Professor Raúl A. Radovitzky

Development of numerical methods

Professor Radovitzky is also developing novel numerical methods aimed at addressing the limitations of existing computational models, including the first discontinuous Galerkin formulation for large deformation of solids. These methods provide convenient means of handling physical discontinuities including shocks, slip bands and fracture as well as constrained problems, e.g. incompressibility and higher order continuity constraints.

This area of research has the potential of addressing some fundamental numerical difficulties in multiscale modeling. One instance arises in the family of non-local or gradient theories of plasticity which have not received an adequate numerical treatment so far. These models establish a clear connection between the dislocation-mediated lattice curvature and the macroscopic deformation. They also naturally introduce a length scale in the continuum equations and can therefore describe size-dependent effects. However, these models have seldom been applied to solve boundary value problems perhaps due to the difficulties introduced by their non-local character leading to higher-order theories. Professor Radovitzky’s work addresses these issues by extending his discontinuous Galerkin formulation to higher-order theories as a means of enforcing weakly the required inter-element C1 continuity and, in particular, developing a strain gradient crystal plasticity model based on his discontinuous Galerkin formulation.

One of the main problems thwarting the myriad of existing constitutive models describing complex material behavior is the lack of appropriate mathematical frameworks providing theoretical guarantees of existence of solutions when these models are used in applications involving boundary value problems. A sufficient condition guaranteeing the sequential weak lower semi-continuity of the energy functional and, thus the existence of infimizing sequences in boundary value problems is the polyconvexity of the strain energy density function. Paradoxically, this rigorous framework has not pervaded into the development of concrete constitutive models, barring a few exceptions.

Professor Radovitzky’s research has developed the first polyconvex anisotropic model for materials with cubic symmetry.  In conjunction with equations of state obtained from ab initio quantum mechanics calculations and the constitutive model for nanocrystals Professor Radovitzky developed, this model is ideally suited for describing the response of nanocrystalline metals under shock loading. This type of model will be used in the assessment of metallic alloys for the National Ignition Facility (NIF).

The NIF Project in Livermore, CA is the largest, most powerful laser in the world and is part of the U.S. Department of Energy’s National Nuclear Security Administration Defense Programs and LLNL missions of ensuring that the nation’s nuclear weapons remain safe, secure, and reliable.

Professor Radovitzky’s research has also developed mesh healing and optimization algorithms enabling the Lagrangian finite element analysis of problems involving unconstrained plastic flows. This capability has demonstrated for the first time the Lagrangian simulation of deep oblique penetration of ductile targets by kinetic-energy rods.  HealMesh, the software library resulting from this effort, has been made into a commercial product and is being used by researchers at the Army Research Laboratory and at the Army Natick Soldier Center.