Fundamental solution method for inverse source problem of plate equation

doi:10.1007/s10483-012-1641-6

摘要/Abstract

摘要： The elastic plate vibration model is studied under the external force. The size of the source term by the given mode of the source and some observations from the body of the plate is determined over a time interval, which is referred to be an inverse source problem of a plate equation. The uniqueness theorem for this problem is stated, and the fundamental solution to the plate equation is derived. In the case that the plate is driven by the harmonic load, the fundamental solution method (FSM) and the Tikhonov regularization technique are used to calculate the source term. Numerical experiments of the Euler-Bernoulli beam and the Kirchhoff-Love plate show that the FSM can work well for practical use, no matter the source term is smooth or piecewise.

关键词: homeomorphism, existence and uniqueness, global inverse theorem, Kirchhoff-Love plate, Euler-Bernoulli beam, elastic, inverse source problem, fundamental solution method(FSM),Tikhonov regularization method, meshless numerical method

Abstract: The elastic plate vibration model is studied under the external force. The size of the source term by the given mode of the source and some observations from the body of the plate is determined over a time interval, which is referred to be an inverse source problem of a plate equation. The uniqueness theorem for this problem is stated, and the fundamental solution to the plate equation is derived. In the case that the plate is driven by the harmonic load, the fundamental solution method (FSM) and the Tikhonov regularization technique are used to calculate the source term. Numerical experiments of the Euler-Bernoulli beam and the Kirchhoff-Love plate show that the FSM can work well for practical use, no matter the source term is smooth or piecewise.

Key words: homeomorphism, existence and uniqueness, global inverse theorem, Kirchhoff-Love plate, Euler-Bernoulli beam, elastic, inverse source problem, fundamental solution method(FSM),Tikhonov regularization method, meshless numerical method

中图分类号:

顾智杰;谭永基. Fundamental solution method for inverse source problem of plate equation[J]. Applied Mathematics and Mechanics (English Edition), 2012, 33(12): 1513-1532.

Zhi-jie GU;Yong-ji TAN. Fundamental solution method for inverse source problem of plate equation[J]. Applied Mathematics and Mechanics (English Edition), 2012, 33(12): 1513-1532.

