An Iterative Substructuring Method for Coupled Fluid-Solid Acoustic - PDF document

An Iterative Substructuring Method for Coupled Fluid-Solid Acoustic Problems Jan Mandel University of Colorado Supported the Office of Naval Research under grant N-00014-95-1-0663, and the National Science foundation under grants ECS-9725504 and DMS-007428. 1

Model problem Γ n ✲ ν Ω e Γ d Γ a Γ Ω f Γ n Pressure-displacement formulation ∆ p + k 2 p = 0 in Ω f , ∂p ∂p p = p 0 on Γ d , ∂ν = 0 on Γ n , ∂ν + ikp = 0 on Γ a . ∇ · τ + ω 2 ρ e u = 0 in Ω e , e ij ( u ) = 1 2( ∂u i + ∂u j τ = λI ( ∇ · u ) + 2 µe ( u ) , ) , ∂x j ∂x i 1 ∂p ν · u = ν · τ · ν = − p, ν × τ · ν = 0 on Γ ∂ν, ρ f ω 2 2

Variational form and discretization Define the spaces V f = { q ∈ H 1 (Ω f ) | q = 0on Γ d } V e = { u ∈ ( H 1 (Ω e )) n } Variational form: Find p , p − p 0 ∈ V f and u ∈ V e such that for all q ∈ V f and v ∈ V e , ∇ p ∇ q + k 2 ρ f ω 2 ( ν · u ) q = 0 � � � � − pq − ik pq − Ω f Ω f Γ a Γ � λ ( ∇· u )( ∇· v )+2 µe ( u ) : e ( v )+ ω 2 � � − ρ e u · v − p ( ν · v ) = 0 Ω e Ω e Γ This variational formulation and discretization by con- forming elements are well known (Morand and Ohayon 1995). 3

Discretized system  − K f + k 2 M f − ik G f − ρ f ω 2 T        p  r  =       − T ∗ − K e + ω 2 M e  u 0  p ∗ K f q = � ∇ p ∇ q, Ω f p ∗ M f q = � pq, Ω f p ∗ G f q = � pq, Γ a u ∗ K e v = � λ ( ∇ · u )( ∇ · v ) + 2 µe ( u ) : e ( v ) , Ω e u ∗ M e v = ω 2 � ρ e u · v, Ω e p ∗ Tv = � p ( ν · v ) . Γ 4

Decomposition Non-overlapping subdomains: N f N e s =1 Ω s s =1 Ω s Ω f = Ω e = � e , � e . Vector of all subdomain dofs:  p 1   u 1  . .     . .     p = . u = . ˆ  , ˆ  ,             p N f u N e   Corresponding partitioned matrices with subdomain blocks defined by subassembly, K 1  0  f . . . . . ...   . . ˆ p s ∗ K s � . .   K f =  , f q = ∇ p ∇ q,      N f  Ω s  0 . . . K f f ( ˆ K e ˆ M f ˆ M e defined similarly)  T 11 T 1 ,N e  . . . . .  ...  p r ∗ T rs v s = . . ˆ �   T = . . p ( ν · v )  ,     T N f , 1 . . . T N f ,N e   ∂ Ω r f ∩ ∂ Ω s  e 5

Intersubdomain continuity Local to global maps N f and N e : K f = N ∗ f ˆ K e = N ∗ e ˆ K f N f , K e N e p = N f p , u = N e u . ˆ ˆ To enforce same values between subdomains: N f B f = ( B 1 B e = ( B 1 e , . . . , B N e f ) , e ) f , . . . , B such that p = 0 ⇐ ⇒ p = N f p for some B f ˆ ˆ p u = 0 ⇐ ⇒ u = N e u for some B e ˆ ˆ u . Decomposed system K f + k 2 ˆ   − ˆ − ω 2 ρ f ˆ B ∗     0 M f T p ˆ ˆ r f   K e + ω 2 ˆ     − ˆ − ˆ  T ∗ B ∗  0 M e 0  u ˆ          e   =            0 0 0  λ f 0 B f                       0 0 0 λ e 0 B e   ( ˆ u , λ f , λ e ) equivalent to the original system p , ˆ via ˆ p = N f p and ˆ u = N e u . 6

Regularized system  ˆ − ω 2 ρ f ˆ  T B ∗     0 A f p ˆ ˆ r f       − ˆ ˆ  T ∗ B ∗  0 u ˆ 0 A e           e   =            0 0 0  λ f 0 B f                      0  0 0 0 λ e B e   where K f + k 2 ˆ A f = − ˆ ˆ M f + ˆ R f K e + ω 2 ˆ A e = − ˆ ˆ M e + ˆ R e p s ∗ R f q s = ik ˆ R f = ( R rs � f ) rs , t � = s σ st pq, � ∂ Ω s f ∩ ∂ Ω t f u s ∗ R e v s = iω ˆ R e = ( R rs � e ) rs , t � = s σ st ( n · u )( n · v ) , � ∂ Ω s e ∩ ∂ Ω t e where σ st = ± 1, σ st = − σ st . f + k 2 ˆ f , then − ˆ If σ st does not change sign on Ω s K s M s f + ˆ R r f is regular (Farhat, Macedo, Lesoinne 2000). Similary for solid subdomains. In computational tests, we assigned σ st by counting, did not try to avoid change of sign. 7

