Implicit Heaviside filter with high continuity based on suitably graded THB splines

doi:10.1007/s11465-021-0670-2

Front. Mech. Eng.

2022, Vol. 17

Issue (1) : 14 https://doi.org/10.1007/s11465-021-0670-2

RESEARCH ARTICLE

Implicit Heaviside filter with high continuity based on suitably graded THB splines

Aodi YANG¹, Xianda XIE¹, Nianmeng LUO¹, Jie ZHANG², Ning JIANG¹, Shuting WANG¹(

)

¹. School of Mechanical Science and Engineering, Huazhong University of Science and Technology, Wuhan 430074, China
². School of Energy and Power Engineering, Huazhong University of Science and Technology, Wuhan 430074, China

Download: PDF(17695 KB) HTML
Export: BibTeX | EndNote | Reference Manager | ProCite | RefWorks

Abstract

The variable density topology optimization (TO) method has been applied to various engineering fields because it can effectively and efficiently generate the conceptual design for engineering structures. However, it suffers from the problem of low continuity resulting from the discreteness of both design variables and explicit Heaviside filter. In this paper, an implicit Heaviside filter with high continuity is introduced to generate black and white designs for TO where the design space is parameterized by suitably graded truncated hierarchical B-splines (THB). In this approach, the fixed analysis mesh of isogeometric analysis is decoupled from the design mesh, whose adaptivity is implemented by truncated hierarchical B-spline subjected to an admissible requirement. Through the intrinsic local support and high continuity of THB basis, an implicit adaptively adjusted Heaviside filter is obtained to remove the checkboard patterns and generate black and white designs. Threefold advantages are attained in the proposed filter: a) The connection between analysis mesh and adaptive design mesh is easily established compared with the traditional adaptive TO method using nodal density; b) the efficiency in updating design variables is remarkably improved than the traditional implicit sensitivity filter based on B-splines under successive global refinement; and c) the generated black and white designs are preliminarily compatible with current commercial computer aided design system. Several numerical examples are used to verify the effectiveness of the proposed implicit Heaviside filter in compliance and compliant mechanism as well as heat conduction TO problems.

Keywords topology optimization truncated hierarchical B-spline isogeometric analysis black and white designs Heaviside filter

Corresponding Author(s): Shuting WANG

About author: Miaojie Yang and Mahmood Brobbey Oppong contributed equally to this work.

Just Accepted Date: 01 March 2022 Issue Date: 29 April 2022

Cite this article:

Aodi YANG,Xianda XIE,Nianmeng LUO, et al. Implicit Heaviside filter with high continuity based on suitably graded THB splines[J]. Front. Mech. Eng., 2022, 17(1): 14.

URL:

https://academic.hep.com.cn/fme/EN/10.1007/s11465-021-0670-2
https://academic.hep.com.cn/fme/EN/Y2022/V17/I1/14

Fig.1 HB and THB curves after the same local refinement of a B-spline curve: (a) a B-spline curve, (b) the HB curve after local refinement of (a), and (c) the THB curve after local refinement of (a).

Fig.2 Example of HB and its three hierarchical levels.

Fig.3 Comparison between simplified and standard hierarchical bases: (a) standard hierarchical basis, (b) simplified hierarchical basis.

Fig.4 Truncated operation for standard and simplified HB bases: (a) standard HB, (b) simplified HB, (c) standard THB, and (d) simplified THB.

Fig.5 Admissible hierarchical design mesh and density field defined in Eq. (12): (a) 2D density design mesh, (b) density field based on THB of design mesh (a).

Fig.6

Fig.7 Updating rule for design variables in local refinement.

Fig.8

Fig.9 Difference between simplified and standard function spaces in local coarsening.

Fig.10 Mapping between fixed IGA analysis mesh and hierarchical design mesh.

Fig.11 Calculation of density value of an arbitrary Gaussian point for fixed IGA analysis mesh.

Fig.12 Different numbers of density variables that influence the red cross density point. Density point affected by nine density variables of control points (a) in uniform mesh and (b) in hierarchical mesh. (c) Density point affected by 10 density variables of control points in hierarchical mesh.

Fig.13 Influence region of different THB basis functions of level 0: THB basis function (a) in the left of hierarchical mesh, (b) in the middle of hierarchical mesh, and (c) in the top–left of hierarchical mesh.

