@Article{NMTMA-16-242,
author = {Zhu , XiaopengWu , Zhizhang and Huang , Zhongyi},
title = {The Direct Method of Lines for Forward and Inverse Linear Elasticity Problems of Composite Materials in Star-Shaped Domains},
journal = {Numerical Mathematics: Theory, Methods and Applications},
year = {2023},
volume = {16},
number = {1},
pages = {242--276},
abstract = {<p style="text-align: justify;">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 semi-discrete 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 Lamé 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.</p>},
issn = {2079-7338},
doi = {https://doi.org/10.4208/nmtma.OA-2021-0184},
url = {http://global-sci.org/intro/article_detail/nmtma/21351.html}
}