Topology optimization [1] is an iterative process which minimizes a predefined objective function by distributing material in a given design domain. Common objective choices, in linear elasticity, are minimization of structural weight, compliance or stress concentration. The material distribution fulfills a predefined set of constraints, e.g. volume or stress constraints. In a density based formulation, the material distribution is represented by a density field which takes values one or zero: one if a point is occupied with material and zero if the point is void. The optimization problem is usually relaxed so the density takes intermediate values and the field is updated using gradients of the objective and the constraints with respect to the density parametrization. Mesh independence of the optimization process is ensured by regularizing the formulation. The regularized solution possesses intermediate values between zero and one which might not have a clear physical interpretation. Therefore, a 0/1 solution is restored by projection techniques which can be used to represent uncertainties in the production process [2]. Taking into account uncertainties in the geometry results in manufacturable black and white designs [3].
Incorporating the uncertainty model in the optimization process results in significant increase of the computational load. In order to re-utilize already developed deterministic solvers, non-intrusive methods are utilized for obtaining the response of the stochastic problem. This results in a set of completely uncoupled deterministic problems which share similar structure. Here we propose an approach to reduce the solution cost for a single stochastic problem as well as for the sequence of the stochastic problems during the optimization process using preconditioned iterative solvers. The idea is to utilize the recently developed multiscale finite element coarse spaces for high contrast elliptic problems [4] and the similarity between the deterministic problems. The coarse spaces are obtained by solving a set of local eigenvalue problems on overlapping patches covering the computational domain. The approach is relatively easy for parallelization, due to the complete independence of the sub-problems, and ensures contrast independent convergence of the iterative state problem solvers. Several modifications for reducing the computational cost in connection to topology optimization are discussed in details.
[1] Bendsoe, M. P. and Sigmund, O. Topology Optimization - Theory, Methods and Applications Springer Verlag, Berlin Heidelberg, 2003
[2] Jansen, M.; Lazarov, B.; Schevenels, M. and Sigmund, O. On the similarities between micro/nano lithography and topology optimization projection methods. Structural and Multidisciplinary Optimization, Springer Berlin Heidelberg, 2013, 48, 717-730
[3] Andreassen, E.; Lazarov, B. and Sigmund, O. Design of manufacturable 3D extremal elastic microstructure. Mechanics of Materials , 2014, 69, 1 - 10
[4] Efendiev, Y.; Galvis, J. and Wu, X.-H. Multiscale finite element methods for high-contrast problems using local spectral basis functions. Journal of Computational Physics, 2011, 230, 937 - 955