We propose a topology optimisation approach that can effectively account for the size effect of periodic composite plates to determine the optimal material distribution for achieving the largest bandgap width. The approach is based on the modified couple stress continuum and uses the relative bandgap width as the objective function, with volume constraints defined as the constraint function. The material properties are represented by the solid isotropic material with penalisation (SIMP) interpolation model, and the optimality criteria (OC) algorithm is employed to update the design variables. To address the significant size effect of the microplate structure, we use the modified couple stress continuum to model the dynamic behaviour of the unit cell. The Melosh–Zienkiewicz–Cheung (MZC) finite element is employed to ensure nodal continuity and achieve high-order elasticity with respect to inter-element continuity. Our results demonstrate that the proposed topology optimisation methodology is capable of effectively designing optimal unit cell configurations that account for size effect and significantly improve the bandgap width. We also investigate the impact of thickness and volume limitations on the optimised unit cell configuration. The obtained results suggest that the proposed topology optimisation framework is a promising approach for designing unit cell geometries with the account for size effect.
BendsøeMPKikuchiN. Generating optimal topologies in structural design using a homogenization method. Comput Methods Appl Mech Eng1988; 71(2): 197–224.
2.
BendsøeMP. Optimal shape design as a material distribution problem. Struct Optim1989; 1(4): 193–202.
3.
SigmundOSøndergaard JensenJ. Systematic design of phononic band-gap materials and structures by topology optimization. Philos Trans A Math Phys Eng Sci2003; 361(1806): 1001–1019.
4.
MiniaciMKrushynskaAGliozziAS, et al. Design and fabrication of bioinspired hierarchical dissipative elastic metamaterials. Phys Rev Appl2018; 10(2): 024012.
5.
MazzottiMFoehrABilalO, et al. Bio-inspired non self-similar hierarchical elastic metamaterials. Int J Mech Sci2023; 241: 107915.
ShaatMGhavanlooEFazelzadehSA. Review on nonlocal continuum mechanics: physics, material applicability, and mathematics. Mech Mater2020; 150: 103587.
10.
NeffPGhibaIDMadeoA, et al. A unifying perspective: the relaxed linear micromorphic continuum. Continuum Mech Thermodyn2013; 26(5): 639–681.
11.
GhibaIDNeffPMadeoA, et al. The relaxed linear micromorphic continuum: existence, uniqueness and continuous dependence in dynamics. Math Mech Solids2014; 20(10): 1171–1197.
12.
MadeoANeffPGhibaID, et al. Band gaps in the relaxed linear micromorphic continuum. J Appl Math Mech2014; 95(9): 880–887.
13.
MadeoANeffPd’AgostinoMV, et al. Complete band gaps including non-local effects occur only in the relaxed micromorphic model. Comp R Mec2016; 344(11–12): 784–796.
14.
MindlinRD. Influence of couple-stresses on stress concentrations. Exp Mech1963; 3(1): 1–7.
15.
YangFChongALamDCC, et al. Couple stress based strain gradient theory for elasticity. Int J Solids Struct2002; 39(10): 2731–2743.
16.
XiaZCongYGuS, et al. Active tuning of vibration for periodic piezoelectric micro systems: a non-local Mindlin plate finite element approach. Mech Mater2022; 174: 104418.
17.
ZhangGGaoXLDingS. Band gaps for wave propagation in 2-D periodic composite structures incorporating microstructure effects. Acta Mech2018; 229(10): 4199–4214.
18.
XiaZZhangGCongY, et al. A non-classical couple stress based Mindlin plate finite element framework for tuning band gaps of periodic composite micro plates. J Sound Vib2022; 529: 116889.
19.
SigmundO. A 99 line topology optimization code written in Matlab. Struct Multidiscip Optim2001; 21(2): 120–127.
20.
BendsoeMPSigmundO. Topology optimization: theory, methods, and applications. Berlin: Springer Science & Business Media, 2003.
21.
StolpeMSvanbergK. An alternative interpolation scheme for minimum compliance topology optimization. Struct Multidiscip Optim2001; 22: 116–124.
22.
XieYMStevenGP. A simple evolutionary procedure for structural optimization. Comput Struct1993; 49(5): 885–896.
23.
XieYMStevenGPXieY, et al. Basic evolutionary structural optimization. London: Springer, 1997.
24.
QuerinOMStevenGPXieYM. Evolutionary structural optimisation (ESO) using a bidirectional algorithm. Eng Comp1998; 15(8): 1031–1048.
25.
Van DijkNPMauteKLangelaarM, et al. Level-set methods for structural topology optimization: a review. Struct Multidiscip Optim2013; 48: 437–472.
26.
RozvanyGZhouM. The COC algorithm, part I: cross-section optimization or sizing. Comput Methods Appl Mech Eng1991; 89(1–3): 281–308.
27.
