An unstructured finite volume solver for a new conservation law in fast transient dynamics

Since the advent of computational mechanics, the numerical modelling of fast structural dynamics has been a major field of interest in industry. Traditionally, a Lagrangian formulation is employed for the numerical simulation of these problems and low order spatial interpolation is preferred for computational workload convenience. The well known second order solid dynamics formulation, where the primary variable is the displacement, is typically discretised in space by using the Finite Element Method (FEM) and discretised in the time domain by means of a Newmark (trapezoidal) time integrator. However, it has been reported that the resulting space-time discretised formulation presents a series of shortcomings. From the time discretisation point of view, the Newmark method has a tendency to introduce high frequency noise in the solution field, especially in the vicinity of sharp spatial gradients. From the space discretisation point of view, the use of isoparametric linear finite elements leads to second order convergence in displacements, but only first order convergence for stresses and strains. It is also known that these elements exhibit locking phenomena in incompressible or nearly incompressible scenarios. Recently, a new mixed methodology [1] has been developed in the form of a system of first order conservation laws, where the linear momentum and the deformation gradient tensor are regarded as the two main conservation variables. The current paper presents the discretisation of these equations by using the Jameson-Schmidt-Turkel (JST) scheme. The paper will as well focus on comparing the obtained results against the ones obtained using other methodologies such as cell centred Finite Volume or Two Step Taylor Galerkin, Stream Upwind Petrov Galerkin (SUPG).