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. In this talk, we will describe a numerical technique for controlling the bifurcation diagram of a nonlinear partial differential equation by varying the shape of the domain. Our aim is 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, called the Moore–Spence system, that characterize the location of the branch points. We will demonstrate the effectiveness of this technique on several numerical experiments on the Allen–Cahn, Navier–Stokes, and hyperelasticity equations.
- Junior Applied Mathematics Seminar