This thesis is concerned with shape optimization problems under non-linear PDE (partial differential equation) constraints. We give a brief introduction to shape optimization and recall important concepts such as shape continuity, shape derivative and the shape differentiability. In order to review existing methods for proving the shape differentiability of PDE constrained shape functions a simple semi-linear model problem is used as constraint. With this example we illustrate the conceptual limits of each method. In the main part of this thesis a new theorem on the differentiability of a minimax function is proved. This fundamental result simplifies the derivation of necessary optimality conditions for PDE constrained optimization problems. It represents a generalization of the celebrated Theorem of Correa-Seeger for the special class of Lagrangian functions and removes the saddle point assumption. Although our method can also be used to compute sensitivities in optimal control, we mainly focus on shape optimization problems. In this respect, we apply the result to four model problems: (i) a semi-linear problem, (ii) an electrical impedance tomography problem, (iii) a model for distortion compensation in elasticity, and finally (iv) a quasi-linear problem describing electro-magnetic fields. Next, we concentrate on methods to minimise shape functions. For this we recall several procedures to put a manifold structure on the space of shapes. Usually, the boundary expression of the shape derivative is used for numerical algorithms. From the numerical point of view this expression has several disadvantages, which will be explained in more detail. In contrast, the volume expression constitutes a numerically more accurate representation of the shape derivative. Additionally, this expression allows us to look at gradient algorithms from two perspectives: the Eulerian and Lagrangian points of view. In the Eulerian approach all computations are performed on the current moving domain. On the other hand the Lagrangian approach allows to perform all calculations on a fixed domain. The Lagrangian view naturally leads to a gradient flow interpretation. The gradient flow depends on the chosen metrics of the underlying function space. We show how different metrics may lead to different optimal designs and different regularity of the resulting domains. In the last part, we give numerical examples using the gradient flow interpretation of the Lagrangian approach. In order to solve the severely ill-posed electrical impedance tomography problem (ii), the discretised gradient flow will be combined with a level-set method. Finally, the problem from example (iv) is solved using B-Splines instead of level-sets.
|School:||Technische Universitaet Berlin (Germany)|
|Source:||DAI-C 81/1(E), Dissertation Abstracts International|
|Keywords:||Shape optimization problems|
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