Bidirectional Evolutionary Topology Optimization for Stress Minimization Based on the Modified Couple Stress Elasticity

ZHANG Manzhe; GU Shuitao; FENG Zhiqiang

doi:10.21656/1000-0887.450038

Volume 46 Issue 1

Jan. 2025

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ZHANG Manzhe, GU Shuitao, FENG Zhiqiang. Bidirectional Evolutionary Topology Optimization for Stress Minimization Based on the Modified Couple Stress Elasticity[J]. Applied Mathematics and Mechanics, 2025, 46(1): 12-28. doi: 10.21656/1000-0887.450038

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Bidirectional Evolutionary Topology Optimization for Stress Minimization Based on the Modified Couple Stress Elasticity

doi: 10.21656/1000-0887.450038

1.
School of Civil Engineering, Chongqing University, Chongqing 400040, P.R.China
2.
School of Mechanics and Engineering, Southwest Jiaotong University, Chengdu 610031, P.R.China

Received Date: 2024-02-22
Rev Recd Date: 2024-03-17
Publish Date: 2025-01-01

Abstract

Abstract

The application of stress-based bidirectional evolutionary structural optimization (BESO) in the context of the modified couple stress elasticity theory was investigated. This methodology allows for structure optimization of homogenized continuum with a microstructural composition of size effects. The classical BESO technique was extended through the introduction of a novel formulation of couple stress based non-classical equivalent stress, and the minimization design was conducted under the constraint of volume criterion. The iterative update of design variables relies on the sensitivity analysis involving direct derivation of the enriched p-norm global stress with couple stress contributions. Since the high-order elasticity is involved, the FEM implementation requires at least the C¹ nodal continuity. Thus, a Lagrangian finite element complemented by additional integration functions was implemented. The method was validated with 3 distinct cases through investigation of the size effects on the stress optimization and the subsequent structure design. The impacts of other parameters including the norm p value and the material volume fraction, were explored. The results demonstrate the potential of the proposed stress-based BESO method in addressing structural optimization of problems involving size effects.
- topology optimization,
- BESO,
- stress-based,
- modified couple stress,
- size effect

(Contributed by FENG Zhiqiang, M.AMM Editorial Board)

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References(42)