Augmented system Key: ˆ u , ˆ T ∗ ˆ p depend on the values of ˆ u , ˆ p on the wet Tˆ interface Γ only. Define ˆ J f , ˆ J e as expanding vector on Γ by zero entries, then ˆ u = ˆ u Γ = J ∗ Tˆ TJ e ˆ u Γ , ˆ e ˆ u , T ∗ ˆ ˆ p = ˆ T ∗ J f ˆ p Γ = J ∗ p Γ , ˆ f ˆ p . Get the augmented system  ˆ − ω 2 ρ f ˆ      B ∗ 0 0 0 A f TJ e p ˆ r ˆ f       ˆ e − ˆ B ∗ T ∗ J f       0 0 A e 0 0 u ˆ                   0 0 0 0 0 λ f 0  B f            =             0 0 0 0 0 λ e 0 B e                   − J ∗       0 0 0 0 p Γ ˆ 0 I       f             − J ∗ 0 0 0 0 0 ˆ u Γ  I      e Proposed method • eliminate ˆ p and ˆ u from the augmented system • solve the resulting reduced system by Generalized Conjugate Residuals • precondition by scaling and projection on a coarse space defined by rigid body modes and plane waves 8

Scaling • multiply the second equation by ω 2 ρ f • symmetric diagonal scaling ˜ ˜ − ˜  B ∗      0 0 0 ˜ ˜ A f TJ e p r f       ˜ ˜ e − ˜ B ∗ T ∗ J f    ˜    0 A e 0 0 u 0              ˜   ˜    0 0 0 0 0 λ f 0  B f            = ,       ˜ ˜       0 0 0 0 0 0 B e λ e                    − J ∗      0 0 0 0 0 ˜ I p Γ       f             − J ∗ 0 0 0 0 0 ˜  I   u Γ    e where ˜ ˜ ˜ A f = D f ˆ A e = ω 2 ρ f D e ˆ T = ω 2 ρ f D f ˆ A f D f , A e D e , TD e , ˜ ˜ B f = E f B f D f , B e = E e B e D e , ˜ r = D f ˆ r , λ f = D f ˜ λ e = ω 2 ρ f D f ˜ p = D f ˜ u = D e ˜ ˆ p , ˆ u , λ f , λ f . 9

Reduced system Eliminating p = ˜ r − ˜ f ˜ λ f + ˜ A − 1 B ∗ ˜ f (˜ TJ e ˜ u Γ ) u = ˜ e ( − ˜ e ˜ λ e + ˜ A − 1 B ∗ T ∗ J f ˜ ˜ p Γ ) gives Fx = b , where B f ˜ ˜ f ˜ − ˜ B f ˜ f ˜  A − 1 A − 1  B ∗ 0 0 TJ e f   B e ˜ ˜ e ˜ B e − ˜ B e ˜ e ˜  A − 1 A − 1 T ∗ J f  0     F = ,   f ˜ f ˜ f ˜ f ˜ A − 1 A − 1 J ∗ B ∗ J ∗  0  I TJ e   f    J e ˜ e ˜ − J e ˜ e ˜  A − 1 A − 1 T ∗ J f 0 B e I   and B f ˜ ˜ A − 1    f ˜  λ f r     0  λ e        x = b = , .     f ˜    A − 1  − J ∗ ˜ f ˜ p Γ r                 ˜ 0 u Γ First diagonal block is FETI-H operator, 2nd analogue for elasticity. Assuming FETI-H operator is well con- ditioned, the reduced system has the form of Fredholm integral equation of 2nd kind, hence well conditioned! 10

Iterative solution Enforce the residual condition Q ∗ ( Fx − b ) = 0 throughout the iterations. For given v use the initial approximation x (0) = v + Qw , w obtained by solving the residual correction equation, Q ∗ ( F ( v + Qw ) − b ) = 0 . Fx = b solved by GCR with left preconditioning by the projection P = I − Q ( Q ∗ FQ ) − 1 Q ∗ F and initial iterate x (0) . Equivalently, GCR applied to PFx = Pb . Iterations run in a subspace: Q ∗ F ( x ( n ) − x (0) ) = 0 11

Selection of coarse space   0 0 0 D f B f Y f   0 D e B e Y e 0 0     Q =    D f J ∗  0 0 f Z f 0       D f J ∗  0 0 0  f Z e Coarse selection for multipliers: Y f = diag( Y s f ), columns of Y s f are discrete repre- sentations of plane waves in a small number of equally distributed directions, or discrete representation of the constant function. Y e = diag( Y s e ), columns of Y s e are discrete represen- tations of elastic plane waves (both pressure and shear) in a small number of equally distributed directions, or discrete representation of the rigid body motions Coarse selection for wet interface: The matrices Z s f and Z s e are chosen in the same way as Y s f and Y s e , with possibly different selection of the number of directions and selection of constant or rigid body modes. Some of the matrices Y s f , Y s e , Z s e , or Y s e may be void. 12