1. www.cimne.com Advanced evolutionary algorithms for transonic drag reduction and high lift of 3D configuration using unstructured FEM 21 May 2007 www.cimne.com Advanced modelling techniques for aerospace SMEs Gabriel Bugeda TANK Zhili Jordi Pons

2. www.cimne.com Index Mesh generation and quality aspects Robust design Contributions of CIMNE to shape optimization problems in aeronautics:

3. www.cimne.com Mesh generation and quality aspects Shape optimization problem: f objective function x vector of design variables g set of restrictions  Deterministic methods  Evolutionary algorithms

4. www.cimne.com 1.Total computational cost of optimization closely related to FE analysis cost per design. 2.Bad quality of FE analysis:  Introduce noise in the convergence  Possible bad final solution. Evolutionary as well as deterministic methods involves the analysis (FEM) of many different designs. Influence of mesh generation: Mesh Generation Mesh generation and quality aspects

5. www.cimne.com Classical strategies for meshing each individual: 1.Adapt a single existing mesh to all the different geometries.  Existing strategies allow adapting an existing mesh for very big geometry modifications preventing too much distortion.  Cheapest strategy  No control of the discretization error. 2.Classical adaptive remeshing for the analysis of each design.  Good quality of results of each design  High computational cost (each design is computed more than once) Mesh generation and quality aspects

6. www.cimne.com Adaption of a mesh to the boundary shape modifications

7. www.cimne.com Representative of population. Generation of an adapted mesh for each design in one step using error sensitivity analysys  Mesh adaptivity based on Shape sensitivity analysis Projection parameters (sensitivity of nodal coordinates and error indicator) Final h-adapted mesh of representative h-adaptive analysis of representative Classical sensitivity analysis Projection to individuals h-adapted mesh for 1 st individual h-adapted mesh for 2 nd individual h-adapted mesh for 3 rd individual h-adapted mesh for P th individual in “one-step” !! Low cost control of discretization error

8. www.cimne.com Geometry: B-spline. Definition points r(i) Parameterization of the problem Sensitivity analysis of the system of equations: Sensitivity analysis of the B-spline expression: Design variables:Coordinates of some definition points B-spline expression: in terms of the coordinates of “polygon definition points” r i. Polygon definition points vector, R : Obtained solving V=NR ( V  imposed conditions at r(i) )

9. www.cimne.com Mesh generation and mesh sensitivity Mesh Generator  Advancing front method  Background mesh defining the size δ at each point. Mesh sensitivitySmoothing of nodal coordinates Mesh Sensitivity  Boundary nodal points: obtained by the B-spline sensitivity analysis.  Internal nodal points: spring analogy (fixed number of smoothing cycles)

10. www.cimne.com Finite element analysis Solution of standard elliptic equations Discretization:

11. www.cimne.com Error estimation Estimation in energy norm of the error: ZZ-estimator Stress recovery: Global least squares smoothing Approximation of total energy norm:

12. www.cimne.com Sensitivity analysis of the error estimator Discrete-Analytical method: Discretized model (element integral expressions) are analytically differentiated with Sensitivities of - displacements - strains - stresses

13. www.cimne.com Sensitivities of smoothed stresses: Sensitivities of error estimator: Sensitivities of the strain energy: Sensitivity analysis of the error estimator

14. www.cimne.com The used evolutionary algorithm Parameter vector of i-th individual of generation t For each individual, a new trial vector is created by setting some of the parameters u p j (t) to:  Parameters to be modified and individuals q, r, s are randomly selected  The new vector u p (t) replaces x p (t) if it yields a higher fitness.  Non accomplished restrictions integrated in objective function using a penalty approach. Evolutionary algorithm: classical Differential Evolution (Storn & Price).

15. www.cimne.com Projection to each design and definition of the adapted mesh Representative of populationp th individual of population Projection using shape sensitivity analysis Mesh coordinates Error estimation Strain energy Generation of h-adapted mesh.  Admissible global error percentage  Mesh optimality criterion: equidistribution of error density  Target error for each element  New element size

16. www.cimne.com Pipe under internal pressure 4 design variables Circular internal shape P=0.9 MPa  vm  2 MPa ||e es || < 1.0% 30 individuals/generation Optimal analytical solution for external surface: Circular shape R opt = 10.66666 Cross section area A opt = 69.725903 Minimize unfeasible designs

17. www.cimne.com Analytical Optimal shape A = 69.725903 Optimal shape obtained (B-spline defined by 3 points) A = 70.049 Pipe under internal pressure 185 generations 30 individuals/generation only 3% individuals required additional remeshing

18. www.cimne.com Pipe under internal pressure 0.46%

19. www.cimne.com Fly-wheel FE model of Initial design space Optimum topologyInitial design space Initial model for further optimization (60 design variables) 8 independent design variables 60 design variables 8 independent design variables  vm  100 MPa ||e es || < 5.0% 15 individuals/generation

