High order numerical methods for a unified theory of fluid and solid mechanics

This dissertation is a contribution to the development of a unified model of continuum mechanics, describing both fluids and elastic solids as a general continua, with a simple material parameter choice being the distinction between inviscid or viscous fluid, or elastic solids or visco-elasto-plastic media. Additional physical effects such as surface tension, rate-dependent material failure and fatigue can be, and have been, included in the same formalism. The model extends a hyperelastic formulation of solid mechanics in Eulerian coordinates to fluid flows by means of stiff algebraic relaxation source terms. The governing equations are then solved by means of high order ADER Discontinuous Galerkin and Finite Volume schemes on fixed Cartesian meshes and on moving unstructured polygonal meshes with adaptive connectivity, the latter constructed and moved by means of a in- house Fortran library for the generation of high quality Delaunay and Voronoi meshes. Further, the thesis introduces a new family of exponential-type and semi- analytical time-integration methods for the stiff source terms governing friction and pressure relaxation in Baer-Nunziato compressible multiphase flows, as well as for relaxation in the unified model of continuum mechanics, associated with viscosity and plasticity, and heat conduction effects. Theoretical consideration about the model are also given, from the solution of weak hyperbolicity issues affecting some special cases of the governing equations, to the computation of accurate eigenvalue estimates, to the discussion of the geometrical structure of the equations and involution constraints of curl type, then enforced both via a GLM curl cleaning method, and by means of special involution-preserving discrete differential operators, implemented in a semi-implicit framework. Concerning applications to real-world problems, this thesis includes simulation ranging from low-Mach viscous two-phase flow, to shockwaves in compressible viscous flow on unstructured moving grids, to diffuse interface crack formation in solids.