Polygonal Mesh-Based Boundary Element Method For Functionally Graded Materials

With the rapid development of modern materials science and improvement of the overall level of technology industry, Functionally Graded Materials (FGMs), as a new class of composite materials with the material properties varying gradually with respect to spatial coordinates, have attracted the attention of many scholars due to its high temperature resistance, corrosion resistance and high strength characteristics, and obtained extensive applications in the engineering. But in the industrial fabrication of functional gradient materials, inevitably there exist multiple cracks, voids and particle inclusions and other internal defects. In order to solve this problem, people proposed to use polygonal mesh elements to simulate the discontinuous problem in cracks and pores. Compared to the traditional triangular and quadrilateral elements, the discretization of polygonal mesh is more flexible and under the same conditions, the computational results are more accurate. Polygonal elements not only can well simulate the complex boundary geometry, but also can well fit the shapes of materials with polygonal elements as basic unit cells. But in the finite element method, to obtain a satisfactory result for these problems, the workload of generating elements is usually very huge and evaluation of related domain integrals over polygonal elements is sometimes difficult.When using the boundary element method (BEM) to solve non-homogeneous and nonlinear problems, there will be domain integrals appearing in the integral equations. Using the internal cell integration method to evaluate these domain integrals has the advantage of fast computation over the boundary element only based method established from the transformation of domain integral to boundary integral. However, there have been not yet people applying polygonal meshes in the domain integrals in BEM. One of reasons is that it is often difficult to evaluate integrals over polygonal meshes directly using the Gauss quadrature. As commonly discussed in references, the triangulation integration scheme induces additional computational costs due to the discretization of a polygonal mesh into a number of small triangles.In this paper, a new method is proposed to evaluate domain integrals in two dimensional elasticity BEM analysis using the polygonal mesh approach. Firstly, the boundary-do main integral equations with variable coefficients are derived, and shape functions for irregular polygonal meshes are constructed for interpolations of geometry and physical quantities based on the rational function interpolation. Then, the domain integrals over polygonal meshes are converted into line integrals along the polygon perimeter by employing the radial integration method, which results in a unified scheme to evaluate various polygonal mesh integrals. Finally, based on the proposed theory and derived formulation, a Fortran code is written and three representative numerical examples are analyzed to demonstrate the effectiveness of the proposed method in the paper.