参考文献

[1] Yang, Y. and Lim, C. W. A new nonlocal cylindrical shell model for axisymmetric wave propaga-tion in carbon nanotubes. Advanced Science Letters, 4(1), 121-131 (2011)
[2] Zhao, X., Ng, T. Y., and Liew, K. M. Free vibration of two-side simply-supported laminatedcylindrical panels via the mesh-free kp-Ritz method. International Journal of Mechanical Sciences,46(1), 123-142 (2004)
[3] Zhou, D., Lo, S. H., and Cheung, Y. K. 3-D vibration analysis of annular sector plates using theChebyshev-Ritz method. Journal of Sound and Vibration, 320(1-2), 421-437 (2009)
[4] Liu, Y., Hon, Y. C., and Liew, K. M. A meshfree Hermite-type radial point interpolation methodfor Kirchhoff plate problems. International Journal for Numerical Methods in Engineering, 66(7),1153-1178 (2006)
[5] Kurpa, L., Pilgun, G., and Amabili, M. Nonlinear vibrations of shallow shells with complexboundary: R-functions method and experiments. Journal of Sound and Vibration, 306(3-5), 580-600 (2007)
[6] Qian, L. F., Batra, R. C., and Chen, L. M. Static and dynamic deformations of thick functionallygraded elastic plates by using higher-order shear and normal deformable plate theory and meshlesslocal Petrov-Galerkin method. Composites Part B: Engineering, 35(6-8), 685-697 (2004)
[7] Sladek, J., Sladek, V., Wen, P. H., and Aliabadi, M. H. Meshless local Petrov-Galerkin (MLPG)method for shear deformable shells analysis. Chinese Journal of Mechanical Engineering, 13(2),103-117 (2006)
[8] Krys’ko, V. A., Papkova, I. V., and Soldatov, V. V. Analysis of nonlinear chaotic vibrations ofshallow shells of revolution by using the wavelet transform. Mechanics of Solids, 45(1), 85-93(2010)
[9] Li, S. Q. and Hong, Y. Quasi-Green’s function method for free vibration of simply-supportedtrapezoidal shallow spherical shell. Applied Mathematics and Mechanics (English Edition), 31(5),635-642 (2010) DOI 10.1007/s10483-010-0511-7
[10] Michaels, J. E. and Pao, Y. H. The inverse source problem for an oblique force on an elastic plate.Journal of the Acoustical Society of America, 77(6), 2005-2011 (1985)
[11] Li, S. M., Miara, B., and Yamamoto, M. A Carleman estimate for the linear shallow shell equationand an inverse source problem. Discrete and Continuous Dynamical Systems, 23(1-2), 367-380(2009)
[12] Alves, C., Silvestre, A. L., and Takahashi, T. Solving inverse source problems using observabilityapplications to the Euler-Bernoulli plate equation. SIAM Journal on Control and Optimization,48(3), 1632-1659 (2009)
[13] Wang, Y. H. Global uniqueness and stability for an inverse plate problem. Journal of OptimizationTheory and Applications, 132(1), 161-173 (2007)
[14] Yang, C. Y. The determination of two heat sources in an inverse heat conduction problem. InternationalJournal of Heat and Mass Transfer, 42(2), 345-356 (1999)
[15] Yang, C. Y. Solving the two-dimensional inverse heat source problem through the linear least-squares error method. International Journal of Heat and Mass Transfer, 41(2), 393-398 (1998)
[16] Fatullayev, A. G. Numerical solution of the inverse problem of determining an unknown sourceterm in a heat equation. Mathematical and Computers in Simulation, 58(3), 247-253 (2002)
[17] Le, N. C. and Lefevre, F. A parameter estimation approach to solve the inverse problem of pointheat sources identification. International Journal of Heat and Mass Transfer, 47(4), 827-841(2004)
[18] Johansson, B. T. and Lesnic, D. A variational method for identifying a spacewise-dependent heatsource. IMA Journal of Applied Mathematics, 72(6), 748-760 (2007)
[19] Yan, L., Fu, C. L., and Yang, F. L. The method of fundamental solutions for the inverse heatsource problem. Engineering Analysis with Boundary Elements, 32(3), 216-222 (2008)
[20] Jin, B. T. and Marin, L. The method of fundamental solutions for inverse source problems as-sociated with the steady-state heat conduction. International Journal for Numerical Method inEngineering, 69(8), 1570-1589 (2007)
[21] Yan, L., Yang, F. L., and Fu, C. L. A meshless method for solving an inverse spacewise-dependentheat source problem. Journal of Computational Physics, 228(1), 123-136 (2009)
[22] Johansson, B. T., Lesnic, D., and Reeve, T. A method of fundamental solutions for the one-dimensional inverse Stefan problem. Applied Mathematical Modelling, 35(9), 4367-4378 (2011)
[23] Chen, C. W., Young, D. L., and Tsai, C. C. The method of fundamental solutions for inverse 2DStokes problems. Computational Mechanics, 37(1), 2-14 (2005)
[24] Marin, L. and Lesnic, D. The method of fundamental solutions for inverse boundary value problemsassociated with the two-dimensional biharmonic equation. Mathematical and Computer Modelling,42(3-4), 261-278 (2005)
[25] Alves, C. J. S., Colaco, M. J., Leitao, V. M. A., Martins, N. F. M., Orlande, H. R. B., and Roberty,N. C. Recovering the source term in a linear diffusion problem by the method of fundamentalsolutions. Inverse Problems in Science and Engineering, 16(8), 1005-1021 (2008)
[26] Love, A. E. H. On the small free vibrations and deformations of elastic shells. PhilosophicalTransactions of the Royal Society, Series A, 17, 491-549 (1888)
[27] Alves, C. J. S. On the choice of source points in the method of fundamental solutions. EngineeringAnalysis with Boundary Elements, 33(12), 1348-1361 (2009)
[28] Kim, S. M. and McCullough, B. F. Dynamic response of plate on viscous Winkler foundation tomoving loads of varying amplitude. Engineering Structure, 25(9), 1179-1188 (2003)
[29] Liew, K. M., Han, J. B., and Xiao, Z. M. Differential quadrature method for Mindlin plates onWinkler foundations. International Journal of Mechanical Sciences, 38(4), 405-421 (1996)