Metamodels have been widely used in engineering design to facilitate analysis and optimization of complex systems that involve computationally expensive simulation programs. The accuracy of metamodels is directly related to the experimental designs used. Optimal Latin hypercube designs are frequently used and have been shown to have good spacefilling and projective properties. However, the high cost in constructing them limits their use. In this paper, a methodology for creating novel Latin hypercube designs via translational propagation and successive local enumeration algorithm (TPSLE) is developed without using formal optimization. TPSLE algorithm is based on the inspiration that a near optimal Latin Hypercube design can be constructed by a simple initial block with a few points generated by algorithm SLE as a building block. In fact, TPSLE algorithm offers a balanced tradeoff between the efficiency and sampling performance. The proposed algorithm is compared to two existing algorithms and is found to be much more efficient in terms of the computation time and has acceptable spacefilling and projective properties.
In engineering, manufacturing companies strive to produce better and cheaper products more quickly. However, engineering systems are fairly large and complicated nowadays. In addition, design requirements are rigorous and stringent for such systems, especially multidiscipline design optimization systems such as aerospace. These engineering analysis and design problems usually involve expensive computer simulations. For example, it is reported that it takes Ford Motor Company about 36–160 h to run one crash simulation [
In recent decades, various sampling designs have been developed for computer experiments. The classical experiment designs containing alphabetical optimal design [
Latin hypercube designs (LHD) play an important role in computer experiments. The Latin hypercube structure allows one to achieve both the spacefilling requirement and the noncollapsing condition. Each column of an
Although aforementioned methods provide effective ways to produce samples with good spacefilling and projective properties, they are computationally inefficient for problems with large dimensions and sample sizes. For example, Ye et al. [
In this paper, we propose a method that is able to quickly construct a good design of experiments given a limited computational resource. There are two major algorithms involved. One is translational propagation algorithm (TP) [
The remainder of the paper is organized as follows. The proposed TPSLE algorithm for obtaining Latin hypercube designs is described in Section
In order to illuminate the algorithm in detail, the basic procedure of TPSLE is introduced first, followed by the application of the novel algorithm for a twodimensional problem. Then a method of generating experimental designs of any size is proposed. Certainly the summary of TPSLE algorithm will be given at last.
The proposed algorithm is based on the inspiration of constructing the
Assuming to construct a Latin hypercube design of
In the example of the
Next, points of initial block are generated by the optimal Latin hypercube design SLE which will be introduced in the next section. Then the entire design space will be filled with the initial block via translational propagation algorithm. Figure
Process of creating the
Step 1
Step 2
Step 3
The greatest advantage of this approach is that there are no calculations to perform once initial block is completed. All operations can be viewed as a simple translation of the block designs in the
In this section a novel algorithm of maximin LHD using SLE is introduced briefly, referred to in [
Sampling comparison with MATLAB function LHSD (
SLE
LHSD
Similarly, assuming to generate an initial Latin hypercube design of
When using SLE algorithm to construct an initial block for TPSLE algorithm, it is noticed that the variable
To generate an improved Latin hypercube design proposed in this paper with any number of points, the first step is to generate a TPSLE that has more points than the required. The experimental design will be completed without resizing the size of TPSLE if the design of
To construct a
Process of resizing to create a
Step 1. New design with size
Step 2. New design with size
Step 3. New design with size
The proposed algorithm is inspired by the tradeoff between performance and efficiency of experimental sampling design. In practice, good Latin hypercube designs are expected to be obtained efficiently because the consuming time is limited. This is particularly critical for large number of points in high dimensions. TPSLE generated from a fairly small optimal Latin hypercube design used as an initial block via translational propagation algorithm is a superior design relatively. Figure
Flowchart of proposed algorithm TPSLE.
Based on the abovementioned TPSLE algorithm, the sampling points illustrated in Figure
A Latin hypercube design with size
The sampling points generated by the TPSLE algorithm meet the two desired features, namely, spacefilling and projective properties. The distributions of the produced sampling points are even in the design space and the projective points in lower dimensions are almost uniform, especially for projecting to each coordinate axis. According to the sampling process of the TPSLE algorithm, the initial block constructed by SLE is used to generate the sampling points via translational propagation, which are quite different from the existing LHD sampling methods. In TPSLE, there are no global objective functions, such as
In recent years, some optimal criteria are employed widely to achieve a good performance in design of computer experiments. The optimal designs constructed by these optimal criteria have been shown to have a good performance. In other words, these optimal criteria can be used as test criteria to test whether the experimental designs have good performance. Four widely used test criteria are considered in this work.
Maximin distance criterion is proposed by Johnson et al. [
Centered
In 1995, Morris and Mitchell [
In optimal LHD algorithms, the AudzeEglais objective function [
To illustrate the performance, that is, spacefilling and projective properties of the sampling points, four aforementioned criteria are employed, namely,
Various sampling designs are generated by three different Latin hypercube design methods including TPSLE, LHSD, and SLE. In order to reduce the randomness of sampling designs, sampling points are generated for 50 times through the TPSLE and SLE algorithm. Meanwhile, 500 times are for LHSD algorithm with the default set in MATLAB. It is noticed that sampling points are generated for 10 times as
Comparison of test criteria among TPSLE, SLE, and LHSD in twodimension.

