Solving Fluid Structure Interaction Problems on Overlapping Grids

Transcription

1 Solving Fluid Structure Interaction Problems on Overlapping Grids Bill Henshaw Center for Applied Scientific Computing, Lawrence Livermore National Laboratory, Livermore, CA, USA. Virginia Tech, Blacksburg Virginia, October 2, Henshaw (LLNL) FSI on overlapping grids VaTech 1 / 46

2 Acknowledgments. Primary collaborators for this effort: Jeff Banks (LLNL), Kyle Chand (LLNL), Don Schwendeman (RPI), Björn Sjögreen (LLNL). Supported by: ASCR Department of Energy, Office of Science, ASCR Applied Math Program. LDRD LLNL: Laboratory Directed Research and Development (LDRD) program. NSF National Science Foundation. Henshaw (LLNL) FSI on overlapping grids VaTech 2 / 46

3 Outline 1 Background: overlapping grids, Overture and CG. 2 Incompressible flows and rigid bodies (split-step schemes). 1 high-order accurate factored scheme and boundary conditions. 2 matrix-free multigrid. 3 moving grid generation. 3 Compressible flow and deforming bodies (partitioned schemes). 1 Deforming Composite Grids (DCG). 2 a one-dimensional model problem. 3 an interface projection scheme (fluid-solid Riemann problem). 4 Compressible flow and light rigid bodies (partitioned schemes). 1 a one-dimensional model problem. 2 the added-mass Newton Euler equations. 3 overcoming the added-mass instability for light solids. Henshaw (LLNL) FSI on overlapping grids VaTech 3 / 46

4 The Overture project is developing PDE solvers for a wide class of continuum mechanics applications. Overture is a toolkit for solving PDE s on overlapping grids and includes CAD, grid generation, numerical approximations, AMR and graphics. The CG (Composite Grid) suite of PDE solvers (cgcns, cgins, cgmx, cgsm, cgad, cgmp) provide algorithms for modeling gases, fluids, solids and E&M. Overture and CG are available from We are looking at a variety of applications: wind turbines, building flows (cgins), explosives modeling (cgcns), fluid-structure interactions (e.g. blast effects) (cgmp+cgcns+cgsm), conjugate heat transfer (e.g. NIF holhraum) (cgmp+cgins+cgad), damage mitigation in NIF laser optics (cgmx). Henshaw (LLNL) FSI on overlapping grids VaTech 4 / 46

5 What are overlapping grids and why are they useful? Overlapping grid: a set of structured grids that overlap. Overlapping grids can be rapidly generated as bodies move. High quality grids under large displacements. Cartesian grids for efficiency. Smooth grids for accuracy at boundaries. Efficient for high-order methods. Henshaw (LLNL) FSI on overlapping grids VaTech 5 / 46

6 What are overlapping grids and why are they useful? Overlapping grid: a set of structured grids that overlap. Overlapping grids can be rapidly generated as bodies move. High quality grids under large displacements. Cartesian grids for efficiency. Smooth grids for accuracy at boundaries. Efficient for high-order methods. Henshaw (LLNL) FSI on overlapping grids VaTech 5 / 46

7 What are overlapping grids and why are they useful? Overlapping grid: a set of structured grids that overlap. Overlapping grids can be rapidly generated as bodies move. High quality grids under large displacements. Cartesian grids for efficiency. Smooth grids for accuracy at boundaries. Efficient for high-order methods. Henshaw (LLNL) FSI on overlapping grids VaTech 5 / 46

8 What are overlapping grids and why are they useful? Overlapping grid: a set of structured grids that overlap. Overlapping grids can be rapidly generated as bodies move. High quality grids under large displacements. Cartesian grids for efficiency. Smooth grids for accuracy at boundaries. Efficient for high-order methods. Henshaw (LLNL) FSI on overlapping grids VaTech 5 / 46

9 What are overlapping grids and why are they useful? Overlapping grid: a set of structured grids that overlap. Overlapping grids can be rapidly generated as bodies move. High quality grids under large displacements. Cartesian grids for efficiency. Smooth grids for accuracy at boundaries. Efficient for high-order methods. Henshaw (LLNL) FSI on overlapping grids VaTech 5 / 46

