschitz boundary and CJ~ = ft\uj'^. Let n be the outer normal vector-field defined a.e. on F or on du;'^ with direction into R^\Q on F and with direction into LJ~ on duj^. We define an open submanifold of dcu^ to be an admissible crack 7. In the two dimensional case this means that any piecewise C^ curve inside Q is an admissible crack, and in the three dimensional case, the main restriction is on the surface orientation, excluding, for instance, the case of a Mobius strip. For smooth manifolds 7 the traces of smooth functions u for XQ € 7 are defined as usual by ^"^(0:0)== lim u{x), d^u{xo) =^ lim n(xo)'Vu{x). If d:^u{xo) = d~u{xo), we define ^^^(^o) = d^uixo). Then each function u G ^^"2(^X7) which satisfies Au e L^(Q\7) has traces u"^ 6 H^^^{^/) and d^u G i/~^/^(7) obtained by continuous extension of the trace operator (cf. [11]). Note that the spaces H^^l'^[^i) can be identified with the spaces defined by taking the restrictions of distributions from H^^l'\d'^^) and that if w G W:^{Q\-i) and iif Aw G L'^i^Vi) then the jumps [u] = u+ - uand [dnu] = d:^u — d~u can be extended by zero to distributions in H'^^l'^{duj'^). We consider the following Dirichlet problem
I
-Au -: 0 1/^ = 0
in fi\7, on 7,
u=f
onF,
The problem (D) is well posed in ^^^2(^X7) for any given / G /f^/^(F), and one can define the jump [dnu] G H~^^'^{^/) across the admissible crack 7, as well as the normal derivative g = dnU G i/~^/^(F). The function / will be called the input data and g the output data. The inverse problem that we consider is to determine a crack 7 from the couple (/, g) of input and output data. We emphasize that here we will consider connected cracks, and we will prove that, using positive (or negative) input data, one measurement is enough to determine a conductive crack. Difficulties with identification. First, we notice that it is rather simple to consider an example where a single measurement is not sufficient to locate a crack. In fact, just consider the input data f{xi,X2) = :ri3:2, and any crack located on the xi or on the X2
On the identification of conductive cracks
215
axis. It is clear that u{xi,X2) = 2:1X2 is a solution of problem (D), no matter what set Q is considered, as long as it contains the cracks. Since the solution is the same, this input data does not allow to distinguish between two different conductive cracks if they are located in the axes. Moreover, supose that the two diflFerent cracks are both located in the xi axis, if the experiments are made using input data /(xi, X2) = {xi — c)x2, where c is any constant, the solution is u{xi, X2) = {xi-c)x2, no matter where the cracks are, since we have u{xi, 0) = 0, for all Xi. This shows that even an infinite number of measurements may be not sufficient to identify the crack, if the the input data is not well chosen. In the previous examples the input data changes sign, and we will now prove that if we provide input data that does not change sign, only one measurement will be enough to identify any conductive crack. To do this we begin by proving a crucial lemma that connects the support of the jump [dnu] with the crack itself. Notice that in the previous examples this support was void, because the solution was in fact analytical inside the all domain Q.
Figure 1: 3D profiles of the solution in two different cracks. One can see the discontinuities of the normal derivative on the crack. Under the hypothesis of positiveness on F it is clear, by the maximum principle, that no analytical solution would be possible. It is sufficient to change the sign of /, considering f{xi,X2) = |3:iX2| to see that the support of the [dnu] coincides with the crack. Two simple experiments with a finite difference method are presented in Fig.l. In both cases we considered Q —] - 1 , I p . The plot on the left shows the results for a crack 7i = [ - | , | ] x { 0 } , and the one on the right was made for a smaller crack 72 = {0} x [—i, 0], orthogonal to 71. In both cases it is clear the discontinuity of the normal derivative along the crack, which shows the relation between the support of [dnu] and the crack itself. In Fig.2, on the left, we plot out the difference between the output data dnU on the edges of the square for 71. It is clear that a 90 degrees rotation would produce a similar result, concerning a crack 73 = {0} X [—|, | ] , therefore it is this difference in F, the edges of the square ] — 1, I p , that allows the distinction between an horizontal or a vertical crack. It is worth noting that this difference is relatively small, in this case it is about 8% of the absolute values measured, and this is probably due to the fact that we impose / = 0 on the crack lines. More clear is the distinction between 72 and 73, as one can see in Fig.2, on the right. The dashed line shows the information on the edges which are parallel to the crack (notice that only one line is needed since the results turn to be the same either on {-1} x [-1,1] or on {1} X [-1,1]). One can see that the difference is more significant on the right, which
216
C.J.S. Alves, T. Ha-Duong and F. Penzel
corresponds also to the difference between 72 and 73 in what concerns the X2 axis. The thick line shows the result on the [—1,1] x {1} edge, where the difference is larger, and the the thiner one on the [-1,1] x {-1} edge, where the difference is smaller. This may still be explained to the fact that the difference between 72 and 73 can be resumed to the segment {0} x [0, \] which is closer to the [-1,1] x {1} edge. ,_,^^____^^
£0
15
^ ^ . - - ^
0.05 -0.1 0.15 -0.2
\
~ \
-
\y
-
-
/ /
Figure 2: The difference between the boundary profiles of dnU of the cracks 71 and 73 (on the left), and of 72 and 73 (on the right). Lemma 1. Let u he the solution of problem (D). If^ is an admissible crack and we consider f ^ 0,f ^ 0 onT, then supp{[dnu]) = 7. Proof. Suppose supp([^n^]) = 7i / 7, then 70 = 7\7i has a non void interior with respect to the topology of 7 and 70 Pi (^\7i) ^ 0. Since in 70 we have [dnu] = 0 and [u] = 0 (in fact u = 0), one can deduce that Aw = 0 in Q\7i. Thus, the maximum principle proves that there can not be a point XQ in ^ \ 7 i such that u{xo) — 0. That contradicts the fact that u is null on 70. • Remark 1. This support lemma also holds for non connected cracks and can be stated even for more general cracks with bifurcations, however we can not avoid the orientation of the crack, since otherwise we would not be able to define [dnu]Theorem 1. Let 71,72 be two admissible connected cracks such that, for an input data / > 0 , / 7^ 0, one has dnUi = dnU2 = ^, where u\,U2 are solutions of (D) for 71 and 72 (respectively). Then 71 — 72. Proof. We consider the open set r^\(7iU72). We define Qc to be the connected component of Q\(7iU72) with r C dO-c (there is only one component in this situation because 71,72 C Q), and we define Q* — Q\Qc (see Fig. 3). We have v = ui - U2 —^ inVtc because of Holmgren's theorem, because ui — U2 = f and dnUi = dnU2 = g on V implies v = 0 in a neighborhood of F inside Qc and by analytical extension i' = 0 on Qci) Suppose Q* = 0. This is the case when n\(7i U 72) = Qc is connected. In this case 1; = 0 on Q\(7i U 72), therefore [^„wi] — [dnU2\ on 71 and on 72. Outside 72 we know that U2 is analytic, meaning that 0 — [^„if2] = [dnU\] on 7i\72 and therefore supp([^n«i]) = 7i n 72- Using the Lemma we know that supp([^nWi]) = 71 and conclude that 7i = 7i n 72. The same argument gives 72 = 71 H 72, thus 71 = 72-
On the identification of conductive cracks
217
Figure 3: The filled area denotes the Q* set. This area can be crossed by 71 defining connected open components. One of this components will be the open set u. a) Suppose Q* y/^ 0. Let 7^ — Q* n 71, i.e. a part of 71 that divides Q* in open connected components and we take uu to be one of that components (if 7^ — 0 this means that uj = n*). Since dcu C Yi U dn\ this means that du = 7* U {dQ* n 71) U {dQ* n 72) and the part 72 = dQ* n 72 can not be void (otherwise du: C 71). Now, since ui = U2 on Q.c we have Ui — U2 — ^ on 72 C d^c and the condition Ui = 0 on 71 imply ui ::= 0 on duj. By uniqueness of the interior Dirichlet problem Ui — 0 in a; C Q\7i and therefore by analyticity Ui = 0 in Q\^fi which implies / = 0. This contradicts the hypothesis and therefore Vt* = 0, which brings us to case i). D Remark 2. An extension of the proof of this theorem to non connected cracks is under current research. SCREEN IDENTIFICATION We now extend this results for the Helmholtz equation, concerning the identification of acoustic screens. The proof of the Theorem 1 can follow the same steps, however, the proof of the support lemma must be different, since the maximum principle is no longer available! We now have
r -{A + k^)u=^0 {H)< u^ = 0
[ u=f
in Q\7, on 7,
onF,
where A; > 0 is the wavenumber. Our proof of the support lemma will have two restrictions: (i) 7 must be connected, and (ii) 0 < k < kc. Here ko is the smallest eigenvalue of the interior Dirichlet problem in Q. Lemma 2. Let u he the solution of problem {H). Suppose 0 < k < ko and that 7 is an admissible connected crack. If we consider / > 0 on F, then supp{[dnu]) — 7. Proof. Suppose again that supp([5„w]) = 71 ^ 7, then 70 — 7\7i still has a non void interior with respect to the topology of 7 and 70 Pi (0\7i) ^ 0. Since in 70 we have [^„w] •=• 0 and [w] = 0, one can deduce that (A + k^)u = 0 in Q\7i. Therefore 70 must be a part of a level line, and therefore it has an analytical extension, 7o where w — 0. It is clear that 70 can not cross F because we assume / > 0. Thus 70
218
C.J.S. Alves, T. Ha-Duong and F. Penzel
must intersect 7, and since 7 is connected this intersection defines an interior open set u, with border du) C 7 U 70, with Dirichlet boundary conditions w = 0. Since we suppose that k < ko and since a; C Q, by the strong monotonicity property of the eigenvalues we conclude that k is not an eigenvalue of the Dirichlet problem in cj, and therefore u — ^ in uj. The result now follows immediately, since by analytic continuation, if = 0 in Q\7, because u C n \ 7 , and this contradicts / > 0. • Remark 3. This proof can not be extended to non connected cracks. Consider the situation where 7 is defined by three components 7^ - [-3, -1] x {1}, 75 = [1,3] x {1}, 7^ = ^B(0,2)n R X R~. Suppose now that the support of [dnu\ is 71 = 7a U 76. Therefore 70 = 7c, and the analytic extension of the circle intersects 71 in the two disjoint parts 7a and 7^ without defining an interior open set uo, since the analytic extension can not cross 71 (see Fig.4). Remark 4- The restriction on k is merely for the sake of uniqueness in u;. An extension of the proof to any k is under current research.
