Robust isogeometric topology optimization for piezoelectric actuators with uniform manufacturability

doi:10.1007/s11465-022-0683-5

Front. Mech. Eng.

2022, Vol. 17

Issue (2) : 27 https://doi.org/10.1007/s11465-022-0683-5

RESEARCH ARTICLE

Robust isogeometric topology optimization for piezoelectric actuators with uniform manufacturability

Jie GAO^1,², Mi XIAO³, Zhi YAN^1,²(

), Liang GAO³(

), Hao LI³

¹. Department of Engineering Mechanics, School of Aerospace Engineering, Huazhong University of Science and Technology, Wuhan 430074, China
². Hubei Key Laboratory for Engineering Structural Analysis and Safety Assessment, Huazhong University of Science and Technology, Wuhan 430074, China
³. State Key Laboratory of Digital Manufacturing Equipment and Technology, Huazhong University of Science and Technology, Wuhan 430074, China

Download: PDF(9446 KB) HTML
Export: BibTeX | EndNote | Reference Manager | ProCite | RefWorks

Abstract

Piezoelectric actuators have received substantial attention among the industry and academia due to quick responses, such as high output force, high stiffness, high accuracy, and precision. However, the design of piezoelectric actuators always suffers from the emergence of several localized hinges with only one-node connection, which have difficulty satisfying manufacturing and machining requirements (from the over- or under-etching devices). The main purpose of the current paper is to propose a robust isogeometric topology optimization (RITO) method for the design of piezoelectric actuators, which can effectively remove the critical issue induced by one-node connected hinges and simultaneously maintain uniform manufacturability in the optimized topologies. In RITO, the isogeometric analysis replacing the conventional finite element method is applied to compute the unknown electro elastic fields in piezoelectric materials, which can improve numerical accuracy and then enhance iterative stability. The erode–dilate operator is introduced in topology representation to construct the eroded, intermediate, and dilated density distribution functions by non-uniform rational B-splines. Finally, the RITO formulation for the design of piezoelectric materials is developed, and several numerical examples are performed to test the effectiveness and efficiency of the proposed RITO method.

Keywords piezoelectric actuator isogeometric topology optimization uniform manufacturability robust formulation density distribution function

Corresponding Author(s): Zhi YAN,Liang GAO

About author:

Tongcan Cui and Yizhe Hou contributed equally to this work.

Just Accepted Date: 22 April 2022 Issue Date: 15 August 2022

Cite this article:

Jie GAO,Mi XIAO,Zhi YAN, et al. Robust isogeometric topology optimization for piezoelectric actuators with uniform manufacturability[J]. Front. Mech. Eng., 2022, 17(2): 27.

URL:

https://academic.hep.com.cn/fme/EN/10.1007/s11465-022-0683-5
https://academic.hep.com.cn/fme/EN/Y2022/V17/I2/27

Fig.1 Several designs with one-node connected hinges: (a) an optimized design of an inverter; (b) structural deformation of optimized design (a); (c) blue-print design of an inverter with uniform manufacturability; (d) structural deformation of optimized design (c). (a), (b), (c), and (d) are reprinted with permission from Ref. [49] from Springer Nature. (e) A design of piezoelectric actuator, reprinted with permission from Ref. [22] from Springer Nature; (f) a design of piezoelectric actuator, reprinted with permission from Ref. [18] from IOP Publishing; (g) a design of piezoelectric actuator, reprinted with permission from Ref. [20] from Elsevier.

Fig.2 Definition of DDF and its corresponding topology. (a) Step 1: control design variables; (b) Step 2: smoothing mechanism; (c) Step 3: threshold projection; (d) Step 4: DDF; (e) Step 5: topology. DDF: density distribution function.

Fig.3 Erode–dilate operator for DDF. DDF: density distribution function.

