An Overview of New Features in

Transcription

1 6. LS-DYNA Anwenderforum, Frankental 2007 Optimierung An Overview of New Features in LS-OPT Version 3.3 Nielen Stander*, Tusar Goel*, David Björkevik** *Livermore Software Tecnology Corporation, Livermore, CA94551, USA **Engineering Researc Nordic AB, Linköping, Sweden Summary: Tis paper summarizes te development status of LS-OPT Version 3.3 and focuses mainly on te following new features: (i) Radial Basis Function Networks, (ii) Multi-objective Optimization (MOO), (iii) Metamodel-based MOO, (iv) User-defined metamodeling, (v) User-defined queuing Keywords: LS-OPT, Multi-Objective Optimization, Combinatorial Optimization, Radial Basis Function Networks, User-defined metamodel J - I - 1

2 Optimierung 6. LS-DYNA Anwenderforum, Frankental Introduction: LS-OPT overview In today s CAE environment it is unusual to make engineering decisions based on a single pysics simulation. A typical user conducts multiple analyses by varying te design and uses te combined results for design improvement. LS-OPT [1] provides an environment for design and is tigtly interfaced to LS-DYNA and LS-PREPOST wit te goal of allowing te user to organize input for multiple simulations and gater and display te results and statistics. More specifically, LS-OPT as capabilities for improving design performance in an uncertain environment and conducting system and material identification. Tese objectives can be acieved troug te use of statistical tools and optimization. Te individual tasks tat can tus be accomplised are: Identify important design variables Find te optimal surface (Pareto front) for multi-objective problems Explore te design space using surrogate design models Identify sources of uncertainty in FE models Visualize statistics of multiple runs Optimize te design wit consideration of uncertainties Conduct robust parameter design Te typical applications are: Multidisciplinary Design Optimization (craswortiness, modal analysis, durability analysis, etc.), system and material identification (biomaterials, metal alloys, concrete, airbag properties, etc.) and process design (metal forming). Te main tecnologies available in LS-OPT are: Experimental Design (DOE). D-Optimal design, Latin Hypercube sampling, Space Filling and oters. DOE allows te user to automatically select a set of different designs to be analyzed. Te main types mentioned ere are eac suited to a different type of analysis: D-Optimal for polynomials and sequential optimization, Latin Hypercube for stocastic analysis and Space Filling for Neural Networks. Metamodels (approximations). Response Surface Metodology and Neural Networks are provided. Bot Feedforward Neural Networks and Radial Basis Function Networks are available. Wit tese tools, te user can explore te design space and quantify te predictability of a response, i.e. identify sources of noisy response. A user-defined metamodel can also be specified by creating a dynamically linked library. Variable screening [4] provides information on te relative importance of design variables. Optimization. Used for automated design improvement. Te Successive Response Surface Metod (SRSM) [5] is te principal iterative tool for finding a converged optimum and is very efficient. A similar metodology is used for finding a converged result using Neural Net updating wit adaptive Space Filling. Multi-objective optimization can be activated by selecting more tan one objective togeter wit te GA core solver. Tis combination will produce a Pareto front. Optimization wit discrete variables is possible, as is combinatorial optimization. A direct multi-objective GA optimizer is used for te latter. Probabilistic analysis includes Reliability Analysis, Outlier Analysis, Robust Parameter Design and Reliability-Based Design Optimization (RBDO)[3]. Reliability analysis allows te user to evaluate te probability of failure wile Outlier Analysis allows te identification of parts of a model tat contribute to noisy response and terefore may affect te overall predictability of te results. Robust Parameter Design and RBDO allow for defining robustness as an objective and te consideration of te probability of failure as a constraint option in optimization. Te outlier analysis uses integrated LS- PREPOST features to visualize structural zones wit unpredictable beavior. J - I - 2