Figure 4: In the case of non connected screens, it is not possible to ensure the existence of a;, as one can see in this example, since the analytical extension of 70 crosses 71 in two non connected components. Theorem 2. Suppose 0 < k < ko and let 71,72 be two admissible connected cracks. If for an input data / > 0, one has dnUi = dnU2 — g, where Ui,U2 are solutions of (H) for 71 and 72 (respectively), then 71 = 72. Proof. Immediate consequence of the proof of Theorem 1 and of Lemma 2. • REFERENCES [I] Isakov, V. (1998). Inverse Problems for Partial Differential Equations. Springer, New York. [2] Kress, R. (1995) Math. Meth. Appl. Sci. 18, 267. [3] Alves, C. J. S. and Ha-Duong, T. (1997) Inverse Problems 13, 1161. [4] Bryan, K. and Vogelius, M. (1992) SIAM J. Math. Anal. 23, 950. [5] Andrieux, S. and Ben Abda, A., (1996) Inverse Problems 12, 553. [6] EUer, M. (1996) Inverse Problems 12, 395. [7] Friedman, A. and Vogelius, M. (1989) Ind. Univ. Math. J. 38, 497. [8] Alessandrini, G. and Valenzuela, A. D. (1996) SIAM J. Contr. Opt. 34, 913. [9] Alessandrini, G. and DiBenedetto, E. (1997) Indiana Univ. Math. J. 46, 1. [10] Kubo, S. (1991). InJnverse Problems in Engineering Sciences, pp. 52-58; M. Yamaguti (Ed.), Springer, New York. [II] Grisvard, P. (1985). Elliptic Problems in Nonsmooth Domains. Pitman, London.
INVERSE PROBLEMS IN ENGINEERING MECHANICS II M. Tanaka, G.S. Dulikravich (Eds.) © 2000 Elsevier Science Ltd. All rights reserved.
219
I N V E R S I O N OF D E F E C T S B Y L I N E A R I Z E D I N V E R S E S C A T T E R I N G METHODS WITH MEASURED WAVEFORMS K. Nakahata and M. Kitahara Department of Civil Engineering, Tohoku University Aoba-yama06, Aoba-Ku, Sendai 980-8579, Japan
ABSTRACT The linearized inverse scattering methods are investigated to reconstruct the shape of defects in the two-dimensional elastic body. The methods are based on the Born and Kirchhoff approximations for unknov^n displacements in the integral representation of the scattered field. To show the performance of the methods, the specimen that has the circular cavity with cracks is prepared and the backscattering data are measured. The measured waveforms are processed and fed into the inverse methods. The results show the capability of the methods to reconstruct the shape and size of defects. KEYWORDS Ultrasonics, inverse scattering method, elastodynamics, measured backscattering data.
INTRODUCTION The size, shape and location of defects are a fundamental information to estimate the residual life of the structural component. The linearized inverse scattering methods to reconstruct the image of defects from the ultrasonic backscattering waveforms are investigated in this paper. The comprehensive review of the linearized inverse scattering methods has been given by Langenberg[l]. The Born inversion method has been studied by Rose et a/.[2,3] and KirchhofP inversion method by Cohen and Bleistein[4,5]. The performance of the methods to reconstruct the shape of defects has been examined by using the numerically calculated backscattering data in Ref.[6] for the 2D problem and in Ref.[7] for the 3D problem. The experimental performance of the methods has been shown in Ref.[8] for an elliptical cavity model and a notch model. In this paper, the aluminum specimen which has the circular cavity with notches is prepared as the combined defect model and the backscattering waveforms from the defect are
220
K. Nakahata andM. Kitahara
acquired by the immersion ultrasonic testing in the pulse-echo transducer configuration. The performance of the linearized inverse scattering methods is examined by using the measured backscattering waveforms.
SCATTERING PROBLEM It is important to distinguish the crack-like defects from the volumetric defects like a circular cavity. Here we consider the combined defect model as shown in Fig.l. The defect D^ exists in the two-dimensional elastic body D. The elastic modulus and mass density are denoted by Cijki and p for the host matrix D\D^ and by Cijki + ^Cijki and p + Sp for the defect D^. The measurement point y is far from the defects in the usual transducer arrangement. In this measurement situation, the L-L pulse echo methods for the ultrasonic testing is adopted. Here, the longitudinal(L) wave is transmitted as the incident wave and the same transducer receives the backscattered longitudinal(L) wave from defects.
Fig.l, Defect D"^ and wave fields in the host matrix
D\D^
LINEARIZED INVERSE METHOD In this study, two inversion methods are applied to reconstruct the shape of defects from the ultrasonic backscattering data. These methods are based on the elastodynamic inverse Born and Kirchhoff approximations. The details of two inversions have been given by Kitahara et a/. [6,7]. Here the brief outline of inversions are summarized. In the usual transducer configuration for the ultrasonic testing, the measurement point y is far from the surface of defects. The far-field expression of scattered wave is obtained as < ^ - ( y ) =. D{kL I y \)Am{if) + D{kT \ y \)B^{y)
(1)
where D{z) — J2/{Tiz)e'^^~''/'^\ In Eq.(l), Am and Bm are scattering amplitudes for longitudinal and transverse waves, respectively. In this study, we use the longitudinal
Inversion of defects
221
scattering amplitude in the actual experimental measurement A^[y) = ^^%ym 4/i
j qi{x)e-"''-y-^dV
(2)
JD
where K = ki/kr and y is the unit vector pointing to the measurement point y. In Eq.(2), qi{x) is the equivalent source which represents the characteristics of the defect. There are two ways to get the explicit form of the equivalent source. From the volume-type integral formulation, we get the expression qi{x) = r{x){Spuj'^Ui{x) - SCijkiUk,i{x)d/dxj}
(3)
where r{x) is the characteristic function of the defect D^ and has a unit value in the defect. From the surface-type integral formulation, the following form of qi can be obtained qi{x) = -j{x)Cijki{nj{x)uk,i{x)
- ni{x)uk{x)d/dxj]
(4)
where 7(x) is the singular function which has a value on the surface of the defect. When the scattered longitudinal wave component is measured, the left-hand side of Eq.(2) is known. If we can solve this integral equation for P or 7 in the equivalent source g^, the shape of the defect can be reconstructed. The problem is that both the geometrical function F (or 7) and the displacement field Um are unknowns in Eq.(2). For this reason two approximations are introduced for displacement fields to linearize Eq.(2). Born Inversion The Born approximation is introduced to the scattering amplitude in Eq.(2). Here the volume type of the equivalent source in Eq.(3) is adopted. The Born approximation is to replace the displacement field u by the incident wave u^ in the defect D^. The incident wave n° is assumed to be the longitudinal plane wave u°(x) = ^Od%xp(z^V.ic)
(5)
where vP is the amplitude, d is the unit polarization vector, k^ is the wave number of the incident wave, and p^ is the unit propagation vector. In the case of L-L pulse echo method, k^ = ki, d = p^ = —y. For voids, the elastic modulus SCijki and mass density Sp are set to be —Cijki and —p. The longitudinal scattering amplitude in Eq.(2) is reduced to the following form
AUh, y) = ^^%^ / r{x)e-'"'^y-^dv. Z
JD
(6)
In Eq.(6), the integral is the Fourier transform r{K) \j^^2k ii ^^^^^ characteristic function with K = 2kLy- If we can obtain the scattering amplitude Am{kL,y), the characteristic function T{x) is reconstructed by the inverse Fourier transform
r(x) = - 4 r
TT 70
r
^0
%iAm{kL,y)e''''y-''kLdk,dy. U KIT
(7)
222
K. Nakahata and M. Kitahara
Kirchhoff Inversion In the Kirchhoff approximation, the surface type of the equivalent source in Eq.(4) is adopted. The Kirchhoff approximation is to replace the displacement field u on the defect surface by the incident wave u^ and the reflected waves at the tangent plane of the surface. In this case, the scattering amplitude in Eq.(2) is reduced to Am{kL,y)
u^ymki = --
(8)
JD
where 7//(x) = ^{x)H{y • n{x)) is the function which has values in the illuminated side of the defect surface and H{') is the Heaviside step function. From Eq.(8), JH{X) is obtained by the inverse Fourier transform 2
r27r
TT^ Jo
Jo
(9)
U^KL
P R O P E R T Y OF B O R N A N D K I R C H H O F F I N V E R S I O N The basic property of two inversion methods is shown first by using the numerically calculated backscattering waveforms. The defect model is shown in Fig.2. In this model, the cracks of length a are attached to a cylindrical cavity of radius a.
o a
(y/a)
(y/a) oJ
- 2 - 1 0 1 2
-
3
-
2
B o m inversion
-
1
0
1
2
3
(x/a)
(x/a)
Kirchhoff inversion
Fig.2. Born and Kirchhoff inversions from numerical waveforms.
Inversion of defects
223
The backscattered waveforms are calculated by the boundary element method in all directions. The Born inversion reconstructs the characteristic function F which takes the value in the defects. The volumetric part is reconstructed well in Fig.2. The Kirchhoff inversion reconstructs the singular function 7 which takes the value only on the surface. The surface of the defects is reconstructed clearly in Fig.2.
EXPERIMENTAL SETUP The experimental setup is shown in Fig.3. The cylindrical aluminum specimen with a defect model is immersed in water. The pulser generates the electric pulse signal. This signal makes transducer vibrate and the ultrasonic waveform is transmitted into water. The scattered waveform from the defect is measured by the same transducer. The waveform transformed to the electric signal is branched to the receiver, and recorded on the digitaloscilloscope as the time-averaged data. The data in time domain are transformed to the frequency domain. The frequency domain data are used as the input for the linearized inverse methods.
