Improving Multiobjective Multidisciplinary Optimization With a Data ...

Proceedings of the ASME 2015 International Design Engineering Technical Conferences & Computers and Information in Engineering Conference IDETC/CIE 2015 August 2-5, 2015, Boston, Massachusetts, USA

DETC2015-47361

IMPROVING MULTIOBJECTIVE MULTIDISCIPLINARY OPTIMIZATION WITH A DATA MINING-BASED HYBRID METHOD Hongyi Xu*

Ching-Hung Chuang

Ren-Jye Yang

Research & Advanced Engineering Ford Motor Company Dearborn, MI, US ABSTRACT Multiobjective, multidisciplinary design optimization (MDO) of complex system is challenging due to the long computational time needed for evaluating new designs’ performances. Heuristic optimization algorithms are widely employed to overcome the local optimums, but the inherent randomness of such algorithms leads to another disadvantage: the need for a large number of design evaluations. To accelerate the product design process, a data mining-based hybrid strategy is developed to improve the search efficiency. Based on the historical information of the optimization search, clustering and classification techniques are employed to detect low quality designs and repetitive designs, and which are then replaced by promising designs. In addition, the metamodel with bias correction is integrated into the proposed strategy to further increase the probability of finding promising design regions within a limited number of design evaluations. Two case studies, one mathematical benchmark problem and one vehicle side impact design problem, are conducted to demonstrate the effectiveness of the proposed method in improving the searching efficiency. Keywords: multiobjective mining, bias correction

optimization,

algorithm,

predictions may mislead the optimization search and generate designs of low performances. Real simulations are needed to confirm the designs (often referred as “virtual designs”) obtained from the metamodel-based optimization. The second category is the efficiency improvement of Direct MDO [5, 6]. With the recent progresses in advanced optimization algorithms [7-9] and High Performance Computing (HPC), Direct MDO has drawn engineers’ attention again as a promising optimization strategy. The basic idea of Direct MDO is to conduct real simulations for each design evaluation. The high fidelity simulations are directly connected to the optimization search engine. Only true responses of new designs, instead of approximated predictions, are used in the optimization search. Implementation of Direct MDO is not an easy task due to the complex design landscape and the extremely long computational time of high fidelity simulations. Heuristic algorithms such as Evolutionary Algorithm (EA) [10], Genetic Algorithm (GA) [11] Simulated Annealing (SA) [12] and Particle Swarm Optimization (PSO) [13] are widely used in searching the global optimal solutions of engineering problems. With sufficient computational resources, the design space is expected to be fully explored to generate a large number of feasible designs of high performances. One common feature of heuristic search algorithms is the randomness in determining the next step of searching (i.e. generating new designs in next generation/iteration). For example, GA generates new designs (children) by applying random crossover and mutation on existing elite designs (parents); Simulated Annealing (SA) generates new designs by randomly perturbing existing designs’ locations in the design space. Such randomness helps to overcome local optimums. However, the downside is the slow convergence rate. Oftentimes, the randomly generated new designs are very close to previous designs in the optimization history. Evaluation of such “near duplicates” is a waste of computation resources, because limited new knowledge about

data

1. INTRODUCTION Multidisciplinary design optimization (MDO) has been widely studied and practiced for solving complex engineering design problems. Existing methods of MDO fall into two categories. The first category is the metamodel-based optimization, which is long-established and widely practiced in engineering design problems [1-3]. However, metamodels may subject to the issue of low accuracy because they are approximations of the real model [4]. Inaccurate metamodel

* Correspondence to: Dr. Hongyi Xu, [email protected]

