Preface Inverse Problems can be found in many topics of engineering mechanics. There are many successful applications in the fields of the inverse problems (non-destructive 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 mechanical system from some of the output from the system. Mathematically, such problems are ill-posed and have to be overcome through development of new computational schemes, regularization techniques, objective functionals, and experimental procedures. Following the first IUTAM Symposium on these topics held in May 1992 in Tokyo, and another in November 1994 in Paris, we thought it would be very fruitful to gather researchers and engineers again for an exchange of the newest research ideas. Consequently, at the International Symposium on Inverse Problems in Mechanics (ISIP '98) held in March 1998 in Nagano, recent developments in the inverse problems in engineering mechanics and related topics were discussed. The following general areas were the subjects of this symposium: mathematical and computational aspects of the 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. Seventy-two papers from Asia, Europe, and North America were presented at ISIP '98 in Nagano, Japan. The final version of the manuscripts of sixty-five papers from these presentations are contained in this volume of the ISIP '98 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 through the publication of this book. As the chairmen of the ISIP '98 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 Pennsylvania State University (USA) and Ecole Polytechnique (France) is heartily appreciated. Also, co-sponsorship by the Japan Society for Computational Methods in Engineering (JASCOME) and helpful support by the staff of Shinshu University for the management of financial support from Monbusho are gratefully acknowledged. June 1998 Masataka TANAKA, Shinshu University / Japan George .S DULIKRAVICH, The Pennsylvania State University / U.S.A. vii Symposium Chairmen Prof. Masataka TANAKA Department of Mechanical Systems Engineering Faculty of Engineering Shinshu University 005 Wakasato, Nagano 380-8553, Japan :leT +81-26-226-4101, Fax: +81-26-224-6515 E-Mail: dtanaka@gipwc.shinshu-u.ac.jp Prof. George .S Dulikravich Department of Aerospace Engineering The Pennsylvania State University University Park, Pennsylvania 16802, USA :leT + 1-814-863-0134, Fax: + 2907-568-418-1 E-Mail: fl7@psu.edu International Scientific Committee Prof. Masa. Tanaka, Shinshu University (Japan), Chair Prof. .G .S Dulikravich, The Pennsylvania State University (USA), Co-Chair Prof. .O .M Alifanov, Moscow Aviation Institute (Russia) Prof. .S Andrieux, Electricit~ de France (France) Prof. .S Aoki, Tokyo Institute of Technology (Japan) Prof. .J Beck, Michigan State Universitry (USA) Prof. .M Bonnet, Ecole Polytechnique (France) Prof. H.D. Bui, Ecole Polytechnique (France) Prof. .H Engl, Johannes-Kepler-University (Austria) Prof. G.M.L. Gladwell, University of Waterloo (Canada) Prof. .D Ingham, University of Leeds (UK) Prof. .A .J Kassab, University of Central Florida (USA) Prof. .M Kitahara, Tohoku University (Japan) Prof. .S Kobayashi, Kyoto University (Japan) Prof. .S Kubo, Osaka University (Japan) Prof. .P Ladev~ze, LMT, ENS Cachan (France) Prof. G.-L. Liu, Shanghai University (China) Prof. A. Louis, University of Saarland (Germany) Prof. .G Maier, Politecnico id Milano (Italy) Prof. .V Modi, Columbia University (USA) .rD .C W.J. Oomens, Eindhoven University of Technology (Netherlands) Prof. .H Sol, Vrije Universiteit Brussel (Belgium) .rD .H Sobieczky, Deutsche Lufl- und Raumfahrt (Germany) Prof. .N Tosaka, Nihon University (Japan) Prof. .R van den Braembussche, yon Karman Institute of Fluid Dynamics (Belgium) Prof. K.A. Woodbury, University of Alabama (USA) Prof. .Z Yao, Tsinghua University (China) Prof. .N Zabaras, Cornell University (USA) vii Symposium Chairmen Prof. Masataka TANAKA Department of Mechanical Systems Engineering Faculty of Engineering Shinshu University 005 Wakasato, Nagano 380-8553, Japan :leT +81-26-226-4101, Fax: +81-26-224-6515 E-Mail: dtanaka@gipwc.shinshu-u.ac.jp Prof. George .S Dulikravich Department of Aerospace Engineering The Pennsylvania State University University Park, Pennsylvania 16802, USA :leT + 1-814-863-0134, Fax: + 2907-568-418-1 E-Mail: fl7@psu.edu International Scientific Committee Prof. Masa. Tanaka, Shinshu University (Japan), Chair Prof. .G .S Dulikravich, The Pennsylvania State University (USA), Co-Chair Prof. .O .M Alifanov, Moscow Aviation Institute (Russia) Prof. .S Andrieux, Electricit~ de France (France) Prof. .S Aoki, Tokyo Institute of Technology (Japan) Prof. .J Beck, Michigan State Universitry (USA) Prof. .M Bonnet, Ecole Polytechnique (France) Prof. H.D. Bui, Ecole Polytechnique (France) Prof. .H Engl, Johannes-Kepler-University (Austria) Prof. G.M.L. Gladwell, University of Waterloo (Canada) Prof. .D Ingham, University of Leeds (UK) Prof. .A .J Kassab, University of Central Florida (USA) Prof. .M Kitahara, Tohoku University (Japan) Prof. .S Kobayashi, Kyoto University (Japan) Prof. .S Kubo, Osaka University (Japan) Prof. .P Ladev~ze, LMT, ENS Cachan (France) Prof. G.-L. Liu, Shanghai University (China) Prof. A. Louis, University of Saarland (Germany) Prof. .G Maier, Politecnico id Milano (Italy) Prof. .V Modi, Columbia University (USA) .rD .C W.J. Oomens, Eindhoven University of Technology (Netherlands) Prof. .H Sol, Vrije Universiteit Brussel (Belgium) .rD .H Sobieczky, Deutsche Lufl- und Raumfahrt (Germany) Prof. .N Tosaka, Nihon University (Japan) Prof. .R van den Braembussche, yon Karman Institute of Fluid Dynamics (Belgium) Prof. K.A. Woodbury, University of Alabama (USA) Prof. .Z Yao, Tsinghua University (China) Prof. .N Zabaras, Cornell University (USA) viii Organizing Committee Prof. Masa. Tanaka, Shinshu University (Japan), Chair Prof. .G .S Dulikravich, The Pennsylvania State University (USA), Co-Chair Prof. .T Matsumoto, Shinshu University (Japan), Secretary Prof. .T Adachi, Tokyo Institute of Technology (Japan) Prof. .T Aizawa, YtisrevinU of Tokyo (Japan) Prof. .S Aoki, Tokyo Institute of Technology (Japan) Prof. H.D. Bui, Ecole Polytechnique (France) Prof. .T Fukui, Fukui University (Japan) Prof. .K Hayami, University of Tokyo (Japan) Prof. .S Hirose, Okayama University (Japan) Prof. .T Honma, Hokkaido University (Japan) Prof. E Imado, Shinshu University (Japan) Prof. .Y Iso, Kyoto University (Japan) Prof. .K Kagawa, Okayama University (Japan) Prof. .N Kamiya, Nagoya University (Japan) Prof. .K Kishimoto, Tokyo Institute of Technology (Japan) Prof. .E Kita, Nagoya University (Japan) Prof. .M Kitahara, Tohoku University (Japan) Prof. .S Kobayashi, Kyoto 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) INVERSE PROBLEMS IN ENGINEERING SCINAHCEM .M Tanaka, G.S. Dulikravich (Eds.) (cid:14)9 1998 Elsevier Science .V.B llA rights reserved. SPECTRAL AND WAVELET METHODS FOR SOLVING AN INVERSE HEAT CONDUCTION PROBLEM LARS ELDt~N and FREDR/K BERNTSSON Department of Mathematics, Linkiiping University 185-S 38 ,gnip'bkniL 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 both sides of a wall, but where one side is inaccessible to measurements. Mathematically it can be formulated as a Cauchy problem for the heat equation, with data given along the line x - 1, where one wants the solution to the left of that line. The problem is ill-posed, in the sense that the solution (if it exists) does not depend continuously on the data. The problem can be stabilized by replacing the time derivative in the heat equation by a wavelet-based approximation or a spectral-based approximation. The resulting problem is an initial value problem for an ordinary differential equation, which can be solved by standard numerical methods, e.g. a Runge-Kutta method. As test problems we take model equations as well as problems from industrial applications. KEYWORDS Inverse heat conduction problem, ill-posed, temperature measurement, wavelet, Fourier analysis, sideways heat equation NOITCUDORTNI In many industrial applications one wishes to determine the