multidisciplinary computation: application to fluid-structure interaction v. selmin...

Multidisciplinary Computation: Application to Fluid-Structure Interaction

V. Selmin

Multidisciplinary Computation and Numerical Simulation

Multidisciplinary Computation and Numerical Simulation

• Multidisciplinary optimisation process

Integration of different disciplines within the design process,Optimisation, Concurrent Engineering

Development of a new generation of numerical tools

New Trends in Design

Drivers: Reduce product development costs and time to market

• Single discipline optimisation process

From analysis/verification to design/optimisation

• From single to multi-physics

Integration of different disciplines, Interfaces between disciplines,Concurrent Engineering

Multidisciplinary Computation

Fluid Structure Interaction: Process Overview

Aerodynamics/CFD

Structure/CSM

CFD Grid

Loads transfer

Displacements

Aero-structural Design Process


Three field approach: basic formulation The coupled transient aeroelastic problem can then be formulated as a three-field problem:1- the fluid2- the structure3- the dynamic meshThe semi-discrete equations governing this tree-way coupled problem can be written as follows

are fictitious mass, damping and stiffness matrices

associated with the fluid moving grid.

is a transfer matrix that describes the action of the structural side of the fluid structure interface on the fluid dynamic mesh.

For example, includes a particular case of the spring based mesh motion scheme

xKxKx

Dx

M

xxfqfq

M

xxRxxxFxx

*t

**2

2*

int2

2

)),,(()(

,),,(,),,(),(),(

tt

tWt

tWtWtWtt

ext

structureonforcesexternalofvector:

forcesstructuralinternalofvector:

matrixstiffness:

matrixdamping:

matrixmass:

vectorntdisplaceme:

fluxesviscousofvector:

fluxesconvectiveofvector:

cellFVtheofvolume:

vectorstatefluid:

velocitypointgrid:

positionpointgrid:

ext

int

f

f

K

D

M

q

R

F

x

x

W

*** ,, KDM

0DM **

*tK


Three field approach: basic formulation The fluid and mesh equations are directly coupled.The fluid and structure equations are coupled by the interface conditions on

The first transmission condition states that the tractions on the wet surface of the structure are in equilibrium with

those on the fluid side .The second condition expresses the compatibility between the velocity fields of the structure and the fluid at the fluid/structure interface.For inviscid flows, this second equation is replaced by the slip wall boundary condition

The equations governing the structure and fluid mesh motions are coupled by the continuity conditions

tt

-p

FS

FS

qq

nΠnnΠ

SF

SF

S

F

S

F

p

/

/

tovectornormal:

boundaryinterfacestructurefluid:

tensorstressstructure:

tensorstressviscousfluid:

pressure:

ntdisplacemestructure:

fieldntdisplacemefluid:

n

Π

Π

q

q

ttMS

MS

qq

qq

nq

nq

ttFS

SF /

SF /


Flow Solver for Moving Meshes Conservation law form:

Unknowns & Flux vector (Eulerian approach):

State equation:

Moving meshes (ALE approach):

0),()(

WWWt

WRF

qΠw

ΠRIww

w

Fw

-hw

p

e

W

t

T

t

0

,,

q

Π

ww

w

:fluxheat

:tensorstress

:ratioheats.spec

,2

:energyinternal

:enthalpytotal

:energytotal

:pressure

),,(:velocity

:densitymass

t

tt

t

ee

peh

e

p

wvu

)()1(

),(

gasperfectep

efp

WWWW xFxFF )(),()(x:velocitiesnode

ALE: Arbitrarian Lagrangian Eulerian


Flow Solver for Moving Meshes Finite Volume DiscretisationDiscretisation of the integral equation:

Discrete equation:

Numerical flux:

Time integration:

0 iiddW

dt

d nR-F

n

n

i

ji

d

d

i

ij

nν

nν

)(),()(),(),,(

..),,,(),,()(

xνRxνxFνxx

νxxxx

ijijijijij

iKjiji

WWW,W

TBWW

0),,( xx WWdt

diii

0),,(11

dtWdtWdt

d n

n

n

n

t

t it

t ii xx


Flow Solver for Moving Meshes Time IntegrationA second-order accurate time-accurate implicit algorithm that is popular in CFD is the second-order backward difference scheme.A generalisation of this algorithm for dynamic mesh can be written as:

where

and denote some linear combination of the mesh configurations and their velocities, i.e.

