The Direct Method of Lines for Forward and Inverse Linear Elasticity Problems of Composite Materials in Star-shaped Domains  

作　　者：Xiaopeng Zhu Zhizhang Wu Zhongyi Huang 

机构地区：[1]Department of Mathematical Sciences,Tsinghua University,Beijing 100084,China [2]Department of Mathematics,The University of Hong Kong,Pokfulam Road,Hong Kong SAR,China

出　　处：《Numerical Mathematics(Theory,Methods and Applications)》2023年第1期242-276,共35页高等学校计算数学学报（英文版）

基　　金：This work was partially supported by the NSFC Projects No.12025104,11871298,81930119.

摘　　要：In this paper,we generalize the direct method of lines for linear elasticity problems of composite materials in star-shaped domains and consider its application to inverse elasticity problems.We assume that the boundary of the star-shaped domain can be described by an explicit C 1 parametric curve in the polar coordinate.We introduce the curvilinear coordinate,in which the irregular star-shaped domain is converted to a regular semi-infinite strip.The equations of linear elasticity are discretized with respect to the angular variable and we solve the resulting semidiscrete approximation analytically using a direct method.The eigenvalues of the semi-discrete approximation converge quickly to the true eigenvalues of the elliptic operator,which helps capture the singularities naturally.Moreover,an optimal error estimate of our method is given.For the inverse elasticity problems,we determine the Lam´e coefficients from measurement data by minimizing a regularized energy functional.We apply the direct method of lines as the forward solver in order to cope with the irregularity of the domain and possible singularities in the forward solutions.Several numerical examples are presented to show the effectiveness and accuracy of our method for both forward and inverse elasticity problems of composite materials.

关 键 词：Composite materials linear elasticity problems inverse elasticity problems starshaped domains method of lines 

分 类 号：O17[理学—数学]