Additive manufacturing (AM) is now capable of fabricating geometrically complex geometries such as a variable-density lattice structure. This ability to handle geometric complexity provides the designer an opportunity to rethink the design method. In this work, a novel topology optimization algorithm is proposed to design variable-density lattice infill to maximize the first eigenfrequency of the structure. To make the method efficient, the lattice infill is treated as a continuum material with equivalent elastic properties obtained from asymptotic homogenization (AH), and the topology optimization is employed to find the optimum density distribution of the lattice structure. Specifically, the AH method is employed to calculate the effective mechanical properties of a predefined lattice structure as a function of its relative densities. Once the optimal density distribution is obtained, a continuous mapping technique is used to convert the optimal density distribution into variable-density lattice structured design. Two three-dimensional (3D) examples are used to validate the proposed method, where the designs are printed by the EOS direct metal laser sintering (DMLS) process in Ti6Al4V. Experimental results obtained from dynamical testing of the printed samples and detailed simulation results are in good agreement with the homogenized model results, which demonstrates the accuracy and efficiency of the proposed method.
Issue Section:
Design Innovation Paper
References
1.
Shapiro
, A.
, Borgonia
, J.
, Chen
, Q.
, Dillon
, R.
, McEnerney
, B.
, Polit-Casillas
, R.
, and Soloway, L., 2016
, “Additive Manufacturing for Aerospace Flight Applications
,” J. Spacecr. Rockets
, 53
(5
), pp. 952
–959
.2.
Zhu
, J.-H.
, Zhang
, W.-H.
, and Xia
, L.
, 2016
, “Topology Optimization in Aircraft and Aerospace Structures Design
,” Arch. Comput. Methods Eng.
, 23
(4
), pp. 595
–622
.3.
Giannatsis
, J.
, and Dedoussis
, V.
, 2009
, “Additive Fabrication Technologies Applied to Medicine and Health Care: A Review
,” Int. J. Adv. Manuf. Technol.
, 40
(1–2
), pp. 116
–127
.4.
Thompson
, S. M.
, Aspin
, Z. S.
, Shamsaei
, N.
, Elwany
, A.
, and Bian
, L.
, 2015
, “Additive Manufacturing of Heat Exchangers: A Case Study on a Multi-Layered Ti–6Al–4V Oscillating Heat Pipe
,” Addit. Manuf.
, 8
), pp. 163
–174
.5.
Rosen
, D. W.
, 2007
, “Computer-Aided Design for Additive Manufacturing of Cellular Structures
,” Comput.-Aided Des. Appl.
, 4
(5
), pp. 585
–594
.6.
Murr
, L.
, Gaytan
, S.
, Medina
, F.
, Lopez
, H.
, Martinez
, E.
, Machado
, B.
, Hernandez, D., Martinez, L., Lopez, M., and Wicker, R., 2010
, “Next-Generation Biomedical Implants Using Additive Manufacturing of Complex, Cellular and Functional Mesh Arrays
,” Philos. Trans. R. Soc. London A
, 368
(1917
), pp. 1999
–2032
.7.
Cheng
, L.
, Zhang
, P.
, Biyikli
, E.
, Bai
, J.
, Robbins
, J.
, and To
, A.
, 2017
, “Efficient Design Optimization of Variable-Density Cellular Structures for Additive Manufacturing: Theory and Experimental Validation
,” Rapid Prototyping J.
, 23
(4
), pp. 660–677.8.
Evans
, A. G.
, Hutchinson
, J.
, and Ashby
, M.
, 1998
, “Multifunctionality of Cellular Metal Systems
,” Prog. Mater. Sci.
, 43
(3
), pp. 171
–221
.9.
Zhao
, C.
, 2012
, “Review on Thermal Transport in High Porosity Cellular Metal Foams With Open Cells
,” Int. J. Heat Mass Transfer
, 55
(13–14
), pp. 3618
–3632
.10.
Ryan
, G.
, Pandit
, A.
, and Apatsidis
, D. P.
, 2006
, “Fabrication Methods of Porous Metals for Use in Orthopaedic Applications
,” Biomaterials
, 27
(13
), pp. 2651
–2670
.11.
