改进的基本解法在薄体各向异性位势Cauchy问题中的应用

该文提出一种改进的基本解法,应用于薄体各向异性位势边界条件识别反问题的研究。基本解法求解反问题所产生的线性系统往往是高度病态的,我们采用截断奇异值分解方法来求解,广义交叉校验准则用来确定正则化参数。正则化方法的使用大大地拓展了源点与真实边界间距离的选取范围,同时有效地降低了解的精度对"距离选择"的敏感度。算例的数值实验表明,该文方法简单、效率高,即使薄体结构的厚度小到纳米级,仍然可获得非常高精度的数值解。该文为二维薄体各向异性位势反问题的研究开辟了新的途径,也拓展了基本解法的应用领域。     

Regularized solutions with a singular point for the inverse biharmonic boundary value problem by the method of fundamental solutions

Two numerical methods for the Cauchy problem of the biharmonic equation are proposed. The solution of the problem does not continuously depend on given Cauchy data since the problem is ill-posed. A small noise contained in the Cauchy data sensitively affects on the accuracy of the solution. Our problem is directly discretized by the method of fundamental solutions (MFS) to derive an ill-conditioned matrix equation. As another method, our problem is decomposed into two Cauchy problems of the Laplace and the Poisson equations, which are discretized by the MFS and the method of particular solutions (MPS), respectively. The Tikhonov regularization and the truncated singular value decomposition are applied to the matrix equation to stabilize a numerical solution of the problem for the given Cauchy data with high noises. The L-curve and the generalized cross-validation determine a suitable regularization parameter for obtaining an accurate solution. Based on numerical experiments, it is concluded that the numerical method proposed in this paper is effective for the problem that has an irregular domain and the Cauchy data with high noises. Furthermore, our latter method can successfully solve the problem whose solution has a singular point outside the computational domain.     

Comparison of regularization methods for solving the Cauchy problem associated with the Helmholtz equation

In this paper, several boundary element regularization methods, such as iterative, conjugate gradient, Tikhonov regularization and singular value decomposition methods, for solving the Cauchy problem associated to the Helmholtz equation are developed and compared. Regularizing stopping criteria are developed and the convergence, as well as the stability, of the numerical methods proposed are analysed. The Cauchy problem for the Helmholtz equation can be regularized by various methods, such as the general regularization methods presented in this paper, but more accurate results are obtained by classical methods, such as the singular value decomposition and the Tikhonov regularization methods. Copyright © 2004 John Wiley & Sons, Ltd.     

Adaptive error estimation technique of the Trefftz method for solving the over-specified boundary value problem

In this paper, numerical solutions are investigated based on the Trefftz method for an over-specified boundary value problem contaminated with artificial noise. The main difficulty of the inverse problem is that divergent results occur when the boundary condition on over-specified boundary is contaminated by artificial random errors. The mechanism of the unreasonable result stems from its ill-posed influence matrix. The accompanied ill-posed problem is remedied by using the Tikhonov regularization technique and the linear regularization method, respectively. This remedy will regularize the influence matrix. The optimal parameter λ of the Tikhonov technique and the linear regularization method can be determined by adopting the adaptive error estimation technique. From this study, convergent numerical solutions of the Trefftz method adopting the optimal parameter can be obtained. To show the accuracy of the numerical solutions, we take the examples as numerical examination. The numerical examination verifies the validity of the adaptive error estimation technique. The comparison of the Tikhonov regularization technique and the linear regularization method was also discussed in the examples.     

Numerical solution of two backward parabolic problems using method of fundamental solutions

This work is devoted to a numerical algorithm based on the method of fundamental solutions (MFS) for solving two backward parabolic problems with different boundary conditions, one with nonlocal Dirichlet boundary conditions, and second one with Robin type boundary conditions. The initial temperature distribution will be identified from the final temperature distribution, which appear in some applied subjects. The Tikhonov regularization method with the L-curve criterion for choosing the regularization parameter is adopted for solving the resulting matrix equation which is highly ill-conditioned. Two numerical examples are provided to show the high efficiency of the suggested method.     