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the design landscape can be provided. Therefore, the “near duplicate” is identified as one of the major limitations to MDO’s efficiency. In order to obtain a high optimal search efficiency, multiple “smart” algorithms have been developed and implemented in commercial software [7, 9, 14]. The new algorithms can automatically switch between “explorative search” and “exploitative search” by adjusting the algorithm parameters to adapt to different design landscapes. There is also increasing interest in applying data mining and machine learning techniques to improve traditional optimization algorithms. Data mining-based methods provide additional knowledge for guiding optimal search. The computational costs of data mining are negligible compared to the high fidelity simulations used in design evaluations. For example, clustering methods are used in finding the nearest neighbor in the process of optimizing zeroskew routes [15], and in progressively searching of neighborhood of existing designs [16]. K-means method are employed to automatically determine the values of algorithm parameters (mutation rate and crossover rate) [17]. A few works have been accomplished utilizing data mining techniques for reducing the number of function evaluations. Promising design regions are identified before calling the time consuming function evaluations. Wang et al. [18] has developed a dynamic clustering-based differential evolution algorithm, where promising design regions are identified based on the current design generation using the hierarchical clustering method. The algorithm allows a gradual change from the explorative search at the early stage to the exploitative search with high precision at the end stage. However, this algorithm does not fully utilize the historical information of the entire optimization process. Only one generation is considered in data mining. Another question is the choice of the number of design clusters. The number of clusters is pre-determined without considering the dynamics of the dataset, so the clustering step lacks of adaptivity. Wang’s work only handles “box-constraint” problems, which is not applicable to design problems of nonlinearly constraints. This paper focuses on achieving fast convergence to the optimal/feasible designs within limited iteration numbers. Reducing the iteration number is critical, because it is approximately proportional to the total turn-around time of optimization, when parallel computing on HPC is utilized. Multiple designs of the same iteration/generation are evaluated at the same time on HPC. In this paper, a data mining-based hybrid strategy is developed to improve the efficiency of multiojective MDO under nonlinear constraints. This strategy fully utilizes the entire optimal search history as the database to identify ineffective new designs such as “near duplicates” before operating the time consuming function evaluation. The proposed strategy is applicable to any traditional heuristic optimization algorithms, to achieve the goal of exploring larger design space and finding better designs within a small number of optimization iterations. The remainder of this paper is organized as follows: Section 2 introduces the data miningbased hybrid strategy. In Section 3, two benchmark problems

are studied to demonstrate the proposed method. Section 4 is the summary. 2. DATA-MINING-BASED HYBRID STRATEGY In this section, a clustering and classification-based method is presented to accelerate the optimization process. Based on the real-time updating design database, the “nearly duplicated designs” generated by optimization algorithms are identified and replaced by “non-duplicates” located in the promising design regions, before operating design evaluations (calling time-consuming advanced simulation codes). This section also presents solutions to how to construct an effective initial design database (“smart starting”) using metamodels. The accuracy of the metamodel is improved by a Gaussian random processbased bias correction technique. 2.1 Basic idea and workflow The similarity between designs is quantified by the distance between them in the normalized design space. Figure 1 describes the definition of “near duplicates”. The existing known designs (“individuals”) are grouped into multiple clusters. The feasible designs and infeasible designs are clustered respectively. Designs in the same cluster are similar in terms of both design variable values and performances. In explorative search, a newly generated design will not be preferred for evaluation if its distance to the nearest cluster centroid is smaller than a predetermined value (e.g. 90% quantile of individual-to-centroid distances). On the other hand, newly generated designs inside the range of feasible design clusters will be preferred in the exploitative search of high precision.

Figure 1: Similar designs ("near duplicates”, black triangles) and “nonduplicated” (black stars). 𝒙𝟏 and 𝒙𝟐 represent design variables.

Given a dataset of previous designs and their correspondent performances (e.g. the initial design population generated by Design of Experiments, DOE), the promising regions for future optimal search are predicted using data mining techniques. However, the non-intelligent traditional optimization algorithms ignore historical information when generating new designs. Take GA as an example, it is possible that some of the new designs generated by crossover and

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often fails to detect feasible designs for highly constraint vehicle safety design problems [5, 6]. We propose to use bias correction-enhanced metamodeling techniques to discover promising designs to enrich the initial design database. The time cost of metamodeling is negligible compared to high fidelity simulations. STEP (A.1) Metamodeling based on DOE set of size 𝑁. If the “crowded DOE points” (too close to its neighbors) exist, they will be identified and reserved from metamodeling in order to avoid local overfitting. STEP (A.2) Bias correction. We have developed a Gaussian Process-based bias correction technique to improve the accuracy of metamodels [19]. The reserved DOE points in Step (A.1) are used to evaluate the discrepancy between the metamodel and the true response. An additive model bias correction is defined:

mutation are quite close to previous designs. GA will still call the simulation codes to evaluate such “near duplicates”. Such design evaluations are unnecessary and inefficient. It is preferred that new designs are located in the unknown design regions so that the knowledge of the design landscape is maximized within a given number of function evaluations. Figure 2 shows how the proposed strategy works in GA. The algorithm checks each newly generated design to determine whether it is a “near duplicate” or not. If yes, the algorithm will go one step back and regenerate a new design. Those two steps are repeated until a “non-duplicate” is generated.