3 6. LS-DYNA Anwenderforum, Frankental 2007 Optimierung Features are available to distribute simulation jobs across a network, using a variety of standard and user-defined queuing systems. In te sections tat follow, a number of new features are discussed, namely Radial Basis Function Networks, User-defined metamodels, a GA for direct combinatorial optimization and multi-objective optimization. 2 New Metamodels Two new metamodels ave been introduced. Te first, Radial Basis Function Networks, as been added to enance te speed of building metamodels wile te User-defined metamodel as been added for te benefit of commercial users wit proprietary metamodel codes as well as for researcers wo want to test new metamodels, using te optimization and viewing features available in LS-OPT. A detailed description of RBFN s follows. 2.1 Radial Basis Function Networks A radial basis function neural network as a distinct 3-layer topology. Te input layer is linear (transparent). Te idden layer consists of non-linear radial units, eac responding to only a local region of input space. Te output layer performs a biased weigted sum of tese units and creates an approximation of te input-output mapping over te entire space Formulation Several forms of radial basis function are considered in te literature, te most common being te Gaussian K 2 2 g( x1,..., xk ) = exp ( xk Wk ) / 2σ (1) k= 1 Te activation of te t Gaussian unit is determined by te Euclidean distance k= 1 ( W 1,..., Wk 1/ 2 2 K r = ( x k W k ) between te input vector = ( x 1,..., x K ) = ) x and RBF center W in K-dimensional space. Te Gaussian is a localized function (peaked at te center and descending outwards see Fig. 1) wit te property tat g 0 as r. Parameter σ controls te smootness properties of te RBF unit. 2 Fig. 1: Radial basis transfer function y = exp[ x ] For a given input vector ( x 1,..., x K ) wit H Gaussian units is given by: x = te output of RBF network wit K inputs and a idden layer J - I - 3

4 Optimierung 6. LS-DYNA Anwenderforum, Frankental 2007 were H Y ( x, W ) = W0 + W f ( ρ) (2) = 1 K 2 0 ( xk Wk ) k= 1 ρ = W ; W 2 0 1/ 2σ = ; f ( ρ) = e ρ Notice tat idden layer parameters W 1,..., W k corresponds to its deviation. Parameters W0 and respectively. represent te center of te t radial unit, wile W 0 W 1,...,WH are te output layer's bias and weigts, A key aspect of RBF networks, as distinct from feedforward neural networks, is tat tey can be interpreted in a way wic allows te idden layer parameters (i.e. te parameters governing te radial functions) to be determined by semi-empirical, unsupervised training tecniques. Accordingly, altoug a radial basis function network may require more idden units tan a comparable feedforward network, RBF networks can be trained extremely quickly. For RBF networks, te training process generally takes place in two distinct stages. First, te centers and deviations of te radial units (i.e. idden layer parameters W 11,...,WHK and W,..., 10 WH 0 ) must be set. All te basis functions are ten kept fixed, wile te linear output layer (i.e., W 0,...,WH ) is optimized in te second pase of training. In contrast, all of te parameters in a FF network are usually determined at te same time as part of a single training (optimization) strategy. Here it is assumed tat te RBF parameters ave already been cosen, and te focus is on te problem of optimizing te output layer weigts. Matematically, te goal of output layer optimization is to minimize a performance function, wic is normally cosen to be te mean sum of squares of te network errors on te training set. If te idden layer's parameters W 10, W11,..., WHK in (2) are kept fixed, ten te performance function P i ( yˆ y ) i i 2 MSE = / P, is a quadratic function of te output layer parameters W 0,...,WH and its minimum can be found in terms of te solution of a set of linear equations (e.g. using singular value decomposition). Te symbol y represents te function values, ŷ te interpolated (approximate) values and P is te number of points Locating centers and radii of te basis functions Te ultimate goal of RBF neural network training is to find a smoot mapping wic captures te underlying systematic aspects of te data witout fitting te noise. Tere are a number of ways to address tis problem. By analogy wit FF network training, one can add to te MSE a regularization term tat consists of te mean of te sum of squares of te optimized weigts. For RBF networks, owever, te most effective regularization metods are probably tose pertaining to selecting radial centers and deviations in te first pase of RBF training. Te commonly eld view is tat RBF centers and deviations sould be cosen so as to form a representation of te probability density of te input data. Te classical approac is to set RBF centers equal to all te input vectors from te training dataset. Te widt parameters σ are typically cosen rater arbitrarily to be some multiple S σ of te average spacing between te RBF centers (e.g. to be rougly twice te average distance). Tis ensures tat te RBF's overlap to some degree and ence give a relatively smoot representation of data. In LS-OPT, instead of using just one value of te widt parameter σ for all RBF's, eac RBF unit's deviation is individually set to te distance to its N σ << N nearest neigbors were N is te number of simulation points. Hence, deviations σ become smaller in densely populated areas of space, J - I - 4

