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

2. 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

3. 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

4. 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

5. 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)

6. 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)

7. 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

8. 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.

9. 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

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

11. 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.

12. 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‘‘ )

13. 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

14. I22 and I23 airplanes from: wikimedia

15. 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

16. 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

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

18. 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

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

20. Results of test case 1
# grams on the wing's tip

21. 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

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

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

24. 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: % of TKE

25. 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

26. 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

27. 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

28. 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‘‘ ]

29. 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

30. 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

31. CFD/CSM
Coupling
Canonical computational domain

32. 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