Zhang
, P.
, Toman
, J.
, Yu
, Y.
, Biyikli
, E.
, Kirca
, M.
, Chmielus
, M.
, and To, A. C., 2015
, “Efficient Design-Optimization of Variable-Density Hexagonal Cellular Structure by Additive Manufacturing: Theory and Validation
,” ASME J. Manuf. Sci. Eng.
, 137
(2
), p. 021004
.12.
Huang
, X.
, Zhou
, S.
, Xie
, Y.
, and Li
, Q.
, 2013
, “Topology Optimization of Microstructures of Cellular Materials and Composites for Macrostructures
,” Comput. Mater. Sci.
, 67
, pp. 397
–407
.13.
Zhang
, J.
, Wang
, Z.
, and Zhao
, L.
, 2016
, “Dynamic Response of Functionally Graded Cellular Materials Based on the Voronoi Model
,” Composites, Part B
, 85
, pp. 176
–187
.14.
Akbarzadeh
, A.
, Fu
, J.
, Chen
, Z.
, and Qian
, L.
, 2014
, “Dynamic Eigenstrain Behavior of Magnetoelastic Functionally Graded Cellular Cylinders
,” Compos. Struct.
, 116
, pp. 404
–413
.15.
Akbarzadeh
, A.
, Fu
, J.
, Liu
, L.
, Chen
, Z.
, and Pasini
, D.
, 2016
, “Electrically Conducting Sandwich Cylinder With a Planar Lattice Core Under Prescribed Eigenstrain and Magnetic Field
,” Compos. Struct.
, 153
, pp. 632
–644
.16.
Wegst
, U. G.
, Bai
, H.
, Saiz
, E.
, Tomsia
, A. P.
, and Ritchie
, R. O.
, 2015
, “Bioinspired Structural Materials
,” Nat. Mater.
, 14
(1
), pp. 23
–36
.17.
Cheng
, L.
, Zhang
, P.
, Biyikli
, E.
, Bai
, J.
, Pilz
, S.
, and To
, A. C.
, 2015
, “Integration of Topology Optimization With Efficient Design of Additive Manufactured Cellular Structures
,” Solid Freeform Fabrication Symposium (SFF
), Austin, TX, Aug. 10–12, pp. 1370–1377.https://pdfs.semanticscholar.org/62dc/b8aea14d71d86cf578ae8e4c63b9281f70e9.pdf18.
Brackett
, D.
, Ashcroft
, I.
, and Hague
, R.
, 2011
, “Topology Optimization for Additive Manufacturing
,” Solid Freeform Fabrication Symposium
(SFF
), Austin, TX, pp. 348
–362
.http://sffsymposium.engr.utexas.edu/Manuscripts/2011/2011-27-Brackett.pdf19.
Wang
, X.
, Zhang
, P.
, Ludwick
, S.
, Belski
, E.
, and To
, A. C.
, 2018
, “Natural Frequency Optimization of 3D Printed Variable-Density Honeycomb Structure Via a Homogenization-Based Approach
,” Addit. Manuf.
, 20
(18), pp. 189–198.20.
Cheng
, L.
, Liu
, J.
, Liang
, X.
, and To
, A. C.
, 2018
, “Coupling Lattice Structure Topology Optimization With Design-Dependent Feature Evolution for Additive Manufactured Heat Conduction Design
,” Comput. Methods Appl. Mech. Eng.
, 332
, pp. 408
–439
.21.
Cheng
, L.
, Liu
, J.
, and To
, A. C.
, 2018
, “Concurrent Lattice Infill With Feature Evolution Optimization for Additive Manufactured Heat Conduction Design
,” Struct. Multidiscip. Optim.
, epub.22.
Bendsøe
, M. P.
, and Kikuchi
, N.
, 1988
, “Generating Optimal Topologies in Structural Design Using a Homogenization Method
,” Comput. Methods Appl. Mech. Eng.
, 71
(2
), pp. 197
–224
.23.
Bendsøe
, M. P.
, and Sigmund
, O.
, 1999
, “Material Interpolation Schemes in Topology Optimization
,” Arch. Appl. Mech.
