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A Parallel Multiobjective Efficient Global Optimization: The Finite Element Method in Optimal Design and Model Development A. C. Berbecea, S. Kreuawan, F. Gillon, and P. Brochet L2EP, ECLille, Université Lille Nord de France, Lille 59650, France Conceiving electromagnetic devices using finite element modeling tools is a complex task and also time-costly. The Efficient Global Optimization method, based on the progressive construction of surrogate models, is studied. The method uses Kriging models and allows for multiobjective optimization. An original infill criterion, combining the surrogate models and an estimate of their error is proposed. Moreover, two techniques for the calculi distribution, adapted to the algorithm, are tested on an eight-core machine. An advantage of the method consists in its capability of providing sufficiently accurate models for each objective and constraint function around the obtained Pareto front. The SMES device of the TEAM problem 22 is used as benchmark. Index Terms—Kriging surrogate model, multiobjective optimization, optimal design, parallel computing, surrogate-assisted optimization, TEAM workshop problem 22.
I. INTRODUCTION
I
N THE DESIGN optimization process, analytical models are ordinarily preferred in order to simulate the behavior of the designed product for time-saving reasons. The finite element analysis (FEA) is used in the late stages of product development just to validate the design. However, integrating the FEA early in the development cycle is the very best way to succeed, as it provides an accurate virtual prototype all along the cycle. However, its direct integration within an optimal design process is difficult because of the large number of model evaluations required by optimization algorithms. The surrogate-assisted optimization strategy appears as a promising tradeoff [1], allowing the integration of high-fidelity models, such as FEA, at a reasonable computational cost. In the past, a large research effort has been focused on the single-objective optimization case, giving birth to a large class of methods like Efficient Global Optimization (EGO). In order to benefit from a parallel computing environment, instead of a single infill point, a set of infill points should be located and then evaluated in parallel using the finite element model (FEM). The algorithm described in [2] identifies multiple infill points by detecting several local maxima of the expected improvement function, but without a prior knowledge of the number of existing local maxima. Another method, allowing identifying a desired number of infill points, is based on modifying the infill criteria [3]. After finding an infill point, a new sample point is imposed at this location, and the infill search is restarted on the augmented sampled data set. [4] proposes a weighted expected improvement criterion for a better balance between exploitation and exploration. However, most of the design engineering problems are complex, presenting many conflicting objectives and constraints. A basic extension of the single-objective EGO to accommodate multiple objectives is done by building independent surrogate models for each design objective [5], [6]. In [7], an enhanced
Manuscript received December 22, 2009; accepted February 10, 2010. Current version published July 21, 2010. Corresponding author: A. C. Berbecea (e-mail:
[email protected]). Color versions of one or more of the figures in this paper are available online at http://ieeexplore.ieee.org. Digital Object Identifier 10.1109/TMAG.2010.2043654
probability of improvement criterion is proposed. A multiobjective infill criterion based on the nondominated concept is introduced in [8]. In this paper, two parallelization strategies for a multiobjective surrogate-assisted optimization (Parallel Multiobjective Efficient Global Optimization, PMEGO) algorithm are proposed and tested on TEAM workshop problem 22 [9]. II. PARALLEL MULTIOBJECTIVE EFFICIENT GLOBAL OPTIMIZATION ALGORITHM This section describes a multiobjective optimization algorithm adapted to parallel computing. The surrogate models used by the algorithm are Kriging models, which provide both the modeled functions and the modeling error. Here are presented the models, the algorithm, the infill point selection criterion, and the two parallelization strategies. A. Kriging Surrogate Model In the Kriging model, the unknown function as the sum of two terms
is expressed
(1) The first term, , is called the regression polynomial model and gives the global trend of the function; the other one, , gives the local deviations from the global trend. This correlation function controls the smoothness of the model. Here, the Gaussian correlation function between points and is chosen (2) is the number of dewhere is the design variable vector, sign variables, and represents the unknown correlation function parameter vector. The parameter has an influence on the smoothness of the Kriging model. Small values of smoothen the Kriging prediction, while for large values of , the Kriging model has accurate predictions around the sampled points over which it is built, and very false predictions elsewhere.
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BERBECEA et al.: PARALLEL MULTIOBJECTIVE EFFICIENT GLOBAL OPTIMIZATION
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Fig. 2. (a) Dominance distance. (b) Neighboring distance.
Fig. 1. PMEGO algorithm flowchart.
The Kriging model predicts the estimated response value . The expression for the Kriging predictor is (3)
point at a time, by using one of the two parallelization strategies described in Section II-D, combined with the pseudo distance criterion described in Section II-C. This allows the distribution of the FEM evaluations on each available core of the machine. Once evaluated, each of these designs is validated against acceptance conditions and must satisfy the constraints. In this case, it is added to the improvement solutions set (IS). In both cases, the point is added to the sample data set (SD). The iteration is then terminated, and the control returns to the fitting of new Kriging models using the SD set. The algorithm continues until the maximum number of high-fidelity model evaluations is attained. At the end, the Pareto front is obtained and also a Kriging model for each objective and constraint function around the Pareto set. C. Infill Point Selection Criterion
where is an estimator for the regression model, a correlation vector between a new location to be estimated and the sample points locations, the correlation matrix between sample points, and a unit vector with length equal to the number of sampled points . The mean squared error (MSE) is the expected value of difference between the true response and the estimated one. Since Kriging interpolates the data, MSE is zero at the sampled points. At unknown points, MSE should be minimized in order to obtain a good approximation. The MSE and the standard error can be expressed as
The information about the predicted standard error is used by the infill criterion to locate the next infill point. The multiobjective infill criterion chosen is the pseudo distance [8]. It is based on the nondominated concept. The pseudo distance is composed of two terms: the dominance distance and the neighboring distance , presented in (6)–(8).
