Proportional Topology Optimization algorithm for two-scale concurrent design of lattice structures

Author affiliations

Authors

  • Minh Ngoc Nguyen \(^1\) Duy Tan Research Institute for Computational Engineering (DTRICE) - Duy Tan University, Ho Chi Minh City 700000, Vietnam
    \(^2\) Faculty of Civil Engineering, Duy Tan University, Da Nang 550000, Vietnam     https://orcid.org/0000-0002-2026-2310
  • Duy Vo \(^1\) Duy Tan Research Institute for Computational Engineering (DTRICE) - Duy Tan University, Ho Chi Minh City 700000, Vietnam
    \(^2\) Faculty of Civil Engineering, Duy Tan University, Da Nang 550000, Vietnam     https://orcid.org/0000-0002-3160-2878
  • Tinh Quoc Bui \(^1\) Duy Tan Research Institute for Computational Engineering (DTRICE) - Duy Tan University, Ho Chi Minh City 700000, Vietnam
    \(^2\) Faculty of Civil Engineering, Duy Tan University, Da Nang 550000, Vietnam     https://orcid.org/0000-0002-6426-6639

DOI:

https://doi.org/10.15625/0866-7136/18368

Keywords:

proportional topology optimization, PTO, two-scale, lattice structures

Abstract

In this paper, the Proportional Topology Optimization (PTO) algorithm is extended for the two-scale concurrent topology optimization, in which both the structure and material cellular micro-structure are subject to design. PTO was originally developed on the concept that the amount of material being distributed to an element would be proportional to the contribution of that element in the objective function. Sensitivity analysis is not required. In a two-scale concurrent topology optimization problem, two sets of design variables are defined, one for macro-structure and one for micro-structure. Here, the objective function is reformulated such that the contribution of each micro-scale design variable can be determined, facilitating the employment of PTO. The macroscopic effective elastic tensor is evaluated by the energy-based homogenization method (EBHM), providing a link between micro-structure and macro-structure. Feasibility and efficiency of the proposed PTO approach are demonstrated via several benchmark examples of both two and three dimensional structures.

Downloads

Download data is not yet available.

Metrics

Metrics Loading ...

References

M. P. Bendsøe and N. Kikuchi. Generating optimal topologies in structural design using a homogenization method. Computer Methods in Applied Mechanics and Engineering, 71, (1988), pp. 197–224. DOI: https://doi.org/10.1016/0045-7825(88)90086-2

H. P. Mlejnek. Some aspects of the genesis of structures. Structural Optimization, 5, (1-2), (1992), pp. 64–69. DOI: https://doi.org/10.1007/BF01744697

M. P. Bendsøe and O. Sigmund. Material interpolation schemes in topology optimization. Archive of Applied Mechanics (Ingenieur Archiv), 69, (9-10), (1999), pp. 635–654. DOI: https://doi.org/10.1007/s004190050248

O. Sigmund. A 99 line topology optimization code written in Matlab. Structural and Multidisciplinary Optimization, 21, (2), (2001), pp. 120–127. DOI: https://doi.org/10.1007/s001580050176

E. Andreassen, A. Clausen, M. Schevenels, B. S. Lazarov, and O. Sigmund. Efficient topology optimization in MATLAB using 88 lines of code. Structural and Multidisciplinary Optimization, 43, (1), (2010), pp. 1–16. DOI: https://doi.org/10.1007/s00158-010-0594-7

Y. M. Xie and G. P. Steven. A simple evolutionary procedure for structural optimization. Computers & Structures, 49, (5), (1993), pp. 885–896. DOI: https://doi.org/10.1016/0045-7949(93)90035-C

X. Huang and Y. M. Xie. Bi-directional evolutionary topology optimization of continuum structures with one or multiple materials. Computational Mechanics, 43, (3), (2009), pp. 393–401. DOI: https://doi.org/10.1007/s00466-008-0312-0

M. Y. Wang, X. Wang, and D. Guo. A level set method for structural topology optimization. Computer Methods in Applied Mechanics and Engineering, 192, (1-2), (2003), pp. 227–246. DOI: https://doi.org/10.1016/S0045-7825(02)00559-5

