The direct method of lines for incompressible material problems on polygon domains

In this paper, we discuss the numerical solutions of the incompressible material problems on a polygon using a semi-discrete method [HH99]. After a suitable transformation of the coordinates, the original boundary value problem (BVP) is reduced to a discontinuous coefficients problem on a rectangle, which is semi-discreted to a BVP of a system of ordinary differential equations (O.D.E’s). After solving the BVP of the system by a direct method, the semi-discrete approximation of the original problem is obtained. It’s worth to point out that the semi-discrete approximation in form of separable variables naturally possesses the singularity of the original problem. Finally, the numerical examples show that our method is feasible and very effective for solving the incompressible material problems with singularities numerically. The use of nearly incompressible materials is common in many engineering applications, such as tires, building and bridge bearings, engine mounts, gaskets etc. The natural rubber is the nearly incompressible material, typically the bulk modulus of rubber is several thousand times of the shear modulus. As the material is undergoing plastic deformations, it is nearly incompressible too. We can use the Stokes equations as a model to deal with the incompressible materials. It is also a model for the incompressible fluids. The stress analysis of incompressible materials becomes very significant. The difficulties for solving the incompressible material problems numerically are: the stress singularity existing at the joint of the interface, the crack-tip or the corner; the incompressibility and the large deformations. To overcome the above difficulties, a great deal of research effort by engineers and mathematicians has been devoted to the development of the FEM(finite element method) for the numerical approximation of incompressible problems. Herrmann [Her65] presented a mixed variational formulation for incompressible isotropic materials. Babuska and Brezzi [Bab73, Bre74] derived the inf-sup condition for the mixed FEM for incompressible problems. Oden et al [OK82, JTOS82] presented general criteria for stability and convergence of mixed and penalty methods(with reduced integration) and applied these criteria to the analysis of elasticity and Stokesian flow problems. Recently, many researchers developed other methods for incompressible problems [AWS95]. For more references, we refer to the paper by Gadala [Gad86]. The singularities of incompressible materials have also been paid attention by researchers [NAHM96]. We know that the singularities at singular points in the incompressible composite material problems are very complex. On each singular point, the singularity is different. Therefore, the standard finite element method and finite difference method can not give satisfied results for incom-