Iterative Solvers for Coupled FluidSolid Scattering

1
Iterative Solvers for Coupled Fluid-Solid
Scattering

• Jan Mandel
• Work presentation
• Center for Aerospace Structures
• University of Colorado at Boulder
• October 10, 2003

2
Outline

• The coupled scattering problem
• the PDEs
• the discrete 22 matrix form
• Solving the discrete equations
• Simply proceed as usual on the matrices, or
• Preconditioner to ignore weak coupling between
  fluid and the solid blocks, or
• Couple existing separate FETI-type methods in the
  fluid and in the solid

3
Model coupled problem
Radiation b.c.
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Coupled equations

• On the wet interface
• The value of the solid displacement u provides
  load for the Helmholtz problem in the fluid
• The value of the fluid pressure p provides load
  for the elastodynamic problem in the solid
• There are no equality constraints on the wet
  interface, for this choice of variables

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Existence of solution

• Solution exists
• Solution is unique up to non-radiating modes in
  the solid vibrations of the solid that have no
  effect on the outside

Only bodies with certain symmetries (such as
sphere) have non-radiating modes almost all
bodies have no non-radiating modes
6
Variational formulation

• Find such that

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Discrete problem

• 2x2 block system of equations
• Coupling matrix is like a boundary load

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About the discrete system

• Coupled problem with vastly different scales,
  easily by 10 orders of magnitude
• scaling is essential
• what is the meaning of the residual?
• Algorithms should be invariant to change of
  physical units
• assured when physical units match
• Coupling between fluid and solid is weak (details
  later)

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Scaling of the discrete problem

• Multiply 2nd equation to make the off-diagonal
  blocks same then,
• Symmetric diagonal scaling to make both diagonal
  blocks of the same magnitude O(1)

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The fluid and the solid are coupled only weakly

• Scaling the fields reveals that the fluid and the
  solid are numerically decoupled when
• (Mandel 2002)
• Completely decoupled in the limit for a stiff
  solid scatterer
• Numerically almost decoupled for practically
  interesting problems (aluminum, water)

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Solving the discrete equations

• Just go ahead on the matrices with no change in
  method, align method interfaces with wet
  interface (or not?)
• Multigrid (known to work)
• Substructuring (requires apportioning the matrix
  T to substructures in some cases not tested
  without)
• Ignore the weak coupling in a preconditioner
• Block diagonal preconditionining (known to
  work)
• Needs regularization in solid to avoid resonance
• FETI-type substructuring
• What multipliers on wet interface?
• Couple FETI for fluid FETI for solid
• Known not good enough for unstructured 3D meshes
• Maybe FETI-DP will be better

12
Multigrid for the coupled problem

• Coarse nodes on wet interface
• Coarse problem needs to be fine enough to express
  waves, albeit crudely
• Krylov smoothing (e.g., GMRES) allows
  significantly coarser coarse problems (Elman 2001
  for Helmholtz Popa 2002 thesis, for coupled)

coarse
fine
fluid
solid
wet interface
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Multigrid performance
Decreasing h, k3h2 const, adding coarse levels,
10 smoothing steps, k10 for h1/32, average
residual reduction smoothing step, domain 1x1
with 0.2x0.2 obstacle in the middle (Popa, 2002)
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Substructuring is based on subassembly of the
block diagonal matrix of Schur complements

• T couples dofs across wet interface
• The extra coupling spoils the subassembly
  property decomposition of global matrix into
  independent local substructure matrices
• The matrix of substructure matrices is no longer
  block diagonal, which is needed for parallelism

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Building a substructuring method

• The matrix of substructure Schur complements
  needs to be block diagonal to get parallelism.
  Some possible approaches
• Keep decomposition into substructures, add dofs
  of the other kind to substructures adjacent to
  the wet interface and apportion the matrix T
• Primal only ignore T in the preconditioner, same
  as block diagonal preconditioner
• Or, duplicate dofs on wet interface, T works
  between the duplicates, and enforce equality of
  duplicates at converged solution
• by Lagrange multipliers
• add new equations to the system

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Substructuring choices

• Respect wet interface as substructure boundary?
  If so,
• Interiors of substructures get only their
  respective fluid or solid dofs
• The reduced problem has both fluid and solid dofs
  for substructures adjacent to the wet interface
• The interface matrix T becomes part of the global
  Schur complement
• To have global matrix equal to assembly of local
  matrices (subassembly property), T needs to be
  apportioned to substructures

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Apportioning the interface matrix T

• Fluid substructures get additional solid dofs on
  the interface
• The local interface matrix is added to the local
  matrix of the fluid substructure
• Assembled system remains same

Interface matrix

Eliminate interiors
Solid substructures
Fluid substructures
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Apportioning the interface matrix T

• Solid substructures get additional fluid dofs on
  the interface
• The local interface matrix is added to the local
  matrix of the fluid substructure
• Assembled system remains same

Interface matrix

Eliminate interiors
Solid substructures
Fluid substructures
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Apportioning the interface matrix T

• Fluid substructures get additional solid dofs,
  AND solid substructures get fluid dofs both
  shared by substructures adjacent across the wet
  interface
• Part of the local interface matrix is added to
  the local matrix of the fluid substructure, part
  to the solid substructure
• The assembled system remains same

Interface matrix

Eliminate interiors
Solid substructures
Fluid substructures
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Substructuring that ignores the wet interface

• Respect wet interface as substructure boundary?
  If not
• Substructures can have both fluid or solid dofs
• The interface matrix T becomes part of the global
  Schur complement
• But T still needs to be apportioned every time
  more than one substructure have a common segment
  of the wet interface
• Efficient iterative substructuring when the
  substructures may have both types of dofs?

