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Development of a topology optimization program considering density and homogeni-zation methods

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Abstract

In most previous studies of topology optimization, commercial programs, such as Optistruct, ANSYS, and MSC Patran, usually were used during implementation. Such commercial programs are not easy to use and entail time and cost. In addition, it is difficult to confirm results with reference to individual stages of optimization. For addressing this disadvantage, a topology optimization program, which is based on the C language, is developed in this study. This is a very convenient and powerful program for users to conduct topology optimization by using all density methods and homogenization methods in compliance with the methodology. For verifying the developed program, first of all, topology optimization was implemented by using density methods to evaluate the strain energy density of a cantilever plate and a simply supported plate, which are simple models. The feasibility of the program was verified through a comparison of the results with those from Optistruct, which is a commercial program. Finally, topology optimization was implemented with regard to the rolling-stock leading-cab, which is an application model. Through the Ls-Dyna program, the collision characteristic was also confirmed. Next, through homogenization methods, crash analysis was implemented in the rollingstock leading-cab. By the use of the internal energy density as deduced from the collision interpretation, topology optimization was implemented. The optimal values, by which the internal energy was maximized per unit weight, of the parameters of homogenization methods were deduced. By the use of Ls-Dyna program for the optimum model, where internal energy is maximized per unit weight, the crash characteristic was confirmed on the basis of the optimization result and the feasibility of the result was verified. This methodology deduces the axis compression deformation by implementing the role of a crash initiator during collision. In addition, the economic advantage of light-weight cars also can be deduced.

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Abbreviations

b:

body-force intensity

C:

compliance

e:

element index

E:

Young’s modulus

E0 :

original Young’s modulus

Eijkl :

elastic module

EH ijkl :

homogenized elastic module

tf :

final time-step

T:

boundary traction

Th:

threshold value for eliminate low-density element for threshold algorithm

U:

internal energy

UH(μ):

homogenized internal energy at in-plane stress density μ

VF:

volume fraction that is specified as a percentage of the original volume for resizing algorithm

V:

allowable volume

V(μ):

volume at in-plane stress density μ

z:

displacement field

β1 :

parameter having effect of limiting the changes in the density for resizing algorithm

β2 :

parameter requiring the change in the density for resizing algorithm

Λ:

optimality criterion that is calculated by sum of product of internal energy at each time step and weighting factor

Λave :

average optimality criterion

μ:

in-plane stress density

μ(e):

in-plane stress density of element e

ρ:

bulk material density

Ω:

structural domain

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Authors and Affiliations

  1. Graduate School of Mechanical Engineering, Sungkyunkwan University, 300 Cheoncheon-dong, Suwon, Kyonggi-do, Korea, 440-746

    Hyun-Jun Kim & Bae-Young Kim

  2. School of Mechanical Engineering, Sungkyunkwan University, 300 Cheoncheon-dong, Suwon, Kyonggi-do, Korea, 440-746

    Myung-Won Suh

Authors
  1. Hyun-Jun Kim
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  2. Bae-Young Kim
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Correspondence to Myung-Won Suh.

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Kim, HJ., Kim, BY. & Suh, MW. Development of a topology optimization program considering density and homogeni-zation methods. Int. J. Precis. Eng. Manuf. 12, 303–312 (2011). https://doi.org/10.1007/s12541-011-0040-9

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