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Derivative Free Methods in Shape Optimization

English title Derivative Free Methods in Shape Optimization
Applicant Harbrecht Helmut
Number 137668
Funding scheme Project funding (Div. I-III)
Research institution Fachbereich Mathematik Departement Mathematik und Informatik Universität Basel
Institution of higher education University of Basel - BS
Main discipline Mathematics
Start/End 01.10.2012 - 30.09.2014
Approved amount 110'598.00
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Lay Summary (English)

Lead
Lay summary

Throughout the last 25–30 years, optimal shape design has become more and more important in engineering applications. Many problems that arise in structural mechanics, fluid dynamics and electromagnetics lead to the minimization of functionals defined over a class of admissible domains, governed by the solutions of boundary value problems or initial boundary value problems. Most of the related optimization methods are based on shape gradients which has two major drawbacks. Firstly, the shape gradients need to be analytically computed which is a nontrivial task. Secondly, the shape gradients need also to be implemented. We thus intend to develop gradient-free optimization methods for the numerical solution of shape optimization problems where the focus is on free boundary problems.

Direct link to Lay Summary Last update: 21.02.2013

Responsible applicant and co-applicants

Employees

Name Institute

Publications

Publication
A second order convergent trial method for free boundary problem in three dimensions
Bugeanu Monica, Harbrecht Helmut (2015), A second order convergent trial method for free boundary problem in three dimensions, in Interfaces Free Bound., 17(4), 517-537.
Stabilization of the trial method for the Bernoulli problem in case of prescribed Dirichlet data
Harbrecht Helmut, Mitrou Giannoula (2015), Stabilization of the trial method for the Bernoulli problem in case of prescribed Dirichlet data, in Math. Meth. Appl. Sci., 38(13), 2850-2863.
Improved trial methods for a class of generalized Bernoulli problems
Harbrecht Helmut, Mitrou Giannoula (2014), Improved trial methods for a class of generalized Bernoulli problems, in J. Math. Anal. Appl., 420(1), 177-194.

Collaboration

Group / person Country
Types of collaboration
Prof. Dr. Johannes Tausch (SMU Dallas) United States of America (North America)
- in-depth/constructive exchanges on approaches, methods or results
- Publication
Prof. Dr. Marc Dambrine (Universite de Pau et des Pays de l’Adour) France (Europe)
- in-depth/constructive exchanges on approaches, methods or results
- Publication
- Research Infrastructure
- Exchange of personnel
Prof. Dr. Reinhold Schneider (TU Berlin) Germany (Europe)
- in-depth/constructive exchanges on approaches, methods or results

Scientific events

Active participation

Title Type of contribution Title of article or contribution Date Place Persons involved
86. GAMM Jahrestagung Talk given at a conference A second order convergent trial method for free boundary problems in three dimensions 23.03.2015 Lecce, Italy Harbrecht Helmut;
Karlsruher PDE-Seminar, Schwerpunkt Partielle Differentialgleichungen Individual talk Improved trial methods for a class of generalized Bernoulli problems 20.11.2014 Karlsruhe Institute of Technology, Germany Harbrecht Helmut;
12th Workshop on fast boundary element methods in industrial applications Talk given at a conference A second order convergent trial method for free boundary problems 25.09.2014 Hirschegg, Austria Harbrecht Helmut;
International Symposium on Applied Analysis in Honour of the 65th Birthday of Michel Chipot and His Retirement Talk given at a conference Shape optimization for free boundary problems 10.06.2014 Zurich, Switzerland Harbrecht Helmut;


Abstract

We develop and implement higher order methods for the derivative free solution of shape optimizationproblem. As model problem we consider a generalized version of Bernoulli's exterior free boundary problem which involves a source term and non-constant boundary data. We derive first and second order shape update rules based on the asymptotic expansion of the solution of either the pure Dirichlet boundary value problem or the mixed boundary value problem. In particular, besides the pointwise boundary update, also variational boundary updates will be considered.Analogies to the Hadamard shape gradient and shape Hessian will be studied in order to prove convergencerates of the optimization scheme.The solution of the state equation is computed by a Nystrom-based p-boundary element method using trigonometric polynomials. It enables in combination with a parametric representation of the free boundary the computation of higher order derivatives of the solutionat the boundary. The developed shape optimization algorithm will be extended to solve constrained shape optimization problems like the electromagnetic shaping or the optimization of elastic bars.
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