TPSLE  LHSD  SLE  


Criteria  Best  Worst  Mean  Best  Worst  Mean  Best  Worst  Mean 
16 

0.177  0.140  0.146  0.178  0.061  0.114  0.198  0.140 


0.059  0.064  0.060  0.045  0.087 

0.058  0.062  0.060  

5.816  7.255  7.025  5.062  8.074  6.790  5.060  7.155 
 

739.75  802.66  749.71  740.87  1398.3  951.87  708.19  754.56 
 


32 

0.113  0.070  0.087  0.095  0.033  0.057  0.156  0.088 


0.031  0.034 

0.025  0.063  0.035  0.030  0.034 
 

8.999  14.713  12.392  10.558  29.996  18.045  6.688  11.314 
 

3957.9  4522.6  4216.8  4465.1  8371.5  5599.4  3820.0  3966.9 
 


64 

0.099  0.035  0.064  0.045  0.014  0.028  0.105  0.035 


0.016  0.017 

0.015  0.045  0.023  0.016  0.018  0.017  

10.467  29.021  18.075  20.458  32.345  26.618  9.732  28.622 
 

19278  21035  20007  24060  44498  30041  19179  19994 
 


128 

0.052  0.017  0.036  0.024  0.008  0.014  0.074  0.025 


0.008  0.009 

0.009  0.028  0.015  0.009  0.011 
 

19.779  58.042  31.482  42.235  129.40  72.667  13.739  40.477 
 

94780  100754  96565  119503  195253  152308  93131  94787 

Comparison of test criteria among TPSLE, SLE, and LHSD in threedimension.

TPSLE  LHSD  SLE  


Criteria  Best  Worst  Mean  Best  Worst  Mean  Best  Worst  Mean 
16 

0.306  0.258 

0.308  0.121  0.210  0.306  0.108  0.246 

0.097  0.115  0.103  0.074  0.132 

0.087  0.095  0.092  

3.358  3.935 

3.323  8.267  4.875  3.284  9.238  4.247  

339.41  382.95 

344.65  557.14  411.31  357.67  438.01  376.71  


32 

0.221  0.153 

0.184  0.084  0.124  0.261  0.077  0.182 

0.051  0.068  0.056  0.045  0.098  0.058  0.050  0.057 
 

4.653  6.716 

5.529  11.926  8.268  3.847  13.064  5.988  

1645.2  1769.0 

1798.5  2689.1  2038.5  1673.7  1882.3  1743.5  


64 

0.147  0.077  0.110  0.106  0.047  0.075  0.206  0.038 


0.029  0.034 

0.029  0.058  0.040  0.031  0.035  0.033  

7.016  13.431  9.704  9.455  21.358  13.639  4.977  26.128 
 

7642.1  8803.8  8148.1  8440.2  10480  9286.4  7617.8  8342.6 
 


128 

0.073  0.038  0.058  0.067  0.027  0.047  0.149  0.014 


0.018  0.021 

0.020  0.048  0.028  0.019  0.023  0.021  

14.028  26.862  18.825  15.130  37.412  22.068  6.744  73.901 
 

34282  38292  35798  37069  44138  40207  33268  39353 

Comparison of test criteria between TPSLE and LHSD in high dimension.
Criteria method  Sampling size 





























0.308  0.205  0.138  0.274  0.210  0.163  0.273  0.225  0.273  0.518  0.635 















3.301  4.971  7.394  3.702  4.811  6.221  3.696  4.486  3.700  1.956  1.380 



272.1 


3049.4 


35147  127547  427341  235684  187425 


1179.8  5145.5 

11116  45053 





According to the comparison study with SLE algorithm and LHSD function with various number of points in twodimension in Table
Similarly, the results of sampling designs in threedimension from Table
3D spacefilling and corresponding 2D projective points generated by TPSLE.
3D spacefilling and corresponding 2D projective points generated by LHSD.
3D spacefilling and corresponding 2D projective points generated by SLE.
For the sake of reflecting good performance of TPSLE further, the test criteria
In a word, we can conclude that better spacefilling and projective properties can be obtained by TPSLE through comparison with LHSD and SLE under different criteria of
In this section, an illustrative comparison among our proposed TPSLE algorithm, LHSD function in MATLAB, and SLE algorithm presented in Zhu et al. [
The time consumptions of sampling designs using different algorithms are compared in Table
Comparison of time(s) among TPSLE, SLE, and LHSD.
Time(s) method  Sampling size  














