Subject: Fast Solvers and Adaptive Hight-Order FEM in Elastoplasticity
Author: Dipl.-Ing. Peter Gruber
Supervision: O.Univ.Prof. Dipl.-Ing. Dr. Ulrich Langer
External referee: Prof. Dr.-Ing. habil. Alexander Düster (Universität Hamburg-Harburg)

This work is concerned with the numerical solution to elastoplastic problems. Since the whole class of all possible elastoplastic problems is far too large for a common treatment, we restrict ourselves to the investigation of problems which are geometrically linear (strain and displacements are related linearly) and which are quasistatic. Further, the isotropic and homogeneous material should obey the Prandtl-Reuß flow, and a linear isotropic hardening principle.

In the first part of this work the pseudo time variable is discretized and a minimization problem in the so called primal formulation is derived. After elimination of the stresses, the only unknown variables are the displacements, plastic strain, and hardening parameters. Both the plastic strain and hardening parameters may be determined exactly and pointwise in dependence on the displacement field. These dependencies, as well as the minimization functional itself, are not differentiable. Surprisingly, after substitution of the exact minimizers (with respect to the plastic strain and the hardening parameters), a new minimization functional, depending now smoothly on the displacement, is obtained. The first derivative of the (strictly convex) functional is known explicitly, thus the solution to the problem is given by the root of this first derivative.

This can be achieved by a Newton or Newton-like method. It turns out, that the second derivative of the minimization functional does not exist. However, the recently developed concept of slant differentiability serves as a remedy, and the local super-linear convergence rate of the slant Newton method can be rigorously shown under certain assumptions. These assumptions are not needed in the spatially discrete case. In other words, the spatially discrete version of the slant Newton iterates converges locally super-linear without any extra assumptions.

The second part of this work is devoted to specially adapted choices of the spatial discretization, which is accomplished by the widely known Finite Element Method (FEM) of low and high order (hp-FEM). While low order FEM is used in regions where the solution has low regularity, the use of high order FEM speeds up the convergence in regions where the solution has high regularity. Several strategies for determining the corresponding regions (for using low or high order FEM) are discussed. Particularly a very new strategy, the boundary concentrated FEM (BC-FEM), or more precisely, a zone concentrated FEM (ZC-FEM), is applied to elastoplastic problems.

The elastoplastic solver, in combination with several adaptive hp-FEM strategies, has been developed within the software framework NETGEN/NGSolve. Numerous experiments affirm the theoretical results of this work, and provide an extent overview regarding the several techniques for spatial discretization.

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