Iterative solvers within sequences of large linear systems in non‐linear structural mechanics

Abstract

This article treats the computation of discretized constitutive models of evolutionary-type (like models of viscoelasticity, plasticity, and viscoplasticity) with quasi-static finite elements using diagonally implicit Runge-Kutta methods (DIRK) combined with the Multilevel-Newton algorithm (MLNA). The main emphasis is on promoting iterative methods, as opposed to the more traditional direct methods, for solving the non-symmetric systems which occur within the DIRK/MLNA. It is shown that iterative solution of the arising sequences of linear systems can be substantially accelerated by various techniques that aim at sharing part of the computational effort throughout the sequence. In this way iterative solution becomes attractive at clearly lower dimensions than the dimensions where direct solvers start to fail for memory reasons. The applications are related to small strain viscoplasticity of a smooth constitutive model for plastics and a finite strain plasticity model with non-linear kinematic hardening developed for metals.

Monolithic and partitioned coupling schemes for thermo-viscoplasticity

“…According to the proposal in [48,28,38], we perform a linear extrapolation of the last equilibrium states, in this case adapted to the requirements of thermo-mechanically coupled problems,…”

High‐order finite elements compared to low‐order mixed element formulations

“…u (0) n+1 = u n is a frequent choice, see [1,2,52,55], where u n symbolizes the final displacements of the last equilibrium state and load-step t n . We investigate a linear extrapolation of the last two known solutions (nodal displacements) for h-elements in [23], i.e.…”

“…is chosen. In [23] it turned out that not only the number of iterations drastically decreases but also that non-converging problems using u (0) n+1 = u n lead to only a few iterations when (10) is chosen. This has to be investigated for p-version finite elements as well, since the unknowns of the chosen high-order polynomials are not displacements anymore but polynomial coefficients.…”

“…In this article, a p-version finite element formulation based on hierarchic shape functions whose bases are integrated Legendre-polynomials, see [49] and [13], is not only compared with linear and quadratic hexahedral or tetrahedral elements but also with mixed hexahedral elements such as Q1P0-and Q2P1-elements proposed by [45] or [46]. Moreover, some improvements have been selected to accelerate the computation, such as estimating the starting vector of the Newton method: see, for example, [23], to apply the Newton-Chord method: see [28] or, in the context of finite elements, the modified Newton method, see [1]. This has a particularly beneficial effect on the p-version FEM with regard to the computational effort required to assemble element matrices.…”

Preconditioner updates for solving sequences of linear systems in matrix-free environment

“…When applying the Multilevel-Newton algorithm, every time-step contains an inner loop that requires the solution of nonlinear systems, which in turn leads to a sequence of linear systems. For more details on the parameters of the material and of the Multilevel-Newton algorithm which were used, we refer to the description of the first application in [48]. We consider here a sequence of linear systems from a randomly chosen time-step in the middle of the simulation process.…”