Domain	L	H	NURBS orders	Control points	IGA elements	$V max$
1	$10 ? 2$	$10 ? 2$	[3, 3]	102 × 102	100 × 100	0.26
2	$10 ? 2$	$10 ? 2$	[3, 3]	102 × 102	100 × 100	0.20
3	$10 ? 2$	$10 ? 2$	[3, 3]	102 × 102	100 × 100	0.30

Tab.1 Related parameters in the three structural design domains

Fig.4 Boundary and load conditions of (a) structural design domain 1, (b) structural design domain 2, and (c) structural design domain 3.

Fig.5 Color setting in DDF, topology, and polarization.

Fig.6 Optimized designs of the first domain for three different cases: optimized DDF of structural design domain 1 with stiffness equal to (a1) 0.01, (a2) 0.005, and (a3) 0.0015; optimized topology of structural design domain 1 with stiffness equal to (b1) 0.01, (b2) 0.005, and (b3) 0.0015; (c1), (c2), and (c3) layout of polarization in optimized topology in (b1), (b2), and (b3), respectively.

Fig.7 Iterative histories of objective function and volume fraction in optimization of Fig. 6(b3).

Case	$k u u 1$ , $k u u 2$ , $k u u 3$ , and $k u u 4$ in Fig.4(b)	$k u u$ in Fig.4(c)	$u out$ /10^?7
Case		$k u u$ in Fig.4(c)	In Fig.4(b)	In Fig.4(c)
Case 1	0.010	0.010	4.54	14.8
Case 2	0.005	0.005	5.57	19.6
Case 3	0.001	0.002	23.00	31.4

Tab.2 Spring stiffness and optimized displacement of piezoelectric actuator in Figs. 4(b) and 4(c)

Fig.8 Optimized designs of the second design domain for three different cases: (a1) optimized topology of structural design domain 2 with stiffness equal to 0.01, (a2) layout of polarization in optimized topology in (a1), (b1) optimized DDF of structural design domain 2 with stiffness equal to 0.005, (b2) layout of polarization in optimized topology in (b1), (c1) optimized DDF of structural design domain 2 with stiffness equal to 0.001, and (c2) layout of polarization in optimized topology in (c1).

Fig.9 Optimized designs of the third design domain for three different cases: (a1) optimized topology of structural design domain 3 with stiffness equal to 0.01, (a2) layout of polarization in optimized topology in (a1), (b1) optimized DDF of structural design domain 3 with stiffness equal to 0.005, (b2) layout of polarization in optimized topology in (b1), (c1) optimized DDF of structural design domain 3 with stiffness equal to 0.002, and (c2) layout of polarization in optimized topology in (c1).

Tab.3 Output displacement and volume fraction of the three designs

Fig.10 Robust designs of the first domain using RITO formulation: optimized DDF of (a1) eroded design, (a2) intermediate design, and (a3) dilated design; optimized topology of (b1) eroded design, (b2) intermediate design, and (b3) dilated design; (c1), (c2), and (c3) corresponding layout of polarization in optimized topology in (b1), (b2), and (b3), respectively.

Fig.11 Convergent histories of objective function and volume fraction in RITO optimization for the first design domain.

Fig.12 Robust designs of the second domain using RITO formulation: (a1) optimized topology of eroded design, (a2) corresponding layout of polarization in optimized topology in (a1), (b1) optimized topology of intermediate design, (b2) corresponding layout of polarization in optimized topology in (b1), (c1) optimized topology of dilated design, and (c2) corresponding layout of polarization in optimized topology in (c1).

Fig.13 Robust designs of the third domain using RITO formulation: (a1) optimized topology of eroded design, (a2) corresponding layout of polarization in optimized topology in (a1), (b1) optimized topology of intermediate design, (b2) corresponding layout of polarization in optimized topology in (b1), (c1) optimized topology of dilated design, and (c2) corresponding layout of polarization in optimized topology in (c1).

