The direct method of lines for elliptic problems in star-shaped domains

Wu, Zhizhang; Huang, Zhongyi; Wang, Weicheng; Yang, Yi

doi:10.1016/j.cam.2017.06.028

Cited by 3 publications

(1 citation statement)

References 31 publications

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“…The direct method of lines was introduced to deal with elliptic problems with singularities on a polygonal domain [26,27] and boundary value problems on an unbounded domain [24,25]. Recently, the direct method of lines was generalized in [50] for solving the Laplace's equation with singularities in star-shaped domains.…”

Section: Introductionmentioning

confidence: 99%

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

Zhu¹,

null²,

Hu³

2023

NMTMA

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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 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.

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Section: Introductionmentioning

confidence: 99%

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

Zhu¹,

null²,

Hu³

2023

NMTMA

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Starshaped sets

et al. 2020

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This is an expository paper about the fundamental mathematical notion of starshapedness, emphasizing the geometric, analytical, combinatorial, and topological properties of starshaped sets and their broad applicability in many mathematical fields. The authors decided to approach the topic in a very broad way since they are not aware of any related survey-like publications dealing with this natural notion. The concept of starshapedness is very close to that of convexity, and it is needed in fields like classical convexity, convex analysis, functional analysis, discrete, combinatorial and computational geometry, differential geometry, approximation theory, PDE, and optimization; it is strongly related to notions like radial functions, section functions, visibility, (support) cones, kernels, duality, and many others. We present in a detailed way many definitions of and theorems on the basic properties of starshaped sets, followed by survey-like discussions of related results. At the end of the article, we additionally survey a broad spectrum of applications in some of the above mentioned disciplines.

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The direct method of lines for forward and inverse problems of nearly-incompressible composite materials in star-shaped domains

Xiaopeng,

Zhongyi

2024

Sci. Sin.-Math.

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The direct method of lines for elliptic problems in star-shaped domains

Cited by 3 publications

References 31 publications

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

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

Starshaped sets

The direct method of lines for forward and inverse problems of nearly-incompressible composite materials in star-shaped domains

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