# A new level-set based approach to shape and topology optimization under geometric uncertainty

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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.

## Keywords

Topology optimization Geometric uncertainty Level set method Shape optimization## Nomenclature

*a*(*x*,*ω*)A continuous random field

- \(\overline{a} (x)\)
The mean function of

*a*(*x*,*ω*)*a*_{i}(*x*)The

*i*-th eigenfunction*D*_{Ω}( ∙ )The operator of shape derivative

*J*(Ω(**z**),**u**)Performance functional

*l*_{i·j}The

*j*-th node of the*i*-th variable- \(\emph{w}_{i\cdot j}\)
The

*j*-th weight of the*i*-th variable**M**The number of random variables

**n**Normal vector

*p*(**z**)Joint probability density function of the random variables

**z****u**State/displacement vector

**u**(**X**,*t*) or**U**(**x**,*t*)Displacement vector connecting the positions of a particle in the undeformed configuration to its counter point in the deformed configuration to its counter point in the deformed configuration

- \(\nabla_{\rm X}\)
**u** The material displacement gradient tensor

- \(\nabla_{\rm x}\)
**U** The spatial displacement gradient tensor

**R**Rotation tensor

*t*Time

**U**The right stretch tensor

**V**The left stretch tensor

**V**(**X**)Velocity at a point of the boundary

*V*_{n}(**X**)Normal velocity

*V*_{τ}(**X**)Tangential velocity

**X**Material coordinate

**z**Random variables in the system

*μ*The mean of a response/random variable

*λ*_{i}The ith eigenvalue

*ξ*_{i}(*ω*)Independent random variables with zero mean and unit variance

*ϕ*(*x*)Level set function

- \(\boldsymbol{\Psi} \)(
**x**,*t*) A mapping between the initial and deformed domains

*τ*The pseudo time

*σ*Standard deviation of a response/random variable

- Ω
Geometry

- Ω(
**z**) or Ω_{z} Random geometry

## Notes

### Acknowledgements

We would like to thank the anonymous reviewers for their helps in greatly improving the quality of this manuscript.

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