1. www.cimne.com Developments on Shape Optimization at CIMNE October 2006 www.cimne.com Advanced modelling techniques for aerospace SMEs

2. www.cimne.com Geometry: B-spline. Definition points r(i) Shape parametrization 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) )

3. www.cimne.com Shape parametrization Design variables:shape parameters (example of FANTASTIC ship hull)

4. www.cimne.com Design variables:deformation of patches defined with a C1 continuity interpolation function over the bulb of a ship hull Shape parametrization

5. 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

6. 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 methods involves the analysis (FEM) of many different designs. Influence of mesh generation: Mesh Generation Mesh generation and quality aspects

7. 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

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

9. 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

10. www.cimne.com Generation of an adapted mesh for each design in one step using error sensitivity analysys 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)

11. 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) )

12. 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)

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

14. 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:

15. 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

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

17. 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).

18. 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

19. 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

20. 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

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

22. www.cimne.com Gravity Dam Optimization of internal boundary 10 desing variables  vm  2.75 MPa ||e es || < 3.0% 30 individuals/generation

23. www.cimne.com Gravity Dam Original Individual

24. www.cimne.com Optimized shape Original shape Gravity Dam Original Individual 120 generations 30 individuals/generation only 5% individuals required additional remeshing

25. www.cimne.com Gravity Dam Average Individual in Generation 28 Reference mesh

26. 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

27. 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)

28. 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

29. www.cimne.com Thank you very much