A vertex centred finite volume method for solid dynamics.

With the increase of the computational power over the last decades, computational solid dynamics has become a major field of interest in many industrial applications (as for example aerospace, automotive industry, biomedical engineering or manufacturing). Traditionally, these types of simulations have been carried out using the Finite Element Method (FEM) in conjunction with displacement based formulations, where the displacements are treated as the main problem variables. In the context of solid mechanics, the use of low order (linear) tetrahedral (or triangular) elements is always preferred, primarily due to the complexity of the constitutive models involved (i.e., the number of evaluations needs to be kept as low as possible) as well as the automatic tetrahedral (or triangular) mesh generators available. However, the combination of low order elements with FEM displacement based formulations presents a series of shortcomings, namely: one order of accuracy less for stresses (or strains) than for displacements (or velocities), poor behaviour in bending dominated scenarios, volumetric and shear locking or the appearance of spurious pressure modes. Furthermore, the time integration is usually performed using Newmark integrators, which tend to introduce high frequency noise in the vicinity of sharp gradients. Recently, a new Lagrangian mixed methodology has been presented for the simulation of fast transient dynamics problems. This methodology is in the form of a system of first order conservation laws, where the linear momentum, p, and the deformation gradient tensor, F, are regarded as the two main conservation variables. When thermo-mechanical constitutive models are involved, the formulation is complemented with the first law of thermodynamics (conservation of energy). It has been proven that this formulation circumvents the drawbacks of the low order displacement based FEM methodologies mentioned above. The formulation, presented as a set of conservation laws, allows for standard Computational Fluid Dynamcis (CFD) spatial discretisations. So far, successful implementations have been carried out using cell centred upwind Finite Volume method, two-step Taylor Galerkin, Finite Element Petrov Galerkin (PG), and Hybridizable Discontinuous Galerkin (HDG). The objective of this thesis is to present a new spatial discretisation in order to solve large scale real life problems. To do so, the Jameson-Schmidt-Turkel (JST) scheme, widely know within the CFD community, will be chosen. The JST scheme is a vertex centred finite volume, that combines the use of central differences with an artificial dissipation term. The scheme obtains second order spatial accuracy without the need of linear reconstruction. Furthermore, the artificial dissipation term includes a shock capturing sensor, very suitable in the context of fast dynamics. The scheme can be implemented in an edge-based framework, which combined with the vertex centred storage of the variables results into a computationally efficient scheme. The JST spatial discretisation will be combined with a Total Variation Diminishing (TVD) two-stage Runge-Kutta time integrator. These spatial and temporal discretisations will be adapted to the problem at hand. Specifically, compatibility conditions (involutions) will have to be satisfied by the discrete scheme. Furthermore, numerical corrections will be introduced in order to ensure the conservation of linear and angular momenta. The framework results in a low order computationally efficient solver for solid dynamics, which proves to be very competitive in nearly incompressible scenarios and bending dominated applications. The thesis will present numerical results for one dimension, two dimensions (triangular meshes) and three dimensions (tetrahedral meshes). The problems are chosen in order to prove the order of accuracy, robustness and conservation properties of the algorithm