Fig.14 Comparison of influence region of functions for THB of different levels: (a) level 0, (b) level 1, and (c) level 2.

Fig.15 Local refinement of THB when p = 2: (a) initial THB basis with one level, (b) refined THB basis with two levels, and (c) refined THB basis with three levels.

Fig.16 Comparisons of approximated Heaviside functions under different degrees of THB basis: (a) THB basis with p = 2 and (b) Heaviside function approximated by THB basis with p = 2. (c) THB basis with p = 3 and (d) Heaviside function approximated by THB basis with p = 3. (e) THB basis with p = 4 and (f) Heaviside function approximated by THB basis with p = 4. (g) THB basis with p = 5 and (h) Heaviside function approximated by THB basis with p = 5.

Fig.17 Problem configuration of 2D Michell beam.

Tab.1 Comparisons between different degrees of THB basis for hierarchical design mesh subjected to m = 2 admissible constraints

Fig.18 Four optimized results by the proposed adaptive ITO methods with different degrees of THB basis: optimized result with (a) p = 2, (b) p = 3, (c) p = 4, and (d) p = 5.

Fig.19 Influence region of a basis function in optimization result when p = 5: (a) optimized result, (b) magnifying result marked by a yellow rectangle in (a), (c) local hierarchical design mesh of (b), (d) active basis function of level 2 that influences mesh in (c).

Tab.2 Comparisons between different admissible restrictions imposed on hierarchical design mesh with degree p of THB basis equal to 2

Fig.20 Optimized designs and admissible hierarchical design meshes for m = 2, m = 3, and m = 4.

Fig.21 Influence regions of two active THB basis functions with m = 3 admissible constraints: (a) optimized result with p = 2, (b) magnifying result marked by yellow rectangle of (a), (c) local hierarchical design mesh associated with (b), (d) active basis function of level 2 that takes effect on the mesh in (c), (e) another active basis function of level 2 that influences mesh depicted in (c), and (f) magnifying result marked by a blue rectangle in (a).

Fig.22 Variation of structural topologies for Michell structure, where N is the number of levels of hierarchical design mesh, and num_variables is the number of design variables (a) for initial design, (b) for iterations = 23, (c) for iterations = 50, (d) for iterations = 91, and (e) for iterations = 192.

Fig.23 Admissible hierarchical design meshes at four stages of optimization.

Fig.24 Convergence histories of objective and volume fraction constraint functions.

Fig.25 3D density field and its CAD model for Michell structure: (a) 3D density field for optimal design, (b) CAD model of Michell structure.

Fig.26 Comparison of stress analysis of two optimized Michell structures: (a) optimized result by method in Ref. [68], (b) optimized result by our method.

Fig.27 Comparison of gray coefficient of two optimized designs: (a) optimized result based on suitably graded THB [69],

M n d = 18.68 %

, (b) optimized result of this section

M n d = 4.17 %

Tab.3 Comparisons of optimization results of three cases

Fig.28 Comparisons of optimal results of three cases (listed in Table 3) for Michell structure: (a) case 1, (b) case 2, and (c) case 3.

Fig.29 Comparison of updating time in design variables under three cases (listed in Table 3) for Michell structure.

Tab.4 Comparisons of optimization results using two marking strategies

Fig.30 Comparisons of optimized results using two marking strategies: (a) optimal result using marking strategy 1, (b) optimal result using marking strategy 2.

Fig.31 Initial conditions of 3D cantilever problem.

Fig.32 Initial design, three intermediate designs, and final design for 3D cantilever (a) for initial design, (b) for iterations = 15, (c) for iterations = 28, (d) for iterations = 55, and (e) for iterations = 83.

Fig.33 Variations of objective function, volume fraction, and hierarchical density design mesh for 3D cantilever.

Fig.34 Initial conditions of 2D compliant mechanism.

Fig.35 Initial design, two intermediate designs, and final design for compliant mechanism: (a) for initial design, (b) for iterations = 35, (c) for iterations = 76, and (d) for final design.

Fig.36 Variations of objective function, volume fraction constraint fraction, and adaptive design mesh.

Fig.37 Initial conditions of heat conduction TO problem.

