Method of Fundamental Solutions for Stokes' First and Second Problems

Abstract

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.

A Method of Fundamental Solutions in Poroelasticity to Model the Stress Field in Geothermal Reservoirs

“…We can only give a few examples by mentioning applications to the fields of potential theory, potential flow, and Stokes flow [47,145,278,279,299,302], the biharmonic equation [147], the Helmholtz equation [24], the modified Helmholtz equation [187], elastostatics [76,148,150,196,233], Signorini problems [226], fracture mechanics [124,149], the wave equation and acoustics [12,122,162], heat conduction [43,140,141,143], diffusion 1 [46,133,296,297,300], Stefan problems [44,142], Brinkman flows [275], oscillatory and porous buoyant flow [277], diffusion-reaction equations [22], calculation of eigenfrequencies and eigenmodes [9], radiation and scattering problems [75], acoustic wave scattering on poroelastic scatterers [211], microstrip antenna analysis [252], or to two-dimensional unsteady Burger's equations 2 [298].…”

The Modified Collocation Trefftz Method and Exponentially Convergent Scalar Homotopy Algorithm for the Inverse Boundary Determination Problem for the Biharmonic Equation

“…In order to avoid the mesh generation and numerical integration, many meshless methods are proposed recently, such as the MFS [3,6,[13][14][15][16], the radial basis function collocation methods [17][18][19], the MCTM [20][21][22][23][24][25], etc. For the current problem, since positions of some boundary points are unknown, it is then nature for us to adopt the boundary-type meshless scheme to solve the problem.…”

The Eulerian–Lagrangian method of fundamental solutions for two-dimensional unsteady Burgers’ equations

“…There are several meshless methods developed in the past decade and some available methods are the multiquadrics (MQ) method [15,16], the meshless local Petrov-Galerkin (MLPG) method [17][18][19][20] and the method of fundamental solutions (MFS) [21][22][23][24][25][26][27][28][29]. Hon and Mao [15] applied the MQ method to the one-dimensional unsteady Burgers' equation, while Li et al [16] used the MQ method to solve two-dimensional problems.…”

“…Under the novel concept of time-space unification, Young et al [28,29] solved the time-dependent diffusion equations by the diffusion fundamental solution and MFS which can avoid the Laplace transform or finite difference method in discretizing the time state. The time-dependent MFS is further applied to the Stokes' first and second problems in a semi-infinite domain by Hu et al [22].…”