An extension of the projected gradient method to a Banach space setting with application in structural topology optimization

For the minimization of a nonlinear cost functional $j$ under convex constraints the relaxed projected gradient process $\varphi_{k+1} = \varphi_{k} + \alpha_k(P_H(\varphi_{k}-\lambda_k \nabla_H j(\varphi_{k}))-\varphi_{k})$ is a well known method. The analysis is classically performed in a Hilbert space $H$. We generalize this method to functionals $j$ which are differentiable in a Banach space. Thus it is possible to perform e.g. an $L^2$ gradient method if $j$ is only differentiable in $L^\infty$. We show global convergence using Armijo backtracking in $\alpha_k$ and allow the inner product and the scaling $\lambda_k$ to change in every iteration. As application we present a structural topology optimization problem based on a phase field model, where the reduced cost functional $j$ is differentiable in $H^1\cap L^\infty$. The presented numerical results using the $H^1$ inner product and a pointwise chosen metric including second order information show the expected mesh independency in the iteration numbers. The latter yields an additional, drastic decrease in iteration numbers as well as in computation time. Moreover we present numerical results using a BFGS update of the $H^1$ inner product for further optimization problems based on phase field models

Preconditioning for Allen-Cahn variational inequalities with non-local constraints

The solution of Allen-Cahn variational inequalities with mass constraints is of interest in many applications. This problem can be solved both in its scalar and vector-valued form as a PDE-constrained optimization problem by means of a primal-dual active set method. At the heart of this method lies the solution of linear systems in saddle point form. In this paper we propose the use of Krylov-subspace solvers and suitable preconditioners for the saddle point systems. Numerical results illustrate the competitiveness of this approach

Allen-Cahn and Cahn-Hilliard variational inequalities solved with Optimization Techniques

Parabolic variational inequalities of Allen-Cahn and Cahn- Hilliard type are solved using methods involving constrained optimization. Time discrete variants are formulated with the help of Lagrange multipliers for local and non-local equality and inequality constraints. Fully discrete problems resulting from finite element discretizations in space are solved with the help of a primal-dual active set approach. We show several numerical computations also involving systems of parabolic variational inequalities

Relating phase field and sharp interface approaches to structural topology optimization

A phase field approach for structural topology optimization which allows for topology changes and multiple materials is analyzed. First order optimality conditions are rigorously derived and it is shown via formally matched asymptotic expansions that these conditions converge to classical first order conditions obtained in the context of shape calculus. We also discuss how to deal with triple junctions where e.g. two materials and the void meet. Finally, we present several numerical results for mean compliance problems and a cost involving the least square error to a target displacement

Optimal control of Allen-Cahn systems

Optimization problems governed by Allen-Cahn systems including elastic effects are formulated and first-order necessary optimality conditions are presented. Smooth as well as obstacle potentials are considered, where the latter leads to an MPEC. Numerically, for smooth potential the problem is solved efficiently by the Trust-Region-Newton-Steihaug-cg method. In case of an obstacle potential first numerical results are presented

Phase-field approaches to structural topology optimization

The mean compliance minimization in structural topology optimization is solved with the help of a phase field approach. Two steepest descent approaches based on L2- and H-1 gradient flow dynamics are discussed. The resulting flows are given by Allen-Cahn and Cahn-Hilliard type dynamics coupled to a linear elasticity system. We finally compare numerical results obtained from the two different approaches

Sharp interface limit for a phase field model in structural optimization

We formulate a general shape and topology optimization problem in structural optimization by using a phase field approach. This problem is considered in view of well-posedness and we derive optimality conditions. We relate the diffuse interface problem to a perimeter penalized sharp interface shape optimization problem in the sense of $\Gamma$-convergence of the reduced objective functional. Additionally, convergence of the equations of the first variation can be shown. The limit equations can also be derived directly from the problem in the sharp interface setting. Numerical computations demonstrate that the approach can be applied for complex structural optimization problems

Relating phase field and sharp interface approaches to structural topology optimization

A phase field approach for structural topology optimization which allows for topology changes and multiple materials is analyzed. First order optimality conditions are rigorously derived and it is shown via formally matched asymptotic expansions that these conditions converge to classical first order conditions obtained in the context of shape calculus. We also discuss how to deal with triple junctions where e.g. two materials and the void meet. Finally, we present several numerical results for mean compliance problems and a cost involving the least square error to a target displacement

Multi-material phase field approach to structural topology optimization

Multi-material structural topology and shape optimization problems are formulated within a phase field approach. First-order conditions are stated and the relation of the necessary conditions to classical shape derivatives are discussed. An efficient numerical method based on an H1-gradient projection method is introduced and finally several numerical results demonstrate the applicability of the approach