𝑦 𝑒 (𝑥) = 𝑦 𝑚 (𝑥) + 𝛿(𝑥) + 𝜀

(1)

where 𝑦 𝑒 is the true response (e.g. physical observations or high fidelity simulation results); 𝑦 𝑚 denotes the response from low fidelity models or metamodels; 𝛿(𝑥) is the bias function, which describes the discrepancy between the true response and the model response; 𝜀 is the observation error, which is not considered in this work. 𝛿(𝑥), the bias function, is modeled using a Gaussian Process (GP) model. GP model is widely used in computational design to predict the unknown responses based on noisy observations [20]. We define 𝑿 = (𝒙(1) , 𝒙(2) , … , 𝒙(𝑁) )𝑇 as the design variable vector of the DOE points, and their correspondent response errors as 𝑌 = (𝑦 (1) , 𝑦 (2) , … , 𝑦 (𝑁) ). 𝑿 and 𝑌 are the observed values from DOE. The basic idea of GP modeling is to consider the actual response 𝑦(𝒙) as the realizations of a GP with prior mean 𝒉𝑻 (𝒙)𝜷 and prior covariance matrix 𝛼 2 𝑅(𝒙, 𝒙′). 𝒉𝑻 (𝒙) is a vector of the known functions of 𝒙, i.e. constant, linear, etc. 𝜷 is a vector of coefficients associated with 𝒉𝑻 (𝒙) for a polynomial regression. 𝛼 is a constant. 𝑅(𝒙, 𝒙′) can be defined as:

Figure 2: Flowchart of clustering and classification-based strategy. The two new features are marked by dashed green boxes A and B.

There are two important new features in this strategy, as marked by green dashed boxes. The first one (A) is a smart initial database which is designed to detect as many promising design regions as possible. The second new feature (B) is a data mining step. This step identifies “near duplicates” and replaces them with new “non-duplicated” designs before calling the expensive simulation codes for design evaluation.

𝑅(𝒙, 𝒙′ ) = ∏𝑑𝑘=1 exp[𝑤𝑘 (𝑥𝑘 − 𝑥′𝑘 )2 ]

(2)

𝑅(𝒙, 𝒙′ ) = 𝑐𝑜𝑣(𝑦(𝒙), 𝑦(𝒙′)) is the covariance function between the two designs 𝒙 and 𝒙′. 𝛼 represents the variance of the Gaussian random process; and 𝑤𝑘 controls the smoothness of the response surface. Based on the DOE observations, the values of the hyperparameters 𝜷, 𝛼 and 𝑤, are solved by Maximum Likelihood Estimation (MLE). Given the observed values 𝑿 and 𝑌 , the posterior distribution of any new (unknown) point 𝒙 will be obtained using the Bayesian method. Therefore, the mean of the response on an unknown point 𝒙 (the mean of the posterior distribution) is predicted:

2.2 Construction of the initial design database This section corresponds to the dashed green box (A) in Figure 2. As the iterative search process goes, the design database keeps being enriched with added new designs of known performances. The increasing design data leads to a better decision on determining which design regions are searched in the next iteration. However, we do not have sufficient data at the very beginning of the optimization process. The initial DOE dataset, usually generated from uniform space filling method (e.g. Uniform Latin Hypercube Sampling), may not be large enough to cover all possible promising design regions. For example, the initial DOE set

̂ + 𝒓𝑇 (𝒙)𝑹−1 (𝑌 − 𝐇𝜷 ̂) 𝐸[𝑦(𝒙)|𝑌] = 𝒉𝑇 (𝒙)𝜷

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(3)

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where 𝐇 = [ℎ𝑇 (𝑥1 ), … , ℎ𝑇 (𝑥2 ), … , ℎ𝑇 (𝑥𝑁 )]𝑇 is a known function of 𝒙. When multiplying 𝜷, the two form a polynomial regression of the mean of the response. 𝑹 is a 𝑁 × 𝑁 matrix, where the element in 𝑖 th row, 𝑗th column is 𝑅(𝒙𝒊 , 𝒙𝒋 ) . 𝒓(𝒙) is a 𝑁 × 1 vector, whose 𝑖 th element is 𝑅(𝒙, 𝒙𝒊 ) . ̂ = [𝐇 𝑻 𝑹−𝟏 𝐇]−1 𝐇 𝑻 𝑹−𝟏 𝑌 is the vector of the generalized 𝜷 least squares estimation of the polynomial regression. (A.1) and (A.2) together fully utilize all the 𝑁 DOE points.

