Shape optimisation for a class of semilinear variational inequalities with applications to damage models
- Heinemann, Christian
- Sturm, Kevin
2010 Mathematics Subject Classification
- 49K40 49J27 49J40 49Q10 35J61 49K20 74R05 74B99
- shape optimisation, semilinear elliptic variational inequalities, optimisation problems in Banach spaces, obstacle problems, damage phase field models, elasticity
The present contribution investigates shape optimisation problems for a class of semilinear elliptic variational inequalities with Neumann boundary conditions. Sensitivity estimates and material derivatives are firstly derived in an abstract operator setting where the operators are defined on polyhedral subsets of reflexive Banach spaces. The results are then refined for variational inequalities arising from minimisation problems for certain convex energy functionals considered over upper obstacle sets in $H^1$. One particularity is that we allow for dynamic obstacle functions which may arise from another optimisation problems. We prove a strong convergence property for the material derivative and establish state-shape derivatives under regularity assumptions. Finally, as a concrete application from continuum mechanics, we show how the dynamic obstacle case can be used to treat shape optimisation problems for time-discretised brittle damage models for elastic solids. We derive a necessary optimality system for optimal shapes whose state variables approximate desired damage patterns and/or displacement fields.