Iterative Solvers for Coupled
Fluid-Solid Scattering
Jan Mandel
Work presentation
Center for Aerospace Structures
University of Colorado at Boulder
October 10, 2003
Outline
•
The coupled scattering problem
– the PDEs
– the discrete 2×2 matrix form
•
Solving the discrete equations
1. Simply proceed as usual on the matrices, or
2. Preconditioner to ignore weak coupling
between fluid and the solid blocks, or
3. Couple existing separate FETI-type methods
in the fluid and in the solid
Model coupled problem
p
n
 0 Neumann boundary condition
p  k 2 p  0  f fluid
 wet interface
e solid
p  p0
n
     2  eu  0
boundary
  I (  u)  2e(u)
condition
1 ui u j
eij (u)  (

),
2 u j xi
n
n
n u 
1 p
continuity
 f  2 n
nn  p
balance of forces
n    n  0 on  free slip
p
n
 0 Neumann boundary condition
p
n
Radiation b.c.
Dirichlet
n
 ikp  0
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
Existence of solution
• Solution exists 
p, u H1 
f H1 
e 3
• Solution is unique up to non-radiating
modes in the solid = vibrations of the solid
that have no effect on the outside
     2  eu  0 in  e
   0 on 
u   0 on 
Only bodies with certain symmetries (such as sphere) have non-radiating
modes; “almost all” bodies have no non-radiating modes
Variational formulation
Find ( p, u) such that


pq  k 2
f

pq  ik
f
  p (  v  )

q  V f and  v  Ve ,

p  q 
a
2


 f (  u  )q
0


2



(


u
)(


v
)

2

e
(
u
)
:
e
(
v
)



  eu  v  0
e
e
Discrete problem
 K f  k 2M f  ikG f

T

T

  f  2T  p r 
   
2
 K e   M e  u 0
• 2x2 block system of equations
• Coupling matrix is like a boundary load:
pT Tv   p(n  v)
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)
Scaling of the discrete problem
 K f  k 2M f  ikG f

T

T

  f  2T
 K e   2M e
 p r 
   
 u 0
• Multiply 2nd equation to make the offdiagonal blocks same; then,
• Symmetric diagonal scaling to make
both diagonal blocks of the same
magnitude O(1)
The fluid and the solid are coupled
only weakly
• Scaling the fields reveals that the fluid and the
solid are numerically decoupled when
|| T || khcf  
1/ 2
f
1/ 2
 1
(Mandel 2002)
• Completely decoupled in the limit for a stiff solid
scatterer
• Numerically almost decoupled for practically
interesting problems (aluminum, water) ||T||103
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
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)
fine
coarse
fluid
wet interface
solid
Multigrid performance
Decreasing h, k3h2 =const, adding coarse levels, 10 smoothing steps,
k=10 for h=1/32, average residual reduction smoothing step, domain 1x1
with 0.2x0.2 obstacle in the middle (Popa, 2002)
Substructuring is based on
subassembly of the block diagonal matrix
of Schur complements
 N f 1  S f 1
N 
S   f 2  t
 N e1   T22

 t
N
 e 2   T22
T11
Sf2
T21
T22t
S e1
T22t
T12   N f 1 
T22   N f 2 
, Tij 
p (n  u )

  N e1 
   i  j


Se2   N e2 
• 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
Building a substructuring method
•
The matrix of substructure Schur complements
needs to be block diagonal to get parallelism.
Some possible approaches:
1. Keep decomposition into substructures, add dofs of
the other kind to substructures adjacent to the wet
interface and apportion the matrix T
2. Primal only: ignore T in the preconditioner, same as
block diagonal preconditioner
3. Or, duplicate dofs on wet interface, T works
between the duplicates, and enforce equality of
duplicates at converged solution
1. by Lagrange multipliers
2. add new equations to the system
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
Interface matrix
T
Eliminate interiors
Fluid substructures
Solid substructures
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
= fluid dofs
=solid dofs
T
Eliminate interiors
Fluid substructures
Solid substructures
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
T
Eliminate interiors
Fluid substructures
Solid substructures
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
T
Eliminate interiors
Fluid substructures
Solid substructures
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?
T
=solid dofs
Eliminate interiors
= fluid dofs
Apportioning
needed
Solid substructures
Fluid substructures
Interface matrix
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
Block diagonal preconditioning
 Pf


  D1
Pe  
Precond.
  K f  k 2M f  ikG f

D2  
 TT
Scaling
  f  2T   D3

2
 K e   Me  

D4 
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 T only
– Need to provide artificial damping without changing
the solution
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)
“Regularization” of the matrix of the
solid for block diagonal preconditioning
 I
i Tt

2

  K f  k M f  ikG f


I 
 TT
 K f  k 2 M f  ikG f

T
 T  ...

 f T 

2
 K e   Me 
2

2
2 t 
 K e   M e  i f  T T
 f T
2
  e (  2  )
  0
The form of the coefficient follows from
|| T ||1
analogy with a radiation condition that
does not reflect normal shear waves and
from the requirement of correct physical
units (Mandel 2002).
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
Dual approaches=FETI
Apportion T and Tt to local matrices and simply use
FETI on the subassembled system
1.
–
–
–
2.
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
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
Interface matrix
Eliminate interiors
T
Lagrange
multipliers
Fluid substructures
Wet interface substructures
Solid substructures
But this did not work very well… maybe missing coarse for interface?
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
T
Lagrange
multipliers
Fluid substructures
Wet interface substructures
Solid substructures
Augmented system
Original equations
Dofs equal between
substructures
Duplicate dofs on wet
interfaces equal
S f


B f


J t
 f

B tf
Se
Bet
T tJ f
Be
I
J et
J tf , J et
Now eliminate the original variables
 TJ e   p   r 
   
  u  0 
   f  0 
    
  u  0
  p   0 
   
 I   u  0
select wet interface dofs
p, u
….
Reduced system after eliminating
original primary variables
0
Feti operators for fluid and solid
 B f S B


  J tf S f 1 B tf


1
f
0
t
f
1
e
 Be S B
t
e
1
e
t
Be S T J f
I
1
e
J S B
t
e
t
e
1
e
t
JeS T J f
B f S TJ e    f 
 
  e   ...
t
1
J f S f TJ e   p 
 
 I   u 
1
f
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.
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
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?
Convergence of GMRES
|| rn ||
min
deg( p )  n , p ( 0) 1
|| p( A)r0 ||
• 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
Preconditioning by coarse problem
with waves focuses the spectrum
2d reduced operator, fluid only
Same, preconditioned by coarse
problem from plane waves
Preconditioning by coarse problem
with waves focuses the spectrum
2d reduced operator, elastic only
Same, preconditioned by coarse
problem from plane waves
Spectrum of preconditioned reduced
augmented system, 2d structured mesh
Coupled overlaid by fluid and solid
Coupled only
Blue=fluid, green=solid, red=coupled
Spectrum of preconditioned reduced
augmented system, 3d unstructured mesh
Coupled overlaid by fluid and solid
Coupled only
Blue=fluid, green=solid, red=coupled
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