High-dimensional FSI system and Low-Dimensional Modelling Marek Morzyński Witold Stankiewicz Robert Roszak Bernd R. Noack Gilead Tadmor

Overview Elements and of High- Dimensional Aeroelastic System Loosely coupled aeroelastic system Computational aspects Elements of the system Solutions ROM with moving boundaries and ALE ROM in design and flow control ROM for AE – sketch of challenges and ideas

ROM AE model - motivation Need of ROM in design AIAA 2008, Rossow, Kroll Need of online capable ROMs in feedback flow control Aeroservoelasticity Aeroelastic control (Piezo-control of flutter, wing morphing, smart structures) MicroAerialVehicles (maneuverability) Aero Data Production A380 wing 50 flight points 100 mass cases 10 a/c configurations 5 maneuvers 20 gusts (gradient lengths)‏ 4 control laws ~20,000,000 simulations Engineering experience for current configurations and technologies ~100,000 simulations

High- Dimensional Aeroelastic System – ROM testbed Flow code Structural code Interpolation Fluid forces Forces Structure displacements and velocities Deformed CFD mesh, velocities CFD mesh deformation t=t+t convergence yes no Tau Code Spring analogy In-house and AE tools MF3 (in-house), Calculix, Nastran

Computational aspects – Euler code Mesh: 10 mio elements CPU Power: 16 cores Flow code Structural code Interpolation Fluid forces Forces Structure displacements and velocities Deformed CFD mesh, velocities CFD mesh deformation t=t+t convergence yes no t=80s t=10s t=30s t=10s t=4s / 50s One iteration time: 134s (full CSM) / 180s (modal CSM)

Computational aspects - RANS Mesh: 30 mio elements (1 mio: surfaces) CPU Power: 32 cores Flow code Structural code Interpolation Fluid forces Forces Structure displacements and velocities Deformed CFD mesh, velocities CFD mesh deformation t=t+t convergence yes no t=400s t=90s t=220s t=90s t=4s / 50s One iteration time: 850s (full CSM) / 804s (modal CSM)

High-fidelity CFD and CSM solvers CFD - TAU CODE CSM MF3: in-house CSM Tool Finite Element-based Rods, beams, triangles (1st / 2nd order), membranes, shells, tetrahedrons (1st / 2nd order), masses and rigid elements Static analysis Transient (Newmark scheme) Modal analysis MpCCI and EADS AE interfaces Finite volume method solving the Euler and Navier-Stokes equations hybrid grids (tetrahedrons, hexahedrons, prisms and pyramids) Central or upwind-discretisation of inviscid fluxes Runge-Kutta time integration accelerated by multi-grid on agglomerated dual-grids miscellaneous turbulence models Parallelized with MPI Parallel Chimera grids From DLR TAU-code manual

ALE - Motion of boundary and mesh canonical domain Eulerian approach Lagrangian approach Arbitrary Lagrangian-Eulerian (ALE) binds the velocity of the flow u and the velocity of the (deforming) mesh ugrid. For incompressible Navier-Stokes equations the mesh velocity modifies the convective term: With boundary conditions: The fluid mesh can move independently of the fluid particles.

Coupling requirements Alenia SMJ CFD N-S hybrid grid with 1.3 mio nodes and 4.7 mio elements (cells) Alenia SMJ FEM model with 2,815 nodes Aerodynamic mesh 12437 nodes Structural mesh 212 nodes Pressure forces interpolation

Coupling tools The meshes are non-conforming different discretization different shape (whole wing/ torsion box only Non-conservative interpolation Conservative interpolation

Coupling tools MpCCi (Mesh-based parallel Code Coupling Interface), developed at the Fraunhofer Institute SCAI AE Modules, developed in the framework of TAURUS In-house tools, based on bucket search algorithm AE Modules by EADS and in-house modules perform better in the cases, when only torsion box of the wing was modelled on the structural side.

Dynamic Coupling: time integration General aeroelastic equations of motion : [M] x’’ (t) + [D] x’ (t) + [K] x (t) = f (x, x’, x’’, t) Inertial Damping Elastic Aerodynamic forces forces forces forces Structural forces Newmark direct integration method xi+1 = xi + t xi‘ + t2 [ ( 1/2 -  ) xi‘‘ +  xi+1‘‘ ] xi+1‘ = xi ‘ + t [ ( 1 -  ) xi‘‘ +  xi+1‘‘ ] Integration in time in CFD (or CSM) code NEWMARK explicit scheme with  = 0 and  = 0.5 xi+1 = xi + t xi‘ + t2/2 xi‘‘ xi+1‘‘ = ( [M] + t/2 [D] ) -1 { f i+1 - [K] x i+1 - [D] ( xi‘ + t/2 xi‘‘ ) } xi+1‘ = xi‘ + t/2 ( xi‘‘ + xi+1‘‘ )

Fluid mesh deformation Spring analogy All edges of tetrahedra are replaced with springs (torsional, semi-torsional, ortho-semi-torsional, ball-vertex, etc.) The stiffness km of each spring may be constant, or related to element size or distance from boundary Shephard interpolation (Inverse Distance Weighting) Based on the distances di between a given mesh node and boundary nodes: Another possibilities: Elastic material analogy, Volume Splines (Radial Basis Functions), Transfinite Interpolation

