Structure Interactions in a Fully Coupled Form Ali EKEN & Mehmet - - PowerPoint PPT Presentation

The Numerical Simulation of Large-Scale Fluid- Structure Interactions in a Fully Coupled Form

Ali EKEN & Mehmet SAHIN 10th World Congress on Computational Mechanics Astronautical Engineering Department, 8-13 July 2012 — Sao Paulo, Brazil. Faculty of Aeronautics and Astronautics, Istanbul Technical University, 34469, Maslak/Isatanbul, TURKEY

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Contents

• Motivations
• Coupling Strategies - A Review
• Numerical Modelling
• Finite Element Structure Solver
• ALE Based Finite Volume Fluid Solver
• Test Case I: An Oscillating Circular Cylinder
• Fluid-Structure Interaction
• Fluid-Structure Coupling
• Fluid-Structure Solver Validation
• Test Case II: Vortex-induced vibrations of an elastic bar
• Test Case III: 3-D Elastic Solid in a Steady Channel Flow
• Conclusions and Future Work

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Motivations

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FSI - the interaction of some movable or deformable structure with an internal or surrounding fluid flow

• Engineering and biomedical applications.

FSI for a 3d parachute Trimarchi at al., (2011). Membrane wing fluid structure interaction Stanford at al., (2008). Vascular fluid structure interaction Bazilevs at al., (2010).

Coupling Strategies - A Review

Partitioned (Staggered) Methods

• Separate solvers for the fluid and solid domains.
• More freedom in selecting optimized separate solvers.
• Slow convergence for fixed-point iterations.
• May diverge for strong interactions when deformations are large and the

fluid to solid density is close to 1.

• The incompressibility constraint cannot be satisfied with standart

alternating FSI iterations for Dirichlet fluid boundary conditions.

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Coupling Strategies - A Review

Fully Coupled (Monolithic) Methods

• The fluid and solid equations are discretized and solved simultaneously as

a single equation system.

• Requires solution of a large system of coupled nonlinear algebraic

equations.

• Believed to be more robust than partitioned methods , but also believed

to be more computationally expensive for large scale problems.

• Competitive even for weak FSI with improved preconditioners.

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The Computational Method

Finite Element Structure Solver

• St. Venant-Kirchhoff material model for large displacements.
• Isoparametric quadrilateral/hexahedral finite element formulation.
• Generalized- method for time integration.

ALE Fluid Solver

• The incompressible Navier-Stokes equations for the fluid domain.
• Arbitrary Lagrangian-Eulerian formulation based on the side-centered

unstructured finite volume method.

A Fully Coupled Solution Approach

• The fluid and solid equations are discretized and solved simultaneously as

a single nonlinear algebraic equation system for the entire domain.

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Equations of Solid Motion

The governing equations with respect to the initial configuration Galerkin Finite Element Formulation Displacement Boundary Conditions Traction Boundary Conditions

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Finite Element Discretization

The tangent stiffness matrix The Mass Matrix Surface Tractions

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and nonlinear parts

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Linear

The Resulting System of Equations

The modified nonlinear system with the generalized- method The displacements and velocities at the intermediate point Newmark approximations are used for the values at and .

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Equations of Fluid Motion

The incompressible N-S equations on deforming meshes

The momentum equation The continuity equation No-slip wall boundary condition Inflow boundary condition Outflow boundary condition

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Numerical Discretization

Side centered finite volume method

• A Stable numerical scheme with exact mass conservation
• No ad-hoc modifications for pressure-velocity coupling
• Very efficient multigrid solvers are available

(a) Two-dimensional dual volume (b) Three-dimensional dual volume

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Numerical Discretization Cont’d

The contributions for the x-momentum equation for the right element: The time derivation is the volume of the pyramid between points

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Numerical Discretization Cont’d

The convective term due to fluid motion the area vectors of the dual volume triangular surfaces. the velocity vectors defined at the mid-point of each dual volume area.

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The convective term due to mesh motion First-order backward difference for grid velocity

Numerical Discretization Cont’d

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Geometric conservation law satisfied.

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Numerical Discretization Cont’d

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The pressure term 2nd order Taylor series expansion is used for the pressure values at .

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Numerical Discretization Cont’d

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The viscous term Gauss-Green theorem is used for the gradient terms:

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TEST CASE I: An Oscillating Circular Cylinder

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• The location of the cylinder center
• Amplitude and frequency of the oscillation
• Time step
• Reynolds number
• 70,667 quadrilateral elements
• 71,349 vertices
• 354,699 total DOF

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TEST CASE I: An Oscillating Circular Cylinder

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t=20.0s t=21.0s t=22.0s t=23.0s

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The computed u-velocity vector component contours with streamtraces.

