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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

Fluid Structure Interaction SMJ Configuration Multidisciplinary Computation

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

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

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

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

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

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

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

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

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

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

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

Flexible Aircraft Motion Equations Multidisciplinary Computation

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