Pulser-receiver
Fig.3. Experimental setup.
Digital-oscilloscope
pecimen
In this study we use the immersion type transducer whose center frequency is IMHz. The normalized frequency spectra of reflected waveform from the aluminum plate is shown in Fig.4. The frequency range from O.lMHz to 1.5MHz is used for the inversion of defects.
224
K. Nakahata andM. Kitahara 1.0-
— Absolute value - - Real part Imaginary part
y\ 1 ' 1 111/ 1
u 0 "
X vX
I
1
1
1
1
/
\
I
1
1
1
i
I
1
1 1
II II 1J
M
1 1
^
ly*—7-*,-^ \ /
/
V/
~
\ J \\
—
'
—
'
—
1
—
'
—
•
—
'
—
1
—
1
Frequency(MHz) Fig.4. Frequency range of transducer.
DATA P R O C E S S I N G The longitudinal scattering amplitude ym^m in the elastic solid is required for the shape reconstruction from Eqs.(7) and (9). To extract the waveform in the solid, the following data processing[9] is adopted here. The received waveform 0*^(/) in the experimental system is expressed in the frequency domain
o-(/) = /(/)T(/)py(/)H„,(/)£-(/)//,„(/)Ty(/)ii(/)
(10)
where / ( / ) is the input-signal, T{f),W{f),H^s{f),J^sw{f)yR{f) are the effects of the transmitting transducer, water path, transmission from water to the solid, transmission from the solid to water, receiving transducer, respectively. The term E^^{f) represents the scattering effect in the solid. The reference signal 0^^^{f) is measured as shown in Fig.5 and it can be written
O^'Hf) =
nf)T{f)Wif)HUf)E''^H,Uf)Wif)R(f)
(11)
where E^^^ is the reflection coefficient at a planar free surface. It is to be remarked that the measured reference signal in E q . ( l l ) is same as the measured signal in Eq.(lO) except for terms E^^^ and E^^. Therefore the scattered waveform in the elastic soHd is obtained from pref O'V) £-(/) (12) Oref{fy The processed waveform E^'^{f) can be used as the longitudinal scattering amplitude yraAm[kL,y)
in Eqs.(7) and (9).
Inversion of defects
lis
Fig.5. Measured and Reference waveforms.
MEASUREMENT AREA AND ANGLE In the actual NDT, it is not easy to measure the scattering ampHtudes from defects in all directions and frequency range. Here we examine the resolution of defects that the linearized inverse methods have from the following two view points. Measurement area The measurement area is defined in Fig.6. Here the measurement area is chosen in all direction or in the one side of the specimen. Measurement angle The measurement angle A^ is defined in Fig.6. Here the shape reconstruction of defects are performed by the waveforms with the measurement angle of 10° or 20° intervals.
Measurement area
Fig.6. Measurement area and angle.
226
K. Nakahata and M. Kitahara
SHAPE RECONSTRUCTION OF DEFECT We used a specimen made of aluminum as shown in Fig.7. In this specimen, two notches are attached to the circular cavity.
80r
Fig.7. Aluminum specimens for experiment. Figs.8 ~ 10 show the results of the shape reconstruction in the measurement angle of A^ — 10°. Fig.8 is the result from measurements in all directions and the shape of defect is clearly reconstructed. Fig.9 is the case of upper side measurement and Fig. 10 is the left side measurement. In Figs.9 and 10, the side of illuminations is almost reconstructed. Fig. 11 shows the result for the measurement angle of A^ = 20°. The rough shape of the defect is reconstructed in this case.
^G=w
All direction
i5i 10
(mm) 0 -5-
-20 -15
Bom inversion
-10
-5
0
Kirchhoff inversion
Fig.8. Reconstruction by Born and Kirchhoff inversions (measurement area : all direction)
Inversion of defects
227
A0=1O° Focus Upper side
15-|
(mm)
iHi
0-
-5-
•
-10-
10
^i
14 1i t|1111
5-
5
^.--,^=0
(7)
where K is the stiffness matrix of the structure; M is the mass matrix; cOj is the yth natural circular frequency and ^^ is theyth mode shape. The change in square of natural frequency, ^co], can be evaluated in first-order approximation: AcD]^f.AK(l>j{f^M(l>^
(8)
In order to identify the damage locations, the idea of damage signature matching (DSM) technique is adopted [13]: If the changes of the static or dynamic response for all possible damage cases are predicted with an analytical model, the measured response changes can be compared with them one by one. The set of predicted changes, which are well matched with the measured values, can be identified and the corresponding damage case can be considered as the real damage of the structure. However, the magnitude of static or dynamic response changes due to damage depends on both the damage location and extent. The effect of damage severity should be eliminated before using the approach of DSM. Synthetically using the information of natural frequencies and static responses, the following vector is defined as the Damage Signature (DS): {DS}=A«/(A^;)
(9)
where is^co. is the change in anyone of all natural circular frequencies. According to equations (4) and (8), for the case of single damage or multiple damages with the same severity, the scalar a denoting damage extent could be eliminated in equation (9). The DS depends only on the damage location. Then, matching Measured Damage Signatures (MDS) with Predicted Damage Signatures (PDS) can assess the damage location. The MDS of load case i is defined as {MDS,}=[A«,/(A^^)L
(10)
where the subscript m denotes that the measured data are employed. The change in the first order frequency is generally used as the reference value. Similarly, the PDS under load case /, assume element k is damaged, can be calculated from equations (4) and (8) as: {PDS,}=[(A«,y(A«^)J^ (11) where the subscript p denotes that the value is predicted analytically. Finally, the summation of the norms for all load cases is defined as the total discrepancy for the possible damage case to evaluate the possibility of existing damage in element k £.=II|{PDSj-{MDS,,||
(12)
7=1
where nl is the total number of the static load cases. It should be noted that AM in the above equations includes only those degrees of freedom, where measurements have been made. The unmeasured components can be set to be zero or removed from the global vector using an appropriate Boolean matrix. The possible damage location in the structure is indicated by the smallest discrepancy calculated from equation (12). The following damage index Dek is defined for element k\
D.,=\IE,
(13)
242
X. Wang, K Hu and Z.H. Yao
Estimation of damage extent When the possible damaged members are detected by employing the algorithm described above, the estimation of damage extent is the next important step. The present algorithm is based on the minimization of a squared error function, which has the following general form for every load case, where the variables x are linearly constrained between bounds: minimize J(x) = \e(x)f jl
s.t x<x<x
(14)
where e{x) is a measure of discrepancy between an observation and its counterpart computed by the parameterized model. For a certain load case of load vector /?, the measure of error is the difference between the change of the displacements computed from the FE model and the measured response: e,{a) = [K-'p,-{K^^Kyp,]-Mi,
/ = 1,2,...,«/
^K-'/^KK-'p,-^u,
(15)
where AM is the change of the displacements experimentally acquired under the load vector p. The design variable vector a includes the damage extents of all possible damaged elements. If there are NP possible damaged members at the first detection step, from equations (5) and (6), it can be obtained: A7^
K^KK'p
NP
= Y,a,E,K-\B]^k,B,)K-'p
= Y.a,Z,
(16)
where Z, = E.K'iB'^Mi^E^K'p
(17)
By employing the least-square method, the following optimization model can be established. minimize ^ ^ (a,Z/ - Aw ^ j s.t. -1.0*/dxi)ni.
The entire boundary of material m is discretized into boundary elements with A^^^ nodes. When point A is set to be at the z-th nodal point on boundary Tm of material m, Eq. (5) is discretized in the form,
c^^^ + E ^h^^r^ - E ^h^t^ = 0
(m = 1, 2),
(10)
where 0*^ and ql^ are determined by integrals expressed in terms of the fundamental solutions (j)* and q* over elements containing point j . The quadratic element was used in this study. Equation (10) gives a matrix equation for (/> and q at nodal points. The boundary values at nodes, and therefore the distributions of potential (j) are obtained by solving the matrix equation for materials 1 and 2, together with prescribed boundary conditions on the outer boundary concerning 0 or q, and the compatibility of 0 and q written by Eq. (4). NUMERICAL SIMULATIONS Identification
of Two-Dimensional
Delamination
Defect
Numerical simulations of identification were made for two-dimensional delamination defects in bonded dissimilar materials shown in Fig. 1. The axes of orthotropy were taken to coincide with X2X3 (yz). The values of a^""^ and cr^"^^ shown in Table 1 were used. The interface was assumed to be parallel to the y direction. The potential distribution on GH, which was obtained when potential 0 was prescribed to be 1 and 0 at points G and H, was used for the identification of the defect. Location of defect center yc and half defect length a were estimated from the potential distribution.
The boundary element analyses were made to generate measured data of electric potential 0^^^ on the observation surface Fob- The symbol " • " in Fig. 1 shows the location of observation points. To investigate the eflfect of measurement noise in 0^^\ artificial random noise of the level of 0.1%, 0.5% and 1.0% was introduced in the calculated potential values.
An example of the results of identification for the noise level of 0.1%, 0.5% and 1.0% is shown in Table 2. It is seen that a reasonable estimation can be made even for the noise level of 0.5% although the estimation is deteriorated with increase in the noise level.
261
Identification of delamination
H 22. 5
f
22. 5
Ma t e r i a 1
1
O
F ^ II
yo
Ma t e r 1 a 1
3:
2
V ^
y
W= 1 0 0
^
C
Fig. 1 A delamination defect in two-dimensional bonded dissimilar materials used in numerical simulations.
Table 1 Orthotropic electric conductivity used in numerical simulations.
Material 1 776 Material 2| 0.53
0.53 776
3.74 3.74
Table 2 Parameters of a two-dimensional delamination defect estimated from noisy data. Noise[%]
Estimated defect parameters Center yc Ayc/W[%] Half length a Aa/W[%]
0.00
50.000
0.000
0.10
49.391
0.50
49.183
1.00
[Actual value
1
Rs
1
O.OOE+00
7.500
0.000
0.609
7.736
0.236
1.20E-05
0.817
8.553
1.053
1.14E-04
48.988
1.012
9.193
1.693
50.000
0.000
7.500
0.000
4.55E-04 1 0.000 1
262
S. Kubo, T. Sakagami and N. Tanaka
Xc
\
= 35
''illllll
.
