Multidisciplinary Computation and Numerical Simulation for Reduced Flight Test and Wind Tunnel Costs

1. Multidisciplinary Computation and Numerical Simulation Summary V. Selmin

2. Role of the Numerical Simulation Flight Test / Real World Numerical Simulation Wind Tunnel Experiment • Reduce Flight Test and Wind Tunnel costs • Reduce Product Development Costs • Reduce Time to Market Simulation Chain

3. Numerical Simulation Process Real World Physical system Mathematical Model Set of PDE, gas properties Space discretisation (Grid) Numerical scheme (FD,FV,FE) Discretisation Pre-processing of data Execution of numerical code Simulation Approximated Solution Post-processing of data Analysis of results Simulation Chain

4. Simulation Chain CAD representation Grid generation Solver (CFD) • Turn-around time as low as possible • High level of confidence of the numerical simulation • Correct use of the numerical simulation • (Choice of the model and its limitations) Simulation Chain

5. CAD Representation • Features: • Reduce as much as possible hand work to prepare the model for • simulation • CAD repair • Geometry parametrisation at the CAD level • Automatic Design, Optimisation • CAD model recovery • Update geometry to be used for other simulations/applications Simulation Chain

6. Grid Generation • Features: • Highly integrated with the CAD system • Standard inputs (IGES,STEP, …)? • High flexibility for the discretisation of complex geometries • Different element types, anisotropic elements, … • Highly automatised and fast grid generation process • Unstructured grid approach (for complex geometries) • High quality grids • Mesh refinement and mesh deformation capabilities Grid generation

7. Mesh Generation Concept Structured Grids • Mesh structure: • Domain divided into a structured assembly of • quadrilateral cells • Each interior nodal points is surrounded by • exactly the same number of mesh cells (or • elements) • Directions within the mesh can be immediately • identify by associating a curvilinear co-ordinates • system • it is possible to immediately identify the nearest • neigthbours of any node j on the mesh • Advantages: • Large number of algorithms for discretisation are • available • The algorithms can be normally implemented in • a computationallly efficient manner • Disadvantages: • Difficulty to generate grids of regions of general • shapes  multi-block grids • Very high elapsed time necessary to produce a • grid for domains of extremely complex shape Finite difference, finite volume and finite element discretisations Grid generation

8. Mesh Generation Concept Unstructured Grids • Mesh structure: • Computational domain divided into an unstructured • assembly of computational cells • The number of cells surrounding a typical interior • node is not necessarily constant • The nodes and the elements has to be numbered • To get the necessary information on the neightbours • the numerotation of the nodes wich belong to each • element has to be stored • The concept of directionality does not exist anymore • Advantages: • Powerful tool for discretising domains of complex • shapes • Unstructured mesh methods naturally offer the • possibility of incorporating adaptivity • Disadvantages: • Alternative solution algorithms are more limited • Computational implementation places large demands • on both computer memory and CPU Finite volume and finite element discretisations Grid generation

9. Structured Grid Generation • Structured grid generation • Algebraic and Transfinite Interpolations • Grid generation by using PDE (elliptic, hyperbolic, parabolic) • Multi-block structured grid generation concept • Structured grid generation is efficient for simple geometry, but lack • of efficiency for complex geometry. • Strong impact on the solver that has to be adapted at particular • features of the grid like singular points. • New developments (Chimera techniques, ….) transfer the complexity • from the grid generation process to the solver: easier(?) grid generation • process, but more complex solvers. • The technologies based on the unstructured grid generation approach • seems to be more adapted for application of CFD (Computational Fluid • Dynamics) around complex shapes Grid generation

10. Unstructured Grid Generation Advancing Front Method In the classical front advancing method, the nodes coordinates are built at the same time as the elements from the knowledge of the size of the elements that belong to the front and the spacing distribution. Delaunay Method In the Delaunay method, a triangulation of the domain from the knowledge of the boundary discretisation is first performed. Recursively, nodes are added in order to satisfy the imposed spacing distribution and new triangulation is performed in order to taking into account the insertion of new nodes. Grid generation

