Liviu Marin

A survey of applications of the MFS to inverse problems

The method of fundamental solutions (MFS) is a relatively new method for the numerical solution of boundary value problems and initial/boundary value problems governed by certain partial differential equations. The ease with which it can be implemented and its effectiveness have made it a very popular tool for the solution of a large variety of problems arising in science and engineering. In recent years, it has been used extensively for a particular class of such problems, namely inverse problems. In this study, in view of the growing interest in this area, we review the applications of the MFS to inverse and related problems, over the last decade.

An alternating iterative algorithm for the Cauchy problem associated to the Helmholtz equation

In this paper, the iterative algorithm proposed by Kozlov et al. [Comput. Maths. Math. Phys. 31 (1991) 45] for obtaining approximate solutions to the ill-posed Cauchy problem for the Helmholtz equation is analysed. The technique is then numerically implemented using the boundary element method (BEM). The numerical results confirm that the iterative BEM produces a convergent and stable numerical solution with respect to increasing the number of boundary elements and decreasing the amount of noise added into the input data. An efficient stopping regularising criterion is also proposed.

A meshless method for the numerical solution of the Cauchy problem associated with three-dimensional Helmholtz-type equations

In this paper, the application of the method of fundamental solutions to the Cauchy problem associated with three-dimensional Helmholtz-type equations is investigated. The resulting system of linear algebraic equations is ill-conditioned and therefore its solution is regularized by employing the zeroth-order Tikhonov functional, while the choice of the regularization parameter is based on the L-curve method. Numerical results are presented for under-, equally- and over-determined Cauchy problems in a piecewise smooth geometry. The convergence, accuracy and stability of the method with respect to increasing the number of source points and the distance between the source points and the boundary of the solution domain, and decreasing the amount of noise added into the input data, respectively, are analysed.

The method of fundamental solutions for inverse boundary value problems associated with the two-dimensional biharmonic equation

In this paper, the application of the method of fundamental solutions to inverse problems associated with the two-dimensional biharmonic equation is investigated. The resulting system of linear algebraic equations is ill-conditioned and, therefore, its solution is regularized by employing the 0^t^h-order Tikhonov functional, while the choice of the regularization parameter is based on the L-curve method. Numerical results are presented for both smooth and piecewise smooth geometries. The convergence and the stability of the method with respect to increasing the number of source points and the distance between the source points and the boundary of the solution domain, and decreasing the amount of noise added into the input data, respectively, are analysed.

Boundary Element Solution For The Cauchy Problem In Linear Elasticity

In this paper we investigate the solution of the Cauchy problem in linear elasticity using the iterative algorithm proposed by Kozlov et al. [1] for obtaining approximate solutions to ill-posed boundary value problems. The technique is then numerically implemented using the boundary element method (BEM). The numerical results obtained confirm that the iterative BEM produces a convergent and stable numerical solution with respect to increasing the number of boundary elements and decreasing the amount of noise added into the input data. An efficient stopping regularizing criterion is given and in addition the accuracy of the iterative algorithm is improved by using a variable relaxation procedure.

Boundary element solution for the Cauchy problem in linear elasticity using singular value decomposition

Abstract In this paper the singular value decomposition (SVD), truncated at an optimal number, is analysed for obtaining approximate solutions to ill-conditioned linear algebraic systems of equations which arise from the boundary element method (BEM) discretisation of an ill-posed boundary value problem in linear elasticity. The regularisation parameter, namely the optimal truncation number, is chosen according to the discrepancy principle. The numerical results obtained confirm that the SVD+BEM produces a convergent and stable numerical solution with respect to decreasing the mesh size discretisation and the amount of noise added into the input data.

Boundary element method for the Cauchy problem in linear elasticity

In this paper, the iterative algorithm proposed by Kozlov et al. [Comput Maths Math Phys 32 (1991) 45] for obtaining approximate solutions to ill-posed boundary value problems in linear elasticity is analysed. The technique is then numerically implemented using the boundary element method (BEM). The numerical results obtained confirm that the iterative BEM produces a convergent and stable numerical solution with respect to increasing the number of boundary elements and decreasing the amount of noise added into the input data. An efficient stopping regularizing criterion is given and in addition, the accuracy of the iterative algorithm is improved by using a variable relaxation procedure. Analytical formulae for the integration constants resulting from the direct application of the BEM for an isotropic linear elastic medium are also presented.

BEM first-order regularisation method in linear elasticity for boundary identification

In this paper, we study the identification of an unknown portion of the boundary of a two-dimensional domain occupied by an isotropic linear elastic material from additional Cauchy data on the remaining known portion of the boundary. This inverse problem is approached using the boundary element method (BEM) in conjunction with the Tikhonov first-order regularisation procedure. The choice of the regularisation parameter is based on the L-curve criterion. The numerical results obtained show that the proposed method produces a convergent and stable solution.

Boundary element analysis of uncoupled transient thermo-elastic problems with time- and space-dependent heat sources

Abstract A boundary element method (BEM) for the analysis of two- and three-dimensional uncoupled transient thermo-elastic problems involving time- and space-dependent heat sources is presented. The domain integrals are efficiently treated using the Cartesian transformation and the radial integration methods without considering any internal cells. Similar to the dual reciprocity method (DRM), some internal points without any connectivity are considered; however, in contrast to the DRM, any arbitrary mesh-free interpolation method can be used in the present formulation. There is no need to find any particular solutions and the shape functions in the mesh-free interpolation method can be arbitrary and sufficiently complicated. Unlike the DRM, the generated system of equations contains the unknowns only on the boundary. After finding the primary unknowns on the boundary, the temperature, displacement, and stress components at all internal points can directly be found without solving any system of equations. Three examples with different forms of heat sources are presented to demonstrate the efficiency and accuracy of the proposed method. Although the proposed BEM is mathematically more complicated than domain methods, such as the finite element method (FEM), it is more efficient from a modelling viewpoint since only the surface mesh has to be generated in the presented method.

Parameter identification in isotropic linear elasticity using the boundary element method

In this paper, the identification of the Poisson ratio and the shear modulus for an isotropic linear elastic material from boundary measurements is investigated. We consider two cases, namely (i) measurements are possible on the whole boundary, and (ii) measurements are available only on a part of the boundary (Cauchy data). An objective function, based on the boundary element method (BEM) discretisation, is minimised in order to retrieve the material constants and the unknown boundary data.