(4)
(7)
(6)
(5) where is the variance. and depend on using the Maximum Likelihood Estimation.
, which is found (8)
B. Algorithm Fig. 1 presents the flowchart of the algorithm. Time triggers have been added to the flowchart in order to identify the time cost of main blocks. The algorithm starts by sampling the initial points using the Design of Experiments (DoE) method. This set of points is then evaluated in parallel on the available cores using the high-fidelity model, such as the FEM. Each iteration of the algorithm starts by fitting a Kriging surrogate model for each objective and constraint function individually, using the initial set of points. A set of infill points is then determined, instead of generating one
where represents the number of objective functions; the number of nondominated points, which are dominated by the trail point; and respectively the minimum and the maximum of the th objective function; the Kriging prediction of the th objective function; the Kriging predicted standard error associated with ; and the value of the th objective function at the closest point to the trial point. An example is given in Fig. 2 to explain the concept. gives a design vector with the smallest estimated standard error (small ), i.e., accurate Kriging predictions that
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will dominate with the greatest distance the existing nondominated front (NF). Starting from a nondominated front , from the previous iteration, the point will be preferred to another point due to its greater dominate distance, as presented in Fig. 2(a). On the contrary, the will focus to the neighborhood of the trial point. It will give a design vector with a high standard error (high ), i.e., the greatest distance between the trial point and its neighboring nondominated points. It sacrifices prediction accuracy in order to spread the points along the front, filling up the largest gap between two existing nondominated points, as will be preferred to shown in Fig. 2(b). This time, the point because of the greater . The next design vector is located by maximizing (6) using a single-objective optimization algorithm. This step is repeated, guided by one of the two parallelization strategies, until a prescribed number of infill points are obtained, depending on the number of available cores. D. Parallelization Strategies Within the developed algorithm, two parallelization strategies have been implemented and tested. These strategies are based on the techniques proposed for the single-objective optimization case by Sóbester et al. [2] and respectively Schönlau [3]. The two strategies are presented in Fig. 1 and detailed as follows. i) The first strategy, titled here hybrid approach, is: an infill point is chosen based on the infill point selection criterion; , the Kriging predicted value of the new point is added to the SD set; new Kriging models are fit using the SD set. When as many points as available cores are selected, the FEA computations are launched. ii) The second strategy is called weighted criterion: choose , the weighting coefficients, as many as there are available cores; search for a new infill point using the weighted criteria (9) (10) Compared to the first approach, the weighted criterion strategy does not require to fit again models after finding a new infill point. This means that the infill point search step could be done in parallel, allowing then another reduction of optimization time. However, it is clear that the time taken by this step increases with the size of the SD set. III. KRIGING MODELS EXPLOITATION When the optimization process is stopped, the surrogate models derived from it are still very useful. This is due to a particularly interesting feature of this optimization strategy. At each iteration, surrogate models are fed with new information. Usually, these kinds of models are not accurate and cannot be directly used as approximations of the fine model due to the nonlinear character of the different responses and the high dimensionality of the design space. However, this does not mean that the resulting surrogate models are built only to determine
the nondominated solutions of the high-fidelity model. Actually, in the vicinity of the Pareto front, the surrogate models are very accurate, offering a fast estimation of the high-fidelity model responses. Thus, the final surrogate models can be used to determine the sensitivity for each point of the front and obviously as predictions of the objective functions that can be used to compute the Pareto frontier. IV. APPLICATION TO TEAM WORKSHOP PROBLEM 22 In order to illustrate the main features of the Parallel Multiobjective Efficient Global Optimization algorithm, the TEAM workshop problem 22 is chosen as benchmark [9]. A. Optimization of SMES Device Using PMEGO The SMES device consists of two concentric superconducting coils fed with currents that flow in opposite directions. The inner coil is used for storing magnetic energy, while the outer one has the role of diminishing the magnetic stray field. The goal of the optimization problem is to find the design configurations (eight parameters) that give a specified value of stored magnetic energy and a minimal magnetic stray field. Mathematically, this is formulated as (11) with quench condition
, and subjected to the (12)
where is the maximum value of magnetic induction within coil . The device was modeled in 2D using the FEM. A model evaluation implies updating the geometry and mesh, computing the results, and their post-processing. All these are managed by an optimization tool specifically designed. B. Numerical Results The test is implemented on a high-end eight-cores machine (four cores on a CPU) with two Xeon5470 processors. The initialization is done by selecting 80 initial points from a Latin hypercube sampling (LHS) strategy, which are evaluated using the high-fidelity model. Then, the algorithm is launched with a limit of 700 FEM evaluations . Different configurations of available cores (one to eight cores) were considered for the test of the two parallelization strategies (hybrid model and weighted criterion approaches). For each run, the same set of initial points was used. The final results are presented in Fig. 3. For a better visualization of the results, for the weighted criterion, only one configuration result was chosen to be presented (eight cores alfa). From Fig. 3, we remark the L form of the obtained Pareto fronts. Moreover, there is still an important scale difference between the two objective functions, even though these have been scaled within the formulation (11). The elapsed time and the speedup for the SMES FEM evaluation of the TEAM workshop problem 22 using several configurations of available cores are presented in Table I for the iteration step. A similar speedup is obtained for the parallel FEM evaluation at the initialization phase.
