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講者

Developments and applications of the localized method of fundamental solutions

Chia-Ming Fan

Abstract

In this presentation, the recent developments and applications of the localized method of fundamental solutions (LMFS) will be presented. The LMFS, truly free from mesh and numerical quadrature, is one of the most-promising meshless methods. The MFS is evolved from the classical method of fundamental solutions (MFS), so it remains the simplicity and high accuracy of the MFS. Both of interior nodes and boundary nodes are required in the numerical implements of the LMFS. For each node, a linear algebraic equation, which can represent the satisfaction of governing equation or boundary condition, can be derived by implementing the MFS within the subdomain, which includes the central nodes and its neighboring nodes. To enforce the satisfaction of governing equation in each interior node and the satisfaction of boundary condition at each boundary node can yield a sparse system of linear algebraic equations. When the resultant sparse system is efficiently solved, the highly-accurate numerical solutions of the LMFS can be obtained. The proposed LMFS can be applied to large-scale problems in complex domains due to the use of localization. The LMFS has been proposed in 2019 and then was applied to various physical problems, such as the Laplace equation, Helmholtz equation, biharmonic equation, the problem of elasticity, the anisotropic heart conduction problem, inverse problem, inhomogeneous problems, etc. The LMFS will be briefly described in this presentation; meanwhile, the applications of the LMFS will also be introduced.

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