Second-order pyramid element formulations suitable for lumped-mass explicit methods in nonlinear solid mechanics

Pyramid elements can facilitate meshing by simplifying transitions from hexahedral to tetrahedral regions and a second-order basis promotes robust accuracy in unstructured meshes. Over the last decade, second-order C0 hexahedral, tetrahedral, and wedge elements that demonstrate important benefits have emerged for lumped-mass explicit dynamic methods in nonlinear solid mechanics. Whereas various second- and higher-order pyramid element formulations have been developed, none have addressed the necessities for lumped-mass explicit methods. Standard row-summation lumping for the popular 13- and 14-node pyramid elements, for example, produces negative lumped masses at the vertex nodes. Degeneration of hexahedral elements into pyramids is also unsatisfactory for second-order lumped-mass elements, except perhaps for low-interest fill regions. Such issues are resolved herein by developing 19-node second-order C0 pyramid element formulations that provide all-positive row-sum nodal lumped masses and are also compatible with the other suitable second-order lumped-mass elements. Other specific requirements of explicit methods are also addressed, and several uniform and selectively reduced pyramid quadrature rules are investigated. Application to a variety of benchmark problems demonstrates robust performance, including no apparent ill effects arising from using rational shape functions to treat the apex and good accuracy with pyramids placed in critical regions of highly nonlinear inelastic analyses.