All ETDs from UAB

Advisory Committee Chair

David L Littlefield

Advisory Committee Members

James T Baylot

Kent T Danielson

Lee G Moradi

Roman Shterenberg

Document Type


Date of Award


Degree Name by School

Doctor of Philosophy (PhD) School of Engineering


A 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. 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. The PYR19 shape functions were derived by augmenting Bedrosian’s (1992) 13-node pyramid with the face and body centroid nodes. 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. 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.

The goal of this effort was to develop a second-order pyramid to provide additional hex-dominant meshing capabilities for explicit dynamic applications. The PYR19 element is the first second-order pyramid to provide a positive-definite and well-performing diagonal mass matrix, using row-summation lumping, and that is also compatible with the second-order 27-node hexahedral (HEX27), 15-node tetrahedral (TET15), and 21-node wedge (WEDGE21) elements, which have all previously been shown to perform well for explicit dynamics. The addition of PYR19 to this set of elements provides much greater flexibility for hex-dominant meshing, which can greatly reduce the time-to-a-solution when modeling complex geometries.

Two distinct versions of the PYR19 element were implemented and evaluated: a standard displacement-based formulation using uniform numerical quadrature and a selective reduced integration formulation for treating near-incompressibility (e.g., metal plasticity). Benchmark evaluations demonstrated that the most robust performance was achieved using twenty-seven and eight points for the uniform and reduced rules, respectively, but a thirteen-point rule is also acceptable for well-shaped elements.

In summary, this dissertation provides new shape functions for a second-order 19-node pyramid element, along with a routine for extrapolating element variables from the quadrature points to the nodes. Computer subroutines written in Fortran are provided to aid in implementation. Various benchmark evaluations provide guidance regarding numerical integration and demonstrate the element’s suitable performance in real-world applications.

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