Efficient preconditioning of the NavierStokes equations with the free surface condition

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Efficient preconditioning of the Navier-Stokes
equationswith the free surface condition

• Milan D. Mihajlovic
• School of Computer Science
• The University of Manchester

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Acknowledgements

• This is a joint work with Dr Matthias Heil from
  School of Mathematics, Manchester, who initially
  suggested this problem
• Alexandre Klimowicz final year PhD student for
  all the hard work invested in implementation of
  the model in C
• The use of the OOMPHLIB (Object Oriented
  Multi-Physics LIBrary), designed by M.Heil and A.
  Hazel is greatly acknowledged
• (http//www.oomph-lib.org)
• D. Silvester (Manchester) and H. Elman (Maryland)
  people from whom I learned a great deal about
  CFD and whose results in preconditioning of the
  N-S equations motivated this work

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Motivation

• M. Heil modelling of the fluid-solid
  interaction problems in biomechanical engineering

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Model-problem
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Resulting discrete equations
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Standard preconditioning technique
Convergence characteristics of the preconditioned
BiCGSTAB(2) algorithm Exact preconditioning,
, Re0, dt0.2
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Modified preconditioning strategy
Convergence characteristics of the preconditioned
BiCGSTAB(2) algorithm Exact preconditioning,
, Re0, dt0.2
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Main applications

• Phase-change problems (e.g. modelling of
  dendrites)
• Coating flows
• Liquid bridge problems
• Modelling of droplets and bubbles
• Modelling of multiple-fluid flows

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Difficulties in modelling free-surface flows

• Non-linear problems (N-S equations and the
  kinematic BC at the free surface are non-linear)
• Time-dependent problems (evolution of
  free-surface, stability of the flow)
• Important 3D effects
• Changing geometry of the domain with time

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Different approaches in modelling time-dependent
free-surface flows

• Restriction to 2D
• Lubrication approximation (thin liquid film with
  negligible flow variations across the film)
• 3D approximations
• FD-MAC methods
• Volume of fluid
• Level set methods
• FEM (moving grids ALE approach)

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Arbitrary Lagrangian-Eulerian approach moving
grids FE method

• Free surface evolves with the fluid
• The interior grid evolves with respect to
  geometric constraints, rather than with respect
  to the physical flow (the Lagrangian approach)
• Advantages
• Accurate approximation of free surface
• Computational domain restricted to the domain
  occupied by the fluid
• Potential problems
• Not robust for problems where the topology of the
  free surface changes discontinuously with time
  (e.g. the break-up or the merger of droplets)

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Wall deformation
y
x
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The fluid-solid interaction problem
y
p0
x
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The incompressible Navier-Stokes equations

• Non-dimensional form
• Scaling
• Velocity
• Cartesian coordinates
• Time scale
• Pressure

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Free surface conditions

• Position of the free surface is determined by the
  vector
• Kinematic free surface condition
• Single condition - the impermeability of the free
  surface.
• Stress condition (dynamic boundary condition)
  
• There are d such conditions which state that
  pressure vanishes at the free surface.

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Spines and the free surface

• The role of the spines is twofold
• Resolve the free surface position
• Facilitate automatic adjustment of the fluid mesh
  to changes in the fluid domain

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Arbitrary Lagrangian-Eulerian method

• Discrete weak formulation

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Fluid-solid coupling

• The fluid and solid interact via the no-slip
  condition on the wall
• and via the traction that fluid exerts on the
  wall this is manifested through the load term
  in the shell equations
• where is the dimensionless surface tension.

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Discrete system

• The fluid equations are discretised with
  isoparametric Taylor-Hood elements (biquadratic
  approximation for the velocities and bilinear
  approximation for pressures, and biquadratic
  approximation for the free surface).
• Linearisation of the obtained non-linear system
  is done by the Newton-Raphson method.
• Time discretisation adaptive predictor-corrector
  linear multistep method (Gresho, Sani 2000, p.
  805-806). Predictor second-order, variable step
  explicit leapfrog scheme, corrector
  second-order, variable step, implicit BDF2.

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Solution of large, sparse, linear systems

• Direct methods variants of sparse Gaussian
  elimination based on graph reduction techniques
  (frontal, multifrontal, supernodal approaches)
• Advantages accurate, after finite number of
  operations an exact answer is obtained,
  black-box, parallelisable
• Disadvantages non-optimal solvers in terms of
  the execution time and storage requirements. This
  is especially the case for 3D problems.

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Solution of large, sparse, linear systems

• Iterative solution techniques simple
  iterations, smoothing methods, Krylov solvers.
• Advantages optimal computational cost per
  iteration (only matrix-vector products are
  required), optimal storage requirements
• Disadvantages convergence characteristics
  strongly depend on the spectral properties of the
  coefficient matrix (which is usually
  ill-conditioned).
• Remedy preconditioning.

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Preconditioning

• We solve a modified system
• where the modified coefficient matrix
  has more favourable spectral properties than
  .
• Good preconditioner
• Spectrally close to the coefficient matrix
• Relatively cheap to assemble
• Matrix-vector products cheap to
  compute
• Significant improvement of the convergence
  characteristics

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Preconditioning of the Navier-Stokes equations

• Block preconditioners (Elman, Silvester, Wathen)
• is usually achieved by block application of
  MG
• Different approximations of the Schur complement
• BFBt approximation
• Fp approximation

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Fp preconditioner

• Assume commutativity of the underlying
  differential operators
• Matrix interpretation
• Modification to satisfy the Stokes limit
• Not black-box, some problems with stretched grids
• Mesh independent convergence
• Convergence mildly depend on

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BFBt preconditioner

• General argument
• Let

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BFBt preconditioner
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BFBt preconditioner

• Fourier analysis of the BFBt approximation is
  possible for the case of uniform grid, MAC-type
  discretisation, constant wind, and periodic BCs.
• Then it is possible to prove that

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Diagonally scaled BFBt preconditioner

• The idea connects both Fp and BFBt
  preconditioners.

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Diagonally scaled BFBt preconditioner
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Eigenvalue analysis
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Example driven lid cavity problem
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Convergence results
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Convergence results
Direct factorisation of all blocks, GMRES,
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Convergence results
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Convergence results
MG V(1,1) cycles with 4-dir LGS for F and
DJ(8/9) for
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Approximation of the free surface Schur complement
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Approximation of the free surface Schur complement
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Approximation of the free surface Schur complement
Extending Elmans analysis for the N-S case
implies that if the same conditions apply to F
(discretisation on the uniform grid, constant
wind, periodic BCs). Constant stencils in
, can be achieved if the grid is uniform, the
free surface is a straight line (not necessarily
alligned with any of the coordinate axes, and
that all the fluid basis functions are the same
(linear approximation of ).
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Approximation of the free surface Schur complement

• Why the approximation works in the case of the
  free-surface Schur complement?

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Numerical results
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Numerical results
Convergence characteristics of the preconditioned
BiCGSTAB(2) algorithm Exact preconditioning,
, Re0, dt0.2
Re500, dt0.2
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Numerical results
Re500, dt0.2
Re500, dt0.005
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Numerical results
Re500, dt0.2 Exact/MG approximation of F and
(BBt)
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Conclusions

• Interesting extension of the N-S preconditioning
  methodology.
• More testing to demonstrate robustness (different
  problems, adaptive grids, different smoothers for
  convection MG solvers).
• More rigorous treatment of the exact
  approximation case.