A novel multi-dimensional Eulerian approach to computational solid dynamics

Abstract

Many problems in solid dynamics involve moving boundaries, finite elastoplastic deformations, and strong non-linear waves. Continuum modelling of such events is difficult on account of these characteristics, and there exist a number of inadequacies in current numerical algorithms. Furthermore, a comprehensive understanding of certain underlying processes is yet to be achieved which places a limit on the derivation of engineering models to simulate these occurrences. Much needed atomistic studies, capable of revealing much about the governing physical processes, remain limited by current computational resources. This thesis is devoted to targeting these difficulties by proposing new continuum numerical schemes and a means of studying both micro- and macro-scale behaviours via a dynamic coupling of continuum mechanics and molecular dynamics theory. Eulerian shock-capturing schemes have advantages for modelling problems involving complex non-linear wave structures and large deformations in solid media. Various numerical methods now exist for solving hyperbolic conservation laws that have yet to be applied to solid dynamics. A three-dimensional finite-volume scheme on fixed grids is proposed for elastoplastic solids. The scheme is based upon the Godunov flux method and thus requires solution of the Riemann problem. Both exact and approximate solutions are proposed for the special case of non-linear elasticity. An implicit algorithm is developed to allow for resolving rate-dependent inelastic deformations. The methods are tested against exact solutions in one-dimension, and symmetrical polar solutions in two- and three-dimensions. To account for multiple immiscible materials it is necessary to include some means of tracking material boundaries within a numerical scheme. A moving grid scheme is a simple means of accommodating transient boundaries. Interface tracking based on the use of level set functions is an attractive alternative for problems with sliding interfaces since it allows discontinuous velocity profiles at the material boundaries whilst employing fixed grids. Both of these methods are explored in the current context. A series of one-dimensional testcases have been carried out that demonstrate the ability of the numerical schemes to accurately resolve complex boundary conditions between interacting free surfaces. Where singularities occur in a system comprising solid materials, atomistic studies are invaluable for achieving a fundamental insight into the governing physical processes. However where non-linear waves are generated, domain size proves to be a limiting factor in achieving solutions free from numerical artifacts. A domain decomposition multi-scale modelling strategy is developed that couples the Eulerian shock capturing scheme with a molecular dynamics solver. The method is demonstrated for one-dimensional testcases involving strong shear waves and multiple components. Attention is devoted to resolving transient wave propagation free from spurious wave reflections through investigation of the numerical parameters.