Abstract
This paper presents a new strategy to distribute two different materials for multi-material topology optimization. Extended from the bi-directional evolutionary structural optimization (BESO) method for a single material, the multi-material bidirectional evolutionary structural optimization (MBESO) method has been developed, which can effectively handle the topology optimization problems involving two materials like steel and concrete. However, in some special cases, overloading of part of the compressed material occurs in the multi-material structures designed by using the MBESO method. Aimed to solve this critical problem, a simple but effective strategy is proposed in this paper. In steel-concrete composite structures, for instance, the overloaded compressed concrete elements with exceedingly high stress are replaced with steel material. The small amount of steel material added to the highly compressed region can effectively reduce the maximum compressive stress of the concrete material to a safe level. The comparison between the original MBESO method and the improved strategy based on a series of two-dimensional and three-dimensional examples clearly demonstrates the effectiveness of the proposed strategy in enhancing the structural safety and strength of the topologically optimized composite structures. This distinctly different material distribution strategy shows its potential and value in multi-material topology optimization research and applications.
摘要
本文介绍了一种用于多材料拓扑优化中分配两种不同材料的新策略. 基于单一材料的双向渐进结构优化(BESO)法开发的多材料双向渐进结构优化(MBESO)法能够有效的处理诸如钢-混凝土这种双材料结构的拓扑优化问题. 然而, 在一些特殊情况下, 使用MBESO方法设计的多材料结构中, 部分受压材料可能会出现受压超限导致结构不安全的情况. 为了解决这种关键问题, 本文提出了一种简单而有效的材料分布策略. 例如在钢-混凝土结构中, 部分压应力过大的混凝土材料可以用钢材替代. 在压力最大的局部区域布置少量的钢材可以有效地将混凝土材料的最大压应力降至安全水平. 本文通过一系列2D和3D案例的对比研究表明, 相比于原始的MBESO方法, 改进的双材料分布策略可以有效的解决双材料拓扑优化结构中局部压应力过大的情况, 证明了该方法在增强拓扑优化复合结构的安全性方面的有效性, 显示出该材料分配策略在多材料拓扑优化研究和应用中的巨大潜力和价值.
Article PDF
Avoid common mistakes on your manuscript.
References
M. P. Bendsøe, and N. Kikuchi, Generating optimal topologies in structural design using a homogenization method, Comput. Methods Appl. Mech. Eng. 71, 197 (1988).
M. P. Bendsøe, Optimal shape design as a material distribution problem, Struct. Optim. 1, 193 (1989).
A. Rietz, Sufficiency of a finite exponent in SIMP (power law) methods, Struct. Multidisc. Optim. 21, 159 (2001).
M. P. Bendsøe, and O. Sigmund, Material interpolation schemes in topology optimization, Arch. Appl. Mech. 69, 635 (1999).
E. Andreassen, A. Clausen, M. Schevenels, B. S. Lazarov, and O. Sigmund, Efficient topology optimization in MATLAB using 88 lines of code, Struct. Multidisc. Optim. 43, 1 (2011).
M. Y. Wang, X. Wang, and D. Guo, A level set method for structural topology optimization, Comput. Methods Appl. Mech. Eng. 192, 227 (2003).
Y. Mei, and X. Wang, A level set method for structural topology optimization and its applications, Adv. Eng. Software 35, 415 (2004).
G. Allaire, F. Jouve, and A. M. Toader, Structural optimization using sensitivity analysis and a level-set method, J. Comput. Phys. 194, 363 (2004).
Z. Luo, M. Y. Wang, S. Wang, and P. Wei, A level set-based parameterization method for structural shape and topology optimization, Int. J. Numer. Meth. Eng. 76, 1 (2008).
P. Wei, Y. Yang, S. Chen, and M. Y. Wang, A study on basis functions of the parameterized level set method for topology optimization of continuums, J. Mech. Des. 143, 041701 (2021).
L. Chen, J. Wan, X. Chu, and H. Liu, Parameterized level set method for structural topology optimization based on the Cosserat elasticity, Acta Mech. Sin. 37, 620 (2021).
N. Wei, H. Ye, X. Zhang, W. Wang, and Y. Sui, Lightweight topology optimization of graded lattice structures with displacement constraints based on an independent continuous mapping method, Acta Mech. Sin. 38, 421352 (2022).
J. Z. Du, F. W. Meng, Y. H. Guo, and Y. K. Sui, Fail-safe topology optimization of continuum structures with fundamental frequency constraints based on the ICM method, Acta Mech. Sin. 36, 1065 (2020).
Y. M. Xie, and G. P. Steven, A simple evolutionary procedure for structural optimization, Comput. Struct. 49, 885 (1993).
X. Huang, and Y. M. Xie, Evolutionary Topology Optimization of Continuum Structures (Wiley, 2010).
D. Da, L. Xia, G. Li, and X. Huang, Evolutionary topology optimization of continuum structures with smooth boundary representation, Struct. Multidisc. Optim. 57, 2143 (2018).
X. Y. Yang, O. M. Querin, G. P. Steven, and Y. M. Xei, Bidirectional evolutionary method for stiffness optimization, AIAA J. 37, 1483 (1999).
Y. Tang, A. Kurtz, and Y. F. Zhao, Bidirectional evolutionary structural optimization (BESO) based design method for lattice structure to be fabricated by additive manufacturing, Comput.-Aided Des. 69, 91 (2015).
T. Liu, L. Ding, F. Meng, X. Li, and Y. Zheng, Stability analysis of anti-dip bedding rock slopes using a limit equilibrium model combined with bi-directional evolutionary structural optimization (BESO) method, Comput. Geotech. 134, 104116 (2021).
