Advanced evolutionary algorithms for transonic drag reduction and high lift of 3D configuration using unstructured FEM

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

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

3. 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. Mesh generation and quality aspects Evolutionary as well as deterministic methods involves the analysis (FEM) of many different designs. Mesh Generation Influence of mesh generation: • Total computational cost of optimizationclosely related to FE analysis cost per design. • Bad quality of FE analysis: • Introduce noise in the convergence • Possible bad final solution.

5. Mesh generation and quality aspects Classical strategies for meshing each individual: • 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. • 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)

6. Adaption of a mesh to the boundary shape modifications

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

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

9. Mesh generation and mesh sensitivity • Mesh Generator • Advancing front method • Background mesh defining the sizeδat each point. • Mesh Sensitivity • Boundary nodal points: obtained by the B-spline sensitivity analysis. • Internal nodal points: spring analogy (fixed number of smoothing cycles) Smoothing of nodal coordinates Mesh sensitivity

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

11. 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 Projection to each design and definition of the adapted mesh Representative of population pth individual of population Projection using shape sensitivityanalysis

12. Minimize unfeasible designs Pipe under internal pressure 4 design variables Circular internal shape P=0.9 MPa svm 2 MPa ||ees|| < 1.0% 30 individuals/generation • Optimal analytical solution for external surface: • Circular shape Ropt = 10.66666 • Cross section area Aopt = 69.725903

13. 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

14. Pipe under internal pressure 0.46%

15. Initial design space FE model of Initial design space Optimum topology Initial model for further optimization (60 design variables) 8 independent design variables Fly-wheel 60 design variables 8 independent design variables svm 100 MPa ||ees|| < 5.0% 15 individuals/generation

16. OptimumDesign Original Design Fly-wheel 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)

17. 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

18. Robust design Goal: Introducing VARIABILITY (uncertainty) of parameters like Mach numbers or angle of attack in design optimization Outcomes: better control of realistic product performances A product is said to beRobust … 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)

19. Taguchi methods Stochastic optimization Multi-point optimization Fuzzy and probabilistic methods Bounds-based methods Minimax methods Robust design Different robust design methods Two popular methodologies

20. Robust design STOCHASTIC OPTIMIZATION • Modify the objective to directly incorporate the effects of model uncertainties on the design performance • Stochastic analysis of the behaviour of each design • 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

21. 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

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

23. 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: Taguchi method • Reduce effect of variability on design function • Improve the performances

24. Taguchi method Mathematical formulation of Taguchi methods for drag reduction problem • Definition of design problem • Description of robust design problem (2 objectives)

25. 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 EXAMPLE: Robust design optimization problem

26. Taguchi method Pareto non dominated solutions, Nash equilibrium and Single point designed solutions

27. Taguchi method Airfoil list of non-dominated solution, single-point design solution and baseline profile

28. Taguchi method Comparison of airfoils on Pareto front with Nash equilibrium

29. Taguchi method Drag performance of optimized airfoil

30. CONCLUSIONS • 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. • 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.

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