Fig.38 Four designs for 2D heat conduction TO problem: (a) for initial design, (b) for iterations = 21, (c) for iterations = 41, and (d) for final design.

Fig.39 Variations of objective function, volume fraction, and admissible hierarchical design mesh for heat conduction problem solved by proposed adaptive TO approach.

Abbreviations
CAD	Computer aided design
FEM	Finite element method
HB	Hierarchical B-splines
IGA	Isogeometric analysis
ITO	Isogeometric topology optimization
NURBS	Nonuniform rational B-splines
SIMP	Solid isotropic material with penalization
TO	Topology optimization
THB	Truncated hierarchical B-splines
Variables
$A e l$	Volume of the eth element of level l
$A j$	Volume of the jth active element
alpha	Coefficient to control the descent speed of the gradient of design variables
B	B-spline basis function space
$B A l, n e w$	New added active basis functions of level l
$B A, r l, o l d$	The active functions to be refined
$B A, u l, o l d$	Active functions remaining unaltered
c	Compliance
$c k l + 1$	Coefficient of B^l with respect to $B k l + 1$
$c k τ, l + 1$	Truncated coefficient of B^l with respect to $B k l + 1$ for standard HB
$c k τ, l + 1 ~$	Truncated coefficient of B^l with respect to $B k l + 1$ for simplified HB
$C A N − 1, n e w$	Transformation matrix to map the new function space into basis functions of level N − 1
$C A N − 1, o l d$	Transformation matrix to map the old function space into basis functions of level N − 1
$C l$	Transformation matrix to map the active THB basis functions up to level l to the basis functions of level l
$C l l + 1$	Transformation matrix to map $B A l, n e w$ to $B A, r l, o l d$
$C (B i l)$	Children of $B k l + 1$ in B^l+1
C(Q)	Children of Q
$D i k$	Updating factor
E	Elastic modulus
$E 0$	Elastic modulus of solid material
$E min$	Elastic modulus of void material
f	External force
$f i n p u t$	Input force
$J u l$	Index transformation matrix to obtain the indices of $B A, u l, o l d$ in $B A l, n e w$
k_input, k_out	Input and output springs, respectively
K	Stiffness matrix
$K e$	Stiffness matrix of the eth element
$K j 0$	Stiffness matrix of the jth quadrature points
ke	Number of active elements affected by $R A i$
kq	Number of quadrature points in the region supporting of $R A i$
m	Admissible parameter
move	Move limit
$M n d$	Gray coefficient describes the design converged to a black and white (0/1) discrete solution
$M c$	Union of deactivated elements to be reactivated
$M r$	Union of elements to be refined
N	Number of levels of the THB basis function space
$N l$	Dimension of spline space of level l
$N c l$	Number of active control points of level l
$N e l$	Number of active elements of level l
$N c (Q, Q, m)$	Coarsening neighborhood of Q under m admissible constraint
$N r (Q, Q, m)$	Refinement neighborhood of Q under m admissible constraint
NF	Number of density coefficients
p	Degree of B-spline
penal	Penalty factor
$P (B k l + 1)$	Parent of $B i l$ in B^l
P(Q)	Parent of Q
Q	Cartesian mesh
$R 0$ , $R 1$	Active basis functions of levels 0 and 1, respectively
$R A$	An arbitrary active THB basis function
$R A i (ξ, η)$	Value of the ith active basis function at parameter point $(ξ, η)$
$R l$	Active basis functions of level l
$S (Q, k)$	Support extension of Q with respect to level k
T	Temperature
tl	Thermal load
trunc^l+1	Truncated operation
$trun c l + 1 ~ (B i l)$	Truncated operation for simplified HB basis
u	Displacement
$u o u t$	Output displacement
$u e$	Displacement of the eth element
u(x)	Nodal displacement vector
V	Volume of the design domain
$V ¯$	Upper limit of the volume fraction for the solid material
$V (x)$	Material volume
x	Union of material density design variables of all active control points
$x G, j$	Density value of the jth Gaussian quadrature points
α	Value of an active THB basis function
Ξ	Knot vectors
Ω	Parametric domain
$H$	Standard HB basis
$Q$	Hierarchical mesh
$H ~$	Simplified HB basis
$T$	Standard THB basis
$T ~$	Simplified THB basis
ρ	Density
$ρ A$	Density design variable of the active control point of $R A$
$ρ A i$	ith density design variable
$ρ o l d$ , $ρ n e w$	Original and locally refined density fields, respectively
$ρ c n e w$	Locally coarsened density field
$ρ A, r l, o l d$	Design variables with respect to $B A, r l, o l d$
$ρ A, u l, o l d$	Design variables with respect to $B A, u l, o l d$
$ρ e l$	Centroid density of the eth active element of level l
$ρ G j$	Density of the jth quadrature points in the locally supported region
$ρ i k$	ith density design variables at the kth iterative steps
$ρ l o w$ , $ρ u p$	Minimum and maximum values of the densities to determine the blurry regions of the design domain, respectively
$ρ min$	Minimum value of the densities in the coarsening updating rule
$ρ (ξ, η)$	Density at an arbitrary parameter coordinate $(ξ, η)$
$ν$	Poisson’s ratio
β	An arbitrary B-spline basis function
η	Damping factor
λ	Lagrange multiplier
μ	Shift parameter

