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- 1. McCormick School of Engineering Northwestern University Robust Shape &TopologyOptimization under Uncertainty Prof. Wei Chen Wilson-Cook Professor in Engineering Design Integrated DEsign Automation Laboratory (IDEAL) Department of Mechanical Engineering Northwestern University Based on Shikui Chen’s PhD Dissertation http://ideal.mech.northwestern.edu/
- 2. Presentation Outline1. Topology Optimization Level set method Challenge for topology optimization under uncertainty2. Robust Shape and Topology Optimization (RSTO) Framework for RSTO RSTO under Load and Material Uncertainties RSTO under Geometric Uncertainty3. Examples4. Conclusion
- 3. Topology OptimizationWhat is Topology Optimization?• A technique for optimum material distribution in a given Topology design domain.Why do topology optimization?• Able to achieve the optimal design without depending on designers’ a priori knowledge.• More powerful than shape and size optimization. Shape Size ~3~
- 4. Applications of Topology Optimization Load UncertaintyMost of the sate-of-the- pressure distribution on the upper wing surfaceart work in TO isfocused Uncertainty Material on deterministicand purely mechanicalproblems. Aircraft Structure Design (Boeing, 2004) Light Vehicle Frame Design (Mercedes-Benz, 2008) MEMS Design Micro structure ofcomposite material ~4~ ~4~
- 5. Topology Optimization: State of The Art E E0 p SIMP (Rozvany, Zhou and Birker, 1992) Solid Isotropic Material Homogenization with Penalisation (SIMP) (Bendsoe & Kikuchi, 1988) - power law thatGround Structure Method interpolates the Youngs modulus to the scalar selection field
- 6. Dynamic Geometric Model: Level Set Methods ( x) 0 ( x) 0 x ( x) 0 ( x) 0 x ( x) 0 x D Hamilton-Jacobi Equation ( x) 0 Vn ( x) 0 t ( x) 0 D ( x) 0 Implicit representation Benefits – Precise representation of boundaries – Simultaneous shape and topology opt. ( x) 0 – No chess-board patterns – Accurate for geometric variations (M. Wang et al., 2003) ~6~
- 7. Formulation for Robust Shape and Topology Optimizationf (X) Minimize minimize [ f , f ] J * ( , u, z ) ( J ( , u, z )) k ( J ( , u, z ) s.t. g k g 0 Subject to : Volume constraint obj , Perimeter constraint on , divσ (u) f in robust X u 0 on D Robust Design Model (Chen, 1996) σ (u) n g on N Challenges in RSTO: • Modeling and propagation of high-dimensional random-field uncertainty • Sensitivity analysis for probabilistic performances ~7~
- 8. Random Variable and Random Field k X A realization of a weakly correlated random fieldA random variable A realization of a strongly correlated random field ~8~
- 9. Framework for TO under Uncertainty A Uncertainty Quantification (UQ) B Characterization of correlation Uncertainty propagation Dimension reduction in UQ (UP) Random field to random variablesMaterial uncertainty Efficient samplingLoading uncertainty Dimension reductionGeometric uncertainty Update design using TO algorithm in UP C Sensitivity Analysis (SA) for Performance prediction using probabilistic performances finite element simulations Analytical sensitivity analysis for deterministic TO sub-problems Evaluation of probabilistic performances using Gauss Decomposition into deterministic TO quadrature formula Robust & reliable Design sub-problems Chen, S., Chen, W., and Lee, S., “Level Set Based Robust Shape and Topology Optimization under Random Field Uncertainties”, Structural and Multidisciplinary Optimization, 41(4), pg 507, 2010. ~9~
- 10. Module A: Uncertainty Representation •Karhunen-Loeve Expansion •A spectral approach to represent a random field using eigenfunctions of the random field’s covariance function as expansion bases.Random ξ: orthogonal random parametersField g mean function : ith eigenvalue ith eigenvector x - spatial coordinate Significance check - random Select M when s is close to 1 parameter •Truncated K-L Expansion Ghanem and Spanos 1991; Haldar and Mahadevan 2000; Ghanem and Doostan 2006 ~ 10 ~
- 11. Module B: Uncertainty PropagationNumerical Integration with Gaussian Quadrature FormulaApproximate the integration of a function g(ξ) by a weighted sum offunction values at specified pointsUnivariate Dimension Reduction (UDR) Method (Raman andXu, 2004)• Approximate a multivariate function by a sum of multiple univariate functions• Accurate if interactions of random variables ξ are relatively small• Greatly reduce sample points for calculating statistical moments ~ 11 ~
