## Invited Session Wed.1.MA 415

#### Wednesday, 10:30 - 12:00 h, Room: MA 415

**Cluster 19: PDE-constrained optimization & multi-level/multi-grid methods** [...]

### Optimization applications in industry III

**Chair: Dietmar HÃ¶mberg**

**Wednesday, 10:30 - 10:55 h, Room: MA 415, Talk 1**

**Roland Herzog**

Optimal control of elastoplastic processes

**Coauthors: Christian Meyer, Gerd Wachsmuth**

**Abstract:**

Elastoplastic deformations are the basis of many industrial production techniques, and their optimization is of significant importance.

We consider mainly the (idealized) case of infinitesimal strains as well as linear kinematic hardening.

From a mathematical point of view, the forward system in the stress-based form is represented by a time-dependent variational inequality of mixed type.

Its optimal control thus leads to an MPEC (mathematical program with equilibrium constraints) or an equivalent MPCC (mathematical program with complementarity constraints), both of which are challenging for general-purpose nonlinear optimization codes.

In this presentation, we therefore address taylored algorithmic techniques for optimization problems involving elastoplastic deformation processes.

**Wednesday, 11:00 - 11:25 h, Room: MA 415, Talk 2**

**Anton Schiela**

An adaptive multilevel method for hyperthermia treatment planning

**Abstract:**

The aim of hyperthermia treatment as a cancer therapy is to damage deeply seated tumors by heat. This can be done regionally by a microwave applicator and gives rise to the following optimization problem: "Find antenna parameters, such that the damage caused to the tumor is maximized, while healthy tissue is spared''. Mathematically, this is a PDE constrained optimization problem subject to the time-harmonic Maxwell equations, which govern the propagation of the microwaves, and the bio heat transfer equation, a semi-linear elliptic equation, which governs the heat distribution in the human body. Further, upper bounds on the temperature in the healthy tissue are imposed, which can be classified as pointwise state constraints.

In this talk we consider a function space oriented algorithm for the solution of this problem, which copes with the various difficulties. The state constraints are tackled by an interior point method, which employs an inexact Newton corrector in function space for the solution

of the barrier subproblems. Herein, discretization errors are controlled by a-posteriori error estimation and adaptive grid refinement.

**Wednesday, 11:30 - 11:55 h, Room: MA 415, Talk 3**

**Michael Stingl**

Material optimization: From theory to practice

**Coauthors: Fabian Schury, Fabian Wein**

**Abstract:**

A two-scale method for the optimal design of graded materials is presented. On the macroscopic scale we use a variant of the so called free material optimization (FMO) approach, while on the microscopic scale the FMO results (a set of material tensors) are interpreted as periodic two-phase materials.

In FMO an elastic body is optimizied w.r.t. given forces and boundary conditions. Hereby the material properties are allowed to vary from point to point. In contrast to the standard model, continuity of the variation of the material properties is enforced. Moreover constraints on the symmetry of each tensor (e.g., orthotropy, cubic symmetry) as well as special bounds on the stiffness of the material are added. The latter constraints are chosen compatible with the choice of the material on the microscopic level. Approximate solutions methods for this modified FMO problem are presented.

In a discretized finite element setting the result of the macroscopic problem is a set of material tensors. Depending on the macroscopic constraints, these tensors are either directly interpreted as periodic microstructures or accessible through an inverse homogenization approach.