1	M P Bendsøe, N Kikuchi. Generating optimal topologies in structural design using a homogenization method. Computer Methods in Applied Mechanics and Engineering, 1988, 71( 2): 197– 224 https://doi.org/10.1016/0045-7825(88)90086-2
2	M P Bendsøe. Optimal shape design as a material distribution problem. Structural Optimization, 1989, 1( 4): 193– 202 https://doi.org/10.1007/BF01650949
3	O Sigmund. A 99 line topology optimization code written in matlab. Structural and Multidisciplinary Optimization, 2001, 21( 2): 120– 127 https://doi.org/10.1007/s001580050176
4	M Y Wang, X M Wang, D M Guo. A level set method for structural topology optimization. Computer Methods in Applied Mechanics and Engineering, 2003, 192( 1–2): 227– 246 https://doi.org/10.1016/S0045-7825(02)00559-5
5	G Allaire, F Jouve, A M Toader. Structural optimization using sensitivity analysis and a level-set method. Journal of Computational Physics, 2004, 194( 1): 363– 393 https://doi.org/10.1016/j.jcp.2003.09.032
6	Y M Xie, G P Steven. A simple evolutionary procedure for structural optimization. Computers & Structures, 1993, 49( 5): 885– 896 https://doi.org/10.1016/0045-7949(93)90035-C
7	X Guo, W S Zhang, W L Zhong. Doing topology optimization explicitly and geometrically—a new moving morphable components based framework. Journal of Applied Mechanics, 2014, 81( 8): 081009 https://doi.org/10.1115/1.4027609
8	X Guo, W S Zhang, J Zhang, J Yuan. Explicit structural topology optimization based on moving morphable components (MMC) with curved skeletons. Computer Methods in Applied Mechanics and Engineering, 2016, 310 : 711– 748 https://doi.org/10.1016/j.cma.2016.07.018
9	W S Zhang, W Y Yang, J H Zhou, D Li, X Guo. Structural topology optimization through explicit boundary evolution. Journal of Applied Mechanics, 2017, 84( 1): 011011 https://doi.org/10.1115/1.4034972
10	S Y Cai, W H Zhang. An adaptive bubble method for structural shape and topology optimization. Computer Methods in Applied Mechanics and Engineering, 2020, 360 : 112778 https://doi.org/10.1016/j.cma.2019.112778
11	Y Zhou, W H Zhang, J H Zhu, Z Xu. Feature-driven topology optimization method with signed distance function. Computer Methods in Applied Mechanics and Engineering, 2016, 310 : 1– 32 https://doi.org/10.1016/j.cma.2016.06.027
12	L P Jiu, W H Zhang, L Meng, Y Zhou, L Chen. A CAD-oriented structural topology optimization method. Computers & Structures, 2020, 239 : 106324 https://doi.org/10.1016/j.compstruc.2020.106324
13	B Bourdin. Filters in topology optimization. International Journal for Numerical Methods in Engineering, 2001, 50( 9): 2143– 2158 https://doi.org/10.1002/nme.116
14	O Sigmund. Morphology-based black and white filters for topology optimization. Structural and Multidisciplinary Optimization, 2007, 33( 4–5): 401– 424 https://doi.org/10.1007/s00158-006-0087-x
15	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
16	Y J Wang, D J Benson. Geometrically constrained isogeometric parameterized level-set based topology optimization via trimmed elements. Frontiers of Mechanical Engineering, 2016, 11( 4): 328– 343 https://doi.org/10.1007/s11465-016-0403-0
17	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
18	X D Xie, S T Wang, M M Xu, Y J Wang. A new isogeometric topology optimization using moving morphable components based on R-functions and collocation schemes. Computer Methods in Applied Mechanics and Engineering, 2018, 339 : 61– 90 https://doi.org/10.1016/j.cma.2018.04.048
19	X D Xie, S T Wang, M Ye, Z H Xia, W Zhao, N Jiang, M M Xu. Isogeometric topology optimization based on energy penalization for symmetric structure. Frontiers of Mechanical Engineering, 2020, 15( 1): 100– 122 https://doi.org/10.1007/s11465-019-0568-4