, 69
(9–10
), pp. 635
–654
.24.
Bendsoe
, M. P.
, and Sigmund
, O.
, 2013
, Topology Optimization: Theory, Methods, and Applications
, Springer Science & Business Media
, Boston, MA.25.
Xie
, Y. M.
, and Steven
, G. P.
, 1993
, “A Simple Evolutionary Procedure for Structural Optimization
,” Comput. Struct.
, 49
(5
), pp. 885
–896
.26.
Xie
, Y. M.
, and Steven
, G. P.
, 1997
, “Basic Evolutionary Structural Optimization
,” Evolutionary Structural Optimization
, Springer
, London, pp. 12
–29
.27.
Wang
, M. Y.
, Wang
, X.
, and Guo
, D.
, 2003
, “A Level Set Method for Structural Topology Optimization
,” Comput. Methods Appl. Mech. Eng.
, 192
(1–2
), pp. 227
–246
.28.
Allaire
, G.
, Jouve
, F.
, and Toader
, A.-M.
, 2004
, “Structural Optimization Using Sensitivity Analysis and a Level-Set Method
,” J. Comput. Phys.
, 194
(1
), pp. 363
–393
.29.
Liu
, J.
, Ma
, Y.
, Fu
, J.
, and Duke
, K.
, 2015
, “A Novel CACD/CAD/CAE Integrated Design Framework for Fiber-Reinforced Plastic Parts
,” Adv. Eng. Software
, 87
, pp. 13
–29
.30.
Liu
, J.
, Cheng
, L.
, and To
, A. C.
, 2017
, “Arbitrary Void Feature Control in Level Set Topology Optimization
,” Comput. Methods Appl. Mech. Eng.
, 324
, pp. 595
–618
.31.
Li
, Q.
, Steven
, G. P.
, Querin
, O. M.
, and Xie
, Y.
, 1999
, “Shape and Topology Design for Heat Conduction by Evolutionary Structural Optimization
,” Int. J. Heat Mass Transfer
, 42
(17
), pp. 3361
–3371
.32.
Yaji
, K.
, Yamada
, T.
, Kubo
, S.
, Izui
, K.
, and Nishiwaki
, S.
, 2015
, “A Topology Optimization Method for a Coupled Thermal–Fluid Problem Using Level Set Boundary Expressions
,” Int. J. Heat Mass Transfer
, 81
, pp. 878
–888
.33.
Borrvall
, T.
, and Petersson
, J.
, 2003
, “Topology Optimization of Fluids in Stokes Flow
,” Int. J. Numer. Methods Fluids
, 41
(1
), pp. 77
–107
.34.
Qian
, X.
, and Dede
, E. M.
, 2016
, “Topology Optimization of a Coupled Thermal-Fluid System Under a Tangential Thermal Gradient Constraint
,” Struct. Multidiscip. Optim.
, 54
(3
), pp. 531
–551
.35.
Sigmund
, O.
, and Torquato
, S.
, 1997
, “Design of Materials With Extreme Thermal Expansion Using a Three-Phase Topology Optimization Method
,” Proc. SPIE
, 3040
, pp. 52
–60
.36.
Christiansen
, R. E.
, and Sigmund
, O.
, 2016
, “Designing Meta Material Slabs Exhibiting Negative Refraction Using Topology Optimization
,” Struct. Multidiscip. Optim.
, 54
(3
), pp. 469
–482
.37.
Díaaz
, A. R.
, and Kikuchi
, N.
, 1992
, “Solutions to Shape and Topology Eigenvalue Optimization Problems Using a Homogenization Method
,” Int. J. Numer. Methods Eng.
, 35
(7
), pp. 1487
–1502
.38.
Ma
, Z.-D.
, Kikuchi
, N.
, and Cheng
, H.-C.
, 1995
, “Topological Design for Vibrating Structures
,” Comput. Methods Appl. Mech. Eng.
, 121
(1–4
), pp. 259
–280
.39.
Ma
, Z.-D.
, Cheng
, H.-C.
, and Kikuchi
, N.
, 1994
, “Structural Design for Obtaining Desired Eigenfrequencies by Using the Topology and Shape Optimization Method
,” Comput. Syst. Eng.
