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Abstract

It is well known among practitioners that the numerical solution of shape optimization problems constrained by partial differential equations (PDEs) often exhibits several difficulties. In particular, when the PDE is discretized by a finite element method, and the underlying mesh is used to represent the shape of the domain to be optimized directly, one often experiences a degeneracy of the mesh quality as the optimization progresses. The degeneracy manifests itself in some of the mesh cells thinning in the sense that at least one of its heights approaches zero.

Various techniques have been developed to circumvent this major obstacle in computational shape optimization. This thesis offers a new perspective on understanding the particularities of PDE-constrained shape optimization problems when they are treated under the discretize-then-optimize paradigm. We focus on two-dimensional problems, where the PDE is discretized using a finite element method, and the underlying mesh represents the discrete shape. Under these considerations, we make three main contributions. First, we study the set of all possible configurations of node positions a mesh of a given connectivity can attain. Then, using the language of simplicial complexes, we provide theoretical evidence that this set is a smooth manifold, and we term it the \emph{manifold of planar triangular meshes}.

Secondly, we construct two complete Riemannian metrics for the manifold of planar triangular meshes, which avoid all possible self-intersections on a mesh. Moreover, they enjoy certain invariance properties under rigid body motions, and the latter is additionally invariant under uniform mesh refinements. In practice, endowing the manifold of triangular meshes with these metrics allows us to update the meshes following geodesics in any direction and as long as we want without jeopardizing their quality. This property can also be understood as degenerate meshes being infinitely far away from any regular mesh in terms of their geodesic distance. Finally, alongside the newly proposed notion of the complete manifold of planar triangular meshes, we focus on the theoretical and computational aspects of discretized PDE-constrained shape optimization problems. We provide numerical evidence that such problems generally possess no solutions within the manifold of planar triangular meshes, even when the shape functional is bounded below. To overcome this drawback, we introduce a penalty functional which, briefly speaking, controls the mesh quality. When added to the shape functional, it renders well-posed discrete shape optimization problems, \ie, they possess at least one globally optimal solution. Subsequently, we solve the penalized problem using four different variants of the Riemannian steepest descent method. These variants depend on the metric used to transform cotangent vectors into tangent vectors and the metric used to update the meshes. Our numerical experiments reveal that using the proposed complete metrics to navigate the manifold is practically convenient since the optimization scheme does not need explicit monitoring of the mesh quality and can take arbitrarily large steps. Unfortunately, exploiting the properties of the complete metric is computationally challenging since the numerical integration of the respective geodesics is prohibitively expensive. However, we demonstrate that using the proposed Riemannian metric in gradient methods is still beneficial, even when combined with the inexpensive Euclidean retraction. Furthermore, the numerical evidence suggests gradient methods perform well in absence of the mesh quality penalty term when utilizing the complete metric.