A position-based Space-Time formulation for geometrically nonlinear problems

Resumo

Space-Time finite element methods has been developed over years for solving a series of time-dependent

problems like elastodynamics, fluid-structure interaction, fluid flows, advection-diffusion equations and heat trans-
fer problems. The core of this approach is the treatment of time as a dimension of the finite element problem,

leading to space-time finite element discretizations. Single-field or two-fields formulation are possible, where the
first one uses only displacement as unknowns, while the second uses both displacements and velocities as variables.
Some challenges that appear in the Space-Time FEM are the increased size of the equation systems as the precision

in time is increased and the 4D meshes representation. Nevertheless, this approach can lead to higher order accu-
racy in time and direct dynamic spatial re-meshing. On the other hand, time-marching methods are well-known

numerical time integrators that have been applied to discrete systems of differential equations obtained from dif-
ferent spatial discretization techniques, including FEM. Most of them deal with approximations for displacements

and velocities, and the discrete system of differential equations are solved at each discrete time level taking into
account the variable fields from the last time step and the current boundary conditions. Moreover, they can be
formulated to present unconditional stability, to present controlled dissipative properties and different orders of

accuracy. As a disadvantage, dynamic re-meshing procedures are not directly feasible, as it demands the projec-
tion of past time step fields over the new mesh, including projection errors. This work presents a position-based

Space-Time FEM formulation for two-dimensional solids with large displacements, using a total Lagrangian de-
scription. This formulation is naturally isoparametric and designed directly over the large displacement assumption

making the geometric non-linearities intrinsically considered. In order to verify the potential of the formulation, a
comparative analysis with the time-marching method alpha-generalized is carried out.