Parallel Block Preconditioning for Multi-Physics Problems

  • Richard Muddle

Student thesis: Phd


In this thesis we study efficient parallel iterative solution algorithms for multi-physics problems. In particular, we consider fluid structure interaction (FSI) problems, a type of multi-physics problem in which a fluid and a deformable solid interact.All computations were performed in Oomph-Lib, a finite element library for the simulation of multi-physics problems. In Oomph-Lib, the constituent problems in a multi-physics problem are coupled monolithically, and the resulting system of non-linear equations solved with Newton's method. This requires the solution of sequences of large, sparse linear systems, for which optimal solvers are essential. The linear systems arising from the monolithic discretisation of multi-physics problems are natural candidates for solution with block-preconditioned Krylov subspace methods.We developed a generic framework for the implementation of block preconditioners within Oomph-Lib. Furthermore the framework is parallelised to facilitate the efficient solution of very large problems. This framework enables the reuse of all of Oomph-Lib's existing linear algebra infrastructure and preconditioners (including block preconditioners). We will demonstrate that a wide range of block preconditioners can be seamlessly implemented in this framework, leading to optimal iterative solvers with good parallel scaling.We concentrate on the development of an effective preconditioner for a FSI problem formulated in an arbitrary Lagrangian Eulerian (ALE) framework with pseudo-solid node updates (for the deforming fluid mesh). We begin by considering the pseudo-solid subsidiary problem; the deformation of a solid governed by equations of large displacement elasticity, subject to a prescribed boundary displacement imposed with Lagrange multiplier. We present a robust, optimal, augmented-Lagrangian type preconditioner for the resulting saddle-point linear system and prove analytically tight bounds for the spectrum of the preconditioned operator with respect to the discrete problem size.This pseudo-solid preconditioner is incorporated into a block preconditioner for the full FSI problem. One key feature of the FSI preconditioner is that existing optimal single physics preconditioners (such as the well known Navier-Stokes Least Squares Commutator preconditioner) can be employed to approximately solve the linear systems associated with the constituent sub-problems. We evaluate its performance on selected 2D and 3D problems. The preconditioner is optimal for most problems considered. In cases when sub-optimality is detected, we explain the reasons for such behavior and suggest potential improvements.
Date of Award31 Dec 2011
Original languageEnglish
Awarding Institution
  • The University of Manchester
SupervisorMatthias Heil (Supervisor) & Milan Mihajlovic (Supervisor)


  • block preconditioners
  • preconditioners
  • parallel computations
  • multi-physics problems
  • fluid-structure interaction problems

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