Optimal Shape Design for Brittle Materials undergoing Damage: Approximation Schemes and Optimality Systems

• Applicant: Dr. Livia Betz
• Subject Area: Mathematics
• Term: since 2026
• Project identifier: Deutsche Forschungsgemeinschaft (DFG) - Project number 575166624

The proposed project aims at establishing necessary optimality conditions in qualified form for optimal design problems governed by damage models for brittle materials. The proposed problem falls into the vast class of shape and topology optimization problems. Such problems are highly non-convex and are related to optimal control problems, the control variable being the unknown geometry itself. While optimal design problems are usually addressed by methods involving geometrical variations, we intend to use a different, functional variational approach. The transition from geometrical variations to functional ones gives rise to a "classical" control problem (P) where the underlying admissible set consists of a certain family F of shape functions/parametrizations. By proceeding like this, the original shape optimization problem becomes more amenable to methods from optimal control theory, however it still preserves the non-convexity of the admissible set. The mathematical model (state system) consists of a non-linear equation coupled with a viscous evolutionary variational inequality. Letting the unknown geometry aside, the formulation of the governing state equations will give rise to challenges owed to the highly non-smooth character and the complex non-linear nature of the problem. In order to be able to achieve our goal, we propose to proceed as follows: 1. We introduce an approximating control problem for (P), i.e., for the original design problem, where the state system is extended to a fixed reference domain. By proceeding like this, the variable character of the geometry is out of the picture. Moreover, we shall dispense of the non-convexity of the admissible set F, by replacing it with a convex subset of a Hilbert space of functions. 2. We prove the viability of the aforementioned convex approximation scheme. Here we maintain an optimal control point of view that should allow us later to obtain optimality systems for (P). The main result in this step shall be that to each optimal shape we can associate a sequence of global minimizers of the approximating control problem. 3. We derive optimality systems for the main optimization problem (P), by first establishing necessary conditions for the control of its approximating problem and then letting the approximating parameter vanish therein. This should lead to optimality conditions in qualified form consisting of an adjoint system, a gradient inequality and a subdifferential inclusion. These will then be transferred (back) to the original optimal design problem.

• DFG Programme: Research Grants