Partitioned Analysis of Coupled Systems

Transcription

1 Partitioned Analysis of Coupled Systems Hermann G. Matthies, Rainer Niekamp, Jan Steindorf Technische Universität Braunschweig Brunswick, Germany

2 Coupled Problems 2 Coupled problems often combine the models of two or more physical systems, they are multi-physics modells. Different approaches: Monolithical approach, which means one global model of everything Advantages: All encompassing theoretical and numerical treatment Disadvantages: Treatment is ever more complex, every time completely new start, new algorithms, new software, does not scale, not modular Partitioned approach, which means separate models plus coupling Advantages: Complexity constrained to one physical domain, theory may be well worked out, efficient numerical algorithms, existing sophisticated software for each subsystem, modular, scalable Disadvantages: Subsystems have to be coupled together, new numerical and algorithmic problems, coupling software necessary

3 Example: Fluid-Structure-Coupling 3 Fluid incompressible Newtonian fluid (i.e. Navier-Stokes eqns.) in ALE formulation: ϱ f ( v + (v χ) v) div σ + p = r f, 2σ = ν( v + ( v) T ) = 2ν s v, div v = 0, plus boundary conditions. Solid large deformation elastic St. Venant material in Lagrangean formulation: ϱ s ü DIV (F S) = r s, F = I + GRAD u S = λ(tr E)I + 2µE, 2E = (C I), C = F T F, plus boundary conditions. Arbitrary Lagrangean-Eulerian coordinate system: Lχ = β Γ u

4 Example: FSI Interface 4 Conditions on interface Γ I : At spatial location χ(t) = χ 0 + u(χ 0, t) Γ I continuity of velocities: v(χ(t), t) = u(χ 0, t). Variational formulation for velocity condition: Γ I τ I (v(χ(t), t) u(χ 0, t)) dγ I = 0 Conservation of momentum balance of tractions: (σ pi) n = 1 J F SF T n, J = det F. Variational formulation for traction condition treat like any other boundary traction, boundary traction is equal to Lagrange multiplier τ I.

5 Pure Differential Coupling 5 The simplest case is pure differential coupling: The first subsystem as evolution equation in some space X 1 : ẋ 1 = f 1 (x 1, x 2 ), x 1 X 1, The second subsystem as evolution equation in some space X 2 : ẋ 2 = f 2 (x 2, x 1 ), x 2 X 2, Combined nothing but a evolution equation for (x 1, x 2 ) X 1 X 2, direct identification of differential variables in both subsystems. Might have been produced by the partition of monolithic system, or by combination of subsystems with identifiable variables

6 Pure Explicit Coupling 6 Assume that subsystems have been discretised in time (and in space if desired), assume for simplicity same time-step in both subsystems. Approximation at time-step n denoted by x (n) j, (j = 1, 2), with explicit or implicit time-discrete evolution ϕ j, with functions Ψ j (n, t) to approximate evolution of variable x j in [t n 1, t n ]. x (n) 1 = ϕ 1 (x (n 1) 1, Ψ 2 (n, t)), x (n) 2 = ϕ 2 (x (n 1) 2, Ψ 1 (n, t)), Ψ j (n, t) are most easily produced by extrapolation of past values of x (m) j, (j = 1, 2; m = n 1, n 2,...). Simplest case is constant extrapolation pure weak or loose coupling (switching): Ψ j (n, t) x (n 1) j, (j = 1, 2)

7 Explicit Coupling Staggering 7 Advantages: Absolutely simple, can be performed in parallel Disadvantages: Critical time step will appear or may decrease, in case of simple Ψ j (n, t) only first order accurate in t, better extrapolation Ψ j (n, t) decreases stability limit. To achieve better method sacrificing inherent parallelism take as before Ψ 2 (n, t) x (n 1) 2 but then in a Gauss-Seidel fashion to give the basic staggering method: Ψ 1 (n, t) = ϕ 1 (x (n 1) 1, Ψ 2 (n, t)) = x (n) 1.

8 Implicit Coupling 8 Assume that subsystems integrators are implicit, x (n) 1 = φ 1 (x (n) 1, x(n 1) 1, Ψ 2 ), x (n) 2 = φ 2 (x (n) 2, x(n 1) 2, Ψ 1 ), Ψ j (n, t) an interpolation including still unknown x (n) j, simplest case purely constant extrapolation strong or tight coupling: Ψ j (n, t) x (n) j, (j = 1, 2), requires global iteration simplest case as before in time-stepping: in Jacobi or additive Schwarz fashion, or in Gauss-Seidel or multiplicative Schwarz fashion. Advantages: May be globally unconditionally stable, may be higher order in t without compromising stability, same results as monolithical approach. Disadvantages: Requires global iteration, but will converge for small t.