dataset is referred as “database” in this paper. The purpose is to generate 𝑁 new promising designs in generation 𝑘 for evaluation (Figure 4). STEP (B.1) The first step is to conduct cluster analysis on the database. The feasible designs and infeasible designs are divided into multiple clusters respectively, according to their similarity in the normalized design space. 𝑘-means method is employed for clustering. The optimal number of clusters is automatically determined (more details are provided in Section 2.2). STEP (B.2) Generate a new design 𝒙. Classify 𝒙 into an existing cluster. For example, Naive Bayesian Classifier is employed in this paper. STEP (B.3) In case of classifying 𝒙 into an infeasible cluster. Calculate its distance to the cluster centroid. If it is too close (e.g. smaller than the 90% quantile of individual-to-centroid distances), reject this design and go back to Step (B.2) to regenerate a new design; Otherwise, this new design is accepted for evaluation. STEP (B.4) In case of classifying x into a feasible cluster. Calculate its distance to the cluster centroid. If it is far from the centroid (e.g. larger than the 90% quantile of individual-tocentroid distances), accept this design for evaluation. If it is too close to the centroid, it will be accepted with a probability of 𝑝, which means exploitative searching in feasible regions; otherwise, it will be rejected with a probability of 1 − 𝑝, and go back to Step (B.2) to regenerate a new design. The probability 𝑝 is defined as:

Figure 3: Bias correction on metamodels

STEP (A.3) Conduct optimization using the bias corrected metamodels to enrich the database. We obtain a virtual Pareto solution set. A “virtual” design refers to the design from the metamodel, so its true response is unknown. 𝑁 designs are picked from the Pareto set following the rule of maximization of the separation distance:

𝑝 =1−𝑒

−

𝑘 𝜏∙𝑘𝑚𝑎𝑥

(4)

{ 𝑘 is the number of the current iteration, 𝑘max is the maximum number of iterations. This formulation is inspired by the “Cooling Schedule” of SA [12]. Increasing the control coefficient 𝜏 leads to a higher probability of the acceptance of a new design close to an existing cluster, i.e. conducting more local exploitative search. The original purpose of “Colling Schedule” is to balance the two contradictory aspects of the optimization search: the explorative search and the exploitative search. In this work, we borrow this idea to make a balance between “pushing to the unknown design space” (explorative search) and “retaining more designs in known feasible clusters” (exploitative search). At the beginning of the optimization process, there is a higher probability of accepting new designs far from the existing clusters for exploration purpose. This probability will decrease along the optimization process to allow more exploitative searches inside the feasible regions. STEP (B.5) Repeat Step (B.2), (B.3) and (B.4) until N new designs are generated. Then operate design evaluations for these N new designs as generation 𝑘.

̂ } of size 𝑁 from the non-dominated design set Find a subset {𝑿 {𝑿}, s.t.: ̂} MAX: mean of pairwise distance 𝑑(𝒙𝑖 , 𝒙𝑗 ) in set {𝑿

} where 𝒙𝒊 , 𝒙𝒋 represent two different designs; 𝑿 represents a set of designs. A large pairwise distance means a long distance between the two design points. By maximizing the mean pairwise distance (equivalent to the sum of the pairwise distances), we maximize the range of design space covered by the N designs. The true performance of these 𝑁 new designs is evaluated by running real simulations. After this step, a database of 2𝑁 designs points is established. If the computation budget is 𝑁𝑚𝑎𝑥 design evaluations, 𝑁𝑚𝑎𝑥 − 2𝑁 is left for following direct optimization search. 2.3 Improving searching efficiency using data mining methods This section corresponds to the dashed green box B in Figure 2. Before running generation 𝑘 of any heuristic optimization algorithm, all historical designs evaluated in the previous 𝑘 − 1 generations are collected as one dataset. This

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analysis, usually take hours or even days for a single run on HPC. It is much longer than the run-time of k-means clustering.

Figure 4: Workflow of the data mining-based strategy: identification and replacement of “near duplicates”

2.4 Optimal k-means clustering method In Step (B.1) of the proposed strategy, the optimal number of clusters is identified for k -means clustering of design database. A careful choice of the number of clusters in clustering analysis will lead to a small within-cluster variation (high similarity inside the same cluster) and a large betweencluster variation (low similarity between different clusters). Cluster analysis with the optimal number of clusters minimizes the errors of miss-classification of new designs. As shown in Figure 5, the cluster analysis is operated for 𝑐𝑚𝑎𝑥 times with 𝑐 = 1, 2, … , 𝑐𝑚𝑎𝑥 clusters respectively. 𝑐𝑚𝑎𝑥 is the user-defined maximum number of clusters. To assess the appropriateness of the clustering analyses, the averaged Silhouette Index 𝑠̅𝑐 is evaluated for each 𝑐. Silhouette Index measures how similar an individual is to its own cluster [21]: 𝑠𝑐 (𝑡) =

