# Stable high order space-time finite element formulation for large displace- ment elastodynamics

## Palavras-chave:

space-time formulation, positional-based formulation, stable high order time integration, time-marching methods, large displacements## Resumo

Direct time integration methods have been widely used in structural and solid dynamics simulations,

specially in nonlinear problems. Time-marching methods have been applied to discrete systems of differential

equations obtained from different spatial discretization techniques, like finite differences, finite volume, bound-

ary elements and finite elements, with finite elements being currently the most applied method for structural and

solid mechanics. On the other hand, space-time formulations consider time as a dimension of the finite element

discretization, so that the precision in time integration can be increased by using higher order shape functions

in time direction. In this context, this work presents a position-based total Lagrangian space-time finite element

formulation for the solution of two-dimensional elasticity problems with large displacements. By using structured

space-time mesh in time direction, it is possible to divide the space-time domain into space-time slabs, so that such

slabs can be solved progressively with the final nodal positions and velocities from previous slab being applied as

initial conditions to the current one. The adopted space-finite elements are prismatic with a triangular basis corre-

sponding to the spatial discretization and height corresponding to the temporal discretization, so that the space-time

shape functions are given by the product of the Lagrange polynomial shape functions adopted for the triangular

elements of spatial discretization, with Hermite polynomials based shape functions defined along the height of

the prism, for time discretization. The test functions in time direction are modified so that different stability and

precision can be achieved. Through the simulation of selected examples, and the comparison with solutions from

known time-marching methods, the robustness and stability of the proposed formulation is demonstrated.