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# Application of the MLPG Mixed Collocation Method for Solving Inverse Problems of Linear Isotropic/Anisotropic Elasticity with Simply/Multiply-Connected Domains

Tao Zhang1,2, Leiting Dong2,3, Abdullah Alotaibi4, Satya N. Atluri2,5

School of Aeronautic Science and Engineering, Beihang University, Beijing, China.
Center for Aerospace Research & Education, University of California, Irvine, USA.
Corresponding Author. Department of Engineering Mechanics, Hohai University, Nanjing, China.
Email: dong.leiting@gmail.com
Department of Mathematics, King Abdulaziz University, Jeddah, Saudi Arabia.
Distinguished Adjunct-Professor of Multi-Disciplinary Engineering and Computer Science, King Abdulaziz University, Jeddah, Saudi Arabia.

Computer Modeling in Engineering & Sciences 2013, 94(1), 1-28. https://doi.org/10.3970/cmes.2013.094.001

## Abstract

In this paper, a novel Meshless Local Petrov-Galerkin (MLPG) Mixed Collocation Method is developed for solving the inverse Cauchy problem of linear elasticity, wherein both the tractions as well as displacements are prescribed/measured at a small portion of the boundary of an elastic body. The elastic body may be isotropic/anisotropic and simply connected or multiply-connected. In the MLPG mixed collocation method, the same meshless basis function is used to interpolate both the displacement as well as the stress fields. The nodal stresses are expressed in terms of nodal displacements by enforcing the constitutive relation between stress and the displacement gradient tensor at each nodal point. The equations of linear momentum balance are satisfied at each node using collocation method. The displacement as well as traction boundary conditions are also enforced at each measurement location along the boundary where the conditions are over specified on displacement as well as tractions. The current method is very simple because the inverse problem is directly solved in a fashion similar to a direct problem, without resorting to any iterative optimization. The current method is also very general because it can be applied to arbitrary simply/multiply connected bodies composed of arbitrary isotropic/anisotropic material, and it can also be adapted to solve inverse problems of other physics such as heat transfer, electro-magnetics, etc. Several numerical examples demonstrate the effectiveness and robustness of the current method, even when the prescribed displacement/tractions are corrupted with measurement noises. The extension of the current method to solve nonlinear inverse problems will be straightforward within the framework of incremental loading, which will be explored in future studies.

## Keywords

Zhang, T., Dong, L., Alotaibi, A., Atluri, S. N. (2013). Application of the MLPG Mixed Collocation Method for Solving Inverse Problems of Linear Isotropic/Anisotropic Elasticity with Simply/Multiply-Connected Domains. CMES-Computer Modeling in Engineering & Sciences, 94(1), 1–28.

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