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• Inverse problem of determining a time-dependent load source in a simply supported beam. • Missing information is recovered from an additional "local" integral measurement • The existence and uniqueness of a solution to the inverse source problem is proved. • A convergent and stable algorithm is proposed. • Numerical experiments validate the convergence and stability of the proposed procedure. In this paper, the theoretical and numerical determination of a solely time-dependent load distribution is investigated for a simply supported non-homogeneous Euler–Bernoulli beam. The missing source is recovered from an additional "local" integral measurement. The existence and uniqueness of a solution to the corresponding variational problem is proved by employing Rothe's method. This method also reveals a time-discrete numerical scheme based on the backward Euler method to approximate the solution. Corresponding error estimates are proved and assessed by two numerical experiments. [ABSTRACT FROM AUTHOR]

In this paper, we propose an invariant method of fundamental solutions for solving boundary value problems in two-dimensional isotropic linear elasticity. The invariant method of fundamental solutions keeps a very basic natural property the so-called invariance property under trivial coordinate changes in the problem description, e.g. dilations and/or contractions. The problems are solved by the Tikhonov regularization method in conjunction with Morozov’s discrepancy principle. Finally, five examples of boundary value problems are investigated to show the efficiency of this method. The numerical convergence, accuracy, and stability of the proposed method with respect to the number of source points, the distance between the sources and the boundary, and the amount of noise added into the input data, respectively, are also analyzed. [ABSTRACT FROM AUTHOR]

The numerical reconstruction of the missing Dirichlet and Neumann data on an inaccessible part of the boundary in the case of two- and three-dimensional linear isotropic elastic materials from the knowledge of over-prescribed noisy measurements taken on the remaining accessible boundary part is investigated. This inverse problem is solved using the method of fundamental solutions (MFS), whilst its stabilization is achieved through several singular value decomposition (SVD)-based regularization methods, such as the Tikhonov regularization method [48], the damped SVD and the truncated SVD [18]. The regularization parameter is selected according to the discrepancy principle [40], generalized cross-validation criterion [14] and Hansen’s L-curve method [20]. [ABSTRACT FROM AUTHOR]

We investigate the numerical reconstruction of the missing thermal and mechanical boundary conditions on an inaccessible part of the boundary in the case of three-dimensional linear isotropic thermoelastic materials from the knowledge of over-prescribed noisy data on the remaining accessible boundary. We employ the method of fundamental solutions (MFS) and several singular value decomposition (SVD)-based regularization methods, e.g. the Tikhonov regularization method (Tikhonov and Arsenin, 1986), the damped SVD and the truncated SVD (Hansen, 1998), whilst the regularization parameter is selected according to the discrepancy principle (Morozov, 1966),generalized cross-validation criterion (Golub et al., 1979) and Hansen’s L-curve method (Hansen and O’Leary, 1993). [ABSTRACT FROM AUTHOR]

We investigate both theoretically and numerically the so-called invariance property, see e.g. Sun and Ma (2015a,b), of the solution of boundary value problems associated with the anisotropic heat conduction equation (or Laplace–Beltrami’s equation) in two dimensions with respect to elementary transformations of the solution domain, e.g. dilations or contractions. We also show that the standard method of fundamental solutions (MFS) does not satisfy the invariance property. Motivated by these reasons, we introduce, in a natural manner, a modified version of the MFS that remains invariant under elementary transformations of the solution domain and is referred to as the invariant MFS (IMFS). Five two-dimensional examples are thoroughly investigated to assess the numerical accuracy, convergence and stability of the proposed IMFS, in conjunction with the Tikhonov regularization method (Tikhonov and Arsenin, 1986) and Morozov’s discrepancy principle (Morozov, 1966), for Laplace–Beltrami’s equation with perturbed boundary conditions. [ABSTRACT FROM AUTHOR]

We investigate the numerical reconstruction of the missing displacements (Dirichlet data) and tractions (Neumann data) on an inaccessible part of the boundary in the case of two-dimensional linear isotropic elastic materials from the knowledge of over-prescribed noisy measurements taken on the remaining accessible boundary part. This inverse problem is solved using the fading regularization method, originally proposed by Cimetière et al. (2000, 2001) for the Laplace equation, in conjunction with a meshless method, namely the method of fundamental solutions (MFS). The stabilisation of the numerical method proposed herein is achieved by stopping the iterative procedure according to Morozov’s discrepancy principle (Morozov, 1966). [ABSTRACT FROM AUTHOR]