- 12. Single Dimension Gauss Quadrature FormulaeThe k-th order statistical moment of a function of a random variablecan be calculated by a quadrature formula as follows k m k kE g g p ( )d wi g li i 1 wi weights li locations of nodes Provide the highest precision in terms of the integration order Much cheaper than MCS ~ 12 ~
- 13. Tensor Product Quadrature vs. Univariate Dimension ReductionTensor Product Quadrature UDR x2 x2 1 1 1 36 9 36 1 4 1 9 9 9 x1 x1 1 1 1 36 9 36 mi wi j weights yi wi j g X1 , , li j , Xn j 1 1 li j location of mi 2 2 nodes yi wi j g X1 , , li j , , Xn g_ i j 1 ~ 13 ~
- 14. Module C: Shape Sensitivity Analysis for Probabilistic PerformancesExpand the functions of mean and variance using UDR in an additiveformat n D J D J_i n 1 D J , u, μ z (1) i 1 n 1 2 D J D J_i (2) n 2 i 1 J_i i 1 Using adjoint variable method and shape sensitivity analysis (Sokolowski, 1992),we can calculate (1) and (2), and further obtain D J * ( , u, z ) D ( J ( , u, z )) kD ( J ( , u, z)) n n k D J_i n 1 D J ( , u, μ z ) D 2 J_i n i 1 2 i 1 J_i i 1J* u Vn ds Steepest Descent Vn u t Vn | | 0 ~ 14 ~
- 15. Example 1. Bridge Beam with A Random Load at Bottom(1) RSTO under loading uncertainty (2) Deterministic Topology Optimization f fAngle: Uniform distribution [-3pi/4, -pi/4], Domain size: 2 by 1 , f 1magnitude: Gumbel distribution (1, 0.3) ~ 15 ~
- 16. Example 1.RSTO (with A Random Load at Bottom) v.s. DSTO Robust Design Deterministic Design Robust Deterministic 25-point tensor-product quadrature 1410.70 1422.25 E(C) Monte Caro (10000 points) 1400.10 1424.99 25-point tensor-product quadrature 994.86 1030.93 Std(C) Monte Caro (10000 points) 959.86 1042.93 ~ 16 ~
- 17. Example 2 A Micro Gripper under A Random Material Field 1 f outf in 2 f out Chen, S., Chen, W., and Lee, S., 2010, "Level set based robust shape and topology optimization under random field uncertainties," Structural and Multidisciplinary Optimization, 41(4), pp. 507- 524. ~ 17 ~
- 18. Example 2 Robust Design vs. Deterministic Design Parameters Volume Ratio Robust Design Deterministic DesignMaterial Field 1 E 1 0.090 -0.065 -0.07Material Field 2 E 1 0.098 -0.059 -0.055 E 0.3 d 0.5 ~ 18 ~
- 19. Geometric Uncertainty Modeling with A Level Set Model Represents geometric uncertainty by modeling the normal velocity field as a random field; Naturally describes topological changes in the boundary perturbation process; Can model not only uniformly too thin (eroded) or too thick (dilated) structures but also shape- dependent geometric uncertainty Eulerian d ( X) Vn ( X, z ) ( X) 0 Description dt Chen, S. and Chen, W., “A New Level-Set Based Approach to Shape and Topology Optimization under Geometric Uncertainty”, Structural and Multidisciplinary Optimization, 44, 1-18, April 2011 ~ 19 ~
- 20. Module A: Geometric Uncertainty Quantification Extracted boundary points from the level set model N a x, a x i ai x i ~ 20 ~ i 1
- 21. Extending Boundary Velocity to The Whole Design Domain Vn sign( ) V 0Initial velocity on the boundary Extended velocity on the whole domain d (X ) Vn ( X , z) (X ) 0 dt ~ 21 ~
- 22. Challenges in Shape Sensitivity Analysis under Geometric Uncertainty Conventional SSA Problem : How to change to minimize J DJ D Our problem: Need shape gradient of J and J at the same time Challenge: Need shape gradient of J DJ is with respect to Vn D ~ 22 ~
- 23. SSA under Geometric Uncertainties Deformed configuration (perturbed design), t = t x = Ψ(X,t) Path line p x2 u(X) = U(x) Based on largeUnderformed e2 e1 deformation theoryconfiguration x1 P(current design), Using Nanson’s relationt=0 b e3 X2 x3 and Polar decomposition theory, it was proved that E2 E1 DJ DJ X1 Vn ( X ) Vn (x) D D X3 E3 The design velocity field should be mapped along the path line from to ~ 23 ~
- 24. Example: Cantilever Beam Problem Robust Design Deterministic DesignVn X N (0,1), t 0.02 ~ 24 ~
- 25. Configurations of Robust and Deterministic Designs under Geometric UncertaintyRobust Design under Variations Deterministic Design under Variations ~ 25 ~
- 26. Comparison of Deterministic vs. Robust Design A A Std (C ) A A D B C B C D ~ 26 ~
- 27. Robust Designs for Over-Etching and Under-Etching Situations Robust design for the under-etching situation E F Robust design for the F over-etching situation E
- 28. Summary Demonstrated the importance of considering uncertainty in topology optimization A unified, mathematically rigorous and computationally efficient framework to implement RSTO First attempt of level-set based TO under geometric uncertainty (TOGU) Bridge the gap between TO and state-of-the art techniques for design under uncertainty ~ 28 ~

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