1. Multidisciplinary Computation:
Application to Fluid-Structure Interaction
V. Selmin
Multidisciplinary Computation and Numerical Simulation

2. 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 optimisation process
Integration of different disciplines within the design process,
Optimisation, Concurrent Engineering
Development of a new generation of numerical tools
Multidisciplinary Computation and Numerical Simulation

3. Fluid Structure Interaction: Process Overview
Aero-structural Design Process
Aerodynamics/CFD
CFD Grid
Loads transfer
Structure/CSM
Displacements
Multidisciplinary Computation

4. Three field approach: basic formulation
The coupled transient aeroelastic problem can then be formulated as a three-field problem:
1- the fluid
2- the structure
3- the dynamic mesh
The 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
Multidisciplinary Computation

5. 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
Multidisciplinary Computation

6. Flow Solver for Moving Meshes
Conservation law form:
Unknowns & Flux vector (Eulerian approach):
State equation:
Moving meshes (ALE approach):
ALE: Arbitrarian Lagrangian Eulerian
Multidisciplinary Computation

7. Flow Solver for Moving Meshes
Finite Volume Discretisation
Discretisation of the integral equation:
Discrete equation:
Numerical flux:
Time integration:
Multidisciplinary Computation

8. Flow Solver for Moving Meshes
Time Integration
A 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,
Multidisciplinary Computation

9. Flow Solver for Moving Meshes
GCL condition
A 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 Approach
The 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.
Multidisciplinary Computation

10. 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
Multidisciplinary Computation

11. 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.
Multidisciplinary Computation

12. 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
Multidisciplinary Computation

13. 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
Multidisciplinary Computation

14. Fluid Structure Interaction: Simulation Chain
Initial Data
Steady approach
CFD Computation
Loads
Computation
CSM Computation
Displacements
Computation
Unsteady approach
CFD Grid Update
Dt, iter.
Intermediate
Solutions
End of Process
Multidisciplinary Computation

15. Fluid Structure Interaction: Workflow
Input Data
Preprocessing
Solid Model
Geometry
Data Transfer
Information
Computation
Parameters
Structural
Models
Mass
Data
Engine
Data Set
Airflow
CFD
Theoretical Model
CFD
Airflow
Characteristics
Structure
CSM
Displacements
Update
Structural
Response
CFD Grid
Update
CFD Grid
Update
Fluid Structure
Interaction
F/S Interaction
Outputs
Task
Principal dependency
Actor
Multidisciplinary Computation

16. Fluid Structure Interaction: Dataflow
Airflow Simulation: CFD Simulation
CFD
Simulation
Grid
Generation
Airflow
Characteristics
Inputs
Outputs
Outputs
Inputs
CFD Grid
Pressure data
Outputs
Grid Nodes Velocities
Stress Tensor Data
Grid
Update
CFD Computation
Parameters
Simulation
Process
CFD
Solver
Postprocessing
Multidisciplinary Computation

17. Element Connectivities
Fluid Structure Interaction: Dataflow
Airflow Simulation: Grid Generation (CFD)
Grid
Generation
CFD
Simulation
Solid Model
Geometry
Inputs
Outputs
Inputs
Outputs
CAD Model
Element Connectivities
Grid Spacing
Description
Nodes Co-ordinates
Grid Generation
Parameters
Simulation
Process
Surface grid
Generator
Volume Grid
Generator
Multidisciplinary Computation

18. Element Connectivities
Fluid Structure Interaction: Dataflow
Airflow Simulation: Grid Update (CFD)
Grid
Update
Grid
Generation
CFD
Simulation
Inputs
Outputs
Outputs
Inputs
Initial CFD Grid
Element Connectivities
Grid Update
Parameters
Nodes Coordinates
Outputs
Updated Body Surface
Description
Displacements
Update
Simulation
Process
Surface grid
Update
Volume Grid
Update
Multidisciplinary Computation

19. at monitoring stations
Fluid Structure Interaction: Dataflow
Airflow Characteristics
Airflow
Characteristics
Theoretical/CFD
Model
Inputs
Outputs
Outputs
Inputs
Pressure Data
Global Aerodynamic
Coefficients
Structural
Response
Stress Tensor
Aerodynamic Loads
at monitoring stations
CFD grid
Detailed flow
features
Structural
Models
Outputs
Monitoring Stations
Data Transfer
Information
Outputs
CFD to CSM grids
relationships
Simulation
Process
Data Integration
Multidisciplinary Computation

