✬ ✩ Iterative Solvers for Coupled Fluid-Solid Scattering Jan Mandel Center for Computational Mathematics Department of Mathematics University of Colorado at Denver Center for Aerospace Structures Department of Aerospace Engineering University of Colorado at Boulder http://www-math.cudenver.edu/ ∼ jmandel Joint work with Charbel Farhat, Mirela Popa, and Radek Tezaur Supported by the Office of Naval Research under grant N-00014-95-1-0663, and the National Science foundation under grant DMS-007428. University of Kentucky January 15, 2003 ✫ ✪
✬ ✩ University of Colorado Jan Mandel Outline • The coupled scattering problem • Discretization by Finite Elements • Multigrid Method – Multigrid algorithms – Computational results in 2D • Subsctructuring Method – Substructuring by Lagrange multipliers for Helmholtz equation – Extension to elastic scattering – Extension to coupled problem – Computational results in 2D and 3D ✫ ✪
✬ ✩ University of Colorado Jan Mandel Main results • treating a coupled problem with vastly different scales ⇒ algorigthms are invariant to scaling of physical units ⇒ physical units must match ⇒ define residual with care • Multigrid on a properly scaled coupled problem with GMRES as smoother converges well if coarse problem fine enough • extended FETI-H (De La Bourdonnaye, Farhat, Macedo, Magoul` es, Roux, 1998, 2000; Farhat, Macedo, Tezaur, DD11, 1999) to coupled problem ⇒ coupled algorithm reduces to FETI- H in the limit for a very stiff obstacle ✫ ✪
✬ ✩ University of Colorado Jan Mandel Coupled problem Γ n ✲ ν Ω e Γ d Γ a Γ Ω f Γ n • solid obstacle in fluid • time harmonic fluid pressure p ( x, t ) = p ( x ) e iωt = ⇒ Helmholtz equation in the fluid • time harmonic solid displacement u ( x, t ) = u ( x ) e iωt = ⇒ elastodynamic Lam´ e equation in the solid • continuity of displacement & balance of normal forces = ⇒ wet interface conditions ✫ ✪
✬ ✩ University of Colorado Jan Mandel Coupled Problem Fluid Medium -the Helmholtz Equation • isotropic, homogeneous, inviscid • compressible, irrotational • time-harmonic solution of the wave equation • complex amplitude p ( x ): the fluid pressure at location x and time t is p ( x ) e iωt − ∆ p − k 2 p = f in Ω p = p 0 on Γ d ∂p = 0 on Γ n ∂ n ∂p ∂ n + ikp = 0 on Γ a (radiation b.c.) • k = ω c f , ω is wave frequency, c f is speed of sound in acoustic medium • radiation boundary condition - Sommerfeld – does not reflect waves in the normal direction – first-order approximation for radiation to infinity ✫ ✪
✬ ✩ University of Colorado Jan Mandel Coupled Problem Elastic Medium • isotropic, homogeneous • small time-harmonic displacement u ( x ) e iωt of an elastic body • complex displacement amplitude u ( x ) satisfies the Lam´ e equations ∇ · τ + ω 2 ρ e u = 0 in Ω e + boundary conditions, where : density of the elastic medium ρ e − = λI ( ∇ · u ) + 2 µe ( u ) stress tensor τ 1 2( ∇ u + ( ∇ u ) T ) e ( u ) = strain tensor λ, µ − Lam´ e coefficients of elastic medium ✫ ✪
✬ ✩ University of Colorado Jan Mandel Coupled Problem Fluid-Solid Interface 1. continuity across interface 1 ∂p n · u = ρ f ω 2 ∂ n 2. balance of normal forces: n · τ · n = − p 3. nonviscous fluid → zero tangential tension: n × τ · n = 0 ✫ ✪
✬ ✩ University of Colorado Jan Mandel Coupled Problem Summary of the Pressure-Displacement Formulation Helmholtz equation in the fluid region: ∆ p + k 2 p = 0 in Ω f , ∂p ∂p p = p 0 on Γ d , ∂ n = 0 on Γ n , ∂ n + ikp = 0 on Γ a . Lam´ e equation in the elastic region: ∇ · 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 ) , ) , ∂ u j ∂x i Wet interface conditions: 1 ∂p n · u = n · τ · n = − p n × τ · n = 0 on Γ ρ f ω 2 ∂ n On Γ , the value of u provides load for the Helmholtz problem for p and the value of p provides load for the elastodynamic problem for u . ✫ ✪
✬ ✩ University of Colorado Jan Mandel Existence of solution and non-radiating modes Solution exists ( p, u ) ∈ H 1 (Ω f ) × H 1 (Ω e ) 3 and is unique up to non-radiating modes in the solid • the boundary value problem ∇ · τ + ω 2 ρ e u = 0 in Ω e τ · n = 0 on Γ u · n = 0 on Γ has nonzero solution for certain frequencies and geometries - called non-radiating modes . • for this to happen, ω 2 ρ e needs to be an eigenvalue of the pure traction problem and in addition u · n = 0. • bodies with certain symmetries have non-radiating modes (e.q., sphere can sustain torsional oscillations with zero radial component displacement) • almost all elastic bodies do not have non-radiating modes [Harg´ e] ✫ ✪