5 6. LS-DYNA Anwenderforum, Frankental 2007 Optimierung preserving fine detail, and are iger in sparse areas of space, interpolating between points were necessary. RBF networks wit individual radial deviations σ can be particularly useful in situations (suc as te sequentially updated network in LS-OPT) were data tend to cluster in a small subregion of te design space (for example, around te optimum of te underlying system wic RSM is searcing for) and are sparse elsewere. After te first pase of RBF training as been done, tere is no way to compensate for large inaccuracies in radial deviations σ by, say, adding a regularization term to te performance function. If te Gaussians are too spiky, te network will not interpolate between known points, and tus, will lose te ability to generalize. If te Gaussians are very broad, te network is likely to lose fine detail. Te popular approac to find a sub-optimal spread parameter is to loop over several trial values of S σ and N σ and finally select te RBF network wit te smallest GCV (generalized cross validation) error. Tis is somewat analogous to searcing for an optimal number of idden units of a feedforward neural network Efficiency It sould be noted tat RBF networks may ave certain difficulties if te number of RBF units (H) is large. A large number of RBF units increases te computation time spent on optimization of te network output layer and, consequently, te RBF arcitecture loses its main (if not te only) advantage over FF networks fast training. An active learning procedure as terefore been introduced in LS- OPT in wic te RMS training error tresold can be prescribed. Te algoritm loops troug te numbers of centers wic represent te fractions (0.1, 0.25, 0.5 and 1.0) of te total number of training points. If a particular fraction satisfies te tresold, te algoritm exits from te loop. Linear functions ave been added to te RBF s and terefore if a function is exactly linear, it will immediately satisfy te tresold criterion and te RBF s are not constructed. A random selection is used to determine te subset of participating centers. Radial basis function networks actually suffer more from te curse of dimensionality tan feedforward neural networks. To explain tis statement, consider te effect of adding an extra, perfectly spurious input variable to a network. A feedforward network can learn to set te outgoing weigts of te spurious input to zero, tus ignoring it. An RBF network as no suc luxury: data in te relevant lowerdimensional space get smeared out troug te irrelevant dimension, requiring larger numbers of units to encompass te irrelevant variability. 2.2 User-defined metamodel Te user can now integrate an own metamodel wit LS-OPT. Building tools are provided to enable te building of a library tat is dynamically linked to LS-OPT (LS-OPT object code is terefore not required for te building process). Te user metamodel toolkit is distributed as a separate file wic includes a source code template and building tools for Microsoft Windows (Visual C) and Linux. Te following optional parameters can be specified in te LS-OPT input file: 1. Te metamodel pat 2. Any number of numerical parameters 3. A command string Tis feature adds one more user-defined facility to te user-defined experimental design, solver, response extraction and queuing systems already in place. 3 Constrained Multi-Objective Optimization 3.1 Overview Wen two or more objective functions in a design formulation are in conflict, te resulting optimum becomes a ypersurface instead of a single point in te function space. Tis ypersurface is known as te Pareto optimal front. A typical example of conflicting objectives in craswortiness design is te deceleration of te veicle and te intrusion into te veicle. Classical optimization metods are typically able to only find one optimal solution at a time (using for instance te ε -constraint metod or te weigted objective metod). Some metods e.g. te weigted objective metod are not able to J - I - 5