9 Differential-Algebraic Coupling 9 Assume that subsystems are differential-algebraic equations (DAEs) with local differential variables x 1 X 1, local algebraic variables y 1 Y 1 : ẋ 1 = f 1 (x 1, y 1, z), 0 = g 1 (x 1, x 2, y 1, z), same for the second subsystem with local differential variables x 2 X 2, local algebraic variables y 2 Y 2 : ẋ 2 = f 2 (x 2, y 2, z), 0 = g 2 (x 2, x 1, y 2, z). Coupling conditions formulated as algebraic constraints with global algebraic variables z Z: 0 = h(x 1, x 2, y 1, y 2, z).

10 Differential-Algebraic Regularity 10 Assume that each single subsystem, and also global system is an index-1 DAE. This means that the operator matrices D yj g j, [ ] Dyj g j D z g j, (j = 1, 2), D yj h D z h [ Dy g ] D z g D y h D z h have to be regular, where D q is the partial derivative w.r.t. q, and we have set g = (g 1, g 2 ) T and y = (y 1, y 2 ) T. After time discretisation we time-discretise DAEs with an implicit method we have a global system of equations: (x (n) 1, y(n) 1 )T = Φ 1 (x (n) 1, x(n 1) 1, y (n) 1, y(n 1) 1, z (n), z (n 1) ), (x (n) 2, y(n) 2 )T = Φ 2 (x (n) 2, x(n 1) 2, y (n) 2, y(n 1) 2, z (n), z (n 1) ), 0 = h(x (n) 1, x(n) 2, y(n) 1, y(n) 2, z(n) ).

11 Discrete Form of Fluid-Structure-Interaction 11 The Fluid and the ALE-Domain: M f v + N(v χ)v + K f v + B f p = r f + T T f τ I ;, B T f v = 0, K g χ = Au. The Solid: The Interface: M s ü + K s (u)u = r s T T s τ I. T f v = T s u. Equality of interface tractions on coupling interface already included in terms T T f τ I and T T s τ I.

12 The DAE Correspondence 12 The Fluid and the ALE-Domain: (with b := v, the fluid acceleration) and ψ = χ: [ v x 1 =, y χ] 1 = b [ p b, f 1 =, z = τ ψ] I, ψ M f b + N f (v ψ)v + K f v + B f p r f T T f τ I g 1 = B T f M 1 f ( N f(v ψ)v B f p K f v + r f + T T f τ I), K g ψ Aw The Solid: (with a := ẇ = ü, the structural acceleration). [ [ u w x 2 =, y w] 2 = a, f 2 =, z = τ a] I The Interface: g 2 = M s a + K s (u)u r s + T T s τ I ;, h = T f b T s a.

13 Global Equations for Strong Coupling 13 Coupling condition h = 0 usually included with one subsystem in solution process. Set ξ := (x 1, y 1, z) T = (v, χ, b, p, ψ, τ I ) T and ζ := (x 2, y 2 ) T = (u, w, a) T to include interface in first equation, otherwise include z = τ I in ζ and not in ξ. Assume that convergent iterative solvers for subsystems exist: and ξ κ = F 1 (ξ κ 1, ζ), κ = 1, 2,... ; ζ κ = F 2 (ζ κ 1, ξ), κ = 1, 2,... ; Simplest solution process is nonlinear block-jacobi, an additive or parallel Schwarz procedure: ξ κ = F ν 1 1 (ξ κ 1, ζ κ 1 ), ζ κ = F ν 2 2 (ζ κ 1, ξ κ 1 ) ;

14 Nonlinear block-gauss-seidel 14 Almost as simple is nonlinear block-gauss-seidel, a multiplicative or serial Schwarz procedure: ξ κ = F ν 1 1 (ξ κ 1, ζ κ 1 ), and with newly computed ξ κ, do ζ κ = F ν 2 2 (ζ κ 1, ξ κ ). Theorem:[Arnold, Günther] In block-g-s, let L be Lipschitz-constant of Ψ j, and let ( ) 1 α = max (D y 2 g 2 ) 1 D z g 2 D y1 h (D y1 g 1 ) 1 D z g 1 Dy2 h, t [0,T ] the iteration only converges if α < 1, and if at least κ iterations are performed so that Lα κ < 1, and the total global time-step error δ is bounded by δ < C(µ max{0,κ 2} ψ(x) + µ κ 1 ψ(y)) + ε 1 (x) + ε 2 (y), ψ is extrapolation error, ε j is subsystem integrator error, and µ = α + O( t) < 1.