G. Allaire, F. Jouve, and A.-M. Toader. Structural optimization using sensitivity analysis and a level-set method. Journal of Computational Physics, 194, (1), (2004), pp. 363–393. DOI: https://doi.org/10.1016/j.jcp.2003.09.032

A. Takezawa, S. Nishiwaki, and M. Kitamura. Shape and topology optimization based on the phase field method and sensitivity analysis. Journal of Computational Physics, 229, (7), (2010), pp. 2697–2718. DOI: https://doi.org/10.1016/j.jcp.2009.12.017

J. A. Norato, B. K. Bell, and D. A. Tortorelli. A geometry projection method for continuum-based topology optimization with discrete elements. Computer Methods in Applied Mechanics and Engineering, 293, (2015), pp. 306–327. DOI: https://doi.org/10.1016/j.cma.2015.05.005

X. Guo, W. Zhang, and W. Zhong. Doing topology optimization explicitly and geometrically—a new moving morphable components based framework. Journal of Applied Mechanics, 81, (8), (2014). DOI: https://doi.org/10.1115/1.4027609

W. Zhang, D. Li, J. Zhou, Z. Du, B. Li, and X. Guo. A Moving Morphable Void (MMV)-based explicit approach for topology optimization considering stress constraints. Computer Methods in Applied Mechanics and Engineering, 334, (2018), pp. 381–413. DOI: https://doi.org/10.1016/j.cma.2018.01.050

X. Yan, X. Huang, Y. Zha, and Y. M. Xie. Concurrent topology optimization of structures and their composite microstructures. Computers & Structures, 133, (2014), pp. 103–110. DOI: https://doi.org/10.1016/j.compstruc.2013.12.001

X. Gao and H. Ma. A modified model for concurrent topology optimization of structures and materials. Acta Mechanica Sinica, 31, (2015), pp. 890–898. DOI: https://doi.org/10.1007/s10409-015-0502-x

H. Li, Z. Luo, L. Gao, and Q. Qin. Topology optimization for concurrent design of structures with multi-patch microstructures by level sets. Computer Methods in Applied Mechanics and Engineering, 331, (2018), pp. 536–561. DOI: https://doi.org/10.1016/j.cma.2017.11.033

J. Gao, Z. Luo, L. Xia, and L. Gao. Concurrent topology optimization of multiscale composite structures in Matlab. Structural and Multidisciplinary Optimization, 60, (6), (2019), pp. 2621–2651. DOI: https://doi.org/10.1007/s00158-019-02323-6

C. Zhuang, Z. Xiong, and H. Ding. Bézier extraction based isogeometric approach to multiobjective topology optimization of periodic microstructures. International Journal for Numerical Methods in Engineering, 122, (2021), pp. 6827–6866. DOI: https://doi.org/10.1002/nme.6813

J. Guedes and N. Kikuchi. Preprocessing and postprocessing for materials based on the homogenization method with adaptive finite element methods. Computer Methods in Applied Mechanics and Engineering, 83, (2), (1990), pp. 143–198. DOI: https://doi.org/10.1016/0045-7825(90)90148-F

O. Sigmund. Materials with prescribed constitutive parameters: An inverse homogenization problem. International Journal of Solids and Structures, 31, (17), (1994), pp. 2313–2329. DOI: https://doi.org/10.1016/0020-7683(94)90154-6

E. Andreassen and C. S. Andreasen. How to determine composite material properties using numerical homogenization. Computational Materials Science, 83, (2014), pp. 488–495. DOI: https://doi.org/10.1016/j.commatsci.2013.09.006

L. Xia and P. Breitkopf. Design of materials using topology optimization and energy-based homogenization approach in Matlab. Structural and Multidisciplinary Optimization, 52, (6), (2015), pp. 1229–1241. DOI: https://doi.org/10.1007/s00158-015-1294-0

Y. Lu and L. Tong. Concurrent topology optimization of cellular structures and anisotropic materials. Computers & Structures, 255, (2021), p. 106624.

