Level set based robust shape and topology optimization under random field uncertainties

A level-set-based method for robust shape and topology optimization (RSTO) is proposed in this work with consideration of uncertainties that can be represented by random variables or random fields. Uncertainty, such as those associated with loading and material, is introduced into shape and topology optimization as a new dimension in addition to space and time, and the optimal geometry is sought in this extended space. The level-set-based RSTO problem is mathematically formulated by expressing the statistical moments of a response as functionals of geometric shapes and loading/material uncertainties. Spectral methods are employed for reducing the dimensionality in uncertainty representation and the Gauss-type quadrature formulae is used for uncertainty propagation. The latter strategy also helps transform the RSTO problem into a weighted summation of a series of deterministic topology optimization subproblems. The above-mentioned techniques are seamlessly integrated with level set methods for solving RSTO problems. The method proposed in this paper is generic, which is not limited to problems with random variable uncertainties, as usually reported in other existing work, but is applicable to general RSTO problems considering uncertainties with field variabilities. This characteristic uniquely distinguishes the proposed method from other existing approaches. Preliminary 2D and 3D results show that RSTO can lead to designs with different shapes and topologies and superior robustness compared to their deterministic counterparts.