Tab.4 Numerical results containing the output displacement and final volume fraction of Figs. 12 and 13

Tab.5 Comparison of output displacements between deterministic designs and robustly blue-print designs for three domains

Fig.14 Comparison between deterministic design by ITO and robustly blue-print design by RITO for the three domains: (a1) deterministic design of the first design domain, (a2) blue-print design of the first design domain, (b1) deterministic design of the second design domain, (b2) blue-print design of the second design domain, (c1) deterministic design of the third design domain, and (c2) blue-print design of the third design domain.

Setting	Parameters of threshold projection
Setting 1	$η e o = 0.80, η i d = 0.5, η d o = 0.20$
Setting 2	$η e o = 0.75, η i d = 0.5, η d o = 0.25$
Setting 3	$η e o = 0.70, η i d = 0.5, η d o = 0.30$

Tab.6 Three different settings of parameter of threshold projection

Fig.15 Blue-print designs of the three design domains under three different settings: blue-print design of design domain 1 under (a1) setting 1, (a2) setting 2, and (a3) setting 3; blue-print design of design domain 2 under (b1) setting 1, (b2) setting 2, and (b3) setting 3; blue-print design of design domain 3 under (c1) setting 1, (c2) setting 2, and (c3) setting 3.

Abbreviations
CAMD	Continuous approximation of material distribution
DDF	Density distribution function
FEM	Finite element method
IGA	Isogeometric analysis
ITO	Isogeometric topology optimization
MEMS	Micro-electro-mechanical system
MMA	Method of moving asymptotes
NURBS	Non-uniform rational B-splines
OC	Optimality criteria
PEMAP	Piezoelectric material with penalization
PEMAP-P	Piezoelectric material with penalization and polarization
PZT	Lead zirconate titanate
RITO	Robust isogeometric topology optimization
SIMP	Solid isotropic material penalization
Variables
$B u$	Strain?displacement matrix
$c E$	Stiffness tensor in constant electrical field
$D$	Electrical displacement
$e$	Number of the finite element
$e$	Piezoelectric coefficient matrix
$E$	Electrical field
$E 0, e 0, ε 0$	Stiffness, electromechanical coupling, and dielectric coefficients of piezoelectric solids, respectively
$E min$ , $e min$ , $ε min$	Minimum values of stiffness, electromechanical coup-ling, and dielectric coefficients of the voids, respectively
f	Global force imposed at the design domain
$f b$	Body force
$f d$	Dummy load
$f e$	Force in the $e$ th finite element
$f s$	Surface traction
$G 1$	Volume constraint for the eroded, intermediate, and dilated topologies
$G 2$	Volume constraint in the intermediate design
$G (Φ)$	Volume constraint function
$h$	Thickness of the piezoelectric plate
$i$	Number of control point in the first parametric direction
$j$	Number of control point in the second parametric direction
$J$	Objective function
$J 1$	Jacobi matrix from the parametric space to physical space
$J 2$	Jacobi matrix from the bi-unit parent element space to parametric space
$k u u$	Spring stiffness at the output location
$k u u$	Mechanical stiffness matrix
$k u ?$	Piezoelectric coupling matrix
$k ? ?$	Dielectric stiffness matrix
$m$ , $n$	Total number of control points in the parametric directions $η$ and $ξ$ , respectively
$M j, q$	B-spline basis functions in the second parametric direction
$N i, p$	B-spline basis functions in the first parametric direction
p	Degree of NURBS basis functions in the first parametric direction
$p u u$ , $p u ?$ , $p ? ?$ , $p p o$	Penalization parameters for stiffness, piezoelectricity, dielectric, and polarization, respectively
q	Degree of NURBS basis functions in the second parametric direction
$q c$	Surface charge density accumulated on the electrodes
$q e$	Charge density in the $e$ th finite element
$R i, j p, q$	NURBS basis functions in 2D
S	Mechanical strain
T	Mechanical stress
u	Displacement field
u_e	Displacement field in the eth finite element
u_i,j	Displacement at the (i,j)th control point
u_out	Output displacement at the specified locations of the design domain
V_max	Maximum material consumption
$ω i, j$	Positive weight at the (i,j)th control point
$ξ, ζ$	First and second parametric directions, respectively
$ε S$	Permittivity coefficient matrix
?	Electric potential
?_e	Electric potential in the eth finite element
$Ω e$	Physical design domain of the eth IGA element
$Ω ~$	Bi-unit parent element
$φ$	Control design variable
$φ min$	Positive integer to avoid the occurrence of numerical singularity
$φ^e o, φ^i d$ , $φ^d o$	Eroded, intermediate, and dilated control design variables, respectively
$φ ~ i, j$	$(i, j)$ th smoothed control design variable
$β$ , $η$	First and second parameters in the threshold projection, respectively
$η e o, η i d, η d o$	Different values of the parameter $η$ to define the erode, intermediate and dilate operators in threshold projection, respectively
$λ$	Adjoint vector of the dummy load
$Φ$	Density distribution function
$Φ e o, Φ i d$ , $Φ d o$	Eroded, intermediate, and dilated DDFs, respectively
$Φ i s o$	Value of the iso-contour of the DDF
$Φ t o p$	Structural topology
$Φ t o p e o, Φ t o p i d$ , $Φ t o p d o$	Eroded, intermediate, and dilated topologies, respectively
$ψ$	Second type of design variables for the polarization
$Ψ$	Continuous function for the second type of design variable
$Ψ e o, Ψ i d, Ψ d o$	Eroded, intermediate, and dilated continuous functions for the second type of design variable, respectively
$Ψ t o p e o, Ψ t o p i d$ , $Ψ t o p d o$	Optimized distributions of the polarization in three eroded, intermediate, and dilated designs, respectively