Regularized MFS-Based Boundary Identification in Two-Dimensional Helmholtz-Type Equations

We study the stable numerical identification of an unknown portion of the boundary on which a given boundary condition is provided and additional Cauchy data are given on the remaining known portion of the boundary of a two-dimensional domain for problems governed by either the Helmholtz or the modified Helmholtz equation. This inverse geometric problem is solved using the method of fundamental solutions (MFS) in conjunction with the Tikhonov regularization method. The optimal value for the regularization parameter is chosen according to Hansen's L-curve criterion. The stability, convergence, accuracy and efficiency of the proposed method are investigated by considering several examples.     

A meshless method for the stable solution of singular inverse problems for two-dimensional Helmholtz-type equations

We investigate a meshless method for the stable and accurate solution of inverse problems associated with two-dimensional Helmholtz-type equations in the presence of boundary singularities. The governing equation and boundary conditions are discretized by the method of fundamental solutions (MFS). The existence of boundary singularities affects adversely the accuracy and convergence of standard numerical methods. Solutions to such problems and/or their corresponding derivatives may have unbounded values in the vicinity of the singularity. Moreover, when dealing with inverse problems, the stability of solutions is a key issue and this is usually taken into account by employing a regularization method. These difficulties are overcome by combining the Tikhonov regularization method (TRM) with the subtraction from the original MFS solution of the corresponding singular solutions, without an appreciable increase in the computational effort and at the same time keeping the same MFS discretization. Three examples for both the Helmholtz and the modified Helmholtz equations are carefully investigated.     

Boundary knot method for the Cauchy problem associated with the inhomogeneous Helmholtz equation

In this paper, the boundary knot method is extended to the solution of inhomogeneous equations, and it is applied to the Cauchy problem associated with the inhomogeneous Helmholtz equation. Here, we assume that the boundary condition is specified only on a part of the boundary, and the boundary conditions on the remaining part of the boundary are to be determined with the assistance of additional data. Since the resulting matrix equation is highly ill-conditioned, a regularized solution is obtained by employing the truncated singular value decomposition to solve the matrix equation arising from the boundary knot method, with the regularization parameter determined by the L-curve method. Numerical results are presented for several examples with smooth and piecewise smooth boundaries. The numerical verification shows that the proposed numerical scheme is accurate, stable with respect to data noise, and convergent with respect to decreasing the amount of noise in the data.     

Boundary knot method for some inverse problems associated with the Helmholtz equation

The boundary knot method is an inherently meshless, integration‐free, boundary‐type, radial basis function collocation technique for the solution of partial differential equations. In this paper, the method is applied to the solution of some inverse problems for the Helmholtz equation, including the highly ill‐posed Cauchy problem. Since the resulting matrix equation is badly ill‐conditioned, a regularized solution is obtained by employing truncated singular value decomposition, while the regularization parameter for the regularization method is provided by the L‐curve method. Numerical results are presented for both smooth and piecewise smooth geometry. The stability of the method with respect to the noise in the data is investigated by using simulated noisy data. The results show that the method is highly accurate, computationally efficient and stable, and can be a competitive alternative to existing methods for the numerical solution of the problems. Copyright © 2005 John Wiley & Sons, Ltd.     

数值求解Symm积分方程的正则化MINRES方法

Symm积分方程在位势理论中具有重要应用,它是Hadamard意义下的不适定问题。离散该方程将产生对称线性不适定系统。基于GCV准则,并应用截断奇异值分解,本文提出数值求解Symm积分方程的正则化MINRES方法。与Tikhonov正则化方法相比,在数据出现噪声的情况下,新方法能有效地求得Symm积分方程的数值解。     

A spectral alternating method for elastostatic problems with multiple spherical cavities

The problem of an infinite elastic solid containing an arbitrary number of non-overlapping spherical cavities of arbitrary sizes and locations and with arbitrary boundary tractions is considered. The numerical procedure to solve this boundary value problem is based on a spectral method in which the boundary displacements and tractions are represented by truncated series of surface spherical harmonics. By using an alternating method, the problem containing multiple spherical cavities is replaced by a sequence of problems for a single spherical cavity with the boundary conditions adjusted iteratively to account for the cavity interactions. A least squares approximation is used to determine the unknown coefficients of surface spherical harmonics representing these cavity interactions. Several examples are given to illustrate the approach. An erratum to this article can be found at     