𝑏(𝑡)−𝑎(𝑡) 𝑚𝑎𝑥 [ 𝑎(𝑡),𝑏(𝑡) ]

Figure 5: Optimal cluster number for k-means clustering

3. BENCHMARK STUDIES In this section, two benchmark studies are conducted to demonstrate the capability of the proposed method. The first study is a mathematical problem, which is featured by highly nonlinear objectives and constraints; the second study is an optimization problem based on the vehicle side impact analysis. 3.1 Mathematical example: two-objective optimization As shown in Figure 6, two highly nonlinear mathematical functions are constructed in the two-dimensional design space as the competing design objectives: Minimize: 𝑓1 (𝑥1 , 𝑥2 ) = 𝑥21 + 𝑥22 − 10 ∙ 𝑐𝑜𝑠(2𝜋𝑥1 ) − 10 ∙ 𝑐𝑜𝑠(2𝜋𝑥2 ) + 20 (6)

(5)

𝑎(𝑡) is the average dissimilarity of the 𝑡th individual with all other data in the same cluster; 𝑏(𝑡) is the lowest average dissimilarity of 𝑡 to any other cluster where 𝑡 is not a member. The averaged Silhouette Index 𝑠̅𝑐 is the mean of all 𝑠𝑐 (𝑡). 𝑠̅𝑐 is a measure of how tightly the individuals are grouped. It is a value from -1 to 1. Higher value indicates more appropriate clustering. Therefore, the optimal number of clusters 𝑐𝑜𝑝𝑡 is determined by finding the maximum value of 𝑠̅𝑐 . Finally, the data is clustered into 𝑐𝑜𝑝𝑡 clusters. The step of k-means clustering requires longer computation time as the number of samples keeps increasing in the design database. However, such increments in the time costs are negligible for two reasons: (1) the design database has a limited size; (2) the high fidelity simulations, such as the vehicle crash

𝑓2 (𝑥1 , 𝑥2 ) =

10∙𝑥1 ∙𝑥2 +10∙𝑥12 +𝑥23 1000

(7)

Subject to: 𝑥2 −

5

𝑥12 4

−1≤0

(8)

𝑠𝑖𝑛(𝑥1 ) − 4 − 𝑥2 ≤ 0

(9)

𝑥1 , 𝑥2 ∈ [ −5, 5 ]

(10)

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Figure 8: Non-dominated designs obtained from NSGA-II with/without the data mining-based strategy (5N case) Figure 6: Benchmark problem 1: objective functions in the feasible design space

Visually, it can be told that the improved algorithm with the data mining-based strategy can find better or comparable optimal designs than the original NSGA-II. Furthermore, the improved NSGA-II generates much wider Pareto frontiers, which provide engineers with more design choices. Such observations can be validated using three quantitative multiobjective optimization indices [23, 24]: (1) Hypervolume-based Accuracy Performance Index (PI). In a minimization problem, the dominated space 𝐻𝐷 is defined as the hypervolume generated by the solution set and the Pareto set’s upper bounds in the space of design objectives (Figure 9). In a similar way, we can define the non-dominated space 𝐻𝑁 by the solution set and the Pareto set’s lower bounds. PI is calculated as:

Two optimization algorithms are studied comparatively: (1) NSGA-II and (2) NSGA-II improved with data miningbased hybrid strategy. In order to study the impact of the proposed strategy, the optimization algorithms’ parameters (e.g. mutation ratio, crossover ratio) are kept the same. Both algorithms use the same initial population that is generated using Optimal Latin Hypercube Sampling [22]. The computation budget is set at 1000 design evaluations in total. Two different population sizes are tested: (1) 10N, which means that population is 20 and generation number is 50; (2) 5N, which means that population size is 10 and generation number is 100. Each optimization setup is run 20 times. The non-dominated designs obtained from all 20 runs together are plotted in Figure 7 and Figure 8.