I22 and I23 airplanes from: wikimedia

Flutter analysis for I-23 airplane Mach number: M = 0.166, 0.2, 0.3, 044 Atmospheric pressure: P = 0.1 MPa Reynolds number: Re = 2e+6 Angle of attack: α = 0.026 Time step: dt = 0.01 s Singular input function: Fz = 2000 N in time t = 0.01 s

Flutter analysis for I-23 airplane Time history for displacement and rotation in control node on wing Simulation: flutter at Ma=0.44 Experiment: flutter at Ma=0.41

Flutter Laboratory IoA and PUT experiment and computations Scale : Length - 1:4 Strouhal number 1:1

Experimental configurations 5 cases – mass added - 50 grams on the wing's tip - 20 grams in the middle of ailerons - 30 grams on vertical stabilizer + 20 grams on tail plane aileron - 20 grams on horizontal stabilizer - configuration

FSI - test case 1 #1 - 50 grams on the wing's tip

Results of test case 1 #1 - 50 grams on the wing's tip

Low-Dimensional FSI algorithm t=t+t convergence yes no Flow ROM Pressure Deformed CFD mesh, velocities Amplitudes of „mesh” modes Interpolation CFD mesh deformation Forces on structure Interpolation Structural code Structure displacements and velocities

Reduced Order Model of the flow Navier-Stokes Equations 1. GALERKIN APROXIMATION 2. GALERKIN PROJECTION 3. GALERKIN SYSTEM

Projection of convective term Arbitrary Lagrangian-Eulerian Approach 1. DECOMPOSITION 2. GALERKIN PROJECTION

ROM for a moving boundary NACA-0012 AIRFOIL DNS with ALE 2-D, viscous, incompressible flow  = 15˚, Re = 100 (related to chord length) displacement of the boundary and mesh velocity: where: T = 5s and Y1 = 1/4 of chord length Inverse Distance Weighted First 8 POD modes: 99.96% of TKE

ROM for a moving boundary Eulerian ROM vs ref. DNS ALE ROM vs DNS Dumping of oscillation typical for sub-critical Re The first two modes

AE mode basis for a flow induced by structure deformations Test-case: bending and pitching LANN wing Fluid answer to separated, modal deformations (varying amplitudes) Fluid answer to combined deformation Pressure field and structure deformation (high-dimensional AE) LANN wing structure

ROM AE: CFD → CSM Coupling We preserve full-dimensional CSM and existing AE coupling tools to interpolate fluid forces on coupling - “wet” - surface; (similarly to Demasi 2008 AIAA) Neighbour search: ae_modules f_cfd2csd Pressure interpolation: ae_modules b_cfd2csd where si (i=1..15) is a distance from CFD node to closest CSM elements High-dimensional fluid forces retrived from the Galerkin Approximation

Coupling and CFD mesh deformation ROM AE: CSM → CFD Coupling and CFD mesh deformation Linear CSM: deformation decomposed onto mesh modes; Galerkin Projection of ALE term is performed during the construction of GM Solution of resulting Galerkin System requires only the input of mesh mode amplitudes Time stepping: the mesh deformation/velocity calculated for next time step with the Newmark scheme ui+1 = ui + t ui‘ + t2 [ ( 1/2 -  ) ui‘‘ +  ui+1‘‘ ] ui+1‘ = ui ‘ + t [ ( 1 -  ) ui‘‘ +  ui+1‘‘ ]

Parametrized Mode Basis (Reynolds number here) Mode interpolation Parametrized Mode Basis (Reynolds number here) OPERATING CONDITIONS II POD modes time-avg. solution =0.25 =0.50 shift-mode =0.75 Eigen-modes steady solution M. Morzynski & al.. Notes on Numerical Fluid Mechanics 2007 Tadmor & al. CISM Book 2011 -fast transients OPERATING CONDITIONS I

Results and Conclusions Advanced platform for FSI ROMs testing open for common research Computations ongoing Treatment of CSM - evolution Linear CSM model Non-linear CSM model Tadmor & al. CISM Book 2011 – control capable AE model Mode parametrization

CFD/CSM Coupling Canonical computational domain

Coupling in Low-Dimensional AE Full-dimensional CSM The algorithm essentially the same as the high-dimensional one Interpolation of pressures/forces required Interpolation of boundary displacements and mesh deformation required: dependent on the chosen approach of boundary motion modelling (acceleration forces / actuation modes / Lagrangian-Eulerian / …) – Tadmor et al., CISM book Modal CSM The aerodynamic forces on the surface of structure might be related to the POD (or any other) decomposition of pressure field Thus: interpolation of pressures/forces not required Mesh deformation (velocity) modes / actuation modes calculated in relation to the eigenmodes of the structure The amplitudes of „mesh” modes calculated from the amplitudes of eigenmodes of structure (time integration?) Thus: interpolation of boundary displacements and mesh deformation not required