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Control of bifurcation structures using shape optimization

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Version 2 2022-09-23, 09:47
Version 1 2021-12-17, 16:07
journal contribution
posted on 2022-09-23, 09:47 authored by Nicolas Boullé, Patrick E Farrell, Alberto Paganini
Many problems in engineering can be understood as controlling the bifurcation structure of a given device. For example, one may wish to delay the onset of instability, or bring forward a bifurcation to enable rapid switching between states. We propose a numerical technique for controlling the bifurcation diagram of a nonlinear partial differential equation by varying the shape of the domain. Specifically, we are able to delay or advance a given branch point to a target parameter value. The algorithm consists of solving a shape optimization problem constrained by an augmented system of equations, the Moore–Spence system, that characterize the location of the branch points. Numerical experiments on the Allen–Cahn, Navier–Stokes, and hyperelasticity equations demonstrate the effectiveness of this technique in a wide range of settings.

Funding

EPSRC Centre for Doctoral Training in Industrially Focused Mathematical Modelling

Engineering and Physical Sciences Research Council

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Gen X: ExCALIBUR working group on Exascale continuum mechanics through code generation.

Engineering and Physical Sciences Research Council

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PRISM: Platform for Research In Simulation Methods

Engineering and Physical Sciences Research Council

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History

Author affiliation

School of Mathematics & Actuarial Science

Version

  • VoR (Version of Record)

Published in

SIAM Journal on Scientific Computing

Volume

44

Issue

1

Publisher

Society for Industrial and Applied Mathematics

eissn

1095-7197

Acceptance date

2021-09-21

Copyright date

2021

Available date

2022-09-23

Language

en

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