joint work with Roland Herzog, Ronny Bergmann
This work aims to present the novel notion of discrete shape manifold, which will allow solving PDE-constrained shape optimization problems using a discretize-then-optimize approach. This manifold will be endowed with a complete Riemannian metric that prevents mesh destruction. The computation of discrete shape derivatives and its relation with the optimize-then-discretize approach will be discussed. Finally, we will present the application of this approach to the solution of simple 2D optimization problems.
joint work with Benjamin Jurgelucks, Andrea Walther
Piezoelectricity defines the relation between electrical and mechanical changes in a specimen and can be described by a damped PDE system. Often material parameters of given piezoelectrics are not known precisely, but are required to be accurate. The solution of an inverse problem provides these values. For this purpose, sensitivities with respect to the material parameters can be calculated. In order to compute all parameters the sensitivities of some specific parameters must be increased. This is achieved by adjusting the electrode shapes with shape and topology optimization techniques.
joint work with Michael Hinze, Harald Garcke, Kei-Fong Lam
In a given domain $\Omega$ we search for a fluid domain $E$, such that an objective depending on $E$ and the velocity and the pressure field in $E$ is minimized. We describe the distribution of the fluid and void domain by a phase field variable $\varphi\in H^1(\Omega) \cap L^\infty(\Omega)$ that encodes the subdomains by $\varphi(x)=\pm1$. Additionaly we use a porosity approach to extend the Navier--Stokes equation from $E$ to $\Omega$. As solution algorithm we apply the variable metric projection type method from [L. Blank and C. Rupprecht, SICON 2017, 55(3)].