15 Eliminating one Variable 15 One may see block-g-s in the following: F ν 1 1 : ζ κ 1 ξ κ, followed by: (ξ becomes internal ) F ν 2 2 : ξ κ ζ κ. In toto, there is a mapping on ζ alone: S : ζ κ 1 ζ κ ζ may be just the variables on interface. Fixed-point of the map S is part of the solution The fixed-point equation may be solved by some other method (e.g. preconditioned, Newton, Quasi-Newton, etc.)

16 Different Possibilities for block-gauss-seidel 16 For different ordering and distribution of constraint, we have 1st fluid plus coupling, 2nd solid: α = M 1 s T T s (T f M 1 T T f ) 1 T s,, where M 1 = M 1 f (M f B f M p B T f M p = (B T f M 1 f B f) 1. )M 1 f is a Schur complement, and 1st structure plus coupling, 2nd fluid: α = M 1 T T f (T sm 1 s T T s ) 1 T f. 1st fluid, 2nd solid plus coupling: α = (T s M 1 s T T s ) 1 (T f M 1 T T f ). 1st solid plus coupling, 2nd fluid: α = (T f M 1 T T f ) 1 (T s M 1 s T T s ). α depends on ϱ f /ϱ s.

17 Block-Newton 17 Desirable is an iteration scheme which will not depend on ordering and distribution of constraint: Block-Newton. In each block-newton iteration following system has to be solved: [ I Dξ F 1 D ζ F 1 D ξ F 2 Symbolic block-gauss elimination: ] [ ] ξκ I D ζ F 2 ζ κ = [ ] ξκ F 1 (ξ κ, ζ κ ). ζ κ F 2 (ζ κ, ξ κ ) ξ = (I D ξ F 1 ) 1 (ξ F 1 (ξ, ζ)) C ζ, with the multiplier matrix C := (I D ξ F 1 ) 1 [D ζ F 1 ]. Further with Schur complement matrix S: S ζ := (I [D ζ F 2 ] [D ξ F 2 ]C) ζ = r, with r := (ζ F 2 (ζ, ξ)) + [D ξ F 2 ]q, q := (I D ξ F 1 ) 1 (ξ F 1 (ξ, ζ)).

18 Solving the Block-Newton System 18 Solution proceeds by Krylov method (Bi-CGstab): Solving a system with (I D ξ F 1 ): Apply iterative solver F 1 Same with C, plus finite differences for [D ζ F 1 ] Solving the Schur-complement system: Use Bi-CGstab. Compute r with iterating subsystem solver F 2 ; compute action of S by finite differences. Theorem:[Mackens, Voss] If the single system solvers are quadratically convergent (or enough iterations are made in the approximative steps), the global iteration is also quadratically convergent.

19 Quasi-Newton 19 Quasi-Newton methods are generalisations of secant method: [H κ ] [ ] ξκ ζ κ = [ ] ξκ F 1 (ξ κ, ζ κ ). ζ κ F 2 (ζ κ, ξ κ ) Easy to solve with H κ (explicit inverse H 1 κ ). H κ changes by low rank only from step to step. H 1 κ = H 1 κ 1 + a κ b T κ a rank one update or a rank two update H 1 κ = H 1 κ 1 + a κ a T κ + +b κ b T κ a κ, b κ are easy to compute from known data

20 Ausflußrand A Simple Example 20 Fester Rand Einflußrand 1.0 Starrer Körper Elastische Struktur Fester Rand

21 Movement and Pressure Distribution 21

22 Tip Displacement Response Weak Coupling y Verschiebungen am Strukturende Simulationszeit t: dt = 0.02

23 Tip Displacement Response Strong Coupling y Verschiebungen am Strukturende Simulationszeit t: dt = 0.02

24 Iteration Count Approximatives Block Newton Block Gauß Seidel Iterationszahl Simulationszeit t: dt = 0.02

25 Solver Calls Approximatives Block Newton Block Gauß Seidel 80 Anzahl von Löseraufrufen Simulationszeit t: dt = 0.02

26 Another Example 26 Material St.Venant: η = 0.2, E = E l Navier-Lamé: η = 0.2, E = E r

27 Iteration Count 27 E l = 10 5, E r = E l = 10 5, E r = 10 6 E l = , E r = 10 6 iter cpu[t] Jacobi Gauss-Seidel BFGS Newton iter cpu[t] Jacobi - GS BFGS Newton iter cpu[t] Jacobi - GS - BFGS Newton

28 Conclusions 28 staggering algorithms may introduce critical time step coupling may introduce algebraic constraints DAEs are different from pure differential coupling block-gauss-seidel depends strongly on ordering in purely differential case may be made convergent with small t in DAE case may be unconditionally unstable Newton-methods are more robust