The following choice has been made for , denoted by , respectively,

0),,( 1111111 xx ni

nnni

nnni

nnni

n WtWWW

),(),( nnni

ni txxx

1

211 ,

1),1(,

1

21

n

nnnn

t

t

xx ,

1111 ,,,,, nnnnnn xxxxxxxxxx

xx , xx ~~ ,

n

nnnn

t

xx

xxx

x11 ~,

2~

0),,(11

dtWdtWdt

d n

n

n

n

t

t it

t ii xx


Flow Solver for Moving Meshes GCL conditionA sufficient condition for the previous time integrator to be mathematically consistent is to predict exactly the state of a uniform flow. This condition whic can be formulated as a Geometric Conservation Law states that

This equation is satisfied only for versy special choice of which depends from the particular time integration scheme and type of grid elements adopted. A more versatile technique in order to satisfy the GCL condition consists in the computation of the defect

which is used to correct the coefficient .

Dual Time-Stepping ApproachThe basic idea of the dual time-stepping approach is to treat the unsteady problem as a steady state problem and to solve it as an artificial unsteady equation:

where

since the artificial time is used as a relaxation parameter to find the solution of the previously described steady problem. Special techniques in order to accelerate convergence are allowed.

0)()(

1111

iKjijij

nni

nni

nni

n t xνx

)(

1111 )~(~

iKjijij

nni

nni

nni

ni tGCL xνx

)~,~,(1

)( 11111111 xxR*

ni

nni

nnni

nnni

nn

WWWWt

W

0)(**

Wt

WR

xx ,

1n

*t


Second Order Structural Solver The governing equation of linear dynamic equilibrium is

that can be rewritten as

where

Time-integration from to using midpoint trapezoidal rule reads

which is second order accurate in time. Note that the previous equations implies that

FqKq

Dq

M

tt 2

2

GQBQ

A t

F

0G

DK

1-0B

M0

01Aq

qQ ,,,

t

nt nnn ttt 1

2111

2/n

nn

n

nn

t

GQQ

BQQ

A

0qqqq

2

11 nn

n

nn

t

forcesexternalofvector:

matrixstiffness:

matrixdamping:

matrixmass:

vectorntdisplaceme:

F

K

D

M

q


Second Order Time Accurate Staggered Procedure The second-order staggered (SOS) algorithm is built as a leap-frog scheme where the fluid sub-system is

computed at half time stations while the structure sub-system is computed at full

time stations . It can be summarised as follows

1- Predict the structural displacement at time

2- Update the position of the fluid grid in order to match the position that the structure whould have if it were

advanced the predicted displecement .

3- Time integrate the fluid subsystem from to using dual time stepping scheme and a fluid time step . If , sub-cycle the flow solver.

4- Transfer the fluid pressure and viscous stress tensor at time to the structure and compute the

corresponding induced structural loads .

5- Time integrate the structure sub-system from to by using the midpoint rule formula.

),t,t,( 2/1n2/1n

),t,t,( 1nn

2/1nt

t

t nnSnn

qqq

22/1

2/1nq

2/1nx

2/1nt nS

2/1n2/1n Δttt

SF ΔtΔt SF ΔtΔt 2/1nt

2/1nGnt n

Sn1n Δttt


Transfer of Aerodynamics Loads to the Structure

• Reduced axis approach only valid for wings of high aspect ratios

• Interpolation from CFD solution to CSM grid not accurate for coarse structure representation

• Association to each structural skin elements of a portion of the CFD surface grid

complex but accurate

• Outside wing box contributions


Transfer of Structure Deformation to the Fluid Mesh

• Reduced axis approach only valid for wings of high aspect ratios

• Interpolation from CSM grid to CFD grid association of each CFD surface node to a CSM

skin element

• Outside wing box nodes treatment


Fluid Structure Interaction: Simulation Chain

Initial Data

CFD Computation

Loads Computation

CSM Computation

CFD Grid Update

Dt, iter.