11. Riemannian Metric Let a symmetric definitive positive tensor M(x) be defined at any point x of the computational domain . It represents a metric tensor if for any segment v the following relation holds: The tensor M(x) can be decomposed as A transformation matrix T can be associated to M that is defined as By using the property of orthogonality of the eigenvectors of tensor M , we obtain and, consequently Grid generation

12. Characteristic Dimension Parameters The geometrical characteristics of an element can be defined in terms of the following mesh parameters. If n is the number of dimensions, the parameters used are a set of n orthogonal directions and n associated element sizes . The transformation T may be defined as the result of superposing n scaling operations with factor in each direction: Therefore by taking into account that may be rewritten according to the following relations hold Grid generation

13. Solution on the refined grid Solution on the initial grid Grid refinement Refined grid Initial Grid Sensor Build the metrics for the grid refinement and selection of the edges to be cut Grid Refinement Concept Computation of the solution on the initial grid Sensor computation Grid refinement: Isotropic Anisotropic Build the refined grid and the interpolated solution Solution on the refined grid Grid generation

14. Isotropic/Anisotopic Mesh Refinement Euler Grid: Anisotropic refinement, quadrilaterals RAE2822 Airfoil, M=0.75, α=3º Grid generation

15. Node Movement Concept Linear spring analogy Spring Analogy: Source Term: Variational Analogy: Grid generation

16. Grid Deformation Aeroelastic Deformation Grid generation

17. Adaptation through Grid Deformation Euler Grid: Adaptation through node movement, quadrilaterals NACA0012 Airfoil, M=0.85, α=1º Grid generation

18. Mesh Generation-Unstructured Grids New mesh generation procedure requirements 1- Anisotropic Mesh Generation Needed for boundary layer discretisation (hybrid grids) and control of the number of nodes (stretching along wing span) 2- High Flexibility in Mesh Generation For surface grid and volumic grid generations 3- Automatic Background Grid Computation From metrics on the surface to metrics in the volume 4- Grid Quality Fast grid generation is important but high quality grid is far more important - A posteriori grid quality improvement tool Grid generation

19. Mesh Generation-Unstructured Grids From metrics on the surface to metrics in the volume 1- Compute the associated mesh metrics starting from the discretisation of the body skin & the external boundary 2- Form the node connectivity by using Delaunay algorithm 3- Build additional nodes and formed links to nodes on the body 4- Compute metrics for the additional nodes 5- Eventually, correct the computed metrics 6- Connect all the nodes by using Delaunay algorithm 7- Metrics in any point by interpolation on the last Delaunay grid Grid generation

20. Mesh Generation-Unstructured Grids Metrics on the surface 1- Computation of the principal curvatures and principal curvature directions on the surface by using the mathematic definition of the surfaces. 2- The principal curvature directions are orthogonal and will defines the spacing directions in plane tangent to the surface at a given points. 3- The spacing along the principal curvature directions will be computed as a fraction of the curvature radius associated to the principal curvatures 4- The spacing has to be limited by threshold values (maximum and minimum spacing allowed), in order to control the spacing . This is particular true when the curvature radius is very large (case of planes). 5- The third direction is computed as the vectorial product of the first two. The spacing in this direction is computed as the minimum of the first two spacings. 6- In general, discontinuities in metrics are present from a surface to its adjacent surfaces. A smoothing procedure is applied in order to regularise the metrics. Grid generation

21. Mesh Generation-Unstructured Grids Grid generation

22. Skin Outer boundary layer Mesh Generation-Hybrid Grids 560000 nodes, 30 layers in the B.L. region. Grid generation

23. CFD Solvers • Features: • Versatility/Flexibility of the approach • Unstructured vs Structured Data Structure • Fast and accurate solvers • Design of numerical schemes! Verification & Validation! • User friendliness and easiness • GUI, postprocessing & user education • Choice of the algorithm and interpretation of results • Knowledge of the methods limitations, user education & • experience CFD solvers

24. Basic Equations in CFD Navier-Stokes Equations If viscous effects are negligible Euler Equations If the flow is isentropic and irrotational Mathematical Complexity Increasing Full Potential Equations If the non-linearity is removed Linearised Potential Equations CFD solvers

25. Spatial Discretisation Structured Grids versus Unstructured Grids Structured grids: Same number of cells around a node Unstructured grids: The number of cells around a node is not the same Spatial Discretisation Finite Difference Discretisation: Taylor-series expansion Finite Volume Discretisation: Integral formulation Divergence theorem CFD solvers

