Volume 10, Issue 6
A Regularized Singular Boundary Method for Inverse Cauchy Problem in Three-Dimensional Elastostatics

Adv. Appl. Math. Mech., 10 (2018), pp. 1459-1477.

Published online: 2018-09

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• 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.

• Keywords

Meshless method, singular boundary method, method of fundamental solutions, elastostatics, inverse problem.

62P30, 65M32, 65K05

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@Article{AAMM-10-1459, author = {Aixia and Zhang and and 19060 and and Aixia Zhang and Yan and Gu and and 19061 and and Yan Gu and Qingsong and Hua and and 19062 and and Qingsong Hua and Wen and Chen and and 19063 and and Wen Chen}, title = {A Regularized Singular Boundary Method for Inverse Cauchy Problem in Three-Dimensional Elastostatics}, journal = {Advances in Applied Mathematics and Mechanics}, year = {2018}, volume = {10}, number = {6}, pages = {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.

}, issn = {2075-1354}, doi = {https://doi.org/10.4208/aamm.OA-2018-0103}, url = {http://global-sci.org/intro/article_detail/aamm/12722.html} }
TY - JOUR T1 - A Regularized Singular Boundary Method for Inverse Cauchy Problem in Three-Dimensional Elastostatics AU - Zhang , Aixia AU - Gu , Yan AU - Hua , Qingsong AU - Chen , Wen JO - Advances in Applied Mathematics and Mechanics VL - 6 SP - 1459 EP - 1477 PY - 2018 DA - 2018/09 SN - 10 DO - http://doi.org/10.4208/aamm.OA-2018-0103 UR - https://global-sci.org/intro/article_detail/aamm/12722.html KW - Meshless method, singular boundary method, method of fundamental solutions, elastostatics, inverse problem. AB -

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.

Aixia Zhang, Yan Gu, Qingsong Hua & Wen Chen. (1970). A Regularized Singular Boundary Method for Inverse Cauchy Problem in Three-Dimensional Elastostatics. Advances in Applied Mathematics and Mechanics. 10 (6). 1459-1477. doi:10.4208/aamm.OA-2018-0103
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