10 What are overlapping grids and why are they useful? Overlapping grid: a set of structured grids that overlap. Overlapping grids can be rapidly generated as bodies move. High quality grids under large displacements. Cartesian grids for efficiency. Smooth grids for accuracy at boundaries. Efficient for high-order methods. Henshaw (LLNL) FSI on overlapping grids VaTech 5 / 46

11 What are overlapping grids and why are they useful? Overlapping grid: a set of structured grids that overlap. Overlapping grids can be rapidly generated as bodies move. High quality grids under large displacements. Cartesian grids for efficiency. Smooth grids for accuracy at boundaries. Efficient for high-order methods. Asymptotic Performance Principle for overlapping grids As grids are refined, total CPU/memory usage can approach that of a Cartesian grid. Henshaw (LLNL) FSI on overlapping grids VaTech 5 / 46

12 Components of an overlapping grid G 1 G 1 G 2 G 2 interpolation ghost unused Left: an overlapping grid consisting of two structured curvilinear component grids, x = G 1 (r) and x = G 2 (r). Middle and right: component grids for the square and annular grids in the unit square parameter space r. Grid points are classified as discretization points, interpolation points or unused points. Ghost points are used to apply boundary conditions. Henshaw (LLNL) FSI on overlapping grids VaTech 6 / 46

13 Incompressible flows and moving bodies. Challenges: efficient solvers (esp. since grids change at every time step). high-order accuracy including the boundary. efficient grid generation. Henshaw (LLNL) FSI on overlapping grids VaTech 7 / 46

14 Incompressible Navier-Stokes. u t +(u )u + p ν u f = 0, t > 0, x Ω u = 0 t > 0, x Ω Divergence damping term: α u is important. Wall boundary conditions: u = 0, u = 0, (pressure BC) x Ω, with numerical boundary condition: p n = n ( ν u )+n f. Use u instead of u for implicit time-stepping. Henshaw (LLNL) FSI on overlapping grids VaTech 8 / 46

15 Incompressible Navier-Stokes. Split-step, velocity-pressure formulation: u t +(u )u + p ν u f = 0, t > 0, x Ω p u : u α u f = 0, t > 0, x Ω Divergence damping term: α u is important. Wall boundary conditions: u = 0, u = 0, (pressure BC) x Ω, with numerical boundary condition: p n = n ( ν u )+n f. Use u instead of u for implicit time-stepping. Henshaw (LLNL) FSI on overlapping grids VaTech 8 / 46

16 Incompressible Navier-Stokes. Split-step, velocity-pressure formulation: u t +(u )u + p ν u f = 0, t > 0, x Ω p u : u α u f = 0, t > 0, x Ω Divergence damping term: α u is important. Wall boundary conditions: u = 0, u = 0, (pressure BC) x Ω, with numerical boundary condition: p n = n ( ν u )+n f. Use u instead of u for implicit time-stepping. Henshaw (LLNL) FSI on overlapping grids VaTech 8 / 46

17 Incompressible Navier-Stokes. Split-step, velocity-pressure formulation: u t +(u )u + p ν u f = 0, t > 0, x Ω p u : u α u f = 0, t > 0, x Ω Divergence damping term: α u is important. Wall boundary conditions: u = 0, u = 0, (pressure BC) x Ω, with numerical boundary condition: p n = n ( ν u )+n f. Use u instead of u for implicit time-stepping. Henshaw (LLNL) FSI on overlapping grids VaTech 8 / 46

18 Incompressible Navier-Stokes. Split-step, velocity-pressure formulation: u t +(u )u + p ν u f = 0, t > 0, x Ω p u : u α u f = 0, t > 0, x Ω Divergence damping term: α u is important. Wall boundary conditions: u = 0, u = 0, (pressure BC) x Ω, with numerical boundary condition: p n = n ( ν u )+n f. Use u instead of u for implicit time-stepping. Henshaw (LLNL) FSI on overlapping grids VaTech 8 / 46

19 PDE based numerical boundary conditions (NBCs) High-order FD schemes with wide stencils require additional NBCs. One-sided difference approximations can be less stable and accurate. NBC Principle Derive NBCs from the PDE and physical boundary conditions. Example (heat equation): u t = ν(u xx + u yy )+f (x, y, t), 0 < x < 1, < y <, u(0, y, t) =g(y, t), u x (1, y, t) =k(y, t). Compatibility conditions are used as NBCs: (Dirichlet boundary condition), (Neumann boundary condition), g t (y, t) =ν(u xx (0, y, t)+g yy (0, y, t)) + f (0, y, t), k t (y, t) =ν(u xxx (1, y, t)+k yy (1, y, t)) + f x (1, y, t). Normal mode theory shows why compatibility conditions can be applied to lower order accuracy. Henshaw (LLNL) FSI on overlapping grids VaTech 9 / 46

