29 research outputs found
Space Mapping for PDE Constrained Shape Optimization
The space mapping technique is used to efficiently solve complex optimization
problems. It combines the accuracy of fine model simulations with the speed of
coarse model optimizations to approximate the solution of the fine model
optimization problem. In this paper, we propose novel space mapping methods for
solving shape optimization problems constrained by partial differential
equations (PDEs). We present the methods in a Riemannian setting based on
Steklov-Poincar\'e-type metrics and discuss their numerical discretization and
implementation. We investigate the numerical performance of the space mapping
methods on several model problems. Our numerical results highlight the methods'
great efficiency for solving complex shape optimization problems
Version 2.0 -- cashocs: A Computational, Adjoint-Based Shape Optimization and Optimal Control Software
In this paper, we present version 2.0 of cashocs. Our software automates the
solution of PDE constrained optimization problems for shape optimization and
optimal control. Since its inception, many new features and useful tools have
been added to cashocs, making it even more flexible and efficient. The most
significant additions are a framework for space mapping, the ability to solve
topology optimization problems with a level-set approach, the support for
parallelism via MPI, and the ability to handle additional (state) constraints.
In this software update, we describe the key additions to cashocs, which is now
even better-suited for solving complex PDE constrained optimization problems
Quasi-Newton Methods for Topology Optimization Using a Level-Set Method
The ability to efficiently solve topology optimization problems is of great
importance for many practical applications. Hence, there is a demand for
efficient solution algorithms. In this paper, we propose novel quasi-Newton
methods for solving PDE-constrained topology optimization problems. Our
approach is based on and extends the popular solution algorithm of Amstutz and
Andr\"a (A new algorithm for topology optimization using a level-set method,
Journal of Computational Physics, 216, 2006). To do so, we introduce a new
perspective on the commonly used evolution equation for the level-set method,
which allows us to derive our quasi-Newton methods for topology optimization.
We investigate the performance of the proposed methods numerically for the
following examples: Inverse topology optimization problems constrained by
linear and semilinear elliptic Poisson problems, compliance minimization in
linear elasticity, and the optimization of fluids in Navier-Stokes flow, where
we compare them to current state-of-the-art methods. Our results show that the
proposed solution algorithms significantly outperform the other considered
methods: They require substantially less iterations to find a optimizer while
demanding only slightly more resources per iteration. This shows that our
proposed methods are highly attractive solution methods in the field of
topology optimization
Asymptotic Analysis for Optimal Control of the Cattaneo Model
We consider an optimal control problem with tracking-type cost functional
constrained by the Cattaneo equation, which is a well-known model for delayed
heat transfer. In particular, we are interested the asymptotic behaviour of the
optimal control problems for a vanishing delay time .
First, we show the convergence of solutions of the Cattaneo equation to the
ones of the heat equation. Assuming the same right-hand side and compatible
initial conditions for the equations, we prove a linear convergence rate.
Moreover, we show linear convergence of the optimal states and optimal controls
for the Cattaneo equation towards the ones for the heat equation. We present
numerical results for both, the forward and the optimal control problem
confirming these linear convergence rates