Numerical Solutions of Direct and Inverse Stokes Problems by the Method of Fundamental Solutions and the Laplacian Decomposition

Abstract

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 boundarytype 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.

Comparison of the Boundary Element Method and the Method of Fundamental Solutions for Analysis of Potential and Elasticity Problems in Convex and Concave Domains

“…Karageorghis and Fairweather [14] employed the MFS for axisymmetric potential problems for the first time. Later, it was successfully applied to a variety of potential and elastic problems, see for example Fam and Rashed [15], Young et al [16], Marin et al [17], Karageorghis et al [18], Fan and Li [19], Mohammadi et al [20]. Suitable arrangement of source points and collocation points in the MFS has been always disputable and under discussion among the researchers.…”

Localized Boundary Knot Method for Solving Two-Dimensional Inverse Cauchy Problems

“…It is worth emphasizing that among the boundary-type meshless methods, the method of fundamental solutions (MFS) proposed by Kupradze and Aleksidze in 1964 [24] is the most popular in the application of inverse problems [25,26] due to its high accuracy. Young [27] studied the condition number of MFS in a Cauchy problem, and Fan [28] further extended the scheme to solve a Cauchy problem involving Stokes equations. Despite the popularity of the method, determining the appropriate location of the source nodes is one of the difficulties that the MFS needs to overcome.…”

On numerical resolution of an inverse Cauchy problem modeling the airflow in the bronchial tree

“…On the other hand, the inverse Cauchy problem governed by Stokes equations was treated using different approaches. In particular, the alternating method was proposed in Bastay et al (2006), Abda et al (2013) and Chakib et al (2018), the technic of fundamental solutions was suggested in Chen et al (2005), Alves and Silvestre (2004) and Fan and Li (2015), and many other approaches were introduced in García et al (2017), Lai et al (2015), Lechleiter and Rienmüller (2013) and Aboulaich et al (2013).…”