Development of a cell centred upwind finite volume algorithm for a new conservation law formulation in structural dynamics

A novel computational methodology is presented for the numerical analysis of fast transient dynamics phenomena in large deformations. The new mixed formulation can be written in the form of a system of first order conservation laws, where the linear momentum, the deformation gradient tensor and the total energy of the system are used as main conservation variables, leading to identical convergence patterns for both displacements and stresses. A cell centred Finite Volume Method is utilised to carry out the spatial discretisation. Naturally, discontinuity of the conservation variables across control volume interfaces leads to a Riemann problem, whose approximate solution is derived. A suitable numerical interface flux is evaluated by means of the Rankine–Hugoniot jump conditions. We take advantage of the conservative formulation to introduce a Total Variation Diminishing shock capturing technique to improve dramatically the performance of the algorithm in the vicinity of sharp solution gradients. A series of numerical examples will be presented in order to demonstrate the capabilities of the scheme. The new formulation is proven to be very efficient in nearly incompressible and bending dominated scenarios in comparison with classical finite element displacement-based approaches. The proposed numerical framework provides a good balance between accuracy and speed of computation.