New Developments in LS-OPT - Robustness Studies

Transcription

1 5. LS-DYNA - Anwenderforum, Ulm 2006 Robustheit / Optimierung I New Developments in LS-OPT - Robustness Studies Ken Craig, Willem Roux 2, Nielen Stander 2 Multidisciplinary Design Optimization Group University of Pretoria, Pretoria, South Africa 2 Livermore Software Technology Corp, Livermore, CA K - I -

2 Robustheit / Optimierung I 5. LS-DYNA - Anwenderforum, Ulm 2006 New Developments in LS-OPT - Robustness Studies Optimization of shell buckling incorporating Karhunen-Loève-based geometrical imperfections Ken Craig Multidisciplinary Design Optimization Group University of Pretoria, Pretoria, South Africa Willem Roux Nielen Stander Livermore Software Technology Corp Livermore, CA 5 th German LS-DYNA Forum Ulm, Germany October 2-3, 2006 Layout Background and motivation Karhunen-Loève expansions for generating random fields Non-linear dynamic analysis of buckling Optimization set-up Cases: Deterministic optimization Stochastic case Optimization of robustness Future work K - I - 2

3 5. LS-DYNA - Anwenderforum, Ulm 2006 Robustheit / Optimierung I Background and motivation Geometrical and material imperfections cannot always be ignored as in deterministic design Imperfections are random fields (stochastic processes) and can be modeled Shell buckling particularly sensitive to imperfections (both in geometry and in boundary conditions) Stochastic analysis (Monte Carlo) required to quantify stochastic variation in non-linear buckling For optimal design when geometric imperfections are present, the stochastic analysis has to be incorporated into the optimization process Karhunen-Loève expansions Expand random field ϖ ϖ ( x) ( x, θ ) = ϖ ( x) λ ξ ( θ ) f ( x) + = Average random field f i eigenfunctions of covariance kernel λ associated eigenvalues ξ uncorrelated random numbers Series representation of a continuous Gaussian process in terms of the spectral expansion of its covariance function i i i i K - I - 3

4 Robustheit / Optimierung I 5. LS-DYNA - Anwenderforum, Ulm 2006 Karhunen-Loève expansions (2) Fredholm equation of 2 nd kind C ( x x ) f ( x ) dx = λ f ( ), 2 i i i x2 D is solved to obtain f Covariance kernel obtained experimentally or specified analytically E.g. Analytical (Exponential) C x x 2 ( x ) =, x2 exp L c Karhunen-Loève expansions (3) Solutions to Fredholm equation not available in general, perform numerical integration One method is to use Galerkin method N Basis functions: f x = d φ x i ( ) ( ) k = Write as error and make orthogonal to basis functions: N N C x x2 dikφk x dx λi dikφk x φ j x2 dx D D k= k= ( ) ( ) ( ) ( ) 0, 2 = ik k K - I - 4

5 5. LS-DYNA - Anwenderforum, Ulm 2006 Robustheit / Optimierung I Karhunen-Loève expansions (4) Eigensystem is obtained N k = dik D Efficient to use orthogonal wavelets as basis functions: N T ( i) fi ( x) = dikψ k ( x) = ψ ( x) D So that C where C ( x, x2 ) φk ( x ) φ j ( x2 ) dxdx2 λi dik φk ( x2 ) φ j ( x2 ) dx2 = 0 N k = AD = ΛBD k = N N, 2 jk j k 2 x2 j= k= T ( x x ) = A ψ ( x ) ψ ( x ) = ψ ( x ) Aψ ( ) A jk = h h j k 0 0 C ( x x ) ψ ( x ) ψ ( x ), 2 j D k 2 dx dx 2 Karhunen-Loève expansions (5) Solve through 2 successive wavelet transforms for 2D random process So that we again get an eigensystem Or Aˆ ( i) T ( i) ( x) AHD ψ ( x) D T ψ = λi ˆ ( i) ˆ ( i) D = λid And finally f i T 2 ( ) ( ) ( ) ˆ i x =ψ x H D K - I - 5

6 Robustheit / Optimierung I 5. LS-DYNA - Anwenderforum, Ulm 2006 Non-linear dynamic analysis of buckling LS-DYNA simulation Shell with cut-outs Random field of geometrical imperfections superimposed Peak normal force and internal energy extracted from simulation Parametric model using Truegrid Hole area thickness Non-linear dynamic analysis of buckling (2) Internal energy Elastoplastic Elastic material model K - I - 6