TEST CASE I: An Oscillating Circular Cylinder

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The comparion of cd and cl plots with results from Wan and Turek (JCP, 2007).

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Fluid Structure Coupling

Fluid domain Solid Domain Fluid-Structure Interface

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Grid velocity Fluid and solid velocities at interface Fluid and solid stresses at interface

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The Fully Coupled System

the common fluid-structure interface. u fluid velocity p pressure d solid displacements q mesh displacement

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The Fully Coupled System

The modified system

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Remove the zero block for preconditioning Three banded matrix Zero block removed

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The Fully Coupled System

Matrix contains contributions from the right and left elements only.

• Maximum three non-zero entries per row
• Significant reduction in computing time and memory requirement

The Iterative Solver

• Restricted additive Schwarz preconditioned GMRES(m) algorithm.
• Block-incomplete factorization coupled with the reverse Cuthill-McKee
• rdering within each partitioned sub-domains
• Sub-iterations to handle non-linearity due to unknown vertex locations

at (n+1).

• PETSc library for preconditioners, Krylov subspace algorithm,

GMRES(m) algorithm and matrix-matrix multiplications.

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Parallelization and Efficiency

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TEST CASE II: Vortex-induced vibrations of an elastic bar

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• An elastic bar behind a rigid circular cylinder
• The control point A (0.6, 0.2).

Two subcases with different inflow speeds:

• FSI1 corresponds to a steady state solution with
• FSI3 corresponds to an unsteady flow solution with

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Hron and Turek, (2006)

TEST CASE II: Vortex-induced vibrations of an elastic bar

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Boundary Conditions:

• Parabolic inflow velocity
• Natural(traction-free) outlet boundary
• FSI boundary

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TEST CASE II: Vortex-induced vibrations of an elastic bar

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The computational mesh:

• 78,921 quadrilateral elements (7053 for solid)
• 79,806 nodes
• 375,216 total DOF

Local refinement on boundary

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TEST CASE II: Vortex-induced vibrations of an elastic bar

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The time variation of the computed tip displacement for FSI3.

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TEST CASE II: Vortex-induced vibrations of an elastic bar

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Displacements at point A:

Present solution for FSI1 Present solution for FSI3

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TEST CASE II: Vortex-induced vibrations of an elastic bar

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t = 5.185 t = 5.140 t = 5.150 t = 5.240 t = 5.275 t = 5.230 A B C D E F

. . . . . .

TEST CASE II: Vortex-induced vibrations of an elastic bar

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• Period of oscillation

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A (t = 5.140) A B (t = 5.150) B C (t = 5.185) C D (t = 5.230) D E (t = 5.240) E F (t = 5.275) F

TEST CASE III: 3D elastic solid in a steady channel flow

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• An elastic beam is immersed in a rectangular channel.
• The configuration is symmetric in the xy-plane.
• The control point A := (0.45, 0.15, 0.15).
• based on an average

inflow velocity.

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Richter, (2012)

TEST CASE III: 3D elastic solid in a steady channel flow

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Boundary Conditions:

• Parabolic inflow velocity
• Natural(traction-free) outlet boundary
• FSI boundary

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TEST CASE III: 3D elastic solid in a steady channel flow

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The computational mesh:

• 362224 hexahedral elements
• 374,523 nodes
• 4,096,651 total DoF

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TEST CASE III: 3D elastic solid in a steady channel flow

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Upwind Downwind

• The computed streamtraces on the solid walls at

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TEST CASE III: 3D elastic solid in a steady channel flow

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• The streamtraces at
• The color shows the magnitude of the u-velocity component.

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TEST CASE III: 3D elastic solid in a steady channel flow

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• The computed isobaric surfaces at
• The color shows the magnitude of pressure.

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TEST CASE III: 3D elastic solid in a steady channel flow

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Displacements at point A.

• Displacement contours at
• The color shows the magnitude of x-displacement.

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Conclusions and Future Work

• A parallel unstructured fluid structure interaction (FSI) code has been

developed, and successfuly tested for a 2D and a 3D benchmark configuration.

• The fluid solver is based on the solution of the incompressible Navier-

Stokes equations using an ALE formulation.

• The solid solver is based on the nonlinear elastic St. Venant Kirchhoff

material model.

• Currently we use only one level preconditioned iterative solver for the

whole coupled system. In the future, we will investigate multi-level monolitic approaches.

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Acknowledgement

The authors gratefully acknowledge the use of the Chimera machine at the Faculty of Aeronautics and Astronautics at ITU, the computing resources provided by the National Center for High Performance Computing of Turkey (UYBHM) under grant number 10752009 and the computing facilities at TUBITAK ULAKBIM, High Performance and Grid Computing Center. The authors also would like to thank Thomas Richter at the University of Heidelberg for performing the 3D validation case.

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