^
X c=15
(a) Defect shape I
•11^1-4 J (b) Defect shape II
Fig. 2 Delamination defects in three-dimensional bonded dissimilar materials used in numerical simulations. Identification
of Three-Dimensional
Delamination
Defect
Numerical simulations of identification were made for elliptical delamination defects in three-dimensional bonded dissimilar materials shown in Fig. 2. The interface was assumed to be perpendicular to one of the axes of orthotropy. The value of af^' shown in Table 1, which corresponded to that of layered CFRP [18], was used. The electric potential distributions on side surfaces for 75 < 2: < 105 were used, which were obtained when 0 == 1 and (/) = 0 were prescribed on lines y = 16.5 and y = 3.5 on the top surface ABCD. The defects were assumed to be eUiptical, and their locations and sizes were estimated.
The boundary element analyses were made to generate measured data of electric potential (/)^^^ on the observation surface. The symbol " •" in Fig. 2 shows the location of observation points. To investigate the effect of measurement noise in (t)^^\ artificial random noise of the level of 0.1%, 0.5% and 1.0% was introduced in the calculated potential values.
To speed up the identification the hierarchical inversion scheme [19] was employed, in which a two-dimensional scanning analysis was followed by a three-dimensional analysis. In the two-dimensional scanning analysis the least residual method was applied to a cross section shown by dotted area in Fig. 3. The residual Rs between (/)(^^ and ^^^^ was evaluated along lines MN and PO on the free surfaces. The potential distribution (jP^'> on side surfaces, which was obtained when potential (j) is prescribed to be 1 and 0 at points Ti and T2 on the top surface, was used for identification of the defect. The calculated electric potential (t)^^\ was obtained by applying the two-dimensional boundary element analysis. The defect
263
Identification of delamination
A
P
^/W A \n
= 1
D
C
^ = 3-5
nn -VdV Jn Jr
(5) = lim -{M^s) - Jr{f2)) = f U'r + {Vnr + c/^r^) n-V] dP s-vo s Jr ^ ^ where n denotes the outer normal vector well defined at almost all points of T, Vn( •) = V( •) • n, K denotes the mean curvature, and (j)'^ and (l)'p indicate the derivatives under a spatially fixed condition during domain perturbations of the distributed functions (j)n and 0 r respectively [4]. Mf2)
SHAPE IDENTIFICATION PROBLEMS Using the definitions and formulae with respect to domain perturbations, shape derivatives for boundary shape identification problems can be derived. For the simplicity, let us consider a strong elliptic boundary value problem of the second order related to a realvalued scalar state function. This problem is described in the strong form: - V • A{x)Vu{x) + ao{x)u{x) = / ( f ) u{x) = uo{x) X e To A{x)Vu{x)
• n(f) = g{x)
(6) (7)
x e ^
f G T \ fo
(8)
uoeUo
(9)
and in variational form: a{u, v) = l{v)
u-uoeU
yv eU
where the bilinear form a( •, •) and the linear form /(•) are defined by a{u, v) = l(v)=
{Vu • AVv + auv) dx Jn f fvdx-\- [ gvdr Jn Jr\ro
(10) (11)
and UQ{X), A{X) = A{x)'^ and ao(f), x G R^, denote given functions. ellipticity, the following conditions are required. 3a > 0 :
ao{x) >a
and
z- A{x)z > a\z\'^
For the strong
Vf G R""
\/z e R"
(12)
To assure existence of a unique solution for state function u, it must be at least that A G (L°°(i?^))'*^'', ao G L°°(i?^), / and ^ G i/^(i?^) and the admissible sets UQ and U are given by Uo = {ue
H\Q)\
U = {ueH\f2)\u(x)
u{x) = 0, f G r \ A } = 0, xeTo,
f udx = Owhen
Jn
(13) f JTO
dP = 0}
(14)
280
H. Azegami
Referring to boundary value on subboundary A shape determination problem in which the state function u is specified with a given function w on s. subboundary / D C T \ fo can be formulated as a minimization problem of a squared error integral: Err,{u-w,u-w)=
(15)
f {u-wfdr
and described by min
Epn (u — w,u — w)
such that
a(u,v) = l{v)
u — UQ e U
(16)
UQ e UQ ^V e U
For the sake of simplicity, let the coefficient functions of up, A, ap, f and g be fixed in R^ during domain perturbations and the velocity 1/ = 0 at To D T \ To and the singular points on r . The shape derivative of the objective functional can be obtained as follows. Applying the Lagrange multiplier method, or the adjoint variable method, the Lagrange functional L(i7, u, v) of this problem is defined by L = Epj^ {u — w,u — w) — a{u, v) + l{v)
(17)
where v e U was introduced as the Lagrange multiplier function, or the adjoint function, with respect to the weak form in Eq. (16). Using the formulae of the material derivative shown in Eqs. (4) and (5), the shape derivative of the Lagrange functional is obtained by L = 2Er^ {u - w, u') - a{u', v) - a{u, v') + l{v') + {Grr,n, V)
(18)
where the linear form {Gr^'^, V) with respect to the velocity V is defined by {Gr^fi,V)=
f
G^ft-Vdr^
j Gun'Vdr+
Gyj = 2{u - w)S/n(u - w) -\- {u- w)'^K,
I _ GgU-VdE
+ f GfU-VdP
{19)
Gu = - V ( w - Up) • AVv - ap{u - up)v,
Gg = Vngv + ^Vn^ + gvi^, Gf = fv
(20)
Considering the invariability of L with respect to any perturbation of v' E U and u' e U at the optimal condition, the governing equations with respect to u and v are given by a{u,v') = l{v') WeU a{u\v) = 2Eroiu-w,u') WeU
(21) (22)
Eq. (21) gives the same solution as the week form of Eq. (9). Eq. (22) is called the adjoint equation of the present problem. Using the solutions u and v of Eqs. (21) and (22), the derivative of the Lagrange functional agrees with that of the objective functional and the linear form {GTC^, V) with respect to the velocity V: L\u,v^ErM,^={Gr^n,V)
(23)
From the fact that the vector function Gr^n is a coefficient function with respect to velocity V that is the derivative of the design function Tg, GrD^i is called the shape gradient function of the present problem. The scalar function Gpo is called the shape gradient density function.
281
Solution to boundary shape identification problems Referring to gradient in suhdomain
Another shape identification problem is considered in which the gradient of the state function V ^ is specified with a given vector function w in subdomain QD C O. Using the objective functional defined by EQ^ {VU -W,VU-W)=
[
{Vu - w) ' {Vu - w) dx
(24)
JQD
this problem is formulated by min
EQJ^ {VU
such that
— w, Vu — w)
a{u, v) = l{v)
u - UQ e U
UQ e UQ \/V e U
(25)
Let us assume the same conditions for UQ, A, ao, f, g and V. The Lagrange functional of this problem is defined by L = EQJ, {VU -W,VU-W)-
a{u, v) + l{v)
(26)
Using the formulae of the material derivative shown in Eqs. (4) and (5), the shape derivative of the Lagrange functional is obtained by L = 2EQ^ {VU - w, Vu') - a{u', v) - a(u, v') + l{v') + {Gn^n, V)
(27)
where the linear form {GQJ^U^ V) with respect to the velocity V is defined by {Gn^n,V)=
[
G^n'Vdr+
G-uj = {Vu - w)' (Vu ~ w), Gg = Vngv + gVnV H- gvi^,
[ Gun'Vdr+
f _ Ggn'Vdr+
f Gffi'Vdr
(28)
Gu = -V{u - UQ) ' AVv - aQ{u - uo)v, (29)
Gf = fv
where dQo denotes the boundary of HoConsidering the invariability of L with respect to any perturbation of v' e U and u' e U at the optimal condition, the governing equations with respect to u and v are given by (30)
a{u,v')=l{v') WeU a{u', v) = 2En^ {Vu - w, Vu')
W €U
(31)
Eq. (31) is the adjoint equation of the present problem. Using the solutions u and v of Eqs. (30) and (31), the derivative of the Lagrange functional agrees with that of the objective functional and the linear form (G^/^^n, V) with respect to the velocity V: L\u,v = Ea,\u,v = {Gann.V) GQ^U is called the shape gradient function of the present problem.
(32)
282
H. Azegami
GRADIENT METHOD IN HILBERT SPACE With shape derivatives the gradient method in Hilbert space has possibility to be applied to reshaping algorithm. Let ^ be a real Hilbert space with scalar product (•, • )$ and norm II • ||, (j>) > aUWl
\/(l>e^
and
6(0, ip) < /?||0|U||^|U
V0, ^ G + C^0) = JW + {Gj, CM)^ + 0(0 = J{cj>) - 6(Z\0, C^0) + o(C) < J{(t>) - aC||zA(^|p + o(C)
(37)
where C is a small positive number and o( •) is the Landau functional, i.e. lim^^o 7 ^ ( 0 = 0Indeed, the second term in the right side of the inequality is strictly negative and the third term can be made very small. TRACTION METHOD Applying the gradient method in Hilbert space to the shape identification problems, a concrete solution can^be proposed by selecting an appropriate Hilbert space. However, the design variables fg and its derivative V were belong to {W^''^{Br')Y which is not a Hilbert space although being a Banach space. Then it is not possible to apply the gradient method in Hilbert space to the shape identification problems directly. A well-advised idea is to select a Hilbert space which includes {W^''^(BP')Y and to find a domain variation V belongs to {W^^'^{BJ')Y. Such a Hilbert space can be found in [H^{BJ')Y that can be defined by D=^[v
e [H\R'')Y
I V[x) = 0, f G {fo n T\TQ
and singular points on T } , and
constraints with respect to domain perturbations > One of the most familiar coercive bilinear forms in {H^{R^)Y continuum problems restricting rigid motions: a{u, ^) = / CijkiUk,iVij dx
(38) is that used in linear elastic (39)
283
Solution to boundary shape identification problems where Cijki € L°°{R^), ij.kj = 1,2, •••,?!, denotes the Hook stiffness tensor. Using a( •, •) for 6( •, •) in Eq. (36), a concrete solution can be presented for determining the velocity V e D by a{y,y) = -{Gr^n,y)
or
- (C^^n,^
Vy€D
(40)
and reshaping with AsV for a given small positive number As. This solution coincides with the traction method that author proposed previously [6, 7, 9]. Whether or not the solution V in Eq. (40) belongs to J9 fl {W^^'^{R'')Y depends on the smoothness of the shape gradient function. The necessary smoothness for the boundary and coefficient functions was discussed in the previous paper [9] using the regularity theorem for elHptic boundary value problems [3, 19]. Reshaped domains by this solution have smoother boundary in differentiability for one time than those by the direct solution moving boundary in proportion with the shape gradient functions [9]. REFERENCES 1.
