Dr Patrick Farrell, Oxford Mathematical Institute

Computing Multiple Equilibria of Structures via Deflation
When Apr 25, 2016
from 02:00 PM to 03:00 PM
Where LR8
Contact Name
Contact Phone 01865-273925
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Nonlinear equations can permit multiple solutions. In solid mechanics, this manifests itself as distinct equilibria (stable or unstable) that satisfy the governing equations. For example, it is well known that a slender beam subject to a compressive force along its axis undergoes buckling: below the critical load, only one solution exists; above the critical load, three solutions exist. This multiplicity and multistability of structures is extremely useful to designers --- if the multiple solutions can be found.

The standard approach to computing multiple solutions of differential equations combines arclength continuation and branch switching, as invented by Keller in 1977 and implemented in popular software packages such as AUTO. This algorithm has been extremely successful and useful in engineering, but suffers from two major drawbacks. The first is that it only computes connected fragments of bifurcation diagrams: for example, if the symmetry of a slender beam is broken by gravity, the standard approach will only identify one solution, even though the equation permits three solutions. The second is that the algorithm relies on the solution of expensive subproblems, which typically limits its applicability to systems of ODEs.

In this talk I will present a new algorithm, called deflated continuation, that overcomes both of these drawbacks. It allows for the computation of disconnected bifurcation diagrams, and scales as far as the underlying equation solver (up to discretisations of PDEs with billions of degrees of freedom). I will demonstrate its applicability to several nonlinear problems in mechanics including the deformation of a hyperelastic beam, nematic and cholesteric liquid crystals, and the optimal design of pipes