In this talk, we will consider a novel strategy for the approximation of optimal shapes. The approach is to use a phase field to provide a near-optimal shape. When adaptively refining the phase field, one ensure a well resolved interface. Upon meeting a desired stopping criteria, the level set of the phase field, which defines a shape is passed onto a sharp interface shape optimisation...
The well-established level-set shape optimisation method is based on the implicit description of domain boundaries as zero-level sets of a level-set function. Within this framework, domains are updated by evolving the level-set function according to a Hamilton-Jacobi equation, which itself comprises shape gradients as velocity terms.
The common approach is to employ standard finite elements...
Shape optimization constrained to partial differential equations is a vibrant field of research with high relevance for industrial-grade applications. Recent developments suggest that using a $p$-harmonic approach to determine descent directions is superior to classical Hilbert space methods. This applies in particular to the representation of kinks and corners in occurring shapes. However,...
A crucial issue in numerically solving PDE-constrained shape optimization problems is avoiding mesh degeneracy. Recently, there were two suggested approaches to tackle this problem: (i) departing from the Hilbert space towards the Lipschitz topology approximated by $W^{1,p^*}$ with $p^*>2$ and (ii) using the symmetric rather than the full gradient to define a norm.
In this talk we will...
The talk discusses the efficiency and robustness aspects of a first-order approach based on ADMM (Alternating Direction Method of Multipliers) for approximating the direction of the steepest descent in $W^{1,\infty}$ with Lipschitz domains. The robustness of an implementation with respect to certain parameters is crucial for the technical application of shape optimization problems. For...
One main goal of topology optimization is to find an optimal distribution of multiple materials in a design domain in such away that it can withstand internal and external loads applied on the structure. Additive manufacturing techniques like 3D-printing are able to produce complex structures and topologies. To guarantee constructability overhangs need either support structures or should be...
Topology Optimization considers the optimization of a domain by changing its geometric properties by either adding or removing material. Typically, topology optimization problems feature multiple local minimizers. In order to guarantee convergence to local minimizers that perform best globally, it is important to identify multiple local minimizers of topology optimization problems. Moreover,...
We aim to solve a topology optimization problem where the distribution of material in the design domain is represented by a density function. To obtain candidates for local minima, we want to solve the first order optimality system via Newton's method. This requires the initial guess to be sufficiently close to the a priori unknown solution. Introducing a stepsize rule often allows for less...
We consider a PDE-constrained design optimization problem with multiple materials, where some material properties are defined by an additional parameter. For this problem we derive the topological derivative i.e. the pointwise sensitivity of the cost functional subject to material changes and use it to update a vector-valued level set function describing the material distribution within the...
