Dual fat boundary method: the fat boundary method in elasticity with an extension of the application scope

Abstract

The fat boundary method (FBM) is a fictitious domain method, proposed to solve Poisson problems in a domain with small perforations. It can achieve higher accuracy around holes, which makes it very suitable to solve elasticity problems because stress concentrations often appear around holes. However, there are some strict restrictions of the FBM limiting the wide range of applications. For example, the original FBM deals with perforated rectangular domain with only Dirichlet boundary conditions. Furthermore, because the global domain is extended to the holes, analytical solutions in holes corresponding to the Dirichlet boundary conditions around holes are required. This limits both the boundary conditions around holes and the shape of holes, because for arbitrary holes it is difficult to get the analytical solutions. This article makes an attempt to break these limitations and apply the FBM to elasticity. Firstly, we review the FBM and introduce Neumann boundary conditions to the rectangular domain. A mathematical proof of the conditional convergence of the algorithm is presented. Furthermore, the FBM is compared with the Lagrange multiplier method to clarify that the FBM is one kind of weak imposition method. Then we apply the FBM to linear elasticity and the dual fat boundary method is proposed to solve problems without analytical solutions in holes. Some numerical examples are presented to verify the method proposed here.