1	M I Frecker. Recent advances in optimization of smart structures and actuators. Journal of Intelligent Material Systems and Structures, 2003, 14( 4–5): 207– 216 https://doi.org/10.1177/1045389X03031062
2	H J M T S Adriaens, W L De Koning, R Banning. Modeling piezoelectric actuators. IEEE/ASME Transactions on Mechatronics, 2000, 5( 4): 331– 341 https://doi.org/10.1109/3516.891044
3	Y K Zhang, Z Tu, T F Lu, S Al-Sarawi. A simplified transfer matrix of multi-layer piezoelectric stack. Journal of Intelligent Material Systems and Structures, 2017, 28( 5): 595– 603 https://doi.org/10.1177/1045389X16651153
4	R Pérez, J Agnus, C Clévy, A Hubert, N Chaillet. Modeling, fabrication, and validation of a high-performance 2-DoF piezoactuator for micromanipulation. IEEE/ASME Transactions on Mechatronics, 2005, 10( 2): 161– 171 https://doi.org/10.1109/TMECH.2005.844712
5	M P Bendsøe, O Sigmund. Topology Optimization: Theory, Methods and Applications. Berlin: Springer, 2003
6	J Gao, Z Luo, H Li, L Gao. Topology optimization for multiscale design of porous composites with multi-domain microstructures. Computer Methods in Applied Mechanics and Engineering, 2019, 344: 451– 476 https://doi.org/10.1016/j.cma.2018.10.017
7	H Liu, H M, Zong T L Shi, Q Xia. M-VCUT level set method for optimizing cellular structures. Computer Methods in Applied Mechanics and Engineering, 2020, 367: 113154 https://doi.org/10.1016/j.cma.2020.113154
8	Q H Li, O Sigmund, J S Jensen, N Aage. Reduced-order methods for dynamic problems in topology optimization: a comparative study. Computer Methods in Applied Mechanics and Engineering, 2021, 387: 114149 https://doi.org/10.1016/j.cma.2021.114149
9	S Chu, C Featherston, H A Kim. Design of stiffened panels for stress and buckling via topology optimization. Structural and Multidisciplinary Optimization, 2021, 64( 5): 3123– 3146 https://doi.org/10.1007/s00158-021-03062-3
10	S Chu, S Townsend, C Featherston, H A Kim. Simultaneous layout and topology optimization of curved stiffened panels. AIAA Journal, 2021, 59( 7): 2768– 2783 https://doi.org/10.2514/1.J060015
11	E C N Silva, J S O Fonseca, N Kikuchi. Optimal design of piezoelectric microstructures. Computational Mechanics, 1997, 19( 5): 397– 410 https://doi.org/10.1007/s004660050188
12	E C N Silva, J S O Fonseca, F M de Espinosa, A T Crumm, G A Brady, J W Halloran, N Kikuchi. Design of piezocomposite materials and piezoelectric transducers using topology optimization—part I. Archives of Computational Methods in Engineering, 1999, 6( 2): 117– 182 https://doi.org/10.1007/BF02736183
13	E C N Silva, S Nishiwaki, J S O Fonseca, N Kikuchi. Optimization methods applied to material and flextensional actuator design using the homogenization method. Computer Methods in Applied Mechanics and Engineering, 1999, 172( 1–4): 241– 271 https://doi.org/10.1016/S0045-7825(98)00231-X
14	M Zhou, G I N Rozvany. The COC algorithm, part II: topological, geometrical and generalized shape optimization. Computer Methods in Applied Mechanics and Engineering, 1991, 89( 1–3): 309– 336 https://doi.org/10.1016/0045-7825(91)90046-9