20 PDE based numerical boundary conditions (NBCs) High-order FD schemes with wide stencils require additional NBCs. One-sided difference approximations can be less stable and accurate. NBC Principle Derive NBCs from the PDE and physical boundary conditions. Example (heat equation): u t = ν(u xx + u yy )+f (x, y, t), 0 < x < 1, < y <, u(0, y, t) =g(y, t), u x (1, y, t) =k(y, t). Compatibility conditions are used as NBCs: (Dirichlet boundary condition), (Neumann boundary condition), g t (y, t) =ν(u xx (0, y, t)+g yy (0, y, t)) + f (0, y, t), k t (y, t) =ν(u xxx (1, y, t)+k yy (1, y, t)) + f x (1, y, t). Normal mode theory shows why compatibility conditions can be applied to lower order accuracy. Henshaw (LLNL) FSI on overlapping grids VaTech 9 / 46

21 Cgins: an efficient solver for the incompressible Navier-Stokes equations A parallel split-step solver is being developed based on: 1 Fourth-order accurate approximate-factored/compact time-stepping scheme for the momentum equations. 2 Fourth-order accurate multigrid solver for the pressure equation. 3 Fast overlapping grid generation for moving geometry. WDH., A Fourth-Order Accurate Method for the Incompressible Navier-Stokes Equations on Overlapping Grids, J. Comput. Phys, (1994). WDH, N.A. Petersson, A Split-Step Scheme for the Incompressible Navier-Stokes Equations, WDH., K. K. Chand, A Composite Grid Solver for Conjugate Heat Transfer in Fluid-Structure Systems, J. Comput. Phys, Henshaw (LLNL) FSI on overlapping grids VaTech 10 / 46

22 Approximate factorization & compact discretizations - a key point is maintaining accuracy at boundaries. Approximate factorization (AF) schemes offer larger timesteps with second order accuracy in time: becomes (I + t 2 (A + B))Un+1 =(I t (A + B))Un 2 (I + t t A)(I B)Un+1 = (I t t A)(I 2 2 B)Un Compact schemes can be integrated into the AF solves Special combined compact schemes have been developed: reduce the number of factors (a r + b 2 rr )u P 1 (D r a + D rr b)u + corrections preserve accuracy at boundaries 4 th and 6 th order accuracy with a 5 point stencil special penta-diagonal solvers that handle wider boundary stencils Henshaw (LLNL) FSI on overlapping grids VaTech 11 / 46

23 Smallest Scale LES (SSLES) model Our high-order accurate schemes all use central differences. Nonlinear dissipation is used to stablize the schemes. It can be proved [HKR] that the minimum scale of the INS locally satisfies ν λ min u loc + c. Setting h = λ min leads to the non-linear dissipation operators: D 2 (u i )=(a 21 + a 22 h u i )h 2 h u i, D 4 (u i )=(a 41 + a 42 h u i )h 4 ( 2 h)u i. 1 These serve as simple LES models (Smagorinsky type). 2 These terms stabilize the scheme even if ν = 0. [HKR1] On the smallest scale for the incompressible Navier-Stokes equations, WDH, H.-O. Kreiss, L.G.M. Reyna, Theoret. Comput. Fluid Dynamics, [HKR2] Smallest scale estimates for the incompressible Navier-Stokes equations, WDH, H.-O. Kreiss, L.G.M. Reyna, Arch. Rational Mech. Anal., Henshaw (LLNL) FSI on overlapping grids VaTech 12 / 46

24 Code verification: an integral part of our development Twilight zone solutions (a.k.a. manufactured solutions) are extensively supported by the Overture framework. h max e p e u e v e w u 1.34e e e e e e e e-02 (15.9) 1.84e-02 (18.6) 1.01e-02 (21.2) 6.43e-03 (19.1) 2.71e-01 (10.3) 3.34e e-03 (11.8) 1.49e-03 (12.3) 7.50e-04 (13.5) 3.01e-04 (21.4) 3.10e-02 (8.7) 1.67e e-05 (12.9) 1.25e-04 (11.9) 6.55e-05 (11.5) 1.74e-05 (17.3) 4.00e-03 (7.8) rate Henshaw (LLNL) FSI on overlapping grids VaTech 13 / 46