26. Physical space Reference space Physical element Reference element Function approximation Reference element Physical element Integral method Integration by parts Weighted residuals Galerkin method PDE discretisation method Numerical integration Gauss method Numerical integration Spatial Discretisation Finite Element Discretisation: CFD solvers

27. Basic Properties Truncation error Difference between the original partial differential equation (PDE) and the discretised equation (DE). Consistency Consistency deals with the extent to which the discretised equations approximate the partial differential equations. A discretised representation of the PDE is said to be consistent if it can be shown that the difference between the PDE and its discretised representation vanishes as the mesh is refined: Stability Numerical stability is a concept applicable in a strict sense only to marching problems. A stable numerical scheme is one for which errors for any source (round-off, truncation, …) are not permitted to grow in the sequence of numerical procedures as the calculation proceeds from one marching step to the next. Convergence of Marching Problems Lax’s Equivalence Theorem: Given a properly posed initial value problem and a discretised approximation to it that satisfies the consistency conditions, stability is necessary and sufficient condition for convergence. Numerical Scheme Verification Von Neumann Method: Amplification and phase errors Modified Equation Method. CFD solvers

28. Finite Volume/Finite Element Node-Centred finite volume Discretisation of the integral equation: Discrete equation: Consistency: Conservative systems: Second-order spatial approximation: CFD solvers

29. Finite Volume/Finite Element Positivity & artificial viscosity Scalar equation: Discrete equation: Condition on scheme positivity: Monotone numerical scheme: Extension to systems of equations: CFD solvers

30. Finite Volume/Finite Element 1- Advanced Finite Volume schemes have been introduced. They include classical Finite Volume and Galerkin Finite Element Methods 2- Augmented dissipation schemes have been developed for generalised finite volume schemes. 3- Flux Difference Splitting and Flux Vector Splitting schemes may be re-interpreted as a central scheme with augmented dissipation operator, but are first-order accurate schemes 4- Augmented dissipation has to be activated where discontinuities are present, but has to vanish in regions where the solution is regular 5- Numerical schemes have been developed in such a way that total enthalpy is constant for iso-energenetic steady flows (needed in order to obtain accurate stagnation temperature value) In particular, the variables on which limitation is performed have to be accurately chosen. 6- The treatment of high enthalpy non-reactive and reactive flows has been discussed. →extension to finite elements schemes is automatic →introduction of limiters and shock sensors CFD solvers

31. Verification and Validation • Euler equations: • The most obvious properties of the exact (unknown) solution that may be used are as follows: • The total enthalpy is uniform in the whole field for an iso-energenetic flow (Bernouilli’s relation). • Entropy is generated only by discontinuities in the flow (shocks and contact discontinuities). • Upstream of the shock, the total pressure is equal to its free stream value . The value of has • been determined as a function of the given values of pressure and Mach number according to the formula: • Comparing this value of to , then provides a check of consistency of p and M (in particular at the wall). • The shock which terminates a supersonic region in transonic flow past a smooth geometry is normal to the wall • (except if the shock is a the trailing edge), and the pressure and the Mach number jumps at the wall in the • numerical solution are compared with the Rankine-Hugoniot jumps for a normal shock. More precisely, taking • the numerical value of the Mach number just upstream of the shock, , as the given upstream value in the • Rankine-Hugoniot relations, we compute the downstream Mach number, , and the ratio of the downstream • static pressure to upstream total pressure, , and these values are compared to the corresponding ones • from the numerical solution. • Navier-Stokes equations: • Comparison with experimental or flight test data. Saint-Venant Analytical solutions if they exist CFD solvers

32. Verification and Validation • Euler equations: • Based on the properties of the exact (unknown) solution. • Bernoulli’s equation • Entropy generation • Saint Venant relations • Rankine-Hugoniot relations • Navier-Stokes equations: • Comparison with experimental or flight test data. • Wind tunnel simulation issues: • Accuracy of the wind tunnel/flight test data • Correction of wind tunnel data: equivalent conditions for numerical simulation • Reynolds number issue. • Correct use of numerical simulation: • Grid generation issues • Turbulence modelling • Use of the most adequate numerical model with respect to the flow to studied • Critical examination of the results obtained from the simulation Analytical solutions if they exist CFD solvers