6 Optimierung 6. LS-DYNA Anwenderforum, Frankental 2007 find te full Pareto front if it is non-convex (it typically finds some values at te extremes). In tis case a more sopisticated procedure is required. Implemented in LS-OPT is te multi-objective genetic algoritm. Many variants of tis algoritm ave been presented in te literature, but tey all ave in common te dual objective of (i) finding optimal solutions to a linear combination of te objective functions and (ii) maximizing te diversity of te solutions to form te Pareto optimal front. Te last goal is entirely specific to multi-objective evolutionary optimization. Te book by Prof. Kalyanmoy Deb [6] is wolly dedicated to te subject of multi-objective optimization and as a detailed discussion of te most important evolutionary algoritms for acieving bot optimality and diversity simultaneously. LS-OPT features te NSGA-II (non-dominated sorting genetic algoritm II) [7] for multi-objective optimization. Te capabilities are te following: 1. Mixed discrete-continuous optimization. Te discrete variables may also be integer variables, e.g. material types. 2. Multi-objective optimization. Te solution converges to te Pareto optimal front. 3. Global optimization. Due to te fact tat random solutions are generated by te GA, te design space is widely (not just locally) explored for new solutions. 4. NSGA-II can be applied bot using direct simulation and metamodel-based analysis. 3.2 Sequential updating procedure Te multi-objective procedure is best applied using eiter a single iteration wit a large number of points but can also be used wit te sequential updating procedure available in LS-OPT. Tis procedure typically starts wit a linear response surface based on a D-optimal experimental design and in eac new iteration adds Space Filling [1] points for te purpose of enricing te metamodel regionally. LS-OPT uses a linear combination of te objectives as a multi-objective to determine a single optimum point wic serves as a center for te new region of interest. Te objective weigting is uniform, but can be set individually by te user. It is terefore a good idea to normalize te objectives. Te Pareto front is automatically updated after eac iteration, so it will be automatically displayed in te viewer (Tradeoff selection) for eac iteration (see example). 3.3 Post-processing Te Pareto curve can be amended in accordance wit design canges (suc as canges in te constraints and teir bounds, canges in te objectives temselves or canges in te metamodel selection any design or analysis canges for tat matter). Te limited Tradeoff feature previously available in te viewer as been rescinded in favor of using te existing GUI Constraints and Objectives panels to enable te user to make tese canges interactively. After making te desired canges, te new Pareto optimal front can be computed using te Optimization selection in te Repair task. 4 Example: Multidisciplinary, constrained Multi-Objective Optimization using sequentially updated Radial Basis Function Networks 4.1 Modeling Te craswortiness simulation considers a model containing approximately elements of a National Higway Transportation and Safety Association (NHTSA) veicle undergoing a full frontal impact. A modal analysis is performed on a so-called body-in-wite model containing approximately elements. Te cras model for te full veicle is sown in Figure 2. Te vibration model is depicted in Figure 3 in te first torsional vibration mode. Te tracking metodology applied to te torsional mode is described in Reference [1]. Te design variables all represent gauges of structural components in te engine compartment of te veicle (Figure 4), parameterized directly in te solver input file. Twelve parts are affected, comprising aprons, rails, sotguns, cradle rails and te cradle cross member. Seven gauge variables namely apron, rail-inner, rail-outer, sotgun-inner, sotgunouter, cradle rail and cradle cross member are selected to represent te parts. LS-DYNA [2] is used for bot te cras and modal analysis simulations, in explicit and implicit analysis modes respectively. J - I - 6

7 6. LS-DYNA Anwenderforum, Frankental 2007 Optimierung 4.2 MDO multi-objective formulation of optimization problem Te following optimization problem is considered: Minimize (Mass, Intrusion) Subject to: Variable Name Lower bound Upper bound Maximum intrusion ( x cras ) mm Stage 1 pulse( x cras ) g - Stage 2 pulse( x cras ) g - Stage 3 pulse( x cras ) g - Torsional mode frequency( x NVH ) 38.27Hz 39.27Hz Lower Bound Table 1 Design constraints Baseline Design Upper Bound Discrete set Vibration Set After Screening Cras Subset After Screening Rail inner {1,1.25,1.5,1.75,2,2.25,2.5,2.75,3} Rail outer {1,1.25,1.5,1.75,2,2.25,2.5,2.75,3} Cradle rails Aprons Sotgun inner Sotgun outer Cradle member cross Table 2 Starting values and bounds on design variables. Te bounds on te design variables are given in Table 2 togeter wit te starting design. Te tree stage pulses are calculated from te SAE filtered (60Hz) acceleration & x&(t ) and displacement x (t) of a left rear sill node in te following fasion: 2 Stage j pulse = & x dx; k = 0.5 for j = 1, k = 1. 0 oterwise; k d d 1 wit te limits (d 1 ;d 2 ) = (0;184); (184;334); (334;Max(displacement)) for i = 1,2,3 respectively, all displacement units in mm and te minus sign to convert acceleration to deceleration. In summary, te optimization problem aims to minimize te mass and intrusion witout compromising te cras and vibration criteria. Te constraints are scaled using te target values to balance te violations of te different constraints. Tis scaling is important in cases were multiple constraints may be violated as in te current problem. Te inner and outer rails are discrete variables. J - I - 7