CEA, J. (1981). Problems of shape optimization. In: Optimization of Distributed Parameter Structures, Vol. 2, pp. 1005-1048, E. J. Haug and J. Cea (Eds.). Sijthoff & Noordhoff, Alphen aan den Rijn.
2.
ZOLESIO, J. P. (1981). The material derivative (or speed) method for shape optimization. In: Optimization of Distributed Parameter Structures, Vol. 2, pp. 1089-1151, E. J. Haug and J. Cea (Eds.). Sijthoff & Noordhoff, Alphen aan den Rijn.
3.
PiRONNEAU, O. (1984). Optimal Shape Design for Elliptic Systems. SpringerVerlag, New York.
4.
SOKOLOWSKI, J. and ZOLESIO, J. P. (1991). Introduction to Shape Optimization: Shape Sensitivity Analysis. Springer-Verlag, New York.
5.
CEA, J (1981). Numerical methods of shape optimal design. In: Optimization of Distributed Parameter Structures, Vol. 2, pp. 1049-1088, E. J. Haug and J. Cea (Eds.). Sijthoff & Noordhoff, Alphen aan den Rijn.
6.
AZEGAMI, H. (1994). A solution to domain optimization problems. Transactions of the JSME, Series A, 60, pp. 1479-1486 (in Japanese).
7.
AzEGAMi, H., SHIMODA, M . , KATAMINE, E . and Wu, Z. C. (1995). A domain optimization technique for elhptic boundary value problems. In Computer Aided Optimization Design of Structures IV, Structural Optimization, pp. 51-58, S. Hernandez, M. El-Sayed and C. A. Brebbia (Eds.). Computational Mechanics Publications, Southampton.
8.
AzEGAMi, H. and Wu, Z. C. (1996). Domain optimization analysis in linear elastic problems (approach using traction method). JSME International Journal, Ser. A, 39, pp. 272-278.
284
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9.
AZEGAMI, H . , KAIZU, S., SHIMODA, M . and KATAMINE, E . (1997). Irregularity of shape optimization problems and an improvement technique. In Computer Aided Optimization Design of Structures V, pp. 309-326, S. Hernandez and C. A. Brebbia (Eds.). Computational Mechanics Publications, Southampton.
10.
AZEGAMI, H . and SUGAI, Y . (1999). Shape optimization with respect to buckling. In Computer Aided Optimization Design of Structures VI, pp. 57-66, S. Hernandez, J. Kassab, and C. A. Brebbia (Eds.). WIT Press, Southampton.
11.
AZEGAMI, H . and KODAMA, K . (1999). Solution of shape optimization problems to maximize deformation under constraints on stiffness and strength. In Proceedings of the First China-Japan-Korea Joint Symposium on Optimization of Structural and Mechanical Systems, pp. 17-23, Y. X. Gu, B. Y. Duan, H. Azegami, and E. M. Kwak, (Eds.). Xidian University Press, Xi'an, China.
12.
Wu, Z. C. and AZEGAMI, H . (1995). Domain optimization analyses offlowfields. In Proceedings of Second China-Japan Symposium on Optimization of Structural and Mechanical Systems, pp. 236-241.
13.
Wu, Z. C. and SOGABE, Y . and AZEGAMI, H . (1997). Shape optimization analysis for frequency response problems of solids with proportional viscous damping. Key Engineering Materials, 145-149, pp. 272- 278.
14.
SHIMODA, M . , AZEGAMI, H . and SAKURAI, T . (1996). Multiobjective shape optimization of linear elastic structures considering multiple loading conditions: Dealing with mean compliance minimization problems. JSME International Journal, Ser. A, 39, pp. 407-414.
15.
SHIMODA, M . , AZEGAMI, H . and SAKURAI, T . (1998). Numerical solution for min-max problems in shape optimization: Minimum design of maximum stress and displacement. JSME International Journal, Ser. A, 41, pp. 1-9.
16.
SHIMODA, M . , AZEGAMI, H . and SAKURAI, T . (1998). Traction method approach to optimal shape design problems. SAE 1997 Transactions, Journal of Passenger Cars, 106, pp. 2355-2365.
17.
KATAMINE, E . and AZEGAMI, H . (1995). Domain optimization analyses of flow fields. In Computational Mechanics '95, Theory and Applications (Proceedings of the International Conference on Computational Engineering Science), pp. 229234.
18.
IHARA, H .
19.
LADYZHENSKAYA, O . A. and URAL' TSEVA, N . N . (1968). Linear and Quasi-
and AZEGAMI, H . and SHIMODA, M . (1999). Solution to shape optimization problems considering material nonlinearity. In Computer Aided Optimization Design of Structures VI, pp. 87-95, S. Hernandez, J. Kassab, and C. A. Brebbia (Eds.). WIT Press, Southampton. linear Elliptic Equations. Academic Press, New York.
INVERSE PROBLEMS IN ENGINEERING MECHANICS II M. Tanaka, G.S. Dulikravich (Eds.) © 2000 Elsevier Science Ltd. All rights reserved.
285
SHAPE OPTIMIZATION OF TRANSIENT RESPONSE PROBLEMS Z.Q. WU\ Y. S 0 G A B E \ Y. ARIMITSU^ and H. AZEGAMI^ ^Department ofMechanical Engineering, Ehime University, 3, Bunkyou, Matsuyama, Japan ^Department ofMechanical Engineering, Toyohashi University of Technology, 1-1, Hibarigaoka, Toyohashi, Japan ABSTRACT A numerical method of shape optimization for transient vibration problems is presented. The transient responses are expressed as summation of frequency responses of each frequency by using Fourier transform. The shape gradient function is derived as a sum of shape gradient function corresponding to each single frequency and expressed in terms of modal parameters that can be obtained from modal analysis for transient responses. The traction method is applied to shape optimization analysis. It is low computational cost because there is no extra sensitivity analysis necessary. Numerical results are attached at the end of this paper to show the vaUdity of this approach. KEY WORDS Optimum Design, Vibration of Continuous System, Modal Analysis, Shape Optimization, Traction Method. 1. INTRODUCTION In general, the computation cost of the shape optimization problem in transient response is fairly higher than that of the stationary problem. The purpose of this study is to develop a simple and low cost numerical method of shape optimization for transient dynamic problems of soHds with proportional viscous damping. A well-known approach for shape optimization problems, which is called the discrete approach, describes a domain shape with parameters of finite numbers and finds out an optimum solution of the parameters by utilizing mathematical programming methods. This approach, however, becomes quite costly in calculating the sensitivity and searching the optimum when the number of design variables increases. An alternative approach, which is called the continuous approach, describes the domain variation with a mapping fiinction and applies the concept of material derivatives to derive the sensitivity, which we call shape gradient fiinction. The continuous approach is proved to be cheaper in calculating the sensitivity than utilizing the discrete approach. The traction method proposed by one of the authors above is a practical application derived from the theoretical implications of continuous approach[4,5]. The advantage of using this
286
Z.Q. Wuetal.
method is that we can obtain a smooth boundary with less distortion of mesh and lower computational cost. In this paper, the traction method is applied to shape optimization problems of transient dynamic responses. The transient responses are expressed as summation of frequency responses combined with power spectrums of each frequency by using Fourier transform. The shape gradient function is derived as a sum of shape gradient functions corresponding to each single frequency. The shape gradient function of a frequency response is calculated using modal parameters that can be obtained from modal analysis for transient responses. Numerical results are given to show the validity of this approach at the end of this paper. MAPPING AND MATERIAL DERIVATIVE METHOD To obtain a shape gradient function that represents the relation of a variation in a domain and the resulting variation in the objective functional, the material derivative method[l] is used.
Fig.l. 1 to 1 mapping Consider an initial domain i? with a boundary F varying to a domain n^ with a boundary r^ . The domain variation can be expressed in a 1 to 1 mapping T^(X):Xef2\-^xen^ , where the index s denotes the history of the variation. Regarding s as time, the velocity field V can be defined as[2] y -
Considering a domain integral functional J^
ds and a boundary integral functional J ^
(1)
of a
distributed function ^^ (x), their derivatives with respect to s are given by (2) (3)
287
Shape optimization of transient response problems ^; = lim —- (^^^^^ - ^,)
where
(4)
where v^ = n-V- and n is an outward unit normal vector[3]. In this paper, the Einstein summation convention and the gradient notation (•) • •=^d{:)l^x- are used in the tensor notation. The notation K denotes the mean curvature. MINIMIZATION OF FREQUENCY RESPONSES Since an arbitrary time-dependent function can be transformed into a summation of frequency response using Fourier transform, we will treat the frequency response problems at first and then apply the results to an arbitrary response in next section. Problems of Frequency Responses Consider a frequency response problem with proportional viscous damping. can be expressed in continuum form as a(u,v) + c{u,v) + h{Uyv) = d{p,v) where u(t,x)
and v(t,x)
Its state equation (5)
u GU, \fv eU
are the displacement vector and its variation respectively, U is
the admissible set defined as U = {VG(H (R,n^))''\v(R,r^^)
= O} .
