Reconstruction of the corrosion boundary for the Laplace equation by using a boundary collocation method
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 [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 , with the domain , by measuring Cauchy data on an accessible portion of the boundary. Therefore, we consider the following Laplace equation in a two-dimensional plate [13], [1], namely,subject to the boundary conditionswhere is the Laplacian operator, is the accessible part of the boundary with
Boundary collocation method
Assuming that is constant, the following harmonic functionis a solution of Eq. (2.1) in the domain . 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 functionswhere is a shifted harmonic function, and are two sets of pairwise distinct points, and
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 aswhere 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)
- et al.
A meshless, integration-free, and boundary-only RBF technique
Comput. Math. Appl.
(2002) - et al.
Improved multiquadric approximation for partial differential equations
Eng. Anal. Bound. Elem.
(1996) - et al.
A numerical computation for inverse boundary determination problem
Eng. Anal. Bound. Elem.
(2000) - et al.
Circumventing the ill-conditioning problem with multiquadric radial basis functions: applications to elliptic partial differential equations
Comput. Math. Appl.
(2000) - et al.
Numerical solution of a boundary detection problem using genetic algorithms
Eng. Anal. Bound. Elem.
(2004) Stable determination of a crack from boundary measurements
Proc. R. Soc. Edinb. Sect. A
(1993)- et al.
Stable determination of boundaries from Cauchy data
SIAM J. Math. Anal.
(1998) - et al.
Uniqueness and stability for an inverse problem of determining a part of boundary
Inverse Problems in Engineering Mechanics (Nagano, Japan)
(1998) - et al.
Numerical solution of a shape optimization problem
- et al.
New insights into the boundary-only and domain-type RBF methods
Int. J. Nonlinear Sci. Numer. Simul.
(2000)
Conditional stability estimation for an inverse boundary problem with non-smooth boundary in
Trans. Am. Math. Soc.
Regularization of Inverse Problem
Cited by (12)
Inverse solutions of temperature, heat flux and heat source by the Green element method
2015, Applied Mathematical ModellingCitation Excerpt :The inverse heat conduction problem (IHCP) arises in many practical applications in the fields of science and engineering where the transport of heat, mass and energy takes place in natural and man-made materials. There are various classes of IHCPs which range from recovery of boundary temperature and heat flux [1–6], estimation of medium parameters [7–10], recovery of the spatial and temporal distributions of heat sources/sinks [11–19], recovery of initial data distributions [11,20–22], and recovery of the shape and location of boundary and medium features [23–26]. In most instances sensor measurements of temperature and heat fluxes are available at some accessible parts of the domain to support the solution of the IHCPs.
The MFS for numerical boundary identification in two-dimensional harmonic problems
2011, Engineering Analysis with Boundary ElementsCitation Excerpt :In Hon and Li [19], the MFS was applied to one- and two-dimensional inverse boundary determination heat conduction problems. Problem (1) was solved as a Cauchy level-set problem using the MFS by Yang et al. [62,63]. The detection of cavities with the method, in various problems arising in electrical impedance tomography was investigated in Borman et al. [9] and Karageorghis and Lesnic [29].
A new investigation into regularization techniques for the method of fundamental solutions
2011, Mathematics and Computers in SimulationCitation Excerpt :In this paper, we consider three commonly used regularization methods based on the SVD to study the stability and accuracy of the MFS. Using the SVD, this study considers three regularization methods, that is, the TR, TSVD and DSVD [31,32]. These filter factors decay slower than the Tikhonov one and thus makes less filtering.
Genetic Algorithm-Based Optimization Approach for Solving a Class of Inverse Problems with Tikhonov Regularization
2023, WSEAS Transactions on MathematicsDirect and inverse problem for geometric perturbation of the Laplace operator in a strip
2019, Boundary Value ProblemsA semi-analytical boundary collocation solver for the inverse Cauchy problems in heat conduction under 3D FGMs with heat source
2019, Numerical Heat Transfer, Part B: Fundamentals