The method of fundamental solutions for inverse boundary value problems associated with the steady‐state heat conduction in anisotropic media

In this paper, the method of fundamental solutions is applied to solve some inverse boundary value problems associated with the steady‐state heat conduction in an anisotropic medium. Since the resulting matrix equation is severely ill‐conditioned, a regularized solution is obtained by employing the truncated singular value decomposition, while the optimal regularization parameter is chosen according to the L‐curve criterion. Numerical results are presented for both two‐ and three‐dimensional problems, as well as exact and noisy data. The convergence and stability of the proposed numerical scheme with respect to increasing the number of source points and the distance between the fictitious and physical boundaries, and decreasing the amount of noise added into the input data, respectively, are analysed. A sensitivity analysis with respect to the measure of the accessible part of the boundary and the distance between the internal measurement points and the boundary is also performed. The numerical results obtained show that the proposed numerical method is accurate, convergent, stable and computationally efficient, and hence it could be considered as a competitive alternative to existing methods for solving inverse problems in anisotropic steady‐state heat conduction. Copyright © 2005 John Wiley & Sons, Ltd.     

The method of fundamental solutions for inverse source problems associated with the steady‐state heat conduction

This paper presents the use of the method of fundamental solutions (MFS) for recovering the heat source in steady‐state heat conduction problems from boundary temperature and heat flux measurements. It is well known that boundary data alone do not determine uniquely a general heat source and hence some a priori knowledge is assumed in order to guarantee the uniqueness of the solution. In the present study, the heat source is assumed to satisfy a second‐order partial differential equation on a physical basis, thereby transforming the problem into a fourth‐order partial differential equation, which can be conveniently solved using the MFS. Since the matrix arising from the MFS discretization is severely ill‐conditioned, a regularized solution is obtained by employing the truncated singular value decomposition, whilst the optimal regularization parameter is determined by the L‐curve criterion. Numerical results are presented for several two‐dimensional problems with both exact and noisy data. The sensitivity analysis with respect to two solution parameters, i.e. the number of source points and the distance between the fictitious and physical boundaries, and one problem parameter, i.e. the measure of the accessible part of the boundary, is also performed. The stability of the scheme with respect to the amount of noise added into the data is analysed. The numerical results obtained show that the proposed numerical algorithm is accurate, convergent, stable and computationally efficient for solving inverse source problems in steady‐state heat conduction. Copyright © 2006 John Wiley & Sons, Ltd.     

Boundary element methods for boundary condition inverse problems in elasticity using PCGM and CGM regularization

For an isotropic linear elastic body, only displacement or traction boundary conditions are given on a part of its boundary, whilst all of displacement and traction vectors are unknown on the rest of the boundary. The inverse problem is different from the Cauchy problems. All the unknown boundary conditions on the whole boundary must be determined with some interior points' information. The preconditioned conjugate gradient method (PCGM) in combination with the boundary element method (BEM) is developed for reconstructing the boundary conditions, and the PCGM is compared with the conjugate gradient method (CGM). Morozov's discrepancy principle is employed to select the iteration step. The analytical integral algorithm is proposed to treat the nearly singular integrals when the interior points are very close to the boundary. The numerical solutions of the boundary conditions are not sensitive to the locations of the interior points if these points are distributed along the entire boundary of the considered domain. The numerical results confirm that the PCGM and CGM produce convergent and stable numerical solutions with respect to increasing the number of interior points and decreasing the amount of noise added into the input data.     