a(u,v)
,b(u,v)
and d(p, v) are the bilinear forms defined as ^(u,v)
= j ^ ^ C^j^iUj^iV^jdx
(6)
Ku>v) = ^^ pu-v^dx
(7)
d{p.v)=\
(8)
p.v^dr
Here the damping term c(u, v) is considered as a proportional viscous damping, that is c(u, v) = ab(u, v) + /h(u, v)
(9)
where a and J3 are coefficients of damping. We introduce the following relations to rewrite the state equation in order to be convenient to use latter. p(t, r,2) = P(r^,) cos cut ^ P(r,, ) (e^^ + e-^^' )/2 u = —^-^
2
^
where P is the amplitude of a harmonic force, ( • )* is a conjugate complex number of ( • ). Assuming the arbitrary variable i; as i; = ve~^^^, then the state equation is obtained as
288
Z.Q.Wuetal {a{u,v) + jcoc{u,v) - co'b(u,v) - d{P,v)}e-'^ + {a(u\v)
- j(Dc(u\v) ~ co^b(u\v) - d(P,v)}e-''^
=0
Using modal analysis method, the solution is given as 00
^ = Z^(r)^(r) ^(.)-
(11)
Y^-^ ' i(^) -CD -^ JCO{P\r) + « )
TE {1,2,3,-,^}
(12)
where u^^^ and /l^^^ is the rth eigenvector and eigenvalue respectively, and there are the following relations. Problems of Mwimizirig Frequency Responses In a general way, a frequency response function can be expressed into a function of u integrating in a domain or over a boundary. That is F{u)^\^au)dx or
(13)
F(u) = \j{u)dr
For example, the absolute value of mean-compliance, the strain energy and the kinetic energy mean-comj ll W = —a(u,u*) and K = ~co^h(u,u*) d{P,u-^u*}. can be expressed as C |2 2 2 respectively. These functions are used frequently to reduce the vibration of whole structure. Even the local objective functions, such as stress and displacement, can also be written in integral form by introducing a 5 -function. Optimization problems of frequency response can be formulated as Given Q,M,P,co find Q^ that minimize F{u) subject to
{{\ + j(DP)a{u,i) + -^{{\-j(op)a{u\v*)-{(o^
and
772 = j ^ dx<M
{co^^-jcoP)b{u,i)-d(P,i)) + jcoP)co^b(u\v*)
- d(P,v*)} = 0
(14)
where Eq.(14) is the constraint on volume. By combining objective function and constraints with Lagrange multiplier, the Lagrange functional L is assembled as
289
Shape optimization of transient response problems L = F{u) - {(1 + j(DP)a{u,i) - ((D' - jcoa)b(u,d) - d{P,i)} -{(o)^ -jcofi)a(a\i*)-(cD^
+j(oa)h{u\i*)-d{P,i*)}
^^^^ +
A{m-M)
where v is the Lagrange multiplier, and is also known as adjoint variable here, and A is the Lagrange multiplier of the constraint function. Shape Gradient Function In case of objective function F{u) = J f(u)dx,
using Eq.(2) and Eq(3), the derivative of L
with respect to s is derived as L - -{(1 + j(Dp)a{u,v') - (o)^ - ja)a)h{u,v') )-(6>' +j(oa)h{u\v*
-{{\-jcop)a{u\v*
d(P,v)} )-d{P,v*
- {(1 + jcDP)a{u',i) - (co' - jCDa)h{u\i) - \
%:u'dx)
-{(\-j(Dp)a{u*
-:^^^' •'^^ du
)} (16)
X)-(co^
+jcDa)b(u* .v*)-\^
+ A(m - M)
+
^^)
lG{u,i)nV^dr
where the function G is G{u,i) = f{u)-2Re[{\
+ jcDP)Cj,^iUk^iU^^j -{co^ ^ jcoP)pu^v,] +A
(17)
where Re[ • ] means the real part of a complex number. Here we call the scalar function G the shape gradient density function and call the vector function G = Gn the shape gradient function. Let the 5th term of Eq.l6 be equal to 0, we can obtain the constraint equation on volume. Let the 7st and the 2nd term of Eq.(16) be equal to 0, we get the state equation from which u can be determined. Let the 5rd and the ^th terms of Eq.( 16) be equal to 0, we get the adjoint equation as (l + ja)P)a(a\i})-(a)^-jcDa)b(A\i)-\
^u'dx=0 ^^s du
(18)
The solution v of this equation can also be described in terms of u^^.^ and /l^,.) which obtained by modal analysis. For example, when the objective function is given with strain energy, v can be described as
r=l
r=\ 2 (/t(^) -CO ) +CO (^P\r) + ^)
Eliminating terms that are equal to 0, one can get the derivative of L as
290
Z.Q.Wuetal L=\
G(a,i)nV^dr
(20)
•^^ s
In case of objective function F(u) = J f(u)dr,
the shape gradient density function can be
derived as G(u,i) = -2Re[(\ + jcoP)Cj^kiUkiU,j - (co' + jcofi)pu,v,]-^ A
(21)
while the adjoint variable v can be solved from the adjoint equation below (\^jo)/^)a(u\v)-((o^
-j(oa)h{u\v)-
f ^udF^O ^^s du
(22)
For example, when the objective function is given with the absolute value of mean-compliance, the shape gradient density function is obtained as G{u,6) = 2Re[(l + jcoP)Cj,j,iUf,,u,j -(a)'+ JcoP)pa,v^] + A V - sign(d(P,u
+ u*)) — u*
(23) (24)
MINIMIZATION OF TRANSIENT RESPONSE In last section, a method to derive the shape gradient function of problems of frequency response is presented. The result will be extended to transient response problems in this section. By using Fourier transform, any excitation can be broken up into a sum of sinusoids or complex exponentials. Hence, by the superposition property of liner systems, the response to a sum of inputs will be the sum of the responses to the inputs applied separately. Consider an arbitrary time-dependent exciting load. It can be separated into an amplitude part P{r^2) ^"d a time function p{t) as pit, r^2) = -^(^^2 )P(0 The autocorrelation function of p(t) is defined as Cpp(r) =lim^
f_ll P(t)p(t + T)dt
(25)
By taking the Fourier transform of the autocorrelation function, one obtains the power spectral density function Spp(6;): Sppico) = r CppiT)e-''*dt J-oo
(26)
By using Fourier transform, a response function of p(t,r^2)^ ^^^ instance, the time average of stain energy
291
Shape optimization of transient response problems —
1 rTli
W = lim—\
aiu(t,n,)Mt,^J)dt
(27)
is converted to W = £a(A,(^,/2J,A\(^,/2J)SppM/^l
(28)
which is a sum of frequency response of each frequency co. Therefore, the shape gradient function G for this objective function can be expressed into G{uico,rj,vio^,rj
= £G(^„(a),i2J,fe„(«,/2J)Spp(«)d[^^j
(29)
where h^{(o,f2g)y h^^(co,Ils) are displacement vector and its adjoint variable respectively when Fourier transform of excitation p(t), p(cD) = 1. It is obvious that the shape gradient function is a sum of shape gradient function for each frequency derived in last section. TRACTION METHOD As shown in last two sections, for an objective functional L(M,yl) with a constraint function, applying the material derivative method and the Lagrange multiplier method, the derivative of L to 5- can be derived into a boundary integral function of a shape gradient function G and a velocity field V as L = l(G,V)=j G^Vdr^^^Gn^Vdr
(30)
The traction method was proposed as the method that solves the velocity field V by[4,5] a(y, w) = -l(G, w)yweU
(31)
where a ( - / ) is the bilinear form for the variational elastic strain energy, U is the admissible set of domain variations. Eq.(31) means that the velocity field is obtained as a displacement of a pseudo-elastic body defined in i7^ by loading of the pseudo-external force in proportion to -G under constraints on displacement of the invariable boundaries. That the solution V can decrease the objective functional in convex problems is assured by the positive definite property of the energy bilinear form a ( - / ) as shown in the following equation[4]. L = l(G, V) - -a(V, V);„ w) - 2ErM\
(j> - M]
= 0 dt = 0
\/w' e Wt
(11)
V0' G %
(12)
that indicate the variational form of the original state equation for temperature (j){x^ t) and the variational form for w{x,t), which we call an adjoint equation, respectively. Under the condition satisfying Eqs.(ll) and (12), the derivative of the Lagrange functional agrees with that of the objective functional and the linear form Iciy) with respect to V: L|^,„ = [j'^ Er^ dt) U,„ = IG{V)
(13)
The coefficient vector function Gn in Eq. (9) have the meaning of a sensitivity function relative to domain variation and are the so-called shape gradient function. The scalar function G is called the shape gradient density function. The shape gradient function can be derived theoretically, thus, domain variation can be analyzed by the traction method [12] [13].