20. www.cimne.com Fly-wheel Original Design Optimum Design 300 generations 15 individuals/generation Weight reduction 1.53  1.445 kg (0.25  0.17 in the design area) (Deterministic: 1.53  1.45 kg)

21. www.cimne.com Conclusions  A strategy for integrating h-adaptive remeshing into evolutionary optimization processes has been developed and tested  Adapted meshes for each design are obtained by projection from a reference individual using shape sensitivity analysis  Quality control of the analysis of each design is ensured  Full adaptive remeshing over each design is avoided  Low computational cost (only one analysis per design)  Numerical tests show The strategy does not affect the convergence of the optimization process Good evaluation of the objective function and the constraints for each different design is ensured

22. www.cimne.com Goal: Introducing VARIABILITY (uncertainty) of parameters like Mach numbers or angle of attack in design optimization Outcomes: better control of realistic product performances Robust design 1. Performs consistently as intended (design) 2. Throughout its life cycle (manufacturing) 3. Under a wide range of user conditions (design) 4. Under a wide range of outside influences (design) A product is said to be Robust …

23. www.cimne.com POSSIBLE GOALS FOR ROBUST DESIGN Maximize worst-case performance (non-probabilistic) Maximize worst-case performance (non-probabilistic) Maximize the consistent improvement of an existing design over the entire range (non-probabilistic) Maximize the consistent improvement of an existing design over the entire range (non-probabilistic) Minimize the performance fluctuation over the entire range (probabilistic) Minimize the performance fluctuation over the entire range (probabilistic) Maximize the overall expected value of the performance (probabilistic) Maximize the overall expected value of the performance (probabilistic) Each method typically results in a different design! Robust design

24. www.cimne.com Taguchi methodsTaguchi methods Stochastic optimizationStochastic optimization Multi-point optimizationMulti-point optimization Fuzzy and probabilistic methodsFuzzy and probabilistic methods Bounds-based methodsBounds-based methods Minimax methodsMinimax methods Two popular methodologies Different robust design methods Robust design

25. www.cimne.com STOCHASTIC OPTIMIZATION Modify the objective to directly incorporate the effects of model uncertainties on the design performance Modify the objective to directly incorporate the effects of model uncertainties on the design performance Stochastic analysis of the behaviour of each designStochastic analysis of the behaviour of each design Robust design Minimize the expected value of the drag over the design lifetime: Minimize the expected value of the drag over the design lifetime: Is drag function Is design vector (geometry, angle of attack) Is uncertain parameter (Mach number) Is probability density function of Mach number

26. www.cimne.com Stochastic Optimisation Geometry + Environment Main Target: define an optimisation procedure under geometry and environmental parameters variability Secondary Target (later): analyse results variability under geometry and environmental parameters variability Problems Solved: –First massive control of TDYN input using STAC –Definition of the next geometry and environmental parameters

27. www.cimne.com Methodology: –Define probabilistic distribution of values for both geometry and environmental parameters. All in the same analysis. Input variables: –Angle of attack, Mach number and Reynolds number –Knot coordinates; two points on upper profile and two points on lower profile Conclusions: –Graphical representation of the [-3σ, +3σ] range and mean value –Mixed effect between geometry and environment do not define any clear relationship. Stochastic Optimisation

28. www.cimne.com On the X-axis the number indicates each analysis: 1.- Evolution of the geometry under optimisation process. Stochastic Optimisation

29. www.cimne.com A disciplined engineering approach (Parameter Design) to find the best combination of design parameters (control factors) for making a system insensitive to outside influences (noise factors) 2 steps in the optimization procedure: 1.Reduce effect of variability on design function 2.Improve the performances Taguchi method

30. www.cimne.com Mathematical formulation of Taguchi methods for drag reduction problem 1.Definition of design problem 2.Description of robust design problem (2 objectives) Taguchi method

31. www.cimne.com EXAMPLE: Robust design optimization problem 1. Find optimal airfoil geometry, which results in minimum drag Cd over a range of free flow Mach numbers while maintaining a given target lift. 2. The thickness and its position is maintained during the optimization. The NACA-2412 is the baseline profile. 3. For this example we assume that the Mach number The Mach number can not fall outside of this interval. 4. We use a inviscid EULER solver to analyze the flow field. Taguchi method

32. www.cimne.com Pareto non dominated solutions, Nash equilibrium and Single point designed solutions Taguchi method

33. www.cimne.com Airfoil list of non-dominated solution, single-point design solution and baseline profile Taguchi method

34. www.cimne.com Comparison of airfoils on Pareto front with Nash equilibrium Taguchi method

35. www.cimne.com Drag performance of optimized airfoil Taguchi method

36. www.cimne.com CONCLUSIONS 1.Robust design optimization is significantly more realistic for designers than the single point design optimization. This Taguchi based uncertainty methodology can identify new shapes with better performance and stability simultaneously maintained within a given range of operation. 2.Compromised solutions are captured by Pareto or Nash strategies. It is shown that a Nash equilibrium solution is also a good initial guess for capturing efficiently a Pareto non- dominated solution.

37. www.cimne.com Thank you very much