Shape optimization in the space of piecewise-smooth shapes for the Bingham flow variational inequality

Abstract

This paper sets up an approach for shape optimization problems constrained by variational inequalities (VI) in an appropriate shape space. In contrast to classical VI, where no explicit dependence on the domain is given, VI constrained shape optimization problems are in particular highly challenging because of two main reasons: Firstly, one needs to operate in inherently non-linear, non-convex and infinite-dimensional shape spaces. Secondly, the problem cannot be solved directly without any regularization techniques in general because, e.g., one cannot expect the existence of the shape derivative for an arbitrary shape functional depending on solutions to VI. This paper introduces a specific shape manifold and presents an optimization technique to handle the non-differentiabilities on this shape manifold. In particular, we formulate an optimization system based on Gâteaux semiderivatives and Eulerian derivatives for a shape optimization problem constrained by the Bingham flow variational inequality. Numerical results show the applicability and efficiency of the proposed approach.

Figures (11)

• Figure 1: Residual norm and accepted step sizes of the iterative scheme using $\rho=0$ and $\rho=10$ with regularized and unregularized $\max$-operator for the solution of the state equation.
• Figure 2: Active set $\gamma \! \left\| \bm{\varepsilon}(\bm{y}) \right\| - g$ at the start of the optimization with regularization of the $\max$-operator and $\rho=0$ (left) and $\rho=10$ (right).
• Figure 3: Active set $\gamma \! \left\| \bm{\varepsilon}(\bm{y}) \right\| - g$ at the start of the optimization without regularization of the $\max$-operator and $\rho=0$ (left) and $\rho=10$ (right).
• Figure 4: Augmented Lagrange functional (left) and absolute change of augmented Lagrange functional $\Delta (\cdot ) = \left|(\cdot )^{i}-(\cdot )^{i-1}\right|$, $i=1,\ldots,20\,000$, (right) using $\rho=0$ with regularized and unregularized $\max$-operator.
• Figure 5: $H^1$-norm of the mesh deformation using $\rho=0$ with regularized and unregularized $\max$-operator.

Theorems & Definitions (1)

• Remark 1