# Adaptive Strategies for hp-FEM in Elastoplasticity

## Dipl.-Ing. Peter Gruber

**April 21, 2009, 11:45 a.m. HS 14**

This work is concerned with the numerical solution of variational problems in elastoplasticity. Since the whole class of all possible elastoplastic problems is too large for a common treatment, we restrict ourselves to the investigation of problems which are quasi-static, and where strain and displacement are related linearly. Further, the homogeneous and isotropic material should obey the Prandtl-Reuß ﬂow and a linear hardening principle. After an implicit Euler discretization in time, it is possible to derive a one-time step minimization problem, which can be solved by a Newton-like method.

Each iteration step represents a linear boundary value problem, which has to be discretized in space and approximately solved on the computer. Due to regularity reasons, the application of a Finite Element Method (FEM) of low and high order ($hp$-FEM) looks promising. Roughly speaking, low order FEM is used in regions of the domain where the solution is expected to have low regularity (plastic zones), while the use of high order FEM speeds up the convergence of the approximate solution in regions where the solution has high regularity (elastic zones).

We discuss a several diﬀerent $hp$-adaptive Strategies, particularly a technique is applied, where the regularity of the solution is estimated by analyzing the expansion of the Finite Element solution with respect to an $L_2$-orthogonal polynomial basis. Also the application of a Boundary Concentrated FEM to elastoplastic problems is discussed and numerically tested.