8 Optimierung 6. LS-DYNA Anwenderforum, Frankental 2007 Fig. 2: Finite element cras model Fig. 3: Finite element vibration model Fig. 4: Tickness design variables (wit exploded view) 4.3 Results Te following plots compare te results of te RBF and FFNN. Te intrusion metamodel is sown in Figure 5. Te plots ave identical scale settings for te vertical axis. Te Pareto optimal front (Figure 6) is based on te dual objectives of predicted intrusion and mass. Figure 7 as been introduced to illustrate te reason for te uneven Pareto optimal front for te FFNN wic occurs as a result of a sudden jump of te optimum wen tigtening te intrusion constraint. Te transition also exists for te RBF but, for some reason, is smooter. Finally, Figure 8 depicts te optimization istory of te mass. Fig. 5: Metamodel of te intrusion as a function of te inner and outer rail ticknesses (a) Radial Basis Function Networks and (b) Feedforward NN. J - I - 8

9 6. LS-DYNA Anwenderforum, Frankental 2007 Optimierung Fig. 6: Pareto optimal front using (a) Radial Basis Function Networks and (b) Feedforward NN Fig. 7: Plot of te Pareto optimal inner sotgun tickness vs. Intrusion to illustrate te reason for te kink in te Feedforward NN, not present in te RBF surrogate. Fig. 8: Mass convergence for (a) Radial Basis Function Networks and (b) Feedforward NN J - I - 9

10 Optimierung 6. LS-DYNA Anwenderforum, Frankental 2007 Te final optimum results after 10 iterations is presented in Table 3. RBF FFNN Rail inner Rail outer Cradle rails Aprons Sotgun inner Sotgun outer Cradle cross member 1 1 Mass Maximum intrusion Stage 1 pulse Stage 2 pulse Stage 3 pulse 20.22* Torsional mode frequency * Max. constraint violation (fraction of constr. value) Tables 3 Optimum results (10 iterations). Te * indicates a violated constraint 5 Blackbox user-defined queuing system Te Blackbox queueing system is anoter flavor of te User-defined queueing system. It can be used wen te computers running te jobs are separated from te computer running LS-OPT by means of a firewall. Te key differences between User-defined and Blackbox are: It is te responsibility of te queueing system or te user provided scripts to transfer input and output files for te solver between te queueing system and te workstation running LS-OPT. LS-OPT will not attempt to open any communications cannel between te compute node and te LS-OPT workstation. Extraction of responses and istories takes place on te local workstation instead of on te computer running te job. LS-OPT will not run local placeolder processes (i.e. extractor/runqueuer) for every submitted job. Tis makes Blackbox use less system resources, especially wen many jobs are run in eac iteration. 6 Conclusion LS-OPT Version 3.3 presents a significant step forward for optimization wit LS-DYNA by providing powerful new metods for combinatorial optimization and multi-objective optimization. New metamodels ave been added to improve speed and flexibility. 7 Acknowledgement Te autors would like to acknowledge te work of Nely Fedorova and Serge Terekoff wo developed te Neural Network and Radial Basis Function Network codes. 8 References 1. Stander, N., Roux, W.J., Goel, T., Eggleston T. and Craig, K.J. LS-OPT Version 3.3 User s Manual, Livermore Software Tecnology Corporation, October Hallquist, J.O. LS-DYNA User s Manual, Version 971. J - I - 10

11 6. LS-DYNA Anwenderforum, Frankental 2007 Optimierung 3. Roux, W.J., Stander, N., Günter, F. and Müllerscön, H. Stocastic analysis of igly nonlinear structures, International Journal for Numerical Metods in Engineering, Vol. 65: , Craig, K.J. and Stander, N., Dooge and Varadappa, S. Automotive craswortiness design using response surface-based variable screening and optimization, Engineering Computations, Vol. 22:38-61, Stander, N. and Craig, K.J. On te robustness of a simple domain reduction sceme for simulationbased optimization, Engineering Computations, Vol. 19: , Deb, K. Multi-Objective Optimization using Evolutionary Algoritms. Wiley, Deb, K and Goel, T. Controlled elitist non-dominated sorting genetic algoritms for better convergence. Proceedings of te First International Conference on Evolutionary Multicriterion Optimization (EMO-2001), pp 67-81, 2001 J - I - 11

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