Desingularised method of double layer fundamental solutions for potential flow problems

On the application of the method of fundamental solutions to boundary value problems with jump discontinuities

On the choice of source points in the method of fundamental solutions

Angular basis functions formulation for 2D potential flows with non-smooth boundaries

This paper describes a combination of the method of fundamental solutions (MFS) and the dual reciprocity method (DRM) as a mesh‐free numerical method (MFS‐DRM model) to solve 2D and 3D nonhomogeneous diffusion problems. Using our method, the homogeneous solutions of the diffusion equations are solved by the MFS, and the DRM, based on the radial basis functions (RBF) of the thin plate splines (TPS), is employed to solve for particular solutions. The present scheme is free from the frequently used Laplace transform and the finite difference discretization method to deal with the time derivative term in the governing equation. By properly placing the source points in the time‐space domain, the solution is advanced in time until a steady state solution (if one exists) is reached. Since the present method does not need mesh discretization and nodal connectivity, the computational effort and memory storage required are minimal as compared to other domain‐oriented numerical schemes such as FDM, FEM, FVM, etc. Test results obtained for 2D and 3D diffusion problems show good comparability with analytical solutions and other numerical solutions, such as those obtained by the MFS‐DRM model based on the modified Helmholtz fundamental solutions. Thus the present numerical scheme has provided a promising mesh‐free numerical tool to solve nonhomogeneous diffusion problems with space‐time unification for diffusion fundamental solutions.

An extended method of time-dependent fundamental solutions for inhomogeneous heat conduction

This paper describes the applications of the method of fundamental solutions (MFS) as a mesh-free numerical method for the Stokes' first and second problems which prevail in the semi-infinite domain with constant and oscillatory velocity at the boundary in the fluid-mechanics benchmark problems. The time-dependent fundamental solutions for the semi-infinite problems are used directly to obtain the solution as a linear combination of the unsteady fundamental solution of the diffusion operator. The proposed numerical scheme is free from the conventional Laplace transform or the finite difference scheme to deal with the time derivative term of the governing equation. By properly placing the field points and the source points at a given time level, the solution is advanced in time until steady state solutions are reached. It is not necessary to locate and specify the condition at the infinite domain such as other numerical methods. Since the present method does not need mesh discretization and nodal connectivity, the computational effort and memory storage required are minimal as compared to the domain-oriented numerical schemes. Test results obtained for the Stokes' first and second problems show good comparisons with the analytical solutions. Thus the present numerical scheme has provided a promising mesh-free numerical tool to solve the unsteady semi-infinite problems with the space-time unification for the time-dependent fundamental solution.

The Method of Fundamental Solution is applied to potential problems. The source and collocation points are supposed to coincide and are located along the boundary. The singularities due to the singularity of the fundamental solution are avoided by several techniques (regularization and desingularization). Both the monopole and the dipole formulations are investigated. The resulting algebraic systems have advantageous properties provided that pure Dirichlet or pure Neumann boundary condition is prescribed. Otherwise, the original problem is converted to a sequence of pure Dirichlet and pure Neumann subproblems, the solutions of which converge rapidly to the solution of the original mixed problem. The iteration is embedded to a multi-level context in a natural way. Thus, the computational cost can be significantly reduced, and the problem of large and ill-conditioned matrices is also avoided.

This paper studies unsteady viscous flows in a capillary using the method of fundamental solutions. The effect on fluid velocity of a thin porous surface layer on the wall of the tube is modelled using a slip boundary condition. Laplace transform is used to handle the time variables in the problem. Fundamental solutions of point force in the transformed domain and in the time domain are derived. This study extends our previous work by developing a sophisticated boundary singularity scheme based on the Method of Fundamental Solution (MFS).

