Control of bifurcation structures using shape optimization
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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.
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