Finite elements in space and time for parallel computing of viscoelastic deformation

Abstract

The efficient parallel computation of time dependent problems, e.g. parabolic problems of viscoelastic material deformation, underlies the “bottleneck” of the serial approach in time. The usual method of lines, also called semidiscretization, leads to an iterative calculation in time, i.e. a sequential solution of the spatial problems for all time steps. Due to that, only one spatial problem can be solved in parallel at a certain time step. For an efficient parallelization, it is necessary to compute the whole problem in a distributed way. Furthermore, both h- and p-adaptive approximation should be possible in time and space.

For these purposes, in addition to the spatial FE-discretization, a continuous finite element discretization in time is used. Thus, one obtains a total algebraic equation system in space and time, whose solution has to be parallelized efficiently, and h- and p-adaptivity in time and space within the frame of the overall Galerkin-process has to be realized.

The present paper treats symmetric and non-symmetric formulations of two different viscoelastic three-parameter models. The new numerical approach concerns first for the Malvern Model (generalized Maxwell Model). The numerical examples for the new non-symmetric formulation and the traditional semidiscretization show the advantage (with respect to convergence to the problem solution) of the new finite element approach with simultaneous discretizations in time and space. But the algebraic systems are bad-conditioned such that parallel iterative solvers with various preconditions are not efficient.

The symmetric formulation for the Malvern Model can be obtained for the one-dimensional case only. A numerical example showed the good iterative solvability of the symmetric formulation. Therefore, in order to obtain a symmetric formulation in the 3D-case the generalized Kelvin–Voigt Model was chosen as an alternative one.

It should be mentioned that the numerical examples show both the effectiveness of parallel computation and the efficiency of h- and p-adaptation (p-adaptation yields the higher rate of convergence than h-adaptation).