# Efficient PDE Constrained Shape Optimization Based on Steklov-Poincaré-Type Metrics

@article{Schulz2015EfficientPC, title={Efficient PDE Constrained Shape Optimization Based on Steklov-Poincar{\'e}-Type Metrics}, author={Volker Schulz and Martin Siebenborn and Kathrin Welker}, journal={SIAM J. Optim.}, year={2015}, volume={26}, pages={2800-2819} }

Recent progress in PDE constrained optimization on shape manifolds is based on the Hadamard form of shape derivatives, i.e., in the form of integrals at the boundary of the shape under investigation, as well as on intrinsic shape metrics. From a numerical point of view, domain integral forms of shape derivatives seem promising, which rather require an outer metric on the domain surrounding the shape boundary. This paper tries to harmonize both points of view by employing a Steklov-Poincar\'e…

## 62 Citations

### Nonlinear Conjugate Gradient Methods for PDE Constrained Shape Optimization Based on Steklov-Poincaré-Type Metrics

- Computer ScienceSIAM J. Optim.
- 2021

The results show that the proposed nonlinear conjugate gradient methods yield a significant improvement over the gradient descent method, and that their performance can be compared to that of the limited memory BFGS methods, making them efficient and attractive gradient based shape optimization algorithms.

### A continuous perspective on modeling of shape optimal design problems

- Mathematics, Computer Science
- 2020

This article considers shape optimization problems as optimal control problems via the method of mappings and focuses on the choice of the set of transformations, which is motivated from a function space perspective.

### A CONTINUOUS PERSPECTIVE ON MODELING OF SHAPE

- Mathematics, Computer Science
- 2020

This article considers shape optimization problems as optimal control problems via the method of mappings and enriched the optimization problem by a nonlinear constraint to guarantee local injectivity of the admissible transformations.

### A novel [[EQUATION]] approach to shape optimisation with Lipschitz domains

- MathematicsESAIM: Control, Optimisation and Calculus of Variations
- 2021

This article introduces a novel method for the implementation of shape optimisation with Lipschitz domains. We propose to use the shape derivative to determine deformation fields which represent…

### Computational Investigations of an Obstacle-Type Shape Optimization Problem in the Space of Smooth Shapes

- MathematicsGSI
- 2019

A regularization strategy leading to novel possibilities to numerically exploit structures, as well as possible treatment of the regularized variational inequality constrained shape optimization in the context of optimization on infinite dimensional Riemannian manifolds are considered.

### PDE-constrained shape optimization: towards product shape spaces and stochastic models

- Computer ScienceArXiv
- 2021

A model problem is constructed, demonstrating how uncertainty can be introduced into the problem and the objective can be transformed by use of the expectation, and numerical experiments in the deterministic and stochastic case are devised, which demonstrate the effectiveness of the presented algorithms.

### Space Mapping for PDE Constrained Shape Optimization

- Computer Science, Mathematics
- 2022

This paper proposes novel space mapping methods for solving shape optimization problems constrained by partial diﬀerential equations (PDEs) in a Riemannian setting based on Steklov-Poincaré-type metrics and discusses their numerical discretization and implementation.

### Pre-Shape Calculus: Foundations and Application to Mesh Quality Optimization

- Computer Science
- 2020

A theoretical framework using pre-shapes to generalize classical shape optimization and -calculus is proposed and tangential directions are featured in pre-shape derivatives, in contrast to classical shape derivatives featuring only normal directions.

### A Discretize-Then-Optimize Approach to PDE-Constrained Shape Optimization

- Computer Science
- 2021

It turns out that the complete metric, combined with a Euclidean retraction, performs well even in the absence of the mesh quality penalty, and the choice of Riemannian metric on the steepest descent method is studied.

### Efficient Techniques for Shape Optimization with Variational Inequalities Using Adjoints

- MathematicsSIAM J. Optim.
- 2020

This paper considers shape optimization problems constrained by variational inequalities of the first kind, so-called obstacle-type problems, and derives existence and closed form of shape derivatives for the regularized problem and proves convergence to a limit object.

## References

SHOWING 1-10 OF 30 REFERENCES

### Shape Optimization by Pursuing Diffeomorphisms

- MathematicsComput. Methods Appl. Math.
- 2015

An algorithm tailored to preserve and exploit the approximation properties of the finite element method, and that allows for arbitrarily high resolution of shapes is presented.

### Discrete Differential-Geometry Operators for Triangulated 2-Manifolds

- Mathematics, Computer ScienceVisMath
- 2002

A unified and consistent set of flexible tools to approximate important geometric attributes, including normal vectors and curvatures on arbitrary triangle meshes, using averaging Voronoi cells and the mixed Finite-Element/Finite-Volume method is proposed.

### On Convergence in Elliptic Shape Optimization

- MathematicsSIAM J. Control. Optim.
- 2007

Existence and convergence of approximate solutions are proved, provided that the infinite dimensional shape problem admits a stable second order optimizer.

### Lagrange method in shape optimization for non-linear partial differential equations : A material derivative free approach

- Mathematics
- 2013

This paper studies the relationship between the material derivative method, the shape derivative method, the min-max formulation of Correa and Seeger, and the Lagrange method introduced by Céa. A…

### Scalable shape optimization methods for structured inverse modeling in 3D diffusive processes

- Computer ScienceComput. Vis. Sci.
- 2015

A novel algorithm is proposed that combines mathematical shape optimization with high-performance computing to fit a parabolic model for drug diffusion through the skin to data measurements and investigates the scalability of the algorithm up to millions of discretization elements.

### Structure of shape derivatives

- Mathematics
- 2002

Abstract. In this paper, we describe the precise structure of second "shape derivatives", that is derivatives of functions whose argument is a variable subset of $ \mathbb{R}^N $. This is done for…

### Domain expression of the shape derivative and application to electrical impedance tomography

- Mathematics
- 2013

The well-known structure theorem of Hadamard-Zol\'esio states that the derivative of a shape functional is a distribution on the boundary of the domain depending only on the normal perturbations of a…

### Boundary element based multiresolution shape optimisation in electrostatics

- BusinessJ. Comput. Phys.
- 2015

### Shapes and Geometries: Analysis, Differential Calculus, and Optimization

- Mathematics
- 1987

This book provides a self-contained presentation of the mathematical foundations, constructions, and tools necessary for studying problems where the modeling, optimization, or control variable is no…