Multiscale uncertainty quantification in solid mechanics using a posteriori error-estimation techniques with applications to additively-manufactured structures / Polyhedral finite elements using harmonic shape functions for nonlinear solid mechanics

Part 1: Two fundamental sources of error and uncertainty in macroscale solid-mechanics modeling are (1) the use of an approximate material model that represents in a mean sense the complex nonlinear processes occurring at the microscale, and (2) the use of homogenization theory with implicit approximations such as a separation-of-scales and the existence of a welldefined representative volume element. Macroscopic material models are typically in error when exercised outside of the calibration regime, and the assumptions in homogenization theory can be violated for additively-manufactured (AM) metallic structures. In order to quantify these errors and uncertainties for solid mechanics, we adopt a posteriori model-form error-estimation techniques. A simple constitutive model is maintained at the macroscale, such as isotropic elasticity or von Mises plasticity, and the errors in engineering quantities of interest are assessed in a post-processing step using a localization process and error bounds. Also, for a given loading scenario, the apparent macroscale properties can be adapted to minimize the goaloriented error and uncertainty.

Part 2: For geometrically complex parts and systems, the time required to develop an analysis capable finite element mesh can still take days to months to develop. Advanced tetrahedral meshing tools can alleviate this burden, but the development of robust and efficient tetrahedral finite-element formulations for applications in large-deformation solid-mechanics is still an active area of research. The recent development of general polyhedral formulations for solid mechanics offers new discretization opportunities. One approach is to use an existing tetrahedral mesh, but then subdivide each tetrahedron using partial rectification to obtain a polyhedral mesh. Two types of polyhedra are formed: (1) an aggregation of sub-tetrahedra attached to the original nodes, and (2) a polyhedron with 12 vertices and 8 faces. Several approaches may then be used to define the shape functions (e.g. using harmonic or maximum entropy barycentric coordinates) and quadrature schemes for the new polyhedra in order to obtain a consistent and stable finite element formulation.

Joe Bishop received his Ph.D. in Aerospace Engineering from Texas A&M University in 1996. His graduate research was in the general areas of the mechanics of composite materials and the mechanisms and mechanics of material damping. From 1997 to 2004 he worked in the Synthesis & Analysis Department of the Powertrain Division of General Motors Corporation, performing thermal-structural analysis of internal combustion engines with a focus on predicting high-cycle fatigue performance of the base engine. He joined Sandia National Laboratories in 2004 in the Engineering Sciences Center. He has worked on diverse research topics such as impact and penetration, pervasive fracture and fragmentation modeling, polyhedral finite elements, geologic CO2 sequestration, and metal-additive manufacturing. His current research interests include multiscale modeling in support of metal-additive manufacturing, experimental methods and computational techniques for determining residual-stress fields, polyhedral finite element formulations and applications, model-form error estimation techniques in finite-element analysis, meshfree technologies for shock-physics applications, and second-generation wavelets.