Iterative Solvers for Coupled Fluid-Solid Scattering

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 equations1. Simply proceed as usual on the matrices, or2. Preconditioner to ignore weak coupling

between fluid and the solid blocks, or3. Couple existing separate FETI-type methods

in the fluid and in the solid

Model coupled problem

n

n

n

fluid02fpkp

0

solid2

ue

e

slipfreennforcesofbalancepnn

continuitynpn

f

on 0

12

u

),(21)(

)(2)(

i

j

j

iij xe

eI

u

uuu

uu 0

ikpnp

Radiation b.c.

n

conditionboundaryDirichlet

0pp

conditionboundary Neumann 0np

conditionboundary Neumann 0np

interface wet

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• Solution is unique up to non-radiating

modes in the solid = vibrations of the solid that have no effect on the outside

p,u H1 f H1 e3

on0on0in02

u

u

ee

Only bodies with certain symmetries (such as sphere) have non-radiating modes; “almost all” bodies have no non-radiating modes

Variational formulation

Find such that

,and

0)(:)(2))(()(

0)(

2

22

ef

e

f

VVq

p

qqpikqpkqp

ee

aff

v

vuveuevuv

u

),( up

Discrete problem

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

02

22 rup

MKTTGMK

eeT

ffff ikk

)( vnpT Tvp

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

• 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)

02

22 rup

MKTTGMK

eeT

ffff ikk

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)

1|| 2/12/1 ffkhc||T

10|| 3||T

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)

fluid solid

coarsefine

wet interface

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

• 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

ji

unpT

NNNN

STTSTT

TTSTTS

NNNN

S ij

e

e

f

f

ett

ettf

f

e

e

f

f

)( ,

2

1

2

1

22222

12222

22212

12111

2

1

2

1

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 multipliers2. add new equations to the system

Substructuring choicesRespect 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

Solid substructuresFluid substructures

Eliminate interiors

Interface matrix T

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


Eliminate interiors

Interface matrix T =solid dofs= fluid dofs

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


Eliminate interiors

Interface matrix T

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


Eliminate interiors

Interface matrix T

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

Interface matrix

T

Apportioningneeded

= fluid dofs

=solid dofs

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

• 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

4

32

22

2

1

DDikk

DD

PP

eeT

ffff

e

f

MKTTGMK

T

Precond. Scaling Scaling

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

1||T

TTMKTTGMK

MKTTGMK

T

||)2(

...

0

22

22

2

22

e

tfee

Tffff

eeT

fffft

iikk

ikkIi

I

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).

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=FETI1. 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

2. 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


Eliminate interiors

Interface matrix T

Wet interface substructures

But this did not work very well… maybe missing coarse for interface?

Lagrange multipliers

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


Eliminate interiors

Interface matrix T

Wet interface substructures

Lagrange multipliers

Augmented system

00000r

up

up

IJIJ

BB

JTBSTJBS

u

f

te

tf

e

f

ftt

ee

etff

te

tf JJ , select wet interface dofs

Dofs equal between substructures

Duplicate dofs on wet interfaces equal

Original equations

Now eliminate the original variables ….up,

Reduced system after eliminating original primary variables

...

11

11

11

11

up

IJTSJBSJTJSJIBSJ

JTSBBSBTJSBBSB

e

f

ft

eetee

te

eftf

tff

tf

ft

eeteee

efftfff

00Feti operators for fluid and solid

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.

0

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

• 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

||)(||min|||| 01)0(,)deg(rApr

pnpn

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

Blue=fluid, green=solid, red=coupled

Coupled onlyCoupled overlaid by fluid and solid

Spectrum of preconditioned reduced augmented system, 3d unstructured mesh

Blue=fluid, green=solid, red=coupled

Coupled onlyCoupled overlaid by fluid and solid

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

Documents

Iterative Solvers for Coupled Fluid-Solid Scattering