, 5
(1
), pp. 77
–89
.40.
Pedersen
, N. L.
, 2000
, “Maximization of Eigenvalues Using Topology Optimization
,” Struct. Multidiscip. Optim.
, 20
(1
), pp. 2
–11
.41.
Du
, J.
, and Olhoff
, N.
, 2010
, “Topological Design of Vibrating Structures With Respect to Optimum Sound Pressure Characteristics in a Surrounding Acoustic Medium
,” Struct. Multidiscip. Optim.
, 42
(1
), pp. 43
–54
.42.
Du
, J.
, and Olhoff
, N.
, 2007
, “Topological Design of Freely Vibrating Continuum Structures for Maximum Values of Simple and Multiple Eigenfrequencies and Frequency Gaps
,” Struct. Multidiscip. Optim.
, 34
(2
), pp. 91
–110
.43.
Niu
, B.
, Yan
, J.
, and Cheng
, G.
, 2009
, “Optimum Structure With Homogeneous Optimum Cellular Material for Maximum Fundamental Frequency
,” Struct. Multidiscip. Optim.
, 39
(2
), pp. 115
–132
.44.
Huang
, X.
, Zuo
, Z.
, and Xie
, Y.
, 2010
, “Evolutionary Topological Optimization of Vibrating Continuum Structures for Natural Frequencies
,” Comput. Struct.
, 88
(5–6
), pp. 357
–364
.45.
Zuo
, Z. H.
, Huang
, X.
, Rong
, J. H.
, and Xie
, Y. M.
, 2013
, “Multi-Scale Design of Composite Materials and Structures for Maximum Natural Frequencies
,” Mater. Des.
, 51
, pp. 1023
–1034
.46.
Xia
, Q.
, Shi
, T.
, and Wang
, M. Y.
, 2011
, “A Level Set Based Shape and Topology Optimization Method for Maximizing the Simple or Repeated First Eigenvalue of Structure Vibration
,” Struct. Multidiscip. Optim.
, 43
(4
), pp. 473
–485
.47.
Gaynor
, A. T.
, and Guest
, J. K.
, 2016
, “Topology Optimization Considering Overhang Constraints: Eliminating Sacrificial Support Material in Additive Manufacturing Through Design
,” Struct. Multidiscip. Optim.
, 54
(5
), pp. 1157
–1172
.48.
Wu
, J.
, Wang
, C. C.
, Zhang
, X.
, and Westermann
, R.
, 2016
, “Self-Supporting Rhombic Infill Structures for Additive Manufacturing
,” Comput.-Aided Des.
, 80
, pp. 32
–42
.49.
Qian
, X.
, 2017, “Undercut and Overhang Angle Control in Topology Optimization: A Density Gradient Based Integral Approach
,” Int. J. Numer. Methods Eng.
, 111
(3), pp. 247–272. 50.
Bensoussan
, A.
, Lions
, J. L.
, and Papanicolaou
, G. C.
, 1979
, “Boundary Layers and Homogenization of Transport Processes
,” Publ. Res. Inst. Math. Sci.
, 15
(1
), pp. 53
–157
.51.
Bensoussan
, A.
, Lions
, J.-L.
, and Papanicolaou
, G.
, 2011
, Asymptotic Analysis for Periodic Structures
, Vol. 374
, American Mathematical Society
, Providence, RI.52.
Willis
, J. R.
, 1981
, “Variational and Related Methods for the Overall Properties of Composites
,” Adv. Appl. Mech.
, 21
, pp. 1
–78
.53.
Bakhvalov
, N. S.
, and Panasenko
, G.
, 2012
, Homogenisation: Averaging Processes in Periodic Media: Mathematical Problems in the Mechanics of Composite Materials
, Vol. 36
, Springer Science & Business Media
, Berlin.54.
Arabnejad
, S.
, and Pasini
, D.
, 2013
, “Mechanical Properties of Lattice Materials Via Asymptotic Homogenization and Comparison With Alternative Homogenization Methods
,” Int. J. Mech. Sci.
, 77
, pp. 249
–262
.55.
