Model reduction and level set methods for shape optimization problems

In this work two topics related to mathematical shape optimization are considered. Topological optimization methods need not know the correct topology (number of connected components and possible holes) of the optimal shape beforehand. Shape optimization can be performed by a topological gradient descent algorithm. Computational applications of topological optimization and level set based shape optimization involve the optimal damping of vibrating structures and an inverse problem of reconstructing a shape based on noisy interferogram measurements. For parametric shape optimization with partial differential constraints the model reduction approach of reduced basis methods is considered. In the reduced basis method a basis of snapshot solutions is used to construct a problem-dependent approximation space that has much smaller dimension than the underlying finite element approximations. The state constraints for optimization are then replaced with their reduced basis approximation, leading to efficient shape optimization methods. Computational examples involve the optimal engineering design of airfoils in potential and thermal flow.