Control of bifurcation structures using shape optimization

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.

20 pages, 11 figures

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Cited by in corpus (2)

• Bifurcation analysis of a two-dimensional magnetic Rayleigh-Bénard problem
• Optimization of Hopf bifurcation points