J. Jia, D. Da, C.-L. Loh, H. Zhao, S. Yin, and J. Xu. Multiscale topology optimization for non-uniform microstructures with hybrid cellular automata. Structural and Multidisciplinary Optimization, 62, (2), (2020), pp. 757–770. DOI: https://doi.org/10.1007/s00158-020-02533-3

P. Liu, Z. Kang, and Y. Luo. Two-scale concurrent topology optimization of lattice structures with connectable microstructures. Additive Manufacturing, 36, (2020). DOI: https://doi.org/10.1016/j.addma.2020.101427

X. Gu, S. He, Y. Dong, and T. Song. An improved ordered SIMP approach for multiscale concurrent topology optimization with multiple microstructures. Composite Structures, 287, (2022). DOI: https://doi.org/10.1016/j.compstruct.2022.115363

G.-C. Luh and C.-Y. Lin. Structural topology optimization using ant colony optimization algorithm. Applied Soft Computing, 9, (2009), pp. 1343–1353. DOI: https://doi.org/10.1016/j.asoc.2009.06.001

D. Sharma, K. Deb, and N. N. Kishore. Domain-specific initial population strategy for compliant mechanisms using customized genetic algorithm. Structural and Multidisciplinary Optimization, 43, (2010), pp. 541–554. DOI: https://doi.org/10.1007/s00158-010-0575-x

G.-C. Luh, C.-Y. Lin, and Y.-S. Lin. A binary particle swarm optimization for continuum structural topology optimization. Applied Soft Computing, 11, (2011), pp. 2833–2844. DOI: https://doi.org/10.1016/j.asoc.2010.11.013

O. Sigmund. On the usefulness of non-gradient approaches in topology optimization. Structural and Multidisciplinary Optimization, 43, (2011), pp. 589–596. DOI: https://doi.org/10.1007/s00158-011-0638-7

D. Guirguis and M. F. Aly. A derivative-free level-set method for topology optimization. Finite Elements in Analysis and Design, 120, (2016), pp. 41–56. DOI: https://doi.org/10.1016/j.finel.2016.06.002

D. Guirguis, W. W. Melek, and M. F. Aly. High-resolution non-gradient topology optimization. Journal of Computational Physics, 372, (2018), pp. 107–125. DOI: https://doi.org/10.1016/j.jcp.2018.06.025

B. Bochenek and K. Tajs-Zielińska. GOTICA - generation of optimal topologies by irregular

cellular automata. Structural and Multidisciplinary Optimization, 55, (2017), pp. 1989–2001. DOI: https://doi.org/10.1007/s00158-016-1614-z

D. C. Da, J. H. Chen, X. Y. Cui, and G. Y. Li. Design of materials using hybrid cellular automata. Structural and Multidisciplinary Optimization, 56, (1), (2017), pp. 131–137. DOI: https://doi.org/10.1007/s00158-017-1652-1

E. Biyikli and A. C. To. Proportional topology optimization: a new non-sensitivity method for solving stress constrained and minimum compliance problems and its implementation in MATLAB. PLOS ONE, 10, (12), (2015). DOI: https://doi.org/10.1371/journal.pone.0145041

H. Wang, W. Cheng, R. Du, S. Wang, and Y. Wang. Improved proportional topology optimization algorithm for solving minimum compliance problem. Structural and Multidisciplinary Optimization, 62, (2), (2020), pp. 475–493. DOI: https://doi.org/10.1007/s00158-020-02504-8

W. Cheng, H. Wang, M. Zhang, and R. Du. Improved proportional topology optimization algorithm for minimum volume problem with stress constraints. Engineering Computations, 38, (2020), pp. 392–412. DOI: https://doi.org/10.1108/EC-12-2019-0560

Z. Ullah, B. Ullah, W. Khan, and S. ul Islam. Proportional topology optimisation with maximum entropy-based meshless method for minimum compliance and stress constrained problems. Engineering with Computers, 38, (6), (2022), pp. 5541–5561. DOI: https://doi.org/10.1007/s00366-022-01683-w

H. Wang, W. Cheng, M. Zhang, R. Du, and W. Xiang. Non-gradient robust topology optimization method considering loading uncertainty. Arabian Journal for Science and Engineering, 46, (2021), pp. 12599–12611. DOI: https://doi.org/10.1007/s13369-021-06010-x