End of Process

Displacements Computation

IntermediateSolutions

Steady approach

Unsteady approach


Fluid Structure Interaction: Workflow

Input DataPreprocessing

Solid ModelGeometry

StructuralModels

MassData

ComputationParameters

StructureCSM

AirflowCharacteristics

Theoretical ModelCFD

AirflowCFD

CFD Grid Update

StructuralResponse

CFD Grid Update

F/S InteractionOutputs

Engine Data Set

Fluid Structure Interaction

DisplacementsUpdate

Task

Principal dependency

Actor

Data TransferInformation


Fluid Structure Interaction: Dataflow

Airflow Simulation: CFD Simulation

CFDSolver

CFD Simulation

Inputs Outputs

SimulationProcess

Postprocessing

CFD Grid

CFD Computation Parameters

Grid Nodes Velocities

Grid Generation

Outputs

Pressure data

Stress Tensor Data

Inputs

Airflow Characteristics

Outputs

Grid Update



Airflow Simulation: Grid Generation (CFD)

Surface gridGenerator

Grid Generation

Inputs Outputs

SimulationProcess

Volume GridGenerator

CAD Model

Grid Generation Parameters

Grid SpacingDescription

Element Connectivities

Nodes Co-ordinates

Inputs

CFD Simulation

Outputs

Solid ModelGeometry



Airflow Simulation: Grid Update (CFD)

Grid Update

Inputs Outputs

SimulationProcess

Initial CFD Grid

Grid Update Parameters

Updated Body Surface Description

Grid Generation

Outputs

Element Connectivities

Nodes Coordinates

Inputs

CFD Simulation

Outputs

Displacements Update

Surface gridUpdate

Volume GridUpdate



Airflow CharacteristicsAirflow

Characteristics

Inputs Outputs

SimulationProcess

Data Integration

Pressure Data

Stress Tensor

Theoretical/CFD Model

Outputs

Global Aerodynamic Coefficients

Aerodynamic Loadsat monitoring stations

Inputs

Monitoring StationsStructuralModels

Outputs

CFD grid Detailed flowfeatures

Structural Response

CFD to CSM gridsrelationships

OutputsData TransferInformation



Structural Response

Structural Response

Inputs Outputs

SimulationProcess

CSM Solver

FE Model

Aerodynamic Loads

Material properties

Structure nodedisplacements

Stresses & strains

Inputs

Inertial Loads

StructuralModels

Outputs

DisplacementsUpdate

Additional Loads

Outputs

Airflow Characteristics

Outputs

Mass Data

Engine Data Set

Outputs


Fluid Structure Integration: Dataflow

Displacements Update

DisplacementsUpdate

Inputs Outputs

SimulationProcess

Postprocessing

CFD grid

Structural nodes Displacements

CFD Surface nodesdisplacement

Inputs

CFD GridUpdate

CSM to CFD gridsrelationships

OutputsData TransferInformation

StructuralResponse

Outputs


Aerodynamic Solver

• Node-centred based Finite Volume spatial discretisation• Blended second- and fourth order dissipation operators• Operates on structured, unstructured and hybrid grids• Time integration based on Multistage Algorithm (5 stages)• Residual averaging and local timestepping• Preconditioning for low Mach number• Pointwise Baldwin-Barth, K-Rt (EARSM) turbulence models• Chimera strategy implementation

• ALE implementation for moving grid• Time accurate simulation provided by using Dual Timestepping

• Scalar, vector and parallel implementation

RANS Solver


Structural Analysis and Optimisation

Based essentially on MSC - NASTRAN software

• SOL101 for static analysis

• SOL200 for structural optimisation based on DOT optimiser (SQP)

• In the case of aeroelastic simulations, use of own software both for direct and modal formulations with extraction of the structural matrices from NASTRAN solutions.

