Fifteen node tetrahedral elements for explicit methods in nonlinear solid dynamics
Despite the ease in meshing and benefits for modeling flexure, curved shapes, etc., second-order tetrahedral elements are not contained in typical explicit solid dynamic programs. This is primarily due to the lack of both a satisfactory consistent nodal loading distribution and mass lumping technique. Row summation lumping, for instance, produces negative vertex node masses for the popular ten node serendipity tetrahedron, which also has zero vertex node loads resulting from a constant pressure on an element face. This has led to piecewise composites of four node tetrahedrons to represent a ten node one in explicit codes. In this paper, truly second-order fifteen node formulations for compressible and for nearly incompressible materials are presented and evaluated. In addition to producing all positive nodal loads from a uniform traction, row summation mass lumping for the fifteen node element is shown to produce all positive nodal masses. Performance is assessed in standard benchmark problems and practical applications using various elastic and elasticplastic material models and involving very large strains/deformations, severe distortions, and contact-impact. Comparisons are also made with several first-order elements and second-order hexahedral formulations. The offered elements performed satisfactorily in all examples. As recently found for second-order hexahedral elements, it is shown that the inclusion of face and centroidal nodes is vital for robust performance with row summation lumping, and high-order quadrature rules are crucial with explicit methods. These second-order elements are shown to be viable for practical applications, especially using todays parallel computers. Whereas the reliable performance is generally attained at significant computational expense compared with first-order and brick types, these elements can be more computationally competitive in flexure and have the desirable trait that they are amenable to automatic tetrahedral meshing software.
nonlinear solid dynamics mass lumping tetrahedron second-order finite elements explicit time integration mass matrix diagonalization