15	M P Bendsøe, O Sigmund. Material interpolation schemes in topology optimization. Archive of Applied Mechanics, 1999, 69( 9–10): 635– 654 https://doi.org/10.1007/s004190050248
16	E C N Silva, N Kikuchi. Design of piezoelectric transducers using topology optimization. Smart Materials and Structures, 1999, 8( 3): 350– 364 https://doi.org/10.1088/0964-1726/8/3/307
17	S Canfield, M Frecker. Topology optimization of compliant mechanical amplifiers for piezoelectric actuators. Structural and Multidisciplinary Optimization, 2000, 20( 4): 269– 279 https://doi.org/10.1007/s001580050157
18	R C Carbonari, E C N Silva, S Nishiwaki. Design of piezoelectric multi-actuated microtools using topology optimization. Smart Materials and Structures, 2005, 14( 6): 1431– 1447 https://doi.org/10.1088/0964-1726/14/6/036
19	M Kögl, E C N Silva. Topology optimization of smart structures: design of piezoelectric plate and shell actuators. Smart Materials and Structures, 2005, 14( 2): 387– 399 https://doi.org/10.1088/0964-1726/14/2/013
20	J E Kim, D S Kim, P S Ma, Y Y Kim. Multi-physics interpolation for the topology optimization of piezoelectric systems. Computer Methods in Applied Mechanics and Engineering, 2010, 199( 49–52): 3153– 3168 https://doi.org/10.1016/j.cma.2010.06.021
21	J F Gonçalves, D M De Leon, E A Perondi. Simultaneous optimization of piezoelectric actuator topology and polarization. Structural and Multidisciplinary Optimization, 2018, 58( 3): 1139– 1154 https://doi.org/10.1007/s00158-018-1957-8
22	A Homayouni-Amlashi, T Schlinquer, A Mohand-Ousaid, M Rakotondrabe. 2D topology optimization MATLAB codes for piezoelectric actuators and energy harvesters. Structural and Multidisciplinary Optimization, 2021, 63( 2): 983– 1014 https://doi.org/10.1007/s00158-020-02726-w
23	S T Yang, Y L Li, X Xia, P Ning, W T Ruan, R F Zheng, X H Lu. A topology optimization method and experimental verification of piezoelectric stick–slip actuator with flexure hinge mechanism. Archive of Applied Mechanics, 2022, 92( 1): 271– 285 https://doi.org/10.1007/s00419-021-02055-4
24	B Yang, C Z Cheng, X Wang, Z Meng, A Homayouni-Amlashi. Reliability-based topology optimization of piezoelectric smart structures with voltage uncertainty. Journal of Intelligent Material Systems and Structures, 2022, 33(15): 1975– 1989 https://doi.org/10.1177/1045389X211072197
25	Y G Wang, Z Kang, X P Zhang. A velocity field level set method for topology optimization of piezoelectric layer on the plate with active vibration control. Mechanics of Advanced Materials and Structures, 2022 (in press) https://doi.org/10.1080/15376494.2022.2030444
26	Z Kang, X M Wang. Topology optimization of bending actuators with multilayer piezoelectric material. Smart Materials and Structures, 2010, 19( 7): 0 75018 https://doi.org/10.1088/0964-1726/19/7/075018
27	R C Carbonari, E C N Silva, G H Paulino. Topology optimization design of functionally graded bimorph-type piezoelectric actuators. Smart Materials and Structures, 2007, 16( 6): 2605– 2620 https://doi.org/10.1088/0964-1726/16/6/065