Hollister
, S. J.
, and Kikuchi
, N.
, 1992
, “A Comparison of Homogenization and Standard Mechanics Analyses for Periodic Porous Composites
,” Comput. Mech.
, 10
(2
), pp. 73
–95
.56.
Andreassen
, E.
, and Andreasen
, C. S.
, 2014
, “How to Determine Composite Material Properties Using Numerical Homogenization
,” Comput. Mater. Sci.
, 83
, pp. 488
–495
.57.
Cheng
, G.-D.
, Cai
, Y.-W.
, and Xu
, L.
, 2013
, “Novel Implementation of Homogenization Method to Predict Effective Properties of Periodic Materials
,” Acta Mech. Sin.
, 29
(4
), pp. 550
–556
.58.
Wang
, Y.
, Xu
, H.
, and Pasini
, D.
, 2017
, “Multiscale Isogeometric Topology Optimization for Lattice Materials
,” Comput. Methods Appl. Mech. Eng.
, 316
, pp. 568–585.59.
Terada
, K.
, Hori
, M.
, Kyoya
, T.
, and Kikuchi
, N.
, 2000
, “Simulation of the Multi-Scale Convergence in Computational Homogenization Approaches
,” Int. J. Solids Struct.
, 37
(16
), pp. 2285
–2311
.60.
Zhang
, W.
, and Sun
, S.
, 2006
, “Scale‐Related Topology Optimization of Cellular Materials and Structures
,” Int. J. Numer. Methods Eng.
, 68
(9
), pp. 993
–1011
.61.
Tantikom
, K.
, Aizawa
, T.
, and Mukai
, T.
, 2005
, “Symmetric and Asymmetric Deformation Transition in the Regularly Cell-Structured Materials—Part I: Experimental Study
,” Int. J. Solids Struct.
, 42
(8
), pp. 2199
–2210
.62.
Shu
, L.
, Wang
, M. Y.
, Fang
, Z.
, Ma
, Z.
, and Wei
, P.
, 2011
, “Level Set Based Structural Topology Optimization for Minimizing Frequency Response
,” J. Sound Vib.
, 330
(24
), pp. 5820
–5834
.63.
Kim
, N. H.
, Dong
, J.
, Choi
, K. K.
, Vlahopoulos
, N.
, Ma
, Z.-D.
, Castanier
, M.
, and Pierre, C., 2003
, “Design Sensitivity Analysis for Sequential Structural–Acoustic Problems
,” J. Sound Vib.
, 263
(3
), pp. 569
–591
.64.
Tcherniak
, D.
, 2002
, “Topology Optimization of Resonating Structures Using SIMP Method
,” Int. J. Numer. Methods Eng.
, 54
(11
), pp. 1605
–1622
.65.
Sundararajan
, V. G.
, 2010
, “Topology Optimization for Additive Manufacturing of Customized Meso-Structures Using Homogenization and Parametric Smoothing Functions
,” Doctoral dissertation
, The University of Texas at Austin, Austin, TXhttps://repositories.lib.utexas.edu/handle/2152/ETD-UT-2010-12-2302.66.
Sigmund
, O.
, 1997
, “On the Design of Compliant Mechanisms Using Topology Optimization*
,” J. Struct. Mech.
, 25
(4
), pp. 493
–524
.67.
Svanberg
, K.
, and Svärd
, H.
, 2013
, “Density Filters for Topology Optimization Based on the Pythagorean Means
,” Struct. Multidiscip. Optim.
, 48
(5
), pp. 859
–875
.68.
Svanberg, K., 2002, “
A Class of Globally Convergent Optimization Methods Based on Conservative Convex Separable Approximations
,” SIAM J. Optim.
, 12
(2), pp. 555–573.69.
Svanberg
, K.
, 1987
, “The Method of Moving a Asymptotes—A New Method for Structural Optimization
,” Int. J. Numer. Methods Eng.
, 24
(2), pp. 359–373.70.
Olhoff
, N.
, Lund
, E.
, and Seyranian
, A.
, 1994
, “Sensitivity Analysis and Optimization of Multiple Eigenvalues in Structural Design Problems
,” AIAA
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