Abstract
The paper addresses a numerical method for boundary identification in a problem governed by Laplace’s equation. The proposed numerical procedure for discrete reconstruction of the unknown boundary from the given temperature data is based on the Trefftz method. In contrast to the procedures described in the reference papers, the present approach requires significantly less and easier computation. The paper undertakes analysis of the resistance of the solution to small perturbations of the prescribed temperature condition at the unknown part of the boundary. We define and then estimate a sensitivity factor which allows quantitative assessment of the relationship between temperature measurement errors and boundary identification errors, even if the exact solution is not known. The included numerical examples demonstrate the effectiveness of the proposed method for boundary reconstruction and present the analysis of numerical stability using a sensitivity factor.
Highlights
There are different kinds of inverse geometry problems like shape and design optimization or identification of defects in materials
We address a problem which consists in identification of the unknown part of a domain boundary
Given a problem described by a differential equation, a numerical reconstruction of the unknown boundary has to be achieved from the available boundary condition
Summary
There are different kinds of inverse geometry problems like shape and design optimization or identification of defects in materials. We address a problem which consists in identification of the unknown part of a domain boundary. Given a problem described by a differential equation, a numerical reconstruction of the unknown boundary has to be achieved from the available boundary condition. There have been different propositions of the efficient solution algorithms to this problem. Seems very suitable for approaching the problem, namely the Trefftz method and the method of fundamental solutions (MFS) which are mathematically equivalent though differ in formulation [1]. A new approach to stability analysis proposed in the paper allows to predict boundary identification errors, calculated on the basis of the known measurement errors
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