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

ITU/FAA Faculty of Aeronautics and Astronautics The Numerical Simulation of Large-Scale Fluid- Structure Interactions in a Fully Coupled Form Ali EKEN & Mehmet SAHIN 10 th World Congress on Computational Mechanics 8-13 July 2012 — Sao Paulo, Brazil. Astronautical Engineering Department, Faculty of Aeronautics and Astronautics, Istanbul Technical University, 34469, Maslak/Isatanbul, TURKEY

ITU/FAA Faculty of Aeronautics and Astronautics 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 WCCM 2012 2

ITU/FAA Faculty of Aeronautics and Astronautics Motivations FSI - the interaction of some movable or deformable structure with an internal or surrounding fluid flow • Engineering and biomedical applications. Membrane wing fluid structure Vascular fluid structure interaction FSI for a 3d parachute interaction Stanford at al., (2008). Trimarchi at al., (2011). Bazilevs at al., (2010). WCCM 2012 3

ITU/FAA Faculty of Aeronautics and Astronautics 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. WCCM 2012 4

ITU/FAA Faculty of Aeronautics and Astronautics 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. WCCM 2012 5

ITU/FAA Faculty of Aeronautics and Astronautics 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 • Th e fluid and solid equations are discretized and solved simultaneously as a single nonlinear algebraic equation system for the entire domain. WCCM 2012 6

ITU/FAA Faculty of Aeronautics and Astronautics Equations of Solid Motion The governing equations with respect to the initial configuration Galerkin Finite Element Formulation Displacement Boundary Conditions Traction Boundary Conditions WCCM 2012 7

ITU/FAA Faculty of Aeronautics and Astronautics Finite Element Discretization The tangent stiffness matrix Linear and nonlinear parts The Mass Matrix Surface Tractions WCCM 2012 8

ITU/FAA Faculty of Aeronautics and Astronautics 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 . WCCM 2012 9

ITU/FAA Faculty of Aeronautics and Astronautics 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 WCCM 2012 10

ITU/FAA Faculty of Aeronautics and Astronautics 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 WCCM 2012 11

ITU/FAA Faculty of Aeronautics and Astronautics 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 WCCM 2012 12

ITU/FAA Faculty of Aeronautics and Astronautics 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. WCCM 2012 13

ITU/FAA Faculty of Aeronautics and Astronautics Numerical Discretization Cont’d The convective term due to mesh motion First-order backward difference for grid velocity Geometric conservation law satisfied. WCCM 2012 14

ITU/FAA Faculty of Aeronautics and Astronautics Numerical Discretization Cont’d The pressure term 2nd order Taylor series expansion is used for the pressure values at . ECCOMAS 2012 15

ITU/FAA Faculty of Aeronautics and Astronautics Numerical Discretization Cont’d The viscous term Gauss-Green theorem is used for the gradient terms: ECCOMAS 2012 16

ITU/FAA Faculty of Aeronautics and Astronautics TEST CASE I: An Oscillating Circular Cylinder • 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 WCCM 2012 17

ITU/FAA Faculty of Aeronautics and Astronautics TEST CASE I: An Oscillating Circular Cylinder t=20.0s t=21.0s t=22.0s t=23.0s The computed u -velocity vector component contours with streamtraces. WCCM 2012 18

ITU/FAA Faculty of Aeronautics and Astronautics TEST CASE I: An Oscillating Circular Cylinder The comparion of c d and c l plots with results from Wan and Turek (JCP, 2007). WCCM 2012 19

ITU/FAA Faculty of Aeronautics and Astronautics Fluid Structure Coupling Fluid domain Grid velocity Solid Domain Fluid and solid stresses at interface Fluid-Structure Interface Fluid and solid velocities at interface WCCM 2012 20

ITU/FAA Faculty of Aeronautics and Astronautics The Fully Coupled System the common fluid -structure interface. u fluid velocity p pressure d solid displacements q mesh displacement WCCM 2012 21

ITU/FAA Faculty of Aeronautics and Astronautics The Fully Coupled System The modified system Remove the zero block for preconditioning Three banded matrix Zero block removed WCCM 2012 22

ITU/FAA Faculty of Aeronautics and Astronautics 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 ordering 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. WCCM 2012 23

ITU/FAA Faculty of Aeronautics and Astronautics Parallelization and Efficiency WCCM 2012 24

ITU/FAA Faculty of Aeronautics and Astronautics TEST CASE II: Vortex-induced vibrations of an elastic bar • An elastic bar behind a rigid circular cylinder • The control point A (0.6, 0.2). Hron and Turek, (2006) Two subcases with different inflow speeds: • FSI1 corresponds to a steady state solution with • FSI3 corresponds to an unsteady flow solution with WCCM 2012 25

ITU/FAA Faculty of Aeronautics and Astronautics TEST CASE II: Vortex-induced vibrations of an elastic bar Boundary Conditions: • Parabolic inflow velocity • Natural(traction-free) outlet boundary • FSI boundary WCCM 2012 26

ITU/FAA Faculty of Aeronautics and Astronautics TEST CASE II: Vortex-induced vibrations of an elastic bar The computational mesh: Local refinement on boundary • 78,921 quadrilateral elements (7053 for solid) • 79,806 nodes • 375,216 total DOF WCCM 2012 27

ITU/FAA Faculty of Aeronautics and Astronautics TEST CASE II: Vortex-induced vibrations of an elastic bar The time variation of the computed tip displacement for FSI3. WCCM 2012 28

ITU/FAA Faculty of Aeronautics and Astronautics TEST CASE II: Vortex-induced vibrations of an elastic bar Displacements at point A : Present solution for FSI1 Present solution for FSI3 WCCM 2012 29

ITU/FAA Faculty of Aeronautics and Astronautics TEST CASE II: Vortex-induced vibrations of an elastic bar t = 5.140 t = 5.150 A B t = 5.185 t = 5.230 C D t = 5.240 t = 5.275 E F WCCM 2012 30