ZhouMRozvanyGI. The COC algorithm, part II: topological, geometrical and generalized shape optimization. Comput Methods Appl Mech Eng1991; 89(1–3): 309–336.
28.
NocedalJWrightSJ. Numerical optimization. New York: Springer, 1999.
29.
DebKGulatiS. Design of truss-structures for minimum weight using genetic algorithms. Finite Elem Anal Des2001; 37(5): 447–465.
30.
RajeevSKrishnamoorthyC. Discrete optimization of structures using genetic algorithms. J Struct Eng1992; 118(5): 1233–1250.
31.
PoggettoVFDBosiaFMiniaciM, et al. Optimization of spider web-inspired phononic crystals to achieve tailored dispersion for diverse objectives. Mater Design2021; 209: 109980.
32.
ChenLGuoYYiH. Optimization study of bandgaps properties for two-dimensional chiral phononic crystals base on lightweight design. Phys Lett A2021; 388: 127054.
JogC. Higher-order shell elements based on a Cosserat model, and their use in the topology design of structures. Comput Methods Appl Mech Eng2004; 193(23–26): 2191–2220.
35.
BruggiMTaliercioA. Maximization of the fundamental eigenfrequency of micropolar solids through topology optimization. Struct Multidiscip Optim2012; 46(4): 549–560.
36.
ChenLWanJChuX, et al. Parameterized level set method for structural topology optimization based on the cosserat elasticity. Acta Mech Sin2021; 37(4): 620–630.
LiuSSuW. Topology optimization of couple-stress material structures. Struct Multidiscip Optim2009; 40(1–6): 319–327.
39.
LiBDuanYYangH, et al. Isogeometric topology optimization of strain gradient materials. Comput Methods Appl Mech Eng2022; 397: 115135.
40.
EringenAC. Nonlocal continuum field theories. New York: Springer, 2004.
41.
OrtigosaRMartínez-FrutosJGilA. A computational framework for topology optimisation of flexoelectricity at finite strains considering a multi-field micromorphic approach. Comput Methods Appl Mech Eng2022; 401: 115604.
42.
GanNWangQ. Topology optimization design related to size effect using the modified couple stress theory. Eng Optim2021; 55(1): 158–176.
43.
DalklintAWallinMBertoldiK, et al. Tunable phononic bandgap materials designed via topology optimization. J Mech Phys Solids2022; 163: 104849.
44.
LopesHNMahfoudJPavanelloR. High natural frequency gap topology optimization of bi-material elastic structures and band gap analysis. Struct Multidiscip Optim2021; 63(5): 2325–2340.
45.
DongHWSuXXWangYS, et al. Topological optimization of two-dimensional phononic crystals based on the finite element method and genetic algorithm. Struct Multidiscip Optim2014; 50(4): 593–604.
46.
VenkayyaV. Optimality criteria: a basis for multidisciplinary design optimization. Comput Mech1989; 5(1): 1–21.
47.
ZienkiewiczOCTaylorRL. The finite element method: solid mechanics, vol. 2. Oxford: Butterworth-Heinemann, 2000.
48.
ZhangBHeYLiuD, et al. A non-classical Mindlin plate finite element based on a modified couple stress theory. Eur J Mech-A/Solids2013; 42: 63–80.
49.
MindlinR. Influence of rotatory inertia and shear on flexural motions of isotropic, elastic plates. J Appl Mech1951; 18: 31–38.
50.
MaHMGaoXLReddyJN. A non-classical mindlin plate model based on a modified couple stress theory. Acta Mech2011; 220(1–4): 217–235.
51.
ZhangGYQuYLGaoXL, et al. A transversely isotropic magneto-electro-elastic Timoshenko beam model incorporating microstructure and foundation effects. Mech Mater2020; 149: 103412.
52.
Gómez-LeónAPlateroG. Floquet–Bloch theory and topology in periodically driven lattices. Phys Rev Lett2013; 110(20): 200403.
53.
HuangXZuoZXieY. Evolutionary topological optimization of vibrating continuum structures for natural frequencies. Comput Struct2010; 88(5–6): 357–364.
54.
MATLAB. 9.7.0.1190202 (R2019b). Natick, MA: The MathWorks Inc., 2018.
55.
LamDCYangFChongA, et al. Experiments and theory in strain gradient elasticity. J Mech Phys Solids2003; 51(8): 1477–1508.
56.
ParkSGaoX. Bernoulli–Euler beam model based on a modified couple stress theory. J Micromech Microeng2006; 16(11): 2355.
Supplementary Material
Please find the following supplemental material available below.
For Open Access articles published under a Creative Commons License, all supplemental material carries the same license as the article it is associated with.
For non-Open Access articles published, all supplemental material carries a non-exclusive license, and permission requests for re-use of supplemental material or any part of supplemental material shall be sent directly to the copyright owner as specified in the copyright notice associated with the article.
