This thesis develops a structure-preserving discretisation for modelling hyperelastic flow in a fully Lagrangian and variational flat space-time in R³ (two-dimensional in space, and one-dimensional in time) via the Mimetic Spectral Element Method (MSEM). Continuum mechanics is used as the primary modelling framework, and its links to differential geometry and duality are made explicit with examples. Deformations are treated as smooth mappings between configurations, with stress, and hence strain, measures understood as metric-dependent mappings. The natural construction of MSEM additionally transfers the symplectic nature of the governing equations onto a high-order discrete mesh, such that Noether invariants, i.e. conservation laws, present in the continuous formulation are upheld at the discrete level.

A generalised, isotropic hyperelastic formulation is constructed as a non-linear, deformation-dependent discrete Hodge operation. Extensions to multiple discrete space-time elements in a spatial sense are formulated through means of hybridisation. Anderson acceleration for fixed-point iterations is utilised as a robust, super-linearly-convergent alternative to stock fixed-point solvers for highly non-linear problems. Numerical experiments demonstrate convergent and conservative behaviour on non-curved geometries under linear mappings, as well as strongly-prescribed and weak boundary conditions. Additionally, the time-dependent solver admits a steady limit, which is used to validate the Hodge operation through the Cook's Membrane benchmark in plane-strain. These results support the notion that hyperelastic flow can indeed be modelled in a structure-preserving and multi-symplectic manner using a Lagrangian formulation on space-time manifolds in R³. ... Read more