25 AFS performance: flow past two cylinders. Grid tcilce64, N g = 22M grid-points Method P-solver N p TPSM TTS RPG PC24 BICGS(5) PC24 MG AFS24 MG Speed-up from baseline PC24-BICGS(5) to AFS24-MG is 16000/ Memory is reduced by a factor of 205/32 6. N p : number of processors. N g : number of grid-points. TPS : CPU (s) time-per-step. TPSM : CPU time (s) per-step, per-million-grid-points, per-processor = N p TPS/(N g/10 6 ). TTS : normalized time-to-solution = TPSM / CFL-number. RPG : memory usage in real-per-grid-point. PC24: predictor-corrector, 2nd-order time, 4th-order space. AFS24: approximate-factored scheme, 2nd-order time, 4th-order space. BICGS(5) : bi-conjugate-gradient-stabilized with ILU(5) preconditionner. Henshaw (LLNL) FSI on overlapping grids VaTech 14 / 46

26 Fourth-order Accurate Parallel Multigrid Solver for the Pressure Equation. Henshaw (LLNL) FSI on overlapping grids VaTech 15 / 46

27 The Ogmg overlapping grid multigrid solver has been extended to 4th-order accuracy and parallel. Ogmg is many times faster than Krylov methods. matrix-free; optimized for Cartesian grids. automatic coarse grid generation. adaptive smoothing variable sub-smooths per component grid. interpolation-boundary smoothing (IBS). Galerkin coarse grid operators (operator averaging). PDE-based numerical boundary conditions for Dirichlet and Neumann problems. WDH., On Multigrid For Overlapping Grids, SIAM J. Sci. Comput. 26, no. 5, (2005) Henshaw (LLNL) FSI on overlapping grids VaTech 16 / 46

28 Automatic coarse grid generation is a key feature. overlap increases interpolation accuracy reduced Henshaw (LLNL) FSI on overlapping grids VaTech 17 / 46

29 Local Fourier analysis significantly improves convergence rates. Over-relaxed Red-Black smoothers and Galerkin coarse grid operators D Poisson, Order 4, RB GS ρ V[1,1] 3G ρ V[1,1] Galerkin 3G ρ 3G V[2,1] ρ 3G V[2,1] Galerkin default rates (ω = 1) optimal rates 3D Poisson, Order 4, RB GS 0.35 ρ 3G V[1,1] ρ 0.3 3G V[1,1] Galerkin ρ 3G V[2,1] 0.25 ρ V[2,1] Galerkin 3G ω ω Three-grid multigrid convergence rates as a function ω. ω : relaxation parameter in Red-Black Gauss-Seidel smoother. ρ 3G : convergence rate per cycle for a 3 grid (i.e. 3 level) MG. V [m, n] : MG V-cycle, m pre-smooths and n-post smooths. Galerkin : Galerkin coarse grid operators. Henshaw (LLNL) FSI on overlapping grids VaTech 18 / 46

30 Accuracy and convergence of the new fourth-order accurate parallel version of Ogmg. Maximum error Dirichlet Neumann Sphere in a Box, Errors versus h Slope 4 maximum residual 10 4 Sphere in a Box, Order 4, V[2,1] CR=0.246, ECR=0.74 CR=0.058, ECR=0.61 no IBS IBS h multigrid cycle Henshaw (LLNL) FSI on overlapping grids VaTech 19 / 46

31 Multigrid is much faster than Krylov based methods. And uses much less memory. Poisson equation, SECOND-order accuracy Poisson equation, FOURTH-order accuracy Ogmg vs Krylov sq2048 cic64 tcilc64 box256 sphere4 sphere8 Ogmg vs Krylov sq2048 cic64 tcilc64 box256 sphere4 sphere8 speed-up memory-factor speed-up memory-factor Performance of Ogmg for solving Poisson s equation to on various grids. Ogmg is compared to a Krylov solver (bi-cg stab, ILU(1)) from PETSc. Henshaw (LLNL) FSI on overlapping grids VaTech 20 / 46

