Abstract : Shape and topology optimization has seen considerable progress in multiple areas recently, in particular in regards to the understanding of non-smoothness and higher order methods. A driving problem class for these new developments have been inverse and reconstruction problems. The goal of this mini-symposium is to bridge the gap between these developments and connect new algorithmic developments with applications.
To this end, the first session focuses on the progression from non-smooth shape optimization problems over Quasi-Newton methods to H^1 and H^2 schemes, while the second part focuses on novel applications and emerging new areas of shape and topology optimization.
[02707] Geometry Segmentation with Total Variation Regularization
Author(s) :
Lukas Baumgärtner (Humboldt Universität zu Berlin)
Ronny Bergmann (NTNU Trondheim)
Roland Herzog (Uni Heidelberg)
Manuel Weiß (Uni Heidelberg (IWR))
Stephan Schmidt (Humboldt-Universität zu Berlin)
Abstract : The total variation has proven as a useful regularizer for various applications in Inverse imaging and Shape optimization problems. For the task of shape segmentation, we consider two models that combine normal vector data of a discrete surface with a total variation penalty that is evaluated in the assignment space and the label space. We show how to solve the model problems using the Chambolle-Pock algorithm and ADMM.
[03021] Topology optimisation with general dilatations via the topological state derivative
Author(s) :
Phillip Baumann (TU Wien)
Idriss Mazari-Fouquer (CEREMADE, Paris Dauphine Université, PSL)
Kevin Sturm (TU Wien)
Abstract : In this work we introduce the topological state derivative, a noval approach to treat PDE-constrained topology optimisation problems. This notion allows to deal with point perturbations as well as more general perturbations like smooth hypersurfaces in a similar way. Furthermore, we draw a connection from the topological state derivative to the asymptotic expansion of the state equation, which is usually derived using boundary layer correctors. Finally, we present numerical results based on these ideas.
[03573] Combining parameterized aerodynamic shape optimization with Sobolev smoothing
Author(s) :
Nicolas R. Gauger (University of Kaiserslautern-Landau)
Stephan Schmidt (Humboldt University Berlin)
Thomas Dick (University of Kaiserslautern-Landau)
Abstract : On the one hand, Sobolev gradient smoothing can considerably improve the performance of aerodynamic shape optimization and prevent issues with regularity. On the other hand, Sobolev smoothing can also be interpreted as an approximation for the shape Hessian. This paper demonstrates, how Sobolev smoothing, interpreted as a shape Hessian approximation, offers considerable benefits, although the parameterization is smooth in itself already. Such an approach is especially beneficial in the context of simultaneous analysis and design, where we deal with inexact flow and adjoint solutions, also called One Shot optimization. Furthermore, the incorporation of the parameterization allows for direct application to engineering test cases, where shapes are always described by a CAD model. The new methodology presented in this paper is used for reference test cases from aerodynamic shape optimization and performance improvements in comparison to a classical Quasi-Newton scheme are shown.
[03630] A combined phase field - Lipschitz method for PDE constrained shape optimization
Author(s) :
Michael Hinze (Universität Koblenz)
Philip Herbert (Heriot-Watt University)
Christian Kahle (Universität Koblenz)
Abstract : Abstract: We present a general shape optimisation framework for PDE constrained shape optimization, which combines phase field methods and the method of mappings in the Lipschitz topology. In a first step the phase field approach determines the topology of the sought shape, and with the zero level set of the phase field simultaneously provides an approximation of the optimal shape. The latter serves as starting point for a sharp interface shape optimization method in the Lipschitz topology. To illustrate our approach we present a selection of PDE constrained shape optimisation problems and compare our findings to results from so far classical Hilbert space methods and recent p-Laplace -approximations.
[04267] Interface Identification constrained by Local-to-Nonlocal Coupling
Author(s) :
Matthias Schuster (Trier University)
Christian Vollmann (Trier University)
Volker Schulz (Trier University)
Abstract : Models of physical phenomena that use nonlocal operators are better suited for some applications than their classical counterparts that employ partial differential operators.
However, the numerical solution of these nonlocal problems can be quite expensive. Therefore, Local-to-Nonlocal couplings have emerged that combine partial differential operators with nonlocal operators.
In this talk, we make use of an energy-based Local-to-Nonlocal coupling that serves as a constraint for an interface identification problem.
[04673] Total Generalized Variation for Geometric Inverse Problems
Author(s) :
Lukas Baumgärtner (Humboldt Universität zu Berlin)
Stephan Schmidt (Humboldt University Berlin)
Roland Herzog (Heidelberg University)
Ronny Bergmann (NTNU, Trondheim)
Manuel Weiß (Uni Heidelberg (IWR))
Jose Vidal-Nunez (University of Alcal ́a de Henares)
Abstract : The total variation of the outer normal vector of a shape is discussed in the context of triangulated meshes embedded in 3D. This non-smooth regularizer requires an advanced algorithm to be used in (inverse) shape optimization problems. A split
Bregman/ADMM method is used for this purpose. There, the non-smooth objective is split into a smooth shape optimization problem and a simple non-smooth problem. The smooth shape problem is solved by a globalized Newton method. Due to the nature of the regularizer, the first and second-order shape derivatives can not be computed by algorithmic differentiation. Therefore, their analytic form is derived and some of their properties are discussed. Numerical results are presented for mesh denoising problems.
An extension, the total generalized variation of the normal, to counteract the so-called staircasing effect is presented.
[04738] Choice of Inner Product in Shape Gradient Descent
Author(s) :
Caitriona Jacqueline McGarry (University of Leicester)
Alberto Paganini (University of Leicester)
Abstract : In optimisation problems with infinite dimensional control spaces, the choice of inner product on the control space affects the gradient, the direction of steepest descent wrt the induced norm.
Shape optimisation is no exception, with the control variable in an infinite dimensional function space. We will study the impact on shape optimisation of endowing this space of vector fields with alternatively an $H^1$ or $H^2$ inner product, especially regarding adding or removing corners of a shape.
[04786] Image and Shape Registration via Transport Equations
Author(s) :
Stephan Schmidt (University of Trier)
Lukas Baumgärtner (Humboldt Universität zu Berlin)
Abstract : We consider the use of transport equations as a model for solving inverse and registration problems. Incorporating shape Hessians to facilitate higher order methods have recently made new classes of problems tractable, ranging from the formation of capillary bridges in particle flows to the detection of motion in medical scans as well as mesh registration problems. Special attention is given on how to treat hyperbolic constraints within the setting of moving shapes and images.