Stable Boundary and Internal Data Reconstruction in Two-Dimensional Anisotropic Heat Conduction Cauchy Problems Using Relaxation Procedures for an Iterative MFS Algorithm

We investigate two algorithms involving the relaxation of either the given boundary temperatures (Dirichlet data) or the prescribed normal heat fluxes (Neumann data) on the over-specified boundary in the case of the iterative algorithm of Kozlov91 applied to Cauchy problems for two-dimensional steady-state anisotropic heat conduction (the Laplace-Beltrami equation). The two mixed, well-posed and direct problems corresponding to every iteration of the numerical procedure are solved using the method of fundamental solutions (MFS), in conjunction with the Tikhonov regularization method. For each direct problem considered, the optimal value of the regularization parameter is chosen according to the generalized cross-validation (GCV) criterion. The iterative MFS algorithms with relaxation are tested for over-, equally and under-determined Cauchy problems associated with the steady-state anisotropic heat conduction in various two-dimensional geometries to confirm the numerical convergence, stability, accuracy and computational efficiency of the method.     

Domain-Decomposition Singular Boundary Method for Stress Analysis in Multi-Layered Elastic Materials

This paper applies an improved singular boundary method (SBM) in conjunction with domain decomposition technique to stress analysis of layered elastic materials. For problems under consideration, the interface continuity conditions are approximated in the same manner as the boundary conditions. The multi-layered coating system is decomposed into multiple subdomains in terms of each layer, in which the solution is approximated separately by the SBM representation. The singular boundary method is a recent meshless boundary collocation method, in which the origin intensity factor plays a key role for its accuracy and efficiency. This study also introduces new strong-form regularization formulas to accurately evaluate the origin intensity factors for elasticity problem. Consequently, we dramatically improve the accuracy and convergence of SBM solution of the elastostatics problems. The proposed domain-decomposition SBM is tested on two benchmark problems. Based on numerical results, we discuss merits of the present SBM scheme over the other boundary discretization methods, such as the method of fundamental solution (MFS) and the boundary element method (BEM).     

Application of symmetric indirect Trefftz method to free vibration problems in 2D

A symmetric indirect Trefftz method is developed to solve the free vibration problem of a 2D membrane. It is proved that in this approach the spurious eigensolution exists, and an auxiliary matrix is constructed to help extraction of the spurious solution using the generalized singular‐value decomposition. In addition to the spurious eigensolution, this regular formulation suffers from its ill‐posed nature, i.e. the numerical instability. In order to deal with the numerical instability, the Tikhonov's regularization method, in conjunction with the generalized singular‐value decomposition, is suggested. The proposed approach has some merits when compared with other regular boundary element formulations reported so far; namely the capacity of representing eigenmodes and the ability to deal with a multiply connected domain of genus 1. Several numerical examples are demonstrated to show the validity of the current approach. Copyright © 2003 John Wiley & Sons, Ltd.     

A method of fundamental solutions for inverse heat conduction problems in an anisotropic medium

Recently, Hon and Wei proposed a method of fundamental solutions for solving isotropic inverse heat conduction problems (IHCP). It provides an efficient global approximation scheme in both spatial and time domains. In this paper, we try to extend the inherently meshless and integration-free method to solve 2D IHCP in an anisotropic medium. First, we acquire the fundamental solution of the governing equation through variables transformation. Then the truncated singular value decomposition and the L-curve criterion are applied to solve the resulting matrix equation which is highly ill-conditioned. Results for several numerical examples are presented to demonstrate the efficiency of the method proposed. The relationship between the accuracy of the numerical solutions and the value of the parameter T is also investigated.     

Simultaneous reconstruction of the time-dependent Robin coefficient and heat flux in heat conduction problems

This paper aims to solve an inverse heat conduction problem in two-dimensional space under transient regime, which consists of the estimation of multiple time-dependent heat sources placed at the boundaries. Robin boundary condition (third type boundary condition) is considered at the working domain boundary. The simultaneous identification problem is formulated as a constrained minimization problem using the output least squares method with Tikhonov regularization. The properties of the continuous and discrete optimization problem are studied. Differentiability results and the adjoint problems are established. The numerical estimation is investigated using a modified conjugate gradient method. Furthermore, to verify the performance of the proposed algorithm, obtained results are compared with results obtained from the well-known finite-element software COMSOL Multiphysics under the same conditions. The numerical results show that the proposed algorithm is accurate, robust and capable of simultaneously representing the time effects on reconstructing the time-dependent Robin coefficient and heat flux.