A Regularized Singular Boundary Method for Inverse Cauchy Problem in Three-Dimensional Elastostatics

A Regularized Singular Boundary Method for Inverse Cauchy Problem in Three-Dimensional Elastostatics

Year:    2018

Author:    Aixia Zhang, Yan Gu, Qingsong Hua, Wen Chen

Advances in Applied Mathematics and Mechanics, Vol. 10 (2018), Iss. 6 : pp. 1459–1477

Abstract

The application of the singular boundary method (SBM), a relatively new meshless boundary collocation method, to the inverse Cauchy problem in three-dimensional (3D) linear elasticity is investigated. The SBM involves a coupling between the non-singular boundary element method (BEM) and the method of fundamental solutions (MFS). The main idea is to fully inherit the dimensionality advantages of the BEM and the meshless and integration-free attributes of the MFS. Due to the boundary-only discretizations and its semi-analytical nature, the method can be viewed as an ideal candidate for the solution of inverse problems. The resulting ill-conditioned algebraic equations is regularized here by employing the first-order Tikhonov regularization technique, while the optimal regularization parameter is determined by the $L$-curve criterion. Numerical results with both smooth and piecewise smooth geometries show that accurate and stable solution can be obtained with a comparatively large level of noise added into the input data.

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Journal Article Details

Publisher Name:    Global Science Press

Language:    English

DOI:    https://doi.org/10.4208/aamm.OA-2018-0103

Advances in Applied Mathematics and Mechanics, Vol. 10 (2018), Iss. 6 : pp. 1459–1477

Published online:    2018-01

AMS Subject Headings:    Global Science Press

Copyright:    COPYRIGHT: © Global Science Press

Pages:    19

Keywords:    Meshless method singular boundary method method of fundamental solutions elastostatics inverse problem.

Author Details

Aixia Zhang

Yan Gu

Qingsong Hua

Wen Chen

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  7. A systematic derived sinh based method for singular and nearly singular boundary integrals

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  12. Triple reciprocity method for unknown function's domain integral in boundary integral equation

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  13. Localized Chebyshev collocation method for solving elliptic partial differential equations in arbitrary 2D domains

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  15. Localized method of fundamental solutions for large-scale modelling of three-dimensional anisotropic heat conduction problems – Theory and MATLAB code

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  16. Simulation of two-dimensional steady-state heat conduction problems by a fast singular boundary method

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  23. The method of transition boundary for the solution of diffraction problems

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  24. A meshless collocation scheme for inverse heat conduction problem in three-dimensional functionally graded materials

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  26. An overview of the method of fundamental solutions—Solvability, uniqueness, convergence, and stability

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  27. The BEM based on conformal Duffy-distance transformation for three-dimensional elasticity problems

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