Formulation of the Audze–Eglais uniform Latin hypercube design of experiments for constrained design spaces

Abstract

Optimal Latin Hypercubes (OLH) created in a constrained design space might produce Design of Experiments (DoE) containing infeasible points if the underlying formulation disregards the constraints. Simply omitting these infeasible points leads to a DoE with fewer experiments than desired and to a set of points that is not optimally distributed. By using the same number of points a better mapping of the feasible space can be achieved. This paper describes the development of a procedure that creates OLHs for constrained design spaces. An existing formulation is extended to meet this requirement. Here, the OLH is found by minimizing the Audze–Eglais potential energy of the points using a permutation genetic algorithm. Examples validate the procedure and demonstrate its capabilities in finding space-filling Latin Hypercubes in arbitrarily shaped 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.