temperature on the surface of a body, where the surface itself is inaccessible for measurements 1. It may also be the case that locating a measurement device (e.g. a thermocouple) on the surface would disturb the measurements so that an incorrect temperature is recorded. In such cases one is restricted to internal measurements, and from these one wants to compute the surface temperature. The measuring situation is illustrated in Figure .1 4 L. Eld~n, E Berntsson Thick wall L\'/, Hot gas or l "qr/ / / / / 1/ liquid I 0 1 Fig. "1 Interior temperature measurements. In a one-dimensional setting, this situation can be modeled as the following ill-posed problem for the heat equation: Determine the temperature u(z, t) for 0 _< z < 1 from temperature and heat flux measurements, g,,,(') := u(1, .) and hm(.) := u~(1, .), when u(z, t) satisfies I (a(:c)u~)~ = ut, O < z < ,1 t _> ,O u(z, )O = ,O 0 _< z _< ,1 (1) ~(1, t) = g~(t), t _> 0, u~(1, t) = hm (t), t __> 0, The coefficient a(x) that represents the heat conducting properties of the medium is assumed to be positive, a(x) _> ao > .0 We refer to (1) as the sideways heat equation .1 It is an ill-posed Cauchy problem for a parabolic equation, see e. g. ,1 2, 3. In principle, the problem (1) might be reformulated as f2 k(t - r)f(r)dr = gin(t), which is an integral equation of the first kind, and standard techniques for ill-posed problems, such as Tikhonov regularization, could be used. However, the kernel function k(t) is explicitly known only in the case a(z) - const. 4. Therefore, for (1) we propose to keep the problem in the differential equation form and solve it essentially as an initial value problem in the space variable ("space-marching", ,1 5, 6, 7, 8, 9, 10, 111). Space-marching methods can be implemented efficiently, if the time-derivative is approximated by a bounded operator 7, 8. In this way we obtain a well-posed initial-boundary value problem for an ordinary differential equation (ODE). The initial value problem can be solved numerically by a standard ODE routine. In this paper we mainly treat the sideways heat equation formulated as in (1), and we discuss the stabilization and numerical solution of this equation. Theoretical results for the quarter plane 1 Also the Inverse Heat Conduction Problem (IHCP) .1 lartcepS dna televaw sdohtem for gnivlos na esrevni taeh noitcudnoc melborp 5 problem with constant coefficients are given in 12, 7, 8, 13, 14. We discuss two different methods for approximating the time derivative and their numerical implementation. Numerical examples are given. APPROXIMATING THE TIME DERIVATIVE We rewrite the sideways heat 0( equation (1) as a first order initial value problem in the space variable 1/a(x)) (u) 0 v ' 0__0, (2) v = O/Ot x with initial values (u(1,t)) (gin(t)) v(1,t) = hm(t) ' and the extra condition u(x, 0) = 0, for 0 < x < .1 The ill-posedness of this problem can be seen by taking the Fourier transform with respect to time. For simplicity, in this discussion we temporarily assume that the coefficient is constant, ooL a(x) = a. Let 1 e-iOu(x t)dt denote the Fourier transform of u(x, t), where we have assumed that u(x, t) is identically zero for t < 0. Then, in frequency space the initial value problem becomes 0 0 "6 ' 0 -5- =: ~j' Spectral and wavelet methods for solving an inverse heat conduction problem 7 we can consider the projection Pj onto Vj, defined by Pjf(t) ,)t(kjpd)kjpd = Z(f, kEZ as a low pass filter. In our context this means that if we expand the solution of the sideways heat equation in terms of the basis of Vj, ZE,~ then no frequencies larger than j~ will be present in it. Thus, the choice of the wavelet space corresponds to the choice of the cut-off level in the spectral approximation above. In 31 we described a wavelet-Galerkin approach for the sideways heat equation. In this method the time differentiation operator is approximated by a bounded operator, which is the differentiation operator Dj in .jV We proved almost optimal error estimates in the case of a quarter