A New Approach to Automatic Generation of an all Pentagonal Finite Element Mesh for Numerical Computations over Convex Polygonal Domains

A new method is presented for subdividing a large class of solid objects into topologically simple subregionssuitablefor automatic finite element meshing withpentagonalelements. It is known that one can improve the accuracy of the finite element solutionby uniformly refining a triangulation or uniformly refining a quadrangulation.Recently a refinement scheme of pentagonal partition was introduced in [31,32,33]. It is demonstrated that the numerical solutionbased on the pentagonal refinement scheme outperforms the solutions based on the traditional triangulation refinement scheme as well as quadrangulation refinement scheme. It is natural to ask if one can create a hexagonal refinement or general polygonal refinement schemes with a hope to offer even further improvement. It is shown in literature that one cannot refine a hexagon using hexagons of smaller size. In general, one can only refine an n-gon by n-gons of smaller size if n = 5. Furthermore, we introduce a refinement scheme of a generalpolygon based on the pentagon scheme. This paper first presents a pentagonalization (or pentagonal conversion) scheme that can create a pentagonal mesh from any arbitrary mesh structure. We also introduce a pentagonal preservation scheme that can create a pentagonal mesh from any pentagonal mesh

A New Approach to Automatic Generation of all Quadrilateral Finite Element Mesh for Planar Multiply Connected Regions

A new approach for the automatic generation and refinement of finite element meshes over multiply connected planar regions has been developed. This paper represents continuation of authors research activities in that area. An algorithm for producing a triangular mesh in a convex polygon is presented in authors recent work. It is used for the finite element triangulation of a complex polygonal region of the plane decomposed into convex polygons. We decompose the convex polygonal regions into simple sub regions in the shape of triangles. These simple regions are then triangulated to generate a fine mesh of triangular elements. We then propose an automatic triangular to quadrilateral conversion scheme.In this scheme, each isolated triangle is split into three quadrilaterals according to the usual scheme, adding three vertices in the middle of the edges and a vertex a the barycentre of the element. To preserve the mesh conformity, a similar procedure is also applied to every triangle of the domain to fully discretize the given complex polygonal domain into all quadrilaterals, thus propagating uniform refinement. This simple method generates a mesh whose elements confirm well to the requested shape by refining the problem domain. We have modified these algorithms and demonstrated their use by generating high quality meshes for some typical multiply connected regions: square domains with regular polygonal holes inside and vice versa. We have also made improvements and modifications to to the above triangulation algorithm of the triangle which can now triangulate a trapezium cut out of a triangle. This new algorithm on the triangulation of a trapezium cut out of a triangle is applied to quadrangulate the planar regions in the shape of a circular annulus and square domain with a square hole inside. We have appended MATLAB programs which incorporate the mesh generation schemes developed in this paper. These programs provide valuable output on the nodal coordinates, element connectivity and graphic display of the all quadrilateral mesh for application to finite element analysi

An Explicit Finite Element Integration Scheme for Linear Eight Node Convex Quadrilaterals Using Automatic Mesh Generation Technique over Plane Regions

This paper presents an explicit integration scheme to compute the stiffness matrix of an eight node linear convex quadrilateral element for plane problems using symbolic mathematics and an automatic generation of all quadrilateral mesh technique , In finite element analysis, the boundary problems governed by second order linear partial differential equations,the element stiffness matrices are expressed as integrals of the product of global derivatives over the linear convex quadrilateral region. These matrices can be shown to depend on the material properties and the matrix of integrals with integrands as rational functions with polynomial numerator and the linear denominator (4+ ) in bivariates over an eight node 2-square (-1 ).In this paper,we have computed these integrals in exact and digital forms using the symbolic mathematics capabilities of MATLAB. The proposed explicit finite element integration scheme is illustrated by computing the Prandtl stress function values and the torisonal constant for the square cross section by using the eight node linear convex quadrilateral finite elements.An automatic all quadrilateral mesh generation techniques for the eight node linear convex quadrilaterals is also developed for this purpose.We have presented a complete program which automatically discritises the arbitrary triangular domain into all eight node linear convex quadrilaterals and applies the so generated nodal coordinate and element connection data to the above mentioned torsion problem. Key words: Explicit Integration, Gauss Legendre Quadrature, Quadrilateral Element, Prandtl’s Stress Function for torsion, Symbolic mathematics,all quadrilateral mesh generation technique