References

[1]	BENDSØE M P, KIKUCHI N. Generating optimal topologies in structural design using a homogenization method[J]. Computer Methods in Applied Mechanics and Engineering, 1988, 71 (2): 197-224. doi: 10.1016/0045-7825(88)90086-2
[2]	BENDSØE M P. Optimal shape design as a material distribution problem[J]. Structural Optimization, 1989, 1 (4): 193-202. doi: 10.1007/BF01650949
[3]	SIGMUND O. On the design of compliant mechanisms using topology optimization[J]. Mechanics of Structures and Machines, 1997, 25 (4): 493-524. doi: 10.1080/08905459708945415
[4]	ROZVANY G I N. Topology optimization in structural mechanics[J]. Structural and Multidisciplinary Optimization, 2001, 21 (2): 89. doi: 10.1007/s001580050173
[5]	ROZVANY G I N. Aims, scope, methods, history and unified terminology of computer-aided topology optimization in structural mechanics[J]. Structural and Multidisciplinary Optimization, 2001, 21 (2): 90-108. doi: 10.1007/s001580050174
[6]	WALLIN M, TORTORELLI D A. Nonlinear homogenization for topology optimization[J]. Mechanics of Materials, 2020, 145 : 103324. doi: 10.1016/j.mechmat.2020.103324
[7]	DBOUK T. A review about the engineering design of optimal heat transfer systems using topology optimization[J]. Applied Thermal Engineering, 2017, 112 : 841-854. doi: 10.1016/j.applthermaleng.2016.10.134
[8]	BRUNS T E. Topology optimization of convection-dominated, steady-state heat transfer problems[J]. International Journal of Heat and Mass Transfer, 2007, 50 (15/16): 2859-2873.
[9]	DVHRING M B, JENSEN J S, SIGMUND O. Acoustic design by topology optimization[J]. Journal of Sound and Vibration, 2008, 317 (3/5): 557-575.
[10]	WADBRO E, BERGGREN M. Topology optimization of an acoustic horn[J]. Computer Methods in Applied Mechanics and Engineering, 2006, 196 (1/3): 420-436.
[11]	DESAI J, FAURE A, MICHAILIDIS G, et al. Topology optimization in acoustics and elasto-acoustics via a level-set method[J]. Journal of Sound and Vibration, 2018, 420 : 73-103. doi: 10.1016/j.jsv.2018.01.032
[12]	MINIACI M, KRUSHYNSKA A, GLIOZZI A S, et al. Design and fabrication of bioinspired hierarchical dissipative elastic metamaterials[J]. Physical Review Applied, 2018, 10 (2): 024012. doi: 10.1103/PhysRevApplied.10.024012
[13]	MAZZOTTI M, FOEHR A, BILAL O R, et al. Bio-inspired non self-similar hierarchical elastic metamaterials[J]. International Journal of Mechanical Sciences, 2023, 241 : 107915. doi: 10.1016/j.ijmecsci.2022.107915
[14]	ERINGEN A. Microcontinuum Field Theories, : Foundations and Solids[M]. New York : Springer, 2012.
[15]	COSSERAT E, COSSERAT F. Theorie des corps d'edormables[Z]. Cornell University Library Historical Math Monographs, 1909.
[16]	REDA H, ALAVI S E, NASIMSOBHAN M, et al. Homogenization towards chiral Cosserat continua and applications to enhanced Timoshenko beam theories[J]. Mechanics of Materials, 2021, 155 : 103728. doi: 10.1016/j.mechmat.2020.103728
[17]	ERINGEN A C, SUHUBI E S. Nonlinear theory of simple micro-elastic solids, Ⅰ[J]. International Journal of Engineering Science, 1964, 2 (2): 189-203. doi: 10.1016/0020-7225(64)90004-7
[18]	MINDLIN R D, TIERSTEN H F. Effects of couple-stresses in linear elasticity[J]. Archive for Rational Mechanics and Analysis, 1962, 11 (1): 415-448. doi: 10.1007/BF00253946
[19]	TOUPIN R A. Elastic materials with couple-stresses[J]. Archive for Rational Mechanics and Analysis, 1962, 11 (1): 385-414. doi: 10.1007/BF00253945
[20]	LAI P, CONG Y, GU S, et al. Size-dependent parametrisation of active vibration control for periodic piezoelectric microplate coupled systems: a couple stress-based isogeometric approach[J]. Mechanics of Materials, 2023, 186 : 104788. doi: 10.1016/j.mechmat.2023.104788
[21]	YANG F, CHONG A C M, LAM D C C, et al. Couple stress based strain gradient theory for elasticity[J]. International Journal of Solids and Structures, 2002, 39 (10): 2731-2743. doi: 10.1016/S0020-7683(02)00152-X
[22]	ROVATI M, VEBER D. Optimal topologies for micropolar solids[J]. Structural and Multidisciplinary Optimization, 2007, 33 (1): 47-59.
[23]	LIU S, SU W. Topology optimization of couple-stress material structures[J]. Structural and Multidisciplinary Optimization, 2010, 40 (1): 319-327.
