49Q10 Optimization of shapes other than minimal surfaces [See also 90C90]
Shape optimization is of interest in many fields of application. In particular, shape optimization problems arise frequently in technological processes which are modelled by partial differential equations (PDEs). In a lot of practical circumstances, the shape under investigation is parametrized by a finite number of parameters, which, on the one hand, allows the application of standard optimization approaches, but, on the other hand, unnecessarily limits the space of reachable shapes. Shape calculus presents a way to circumvent this dilemma. However, so far shape optimization based on shape calculus is mainly performed using gradient descent methods. One reason for this is the lack of symmetry of second order shape derivatives or shape Hessians. A major difference between shape optimization and the standard PDE constrained optimization framework is the lack of a linear space structure on shape spaces. If one cannot use a linear space structure, then the next best structure is a Riemannian manifold structure, in which one works with Riemannian shape Hessians. They possess the often sought property of symmetry, characterize well-posedness of optimization problems and define sufficient optimality conditions. In general, shape Hessians are used to accelerate gradient-based shape optimization methods. This thesis deals with shape optimization problems constrained by PDEs and embeds these problems in the framework of optimization on Riemannian manifolds to provide efficient techniques for PDE constrained shape optimization problems on shape spaces. A Lagrange-Newton and a quasi-Newton technique in shape spaces for PDE constrained shape optimization problems are formulated. These techniques are based on the Hadamard-form of shape derivatives, i.e., on the form of integrals over the surface of the shape under investigation. It is often a very tedious, not to say painful, process to derive such surface expressions. Along the way, volume formulations in the form of integrals over the entire domain appear as an intermediate step. This thesis couples volume integral formulations of shape derivatives with optimization strategies on shape spaces in order to establish efficient shape algorithms reducing analytical effort and programming work. In this context, a novel shape space is proposed.
Large scale non-parametric applied shape optimization for computational fluid dynamics is considered. Treating a shape optimization problem as a standard optimal control problem by means of a parameterization, the Lagrangian usually requires knowledge of the partial derivative of the shape parameterization and deformation chain with respect to input parameters. For a variety of reasons, this mesh sensitivity Jacobian is usually quite problematic. For a sufficiently smooth boundary, the Hadamard theorem provides a gradient expression that exists on the surface alone, completely bypassing the mesh sensitivity Jacobian. Building upon this, the gradient computation becomes independent of the number of design parameters and all surface mesh nodes are used as design unknown in this work, effectively allowing a free morphing of shapes during optimization. Contrary to a parameterized shape optimization problem, where a smooth surface is usually created independently of the input parameters by construction, regularity is not preserved automatically in the non-parametric case. As part of this work, the shape Hessian is used in an approximative Newton method, also known as Sobolev method or gradient smoothing, to ensure a certain regularity of the updates, and thus a smooth shape is preserved while at the same time the one-shot optimization method is also accelerated considerably. For PDE constrained shape optimization, the Hessian usually is a pseudo-differential operator. Fourier analysis is used to identify the operator symbol both analytically and discretely. Preconditioning the one-shot optimization by an appropriate Hessian symbol is shown to greatly accelerate the optimization. As the correct discretization of the Hadamard form usually requires evaluating certain surface quantities such as tangential divergence and curvature, special attention is also given to discrete differential geometry on triangulated surfaces for evaluating shape gradients and Hessians. The Hadamard formula and Hessian approximations are applied to a variety of flow situations. In addition to shape optimization of internal and external flows, major focus lies on aerodynamic design such as optimizing two dimensional airfoils and three dimensional wings. Shock waves form when the local speed of sound is reached, and the gradient must be evaluated correctly at discontinuous states. To ensure proper shock resolution, an adaptive multi-level optimization of the Onera M6 wing is conducted using more than 36, 000 shape unknowns on a standard office workstation, demonstrating the applicability of the shape-one-shot method to industry size problems.