32 Rapid grid generation for moving geometry is critical. Ogen is the overlapping grid generator in Overture. Overlapping grid generation consists of two major steps: 1 construct component grids (Mappings). 2 grid connectivity: cut holes and determine interpolation information using Ogen (this is the step that requires most of the CPU time). In recent work changes have been made to support the generation of large (billion point +) grids. parallel moving-grid flow simulations (Ogen is called at each time step). Grid order of grid points processors cpu (s) accuracy (nodes p/n) Sphere in a box billion 16 (8 2) 136 Re-entry vehicle million 128 (16 8) 1990 Henshaw (LLNL) FSI on overlapping grids VaTech 21 / 46

33 Rapid grid generation for moving geometry is critical. Ogen is the overlapping grid generator in Overture. Overlapping grid generation consists of two major steps: 1 construct component grids (Mappings). 2 grid connectivity: cut holes and determine interpolation information using Ogen (this is the step that requires most of the CPU time). In recent work changes have been made to support the generation of large (billion point +) grids. parallel moving-grid flow simulations (Ogen is called at each time step). Grid order of grid points processors cpu (s) accuracy (nodes p/n) Sphere in a box billion 16 (8 2) 136 Re-entry vehicle million 128 (16 8) 1990 Henshaw (LLNL) FSI on overlapping grids VaTech 21 / 46

34 FSI for compressible fluids and deforming elastic solids (partitioned algorithms). Overcoming the added-mass instablity. Henshaw (LLNL) FSI on overlapping grids VaTech 22 / 46

35 Deforming composite grids (DCG) for Fluid-Structure Interactions (FSI) Goal: To perform coupled simulations of compressible fluids and deforming solids. A mixed Eulerian-Lagrangian approach: Fluids: general moving coordinate system with overlapping grids. Solids : fixed reference frame with overlapping-grids (later: unstructured-grids, or beam/plate models). Boundary fitted deforming grids for fluid-solid interfaces. Strengths of the approach: maintains high quality grids for large deformations/displacements. efficient structured grid methods (AMR) optimized for Cartesian grids. Henshaw (LLNL) FSI on overlapping grids VaTech 23 / 46

36 Deforming Composite Grids for FSI t=1.0 t=1.5 t=2.0 solid fluid interfaces solid Solve Euler equations in the fluid domains on moving grids. Solve equations of linear elasticity in the solid domains. Fluid grids at the interface deform over time (hyperbolic grid generator). Henshaw (LLNL) FSI on overlapping grids VaTech 24 / 46

37 The traditional partitioned algorithm for FSI 1 Advance solid using the stress from the fluid. 2 Advance fluid using the solid position/velocity. Note: traditional scheme has trouble for light solids (e.g. becomes unstable or requires a smaller time-step). this is known as an added mass instability. Added mass: The force required to accelerate a body in a fluid is greater than that in a vacuum (since both body and fluid must be accelerated). Henshaw (LLNL) FSI on overlapping grids VaTech 25 / 46

38 The elastic piston - a 1D FSI model problem solid fluid ρ : density ū : displacement v : velocity σ : stress c p : speed of sound z = ρc p : impedance ρ : density v : velocity σ = p : stress, pressure a : speed of sound z = ρa : impedance Henshaw (LLNL) FSI on overlapping grids VaTech 26 / 46

39 The elastic piston x = 0 x = F (t) t C + x = c pt ū 0 ( x) v 0 ( x) σ 0 ( x) solid (F (τ),τ) (x, t) fluid x = a 0 t ρ 0 v 0 p 0 x The governing equations for the solid and fluid are ū t v = 0 v t σ x / ρ = 0, for x < 0, σ t ρc p 2 v x = 0 where ρe = p/(γ 1)+ρv 2 /2. The interface conditions are v(0, t) =v(f (t), t), ρ t +(ρv) x = 0 (ρv) t +(ρv 2 + p) x = 0, for x > F (t), (ρe) t +(ρev + pv) x = 0 σ(0, t) =σ(f (t), t) p(f (t), t)+p e. Henshaw (LLNL) FSI on overlapping grids VaTech 27 / 46

40 The elastic piston x = 0 x = F (t) t C + x = c pt ū 0 ( x) v 0 ( x) σ 0 ( x) solid (F (τ),τ) (x, t) fluid x = a 0 t ρ 0 v 0 p 0 x The governing equations for the solid and fluid are ū t v = 0 v t σ x / ρ = 0, for x < 0, σ t ρc p 2 v x = 0 where ρe = p/(γ 1)+ρv 2 /2. The interface conditions are v(0, t) =v(f (t), t), ρ t +(ρv) x = 0 (ρv) t +(ρv 2 + p) x = 0, for x > F (t), (ρe) t +(ρev + pv) x = 0 σ(0, t) =σ(f (t), t) p(f (t), t)+p e. Henshaw (LLNL) FSI on overlapping grids VaTech 27 / 46