Y. Wu, W. Qiu, L. Xia, W. Li, and K. Feng, Design of an aircraft engine bracket using stress-constrained bi-directional evolutionary structural optimization method, Struct. Multidisc. Optim. 64, 4147 (2021).
L. He, and M. Gilbert, Rationalization of trusses generated via layout optimization, Struct. Multidisc. Optim. 52, 677 (2015).
D. Li, and I. Y. Kim, Multi-material topology optimization for practical lightweight design, Struct. Multidisc. Optim. 58, 1081 (2018).
A. T. Gaynor, J. K. Guest, and C. D. Moen, Reinforced concrete force visualization and design using bilinear truss-continuum topology optimization, J. Struct. Eng. 139, 607 (2013).
Y. Yang, C. D. Moen, and J. K. Guest, Three-dimensional force flow paths and reinforcement design in concrete via stress-dependent truss-continuum topology optimization, J. Eng. Mech. 141, 04014106 (2015).
P. Liu, Y. Luo, and Z. Kang, Multi-material topology optimization considering interface behavior via XFEM and level set method, Comput. Methods Appl. Mech. Eng. 308, 113 (2016).
P. Liu, L. Shi, and Z. Kang, Multi-material structural topology optimization considering material interfacial stress constraints, Comput. Methods Appl. Mech. Eng. 363, 112887 (2020).
Y. Han, B. Xu, Z. Duan, and X. Huang, Stress-based multi-material structural topology optimization considering graded interfaces, Comput. Methods Appl. Mech. Eng. 391, 114602 (2022).
V. S. Almeida, H. L. Simonetti, and L. O. Neto, Comparative analysis of strut-and-tie models using smooth evolutionary structural optimization, Eng. Struct. 56, 1665 (2013).
H. G. Kwak, and S. H. Noh, Determination of strut-and-tie models using evolutionary structural optimization, Eng. Struct. 28, 1440 (2006).
Y. Liu, J. L. Jewett, and J. V. Carstensen, Experimental investigation of topology-optimized deep reinforced concrete beams with reduced concrete volume, in: Second RILEM International Conference on Concrete and Digital Fabrication, edited by F. Bos, S. Lucas, R. Wolfs, and T. Salet, RILEM Book series, 28, 601 (2020).
Y. Li, and Y. M. Xie, Evolutionary topology optimization for structures made of multiple materials with different properties in tension and compression, Compos. Struct. 259, 113497 (2021).
Y. Li, and Y. M Xie, Evolutionary topology optimization of spatial steel-concrete structures, J. Int. Assoc. Shell Spat. Struct. 62, 102 (2021).
Y. Li, Y. Lai, G. Lu, F. Yan, P. Wei, and Y. M. Xie, Innovative design of long-span steel-concrete composite bridge using multi-material topology optimization, Eng. Struct. 269, 114838 (2022).
C. Le, J. Norato, T. Bruns, C. Ha, and D. Tortorelli, Stress-based topology optimization for continua, Struct. Multidisc. Optim. 41, 605 (2010).
J. Kirby, S. Zhou, and Y. M. Xie, Optimal fail-safe truss structures: New solutions and uncommon characteristics, Acta Mech. Sin. 38, 421564 (2022).
Q. Li, G. P. Steven, and Y. M. Xie, On equivalence between stress criterion and stiffness criterion in evolutionary structural optimization, Struct. Optim. 18, 67 (1999).
O. Sigmund, and J. Petersson, Numerical instabilities in topology optimization: A survey on procedures dealing with checkerboards, mesh-dependencies and local minima, Struct. Optim. 16, 68 (1998).
K. Ravi-Chandar, 2.05 - Dynamic fracture, Compr. Struct. Integr. 2, 285 (2007).
B. S. Aadnøy, and R. Looyeh, Failure criteria, Pet. Rock Mech. 53 (2019).
V. N. Bastun, M. I. Kolyakov, and Y. N. Semko, Strength criterion for materials with different strengths in tension and compression, Strength Mater. 28, 353 (1996).
Acknowledgements
This work was supported by the Australian Research Council (Grant No. FL190100014), the Shanghai Municipal Science and Technology Major Project (Grant No. 2021SHZDZX0100), the National Key Research and Development Program “Inter-governmental Cooperation in International Science and Technology Innovation” (Grant No. 2022YFE0141400), and the National Natural Science Foundation of China (Grant No. U1913603).
Author information
Authors and Affiliations
Contributions
Yu Li conducted the research, investigated the methodology, validated and visualized the results, and wrote the original draft. Philip F. Yuan supervised the research. Yi Min Xie supervised the research, helped develop the concept and methodology, reviewed and edited the manuscript.
Corresponding author
Rights and permissions
Open Access This article is licensed under a Creative Commons Attribution 4.0 International License, which permits use, sharing, adaptation, distribution, and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons licence, and indicate if changes were made. The images or other third party material in this article are included in the article’s Creative Commons licence, unless indicated otherwise in a credit line to the material. If material is not included in the article’s Creative Commons licence and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder. To view a copy of this licence, visit http://creativecommons.org/licenses/by/4.0/.
About this article
Cite this article
Li, Y., Yuan, P.F. & Xie, Y.M. A strategy for improving the safety and strength of topologically optimized multi-material structures. Acta Mech. Sin. 39, 422134 (2023). https://doi.org/10.1007/s10409-023-22134-x
Received:
Accepted:
Published:
DOI: https://doi.org/10.1007/s10409-023-22134-x