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A MATLAB code for the material-field series-expansion topology optimization method

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Abstract

This paper presents a MATLAB implementation of the material-field series-expansion (MFSE) topology optimization method. The MFSE method uses a bounded material field with specified spatial correlation to represent the structural topology. With the series-expansion method for bounded fields, this material field is described with the characteristic base functions and the corresponding coefficients. Compared with the conventional density-based method, the MFSE method decouples the topological description and the finite element discretization, and greatly reduces the number of design variables after dimensionality reduction. Other features of this method include inherent control on structural topological complexity, crisp structural boundary description, mesh independence, and being free from the checkerboard pattern. With the focus on the implementation of the MFSE method, the present MATLAB code uses the maximum stiffness optimization problems solved with a gradient-based optimizer as examples. The MATLAB code consists of three parts, namely, the main program and two subroutines (one for aggregating the optimization constraints and the other about the method of moving asymptotes optimizer). The implementation of the code and its extensions to topology optimization problems with multiple load cases and passive elements are discussed in detail. The code is intended for researchers who are interested in this method and want to get started with it quickly. It can also be used as a basis for handling complex engineering optimization problems by combining the MFSE topology optimization method with non-gradient optimization algorithms without sensitivity information because only a few design variables are required to describe relatively complex structural topology and smooth structural boundaries using the MFSE method.

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Abbreviations

LOBPCG:

Locally optimal block preconditioned conjugate gradient

MBB:

Messerschmitt-Bölkow-Blohm

MFSE:

Material-field series-expansion

MMA:

Method of moving asymptotes

c(x a, x b):

Correlation function that describes the correlation between φ(xa) and φ(xb)

C :

Correlation matrix

E(x):

Interpolated material’s Young’s modulus

l c :

User-specified correlation length

J :

Objective function

K :

System stiffness matrix

N pt :

Number of the material-field points

\(\overline V \) :

Specified material usage

V solid :

Volume of the material

x :

Spatial coordinate of a point within the structural design domain

ε :

Truncation error in the material-field series expansion

β :

Parameter that controls the steepness of the projection function

η :

Vector form of the design variables

η k :

Material-field series-expansion coefficients

φ(x):

Bounded material field

\(\tilde \varphi ({\boldsymbol{x}})\) :

Projected material field

λ k :

Eigenvalues of the correlation matrix

ψ k :

Eigenvectors of the correlation matrix

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Acknowledgements

The authors acknowledge the support of the National Key R&D Program of China (Grant No. 2017YFB0203604), the National Natural Science Foundation of China (Grant Nos. 11902064 and 11772077), and the Liaoning Revitalization Talents Program, China (Grant No. XLYC1807187). The authors are grateful to members of the iDEAS group for their testing of the code.

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Authors and Affiliations

  1. State Key Laboratory of Structural Analysis for Industrial Equipment, Dalian University of Technology, Dalian, 116024, China

    Pai Liu, Xiaopeng Zhang & Yangjun Luo

  2. School of Aeronautics and Astronautics, Dalian University of Technology, Dalian, 116024, China

    Yi Yan & Yangjun Luo

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  1. Pai Liu
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  2. Yi Yan
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  3. Xiaopeng Zhang
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  4. Yangjun Luo
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Correspondence to Yangjun Luo.

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Liu, P., Yan, Y., Zhang, X. et al. A MATLAB code for the material-field series-expansion topology optimization method. Front. Mech. Eng. 16, 607–622 (2021). https://doi.org/10.1007/s11465-021-0637-3

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  • DOI: https://doi.org/10.1007/s11465-021-0637-3

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