Keywords: dome design, conceptual design, genetic algorithm, convex hull.

Using stochastic search algorithms for the conceptual design of geodesic domes is a relatively neglected area of research, with very few papers published in the last 10 years [1,2]. Previous work in this area by Shea and Cagan [2] approaches the problem by creating a 2D truss that is subsequently projected onto a predefined curved surface. Therefore the solution is a 3D object, but the search is conducted in 2D. However while this 'projection' or 2.5D technique reduces the number of problem variables, by constraining the third dimension to be dependent on the planar layout, it also fixes the two most important variables of a dome: surface area and enclosed volume, and thus produces results that are potentially sub-optimal.

This paper describes a new approach that uses a genetic algorithm [3,4], combined with convex hulls [5,6], a technique developed in computational geometry, to design domes in 3D (without the need to project onto a curved surface).

The convex hull of a finite set of points S, is considered to be the convex polyhedron with the smallest volume that encloses S and can be determined by an incremental algorithm [5,6]. An incremental algorithm constructs the convex hull CH, of a finite set of points S, by initially developing the convex hull CH_i-1 of a simple sub-set S_i, of S. Having constructed the convex hull for S_i, the algorithm then adds one point p at a time from S to S_i and updates the convex hull depending on whether p is internal to S_i. If p is internal to S_i then the convex hull is unaltered however if p is external then the convex hull must be updated to accommodate p. To alter the convex hull the algorithm first needs to identify the horizon, which divides the convex hull into visible and invisible regions, a 'cone' of faces with an apex at p is then attached to the horizon.

The genetic algorithm used in this work has a genome that is composed of three sections: user defined loads, location of the dome supports and the location of potential dome vertices and uses several standard evolutionary operators. Also because this work is aimed at the conceptual design stage, the number of initial input parameters has been kept to a minimum. Therefore the user is only required to input the location of any loads and define the size of the circular base. If requested, the user can specify the number and location of the dome supports otherwise the algorithm will search for appropriate support positions during the run. Once the genetic algorithm is started, it initialises the genome with random variables and runs until a predetermined number of generations have been evolved. The algorithm then returns a range of good solutions for the designer to analyse.

Designing in 3D enables the genetic algorithm to search using all the available variables and hence allows for significantly better solutions to be evolved than was previously possible. However it should be noted that because geodesic breakdowns [7] are not explicitly enforced there is no guarantee that the evolved domes will rigidly adhere to geodesic patterns.

1

B. Porter, S.S. Mohamed and T.R. Crossley, "Genetic computation of geodesics on three-dimensional curved surfaces", Genetic Algorithms in Engineering Systems: Innovations and Applications (GALESIA'95), IEEE Conference Publication No.414, University of Sheffield (UK), 12-14th September, 448-453, 1995. doi:10.1049/cp:19951090

2

K. Shea and J. Cagan, "Innovative dome design: Applying geodesic patterns with shape annealing", Artificial Intelligence for Engineering Design, Analysis and Manufacturing, (11), 379-394, 1997.

3

J.H. Holland, "Adaptation in natural and artificial systems", Ann Arbor: The University of Michigan Press, 1975.

4

D.E. Goldberg, "Genetic algorithms in search, optimisation and machine learning", New-York: Addison-Wesley, 1989.

5

J. O'Rourke, "Computational Geometry in C", Second Edition, Cambridge University Press, 1998.

6

M. de Berg, "Computational geometry: algorithms and applications", New York: Springer, 2000.

7

R. Motro, "Review of the development of geodesic domes", From Z.S. Makowski, "Analysis, design and construction of braced domes", Cambridge University Press, 387-412, 1994.

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