# Multi-material topology optimization for practical lightweight design

## Abstract

Topology optimization is one of the most effective tools for conducting lightweight design and has been implemented across multiple industries to enhance product development. The typical topology optimization problem statement is to minimize system compliance while constraining the design space to an assumed volume fraction. The traditional single-material compliance problem has been extended to include multiple materials, which allows increased design freedom for potentially better solutions. However, compliance minimization has the limitations for practical lightweight design because compliance lacks useful physical meanings and has never been a design criterion in industry. Additionally, the traditional compliance minimization problem statement requires volume fraction constraints to be selected a priori; however, designers do not know the optimized balance among materials. In this paper, a more practical method of multi-material topology optimization is presented to overcome the limitations. This method seeks the optimized balance among materials by minimizing the total weight while satisfying performance constraints. This paper also compares the weight minimization approach to compliance minimization. Several numerical examples prove the success of weight minimization and demonstrate its benefit over compliance minimization.

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## Abbreviations

ρ j::

one design variable (i.e. nominal density) relevant to the j-th material in any element;

$${\rho }_{e}^{j}$$::

one design variable (i.e. nominal density) relevant to the j-th material in the e-th element;

ρ j::

all design variables (i.e. a vector of nominal densities) relevant to the j-th material;

$$(\rho ^{j})^{p}, ({\rho }_{e}^{j})^{p}$$::

powers of ρj and $${\rho }_{e}^{j}$$ with the component p;

E j,E (j)::

original elastic modulus of the j-th material;

W j,W (j)::

original weight of any element filled with the j-th material;

$${W_{e}^{j}}, W_{e}^{(j)}$$::

original weight of the e-th element filled with the j-th material;

E (1,⋯ ,j),W (1,⋯ ,j)::

interpolated elastic modulus and weight of the materials from the first to the j-th materials for any element.

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## Acknowledgements

This research was funded by Automotive Partnership Canada and General Motors of Canada. Technical advice and direction were gratefully received from Joe Moore, Balbir Sangha, Manish Pamwar, Derrick Chow, and Chandan Mozumder, at General Motors.

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Correspondence to Il Yong Kim.

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Li, D., Kim, I.Y. Multi-material topology optimization for practical lightweight design. Struct Multidisc Optim 58, 1081–1094 (2018). https://doi.org/10.1007/s00158-018-1953-z

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• DOI: https://doi.org/10.1007/s00158-018-1953-z

### Keywords

• Topology optimization
• Multi-material
• Lightweight
• Weight minimization
• Stiffness constraints
• SIMP