20. Fluid Structure Interaction: Dataflow
Structural Response
Structural
Response
Structural
Models
Inputs
Outputs
Outputs
Inputs
FE Model
Structure node
displacements
Displacements
Update
Airflow
Characteristics
Material properties
Stresses & strains
Outputs
Aerodynamic Loads
Mass
Data
Outputs
Inertial Loads
Outputs
Additional Loads
Engine
Data Set
Simulation
Process
CSM Solver
Multidisciplinary Computation

21. Fluid Structure Integration: Dataflow
Displacements Update
Displacements
Update
Inputs
Outputs
Inputs
Structural
Response
CFD grid
CFD Surface nodes
displacement
CFD Grid
Update
Outputs
Structural nodes
Displacements
Data Transfer
Information
Outputs
CSM to CFD grids
relationships
Simulation
Process
Postprocessing
Multidisciplinary Computation

22. Aerodynamic Solver RANS 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
Multidisciplinary Computation

23. Structural Analysis and Optimisation
Structural Solver
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.
Multidisciplinary Computation

24. Grid Deformation Spring Analogy
Source terms allow to control mesh quality
Multidisciplinary Computation

25. Process Initialisation
Fluid Structure Interaction
Steady Approach
Problem Definition
Process Initialisation
CFD Computation
Loads Computation
CSM Computation
CFD Grid Update
Residual Computation
Residual <
Resmax ?
End of Process No
Yes
Flow solver
Structural solver
Multidisciplinary Computation

26. Fluid Structure Interaction
Unsteady Approach
Structural solver
Flow solver
Coupling
Multidisciplinary Computation

27. Continuity Conditions
Fluid Structure Interaction
Metrics & Nodes Velocities
SOS Coupling Procedure
1- 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
Continuity Conditions
Mathematical Consistency (GCL)
Steady:
Unsteady
Multidisciplinary Computation

28. Fluid Structure Interaction
SMJ Configuration
Multidisciplinary Computation

29. Fluid Structure Interaction
SMJ Test-Case
FE Structural Model
CFD Model
Static cases: Cruise, pull-up manoeuvre & push-down manoeuvre
Dynamic cases: unstable, marginally stable & stable conditions.
Multidisciplinary Computation

30. Static Aeroelasticity
Aeroelastic Simulation: M=0.8, Nz=1, Z=11277 m
M=0.8 , Cl=0.45
Jig Shape
Deformed Shape
Multidisciplinary Computation

31. Static Aeroelasticity
Aeroelastic Simulation: M=0.8, Nz=1, Z=11277 m
Wing Box Deformation
Von Mises Stresses
Multidisciplinary Computation

32. Static Aeroelasticity
Aeroelastic Simulation: M=0.6, Nz=2.5, Z=4500 m
M=0.6 , Cl=0.75
Jig Shape
Deformed Shape
Multidisciplinary Computation

33. Static Aeroelasticity
Aeroelastic Simulation: M=0.6, Nz=-1, Z=4500 m
M=0.6 , Cl=-0.30
Jig Shape
Deformed Shape
Multidisciplinary Computation

34. 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)
Multidisciplinary Computation

35. Dynamic Aeroelasticity
V-G & V-F Diagrams
Multidisciplinary Computation

36. Dynamic Aeroelasticity
Aeroelastic Simulation: M=0.83, z=11300 m
Multidisciplinary Computation

37. Dynamic Aeroelasticity
Aeroelastic Simulation: M=0.83, z=11300 m
Multidisciplinary Computation

38. Dynamic Aeroelasticity
Aeroelastic Simulation: M=0.83, z=7000 m
Multidisciplinary Computation

39. Dynamic Aeroelasticity
Aeroelastic Simulation: M=0.83, z=7000 m
Multidisciplinary Computation

40. Flexible Aircraft Motion Equations
Multidisciplinary Computation

41. Fluid Structure Interaction
Interfaces & Coupling
In 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
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
Correction of linear models
Linearised in time/frequency Euler & Navier-Stokes solvers
Reduced order models
Validation
Need for dedicated and accurate experimental data sets
(much more expensive & difficult to obtain than for a single discipline)
Multidisciplinary Computation