Andreas Karageorghis

The method of fundamental solutions for elliptic boundary value problems

The aim of this paper is to describe the development of the method of fundamental solutions (MFS) and related methods over the last three decades. Several applications of MFS-type methods are presented. Techniques by which such methods are extended to certain classes of non-trivial problems and adapted for the solution of inhomogeneous problems are also outlined.

The method of fundamental solutions for scattering and radiation problems

The development of the method of fundamental solutions (MFS) and related methods for the numerical solution of scattering and radiation problems in fluids and solids is described and reviewed. A brief review of the developments and applications in all areas of the MFS over the last five years is also given. Future possible areas of applications in fields related to scattering and radiation problems are identified.

The method of fundamental solutions for the numerical solution of the biharmonic equation

Abstract The method of fundamental solutions (MFS) is a relatively new technique for the numerical solution of certain elliptic boundary value problems. It falls in the class of methods generally called boundary methods, and, like the well-known boundary integral equation method, is applicable when a fundamental solution of the differential equation is known. In the MFS, the approximate solution is a linear combination of fundamental solutions with singularities placed outside the domain of the problem. The locations of the singularities are either preassigned or determined along with the coefficients of the fundamental solutions so that the approximate solution satisfies the boundary conditions as well as possible. In many applications, these quantities are determined by a least squares fit of the boundary conditions, a nonlinear problem, which is solved using standard software. In this paper, the MFS is formulated for biharmonic problems and is applied to a variety of standard test problems as well as to problems arising in elasticity and fluid flow.

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.

The method of fundamental solutions for the calculation of the eigenvalues of the Helmholtz equation

We investigate the application of the method of fundamental solutions (MFS) for the calculation of the eigenvalues of the Helmholtz equation in the plane subject to homogeneous Dirichlet boundary conditions. We present results for circular and rectangular geometries.

Some Aspects of the Method of Fundamental Solutions for Certain Harmonic Problems

The Method of Fundamental Solutions (MFS) is a boundary-type method for the solution of certain elliptic boundary value problems. The basic ideas of the MFS were introduced by Kupradze and Alexidze and its modern form was proposed by Mathon and Johnston. In this work, we investigate certain aspects of a particular version of the MFS, also known as the Charge Simulation Method, when it is applied to the Dirichlet problem for Laplaces equation in a disk.

The method of fundamental solutions for heat conduction in layered materials

In this paper, we investigate the application of the Method of Fundamental Solutions (MFS) to two-dimensional problems of steady-state heat conduction in isotropic and anisotropic bimaterials. Two approaches are used: a domain decomposition technique and a single-domain approach in which modified fundamental solutions are employed. The modified fundamental solutions satisfy the interface continuity conditions automatically for planar interfaces. The two approaches are tested and compared on several test problems and their relative merits and disadvantages discussed. Finally, we use the domain decomposition approach to investigate bimaterial problems where the interface is non-planar and the modified fundamental solutions cannot be used. Copyright

The method of fundamental solutions for axisymmetric potential problems

In this paper, we investigate the application of the Method of Fundamental Solutions (MFS) to two classes of axisymmetric potential problems. In the first, the boundary conditions as well as the domain of the problem, are axisymmetric, and in the second, the boundary conditions are arbitrary. In both cases, the fundamental solutions of the governing equations and their normal derivatives, which are required in the formulation of the MFS, can be expressed in terms of complete elliptic integrals. The method is tested on several axisymmetric problems from the literature and is also applied to an axisymmetric free boundary problem. Copyright

The method of fundamental solutions for layered elastic materials

In this paper, we investigate the application of the method of fundamental solutions to two-dimensional elasticity problems in isotropic and anisotropic single materials and bimaterials. A domain decomposition technique is employed in the bimaterial case where the interface continuity conditions are approximated in the same manner as the boundary conditions. The method is tested on several test problems and its relative merits and disadvantages are discussed.

The method of fundamental solutions for three-dimensional elastostatics problems

Abstract We consider the application of the method of fundamental solutions to isotropic elastostatics problems in three space dimensions. The displacements are approximated by linear combinations of the fundamental solutions of the Cauchy–Navier equations of elasticity, which are expressed in terms of sources placed outside the domain of the problem under consideration. The final positions of the sources and the coefficients of the fundamental solutions are determined by enforcing the satisfaction of the boundary conditions in a least squares sense. The applicability of the method is demonstrated on two test problems. The numerical experiments indicate that accurate results can be obtained with relatively few degrees of freedom.