Study On Rational Element Method

Dividing heterogeneous material domain into polygonal elements mesh based on its microstructure, it is convenient and efficient to simulate properties of heterogeneous materials. Application of the conventional displacement-based finite element method to an element with n nodes runs into difficulty when n > 4 because its becomes impossible to ensure interelement compatibility of displacements with n-term polynomial representations. Even quadrilateral elements require the introduction of isoparametric techniques to ensure the interelement compatibility. In this dissertation, combined the ideas of inverse distance weighted of Shepard interpolation and considering distribution of nodes in natural neighbor interpolation, the rational function interpolation (RFI) is directly constructed adopting geometric method on a polygonal element. Furthermore using rational function interpolation on polygonal element, a new numerical method, Rational Element Method (REM), for solving partial differential equations based on highly irregular networks is introduced.In this paper the following subjects are investigated.Adopting geometric method, the rational function interpolation is constructed on polygonal element. Some properties for RFI are presented and proved. The computational algebraic expression of RFI is given. Using this expression, it is convenient to program it. Compared with Shepard interpolation, the distribution of nodes is considered in the RFI. Otherwise than natural neighbor Laplace interpolant, the RFI needn't to look for interpolating points. Distinguished from Wachspress type interpolation, there isn't undetermined parameters in RFI, so it is convenient to program. In triangular and rectangle elements, the RFI is equivalent to area coordinates of triangular and bilinear polynomial interpolation in quadrilateral, respectively. The RFI is directly constructed on polygonal element. It doesn't depend on isoperimetric transform.Using rational function interpolation, the surfaces defined on round domain are represented. The numerical tests indicate that the surfaces reconstructed by RFI usinginformation of finite points on the boundary can be wonderfully represented the characters of real surfaces.The approximation temperature distributions on a convex domain are obtained by using rational function interpolation. The approximation temperature values in the inner domain are calculated using temperature values of boundary points. The temperature gradient interpolated by RFI is continuous in the inner domain. It overcomes the defect, which the temperature gradient is non-continuous in the inner domain using conventional finite element interpolation owing to setting nodes in the inner domain. The numerical tests illustrated that the interpolated temperature distribution can be wonderfully represented the characters of real temperature distributions.Adopting geometric method, the rational blending functions are constructed on polygonal element. The rational transfinite interpolation on polygonal element is presented. The special case, rational Coons patch, is obtained. The expressions of rational blending functions are given. Different from the linearity of Coons patch, the blending functions of rational Coons interpolation are nonlinear. The nonlinear can be benefit to represent the characters of complex surfaces. Numerical tests indicated that rational Coons patch is more vivid to represent the characters for complex surfaces than Coons patch.The rational Coons interpolation is applied to interpolate temperature distributions on rectangular domains. The defect of local poor accuracy of temperature distributions interpolated by RFI is improved. Numerical tests indicate that RFI is suited to round domain and rational Coons interpolation is suited to rectangular domains for interpolating temperature distributions.The concept of Delaunay triangulation is extended to Delaunay polygonization. The automatic generating meshes technique for Delaunay polygonization is presented. For given nodes distribution, using Delaunay polygonization technique, the polygonal element meshes can be automatic generated in computational domains. It is convenient and flexible to generate polygonal element meshes on complex domain. Numerical tests indicate that Delaunay polygonization technique is not only to generate polygonal...