28	P H Nakasone, E C N Silva. Dynamic design of piezoelectric laminated sensors and actuators using topology optimization. Journal of Intelligent Material Systems and Structures, 2010, 21( 16): 1627– 1652 https://doi.org/10.1177/1045389X10386130
29	X P Zhang, Z Kang. Dynamic topology optimization of piezoelectric structures with active control for reducing transient response. Computer Methods in Applied Mechanics and Engineering, 2014, 281: 200– 219 https://doi.org/10.1016/j.cma.2014.08.011
30	M Moretti, E C N Silva. Topology optimization of piezoelectric bi-material actuators with velocity feedback control. Frontiers of Mechanical Engineering, 2019, 14( 2): 190– 200 https://doi.org/10.1007/s11465-019-0537-y
31	Z Kang, L Y Tong. Integrated optimization of material layout and control voltage for piezoelectric laminated plates. Journal of Intelligent Material Systems and Structures, 2008, 19( 8): 889– 904 https://doi.org/10.1177/1045389X07084527
32	Z Kang, R Wang, L Y Tong. Combined optimization of bi-material structural layout and voltage distribution for in-plane piezoelectric actuation. Computer Methods in Applied Mechanics and Engineering, 2011, 200( 13–16): 1467– 1478 https://doi.org/10.1016/j.cma.2011.01.005
33	T J R Hughes, J A Cottrell, Y Bazilevs. Isogeometric analysis: CAD, finite elements, NURBS, exact geometry and mesh refinement. Computer Methods in Applied Mechanics and Engineering, 2005, 194( 39–41): 4135– 4195 https://doi.org/10.1016/j.cma.2004.10.008
34	S K Singh, I V Singh. Analysis of cracked functionally graded piezoelectric material using XIGA. Engineering Fracture Mechanics, 2020, 230: 107015 https://doi.org/10.1016/j.engfracmech.2020.107015
35	J Gao, M Xiao, Y Zhang, L Gao. A comprehensive review of isogeometric topology optimization: methods, applications and prospects. Chinese Journal of Mechanical Engineering, 2020, 33( 1): 87 https://doi.org/10.1186/s10033-020-00503-w
36	B Hassani, M Khanzadi, S M Tavakkoli. An isogeometrical approach to structural topology optimization by optimality criteria. Structural and Multidisciplinary Optimization, 2012, 45( 2): 223– 233 https://doi.org/10.1007/s00158-011-0680-5
37	J Gao, L Gao, Z Luo, P G Li. Isogeometric topology optimization for continuum structures using density distribution function. International Journal for Numerical Methods in Engineering, 2019, 119( 10): 991– 1017 https://doi.org/10.1002/nme.6081
38	Y J Wang, D J Benson. Isogeometric analysis for parameterized LSM-based structural topology optimization. Computational Mechanics, 2016, 57( 1): 19– 35 https://doi.org/10.1007/s00466-015-1219-1
39	H Ghasemi, H S Park, T Rabczuk. A level-set based IGA formulation for topology optimization of flexoelectric materials. Computer Methods in Applied Mechanics and Engineering, 2017, 313: 239– 258 https://doi.org/10.1016/j.cma.2016.09.029
40	J Gao, M Xiao, M Zhou, L Gao. Isogeometric topology and shape optimization for composite structures using level-sets and adaptive Gauss quadrature. Composite Structures, 2022, 285: 115263 https://doi.org/10.1016/j.compstruct.2022.115263