20	J Xu, L Gao, M Xiao, J Gao, H Li. Isogeometric topology optimization for rational design of ultra-lightweight architected materials. International Journal of Mechanical Sciences, 2020, 166 : 105103 https://doi.org/10.1016/j.ijmecsci.2019.105103
21	J Gao, M Xiao, L Gao, J H Yan, W T Yan. Isogeometric topology optimization for computational design of re-entrant and chiral auxetic composites. Computer Methods in Applied Mechanics and Engineering, 2020, 362 : 112876 https://doi.org/10.1016/j.cma.2020.112876
22	J Gao, L Wang, M Xiao, L Gao, P G Li. An isogeometric approach to topological optimization design of auxetic composites with tri-material micro-architectures. Composite Structures, 2021, 271 : 114163 https://doi.org/10.1016/j.compstruct.2021.114163
23	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
24	Y J Wang, H Xu, D Pasini. Multiscale isogeometric topology optimization for lattice materials. Computer Methods in Applied Mechanics and Engineering, 2017, 316 : 568– 585 https://doi.org/10.1016/j.cma.2016.08.015
25	E Burman, D Elfverson, P Hansbo, M G Larson, K Larsson. Cut topology optimization for linear elasticity with coupling to parametric nondesign domain regions. Computer Methods in Applied Mechanics and Engineering, 2019, 350 : 462– 479 https://doi.org/10.1016/j.cma.2019.03.016
26	M M Xu, L Xia, S Y Wang, L H Liu, X D Xie. An isogeometric approach to topology optimization of spatially graded hierarchical structures. Composite Structures, 2019, 225 : 111171 https://doi.org/10.1016/j.compstruct.2019.111171
27	Y Bazilevs, V M Calo, J A Cottrell, J A Evans, T J R Hughes, S Lipton, M A Scott, T W Sederberg. Isogeometric analysis using T-splines. Computer Methods in Applied Mechanics and Engineering, 2010, 199( 5–8): 229– 263 https://doi.org/10.1016/j.cma.2009.02.036
28	M A Scott, M J Borden, C V Verhoosel, T W Sederberg, T J R Hughes. Isogeometric finite element data structures based on Bézier extraction of T-splines. International Journal for Numerical Methods in Engineering, 2011, 88( 2): 126– 156 https://doi.org/10.1002/nme.3167
29	M A Scott X Li T W Sederberg T J R Hughes. Local refinement of analysis-suitable T-splines. Computer Methods in Applied Mechanics and Engineering, 2012, 213–216: 206– 222
30	R Kraft. Adaptive and linearly independent multilevel B-splines. In: Proceedings of the 3rd International Conference on Curves and Surfaces. 1997, 2: 209– 218
31	A V Vuong, C Giannelli, B Jüttler, B Simeon. A hierarchical approach to adaptive local refinement in isogeometric analysis. Computer Methods in Applied Mechanics and Engineering, 2011, 200( 49–52): 3554– 3567 https://doi.org/10.1016/j.cma.2011.09.004
32	X D Xie, S T Wang, M M Xu, N Jiang, Y J Wang. A hierarchical spline based isogeometric topology optimization using moving morphable components. Computer Methods in Applied Mechanics and Engineering, 2020, 360 : 112696 https://doi.org/10.1016/j.cma.2019.112696
33	K A Johannessen, T Kvamsdal, T Dokken. Isogeometric analysis using LR B-splines. Computer Methods in Applied Mechanics and Engineering, 2014, 269 : 471– 514 https://doi.org/10.1016/j.cma.2013.09.014
34	K A Johannessen, F Remonato, T Kvamsdal. On the similarities and differences between classical hierarchical, truncated hierarchical and LR B-splines. Computer Methods in Applied Mechanics and Engineering, 2015, 291 : 64– 101 https://doi.org/10.1016/j.cma.2015.02.031
35	T Kanduč, C Giannelli, F Pelosi, H Speleers. Adaptive isogeometric analysis with hierarchical box splines. Computer Methods in Applied Mechanics and Engineering, 2017, 316 : 817– 838 https://doi.org/10.1016/j.cma.2016.09.046
