Keywords: shape optimization, unstructured remeshing, Delaunay triangulation, triangular elements, particle swarm optimization, gradient based optimization.

In this paper, we combine both gradient based and gradient free optimization techniques with a novel unstructured remeshing shape optimization environment. The unstructured remeshing strategy uses triangular elements based on a truss structure analogy [1]. However, the linearly convergent sequential updating scheme previously proposed is replaced by a quadratically convergent scheme. It is then demonstrated that the new scheme may of course also be used to obtain analytical gradients.

Two gradient based optimization algorithms are considered: the Dynamic-Q algorithm proposed by Snyman and Hay [2], and the well known sequential quadratic programming (SQP) algorithm [3]. The gradient free implementation is based on the particle swarm optimization algorithm (PSOA) proposed by Kennedy and Eberhard [4]. However, the velocity rule of the standard algorithm is modified to increase the diversity of the particle searches.

Extensive numerical examples are presented; this includes notes regarding convergence rate, accuracy, efficiency, etc. It is shown that the gradient free PSOA is less likely to be trapped in local minima, albeit at increased computational expense.

In constructing approximate solutions, we implement an unstructured remeshing environment based on mesh generation founded on a truss structure analogy. The domain is discretized using triangular elements, while the domain is defined by a signed distance function using piece-wise linear interpolation functions.

In general, in an unstructured remeshing shape optimization environment, the cost function is noisy or even discontinuous, (amongst others due to the addition and removal of nodes). The frequency and the amplitude of the noise (discontinuities) depends on the elemental dimensions, the number of control points or variables, as well as the order of displacement interpolation (element type). In this paper we study the effects of some of these on convergence.

The implemented algorithms are discussed in sufficient detail to allow the reader to duplicate the results presented in our paper. This includes the modifications made to the PSOA, which increases its rate of convergence notably.

Firstly, extensive results are presented to illustrate the effect of the implemented unstructured meshing strategy. This includes evaluation of mesh (elemental) quality, uniformity, etc.

Secondly, three example problems are presented, namely

1. a cantilever beam subjected to a point load,
2. a Michell-like structure, and
3. a popular spanner problem.

Observations regarding convergence rate, accuracy, efficiency, etc. are presented. For the sake of brevity, these are not presented here. However, it is interesting to point out here that the gradient free PSOA is for some problems able to find superior solutions to the solutions found by the gradient based methods, since the algorithm is less susceptible to become trapped in local minima. The gradient methods, on the other hand, are in not as computationally demanding as the (sequential) PSOA.

1

P.-O. Persson and G. Strang. A simple mesh generator in matlab. To appear in SIAM Review, 2004. doi:10.1137/S0036144503429121

2

J.A. Snyman and A.M. Hay. The Dynamic-Q optimization method: An alternative to SQP? Int. J. Comput. Math. Appl., 2002. doi:10.1016/S0898-1221(02)00281-X

3

M.S. Bazaraa, H.D. Sherali, and C.M. Shetty.Nonlinear Programming Theory and Algorithms. John Wiley, New York, USA, second edition, 1993.

4

J. Kennedy and R.C. Eberhart. Particle swarm optimization. In Proceedings of the 1995 IEEE International Conference on Neural Networks, volume 4, pages 1942-1948, Perth, Australia, IEEE Service Center, Piscataway, NJ, 1995. doi:10.1109/ICNN.1995.488968

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