[24]	SU W, LIU S. Topology design for maximization of fundamental frequency of couple-stress continuum[J]. Structural and Multidisciplinary Optimization, 2016, 53 (3): 395-408. doi: 10.1007/s00158-015-1316-y
[25]	GANGHOFFER J F, GODA I, NOVOTNY A A, et al. Homogenized couple stress model of optimal auxetic microstructures computed by topology optimization[J]. ZAMM-Journal of Applied Mathematics and Mechanics, 2018, 98 (5): 696-717. doi: 10.1002/zamm.201700154
[26]	CHEN W, HUANG X. Topological design of 3D chiral metamaterials based on couple-stress homogenization[J]. Journal of the Mechanics and Physics of Solids, 2019, 131 : 372-386. doi: 10.1016/j.jmps.2019.07.014
[27]	YANG R J, CHEN C J. Stress-based topology optimization[J]. Structural Optimization, 1996, 12 (2): 98-105.
[28]	ROZVANY G I N. On design-dependent constraints and singular topologies[J]. Structural and Multidisciplinary Optimization, 2001, 21 (2): 164-172. doi: 10.1007/s001580050181
[29]	DUYSINX P, SIGMUND O. New developments in handling stress constraints in optimal material distribution[C]// 7 th AIAA/USAF/NASA/ISSMO Symposium on Multidisciplinary Analysis and Optimization. St Louis, MO, USA, 1998. DOI: 10.2514/6.1998-4906.
[30]	BRUGGI M. On an alternative approach to stress constraints relaxation in topology optimization[J]. Structural and Multidisciplinary Optimization, 2008, 36 (2): 125-141. doi: 10.1007/s00158-007-0203-6
[31]	LUO Y, WANG M Y, KANG Z. An enhanced aggregation method for topology optimization with local stress constraints[J]. Computer Methods in Applied Mechanics and Engineering, 2013, 254 : 31-41. doi: 10.1016/j.cma.2012.10.019
[32]	PICELLI R, TOWNSEND S, BRAMPTON C, et al. Stress-based shape and topology optimization with the level set method[J]. Computer Methods in Applied Mechanics and Engineering, 2018, 329 : 1-23. doi: 10.1016/j.cma.2017.09.001
[33]	HUANG X, XIE Y M. Convergent and mesh-independent solutions for the bi-directional evolutionary structural optimization method[J]. Finite Elements in Analysis and Design, 2007, 43 (14): 1039-1049. doi: 10.1016/j.finel.2007.06.006
[34]	XIA L, ZHANG L, XIA Q, et al. Stress-based topology optimization using bi-directional evolutionary structural optimization method[J]. Computer Methods in Applied Mechanics and Engineering, 2018, 333 : 356-370. doi: 10.1016/j.cma.2018.01.035
[35]	FAN Z, XIA L, LAI W, et al. Evolutionary topology optimization of continuum structures with stress constraints[J]. Structural and Multidisciplinary Optimization, 2019, 59 (2): 647-658. doi: 10.1007/s00158-018-2090-4
[36]	ADACHI T, TOMITA Y, TANAKA M. Computational simulation of deformation behavior of 2D-lattice continuum[J]. International Journal of Mechanical Sciences, 1998, 40 (9): 857-866. doi: 10.1016/S0020-7403(97)00127-6
[37]	LE C, NORATO J, BRUNS T, et al. Stress-based topology optimization for continua[J]. Structural and Multidisciplinary Optimization, 2010, 41 (4): 605-620. doi: 10.1007/s00158-009-0440-y
[38]	DUYSINX P, BENDSØE M P. Topology optimization of continuum structures with local stress constraints[J]. International Journal for Numerical Methods in Engineering, 1998, 43 (8): 1453-1478. doi: 10.1002/(SICI)1097-0207(19981230)43:8<1453::AID-NME480>3.0.CO;2-2
[39]	KAHROBAIYAN M H, RAHAEIFARD M, AHMADIAN M T. A size-dependent yield criterion[J]. International Journal of Engineering Science, 2014, 74 : 151-161. doi: 10.1016/j.ijengsci.2013.09.004
[40]	SIGMUND O. A 99 line topology optimization code written in Matlab[J]. Structural and Multidisciplinary Optimization, 2001, 21 (2): 120-127. doi: 10.1007/s001580050176
[41]	彭梦瑶, 顾水涛, 周洋靖, 等. 基于LiToSim平台的疲劳寿命评估LtsFatigue软件开发及应用[J]. 应用数学和力学, 2022, 43 (9): 976-986. doi: 10.21656/1000-0887.420277 PENG Mengyao, GU Shuitao, ZHOU Yangjing, et al. Development and application of fatigue life evaluation software LtsFatigue based on LiToSim[J]. Applied Mathematics and Mechanics, 2022, 43 (9): 976-986. (in Chinese) doi: 10.21656/1000-0887.420277
[42]	叶彦鹏, 顾水涛, 刘敏, 等. 基于LiToSim平台的海上风机过渡段优化软件开发[J]. 应用数学和力学, 2021, 42 (5): 441-451. doi: 10.21656/1000-0887.410354 YE Yanpeng, GU Shuitao, LIU Min, et al. Optimization software development for offshore turbine transition structures based on LiToSim[J]. Applied Mathematics and Mechanics, 2021, 42 (5): 441-451. (in Chinese) doi: 10.21656/1000-0887.410354