36	S T Xia, X P Qian. Isogeometric analysis with Bézier tetrahedra. Computer Methods in Applied Mechanics and Engineering, 2017, 316 : 782– 816 https://doi.org/10.1016/j.cma.2016.09.045
37	C Giannelli, B Jüttler, S K Kleiss, A Mantzaflaris, B Simeon, J Špeh. THB-splines: an effective mathematical technology for adaptive refinement in geometric design and isogeometric analysis. Computer Methods in Applied Mechanics and Engineering, 2016, 299 : 337– 365 https://doi.org/10.1016/j.cma.2015.11.002
38	C Giannelli, B Jüttler, H Speleers. THB-splines: the truncated basis for hierarchical splines. Computer Aided Geometric Design, 2012, 29( 7): 485– 498 https://doi.org/10.1016/j.cagd.2012.03.025
39	X D Xie, S T Wang, Y J Wang, N Jiang, W Zhao, M M Xu. Truncated hierarchical B-spline-based topology optimization. Structural and Multidisciplinary Optimization, 2020, 62( 1): 83– 105 https://doi.org/10.1007/s00158-019-02476-4
40	Y Li, Y M Xie. Evolutionary topology optimization for structures made of multiple materials with different properties in tension and compression. Composite Structures, 2021, 259 : 113497 https://doi.org/10.1016/j.compstruct.2020.113497
41	X P Qian. Topology optimization in B-spline space. Computer Methods in Applied Mechanics and Engineering, 2013, 265 : 15– 35 https://doi.org/10.1016/j.cma.2013.06.001
42	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
43	Y L Gao, Y J Guo, S J Zheng. A NURBS-based finite cell method for structural topology optimization under geometric constraints. Computer Aided Geometric Design, 2019, 72 : 1– 18 https://doi.org/10.1016/j.cagd.2019.05.001
44	G Costa, M Montemurro, J Pailhès. A 2D topology optimisation algorithm in NURBS framework with geometric constraints. International Journal of Mechanics and Materials in Design, 2018, 14( 4): 669– 696 https://doi.org/10.1007/s10999-017-9396-z
45	G Costa, M Montemurro. Eigen-frequencies and harmonic responses in topology optimisation: a CAD-compatible algorithm. Engineering Structures, 2020, 214 : 110602 https://doi.org/10.1016/j.engstruct.2020.110602
46	G Costa, M Montemurro, J Pailhès. NURBS hyper-surfaces for 3D topology optimization problems. Mechanics of Advanced Materials and Structures, 2021, 28( 7): 665– 684 https://doi.org/10.1080/15376494.2019.1582826
47	G Costa, M Montemurro, J Pailhès. Minimum length scale control in a NURBS-based SIMP method. Computer Methods in Applied Mechanics and Engineering, 2019, 354 : 963– 989 https://doi.org/10.1016/j.cma.2019.05.026
48	G Costa, M Montemurro, J Pailhès, N Perry. Maximum length scale requirement in a topology optimisation method based on NURBS hyper-surfaces. CIRP Annals-Manufacturing Technology, 2019, 68( 1): 153– 156 https://doi.org/10.1016/j.cirp.2019.04.048
49	J K Guest, J H Prévost, T Belytschko. Achieving minimum length scale in topology optimization using nodal design variables and projection functions. International Journal for Numerical Methods in Engineering, 2004, 61( 2): 238– 254 https://doi.org/10.1002/nme.1064
50	S L Xu, Y W Cai, G D Cheng. Volume preserving nonlinear density filter based on heaviside functions. Structural and Multidisciplinary Optimization, 2010, 41( 4): 495– 505 https://doi.org/10.1007/s00158-009-0452-7
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	N Pollini, O Amir. Mixed projection- and density-based topology optimization with applications to structural assemblies. Structural and Multidisciplinary Optimization, 2020, 61( 2): 687– 710 https://doi.org/10.1007/s00158-019-02390-9