BERBECEA et al.: PARALLEL MULTIOBJECTIVE EFFICIENT GLOBAL OPTIMIZATION
Fig. 3. TEAM workshop problem 22 optimal design results. TABLE I FEM EVALUATION TIME AND SPEEDUP FOR THE ITERATION STEP
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Fig. 4. Pareto front for the TEAM problem 22 retraced using the Kriging models, zoom over one original nondominated solution.
front; see the arrow in Fig. 3. The size and the color of the buband , bles represent the predicted standard error for respectively. V. CONCLUSION
TABLE II FEM EVALUATION TIME AND SPEEDUP FOR THE ITERATION STEP
Problem solving is fast; geometry construction, meshing, and simulation take about 4.3 s. From the table, we can see a maximum speedup of 4, which is attained for the eight cores configuration, representing just half, instead of an ideal eight. This fact is due to the configuration of the machine used. The obtained nondominated fronts are compared using the metrics proposed in [9]. The generational (GD) and reverse generational distance (RGD), spacing (S), and error ratio (ER) were calculated for the different configurations, and their comparison is presented in Table II. The values of the calculated metrics in Table II reflect the diversity of the obtained Pareto fronts, without drawing a best solution. C. Exploitation of the Final Surrogate Models A genetic algorithm was used with the Kriging models developed through one of the above-mentioned optimization runs in order to finely compute the Pareto front of the TEAM workshop problem 22. The search space is limited to a small -dimension hypercube around a point that interests the designer. Fig. 4 presents a zoom over one of the points on the original Pareto
A multiobjective optimization algorithm that allows the use of finite element models has been presented. It computes accurately the Pareto frontier with a minimum number of FEM evaluations. Furthermore, parallelization is available, and two strategies have been proposed and tested on the TEAM workshop problem 22. In both cases, parallelization leads to a significant reduction of time, the more the high-fidelity model is time-consuming. Using an eight-core machine, a speedup of 4 is achieved. The weighted criterion is preferred to the hybrid model approach because it offers the possibility to parallelize the infill point search phase, which is particularly time-consuming. The surrogate models set up during the optimization provide the sensibility of the design and a straightforward means to build a Pareto frontier. REFERENCES [1] D. R. Jones, “A taxonomy of global optimization methods based on response surfaces,” J. Global Optim., vol. 21, pp. 345–383, 2001. [2] A. Sóbester, S. J. Leary, and A. J. Keane, “A parallel updating scheme for approximating and optimizing high fidelity computer simulations,” Struct. Multidiscip. Optim., vol. 27, pp. 371–383, 2004. [3] M. Schönlau, “Computer experiments and global optimization,” Ph.D. dissertation, University of Waterloo, Waterloo, ON, Canada, 1997. [4] A. Sóbester, S. J. Leary, and A. J. Keane, “On the design of optimization strategies based on global response surface approximation models,” J. Global Optim., vol. 33, pp. 31–59, 2005. [5] P. N. Koch, T. W. Simpson, J. K. Allen, and F. Mistree, “Statistical approximations for multidisciplinary design optimization: The problem of size,” J. Aircraft, vol. 36, no. 1, pp. 275–286, 1999. [6] B. Wilson, D. Cappelleri, W. T. Simpson, and M. Frecker, “Efficient Pareto frontier exploration using surrogate approximations,” Optim. Eng., vol. 2, pp. 31–50, 2001. [7] G. I. Hawe and J. K. Sykulski, “An enhanced probability of improvement utility function for locating Pareto optimal solutions,” in Proc. COMPUMAG, Aachen, Germany, Jun. 2007, pp. 965–966. [8] S. Kreuawan, “Modeling and optimal design in railway applications,” Ph.D. dissertation, Ecole Centrale de Lille, Lille, France, 2008 [Online]. Available: http://tel.archives-ouvertes.fr/docs/00/38/48/43/PDF/ Thesis_SKreuawan_v16052009.pdf [9] P. Alotto, U. Baumgartner, F. Freschi, M. Jaindl, A. Köstinger, Ch. Magele, W. Renhart, and M. Repetto, “SMES optimization benchmark extended: Introducing Pareto optimal solutions into TEAM22,” IEEE Trans. Magn., vol. 44, no. 6, pp. 1066–1069, Jun. 2008.
