Control of bifurcation structures using shape optimization

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.