Robust topology optimization of multi-material lattice structures under material and load uncertainties

doi:10.1007/s11465-019-0531-4

Front. Mech. Eng.

2019, Vol. 14

Issue (2) : 141-152 https://doi.org/10.1007/s11465-019-0531-4

RESEARCH ARTICLE

Robust topology optimization of multi-material lattice structures under material and load uncertainties

Yu-Chin CHAN, Kohei SHINTANI, Wei CHEN(

)

Department of Mechanical Engineering, Northwestern University, Evanston, IL 60208, USA

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Abstract

Enabled by advancements in multi-material additive manufacturing, lightweight lattice structures consisting of networks of periodic unit cells have gained popularity due to their extraordinary performance and wide array of functions. This work proposes a density-based robust topology optimization method for meso- or macro-scale multi-material lattice structures under any combination of material and load uncertainties. The method utilizes a new generalized material interpolation scheme for an arbitrary number of materials, and employs univariate dimension reduction and Gauss-type quadrature to quantify and propagate uncertainty. By formulating the objective function as a weighted sum of the mean and standard deviation of compliance, the tradeoff between optimality and robustness can be studied and controlled. Examples of a cantilever beam lattice structure under various material and load uncertainty cases exhibit the efficiency and flexibility of the approach. The accuracy of univariate dimension reduction is validated by comparing the results to the Monte Carlo approach.

Keywords robust topology optimization lattice structures multi-material material uncertainty load uncertainty univariate dimension reduction

Corresponding Author(s): Wei CHEN

Just Accepted Date: 13 December 2018 Online First Date: 16 January 2019 Issue Date: 22 April 2019

Cite this article:

Yu-Chin CHAN,Kohei SHINTANI,Wei CHEN. Robust topology optimization of multi-material lattice structures under material and load uncertainties[J]. Front. Mech. Eng., 2019, 14(2): 141-152.

URL:

https://academic.hep.com.cn/fme/EN/10.1007/s11465-019-0531-4
https://academic.hep.com.cn/fme/EN/Y2019/V14/I2/141

Density variables	Young’s modulus
$ρ e ? 1 = 0$ , $ρ e ? 2 = 0$	$E 0$
$ρ e ? 1 = 1$ , $ρ e ? 2 = 0$	$E 1$
$ρ e ? 1 = 1$ , $ρ e ? 2 = 1$	$E 2$

Tab.1 Expansion of the proposed interpolation scheme for three phases (M = 2)

Fig.1 (a) Boundary conditions; (b) initial lattice unit cells

Tab.2 Material properties of the 4-phase examples

Fig.2 Optimization history of the objective function under material uncertainty when b = 10

Fig.3 Pareto curve under material uncertainty with different objective function weights b

Fig.4 Optimal solutions under material uncertainty when (a) b = 0 (deterministic) and (b) b = 10 (robust)

Tab.3 Mean and standard deviation of the compliance (N?mm) results under material uncertainty using UDR with 8 quadrature nodes and Monte Carlo with 10000 samples

Fig.5 Boundary conditions with the random external load, which varies in both magnitude (f) and angle (q)

Fig.6 Optimization history for the objective function under load uncertainty when b = 10

Fig.7 Pareto curve for different objective function weights under load uncertainty (a) including b = 0 (deterministic), and (b) showing robust solutions only (b = 1 to b = 10)

Fig.8 Optimal solution under load uncertainty when b = 0 (deterministic)

Fig.9 Optimal solutions under load uncertainty when (a) b = 1 and (b) b = 10

Tab.4 Compliance (N?mm) results under load uncertainty using UDR with 8 quadrature nodes and Monte Carlo with 10000 samples

Fig.10 Optimization history for the objective function under both material and load uncertainty when b = 10

Fig.11 Pareto curve for different objective function weights under both material and load uncertainty (a) including b = 0 (deterministic), and (b) showing robust solutions only (b = 1 to b = 10)

Fig.12 Optimal solution under both material and load uncertainty when b = 0 (deterministic)

Fig.13 Optimal solutions under both material and load uncertainty when (a) b = 1 and (b) b = 10

Tab.5 Compliance (N?mm) results under material and load uncertainties using UDR with 8 quadrature nodes and Monte Carlo with 10000 samples

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