Bifurcation tracking on moving meshes and with consideration of azimuthal symmetry breaking instabilities

TL;DR

This paper introduces a black-box numerical method for bifurcation tracking in multi-physics problems, including those with moving meshes and azimuthal symmetry-breaking instabilities, enabling rapid phase diagram analysis.

Contribution

The authors develop a generic, efficient bifurcation tracking approach that handles moving domain problems and symmetry-breaking instabilities using symbolic differentiation and just-in-time code generation.

Findings

Validated bifurcation detection on fluid dynamics problems

Enabled rapid phase diagram computation within minutes

Extended method to symmetry-breaking instability analysis

Abstract

We present a black-box method to numerically investigate the linear stability of arbitrary multi-physics problems. While the user just has to enter the system's residual in weak formulation, i.e. by a finite element method, all required discretized matrices are automatically assembled based on just-in-time generated and compiled highly performant C code. Based on this method, entire phase diagrams in the parameter space can be obtained by bifurcation tracking and continuation within minutes. Particular focus is put on problems with moving domains, e.g. free surface problems in fluid dynamics, since a moving mesh introduces a plethora of complicated nonlinearities to the system. By symbolic differentiation before the code generation, however, these moving mesh problems are made accessible to bifurcation tracking methods. In a second step, our method is generalized to investigate symmetry-breaking instabilities of axisymmetric stationary solutions by effectively utilizing the symmetry of the base state. Each bifurcation type is validated on the basis of results reported in the literature on versatile fluid dynamics problems.