## Nonsmooth Multi-Level Optimization Algorithms for Energetic Formulations of Finite-Strain Elastoplasticity

### Description

Energetic formulations of finite-strain elastoplasticity are an instance of the general theory of rate-independent systems introduced by Mielke and Roubícek. They generalize the primal formulation of small-strain elastoplasticity, where the variables are the displacements, plastic strain, and possibly hardening variables. As they do not involve derivatives, nonsmooth phenomena can be modeled in a particularly elegant way.

In the energetic formulation, time-discrete elastoplastic problems are sequences of minimization problems, which makes them amenable to optimization algorithms. The increment minimization problems combine various difficulties: They are highly nonlinear, nonconvex and nonsmooth, and some of the independent variables (the plastic strain P) take values in a Lie group (typically SL(d)). On the positive side, after discretization the nonsmooth terms are block-separable, i.e., they can be written as sums of nonsmooth functions with small disjoint sets of independent variables. This fact can be exploited by optimization algorithms.

In this project we plan to develop efficient optimization solvers for energetic formulations of finite-strain elastoplasticity. Motivated by the specific problem structure we will use proximal Newton methods, which reduce the given nonconvex nonsmooth problems to sequences of convex, but still nonsmooth subproblems. Then, these subproblems are solved efficiently with the help of a nonsmooth multigrid method. This overcomes a well-known limitation of proximal Newton solvers, which typically lack efficient solvers for the subproblems. We study the new proximal Newton algorithms both in an algebraic setting and in function spaces.

We will investigate two alternative approaches for restricting the plastic strains P to SL(d). On the one hand, we will consider the strains P as elements of the Euclidean space of d-by-d matrices, and subject them to the nonlinear equality constraint det P = 1. For this formulation we will construct nonsmooth composite step methods. As a complementary approach, we will generalize the multilevel proximal Newton methods to the setting of optimization problems posed on manifolds. The relative merits and shortcomings of these approaches will be compared in a series of benchmarks.

### Publications

Bastian Pötzl, Anton Schiela, Patrick Jaap: Second Order Semismooth Proximal Newton Methods in Hilbert Spaces, Computational Optimization and Applications, 2022 (SPP1962-158).

### Preprints

Bastian Pötzl, Anton Schiela, Patrick Jaap: Inexact Proximal Newton Methods in Hilbert Spaces (SPP1962-192, 04/2022, [bib])

Bastian Pötzl, Anton Schiela, Patrick Jaap: Second Order Semismooth Proximal Newton Methods in Hilbert Spaces (SPP1962-158, 03/2021, [bib])