Joe Bishop received his Ph.D. in Aerospace Engineering from Texas A&M University in 1996. His graduate research was in the mechanics of composite materials and mechanisms of material damping. From 1997 to 2004 he worked in the Powertrain Division of General Motors Corporation performing thermal-structural analysis of internal combustion engines. He joined Sandia National Laboratories in 2004 in the Engineering Sciences Center. He has worked on diverse topics in computational mechanics including pervasive-fracture, impact and penetration, geomechanics, additive manufacturing, inverse methods, polyhedral finite elements, meshfree methods, and multiscale modeling.
There is now a long history of using both element-based and element-free discretizations for solving governing field equations. By combining these two approaches, we can obtain a new, versatile, multiscale, discretization technique. In many engineering applications, domains of interest are geometrically complex containing numerous small features. These features are typically removed in a manual process to facilitate a conventional element-based meshing process. This manual defeaturing process is dependent upon the goals of the simulation and typically involves subjective heuristics. A quasi-meshfree method has been developed that separates the discretization of the domain from that of the solution of the governing partial differential equation (PDE). In this approach, a fine-scale triangulation of the domain is used to first discretize the fully featured geometry but then a coarse-scale element-free discretization is used to approximate the solution of the governing equations. The fine-scale triangulation can be of poor quality and extremely refined since it is not used directly to approximate the solution. The coarse-scale element-free basis can be adapted through refinement or coarsening without the need to alter the fine-scale triangulation or for other geometric considerations. The element-free basis functions are constructed using a traditional moving-least-squares procedure, but the initial weighting functions are constructed using manifold geodesics for general applicability to non-convex domains. A novel coarse-scale quadrature scheme is also developed for integrating the weak form of the governing PDE. Applications in transient, large-deformation, elasto-plasticity are presented to demonstrate the overall method. While
developed for solving PDEs in H 1 , the methodology should be
applicable for solving PDEs in H(div) and H(curl) as well.