53	X D Huang. Smooth topological design of structures using the floating projection. Engineering Structures, 2020, 208 : 110330 https://doi.org/10.1016/j.engstruct.2020.110330
54	X D Huang. On smooth or 0/1 designs of the fixed-mesh element-based topology optimization. Advances in Engineering Software, 2021, 151 : 102942 https://doi.org/10.1016/j.advengsoft.2020.102942
55	Y Q Wang, Z Kang, Q Z He. An adaptive refinement approach for topology optimization based on separated density field description. Computers & Structures, 2013, 117 : 10– 22 https://doi.org/10.1016/j.compstruc.2012.11.004
56	Y Q Wang, J J He, Z Luo, Z Kang. An adaptive method for high-resolution topology design. Acta Mechanica Sinica, 2013, 29( 6): 840– 850 https://doi.org/10.1007/s10409-013-0084-4
57	Y Q Wang, Z Kang, Q Z He. Adaptive topology optimization with independent error control for separated displacement and density fields. Computers & Structures, 2014, 135 : 50– 61 https://doi.org/10.1016/j.compstruc.2014.01.008
58	A Buffa, C Giannelli. Adaptive isogeometric methods with hierarchical splines: error estimator and convergence. Mathematical Models and Methods in Applied Sciences, 2016, 26( 1): 1– 25 https://doi.org/10.1142/S0218202516500019
59	C Bracco, C Giannelli, R Vázquez. Refinement algorithms for adaptive isogeometric methods with hierarchical splines. Axioms, 2018, 7( 3): 43 https://doi.org/10.3390/axioms7030043
60	M Carraturo, C Giannelli, A Reali, R Vázquez. Suitably graded THB-spline refinement and coarsening: towards an adaptive isogeometric analysis of additive manufacturing processes. Computer Methods in Applied Mechanics and Engineering, 2019, 348 : 660– 679 https://doi.org/10.1016/j.cma.2019.01.044
61	C De Boor. On calculating with B-splines. Journal of Approximation Theory, 1972, 6( 1): 50– 62 https://doi.org/10.1016/0021-9045(72)90080-9
62	E M Garau, R Vázquez. Algorithms for the implementation of adaptive isogeometric methods using hierarchical B-splines. Applied Numerical Mathematics, 2018, 123 : 58– 87 https://doi.org/10.1016/j.apnum.2017.08.006
63	J C Xu, Z M Zhang. Analysis of recovery type a posteriori error estimators for mildly structured grids. Mathematics of Computation, 2004, 73( 247): 1139– 1152 https://doi.org/10.1090/S0025-5718-03-01600-4
64	R H Jari, L Mu. Application for superconvergence of finite element approximations for the elliptic problem by global and local L2-projection methods. American Journal of Computational Mathematics, 2012, 2( 4): 249– 257 https://doi.org/10.4236/ajcm.2012.24034
65	P Costantini, C Manni, F Pelosi, M L Sampoli. Quasi-interpolation in isogeometric analysis based on generalized B-splines. Computer Aided Geometric Design, 2010, 27( 8): 656– 668 https://doi.org/10.1016/j.cagd.2010.07.004
66	M Z Li, L J Chen, Q Ma. A meshfree quasi-interpolation method for solving burgers’ equation. Mathematical Problems in Engineering, 2014, 2014 : 492072 https://doi.org/10.1155/2014/492072
67	Z D Ma, N Kikuchi, H C Cheng. Topological design for vibrating structures. Computer Methods in Applied Mechanics and Engineering, 1995, 121( 1–4): 259– 280 https://doi.org/10.1016/0045-7825(94)00714-X
68	F Ferrari, O Sigmund. A new generation 99 line matlab code for compliance topology optimization and its extension to 3D. Structural and Multidisciplinary Optimization, 2020, 62( 4): 2211– 2228 https://doi.org/10.1007/s00158-020-02629-w
69	X D Xie, A D Yang, N Jiang, W Zhao, Z S Liang, S T Wang. Adaptive topology optimization under suitably graded THB-spline refinement and coarsening. International Journal for Numerical Methods in Engineering, 2021, 122( 20): 5971– 5998 https://doi.org/10.1002/nme.6780