41 The solution of the fluid-solid Riemann (FSR) problem defines our interface projection. x = c pt v0 σ 0 x = 0 x = F (t) v1 σ 1 solid t ρ 1 v 1 p 1 fluid x =(v 1 + a 1 )t x = St ρ 0 v 0 p 0 x =(v 0 + a 0 )t special case of elastic piston problem - constant initial conditions in fluid and solid. the fluid may have a shock or expansion fan on the C + characteristic. x Henshaw (LLNL) FSI on overlapping grids VaTech 28 / 46

42 Solution of the linearized fluid-solid Riemann problem Characteristic relations: Solid: z v σ = z v 0 σ 0, on d x/dt = ±c p, Fluid: zv σ = zv 0 σ 0, on dx/dt = v 0 ± a 0, where z = ρc p and z = ρa 0 are the acoustic impedances. The state next to the interface is an impedance weighted average of the fluid and solid states: v 1 = v 1 = z v 0 + zv 0 z + z + σ 0 σ 0 z + z, σ 1 = σ 1 = z 1 σ 0 + z 1 σ 0 z 1 + z 1 + v 0 v 0 z 1 + z 1. The solution to the full nonlinear problem can also be determined. Henshaw (LLNL) FSI on overlapping grids VaTech 29 / 46

43 The FSI-DCG time-stepping algorithm 1 Advance solid to give provisional interface values ( v 0, σ 0 ). 2 Advance fluid to give provisional interface values (v 0,σ 0 ). 3 project the interface values: Notes: v I = z v 0 + zv 0 z + z + σ 0 σ 0 z + z, σ I = z 1 σ 0 + z 1 σ 0 z 1 + z 1 + v 0 v 0 z 1 + z 1 One can prove that the new scheme remains stable under the usual CFL time step restriction, even for very light solids [Banks/Sjögreen 2011]. The standard FSI scheme uses the heavy solid limit, z z, v I = v 0 σ I = σ 0 and is unstable for light solids (i.e. requires a small time step). Henshaw (LLNL) FSI on overlapping grids VaTech 30 / 46

44 The FSI-DCG time-stepping algorithm 1 Advance solid to give provisional interface values ( v 0, σ 0 ). 2 Advance fluid to give provisional interface values (v 0,σ 0 ). 3 project the interface values: Notes: v I = z v 0 + zv 0 z + z + σ 0 σ 0 z + z, σ I = z 1 σ 0 + z 1 σ 0 z 1 + z 1 + v 0 v 0 z 1 + z 1 One can prove that the new scheme remains stable under the usual CFL time step restriction, even for very light solids [Banks/Sjögreen 2011]. The standard FSI scheme uses the heavy solid limit, z z, v I = v 0 σ I = σ 0 and is unstable for light solids (i.e. requires a small time step). Henshaw (LLNL) FSI on overlapping grids VaTech 30 / 46

45 The FSI-DCG time-stepping algorithm 1 Advance solid to give provisional interface values ( v 0, σ 0 ). 2 Advance fluid to give provisional interface values (v 0,σ 0 ). 3 project the interface values: Notes: v I = z v 0 + zv 0 z + z + σ 0 σ 0 z + z, σ I = z 1 σ 0 + z 1 σ 0 z 1 + z 1 + v 0 v 0 z 1 + z 1 One can prove that the new scheme remains stable under the usual CFL time step restriction, even for very light solids [Banks/Sjögreen 2011]. The standard FSI scheme uses the heavy solid limit, z z, v I = v 0 σ I = σ 0 and is unstable for light solids (i.e. requires a small time step). Henshaw (LLNL) FSI on overlapping grids VaTech 30 / 46

