A General Procedure to Formulate 3D Elements for Finite Element Applications

Abstract

This paper presents a general procedure to formulate and implement 3D elements of arbitrary order in meshes with multiple element types. This procedure includes obtaining shape functions and integration quadrature and establishing an approach for checking the generated element’s compatibility with adjacent elements’ surfaces. This procedure was implemented in Matlab, using its symbolic and graphics toolbox, and complied as a GUI interface named ShapeGen3D to provide finite element users with a tool to tailor elements according to their analysis needs. ShapeGen3D also outputs files with the element formulation needed to enable users to implement the generated elements in other programming languages or through user elements in commercial finite element software. Currently, finite element (FE) users are limited to employing element formulation available in the literature, commercial software, or existing element libraries. Thus, the developed procedure implemented in ShapeGen3D offers FEM users the possibility to employ elements beyond those readily available. The procedure was tested by generating the formulation for a brick element, a brick transition element, and higher-order hexahedron and tetrahedron elements that can be used in a spectral finite element analysis. The formulation obtained for the 20-node element was in perfect agreement with the formulation available in the literature. In addition, the results showed that the interpolation condition was met for all the generated elements, which provides confidence in the implementation of the process. Researchers and educators can use this procedure to efficiently develop and illustrate three-dimensional elements.

Keywords

• higher-order elements
• hexahedrons
• tetrahedrons
• integration quadrature
• custom elements
• transition elements