33. 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 optimisation process • Integration of different disciplines within the design process, • Optimisation, Concurrent Engineering Multidisciplinary Computation and Optimisation

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

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

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

37. Fluid Structure Interaction: Dynamic Aeroelasticity Aeroelastic Simulation: M=0.83, z=7000 m Multidisciplinary Computation and Optimisation

38. 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 • Correction of linear models • Linearised in time/frequency Euler & Navier-Stokes solvers • Reduced order models Fluid Structure Interaction: Summary • 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) Multidisciplinary Computation and Optimisation

39. Problem Definition Problem Definition Initial Models Database Initial Models Database Multi- Models Generation Multi- Models Generation Multidisciplinary Computations Optimisation Algorithm Multidisciplinary Computations Optimisation Algorithm Objective Constraints Objective Constraints Optimised Models Database Optimised Models Database Multidisciplinary Design Optimisation Mathematical tools, such as sensitivity analysis, modelling methods, and optimisation solvers, provide a mechanism by which working together can be accomplished. Multidisciplinary Computation and Optimisation

40. Multidisciplinary Design Optimisation • THAT MEANS • Automatic overall process management and monitoring • Automatic generation of models related to different disciplines • Parametrisation: Design Variables • Capability of executing single discipline solvers on heterogeneous platforms • Efficient and robust optimisation strategy • Definition of objective function and constraints for multidisciplinary problems • Education of engineer (no more thinking single discipline but multi disciplines) • Working Groups or Multidisciplinary Engineers As an Example Aerodynamics + Structural Mechanics + Aeroelasticity Multidisciplinary Computation and Optimisation

41. Objective: Design points: Constraints: Transonic Multi Point Airfoil Optimisation Multidisciplinary Computation and Optimisation

42. Airfoil geometry Thickness Distribution Camberline M=0.680, a=1.8 M=0.734, a=2.8 M=0.754, a=2.8 Transonic Multi Point Airfoil Optimisation Multidisciplinary Computation and Optimisation

43. Supersonic Commercial Aircraft Optimisation Supersonic Commercial Aircraft: Wing optimisation Multidisciplinary Computation and Optimisation

44. Angle of attack • Sweep Angle • Root thickness • Inboard thickness • Crank thickness • Tip thickness Multidisciplinary Design Optimisation Design Variables • Skin thickness • Ribs thickness • Spars thickness Multidisciplinary Computation and Optimisation

45. Multidisciplinary Design Optimisation Cp distribution: MDO Multidisciplinary Computation and Optimisation

46. Upper side Upper side Lower side Lower side Initial shape Optimised shape Multidisciplinary Design Optimisation Skins Thickness Multidisciplinary Computation and Optimisation

47. Gradient based methods • Finite Difference • Adjoint formulation • Automatic differentiation • Evolution strategies • Genetic algorithms • NN & Fuzzy logics • Games theory Multidisciplinary Design Optimisation Optimisation Problem Definition The optimisation problem – objective and constraint functions – have to be accurately defined Optimisation Algorithms Deterministic Methods Stochastic Methods Unconstrained-Constrained problems issues Parametrisation Issues Impact on the optimisation problem solution Impact on the users (basic knowledge, experience) Treatment of complex shapes issues CFD Grid Deformation Issues Maintain the same grid quality during the overall optimisation process. Automated Optimisation System Issues Integration of the system issues Flexibility Robustness of the system Control during the optimisation process Multidisciplinary Computation and Optimisation

48. Involvement of large intellectual, human and financial resources Education Think on a multidisciplinary basis Multidisciplinary Design Optimisation • Methods • Automatic overall process management and monitoring • Automatic generation of models related to different disciplines • Ensure adequate accuracy of interfaces between disciplines • Efficient and robust analysis & optimisation strategies • Genuine treatment of constraints • Needs • Greater collaboration & integration between pure mathematics, applied mathematics and • engineering sciences (New good ideas) • Data/methods for the verification & validation of tools/strategies • Spread/enforce MDO philosophy within Companies & Industry Multidisciplinary Computation and Optimisation