1. CONTROL OF BIFURCATION STRUCTURES USING SHAPE OPTIMIZATION
- Author
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Nicolas Boullé, Alberto Paganini, Patrick Farrell, and Apollo - University of Cambridge Repository
- Subjects
Applied Mathematics ,MathematicsofComputing_NUMERICALANALYSIS ,bifurcation analysis ,Moore--Spence system ,010103 numerical & computational mathematics ,Numerical Analysis (math.NA) ,01 natural sciences ,010101 applied mathematics ,Computational Mathematics ,Optimization and Control (math.OC) ,shape optimization ,FOS: Mathematics ,65P30, 65P40, 37M20, 65K10 ,Mathematics - Numerical Analysis ,0101 mathematics ,Mathematics - Optimization and Control - Abstract
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., 20 pages, 11 figures
- Published
- 2023
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