41	W B Hou, Y D Gai, X F Zhu, X Wang, C Zhao, L K Xu, K Jiang, P Hu. Explicit isogeometric topology optimization using moving morphable components. Computer Methods in Applied Mechanics and Engineering, 2017, 326: 694– 712 https://doi.org/10.1016/j.cma.2017.08.021
42	W S Zhang, D D Li, P Kang, X Guo, S K. Youn Explicit topology optimization using IGA-based moving morphable void (MMV) approach. Computer Methods in Applied Mechanics and Engineering, 2020, 360: 112685 https://doi.org/10.1016/j.cma.2019.112685
43	Z P Wang, L H Poh, J Dirrenberger, Y L Zhu, S Forest. Isogeometric shape optimization of smoothed petal auxetic structures via computational periodic homogenization. Computer Methods in Applied Mechanics and Engineering, 2017, 323: 250– 271 https://doi.org/10.1016/j.cma.2017.05.013
44	J Gao, H P Xue, L Gao, Z Luo. Topology optimization for auxetic metamaterials based on isogeometric analysis. Computer Methods in Applied Mechanics and Engineering, 2019, 352: 211– 236 https://doi.org/10.1016/j.cma.2019.04.021
45	C Wang, T T Yu, G J Shao, T Q Bui. Multi-objective isogeometric integrated optimization for shape control of piezoelectric functionally graded plates. Computer Methods in Applied Mechanics and Engineering, 2021, 377: 113698 https://doi.org/10.1016/j.cma.2021.113698
46	B L Zhu, X M Zhang, H C Zhang, J W Liang, H Y Zang, H Li, R X Wang. Design of compliant mechanisms using continuum topology optimization: a review. Mechanism and Machine Theory, 2020, 143: 103622 https://doi.org/10.1016/j.mechmachtheory.2019.103622
47	S Koppen, M Langelaar, F van Keulen. A simple and versatile topology optimization formulation for flexure synthesis. Mechanism and Machine Theory, 2022, 172: 104743 https://doi.org/10.1016/j.mechmachtheory.2022.104743
48	R X Wang, X M Zhang, B L Zhu, F H Qu, B C Chen, J W Liang. Hybrid explicit–implicit topology optimization method for the integrated layout design of compliant mechanisms and actuators. Mechanism and Machine Theory, 2022, 171: 104750 https://doi.org/10.1016/j.mechmachtheory.2022.104750
49	O Sigmund. Manufacturing tolerant topology optimization. Acta Mechanica Sinica, 2009, 25( 2): 227– 239 https://doi.org/10.1007/s10409-009-0240-z
50	Q Xia, T L Shi. Topology optimization of compliant mechanism and its support through a level set method. Computer Methods in Applied Mechanics and Engineering, 2016, 305: 359– 375 https://doi.org/10.1016/j.cma.2016.03.017
51	F W Wang, B S Lazarov, O Sigmund. On projection methods, convergence and robust formulations in topology optimization. Structural and Multidisciplinary Optimization, 2011, 43( 6): 767– 784 https://doi.org/10.1007/s00158-010-0602-y
52	J Z Luo, Z Luo, S K Chen, L Y Tong, M Y Wang. A new level set method for systematic design of hinge-free compliant mechanisms. Computer Methods in Applied Mechanics and Engineering, 2008, 198( 2): 318– 331 https://doi.org/10.1016/j.cma.2008.08.003
53	G A da Silva, A T Beck, O Sigmund. Topology optimization of compliant mechanisms with stress constraints and manufacturing error robustness. Computer Methods in Applied Mechanics and Engineering, 2019, 354: 397– 421 https://doi.org/10.1016/j.cma.2019.05.046