plane. Also a recipe for choosing jV was given, depending on the error level of the data. SOLVING THE ODE PROBLEM In the numerical solution of the initial value problem (5) it is important to have information about the behaviour of the analytic solution of the ordinary differential equation. Since the system is linear, the behaviour is, to a large extent, determined by the eigenvalues of the matrix 0 1/a(x)I) B(x)= D 0 ' where D is the matrix corresponding to time derivative. In Fig. 2 we illustrate the eigenvalues of -hB a(z) for = i and D = De, the spectral approximation of the derivative (6), and h is the step length in space. The eigenvalues of-hB in the left half plane correspond to solution components that decrease when z goes from 1 to 0, whereas those in the right half plane correspond to increasing solution components. In Fig. 2 we have also plotted the stability region of an explicit fourth order Runge-Kutta method. This is a standard method for solving numerically initial value problems for ODE's 19. The significance of the stability region is that the numerical method gives a non-increasing solution for solution components corresponding to eigenvalues inside the region, and an increasing solution for components corresponding to eigenvalues outside the stability region. Thus, if h is taken small enough, then all the eigenvalues in the left half plane will be inside the stability region, and the numerical solution has the correct properties with respect to increasing and decreasing solution components. In 8 it is shown that for realistic problems a standard Runge-Kutta method, with automatic step length control, is adequate. 8 .L ,n~dlE E Berntsson 2 1 x x x xxxxxxxx~ 0 x x x .xxxX ~ -1 -2 % . . . . . . o'., Fig. 2: Eigenvalues in the complex plane of the matrix -hB, for D = De, cut off level c~ = 90 and step length h = 0.2. The closed curve is the stability region of a fourth order Runge-Kutta method. NUMERICAL EXAMPLES Here we present a few numerical experiments conducted using our methods. These tests are not to be considered as a comprehensive evaluation, but rather they are intended to demonstrate the usefulness of the approach. In the first numerical experiment we solved the problem (1) using the coefficient function 1 a(x) = ~(1 + e-Z), x > .0 The test was performed in the following way: We selected a solution u(0, t) = f(t), 0 < t < 1 and computed data functions u(1, t) = g(t) and u~(1, t) = h(t) by solving a well-posed quarter plane problem for the heat equation using a finite difference scheme. Then we added a normally distributed perturbation of variance 10 -3 to each data function, giving vectors ~,G and H,~. From the perturbed data G,~, Hm we reconstructed the surface temperature, and compared the result with the known solution. The results are displayed in Fig. 3. In a certain industrial application one is interested in finding the temperature on the surface of a particle board 14. In this particular example it is difficult to measure the surface temperature directly, and instead a thermocouple placed inside the board records the temperature history, u(L, t), at a distance of 2.9 mm from the surface, and at 8192 points in time. In this application it is natural to assume that the temperature is symmetric with respect to the center of the board. Therefore we can compute u~(L, t), by solving a well-posed problem for the heat equation. Thus no heat-flux measurements are required. The computational results are displayed in Fig. 4. Here we present only the results from the wavelet method, but similar results are obtained using the spectral method. ACKNOWLEDGEMENT The work of Fredrik Berntsson is supported by a grant from the National Graduate School for Scientific Computing.

Inverse Problems in Engineering Mechanics. International Symposium on Inverse Problems in Engineering Mechanics 1998 (ISIP '98) Nagano, Japan

Similar Inverse Problems in Engineering Mechanics. International Symposium on Inverse Problems in Engineering Mechanics 1998 (ISIP '98) Nagano, Japan