Control of bifurcation structures using shape optimization

Journal article

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.

Authors: Boullé, N; Farrell, PE; Paganini, A

• Publication status: Published
• Peer review status: Peer reviewed
• Publisher: Society for Industrial and Applied Mathematics
• Journal: SIAM Journal on Scientific Computing
• Volume: 44
• Issue: 14
• Pages: A57–A76
• Publication date: 2022-01-05
• Acceptance date: 2021-09-21
• DOI: 10.1137/21M1418708
• Language: English
• Keywords: FFR; rationalization