It is well known that the method of fundamental solutions (MFS) is a numerical method of exponential convergence. In other words, the logarithmic error is proportional to the node number of spatial discretization. In this study, the exponential convergence of the MFS is demonstrated by solving the Laplace equation in domains of rectangles, ellipses, amoeba-like shapes, and rectangular cuboids. In the solution procedure, the sources of the MFS are located as far as possible and the instability resulted from the ill-conditioning of system matrix is avoided by using the multiple precision floating-point reliable (MPFR) library. The results converge faster for the cases of smoother boundary conditions and larger area/perimeter ratios. For problems with discontinuous boundary data, the exponential convergence is also accomplished using the enriched method of fundamental solutions (EMFS), which is constructed by the fundamental solutions and the local singular solutions. The computation is scalable in the sense that the required time increases only algebraically.

A boundary-type method for solving the Laplace problems using the modified method of fundamental solutions (MMFS) is proposed. The present method (MMFS) implements the sin- gular fundamental solutions to evaluate the solu- tions, and it can locate the source points on the real boundary as contrasted to the conventional MFS, where a fictitious boundary is needed to avoid thesingularityofdiagonal term ofinfluence matrices. The diagonal term of influence matrices for arbitrary domain can be novelly determined by relating the MFS with the indirect BEM and are also solved forcircular domain analyticallyby using separable kernels and circulants. The major difficultyofthecoincidenceofthesourceand col- locationpointsintheconventionalMFS isthereby overcome. The off-diagonal coefficients of influ- ence matrices can be easily determined by using thetwo-pointfunction. Theill-posednatureofthe conventional MFS then disappears. Finally, we provide numerical evidences that the presentmethod improves theaccuracy of the solu- tion after comparing with the conventional MFS, in particular for complicated boundaries in which the conventional MFS may encounter difficulties. Good agreements are observed as comparing with analytic or other numerical solutions.

Calculating transport of water from a conduit to the porous matrix by boundary distributed source method

Localized method of fundamental solutions for large-scale modelling of three-dimensional anisotropic heat conduction problems – Theory and MATLAB code

In this study, both of direct and inverse Stokes problems are stably and accurately analyzed by the method of fundamental solutions (MFS) and the Laplacian decomposition. In order to accurately resolve the Stokes problem, the Laplacian decomposition is adopted to convert the Stokes equations into three Laplace equations, which will be solved by the MFS, with an augmented boundary condition. To enforce the satisfactions of continuity equation along whole boundary as an augmented boundary condition will guarantee the satisfactions of mass conservation inside the computational domain. The MFS is one of the most promising boundary-type meshless methods, since the time-consuming tasks of mesh generation and numerical quadrature can be avoided as well as only boundary nodes are needed for numerical implementations. The numerical solutions of the MFS are expressed as linear combinations of fundamental solutions of Laplace equation and the sources are located out of the computational domain to avoid numerical singularity. The numerical solutions for velocity components, pressure and their gradient terms can be obtained by simple summation due to the simplicity of the MFS. Several numerical examples of direct and inverse Stokes problems are analyzed by the proposed boundary-type meshless numerical scheme. The simplicity and the accuracy of the proposed method are verified by numerical experiments and comparisons. Moreover, different levels of noise are added into boundary conditions of inverse Stokes problems to validate the stability of the proposed numerical scheme.

The under-determined version of the MFS: Taking more sources than collocation points

The method of fundamental solutions (MFS) is a meshless method for solving boundary value problems with some partial differential equations. It allows to obtain highly accurate approximations for the solutions assuming that they are smooth enough, even with small matrices. As a counterpart, the (dense) matrices involved are often ill-conditioned which is related to the well known uncertainty principle stating that it is impossible to have high accuracy and good conditioning at the same time. In this work, we propose a technique to reduce the ill conditioning in the MFS, assuming that the source points are placed on a circumference of radius R. The idea is to apply a suitable change of basis that provides new basis functions that span the same space as the MFS’s, but are much better conditioned. In the particular case of circular domains, the algorithm allows to obtain errors close to machine precision, with condition numbers of order O(1), independently of the number of points sources and R.