Formulation of the Audze–Eglais uniform Latin hypercube design of experiments for constrained design spaces
Introduction
Surrogate or Metamodelling is a common task in Engineering Design when underlying computer simulations are computationally expensive. The choice of the parameter values for the designs that the model is built with has a considerable impact on the accuracy of the extracted output. This set of designs is called Design of Experiments (DoE) and plenty of papers have been published in this field which resulted in numerous ways to create them as space-filling and efficiently possible.
A popular approach was proposed by McKay et al. [1] and Iman and Conover [2] called the Latin Hypercube (LH) sampling method. The most important property is that no two experiments share the same coordinate. This characteristic is denoted as Non-collapsingness. The importance of this is based on the fact that some parameters have only a minor influence on the result of a simulation. Thus, running two collapsing experiments might lead to similar results and hence no new information, ultimately to unnecessary computational costs.
The random generation of LH leads to Non-collapsing DoEs but they usually have a poor space-filling quality. Numerous publications can be found on Optimal Latin Hypercubes (OLH). OLHs are LHs that are characterized by an optimal distribution of the placed experiments in the sense of a chosen criterion. Again, several different approaches arose, for instance minimization of the integrated mean square error (IMSE) by Sacks et al. [3], maximization of entropy by Shewry and Wynn [4], the maximin distance criterion by Johnson et al. [5] or the minimization of the Audze Eglais potential energy by Bates et al. [6], [7]. Liefvendahl and Stocki [8] studied the performance in terms of efficiency and space-fillingness of the last two approaches and concluded that the approach based on the potential energy minimization outperforms the maximin distance criterion. Publications following this recommendation include Panda and Manohar [9] and Cook and Skadron [10].
Commonly OLHs are created by assuming a complete n-dimensional design space even though this might not reflect reality. Thus, this causes experiments to be created that might not be feasible or do not have a physical meaning. This means for both, the creation of the DoEs and the response surface, unnecessary computational costs. Simply omitting infeasible points leads to a DoE with fewer experiments than desired and to a set of points that is not optimally distributed.
Stinstra et al. [11] developed a procedure for the creation of DoEs for constrained design spaces. But the Non-collapsingness criterion (NC) is not considered.
The purpose of the presented work is the development of a novel procedure that creates OLHs for an arbitrarily constrained design space. The formulation by Bates et al. [6], [7] will be extended to meet this requirement. Here, the OLH is found by minimizing the potential energy of the points according to Audze and Eglais [12] using a permutation genetic algorithm. The validity of the approach is demonstrated using several 2D examples.
Section snippets
Unconstrained formulation
The original formulation, for the creation of space-filling designs in unconstrained design spaces, by Bates et al. [6], [7], which this work is based on, is using the Audze–Eglais [12] method which follows a physical analogy: A system of mass-points exert repulsive forces on each other leading to potential energy in the system. In case of a minimum of the potential energy the points are in equilibrium. The analogous potential energy U for the creation of the OLH is defined as:
Effect of large infeasible areas
As described before, the entries in have to be increased in case where due to the constraints insufficient possibilities exist to place feasible points without violating the NC. In general, one can choose the minimum number of design space divisions for each dimension m for which the algorithm is capable of finding an initial feasible population. In general, the lower the feasibility ratio becomes, i.e. the higher the number of infeasible grid points in the design space becomes,
Results
Several 2D example problems have been solved to show the capabilities of the developed procedure. The results will be discussed in the following.
Conclusions
The aim of the present work was to extend the formulation for the creation of OLHs in unconstrained design spaces by Bates et al. [6], [7] to arbitrarily constrained design spaces of any dimension. This has been achieved. Amendments of the original genetic algorithm comprise the possibility to penalize infeasible experiments. The requirement that the number of points equals the number of design space divisions was removed and the algorithm was extended accordingly. Consequently, the developed
Acknowledgment
The authors gratefully acknowledge the financial support from Asset International Ltd and KWH Pipe Ltd.
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