[1]	Jie GAO, Mi XIAO, Zhi YAN, Liang GAO, Hao LI. Robust isogeometric topology optimization for piezoelectric actuators with uniform manufacturability[J]. Front. Mech. Eng., 2022, 17(2): 27-.
[2]	Yingjun WANG, Liang GAO, Jinping QU, Zhaohui XIA, Xiaowei DENG. Isogeometric analysis based on geometric reconstruction models[J]. Front. Mech. Eng., 2021, 16(4): 782-797.
[3]	Pai LIU, Yi YAN, Xiaopeng ZHANG, Yangjun LUO. A MATLAB code for the material-field series-expansion topology optimization method[J]. Front. Mech. Eng., 2021, 16(3): 607-622.
[4]	Kai LONG, Xiaoyu YANG, Nouman SAEED, Ruohan TIAN, Pin WEN, Xuan WANG. Topology optimization of transient problem with maximum dynamic response constraint using SOAR scheme[J]. Front. Mech. Eng., 2021, 16(3): 593-606.
[5]	Liang XUE, Jie LIU, Guilin WEN, Hongxin WANG. Efficient, high-resolution topology optimization method based on convolutional neural networks[J]. Front. Mech. Eng., 2021, 16(1): 80-96.
[6]	Peng WEI, Wenwen WANG, Yang YANG, Michael Yu WANG. Level set band method: A combination of density-based and level set methods for the topology optimization of continuums[J]. Front. Mech. Eng., 2020, 15(3): 390-405.
[7]	Xianda XIE, Shuting WANG, Ming YE, Zhaohui XIA, Wei ZHAO, Ning JIANG, Manman XU. Isogeometric topology optimization based on energy penalization for symmetric structure[J]. Front. Mech. Eng., 2020, 15(1): 100-122.
[8]	Emmanuel TROMME, Atsushi KAWAMOTO, James K. GUEST. Topology optimization based on reduction methods with applications to multiscale design and additive manufacturing[J]. Front. Mech. Eng., 2020, 15(1): 151-165.
[9]	Mariana MORETTI, Emílio C. N. SILVA. Topology optimization of piezoelectric bi-material actuators with velocity feedback control[J]. Front. Mech. Eng., 2019, 14(2): 190-200.
[10]	Markus J. GEISS, Jorge L. BARRERA, Narasimha BODDETI, Kurt MAUTE. A regularization scheme for explicit level-set XFEM topology optimization[J]. Front. Mech. Eng., 2019, 14(2): 153-170.
[11]	Yu-Chin CHAN, Kohei SHINTANI, Wei CHEN. Robust topology optimization of multi-material lattice structures under material and load uncertainties[J]. Front. Mech. Eng., 2019, 14(2): 141-152.
[12]	Long JIANG, Yang GUO, Shikui CHEN, Peng WEI, Na LEI, Xianfeng David GU. Concurrent optimization of structural topology and infill properties with a CBF-based level set method[J]. Front. Mech. Eng., 2019, 14(2): 171-189.
[13]	Junjie ZHAN, Yangjun LUO. Robust topology optimization of hinge-free compliant mechanisms with material uncertainties based on a non-probabilistic field model[J]. Front. Mech. Eng., 2019, 14(2): 201-212.
[14]	Jikai LIU, Qian CHEN, Xuan LIANG, Albert C. TO. Manufacturing cost constrained topology optimization for additive manufacturing[J]. Front. Mech. Eng., 2019, 14(2): 213-221.
[15]	Manman XU, Shuting WANG, Xianda XIE. Level set-based isogeometric topology optimization for maximizing fundamental eigenfrequency[J]. Front. Mech. Eng., 2019, 14(2): 222-234.

Viewed

Full text

Abstract

Cited

Shared

Discussed