Abstract: We study the reconstruction of the missing thermal and mechanical fields on an inaccessible part of the boundary for two-dimensional linear isotropic thermoelastic materials from over-prescribed noisy (Cauchy) data on the remaining accessible boundary. This problem is solved with the method of fundamental solutions (MFS) together with the method of particular solutions (MPS) via the MFS-based particular solution for two-dimensional problems in uncoupled thermoelasticity developed in Marin and Karageorghis (2012a, 2013). The stabilisation/regularization of this inverse problem is achieved by using the Tikhonov regularization method (Tikhonov and Arsenin, 1986), whilst the optimal value of the regularization parameter is selected by employing Hansen’s L-curve method (Hansen, 1998). [Copyright &y& Elsevier]

Abstract: We investigate two numerical procedures for the Cauchy problem in linear elasticity, involving the relaxation of either the given boundary displacements (Dirichlet data) or the prescribed boundary tractions (Neumann data) on the over-specified boundary, in the alternating iterative algorithm of . The two mixed direct (well-posed) problems associated with each iteration are solved using the method of fundamental solutions (MFS), in conjunction with the Tikhonov regularization method, while the optimal value of the regularization parameter is chosen via the generalized cross-validation (GCV) criterion. An efficient regularizing stopping criterion which ceases the iterative procedure at the point where the accumulation of noise becomes dominant and the errors in predicting the exact solutions increase, is also presented. The MFS-based iterative algorithms with relaxation are tested for Cauchy problems for isotropic linear elastic materials in various geometries to confirm the numerical convergence, stability, accuracy and computational efficiency of the proposed method. [ABSTRACT FROM AUTHOR]

Abstract: We investigate the stable numerical reconstruction of an unknown portion of the boundary of a two-dimensional domain occupied by an isotropic linear elastic material from a prescribed boundary condition on this part of the boundary and additional displacement and traction measurements (i.e. Cauchy data) on the remaining known portion of the boundary. This inverse geometric problem is approached by combining the method of fundamental solutions (MFS) and the Tikhonov regularization method, whilst the optimal value of the regularization parameter is chosen according to the discrepancy principle. Various geometries are considered, i.e. convex and non-convex domains with a smooth or piecewise smooth boundary, in order to show the numerical stability, convergence, consistency and computational efficiency of the proposed method. [ABSTRACT FROM AUTHOR]

Abstract: We study the stable numerical identification of an unknown portion of the boundary on which either a Dirichlet or a Robin boundary condition is provided, while additional Cauchy data are given on the remaining known part of the boundary of a two-dimensional domain, in the case of steady state anisotropic heat conduction problems. 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 in both smooth and piecewise smooth geometries. [Copyright &y& Elsevier]

Abstract: We investigate a meshless method for the accurate and non-oscillatory solution of problems associated with two-dimensional Helmholtz-type equations in the presence of boundary singularities. The governing equation and boundary conditions are approximated by the method of fundamental solutions (MFS). It is well known that the existence of boundary singularities affects adversely the accuracy and convergence of standard numerical methods. The solutions to such problems and/or their corresponding derivatives may have unbounded values in the vicinity of the singularity. This difficulty is overcome by subtracting from the original MFS solution the corresponding singular functions, without an appreciable increase in the computational effort and at the same time keeping the same MFS approximation. Four examples for both the Helmholtz and the modified Helmholtz equations are carefully investigated and the numerical results presented show an excellent performance of the approach developed. [Copyright &y& Elsevier]

Abstract: In this paper, yet another iterative procedure, namely the minimal error method (MEM), for solving stably the Cauchy problem in linear elasticity is introduced and investigated. Furthermore, this method is compared with another two iterative algorithms, i.e. the conjugate gradient (CGM) and Landweber–Fridman methods (LFM), previously proposed by Marin et al. [Marin, L., Háo, D.N., Lesnic, D., 2002b. Conjugate gradient-boundary element method for the Cauchy problem in elasticity. Quarterly Journal of Mechanics and Applied Mathematics 55, 227–247] and Marin and Lesnic [Marin, L., Lesnic, D., 2005. Boundary element-Landweber method for the Cauchy problem in linear elasticity. IMA Journal of Applied Mathematics 18, 817–825], respectively, in the case of two-dimensional homogeneous isotropic linear elasticity. The inverse problem analysed in this paper is regularized by providing an efficient stopping criterion that ceases the iterative process in order to retrieve stable numerical solutions. The numerical implementation of the aforementioned iterative algorithms is realized by employing the boundary element method (BEM) for two-dimensional homogeneous isotropic linear elastic materials. [Copyright &y& Elsevier]

