Meshless Singular Boundary Methods For Biharmonic And Stokes Problems

Biharmonic problem and Stokes flow problem are two important boundary value problems in physics and mathematics, so these two problems have come to a wide range of research and development in the past decades.In recent years, meshless method has been widely used in these two problems, including the method of fundamental solutions, the boundary knot method and so on. The singular boundary method(SBM) is a novel boundary type meshless method that is the same as the method of fundamental solutions, both of which are expressed as a linear combination of the fundamental solutions. It does not need to select the virtual boundary, overcoming the difficulty in choosing the virtual boundary problem of the method of fundamental solutions, which is different from the method of fundamental solutions. So the singular boundary method is suitable for solving the above two problems.In this paper, the singular boundary method is developed to solve the Biharmonic problem. The singular boundary method(SBM) needs to choose the sample nods and then uses an inverse interpolation technique(IIT) to obtain the source intensity factor. The singular boundary method avoids the fictitious boundary used in the method of fundamental solutions(MFS). However, the numerical results show that the accuracy may be sensitive to the placement of these sample nodes.Thus, the improved singular boundary method(ISBM) is developed in Biharmonic problem and Stokes problem. In the improved singular boundary method, the calculating formula of the source intensity factor has been developed which uses the subtracting and adding-back techniquewithout sample nodes. Sothe accuracy and the stability of the singular boundary method are greatly improved.Finally, with some numerical examples comparing MFS, SBM and ISBM, it is concluded that the accuracy and stability of the improved singular boundary method are better.