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A discrete material optimization method with a patch strategy based on stiffness matrix interpolation

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Abstract

A discrete material optimization method with a patch strategy based on the stiffness matrix interpolation is proposed, and a comprehensive technical process of patch discrete material optimization using existing finite element software was developed. This method employs the stiffness matrix instead of the constitutive matrix for material interpolation, which facilitates the optimization process integrated with the existing finite element software. The element stiffness matrix can be derived directly from the finite element analysis, which can not only ensure the correctness of the data, but also reduce the programming work of solving the numerical integration of the composite constitutive matrix. The mathematical model of the patch discrete material optimization is established, which takes the artificial density as the design variable, the minimum compliance as the objective function, and the sequential quadratic programming (SQP) algorithm as the optimization solver. Numerical examples show that by seeking a balance between the number of regions and practical production, the performance of the composite could be further improved using the discrete material optimization method with a patch strategy. Besides, the convergence rate of the optimization is increased by introducing the sum constraints of the design variables and the value functions. The effectiveness and the feasibility of the method were verified.

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Abbreviations

D :

Elastic matrix of the element

K :

Overall stiffness matrix of the structure

B :

Strain matrix

α :

Penalty parameter

X :

Design variable array

C :

Compliance of the structure

X i,j :

Design variables of material j at region i

ω i,j :

Weight function concerning the ith candidate material

f(x):

Objective function

h(x), g(x):

Constraint functions

∇():

Gradient vector of each function

W :

Hessian matrix of the Lagrange function

ϕ(x):

1-norm value function

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Acknowledgments

This work was supported by National Natural Science Foundation of China (Grant no. 51865041 and 52165035), the Natural Science Foundation of Inner Mongolia Autonomous Region of China (Grant no. 2020MS05022 and 2019MS05070) and the State Key Laboratory of Fluid Power and Mechatronic Systems (Grant no. SKLoFP_ZZ_2102).

Author information

Authors and Affiliations

  1. School of Mechanical Engineering, Inner Mongolia University of Technology, Hohhot, 010051, China

    Pengwen Sun, Jie Zhang & Lanting Zhang

  2. Inner Mongolia Key Laboratory of Advanced Manufacturing Technology, Hohhot, 010051, China

    Pengwen Sun & Lanting Zhang

  3. Department of Engineering Mechanics, Dalian University of Technology, Dalian, 116024, China

    Penghui Wu

  4. The Fourth Branch, Inner Mongolia First Machinery Group Co., Ltd., Baotou, 014030, China

    Jiandong Li

  5. State Key Laboratory of Fluid Power and Mechatronic Systems, Zhejiang University, Hangzhou, 310027, China

    Weifei Hu

Authors
  1. Pengwen Sun
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  2. Jie Zhang
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  3. Penghui Wu
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  4. Jiandong Li
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  5. Lanting Zhang
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Corresponding author

Correspondence to Pengwen Sun.

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Pengwen Sun is a Professor of Mechanical Engineering, Inner Mongolia University of Technology, Hohhot, P. R. China. He received his Ph.D. in Power Machinery and Engineering from Beijing Institute of Technology. His research interests include structure design and optimization, wind turbine blade.

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Sun, P., Zhang, J., Wu, P. et al. A discrete material optimization method with a patch strategy based on stiffness matrix interpolation. J Mech Sci Technol 36, 797–807 (2022). https://doi.org/10.1007/s12206-022-0127-5

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  • DOI: https://doi.org/10.1007/s12206-022-0127-5

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