𝑃𝐼 = 𝐻

𝐻𝐷

(11)

𝐷 +𝐻𝑁

Figure 9: The definition of the dominated space 𝑯𝑫 and the nondominated space 𝑯𝑵

(2) Pareto Spread Index (PS). Figure 7: Non-dominated designs obtained from NSGA-II with/without the data mining-based strategy (10N case)

𝑃𝑆 =

6

∏𝑁 𝑖=1|𝑚𝑎𝑥(𝑓𝑖 )−𝑚𝑖𝑛 (𝑓𝑖 )| ∏𝑁 𝑖=1|𝑓𝑖 𝑚𝑎𝑥 −𝑓𝑖 𝑚𝑖𝑛 |

(12)

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max(𝑓𝑖 ) is the maximum value of the non-dominated designs’ performances in the 𝑖th dimension; min(𝑓𝑖 ) is the minimum value. 𝑓𝑖 𝑚𝑎𝑥 is the upper bound of the nondominated designs’ performances in the ith dimension; 𝑓𝑖 𝑚𝑖𝑛 is the lower bound. If the upper and lower bounds are unknown, 𝑓𝑖 𝑚𝑎𝑥 and 𝑓𝑖 𝑚𝑖𝑛 are assigned with the maximum and minimum of all 𝑓𝑖 values that are generated by the two methods under comparison. High PS values are preferred as it indicates widespread Pareto solutions in the objective space. (3) Space Distribution Index (SD). 1

𝑆𝐷 = √|𝑆−1| ∑𝑆𝑖=1(𝑑𝑖 − 𝑑̅ )

2

The two methods are compared with respect to the three indices in Figure 10 and Figure 11. For all three indices, higher values are preferred. In both cases, the data mining-based strategy leads to wider and more evenly distributed Pareto Frontier (higher PS and SD). The data mining-based method provides more high-performance design choices for designers to make decisions. Statistical tests are conducted to further validate the improvements of the proposed method. The collection of NSGA-II metric values are demoted as {PI1 , PS1 , SD1 }, and the collection of data mining-based strategy metric values are denoted as {PI2 , PS2 , SD2 }. The hypothesis testing problems are formulated as:

(13)

H0: PI1 (PS1 or SD1 ) > PI2 (PS2 or SD2 );

𝑑𝑖 is the nearest neighbor distance of 𝑖th point on the Pareto frontier; 𝑑̅ is the mean of 𝑑𝑖 . 𝑆 is the number of points on the Pareto frontier. Higher SD value indicates more evenly distributed design points on the Pareto frontier.

H1: PI1 (PS1 or SD1 ) ≤ PI2 (PS2 or SD2 ). With t-test of confidence level 0.95, we reject H0 for all three indices. It indicates that the data mining-based strategy has a statistically significant higher PI, PS and SD values. The p-values of the testing are listed in Table 1. Table 1: P-values of hypothesis testing. If p-value < 0.05, fail to reject H0

Figure 10: Comparison of multiobjective optimization indices, 10N case (higher index value is preferred)

3.2 Multiobjective MDO of side impact problem The second benchmark case study aims to minimizing the weight and abdomen load simultaneously in a vehicle side impact problem (Figure 12). The National Highway Traffic Safety Administration (NHTSA) side impact test configuration is used. The dummy performance is the main concern in side impact. This problem includes 9 system design variables (structure dimension parameters) and 11 responses. The design variables are summarized as follows: x1: thickness of front reinforcement upper and lower, x2: Length of reinforcement lower, x3: thickness of rear reinforcement, x4: thickness of sliding door inner, x5: thickness of B-pillar inner seat belt trap, x6: thickness of B-pillar outer lower 1/3, x7: thickness of B-pillar lower bulkheads, x8: thickness of Rocker outer reinforcement, x9: thickness of rear rail bracket.

Figure 11: Comparison of multiobjective optimization indices, 5N case (higher index value is preferred)

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The optimization problem [25] is formulated as:

H0: PI1 (SD1 ) > PI2 (SD2 );

Minimize: Weight, Abdomen Load

H1: PI1 (SD1 ) ≤ PI2 (SD2 ).

Subject to:

With t-test of confidence level 0.95, we reject H0 for both PI and SD. It indicates that the data mining-based strategy has a statistically significant higher PI and SD values. The p-values of the testing are listed in Table 2.

Rib deflection constraints Dup ≤ 32 mm, Dmid ≤ 32 mm, Dlow ≤ 32 mm,

Table 2: P-values of hypothesis testing. If p-value < 0.05, fail to reject H0

Viscous Criterion (thoracic injury) constraints VCup ≤ 0.32 m/s, VCmid ≤ 0.32 m/s, VClow ≤ 0.32 /ms, Pubic Symphysis Force FPS ≤ 4.0 KN, Velocity of B Pillar at Middle Point VB - Pillar ≤ 9.9 m/s, Velocity of Front Door at B Pillar Vdoor ≤ 15.69 m/s.

