Reconstruction of the corrosion boundary for the Laplace equation by using a boundary collocation method

doi:10.1016/j.matcom.2008.11.019

Mathematics and Computers in Simulation

Volume 79, Issue 7, March 2009, Pages 2148-2156

https://doi.org/10.1016/j.matcom.2008.11.019 Get rights and content

Abstract

In this paper, we consider the identification of a corrosion boundary for the two-dimensional Laplace equation. A boundary collocation method is proposed for determining the unknown portion of the boundary from the Cauchy data on a part of the boundary. Since the resulting matrix equation is badly ill-conditioned, a regularized solution is obtained by employing the Tikhonov regularization technique, while the regularization parameter is provided by the generalized cross-validation criterion. Numerical examples show that the proposed method is reasonable and feasible.

Introduction

The non-destructive evaluation of a corrosion boundary continues to attract interesting in engineering and mathematics. Unfortunately, in many practical situations, the known boundary data are not complete, and corrosion takes place on an inaccessible part of the boundary. Our task is to determine the shape of the corrosive boundary from the measured data. This is an inverse boundary problem, and it is extremely ill-posed, namely a small perturbation in the measured data may cause a dramatically large error in the solution.

This inverse problem has recently received much attention and both stability and identifiability issues have been investigated. Bukhgeim et al. [3] gave a very weak conditional stability estimate of a logarithmic rate under a regularity assumption on the unknown boundary, which was then extended to $R^{3}$ [7]. Conditional stability of various forms have also been extensively studied [2], [21].

Several numerical methods have been proposed for the inverse boundary determination problem. Mclver [16] and Michael et al. [18] devised a non-destructive evaluation technique by applying an alternating current to the accessible boundary and determining the corroded boundary by observing the output current fields. Charton et al. [4] and Mera et al. [17] proposed a variational technique based on parameterizing the unknown boundary using a function specification method. Kaup and Santosa [15] employed the Tikhonov regularization method in connection with the L-curve method to solve this problem. Recently, Hon and Wu [13] developed a meshfree method based on radial basis interpolation to determine the unknown boundary. They tested the proposed method and showed that provided a good approximation of the solution with exact data. However, the result was unstable while using noisy data due to the ill-conditioning of the interpolation matrix.

In the present paper, we use the boundary collocation method combined with the discrete Tikhonov regularization technique for solving the inverse boundary problem using noisy data. This meshfree method emerges as a competitive alternative to the mesh-dependent method, including the radial basis functions (RBF) method [19], [24], the method of fundamental solutions (MFS) [9] and the boundary knot method (BKM) [5], [6]. These meshfree methods require neither domain discretization, as in the finite element method (FEM) and finite difference method (FDM), nor boundary discretization, as in the boundary element method (BEM), thus they improve the computational efficiency and they can be easily extended to solve high order and high dimensional differential equations. In this paper, the boundary collocation method is employed to discretize the equation, and the discrete Tikhonov regularization method is used to solve the resulting matrix equation to obtain a numerically stable scheme. Results for several numerical examples are presented to demonstrate the effect of the method.

The paper is organized as follows. In Section 2, we formulate the problem mathematically. The boundary collocation method is described in Section 3. The regularization techniques for solving the matrix equation and the rule for choosing a regularization parameter are described in Section 4. In Section 5, we present and discuss the results for several numerical examples. Section 6 provides a summary and concluding remarks.

Section snippets

Mathematical formulation of the problem

In this paper the boundary collocation method is used to determine the material loss occurring on the inaccessible boundary $γ \subset \partial Ω$ , with the domain $Ω \subset R^{2}$ , by measuring Cauchy data on an accessible portion of the boundary. Therefore, we consider the following Laplace equation in a two-dimensional plate $Ω$ [13], [1], namely, $Δ u (x, y) = 0, in Ω,$ subject to the boundary conditions $u (x, y) = f (x, y), \frac{\partial u (x, y)}{\partial ν} = g (x, y), on Γ,$ $u (x, y) = 0, on γ,$ where $Δ$ is the Laplacian operator, $Γ$ is the accessible part of the boundary $\partial Ω$ with

Boundary collocation method

Assuming that $β > 0$ is constant, the following harmonic function $ϕ (x, y) = e^{- β x^{2} + β y^{2}} \cos (2 β x y)$ is a solution of Eq. (2.1) in the domain $Ω$ . Inspired by [13], we assume that an approximation to the solution of problems (1), (2), (3) can be expressed by the following linear combination of harmonic functions $ũ (x, y) = Σ_{j = 1}^{n} λ_{j} G_{j} (x, y) + Σ_{j = n + 1}^{n + m} μ_{j} \frac{\partial G_{j} (x, y)}{\partial ν}, (x, y) \in Ω,$ where $G_{j} (x, y) = ϕ (x - x_{j}, y - y_{j})$ is a shifted harmonic function, ${(x_{j}, y_{j})}_{j = 1}^{n} \subset Γ$ and ${(x_{j}, y_{j})}_{j = n + 1}^{n + m} \subset Γ$ are two sets of pairwise distinct points, and ${λ_{j}, j$

