Control of bifurcation structures using shape optimization

TL;DR

This paper introduces a numerical shape optimization method to control bifurcation structures in nonlinear PDEs, enabling delay or advancement of bifurcation points for improved device stability and switching.

Contribution

It presents a novel shape optimization algorithm constrained by the Moore–Spence system to manipulate bifurcation points in nonlinear PDEs.

Findings

Successfully delays bifurcation points in various equations

Effectively advances bifurcation points to target parameters

Demonstrates versatility across multiple nonlinear systems

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.