46 Verification: elastic piston numerical results Computed solution for a smoothly receding piston. Fluid Solid grid N ρ r v r p r ū r v r σ r G e-3 2.7e-3 1.6e-3 4.9e-4 9.3e-5 1.2e-4 G e e e e e e G e e e e e e G e e e e e e rate Max-norm errors for very light solid: ρ = 10 5, z/ z = Fluid Solid grid N ρ r v r p r ū r v r σ r G e-4 9.6e-4 8.5e-4 3.0e-4 1.3e-5 1.1e-5 G e e e e e e G e e e e e e G e e e e e e rate Max-norm errors for very heavy solid: ρ = 10 5, z/ z = Max-norm errors are converging to second-order accuracy. Scheme is stable for large and small impedance ratios. Henshaw (LLNL) FSI on overlapping grids VaTech 31 / 46

47 The superseismic shock FSI problem y shock w 1 ξ w 0 fluid θ S solid x interface w 2 s-wave η s w 1 η p p-wave w 0 a fluid shock moving from left to right deflects a solid interface. an FSI problem with an analytic traveling wave solution. Henshaw (LLNL) FSI on overlapping grids VaTech 32 / 46

48 Superseismic shock: grids and computed solution ρ v 2 The red grid for the solid domain is shown adjusted for the displacement. Henshaw (LLNL) FSI on overlapping grids VaTech 33 / 46

49 L 1 -norm errors and convergence rates for the superseismic shock Solid Fluid Grid ū r v r σ r ρ r v r T r G (4) ss 8.6e-5 2.8e-3 7.9e-3 4.6e-3 2.2e-2 1.1e-2 G (8) ss 3.6e e e e e e G (16) ss 1.5e e e e e e G (32) ss 5.6e e e e e e rate Due to the discontinuities, the L 1 -norm errors do not converge at second-order. The solid variables v and σ converge at the expected rates of 2/3 (due to spreading of the linear discontinuities). In isolation the fluid domain should converge at first order. Henshaw (LLNL) FSI on overlapping grids VaTech 34 / 46

50 FSI for compressible fluids and light rigid bodies (partitioned algorithms). Overcoming the added-mass instability. Henshaw (LLNL) FSI on overlapping grids VaTech 35 / 46

51 Light rigid body in a fluid - model problem w b /2 0 w b /2 x fluid (acoustics) solid (rigid body) fluid (acoustics)... 3 vl vr σ L Fluid domains (acoustics): t i vk σ k v b σ R 0 1 ρ vk ρc 2 = 0, k=l,r, 0 x σ k Rigid body: (Question: what happens when m b 0?) i Interface conditions: m b v b = F, F = σ R x=wb /2 σ L x= wb /2 v L x= wb /2 = v b, v R x=wb /2 = v b. Henshaw (LLNL) FSI on overlapping grids VaTech 36 / 46

52 The fluid-solid Riemann problem for a rigid body. x = r b (t) t v b,σ b C : σ + zv = σ0 + zv0 f b body fluid v 0,σ 0 x From characteristics we obtain the fluid force on the body: σ(r b, t) =σ(r b +, t )+z v(r b +, t ) v b (t). (1) where z = ρc is the fluid impedence. Equation (1) defines our stress projection. Note the dependence of the stress on the velocity of the body v b. Henshaw (LLNL) FSI on overlapping grids VaTech 37 / 46

53 Light rigid body in a fluid - algorithm First order upwind scheme n+1 vk σ k i = n vk σ k i n + tr k Λ vk k R 1 k D σ k i n + tr k Λ + vk k R 1 k D +, σ k i Partitioned time-stepping algorithm (new scheme: α = z, traditional: α = 0), n+1 vk 1 Advance fluid domans: σ k i 2 Set F n+1 = σ n+1 R,1 n+1 + α(vr,1 v n+1 b ) σ n+1 n+1 L, 1 α(vb v n+1 L, 1 ), 3 Solve for rigid body: m b v n+1 b = m b v n b + tf n+1, 1 v n+1 b = m b + 2 tα m b vb n + t σ n+1 R,1 σn+1 n+1 L, 1 + α(vr,1 + v n+1 L, 1. ) 4 Set ghost values from projection. Note the added mass term 2 tα. Henshaw (LLNL) FSI on overlapping grids VaTech 38 / 46

54 Normal mode stability analysis Theorem The first-order upwind scheme with projection parameter α = z is stable for λ = c t x < 1 and any m b 0. Theorem The first-order upwind scheme with projection parameter α = 0 has the time-step restriction λ<1 and t < m b (4 λ)/(zλ). Note 1: The projection scheme (α = z) remains stable with the usual time-step restriction for any m b 0. Note 2: The traditional scheme (α = 0) is formally stable for m b > 0 but the time-step t goes to zero at m b 0. Henshaw (LLNL) FSI on overlapping grids VaTech 39 / 46