Abstract: In this paper, we investigate the application of the method of fundamental solutions (MFS) to two-dimensional steady-state heat conduction problems for both isotropic and anisotropic, single and composite (bi-materials), nonlinear functionally graded materials (FGMs). In the composite case, the interface continuity conditions are approximated in the same manner as the boundary conditions. The method is tested on several examples and its relative merits and disadvantages are discussed. [Copyright &y& Elsevier]

Abstract: Helmholtz-type equations arise naturally in many physical applications related to wave propagation, vibration phenomena and heat transfer. These equations are often used to describe the vibration of a structure, the acoustic cavity problem, the radiation wave, the scattering of a wave and heat conduction in fins. In this paper, the numerical recovery of a single and two circular cavities in Helmholtz-type equations from boundary data is investigated. The boundary element method (BEM), in conjunction with a constrained least-squares minimisation, is used to solve this inverse geometric problem. The accuracy and stability of the proposed numerical method with respect to the distance between the cavities and the outer boundary of the solution domain, the location and size of the cavities, and the distance between the cavities are also analysed. Unique and stable numerical solutions are obtained. [Copyright &y& Elsevier]

Abstract: The application of the method of fundamental solutions to the Cauchy problem for steady-state heat conduction in two-dimensional functionally graded materials (FGMs) is investigated. The resulting system of linear algebraic equations is ill-conditioned and, therefore, regularization is required in order to solve this system of equations in a stable manner. This is achieved 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 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. [Copyright &y& Elsevier]

Abstract: 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. [Copyright &y& Elsevier]

Abstract: In this paper, the application of the method of fundamental solutions to the Cauchy problem associated with two-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 first-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. [Copyright &y& Elsevier]

In this paper, the application of the method of fundamental solutions to the Cauchy problem in two-dimensional isotropic linear elasticity is investigated. The resulting system of linear algebraic equations is ill-conditioned and therefore its solution is regularised by employing the first-order Tikhonov functional, while the choice of the regularisation parameter is based on the L-curve method. Numerical results are presented for both smooth and piecewise smooth geometries, as well as for constant and linear stress states. 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. [Copyright &y& Elsevier]

We study the numerical reconstruction of the missing thermal boundary data on a portion of the boundary occupied by an anisotropic solid in the case of the steady-state heat conduction equation from the knowledge of both the temperature and the normal heat flux (i.e. Cauchy data) on the remaining and accessible part of the boundary. This inverse problem is known to be ill-posed and hence a regularization procedure is required. Herein we develop a solver for this problem by exploiting two sources of regularization, namely the smoothing nature of the corresponding direct problems and a priori knowledge on the solution to the inverse problem investigated. Consequently, this inverse problem is reformulated as a control one which reduces to minimising a corresponding functional defined on a fractional Sobolev space on the inaccessible part of the boundary. This approach yields a gradient-based iterative algorithm that consists, at each step, of the resolution of two direct problems and three corresponding adjoint problems in accordance with the function space where the control is sought. The theoretical convergence of the algorithm is studied by deriving an iteration-dependent admissible range for the parameter. Numerical experiments are realized for the two-dimensional case by employing the finite-difference method, whilst the numerical solution is stabilised/regularized by stopping the iterative process based on three criteria. [ABSTRACT FROM AUTHOR]

In this paper, we combine the fading regularization method with the method of fundamental solutions (MFS) and investigate its application to the Cauchy problem for the two-dimensional Helmholtz equation. We present a numerical reconstruction of the missing data on an inaccessible part of the boundary from the knowledge of overprescribed noisy data taken on the remaining accessible boundary part for both smooth and piecewise smooth two-dimensional geometries. The accuracy, convergence, stability and efficiency of the proposed numerical algorithm, as well as its capability to deblur the noisy data, are validated by three numerical examples. [ABSTRACT FROM AUTHOR]

Abstract: We investigate the numerical reconstruction of smooth star-shaped voids (rigid inclusions and cavities) which are compactly contained in a two-dimensional isotropic linear elastic body from a single non-destructive measurement of both the displacement and traction vectors (Cauchy data) on the external boundary. The displacement vector satisfying the Lamé system in linear elasticity is approximated using the meshless method of fundamental solutions (MFS). The fictitious source points are located both outside the (known) outer boundary of the body and inside the (unknown) void. The inverse geometric problem is then reduced to finding the minimum of a nonlinear least-squares functional that measures the gap between the given and computed data, penalized with respect to both the MFS constants and the derivative of the radial polar coordinates describing the position of the star-shaped void. The interior source points are anchored and move with the void during the iterative reconstruction procedure. The stability of the numerical method is investigated by inverting measurements contaminated with noise. [Copyright &y& Elsevier]