Figure 12: Benchmark problem 2: side impact problem

Again, this design problem is solved using both the original NSGA-II and the improved NSGA-II by the data mining-based hybrid strategy respectively. The same optimization algorithm parameters (e.g. mutation ratio, crossover ratio, etc.) as well as the same initial population (generated from Optimal Latin Hypercube Sampling) are used for the two algorithms. The computation budget is 1000 design evaluations (simulations), which is finished in 20 generations with a population size of 50 (5.56N). 20 independent optimization runs are conducted using each algorithm. We combine the non-dominated designs obtained by the two methods, and then make comparisons in Figure 13 and Figure 14. Significant improvements in Performance and Spacing Distribution are achieved by the data mining-based strategy. Statistical tests are conducted to further validate the improvements of the proposed data mining strategy. The collection of NSGA-II metric values are demoted as {PI1 , PS1 , SD1 }, and the collection of data mining-based strategy metric values are denoted as {PI2 , PS2 , SD2 }. The hypothesis testing problems are formulated as:

Figure 13: Non-dominated designs obtained using NSGA-II with/without the data mining-based strategy

Figure 14: Comparison of multiobjective optimization indices

Figure 15 shows the history of the optimization process. The non-dominated designs at generation 5, 10 and 15 of both

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generation increases, there are more and more designs collected in the database so that the data mining-based strategy gradually shows its strength. Eventually it finds better designs than the standard NSGA-II.

methods are plotted and compared in all three subplots. In each subplot, the final non-dominated designs are also plotted in light crosses and dark squares as references. The data miningbased strategy has no significant advantage at generation 5, because the database is too small to be informative. As the

Figure 15: Example of optimization history. The intermediate non-dominated designs are linked by lines for illustration purpose. The “in progress” values of design performances are extracted after the design evaluation box in Figure 2 is finished.

Applying the data mining technique definitely requires additional computational resources. However, such increments in computational time are negligible in Direct MDO using high fidelity simulation models. For example, the data mining process on 1000 designs takes less than 1 minute on a local server of Intel(R) Core(TM) i7-3740QM CPU @ 2.70GHz, 2.70GHz. In contrast, one high fidelity vehicle crash analysis takes more than 10 hours with 48 CPUs on HPC, even if we exclude the unpredictable queue time resulted from the dynamic allocation of CPUs and software licenses. Therefore, we propose to measure the computational efficiency using “batch size” instead of the real turn-around time [6]. “Batch size” refers to the number of simulations submitted to HPC simultaneously for parallel computing. A larger batch size indicates shorter time to finish the computation under the same budget. In this sense, the proposed method has the same efficiency as the NSGA-II in the two case studies.

4. SUMMARY In this paper, a data mining-based hybrid strategy is developed to improve the efficiency of heuristic optimization algorithms. A design database of optimization history is established and updated during the optimization process. Data mining techniques are applied on the design database to identify and replace the ineffective designs (“near duplicates”) before design evaluation. Furthermore, we employ the bias correction-enhanced metamodeling technique to establish a smart initial design database. The proposed strategy is implemented in two studies. A summary is outlined as below. (1) “Near duplication” of previous designs limits the efficiency of the heuristic optimization algorithms. The proposed strategy screens the “near duplicates” using the knowledge obtained from mining the previous designs. Design

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performances are significantly improved with the same number of design evaluations. (2) The clustering and classification methods require much less computational resources than the advanced simulations. Therefore, the data mining strategy will not significantly increase the computational burden. The limitations of this work are summarized as future work. First, even though this paper demonstrates a data miningbased strategy to improve the performances of NSGA-II algorithm. A comprehensive understanding of this strategy will require a wide range of benchmark studies covering multiple algorithms and different types of design problems with single/multiple design objectives. Second, different clustering and classification methods should be discussed and compared to identify the best choices for different types of design data.

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[12]

[13]

[14]