299
Solution to shape determination problem
T E M P E R A T U R E GRADIENT SQUARE ERROR INTEGRALS MINIMIZATION PROBLEM In previous section, we formulated a prescribed problem for time-histories of temperature distributions (/)(r/;, [0,T]) on the prescribed subboundaries FD Q A in unsteady heat-conduction fields and derived the shape gradient function for the problem. In a similar manner, we can derive the shape gradient function for a prescribed problem where the time-histories of temperature gradient distributions V(J){QD^ P J ^ ] ) ^^^ specified with 9D{^D, [0?^]) in the prescribed subdomains QD C /2, as shown in Fig.l(b). This problem is formulated as Given i? and A;, p, c, / , q, (^, (j)o : fixed in space
(14)
find Hs or f,(i7) G D that minimize / EQ^dt= Jo = r Jo
subject to j
(15)
/ EQ^^iVcj) — go, V^ — gD)dt Jo
( V ^ - go) • {Vct> - go) dx dt
I
(16)
JQD
{a((/), w) + 6((/),t, w) - c{w) - d{w)] dt = 0
(t)^^t
V^ G W, (17)
For simplicity, we assumed that the subboundaries / Q and the prescribed subdomains QD are invariable, i.e., fs{rQ) = FQ and fs{QD) = ^DThe linear form Iciy) of the velocity function V and the shape gradient function for this problem are obtained in the manner of the previous section as lG{y)= G=
(18)
f Gfi'VdF
Jri
(19)
l—k(j)jWj — pc(j)^t'^ + fw-{-\/niq'w)-{-{qw)K\dt
where Vn{') = ^{-)-n and K, denotes the mean curvature. (/)(x, t) is the solution of the original state E q . ( l l ) and w{x,t) is determined by the adjoint equation: £{a{cl>\ w) + b{% w) - 2E^^{V(P\ V(/> - go)} dt = 0
V(^' G H^t
(20)
MUMERICAL SOLUTION TECHNIQUE Traction
Method
The traction method is a procedure for determining the amount of domain variation (velocity function V) that reduces the objective functional, based on the governing Eq. (21). This method uses the gradient method in a Hilbert space, a technique that is also employed in distributed parameter optimal control problems. a'^iV^y)
=-lG{y),
^yeD
(21)
300
E. Katamine, H. Azegami and Y. Matsuura ri=ro Initial design
Un-steady state heat conduction field analysis
I
(a) Coolant flow passage in wing D
Un-steady state adjoint heat conduction field analysis •^
End )
Calculation of shape gradient function Velocity analysis Updating shape with velocity
(b) Nozzle
Fig. 2 Numerical procedure where the bilinear form a^{V,y) defined in the domain i7o as
Fig. 3 Shape determination problem gives the strain energy of the pseudo linear elastic body
a^{u,v)
= J
AijkiUk,iVij dx,
(22)
where u^ v and Aijki are displacement, variational displacement and an elastic tensor, respectively. The governing Eq.(21) indicates that the velocity field t? is a displacement field when negative shape gradient functions —Gn act on the boundaries as an external force. In other words, with the traction method, domain variation is a displacement field when the shape gradient functions act as an external force in a pseudo elastic problem. Accordingly, Eq.(21) can be solved using a solution to ordinary linear-elastic problems, thus confirming the general applicabihty of the traction method. Numerical
Procedure
A flow chart of the shape optimization system is shown in Fig. 2. The main elements of the system include two unsteady heat conduction analyses, calculation of the shape gradient function, a velocity analysis based on the traction method and shape updating. The shape gradient function is evaluated using the two heat conduction analyses in which the distributions of temperature (j){x, t) and adjoint temperature w{x^ t) are analyzed. The time-histories of temperature distributions are evaluated using the finite element method for space integral and the Crank-Nicolson method for time integral. In the solution of the original state E q . ( l l ) , the temperature (j){x,t) is analyzed using the initial condition (/)(x, 0) = (t>o{x) in the time direction from t = Q to t = T. On the other hand, in the solution of the adjoint Eq.(12) or (20), the adjoint temperature w{x,t) is analyzed using the initial condition w{x, T) = WT{X) = 0 in the time direction from t = Ttot=^0. Then, the shape gradient function is calculated using the results. The domain variation V on the Eq. (21) is analyzed using the finite element method. Shape optimization analysis is performed by repeatedly executing these elements sequentially To perform an analysis, a doniain variation coefficient As is set which adjusts the magnitude of the domain variation AsV per iteration.
301
Solution to shape determination problem
t = 2.5 X 10-2 i
t = 5.0 X 10"^ sec
t = 5.0 X 10-2 sec
Fig. 4 Objective temperature distributions of coolant flow passage problem deg
20 30 40 50 60 70 80 90 100 110
EZII
t = 5.0 X 10-^ sec
^ -= 2.5 X 10-2 sec Initial Domain
t = 5.0 X 10-2 sec deg
I 20 ] 30 40 50 CZZ! 60 nz2 70 80 90 100 110
t - 5.0 X 10-^ sec
^ = 2.5 X 10-2 sec Converged Domain
t = 5.0 X 10-2 sec
Fig. 5 Results of temperature distributions in the coolant flow passage problem N U M E R I C A L RESULTS We present the results of two numerical analyses for 2D shape determination problems using the traction method and shape gradient function derived in previous sections. Coolant
flow
passage
in
wing
We analyzed a shape determination problem of coolant flow passage in wing for the temperature prescribed problem as shown in Fig.3(a). The outer surface boundary of wing was assumed as the prescribed subboundary / D = / \ . The shape shown in Fig.4 was made to be the objective shape, and the temperature distribution in the outer surface boundary was assumed to be the prescribed temperature distribution 4>D{^I t)- The design boundary is the left-side boundary of coolant flow passage. For simplicity, the case we considered had the following conditions : length of wing / = 0.25 m, specified temperature (^ = 20 deg, heat-conductivity coeflficient k = 0.204 k W / m deg, heat-flux q = 150 kW/m2, density p = 2710 kg/m^, capacity c = 0.896 kJ / kg deg, initial temperature distribution (/)o(x) = (j)Q — 20 deg, specified period of time T = 0.05 sec. Numerical results of this problem are shown in Figs. 5 and 6. Figure 5 shows a comparison of the shapes and temperature distributions between the initial domain and the converged domain. Figure 6 shows the meshs, the time histories of temperature at point A in Fig.3 (a), and the iterative history ratio of the objective functional normalized with the initial
302
E. Katamine, H. Azegami and Y. Matsuura
value. Based on these results, it was confirmed that the time histories of temperature in the converged domain exhibited agreement with the time histories in the objective domain, and the value of the objective functional approached zero. According to the numerical results of this basic problem, we confirmed the vahdity of the present method.
Objective domain
A Objective domain 0 Initial domain D Converged domain 0.02
. .0.031 . , . 1 , .
Time [sec]
Time histories of temperature at point A j
Initial domain
1
1 ° Objective Functional |
]
'°° 1
°°< 10
Converged domain Mesh
20
30
40
Number of Iteration
Iteration history of objective functional
Fig. 6 Results of the coolant flow passage problem Nozzle In the prescribed problem for temperature gradient distribution, we analyzed a thickness shape problem in nozzle as shown in Fig.3(b). Considering that the two halves are symmetrical, the upper half of domain A-B-C-D was analyzed. The subdomain in the neighborhood of inner wall in nozzle was assumed as the prescribed subdomain Qr,. The shape and temperature gradient distributions shown in Fig.7 were given as the objective shape, and the temperature gradient distributions, respectively. The design boundary is the outer side boundary A-D. The case we considered had the following conditions : length of wing / = 0.06 m, specified temperature ^ = 100 deg, heat-conductivity coefficient k = 0.204 k W / m deg, heat-flux q = 0 kW/m^, density p = 2710 kg/m^, capacity c = 0.896 kJ / kg deg, initial temperature distribution 0o(^) = 0o = 20 deg, specified period of time T = 0.001 sec. Numerical results of this problem are shown in Figs. 8 and 9 in the same way as the above results. We confirmed that the values of the objective functional approached zero, and the converged domains analyzed by the proposed method exhibited good agreement with the objective domain. CONCLUSIONS This paper derived the shape gradient functions with respect to the shape identification problems of unsteady heat-conduction fields to control temperature distributions and temperature gradient distributions to prescribed distributions. The validity of the traction method using the derived shape gradient functions was confirmed by the numerical results.
Solution to shape determination problem
w^fimwjir**-
[deg/m] , -6300 ! -5600 , -4900 -4200 -3500 EZZ3 -2800 , , -2100 ^ -1400 = r -700
y*->"1^'
t = 1.0x 10-^ sec
303
t = 5.0 X 10"^ sec
t = 1.0 x 10"^ sec
Fig. 7 Objective temperature gradient distributions of nozzle problem [deg/m] -6300 -5600 -4900 -4200 -3500 E=Z] -2800 cm -2100 -1400 -700
t - 1.0 X 10-^ sec
t = 1.0 X 10"^ sec
t - 5.0 X 10-4 sec Initial Domain
t = 1.0x 10-^ sec
t - 5.0 X 10"^ sec
t - 1.0 x 10"^ sec
[deg/m] -6300 -5600 -4900 o:^ -4200 -3500 -2800 -2100 -1400 H ^ -700
Converged Domain Fig. 8 Results of temperature gradient distributions in the nozzle problem ACKNOWLEDGEMENTS This study was financially supported by the Sasakawa Scientific Research Grant from The Japan Science Society. REFERENCES 1. Nakamura M., Tanaka M. and Ishikawa H. (1992). Inverse Analysis Using BEM to Estimate Unknown Boundary Values in Transient Heat Conduction Problems, Trans, of Jpn. Sac. of Mech. Engs., (in Japanese), 58-555, A, 2206-2221. 2. Bai Q. and Pujita Y. (1997). A Finite Element Analysis for Inverse Heat Conduction Problems, Trans, of Jpn. Soc. of Mech. Engs., (in Japanese), 63-608, B, 1320-1326. 3. Kubo S., Ohnaka K. and Ohji K. (1988). Identification of Heat-Source and Force Using Boundary Integrals, Trans, of Jpn. Soc. of Mech. Engs., (in Japanese), 54-503, A, 13291334. 4. Momose K. and Kimoto H. (1995). Green's Function Approach to Optimal Arrangement of Heat Sources, Trans, of Jpn. Soc. of Mech. Engs., (in Japanese), 61-585, B, 1762-1767. 5. Tanaka M., Nakamura M. and Shiozaki A. (1993). A Boundary-Element Inverse Analysis Procedure for Estimation of Thermal Properties in Transient Heat Conduction, J. Soc. Mat. Jpn., (in Japanese), 42-477, 708-713.