55 The Newton-Euler equations for rigid body dynamics The equations of motion for the rigid body are the Newton-Euler equations: ẋ b = v b, m b v b = F, A ω = WAω + T, Ė = WE. m b : mass of the body, x b (t) R 3 : position of the center of mass, v b (t) R 3 : is the velocity of the center of mass, ω(t) R 3 : is the angular velocity, A(t) R 3 3 : moment of inertia matrix, E(t) R 3 3 : matrix with columns being the principle axes of inertia, W (t) R 3 3 : angular velocity matrix, W a = ω a, F(t) R 3 : resultant force on the body, T (t) R 3 : resultant torque the body. Henshaw (LLNL) FSI on overlapping grids VaTech 40 / 46

56 Extension to 3D: added mass matrices For a point r on the rigid body in 3D, the interface projection is p(r, t)n = p f n + z f n T v f v(r, t) n. Force and torque: (y = r x b (t), v(r, t) =v b (t) y(r, t) ω(t)) F = p(r, t)n ds = A vv v b A vω ω + F, r B T = y ( p(r, t)n) ds = A ωv v b A ωω ω + T r B where A ij are the added-mass matrices A vv = z f nn T ds, A vω = z f n(y n) T ds, (2) B B A ωv = z f (Y n)n T ds A ωω = z f Y n(y n) T ds, (3) B B and where Y is the matrix corresponding to y. Henshaw (LLNL) FSI on overlapping grids VaTech 41 / 46

57 Extension to 3D: added mass matrices For a point r on the rigid body in 3D, the interface projection is p(r, t)n = p f n + z f n T v f v(r, t) n. Force and torque: (y = r x b (t), v(r, t) =v b (t) y(r, t) ω(t)) F = p(r, t)n ds = A vv v b A vω ω + F, r B T = y ( p(r, t)n) ds = A ωv v b A ωω ω + T r B where A ij are the added-mass matrices A vv = z f nn T ds, A vω = z f n(y n) T ds, (2) B B A ωv = z f (Y n)n T ds A ωω = z f Y n(y n) T ds, (3) B B and where Y is the matrix corresponding to y. Henshaw (LLNL) FSI on overlapping grids VaTech 41 / 46

58 Multi-dimensional algorithm Scheme: 1 second-order accurate Godunov based scheme for the (reactive) Euler equations. 2 Implicit Runge-Kutta (DIRK) time-stepping for added-mass rigid-body equations. Notes: 1 Second-order accurate in the max norm for smooth solutions (with limiter turned off). 2 First-order accurate in the discrete L 1 -norm for problems with shocks. 3 Adaptive mesh refinement. 4 Equations are solved in the moving grid coordinate system. Henshaw (LLNL) FSI on overlapping grids VaTech 42 / 46

59 Shock driven zero mass ellipse t = t = t = p p p Henshaw (LLNL) FSI on overlapping grids VaTech 43 / 46

60 Shock driven zero mass ellipse 2.5 x 1 Shock driven ellipse, M=0 2 x 2 v 1 v w t Time histories of rigid-body properties. Colour: grid G re (8), black: fine grid G re (32) Grid G (j) h j e ρ (j) r e (j) u r e (j) v r e (j) p r G re (8) 1/40 2.1e-3 9.3e-4 9.6e-4 2.1e-3 G re (16) 1/80 9.9e e e e G re (32) 1/ e e e e rate A posteriori estimated errors (L 1 -norm) and convergence rates for the accelerated ellipse at t = 1.0. Henshaw (LLNL) FSI on overlapping grids VaTech 44 / 46

61 Shock driven zero mass ellipse Coarse grid G re (32) (left), medium grid G re (16 4) (middle), fine (AMR) grid G re (8 4 4) (right). Henshaw (LLNL) FSI on overlapping grids VaTech 45 / 46

62 Summary We have been developing efficient algorithms for modeling 1 incompressible flows and rigid body motion, 2 compressible flows and rigid/deforming bodies. The approach is based on 1 overlapping grids for flexible representation of geometry, 2 high-order accurate algorithms, 3 accurate treatment of interfaces and boundary conditions. 4 solutions to fluid-solid Riemann problems to overcome added-mass instabilities. Henshaw (LLNL) FSI on overlapping grids VaTech 46 / 46