Regularization method

The main difficulty with radial basis function collocation techniques is that the interpolation matrix A may be highly ill-conditioned [14]. For the solution of forward problems with exact data, this does not pose any great challenge. Standard methods may be able to yield accurate results. However, for inverse problems, the situation is delicate. Inverse problems are usually ill-posed, and the ill-posedness carries over into the discrete formulations. Most numerical methods for treating

Numerical verification

In order to test the convergence and the stability of the method proposed, without any loss of generality, we consider the solution domain shown in Fig. 1. Different domains, smooth or non-smooth, can be investigated by the same algorithm but they are not considered in this paper. In order to present the performance of the boundary collocation method in conjunction with the TR method, we first defined the relative root mean square error as $rel = \frac{\sqrt{Σ_{i = 1}^{N} {(s_{i} - {\tilde{s}}_{i}^{*})}^{2}}}{\sqrt{Σ_{i = 1}^{N} {(s_{i})}^{2}}},$ where N is the total

Conclusions

In this paper, we study an efficient, accurate, convergent and stable scheme for solving the identification of a corrosion boundary. The present numerical procedure was based on the boundary collocation method, in conjunction with the TR method. Numerical results for both exact and noisy data have been presented. The numerical results indicate that the proposed scheme provides accurate and stable approximations of the unknown boundary.

Acknowledgments

The authors gratefully thank the referees for their valuable constructive comments which improve greatly the quality of the paper. The project is supported by the NNSF of China (Nos. 10671085 and 10571079) and the program of NCET.

References (24)

W. Chen et al.
A meshless, integration-free, and boundary-only RBF technique
Comput. Math. Appl.
(2002)
M.A. Golberg et al.
Improved multiquadric approximation for partial differential equations
Eng. Anal. Bound. Elem.
(1996)
Y.C. Hon et al.
A numerical computation for inverse boundary determination problem
Eng. Anal. Bound. Elem.
(2000)
E.J. Kansa et al.
Circumventing the ill-conditioning problem with multiquadric radial basis functions: applications to elliptic partial differential equations
Comput. Math. Appl.
(2000)
N.S. Mera et al.
Numerical solution of a boundary detection problem using genetic algorithms
Eng. Anal. Bound. Elem.
(2004)
G. Alessandrini
Stable determination of a crack from boundary measurements
Proc. R. Soc. Edinb. Sect. A
(1993)
E. Beretta et al.
Stable determination of boundaries from Cauchy data
SIAM J. Math. Anal.
(1998)
A.L. Bukhgeim et al.
Uniqueness and stability for an inverse problem of determining a part of boundary
Inverse Problems in Engineering Mechanics (Nagano, Japan)
(1998)
M. Charton et al.
Numerical solution of a shape optimization problem
W. Chen et al.
New insights into the boundary-only and domain-type RBF methods
Int. J. Nonlinear Sci. Numer. Simul.
(2000)

J. Cheng et al.

Conditional stability estimation for an inverse boundary problem with non-smooth boundary in $R^{3}$

Trans. Am. Math. Soc.

(2001)

H.W. Engle et al.

Regularization of Inverse Problem

(1996)

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Reconstruction of the corrosion boundary for the Laplace equation by using a boundary collocation method

Abstract

Introduction

Section snippets

Mathematical formulation of the problem

Boundary collocation method

Regularization method

Numerical verification

Conclusions

Acknowledgments

Comput. Math. Appl.

Eng. Anal. Bound. Elem.

Eng. Anal. Bound. Elem.

Comput. Math. Appl.

Eng. Anal. Bound. Elem.

Stable determination of a crack from boundary measurements

Proc. R. Soc. Edinb. Sect. A

Stable determination of boundaries from Cauchy data

SIAM J. Math. Anal.

Uniqueness and stability for an inverse problem of determining a part of boundary

Inverse Problems in Engineering Mechanics (Nagano, Japan)

Numerical solution of a shape optimization problem

New insights into the boundary-only and domain-type RBF methods

Int. J. Nonlinear Sci. Numer. Simul.

Conditional stability estimation for an inverse boundary problem with non-smooth boundary in R3

Trans. Am. Math. Soc.

Regularization of Inverse Problem

Conditional stability estimation for an inverse boundary problem with non-smooth boundary in $R^{3}$