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E. Katajnine, H. Azegami and Y. Matsuura
6 a
Objective domain
i
2
^ sJ^
A Objective domain | • o Initial domain H D Converged domain |
~T .. 1.." Time [sec]
"""^
Time histories of temperature gradient in xi direction at point P Initial domain
" ' T " 'j 1 o Objective Functional ]
2 °-* 2 0.6
\
0
0.4 0
0.2
Converged domain Mesh
..
f'^joL^.^-K-ff
Number of Iteration
Iteration history of objective functional Fig. 9 Results of the nozzle problem
6. Barone M. R. and Caulk D. A. (1982). Optimal Arrangement of Hole in a Two-Dimensional Heat Conductor by a Special Boundary Integral Method, Int. J. Num. Meth. Eng.^ 18, 675685. 7. Kennon S. R. and Duhkravich G.S. (1986). Inverse Design of Multiholed Internally Cooled Turbine Blades, Int. J. Num. Meth. Eng., 22, 363-375. 8. Yoshikawa F., Nigo S., Kiyohara S., Taguchi S., Takahashi H. and Ichimiya M. (1987). Estimation Refractory Wear and Solidified Layer Distribution in the Blast Furnace Hearth and Its AppUcation to the Operation, Tetsu-to-Hagane, (in Japanese), 73-15, 2068-2075. 9. Shau R., Batista J. and Carey G. F. (1990). An Improved Algorithm for Inverse Design of Thermal Problems With Multiple Materials, Tran. ASME J. Heat Transfer, 112, 274-279. 10. Meric, R. A. (1995). Differential and Integral Sensitivity Formulations and Shape Optimization by BEM, Engineering Analysis with Boundary Elements, 15, 181-188. 11. Katamine E., Azegami H. and Kojima M. (1999). Boundary Shape Determination on SteadyState Heat Conduction Fields, Proc. of the First China-Japan-Korea Joint Symposium on Optimization of Structural and Mechanical Systems, edited by Y. X. Gu, B.Y. Duan, H.Azegami and B.M. Kwak, Xidian University Press, 33-40. 12. Azegami H. (1994). Solution to Domain Optimization Problems, Trans, of Jpn. Soc. of Mech. Engs., (in Japanese), 60-574, A, 1479-1486. 13. Azegami H., Kaizu S., Shimoda M. and Katamine E. (1997). Irregularity of Shape Optimization Problems and an Improvement Technique, Computer Aided Optimum Design of Structures V, edited by S. Hernandez and C. A. Brebbia, Computational Mechanics Publications, Southampton, 309-326.
Parameter Identification in Solid Mechanics
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INVERSE PROBLEMS IN ENGINEERING MECHANICS II M. Tanaka, G.S. Dulikravich (Eds.) © 2000 Elsevier Science Ltd. All rights reserved.
307
MULTI-OBJECTIVE PARAMETER IDENTIFICATION OF UNIFIED MATERIAL MODELS Tomonari Furukawa, Shinobu Yoshimura and Genki Yagawa Department of Quantum Engineering and Systems Science University of Tokyo 7-3-1 Kongo, Bunkyo-ku, Tokyo 113-8656 Japan Ph: +81-3-5802-5334, E-mail:
[email protected] ABSTRACT Although the regular! sation increased the popularity of parameter identification due to its capability of deriving a stable solution, the significant problem is that the solution depends upon the regularisation parameters chosen. This paper presents a technique for deriving solutions without the use of the parameters, and further an optimisation method, which can work efficiently for problems of concern. Numerical examples show that the technique can efficiently search for appropriate solutions. INTRODUCTION With the increase of safety design requirements, an inelastic material model, which can describe accurate material behaviours, is indispensable. Originally, such a model was developed to describe a particular material behaviour although various material behaviours such as cyclic, stress-relaxation, creep behaviours are observed in real life. A number of material models have been proposed in the last decades accordingly [1 and references therein]. In order to cover a broader range, recently developed models include terms describing plasticity and viscosity, which are the two major material behaviours, and, unified models, which unify both the plastic and viscous terms as viscoplastisity, have received considerable attention. In accordance, other terms are added to these models for accuracy, and the models resultantly have complex formulations with a number of material parameters. To use the models, this gives rise to developing a technique for identifying material parameters for these models. Considering the characteristics of the models, necessary for such a technique is that the technique can handle: (1) various material models (2) a number of material parameters, (3) various experiments such as cyclic, stress-relaxation and creep tests. The promising formulation for general nonlinear identification problems is the method of least squares where a solution is found by minimising an objective function, the residual between experimental and computed outputs, using an optimisation method. Mahnken and Stein [2] developed a technique for identifying a number of parameters for various models using a gradient-based optimisation method. This technique can however find a solution when the objective function is not complex. The first robust technique was then developed by Furukawa and Yagawa [3] where a Continuous Evolutionary Algorithm (CEA) [4] was used as an optimisation method. Nevertheless, the technique handles plural experimental data by introducing weighting
308
T. Furukawa, S. Yoshimura and G. Yagawa
factors, and the solution is largely dependent upon the weighting factors to be chosen. In this paper, a technique for identifying parameters from various experimental data is first proposed. In this technique, weighting factors are not introduced, and the identification problem is formulated with a multi-objective function. A robust multi-objective optimisation method termed Multi-objective Continuous Evolutionary Algorithms (MCEAs) [5], is then proposed for use to find the solutions for this class of problems efficiently. PARAMETER IDENTinCATION Suppose that we have a set of experimental data [u-*, v,*], where U.*G U and V.*G V , and the corresponding model v having parameters x e X , the experimental data can be related to the model by v(u-*,x) + e. = v- *, (1) where e, represents the sum of the model errors and measurement errors: e. =e,'"^+epp. (2) The parameter identification is typically defined to idendfy the continuous vector in engineering problems X c /?", given a set of continuous experimental data, U,V ^R". In order to solve it, a parameter identification problem is often converted to the minimisation of a continuous functional [6]: / ( x ) -^ min , (3) X
where the f : R" ^ R is most commonly the residuals between the computed and model outputs: (4) / ( x ) = £||(v(u,*,x)-v,*)|| . The parameter set minimising such an objective function is to be found within a search space: X„,in^X<X,3,,
(5)
where K i „ , x „ , J = X . UNIHED MATERIAL MODELS Input and output for identification Constitutive models describe the stress-strain relationship, and are constructed with material experiments, given the time-variant behaviour of one of the strain and stress as input and getting the time-variant behaviour of the other as output. Let the input and the output be expressed as g and h respectively, i.e., [g,h]e {[£,o],[o,£]}. If the time-variant input is given by g = git)^ (6) Constitutive models can then derive the output with respect to the input: h = h(g;x), C^) where x represents the material parameters. The output is also related to time via the input, and can be rewritten implicitly as:
Multi-objective parameter identification
309
h = h\t\x). (^) Cyclic behaviours, for example, are often derived by controlling the strain, and the constitutive models describing the behaviours are therefore constructed by defining Eq. (4) with [w,v] = [£,o]. Meanwhile, the stress relaxation behaviours and the creep behaviours represent the time-variant output with constant strain and stress respectively, i.e., [M,v] = [r,o] for stress relaxation behaviours and [w, v] = [r,£] for creep behaviours. Often, the constitutive models are constructed by separating the effect of elastic behaviour from the overall behaviours. The total strain is thus the sum of the elastic strain e' and the inelastic strain a'", and is given by: e^e'+e'". (9) The stress is hence derived from the strain: (7 = E(e-e'"), (10) and vice versa, where E is Young's modulus representing a linear coefficient, and the key of the constitutive modelling is how to define the inelastic strain £'". Unified models The most distinguished inelastic material behaviours are the viscous and plastic behaviours. Unified models describe the behaviours by a unified viscoplastic strain: £'" =£' -\-£P =£'P,
(11)
where £^\ £" and £ "^ represent the plastic, viscous and viscoplastic strains respectively. Plastic hardening materials perform plastic deformations only upon increasing the stress level and the yield condition for plastic deformations changes during the loading process. The performance of such materials thus depends on the previous states of stress and strain. In such path-dependent cases, the inelastic range of materials is in general expressed by means of the thermodynamic forces associated with the two internal variables, back stress representing kinematic hardening x ^"^ ^^^S stress representing isotropic hardening R : (12) f = J(o-z)-R-k{[ a,, h ^ -a 3. h „ - p^,- a^,- (h„- h,3)]"'+ ^ [ ttj. h ^ - a 3. h „ - p^3. a ^ . (h„-h,3)]°} [h„]"' = [hg"+(At/2)-{[ a,. h „ - p^. a^. (h„- hjf*\
[ a,- h „ - p,. a^- (h„- h J]°} (13).
362
M. Kanoh, T. Hosokawa and T. Kuroki
Where At is the time differential and [h^J
stands for the value of h^^ on (n+l)«At time.
CONSTRAINTS In order to stabilize the solutions against the oscillation in the back analysis of the tank model, we consider the following constraints as (14).
g(p) = 0 Differentiating the Equation (14) with respect to p and linearizing it, we have E^Ap + g(p)=0 where E
(15),
is the transposed coefficient matrix of E, and E should satisfy Equation (16). E=[dg/dpi ag/dP2 • • • ^g/^Pn]
(1^)-
DETERMINATION OF Ap Solving the observation Equation (7) with the constraints shown as Equation (14), we determined Ap. Using Lagrange's method of the indeterminate muhiplier, we tried to minimize ^ described as ^ = (HAp + D '
W(H/^
, + L) + 2 KT' /^T ( E ' Ap + g(p))
(17),
where W is the weight matrix and K is the indeterminate coefficient vector. Differentiating Equation (17) with respect to Ap and letting it be equal to zero, we have
d Jl
--A'
D2
-.*
•
S Ml
^ 1 2 3 4 5 6 7 8 9 1
Figure 6. Correlation dimension as a function of m- PUMA robot 1. We introduce the first feature vector using the whole signal s without taking into account that it is a deterministic nonlinear chaotic one. In order to chose features for it, i.e. to reduce the number of coordinates, the Karhunen-Loeve transform is applied to the signal s, which is presented as
s=i/jO^-
(8)
7=1
where and is formulated into the following analytical form, A{x, y, z) = - i - A j / ^ J [ A ^ < ( e , , , , +0) - A^{(, TJ, -0)]^{X,
y, z; (, 77,0)}.^^'^^;
where 'i!{x,y,z;^,r],C)
= cosh-'
"^ ~ ^
(5)
The domain of integration is denoted by r+ where (x - f )^ - {y — vY ^ 0- I^ is shown as a shadowed area on a wing surface in the right-hand-side sketch of Fig. 2. Then the x
402
K. Matsushima
partial derivatives in Eq.(4) is conducted and integral by parts with respect to ^ is done to avoid singularity. Evevtually, one obtains
z{x-i) 27r 27r J Jr^ Jr. [(y - 7?)2 + z^]r,
(AMi, rj, +0) - A