M. N. Nguyen and T. Q. Bui. Multi-material gradient-free proportional topology optimization analysis for plates with variable thickness. Structural and Multidisciplinary Optimization, 65, (3), (2022). DOI: https://doi.org/10.1007/s00158-022-03176-2

M. N. Nguyen, M. T. Tran, H. Q. Nguyen, and T. Q. Bui. A multi-material Proportional Topology Optimization approach for compliant mechanism problems. European Journal of Mechanics - A/Solids, 100, (2023). DOI: https://doi.org/10.1016/j.euromechsol.2023.104957

R. Tavakoli and S. M. Mohseni. Alternating active-phase algorithm for multimaterial topology optimization problems: a 115-line MATLAB implementation. Structural and Multidisciplinary Optimization, 49, (4), (2014), pp. 621–642. DOI: https://doi.org/10.1007/s00158-013-0999-1

M. N. Nguyen and T. Q. Bui. Concurrent multiscale topology optimization: A hybrid approach. Vietnam Journal of Mechanics, 44, (3), (2022), pp. 266–279. DOI: https://doi.org/10.15625/0866-7136/17331

M. N. Nguyen and T. Q. Bui. A meshfree-based topology optimization approach without calculation of sensitivity. Vietnam Journal of Mechanics, 44, (1), (2022), pp. 45–58. DOI: https://doi.org/10.15625/0866-7136/16679

M. T. Tran, M. N. Nguyen, T. Q. Bui, and H. Q. Nguyen. An enhanced proportional topology optimization with virtual elements: Formulation and numerical implementation. Finite Elements in Analysis and Design, 222, (2023). DOI: https://doi.org/10.1016/j.finel.2023.103958

O. Sigmund. Morphology-based black and white filters for topology optimization. Structural and Multidisciplinary Optimization, 33, (2007), pp. 401–424. DOI: https://doi.org/10.1007/s00158-006-0087-x

J. K. Guest, A. Asadpoure, and S.-H. Ha. Eliminating beta-continuation from Heaviside projection and density filter algorithms. Structural and Multidisciplinary Optimization, 44, (2011), pp. 443–453. DOI: https://doi.org/10.1007/s00158-011-0676-1

L. Li and K. Khandelwal. Volume preserving projection filters and continuation methods in topology optimization. Engineering Structures, 85, (2015), pp. 144–161. DOI: https://doi.org/10.1016/j.engstruct.2014.10.052

Y. Lu and L. Tong. Concurrent topology optimization of cellular structures and anisotropic materials. Computers & Structures, 255, (2021). DOI: https://doi.org/10.1016/j.compstruc.2021.106624

B. Li, Y. Duan, H. Yang, Y. Lou, and W. H. Müller. Isogeometric topology optimization of strain gradient materials. Computer Methods in Applied Mechanics and Engineering, 397, (2022). DOI: https://doi.org/10.1016/j.cma.2022.115135

S. Liu and W. Su. Topology optimization of couple-stress material structures. Structural and Multidisciplinary Optimization, 40, (1-6), (2009), pp. 319–327. DOI: https://doi.org/10.1007/s00158-009-0367-3

N. Gan and Q. Wang. Topology optimization design related to size effect using the modified couple stress theory. Engineering Optimization, 55, (1), (2021), pp. 158–176. DOI: https://doi.org/10.1080/0305215X.2021.1990911

Downloads

Published

30-06-2023

How to Cite

[1]
M. N. Nguyen, D. Vo and T. Q. Bui, Proportional Topology Optimization algorithm for two-scale concurrent design of lattice structures, Vietnam J. Mech. 45 (2023) 164–182. DOI: https://doi.org/10.15625/0866-7136/18368.

Issue

Vol. 45 No. 2 (2023)

Section

Research Article

Categories

License

Creative Commons License

This work is licensed under a Creative Commons Attribution-ShareAlike 4.0 International License.

Crossref
Scopus
Google Scholar
Europe PMC

Most read articles by the same author(s)