7 5. LS-DYNA - Anwenderforum, Ulm 2006 Robustheit / Optimierung I Non-linear dynamic analysis of buckling (3) Peak force Elastoplastic Elastic material model Stress on buckled shape at 2.4ms Elastoplastic Elastic material model K - I - 7

8 Robustheit / Optimierung I 5. LS-DYNA - Anwenderforum, Ulm 2006 Optimization set-up Design variables: thickness and hole area Constraints on maximum peak force and internal energy considered Run Monte Carlo analysis at each experimental point to obtain average peak force and internal energy Use LS-OPT for Successive Response Surface Method (SRSM) and Monte Carlo analysis Optimization cases Case : Deterministic optimization Min Mass, s.t. Peak Force < 800, Internal Energy > 0.3 Case 2: Stochastic optimization Min Mean Mass, s.t. Avg Peak Force < 800, Avg Internal Energy > 0.3 Case 3: Robust optimization Min COV (Peak Force), s.t. Avg Peak Force < 800, Avg Internal Energy > 0.3 COV = Coef of Variation = σ/μ K - I - 8

9 5. LS-DYNA - Anwenderforum, Ulm 2006 Robustheit / Optimierung I Optimization flow chart (stochastic) LS-OPT in Metamodel- Based Optimization Mode LS-OPT in Monte Carlo Analysis Mode Starting design Generate 7 n full-factorial experimental design Run Monte Carlo analysis (96 samples) Calculate average responses Repeat for all exp points Construct metamodels for each averaged response Optimizer New design No Converged? Yes Stop Case : Deterministic optimization Optimum Design variables Objective Min Mass, s.t. Peak Force < 800, Internal Energy > 0.3 K - I - 9

10 Robustheit / Optimierung I 5. LS-DYNA - Anwenderforum, Ulm 2006 Case : Deterministic optimization (2) Peak force constraint Internal energy constraint Min Mass, s.t. Peak Force < 800, Internal Energy > 0.3 Case 2: Stochastic optimization Mean Mass Initial Robust Stochastic Deterministic Min Mean Mass, s.t. Avg Peak Force < 800, Avg Internal Energy > 0.3 Active constraint K - I - 0

11 5. LS-DYNA - Anwenderforum, Ulm 2006 Robustheit / Optimierung I Case 2: Stochastic optimization (2) Mean Internal Energy Initial Robust Stochastic Min Mean Mass, s.t. Avg Peak Force < 800, Avg Internal Energy > 0.3 Active constraint Case 2: Stochastic optimization (3) Mean Peak Force Initial Robust Stochastic Min Mean Mass, s.t. Avg Peak Force < 800, Avg Internal Energy > 0.3 Active constraint K - I -

12 Robustheit / Optimierung I 5. LS-DYNA - Anwenderforum, Ulm 2006 Case 3: Robust optimization Quadratic Initial Robust Stochastic COV Peak Force Neural Net Min COV(Peak Force), s.t. Avg Peak Force < 800, Avg Internal Energy > 0.3 Active constraint Active constraint Notice different baseline performance Summary results Case (Deterministic) Case 2 (Mean Mass) Case 3 (COV peak force) Initial Final Initial Quadratic NN Initial Quadratic NN Thickness [mm] Hole area [m 2 ] Average mass [kg] Average peak force [N] Average internal energy Standard deviation Coefficient of variation Peak force Internal energy Peak force Internal energy NN fits similar Mass reduced Similar result (small holes, thin) COV improved at cost of mass increase (large holes but thick) K - I - 2

13 5. LS-DYNA - Anwenderforum, Ulm 2006 Robustheit / Optimierung I Geometrical imperfections of optima 96 random fields for Monte Carlo runs Stochastic optimum Robust optimum Conclusions Feasible, but expensive, to include stochastic and robustness effects into optimization process Inclusion of stochastic effects did not modify deterministic optimum significantly Robust optimization led to a much heavier design Robust and stochastic optimization process fully automated using LS-OPT in Metamodel mode to call LS-OPT in Monte Carlo Analysis mode K - I - 3

14 Robustheit / Optimierung I 5. LS-DYNA - Anwenderforum, Ulm 2006 Future work Extend simulation to consider postbuckling performance Apply to more realistic examples (require measured geometric imperfections) Post-buckling K - I - 4