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 - motivationNeed of ROM in design

AIAA 2008, Rossow, Kroll 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

Need of online capable ROMs in feedback flow control

Aeroservoelasticity

Aeroelastic control (Piezo-control of flutter, wing morphing, smart structures)

MicroAerialVehicles (maneuverability)

High- Dimensional Aeroelastic System – ROM testbed

Tau Code

MF3 (in-house),Calculix, Nastran

In-house and AE tools

Spring analogy

Flow code

Structural code

Interpolation

Fluid forces

Forces

Structure displacements and velocities

Deformed CFD mesh, velocities

CFD mesh deformation

Interpolation

t=t+t

convergence

yes

no

Computational aspects – Euler code

Flow code

Structural code

Interpolation

Fluid forces

Forces

Structure displacements and velocities

Deformed CFD mesh, velocities

CFD mesh deformation

Interpolation

t=t+t

convergence

yes

not=80s

t=10s

t=10s

t=4s / 50s

t=30s

One iteration time: 134s (full CSM) / 180s (modal CSM)

Mesh:10 mio elements

CPU Power:16 cores

Computational aspects - RANSRANS

Flow code

Structural code

Interpolation

Fluidforces

Forces

Structure displacements and velocities

Deformed CFD mesh, velocities

CFD mesh deformation

Interpolation

t=t+t

convergence

yes

not=400s

t=90s

t=90s

t=4s / 50s

t=220s

One iteration time: 850s (full CSM) / 804s (modal CSM)

Mesh:30 mio elements(1 mio: surfaces)

CPU Power:32 cores

High-fidelity CFD and CSM High-fidelity CFD and CSM solverssolvers

CFD - TAU CODE

• 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

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

From DLR TAU-code manual

ALE - Motion of boundary and mesh ALE - Motion of boundary and mesh canonical domaincanonical domain

With boundary conditions:

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:

The fluid mesh can move independently of the fluid particles.

0Re

1 =upuuu ),,,( txxxFxKxCxM Eulerian approach Lagrangian approach

Coupling requirementsCoupling 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)

XY

Z

Aerodynamic mesh12437 nodes

Structural mesh212 nodes

Pressure forces interpolation

Coupling toolsCoupling toolsThe meshes are non-conforming•different discretization•different shape (whole wing/torsion box only

Non-conservative interpolation

Conservative interpolation

Coupling toolsCoupling 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.

Newmark direct integration method

[M] x’’ (t) + [D] x’ (t) + [K] x (t) = f (x, x’, x’’, t)

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‘‘ )

NEWMARK explicit scheme

with = 0 and = 0.5

Inertial Damping Elastic Aerodynamic forces forces forces forces

Structural forces

Integration in time in CFD (or CSM) codexi+1 = xi + t xi‘ + t2 [ ( 1/2 - ) xi‘‘ + xi+1‘‘ ]

xi+1‘ = xi ‘ + t [ ( 1 - ) xi‘‘ + xi+1‘‘ ]

Dynamic Coupling: time integrationDynamic Coupling: time integration

General aeroelastic equations of motion :

Fluid mesh deformationFluid mesh deformation

• 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

• Another possibilities:Elastic material analogy, Volume Splines

(Radial Basis Functions), Transfinite Interpolation

• Spring analogy

• Shephard interpolation(Inverse Distance Weighting) Based on the distances di between a given mesh node and boundary nodes:

I22 and I23 airplanesfrom: wikimedia

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

Flutter analysis for I-23 airplaneFlutter analysis for I-23 airplane

Simulation: flutter at Ma=0.44

Experiment: flutter at Ma=0.41

Time history for displacement and rotation Time history for displacement and rotation in control node on wingin control node on wing

Flutter Laboratory IoA and PUT

experiment 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 algorithmLow-Dimensional FSI algorithm

Flow ROM

Structural code

Interpolation

Pressure

Forces on structure

Structure displacements and velocities

Deformed CFD mesh, velocities

CFD mesh deformation

Interpolation

t=t+t

convergence

yes

no

Amplitudes of „mesh” modes

Reduced Order Model of the flowReduced Order Model of the flow

N

iii

N uauu1

0][

kj

N

j

N

kijkj

N

jiji aaqala

0 00Re

1

0Re

1 =upuuu

1. GALERKIN APROXIMATION

2. GALERKIN PROJECTION

3. GALERKIN SYSTEM

0, ][ N

i uu

Navier-Stokes Equations

Projection of convective termProjection of convective term

G

ii

N

i

GGgrid uau

1

kGji

G uuuqijk

,

0)(Re

1 =upuuuu grid

1. DECOMPOSITION

2. GALERKIN PROJECTION

GN

jk

Gj

N

k

Gijk

N

jkj

N

kijk

gridiigridi

aaqaaq

uuuuuuuuuu

1 00 0

,,)(,

Arbitrary Lagrangian-Eulerian Approach

ROM for a moving boundaryROM for a moving boundaryDNS with ALE

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

NACA-0012 AIRFOIL

Inverse Distance Weighted

First 8 POD modes: 99.96% of TKE

ROM for a moving boundaryROM for a moving boundary

ALE ROM vs DNSEulerian ROM vs ref. DNS

Dumping of oscillationtypical for sub-critical Re

The first two modes

AE mode basisAE mode basis

• Test-case: bending and pitching LANN wing

• Fluid answer to separated, modal deformations (varying amplitudes)

• Fluid answer to combined deformation

LANN wing structure

Pressure field and structure deformation

(high-dimensional AE)

for a flow induced by structure deformations

ROM AE: ROM AE: CFD → CSM CouplingCFD → CSM Coupling

• High-dimensional fluid forces retrived from the Galerkin Approximation

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

• We preserve full-dimensional CSM and existing AE coupling tools to interpolate fluid forces on coupling - “wet” - surface;

(similarly to Demasi 2008 AIAA)

ROM AE: ROM AE: CSM → CFDCSM → CFDCoupling and CFD mesh deformationCoupling 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

G

ii

N

i

GGgrid uau

1

kGji

G uuuqijk

,

ui+1 = ui + t ui‘ + t2 [ ( 1/2 - ) ui‘‘ + ui+1‘‘ ]

ui+1‘ = ui ‘ + t [ ( 1 - ) ui‘‘ + ui+1‘‘ ]

Mode interpolation Mode interpolation Parametrized Mode Basis (Reynolds number here)

steady solution

time-avg. solution

shift-mode

OPERATING CONDITIONS II

=0.50

=0.25

=0.75

POD modes

Eigen-modes

OPERATING CONDITIONS I

M. Morzynski & al.. Notes on Numerical Fluid Mechanics 2007

Tadmor & al. CISM Book 2011 -fast transients

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/CSMCFD/CSM

Coupling Canonical computational domain

Coupling in Low-Dimensional Coupling in Low-Dimensional AEAE

• Full-dimensional CSM

• Modal 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

• 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