Eliminate interiors
Apportioning needed
Solid substructures
Fluid substructures
Interface matrix

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Substructuring with apportioned T

• Once the problem is written as subassembly of
  local substructure matrices, all existing
  substructuring methods can proceed (primal BDD,
  or dual, Lagrange multipliers FETI)
• Basis for futher developments
• But specific methods not tested

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Block diagonal preconditioning
Precond. Scaling

Scaling

• Preconditioner can use existing solvers for fluid
  and solid separately
• The 2nd diagonal block (solid) will be singular
  at resonance frequences
• Damping for solid provided via only
• Need to provide artificial damping without
  changing the solution

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Avoiding resonance for block diagonal
preconditioning

• Regularization Add to the equations in solid a
  complex linear combination of equations in fluid,
  coefficients determined by analogy with
  radiation boundary conditions in solid (Mandel,
  Popa 2003)
• Needed also for FETI-type methods when there is
  only one substructure for solid (Mandel 2002)

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Regularization of the matrix of the solid for
block diagonal preconditioning
The form of the coefficient follows from analogy
with a radiation condition that does not reflect
normal shear waves and from the requirement of
correct physical units (Mandel 2002).
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The effect of the regularization of the solid in
block diagonal preconditioning

• Residual reduction by 3 GMRES iterations with
  block diagonal preconditioning by independent
  solvers in the fluid and in the solid, mesh
  200x200

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Dual approachesFETI

• Apportion T and Tt to local matrices and simply
  use FETI on the subassembled system
• Not tested
• Note cannot have multipliers to enforce equality
  of fluid pressure and solid displacement at wet
  interface nothing needs to equal there
• Substructure adjacent to wet interface will have
  both fluid and solid dofs
• Goal To use methods known to work for solid and
  fluid separately
• duplicate dofs on wet interface to have only
  substructures that have only fluid or only solid
  dofs
• Enforce equality of the duplicates by Lagrange
  multipliers or additional equations
• The matrix T forms other blocks in the system

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Variant 1 FETI with interface segments as new
substructures

• Duplicate dofs on wet interface just like dofs
  are duplicated on substructure interfaces in
  standard subassembly create new substructures
  with the duplicate dofs on the wet interface
• Have 3 types of substructures solid, fluid, wet
  interface
• Enforce equality of duplicated dofs by Lagrange
  multipliers
• Eliminate dofs, keep Lagrange multipliers.get
  FETI

Lagrange multipliers
But this did not work very well maybe missing
coarse for interface?
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Variant 2 FETI with system augmentation

• Goal exploit the numerically weak coupling
  between the fluid and the solid a method that
  converges like fluid and solid separately
• Inspired by FETI-DP, leave something primal
  around
• Duplicate dofs on wet interface
• Keep duplicates in the system
• Keep equations enforcing equality of duplicated
  dofs in the system
• Eliminate substructure dofs, keep Lagrange
  multipliers and the duplicated primal dofs on the
  wet interface

Interface matrix

Eliminate interiors
Lagrange multipliers
Fluid substructures
Wet interface substructures
Solid substructures
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Augmented system

Original equations
Duplicate dofs on wet interfaces equal
select wet interface dofs
Now eliminate the original variables
.
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Reduced system after eliminating original primary
variables
0

Feti operators for fluid and solid
0
0
In the limit for stiff obstacle, the reduced
system becomes triangular. The diagonal blocks
are FETI operators for fluid and solid, and
identity. The spectrum of the reduced operator
becomes union of the spectra of the two FETI
operators, and the number one.
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Coarse problem

• Variational coarse correction in the usual way,
  using plane waves or eigenfunctions
• For better convergence the wet interface
  components also need have coarse space functions
• Setup and solution of the coarse problem is a
  dominant cost
• Coarse space needs to be large enough for
  convergence

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Convergence

• OK in 2D, structured meshes
• About same as max of iterations for solid or
  fluid separately
• Not so good in 3D unstructured meshes
• About as sum of the iterations for fluid and
  solid separately

Why?
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Convergence of GMRES

• GMRES convergence depends on
• clustering of the spectrum
• Estimates exist for spectrum on one side of
  origin
• Convergence better when eigenvalues are clustered
  away from origin bad when eigenvalues scattered
  around
• condition of the matrix of eigenvectors
• But there is no reason why the convergence of
  GMRES for a block diagonal matrix should be
  rigorously bounded by a formula involving
  iteration counts when GMRES is applied to the
  blocks separately
• Even if the spectrum of the matrix is the union
  of the spectra of the blocks

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Preconditioning by coarse problem with waves
focuses the spectrum
2d reduced operator, fluid only
Same, preconditioned by coarse problem from plane
waves
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Preconditioning by coarse problem with waves
focuses the spectrum
2d reduced operator, elastic only
Same, preconditioned by coarse problem from plane
waves
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Spectrum of preconditioned reduced augmented
system, 2d structured mesh
Coupled only
Coupled overlaid by fluid and solid
Bluefluid, greensolid, redcoupled
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Spectrum of preconditioned reduced augmented
system, 3d unstructured mesh
Coupled only
Coupled overlaid by fluid and solid
Bluefluid, greensolid, redcoupled
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Conclusions for coupled FETI

• Spectrum of the coupled problem is almost exactly
  the union of the spectrum of the fluid and the
  solid, because the coupling is weak
• For the 3d unstructured problem, the union is not
  well clustered away from origin
• While for 2d structured the spectra fit well
  together
• Unknown if the culprit is 3d or unstructured
• This problem will be shared by every method that
  runs FETI-H for fluid and for solid together
• For FETI-DP the situation may be different