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arXiv · 2105.14884

Control of bifurcation structures using shape optimization

Nicolas Boullé·Patrick E. Farrell·Alberto Paganini
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Abstract

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.

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Nicolas Boullé, Patrick E. Farrell, Alberto Paganini. 2021-05-31. Control of bifurcation structures using shape optimization. https://doi.org/10.1137/21m1418708

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