Keywords: design of experiments, latin hypercube sampling, constrained design spaces, non-regular design spaces, space-filling, Delaunay triangulation.

Space-filling design strategies known as design of experiments (DoE) constitute an essential part of any experimentation. This paper concerns one particular domain of constrained design spaces. The most frequent example is the case of a mixture experiment, where individual inputs form a unity volume or unity weight [1]. This single condition leads to the simplex space; further limits on individual inputs then form a polytope, still convex but generally an irregular space. Therefore, all traditional DoEs [1] that are constructed for hypercube spaces cannot be applied here. Although the problem of mixture experiments has been known for decades, the progress of the methods for DoEs does not follow current developments.

In this paper a different approach based on Delaunay triangulation (DT) of an admissible domain and a utilization of the properties of the Distmesh tool (DM) [2] is presented. In the authors' method the domain described by corner vertices is triangulated using DT and the desired number of random points is generated inside. Then the DM tool is applied. The Distmesh tool is a heuristic smoothing algorithm for generating uniform meshes that is based on a simple dynamic system of an expanding pin-jointed structure. Those trusses that are too short cause repulsive forces that move apart nodes that are too close creating uniformly spaced points.

The results are compared with seven constrained examples in two dimensions and one three dimensional example presented in [3], namely a placing of design points in a triangle, parallelogram, pentagon, hexagon, heptagon, octagon, irregular hexagon and prism. Although not designed directly for space-filling optimization, our procedure is able to outperform the reference algorithm even from the D-optimal point of view; however, only in two-dimensions as is shown in the paper.

1

D.C. Montgomery, "Design and Analysis of Experiments", 5th Edition, Wiley, 2000.

2

P.-O. Persson, G. Strang, "A simple mesh generator in MATLAB", SIAM Review, 46(2), 329-345, 2004. doi:10.1137/S0036144503429121

3

M. Hofwing, N. Strömberg, "D-optimality of non-regular design spaces by using a Bayesian modification and a hybrid method", Structural and Multidisciplinary Optimization, 42, 73-88, 2010. doi:10.1007/s00158-009-0464-3

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