REFERENCES [1] W. J. Hill and W. G. Hunter, "A Review of Response Surface Methodology - a Literature Survey," Technometrics, vol. 8, pp. 571-&, 1966. [2] R. Jin, W. Chen, and T. W. Simpson, "Comparative studies of metamodelling techniques under multiple modelling criteria," Structural and Multidisciplinary Optimization, vol. 23, pp. 1-13, Dec 2001. [3] T. W. Simpson, J. D. Peplinski, P. N. Koch, and J. K. Allen, "Metamodels for computer-based engineering design: survey and recommendations," Engineering with Computers, vol. 17, pp. 129-150, 2001. [4] H. Fang, M. Rais-Rohani, Z. Liu, and M. F. Horstemeyer, "A comparative study of metamodeling methods for multiobjective crashworthiness optimization," Computers & Structures, vol. 83, pp. 2121-2136, Sep 2005. [5] M. Majcher, H. Xu, Y. Fu, C. Chuang, and R. Yang, "A Comparative Benchmark Study of using Different Multi-Objective Optimization Algorithms for Restraint System Design," in SAE 2014 World Congress & Exhibition, Detroit, Michigan, USA, 2014. [6] H. Xu, M. T. Majcher, C.-H. Chuang, Y. Fu, and R.-J. Yang, "Comparative Benchmark Studies of Response Surface Model-Based Optimization and Direct Multidisciplinary Design Optimization," SAE Technical Paper2014. [7] A. Van der Velden and S. I. M. U. L. I. A. Director, "Isight Design Optimization Methodologies," in ASM Handbook 22, ed, 2010. [8] S. Poles, E. Rigoni, and T. Robic, "MOGA-II performance on noisy optimization problems," presented at the International Conference on Bioinspired Optimization Methods and their applications. BIOMA., Ljubljana, Slovena 2004. [9] N. Chase, M. Rademacher, E. Goodman, R. Averill, and R. Sidhu. A Benchmark Study of Multi-Objective Optimization Methods [Online]. [10] E. Elbeltagi, T. Hegazy, and D. Grierson, "Comparison among five evolutionary-based optimization

[15]

[16]

[17]

[18]

[19]

[20]

[21]

[22]

[23]

[24]

10

algorithms," Advanced Engineering Informatics, vol. 19, pp. 43-53, Jan 2005. D. E. Goldberg, Genetic Algorithms in Search, Optimization & Machine Learning: Addison-Wesley, 1989. S. Kirkpatrick, C. D. Gelatt, and M. P. Vecchi, "Optimization by simmulated annealing," science, vol. 220, pp. 671-680, 1983. R. C. Eberhart and Y. Shi, "Particle swarm optimization: developments, applications and resources," in Evolutionary Computation, 2001. Proceedings of the 2001 Congress on, 2001, pp. 81-86. H. E. Radhi and S. M. Barrans, "Comparison between Multiobjective Population-Based Algorithms in Mechanical Problem," Applied Mechanics and Materials, vol. 110, pp. 2383-2389, 2012. M. Edahiro, "A clustering-based optimization algorithm in zero-skew routings," in Proceedings of the 30th international Design Automation Conference. ACM, 1993. P. M. Ortigosa, I. García, and M. Jelasity, "Reliability and performance of UEGO, a clustering-based global optimizer," Journal of Global Optimization, vol. 19, pp. 265-289, 2001. J. Zhang, H. H. Chung, and W. L. Lo, "Clusteringbased adaptive crossover and mutation probabilities for genetic algorithms," Evolutionary Computation, IEEE Transactions, vol. 11, pp. 326-335, 2007. Y. J. Wang, J. S. Zhang, and G. Y. Zhang, "A dynamic clustering based differential evolution algorithm for global optimization " European Journal of Operational Research, vol. 183, pp. 56-73, 2007. L. Shi, R.-J. Yang, and P. Zhu, "An Adaptive Response Surface Method Using Bayesian Metric and Model Bias Correction Function," Journal of Mechanical Design, vol. 136, p. 031005, 2014. P. D. Arendt, D. W. Apley, and W. Chen, "Quantification of Model Uncertainty: Calibration, Model Discrepancy, and Identifiability," Journal of Mechanical Design, vol. 134, Oct 2012. P. J. Rousseeuw, "Silhouettes: a graphical aid to the interpretation and validation of cluster analysis," Journal of computational and applied mathematics, vol. 20, pp. 53-65, 1987. F. Xiong, Y. Xiong, W. Chen, and S. Yang, "Optimizing Latin hypercube design for sequential sampling of computer experiments," Engineering Optimization, vol. 41, pp. 793-810, 2009. J. Wu and S. Azarm, "Metrics for quality assessment of a multiobjective design optimization solution set," Journal of Mechanical Design, vol. 123, pp. 18-25, 2001. T. Okabe, Y. Jin, and B. Sendhoff, "A critical survey of performance indices for multi-objective optimisation," in Evolutionary Computation, 2003. CEC'03. The 2003 Congress on, 2003, pp. 878-885.

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[25]

R. Yang and L. Gu, "Experience with approximate reliability-based optimization methods," Structural and Multidisciplinary Optimization, vol. 26, pp. 152159, 2004.

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