An algorithm for the computation of multiple Hopf bifurcation points based on Pade approximants

Grégory Girault (1) , Yann Guevel (1) , Jean-Marc Cadou (1)

1 LIMATB - Laboratoire d'Ingénierie des Matériaux de Bretagne

Recently, a numerical method was proposed to compute a Hopf bifurcation point in fluid mechanics. This numerical method associates a bifurcation indicator and a Newton method. The former gives initial guesses to the iterative method. These initial values are the minima of the bifurcation indicator. However, sometimes, these minima do not lead to the convergence of the Newton method. Moreover, as only a single initial guess is obtained for each computation of the indicator, the computational time to obtain a Hopf bifurcation point can be quite long. The present algorithm is an enhancement of the previous one. It consists in automatically computing several initial guesses for each indicator curve. The majority of these initial values leads to the convergence of the Newton method. This method is evaluated through the problem of the lid-driven cavity with several aspect ratios in the framework of the finite element analysis of the 2D Navier-Stokes equations. The results prove the efficiency and the robustness of the proposed algorithm.

Mots clés: viscous flows; stability; Navier–Stokes; incompressible flow; generalize FEM; nonlinear solvers

Domaines: Mécanique [physics.med-ph]

Grégory Girault, Yann Guevel, Jean-Marc Cadou. An algorithm for the computation of multiple Hopf bifurcation points based on Pade approximants. International Journal for Numerical Methods in Fluids, 2012, 68 (9), pp.1189-1206. ⟨10.1002/fld.2605⟩. ⟨hal-00715976⟩