TPSLE  0.0042  0.0100  0.0266  0.0039  0.0098  0.0579  0.0061  0.0050  0.0044  12.34  19.85  279.24 
SLE  0.0058  0.0902  0.3802  0.0531  3.9341  35.948  1023.8  
LHSD  0.0040  0.0043  0.0048  0.0040  0.0054  0.0055  0.0052  0.0058  0.0080  12.48  12.77  12.83 
In this section, five mathematical examples listed in Appendix
Sampling designs are very important for constructing metamodels. Poor sampling designs not only lead to poor accuracy of metamodels, but also reduce the efficiency. In this paper, five widely accepted mathematical examples are employed to test the accuracy of metamodels that are built with different sampling methods, that is, LHSD and TPSLE. As one of the most effective approximation methods, radial basis functions’ (RBF) [
To make a fair comparison of two methods, the total number of sampling points (
Comparison of metamodels accuracy between TPSLE and LHSD.
Function  Number of variables  NRMSE  NMAX  

TPSLE  LHSD  TPSLE  LHSD  
BR  2  0.0519  0.0973  0.0519  0.0973 
AF  2  0.274  0.4012  0.274  0.4012 
PEAKS  2  0.1153  0.1919  0.1153  0.1919 
HN  3  0.3929  0.5755  0.3929  0.5755 
MATH [ 
5  0.0383  0.0417  0.0383  0.0417 
From the results shown in Table
The performance of TPSLE algorithm is tested by a typical mechanical design optimization problem involving three design variables, that is, pressure vessel design. This problem is modified from the original problem recorded in [
Diagram of pressure vessel design.
The objective function is the combined cost of materials, forming and welding of the pressure vessel. The mathematical model of the optimization problem is expressed as
The ranges of the design variables
The problem formulated above is a simple nonlinear constrained problem. Now assume the objective function defined by (
Optimal results of pressure vessel design.
Method  Number of function evaluations  Number of design iterations  Variable design  Objective value 

LHSD  16  11.8  [1.00, 44.50, 149.02]  7682.7 
TPSLE  16  7.28  [1.03, 53.57, 72.34]  7044.0 
In this paper, a methodology for creating novel Latin hypercube designs via translational propagation algorithm (TPSLE) is proposed. The approach is inspired by the idea that a simple initial block with a few points generated by a novel algorithm SLE can be used as a building block to construct a near optimal Latin hypercube design. TPSLE algorithm offers a balanced tradeoff between the efficiency and performance, that is, spacefilling and the projective properties. The greatest advantage of the proposed methodology is that it requires virtually no computational time. In fact, no global objective functions are employed to optimize in TPSLE algorithm which is quite different from the existing LHD sampling methods. The performance of the sampling points generated by TPSLE is studied through comparison with other sampling methods under different test criteria, and the efficiency of TPSLE and other sampling methods is compared.
It is found that the spacefilling and projective properties of sampling points using TPSLE are better than LHSD in most cases. In addition, though TPSLE algorithm is not as good as SLE in terms of performance of sampling points, the efficiency of TPSLE is further superior to SLE. TPSLE is a novel LHD sampling algorithm with acceptable spacefilling and projective properties, and the efficiency of sampling algorithm TPSLE is superior. For the sake of examining the validity of the proposed TPSLE sampling algorithm further, five typical mathematical examples and one mechanical design optimization problem have been tested. The assessment measures for accuracy of metamodels, that is, NRMSE and NMAX, are employed. In contrast to the traditional sampling methods LHSD, TPSLE results in more accurate metamodels. Furthermore, TPSLE is superior in solving engineering design optimization problem on exploring global minimum of metamodels.
The proposed sampling algorithm TPSLE is a wise tradeoff between performance and efficiency of sampling design and significantly outperforms the conventional sampling methods. However, there are still some shortcomings in TPSLE algorithm. Firstly, the performance of sampling points in high dimension is not good sometimes. Secondly, the sampling algorithm TPSLE cannot construct sampling points with arbitrary size directly. The problems mentioned above need to be resolved in future work.
Branin function (BR),
Root mean squared error (RMSE):
Normalized root mean squared error (NRMSE):
Maximum absolute error:
The authors declare that there is no conflict of interests regarding the publication of this paper.
The authors would like to thank everybody for their encouragement and support. The support grants from the National Science Foundation (CMMI51375389 and 51279165) are greatly acknowledged.