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

A new shape optimization approach for fracture propagation

Within this work, we present a novel approach to fracture simulations based on shape optimization techniques. Contrary to widely-used phase-field approaches in literature the proposed method does not require a specified 'length-scale' parameter defining the diffused interface region of the phase-field. We provide the formulation and discuss the used solution approach. We conclude with some numerical comparisons with well-established single-edge notch tension and shear tests.Comment: 10 pages, 6 figures, Accepted for publication in: Proceedings in Applied Mathematics and Mechanics 202

Stochastic augmented Lagrangian method in shape spaces

In this paper, we present a stochastic Augmented Lagrangian approach on (possibly infinite-dimensional) Riemannian manifolds to solve stochastic optimization problems with a finite number of deterministic constraints. We investigate the convergence of the method, which is based on a stochastic approximation approach with random stopping combined with an iterative procedure for updating Lagrange multipliers. The algorithm is applied to a multi-shape optimization problem with geometric constraints and demonstrated numerically

Shape optimization for interface identification in nonlocal models

Shape optimization methods have been proven useful for identifying interfaces in models governed by partial differential equations. Here we consider a class of shape optimization problems constrained by nonlocal equations which involve interface-dependent kernels. We derive a novel shape derivative associated to the nonlocal system model and solve the problem by established numerical techniques

Optimization of piecewise smooth shapes under uncertainty using the example of Navier-Stokes flow

We investigate a complex system involving multiple shapes to be optimized in a domain, taking into account geometric constraints on the shapes and uncertainty appearing in the physics. We connect the differential geometry of product shape manifolds with multi-shape calculus, which provides a novel framework for the handling of piecewise smooth shapes. This multi-shape calculus is applied to a shape optimization problem where shapes serve as obstacles in a system governed by steady state incompressible Navier-Stokes flow. Numerical experiments use our recently developed stochastic augmented Lagrangian method and we investigate the choice of algorithmic parameters using the example of this application

Crack propagation in anisotropic brittle materials: from a phase-field model to a shape optimization approach

The phase-field method is based on the energy minimization principle which is a geometric method for modeling diffusive cracks that are popularly implemented with irreversibility based on Griffith's criterion. This method requires a length-scale parameter that smooths the sharp discontinuity, which influences the diffuse band and results in mesh-sensitive fracture propagation results. Recently, a novel approach based on the optimization on Riemannian shape spaces has been proposed, where the crack path is realized by techniques from shape optimization. This approach requires the shape derivative, which is derived in a continuous sense and used for a gradient-based algorithm to minimize the energy of the system. Due to the continuous derivation of the shape derivative, this approach yields mesh-independent results. In this paper, the novel approach based on shape optimization is presented, followed by an assessment of the predicted crack path in anisotropic brittle material using numerical calculations from a phase-field model

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