54	R Lerch. Simulation of piezoelectric devices by two-and three-dimensional finite elements. IEEE Transactions on Ultrasonics, Ferroelectrics, and Frequency Control, 1990, 37( 3): 233– 247 https://doi.org/10.1109/58.55314
55	Z Kang, Y Q Wang. Structural topology optimization based on non-local Shepard interpolation of density field. Computer Methods in Applied Mechanics and Engineering, 2011, 200( 49–52): 3515– 3525 https://doi.org/10.1016/j.cma.2011.09.001
56	D Trillet, P Duysinx, E Fernández. Analytical relationships for imposing minimum length scale in the robust topology optimization formulation. Structural and Multidisciplinary Optimization, 2021, 64( 4): 2429– 2448 https://doi.org/10.1007/s00158-021-02998-w
57	Y Q Wang, F F Chen, M Y Wang. Concurrent design with connectable graded microstructures. Computer Methods in Applied Mechanics and Engineering, 2017, 317: 84– 101 https://doi.org/10.1016/j.cma.2016.12.007
58	Q H Li, R Xu, Q B Wu, S T Liu. Topology optimization design of quasi-periodic cellular structures based on erode–dilate operators. Computer Methods in Applied Mechanics and Engineering, 2021, 377: 113720 https://doi.org/10.1016/j.cma.2021.113720
59	M Xiao, X L Liu, Y Zhang, L Gao, J Gao, S Chu. Design of graded lattice sandwich structures by multiscale topology optimization. Computer Methods in Applied Mechanics and Engineering, 2021, 384: 113949 https://doi.org/10.1016/j.cma.2021.113949
60	Y G Wang, Z Kang. A level set method for shape and topology optimization of coated structures. Computer Methods in Applied Mechanics and Engineering, 2018, 329: 553– 574 https://doi.org/10.1016/j.cma.2017.09.017
61	Y Zhang, M Xiao, L Gao, J Gao, H Li. Multiscale topology optimization for minimizing frequency responses of cellular composites with connectable graded microstructures. Mechanical Systems and Signal Processing, 2020, 135: 106369 https://doi.org/10.1016/j.ymssp.2019.106369
62	Y Zhang, L Zhang, Z Ding, L Gao, M Xiao, W H Liao. A multiscale topological design method of geometrically asymmetric porous sandwich structures for minimizing dynamic compliance. Materials & Design, 2022, 214: 110404 https://doi.org/10.1016/j.matdes.2022.110404
63	K Svanberg. The method of moving asymptotes—a new method for structural optimization. International Journal for Numerical Methods in Engineering, 1987, 24( 2): 359– 373 https://doi.org/10.1002/nme.1620240207
64	L Hägg, E Wadbro. On minimum length scale control in density based topology optimization. Structural and Multidisciplinary Optimization, 2018, 58( 3): 1015– 1032 https://doi.org/10.1007/s00158-018-1944-0

[1]	Meiju YANG,Chunxia LI,Guoying GU,Limin ZHU. A rate-dependent Prandtl-Ishlinskii model for piezoelectric actuators using the dynamic envelope function based play operator[J]. Front. Mech. Eng., 2015, 10(1): 37-42.
[2]	Jinhao QIU, Hongli JI, Kongjun ZHU. Semi-active vibration control using piezoelectric actuators in smart structures[J]. Front Mech Eng Chin, 2009, 4(3): 242-251.

Viewed

Full text

Abstract

Cited

Shared

Discussed