✬ ✩ University of Colorado Jan Mandel Variational Formulation Find ( p, p Γ ), ( p − p 0 , p Γ ) ∈ V f and ( u , n · u Γ ) ∈ V e such that � � � � q + k 2 ρ f ω 2 ( n · u Γ )¯ ∇ p ∇ ¯ p ¯ p Γ ¯ q Γ = 0 − q − ik q Γ − Ω f Ω f Γ a Γ � � v )) + ω 2 − ( λ ( ∇ · u )( ∇ · ¯ v ) + 2 µ e ( u ) : e ( ¯ ρ e u · ¯ v − Ω e Ω e � p Γ ( n · ¯ v Γ ) = 0 Γ ∀ ¯ q, ¯ q Γ ∈ V f and ∀ ¯ v , n · ¯ v Γ ∈ V e , where p ∈ H 1 (Ω f ) , p Γ ∈ H 1 V f = { ( p, p Γ ) | 2 (Γ) | p = 0 on Γ d } 1 u ∈ ( H 1 (Ω e )) 3 , u Γ ∈ ( H 2 (Γ)) 3 } , V e = { ( u , n · u Γ ) | Same wet interface term in both equations = ⇒ symmetric system Industry standard (Morand and Ohayon 1995) ✫ ✪
✬ ✩ University of Colorado Jan Mandel Discretized system Finite elements = ⇒ 2 × 2 symmetric block 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 T is the coupling matrix: p ∗ Tv = � p ( ν · v ) Γ T is transfer of load = ⇒ lower order, weak coupling ✫ ✪
✬ ✩ University of Colorado Jan Mandel Scaling Since λ and µ are large ⇒ use a scaling of the form u = s u ′ and v = s v ′ , where s is a scalar, to control leading terms Discretization and Error Bound Iterative Soution leading terms differ by factor of k 2 comparable leading terms consistent with definition of norm scale to O (1) diagonal 1 1 s = � s = � . c ρ f max { λ, 2 µ } ck ρ f max { λ, 2 µ } � � � � − S f + k 2 M f + ik G f − ρ f ω 2 s T p = R − ρ f ω 2 s T t − ρ f ω 2 s 2 S e + ρ e ρ f ω 4 s 2 M e u ✫ ✪
✬ ✩ University of Colorado Jan Mandel Finite element error estimates If the solution ( p, u ) ∈ H 1+ α (Ω f ) × H 1+ α (Ω e ) 3 and h is small enough then the discretization error satisfies � p − p h � 2 H 1 (Ω f ) + k 2 � p − p h � 2 L 2 (Ω f ) + k 2 ( � u − u h � 2 H 1 (Ω e ) 3 + k 2 � u − u h � 2 L 2 (Ω e ) 3 ) ≤ C ( k ) h 2 α Proof: G˚ arding inequality, scales of spaces, regularity, approximation. PhD thesis of M. Popa, 2002 ✫ ✪
✬ ✩ University of Colorado Jan Mandel Model Problem Γ n ✲ n Ω e Γ d Γ a Γ Ω f Γ n • 2D channel filled with water, unit square • excitation on the left (Dirichet boundary condition) on Γ d • sound-hard sides(Neumann boundary condition) on Γ n • outgoing radiation on the right (absorbing boundary condition) on Γ a • aluminum scatterer in the middle • discretization by uniform mesh, P1 elements ✫ ✪
✬ ✩ University of Colorado Jan Mandel Numerical Solution Square scatterer size 0.2 m in the middle Γ n Γ a Γ n 1.5 x−axis Γ a 0 30 Γ d −1.5 Γ d Γ n Γ n 30 60 60 x−axis 30 30 60 60 y−axis y−axis Fluid pressure Solid displacement ✫ ✪
✬ ✩ University of Colorado Jan Mandel Numerical Solution Obstacle with a gap Γ a Γ a Γ n Γ n 3 2 1 0 0 −1 −2 −3 Γ n Γ n Γ d Γ d 60 60 20 40 30 40 30 x−axis 20 x−axis y−axis y−axis 60 60 Γ n Γ n 20 20 Γ d Γ a x−axis Γ d x−axis Γ a 40 40 Γ n Γ n 60 60 20 40 60 20 40 60 y−axis y−axis gap on x axis gap on y axis ✫ ✪
✬ ✩ University of Colorado Jan Mandel Multigrid for the Coupled Problem • coarsening: standard variational bilinear interpolation, mesh ratio 2 • smoothing – GMRES, BICG-STAB – Preconditioners ∗ inverse of lower triangular part of A ∗ inverse of nodal block diagonal – Gauss-Seidel • for stable smoothers can increase number of steps without causing divergence • all smoothers work well for fluid or solid part separately • wet interface aligned with coarsest grid, otherwise nothing special ✫ ✪
✬ ✩ University of Colorado Jan Mandel Multigrid convergence, scatterer size 0.2 Decreasing h , increasing k , k 3 h 2 constant Adding coarse meshes GMRES as smoother BICG−STAB as smoother −0.09 residual reduction 10 −0.1 residual reduction 10 −0.2 −0.13 10 10 −0.3 10 −0.17 10 5 2e−8 5 4 2e−82e−7 2e−7 4 2e−6 3 2e−6 2e−5 2e−5 3 h multigrid levels h multigrid levels 2 2 GMRES preconditioned by inv GAUSS−SEIDEL as smoother lower triangular part of A as smoother 10 10 residual reduction residual reduction −0.2 10 5 10 −0.3 10 0 10 −0.4 10 −5 10 5 2e−8 2e−8 5 4 2e−7 2e−7 4 2e−6 h 3 2e−5 ✫ 2e−6 ✪ 3 h multigrid levels multigrid levels 2e−5 2 2
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