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 bifurcation
point to a given parameter value, often to within machine precision. 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 bifurcation points. Numerical experiments on the Allen--Cahn,
Navier--Stokes, and hyperelasticity equations demonstrate the effectiveness of
this technique in a wide range of settings.