FKutu

Dtu

M

2

2

Structural Solver

qu


Grid Deformation

Spring Analogy

)x(x,0S)(

ijijijiiKjij KFF

Source terms allow to control mesh quality



Steady Approach

Structural solver

Flow solver

)(wFKu

)()( wFwRt

w

0

Problem Definition

Process Initialisation

CFD Computation

Loads Computation

CSM Computation

CFD Grid Update

Residual Computation

Residual <Resmax ?

End of ProcessNo Yes



Unsteady Approach

Structural solverFlow solver Coupling


MSMS uuuu , Continuity Conditions

Mathematical Consistency (GCL)

Steady:

Unsteady

0 Cd

01111

C

mmmmmm dxt 11 mm

Metrics & Nodes Velocities

m

mmmm

t

xxx,

xxx):xR(w,x,

11

2

21nm

21ntn

nnn u

tuu

221

21nx 21nu21nt nnn ttt 2121

21nt 21nGnt nnn ttt 1

SOS Coupling Procedure1- Predict the structural displacement at time

2- Update the position of the fluid grid in order to match

3- Time-integrate the fluid subsystem from to

4- Transfer the fluid stresses at time to the structure and compute

5- Time-integrate the structure sub-system from to using the mid-point rule formula




SMJ Configuration



SMJ Test-Case

FE Structural ModelCFD Model

Static cases: Cruise, pull-up manoeuvre & push-down manoeuvreDynamic cases: unstable, marginally stable & stable conditions.


Static Aeroelasticity

Aeroelastic Simulation: M=0.8, Nz=1, Z=11277 m

M=0.8 , Cl=0.45

Jig Shape Deformed Shape



Aeroelastic Simulation: M=0.8, Nz=1, Z=11277 m

Wing Box Deformation

Von Mises Stresses



Aeroelastic Simulation: M=0.6, Nz=2.5, Z=4500 m

M=0.6 , Cl=0.75




Aeroelastic Simulation: M=0.6, Nz=-1, Z=4500 m

M=0.6 , Cl=-0.30



Dynamic Aeroelasticity

SMJ Modal Shapes

Mode 1 (2.01 Hz) Mode 2 (2.82 Hz) Mode 3 (3.70 Hz)

Mode 4 (5.26 Hz) Mode 5 (5.73 Hz) Mode 6 (7.38 Hz)



V-G & V-F Diagrams


Dynamic AeroelasticityAeroelastic Simulation: M=0.83, z=11300 m


Dynamic AeroelasticityAeroelastic Simulation: M=0.83, z=7000 m



Aeroelastic Simulation: M=0.83, z=7000 m


Flexible Aircraft Motion Equations



Methods• Linear methods are non conservative for transonic flows (nonlinear effects)• Time dependent methods are too expensive to be used for day to day design work

Validation Need for dedicated and accurate experimental data sets (much more expensive & difficult to obtain than for a single discipline)

• Correction of linear models• Linearised in time/frequency Euler & Navier-Stokes solvers• Reduced order models

Interfaces & CouplingIn the past, God invented the partial differential equations. He was very proud of him. Then, the devil introduced the boundary conditions. Today, the technology can be considered to be mature when referred to single disciplines.For the solution of multidisciplinary problems, the devil is now represented by the interfaces.

Jacques-Louis Lions

multidisciplinary computation: application to fluid-structure interaction v. selmin...

Documents

fluid structure interface

fluid dynamic mesh

fluidstructure interactionv

structure equations

fluid mesh motions

mesh equations

state equation

mesh configurations