Keywords: parallel computing, optimisation, topology, multiscale, homogenization, domain decomposition, bone.

In this work a computational procedure for multi-scale topology optimization problem using parallel computing techniques is developed. The two-scale optimization model is used to obtain the best structure and material distribution simultaneously, considering the minimum compliance criterion [1]. The design space at each scale is defined through density design variables characterizing regions with and without material and it is considered that the structure is made of a cellular material characterized by a microstructure whose design may vary within the structure domain, although a local periodicity is assumed. The asymptotic homogenisation theory is used to obtain the equivalent material properties, at macro scale, for the specific local microstructures.

The simultaneous design of structure and its material leads to massive computations, especially in three-dimensional applications. The two-scale model implies the iterative solution of one problem at macro scale (global problem) and many problems at local scale characterizing the material microstructure (local problem). In our model, the number of local problems depends directly on the number of global model finite elements making the computational time extremely sensitive to the macro discretization used. Typically, the high number of finite elements in the structural domain discretization, required in most practical engineering problems, makes prohibitive the serial mode running, thus making the use of parallel computing techniques an attractive solution to reduce the computational time.

Since the local optimisation problems can be uncoupled, the problem can be efficiently solved by computational means using parallel computing techniques. The following procedure is used in this work: in each iteration of the structural design algorithm, the material design update is a task performed by several processors of a parallel machine; each processor has assigned a sub-domain of the overall structure domain in order to update the material design therein.

In this work the algorithmic strategy to solve the two-scale optimisation problem is presented in a suitable way for parallelization. It is based on the MMA (method of moving asymptotes) to update the macroscopic design variables and on the CONLIN (CONvex LINearization) to update the microstructure design variables. In terms of parallel computing facilities it uses an IBM Cluster 1350 computing system comprising 70 computing nodes each with 2 dual core processors, for a total of 280 cores, interconnected with 2 Gigabit Ethernet networks. It is shown by means of performance measures (speed-up and efficiency plots) that the present two-scale optimisation problem has a very good scalability. Finally a very demanding computational "bone" problem shows the applicability of the model and methodology presented.

1

P. Coelho, P.R. Fernandes, J.M. Guedes and H.C. Rodrigues, "A hierarchical model for concurrent material and topology optimisation of three-dimensional structures", Struct Multidis Optim, 35, 107-115, 2008. doi:10.1007/s00158-007-0141-3

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