A new level-set based approach to shape and topology optimization under geometric uncertainty |
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Authors: | Shikui Chen Wei Chen |
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Affiliation: | (1) Department of Mechanical Engineering, Northwestern University, 2145 Sheridan Road, Evanston, IL 60208, USA;(2) Altair Engineering, Inc., Irvine, CA, USA |
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Abstract: | Geometric uncertainty refers to the deviation of the geometric boundary from its ideal position, which may have a non-trivial
impact on design performance. Since geometric uncertainty is embedded in the boundary which is dynamic and changes continuously
in the optimization process, topology optimization under geometric uncertainty (TOGU) poses extreme difficulty to the already
challenging topology optimization problems. This paper aims to solve this cutting-edge problem by integrating the latest developments
in level set methods, design under uncertainty, and a newly developed mathematical framework for solving variational problems
and partial differential equations that define mappings between different manifolds. There are several contributions of this
work. First, geometric uncertainty is quantitatively modeled by combing level set equation with a random normal boundary velocity
field characterized with a reduced set of random variables using the Karhunen–Loeve expansion. Multivariate Gauss quadrature
is employed to propagate the geometric uncertainty, which also facilitates shape sensitivity analysis by transforming a TOGU
problem into a weighted summation of deterministic topology optimization problems. Second, a PDE-based approach is employed
to overcome the deficiency of conventional level set model which cannot explicitly maintain the point correspondences between
the current and the perturbed boundaries. With the explicit point correspondences, shape sensitivity defined on different
perturbed designs can be mapped back to the current design. The proposed method is demonstrated with a bench mark structural
design. Robust designs achieved with the proposed TOGU method are compared with their deterministic counterparts. |
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