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

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

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

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

**High-fidelity CFD and CSM solvers** CFD - TAU CODE CSMMF3: 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 • Finitevolumemethodsolvingthe Euler and Navier-Stokesequations • hybridgrids (tetrahedrons, hexahedrons, prisms and pyramids) • Central orupwind-discretisation of inviscidfluxes • Runge-Kutta time integration • accelerated by multi-grid on agglomerateddual-grids • miscellaneousturbulencemodels • Parallelizedwith 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) bindsthe 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 FEM model with 2,815 nodes Alenia SMJ CFD N-S hybrid grid with 1.3 mio nodes and 4.7 mio elements (cells) Aerodynamic mesh 12437 nodes Structural mesh 212 nodes Pressure forces interpolation

**Coupling tools** Themeshesarenon-conforming • differentdiscretization • differentshape (wholewing/torsionboxonly Non-conservative interpolation Conservative interpolation

**Coupling tools** • MpCCi (Mesh-based parallel Code Coupling Interface), developed atthe 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 • Shephardinterpolation(InverseDistanceWeighting) Based on thedistancesdibetween a givenmeshnode and boundarynodes: • Anotherpossibilities: Elasticmaterialanalogy, VolumeSplines (RadialBasisFunctions), TransfiniteInterpolation

**I22 and I23 airplanesfrom: 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 rotationincontrolnode on wing Simulation: flutterat Ma=0.44 Experiment: flutterat Ma=0.41

**Flutter Laboratory IoA and PUTexperiment and computations** • Scale : • Length - 1:4 • Strouhal number 1:1

**Experimentalconfigurations** • 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 yes Flow ROM convergence no 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** ArbitraryLagrangian-EulerianApproach 1.DECOMPOSITION 2. GALERKIN PROJECTION

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

**ROM for a moving boundary** ALE ROM vs DNS Eulerian ROM vs ref. DNS Dumping of oscillationtypical 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 forcesretrived from the Galerkin Approximation

**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‘‘ ]

**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