Polygonal And Polyhedral Finite Element Method Based On Radial Integration Method

Along with the diversification of numerically simulated problems, element types used in finite element method are not limited to regular elements such as triangular, tetrahedral, quadrilateral and hexahedral elements. As new kinds of elements, polygonal and polyhedral elements are able to have arbitrary number of faces and edges, and these offer them more flexibility in mesh generation for complex geometries than regular elements. They can be more easily used to simulate thermal and mechanical behaviors of materials, and are more suitable for solving some special problems, such as fracture and crack problems. However, it is very difficult to construct polynomial interpolation functions for arbitrary polygonal and polyhedral elements because their geometries are arbitrary. So it is hard to calculate the integrals of the element stiffness matrix and body load vector. To evaluate the integrals over the polygonal and polyhedral elements, the following researches are carried out in this paper:Firstly, the radial integration method (RIM) is used to evaluate the element stiffness matrices and body load vectors over polygonal elements, in which the Wachspress interpolation is chosen as the shape function of polygonal elements. The RIM is able to transform the integrals of the element stiffness matrices and body load vectors over complex polygonal elements into a sum of line integrals along the edges of elements. This method can avoid the problem caused by irregular polygonal integral domain non-polynomial shape function. In the calculation, polygonal elements are separated into regular elements (triangular and quadrilateral elements) and other irregular elements with arbitrary number of sides. For regular elements, the stiffness matrices and body load vectors are evaluated with the Gaussian integration method directly. For irregular elements, the two-dimensional surface integrals are transformed into one-dimensional line integrals along the edges of the elements by using RIM.Secondly, the RIM is used to evaluate the element stiffness matrices and body load vectors over polyhedral elements, in which the Floater interpolation is chosen as the shape function of polyhedral elements. RIM is used twice to evaluate integrals in polyhedral elements:first, RIM is used to transform the volume integrals into surface integrals over the boundary of the polyhedral elements. Then the integral surfaces are separated into regular and irregular surfaces similar to the separation of polygonal elements. The irregular surfaces are transformed into line integrals along the edges of the surface by using RIM again. Eventually, the volume integrals in polyhedral with irregular integral surfaces are transformed into a sum of line integrals.Finally, two polygonal examples and three polyhedral examples are used to demonstrate the efficiency and accuracy of the proposed the method. A 2D patch test is used to demonstrate the accuracy in arbitrary polygonal elements; the analysis of plane with a hole is used to test the accuracy of RIM for quadrilateral elements; a 3D cantilever beam is used to demonstrate the accuracy affected by the number of integral points; a 3D patch test is used to demonstrate the accuracy in arbitrary polyhedral elements; the truncated cuboctahedra structure has a complex geometry, so it is used to demonstrate the accuracy of the proposed method in evaluating complex polyhedral elements. Numerical examples are used to demonstrate the accuracy and efficiency of the proposed method and the computational results show that the proposed method is more accurate than triangulation approach under the use of the same number of integration points. The advantages of the proposed method are that the polygonal and polyhedral elements need not to be partitioned into triangles or tetrahedrons and is easily coded, strongly universal, and with high computational accuracy.