![]() The goal of this effort was to develop a second-order pyramid to provide additional hex-dominant meshing capabilities for explicit dynamic applications. A post-processing nodal extrapolation routine was also developed for PYR19 elements using a least-squares fit approach to facilitate plotting of element variables, computed at integration points, using typical visualization software. These functions were implemented into a general-purpose nonlinear explicit dynamics finite element code along with all numerical details (e.g., quadrature, time increment estimation, artificial viscosity, etc.) typically required for full application modeling. The PYR19 shape functions were derived by augmenting Bedrosian’s (1992) 13-node pyramid with the face and body centroid nodes. ![]() The pyramid has a quadrilateral base and four triangular sides, and has five vertex nodes, eight mid-edge nodes, five face centroid nodes, and a single body centroid node. No similar modular collection applicable to a range of FEM work, whether symbolic or numeric, has been published before.Ī second-order isoparametric 19-node pyramid finite element (PYR19) was developed that is suitable for use in nonlinear solid mechanics, especially with lumped mass explicit dynamics. Floating point accuracy can be selected arbitrarily. ![]() It can also be used as generator for low-level floating-point code modules in Fortran or C. The collection may be used “as is” in support of symbolic FEM work thus avoiding contamination with floating arithmetic that precludes simplification. Some gaps as regard region geometries and omission of non-product rules are noted in the conclusions. The collection embodies most FEM-useful formulas of low and moderate order for the seven regions noted above. For certain regions: quadrilaterals, wedges and hexahedra, only product rules were included to economize programming. Some unpublished non-product rules for pyramid regions were found and included in the collection. A larger class of formulas, previously known only numerically, were directly obtained through symbolic computations. A few quadrature rules were extracted from sources in the FEM and computational mathematics literature, and placed in symbolic form using Mathematica to generate own code. The latter is useful for computer-algebra aided FEM work that carries along symbolic variables. Information can be returned in floating-point (numerical) form, or in exact symbolic form. Seven regions are considered: line segments, triangles, quadrilaterals, tetrahedral, wedges, pyramids and hexahedra. This paper presents a set of Mathematica modules that organizes numerical integration rules considered useful for finite element work.
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