Abstract
This paper presents formulations for hybrid casting and additive manufacturing (AM) in the density-based topology optimization. A location-based Heaviside function is introduced to represent the parting surface. The optimized part on two sides of the parting surface can be fabricated with casting, additive manufacturing, or both. Through the location-based Heaviside function and density gradient, two global constraints are formulated to remove undercuts and overhangs for casting and AM, respectively, inside the design domain. Since density gradient vanishes on the design domain boundary, two extra density-based global constraints are developed to control the overhangs and undercuts outside the design domain. Due to the smoothed parameterization of the parting surface, we are able to optimize the part and partition surface (including location and parting direction) simultaneously for hybrid casting and additive manufacturing. The proposed formulations for hybrid manufacturing processes are validated through 2D and 3D numerical examples. The proposed approach further enlarges the design space with manufacturing constraints, and has the potential to be used in the design for hybrid and multi-component manufacturing.
Issue Section:
Research Papers
References
1.
Gibson
, I.
, Rosen
, D.
, and Stucker
, B.
, 2014
, Additive Manufacturing Technologies: 3D Printing, Rapid Prototyping, and Direct Digital Manufacturing
, Springer
, New York
.2.
Liu
, J.
, and Ma
, Y.
, 2016
, “A Survey of Manufacturing Oriented Topology Optimization Methods
,” Adv. Eng. Softw.
, 100
, pp. 161
–175
. 3.
Poulsen
, T. A.
, 2003
, “A New Scheme for Imposing a Minimum Length Scale in Topology Optimization
,” Int. J. Numer. Meth. Eng.
, 57
(6
), pp. 741
–760
. 4.
Guest
, J. K.
, Prévost
, J. H.
, and Belytschko
, T.
, 2004
, “Achieving Minimum Length Scale in Topology Optimization Using Nodal Design Variables and Projection Functions
,” Int. J. Numer. Meth. Eng.
, 61
(2
), pp. 238
–254
. 5.
Wang
, F.
, Lazarov
, B. S.
, and Sigmund
, O.
, 2011
, “On Projection Methods, Convergence and Robust Formulations in Topology Optimization
,” Struct. Multidiscipl. Optim.
, 43
, pp. 767
–784
. 6.
Qian
, X.
, and Sigmund
, O.
, 2012
, “Topological Design of Electromechanical Actuators With Robustness Toward Over-and Under-Etching
,” Comput. Meth. Appl. Mech. Eng.
, 253
, pp. 237
–251
. 7.
Zhou
, M.
, Lazarov
, B. S.
, Wang
, F.
, and Sigmund
, O.
, 2015
, “Minimum Length Scale in Topology Optimization by Geometric Constraints
,” Comput. Meth. Appl. Mech. Eng.
, 293
, pp. 266
–282
. 8.
Guest
, J. K.
, 2009
, “Imposing Maximum Length Scale in Topology Optimization
,” Struct. Multidiscipl. Optim.
, 37
(5
), pp. 463
–473
. 9.
Wang
, Y.
, Zhang
, L.
, and Wang
, M. Y.
, 2016
, “Length Scale Control for Structural Optimization by Level Sets
,” Comput. Meth. Appl. Mech. Eng.
, 305
, pp. 891
–909
. 10.
Allaire
, G.
, Jouve
, F.
, and Michailidis
, G.
, 2016
, “Thickness Control in Structural Optimization Via a Level Set Method
,” Struct. Multidiscipl. Optim.
, 53
(6
), pp. 1349
–1382
. 11.
Zhou
, M.
, Fleury
, R.
, Shyy
, Y.
, Thomas
, H.
, and Brennan
, J.
, 2002
, “Progress in Topology Optimization With Manufacturing Constraints
,” 9th AIAA/ISSMO Symposium on Multidisciplinary Analysis and Optimization
, Atlanta, GA
, Sept. 4–6
, p. 5614
.12.
Xia
, Q.
, Shi
, T.
, Wang
, M. Y.
, and Liu
, S.
, 2010
, “A Level Set Based Method for the Optimization of Cast Part
,” Struct. Multidiscipl. Optim.
, 41
(5
), pp. 735
–747
. 13.
Xia
, Q.
, Shi
, T.
, Wang
, M. Y.
, and Liu
, S.
, 2011
, “Simultaneous Optimization of Cast Part and Parting Direction Using Level Set Method
,” Struct. Multidiscipl. Optim.
, 44
(6
), pp. 751
–759
. 14.
Gersborg
, A. R.
, and Andreasen
, C. S.
, 2011
, “An Explicit Parameterization for Casting Constraints in Gradient Driven Topology Optimization
,” Struct. Multidiscipl. Optim.
, 44
(6
), pp. 875
–881
. 15.
Sato
, Y.
, Yamada
, T.
, Izui
, K.
, and Nishiwaki
, S.
, 2017
, “Manufacturability Evaluation for Molded Parts Using Fictitious Physical Models, and Its Application in Topology Optimization
,” Int. J. Adv. Manuf. Technol.
, 92
(1–4
), pp. 1391
–1409
. 16.
Li
, Q.
, Chen
, W.
, Liu
, S.
, and Fan
, H.
, 2018
, “Topology Optimization Design of Cast Parts Based on Virtual Temperature Method
,” Computer-Aided Design
, 94
, pp. 28
–40
. 17.
Zhou
, H.
, Zhang
, J.
, Zhou
, Y.
, and Saitou
, K.
, 2019
, “Multi-Component Topology Optimization for Die Casting (MTO-D)
,” Struct. Multidiscip. Optim.
, 60
, pp. 2265
–2279
. 18.
Guest
, J. K.
, and Zhu
, M.
, 2012
, “Casting and Milling Restrictions in Topology Optimization Via Projection-Based Algorithms
,” ASME 2012 International Design Engineering Technical Conferences and Computers and Information in Engineering Conference
, Chicago, IL
, Aug. 12–15
, pp. 913
–920
.19.
Li
, H.
, Li
, P.
, Gao
, L.
, Zhang
, L.
, and Wu
, T.
, 2015
, “A Level Set Method for Topological Shape Optimization of 3D Structures With Extrusion Constraints
,” Comput. Meth. Appl. Mech. Eng.
, 283
, pp. 615
–635
. 20.
Mirzendehdel
, H. Y.
, Behandish
, M.
, and Nelaturi
, S.
, 2020
, “Topology Optimization With Accessibility Constraint for Multi–Axis Machining
,” Computer-Aided Design
, 122
, p. 102825
. 21.
Lee
, H. Y.
, Zhu
, M.
, and Guest
, J. K.
, 2022
, “Topology Optimization Considering Multi-axis Machining Constraints Using Projection Methods
,” Comput. Meth. Appl. Mech. Eng.
, 390
, p. 114464
. 22.
Gasick
, J.
, and Qian
, X.
, 2021
, “Simultaneous Topology and Machine Orientation Optimization for Multiaxis Machining
,” Int. J. Numer. Meth. Eng.
, 122
, pp. 7504
–7535
. 23.
Deng
, H.
, Vulimiri
, P. S.
, and To
, A. C.
, 2022
, “CAD-Integrated Topology Optimization Method With Dynamic Extrusion Feature Evolution for Multi-axis Machining
,” Comput. Meth. Appl. Mech. Eng.
, 390
, p. 114456
. 24.
Brackett
, D.
, Ashcroft
, I.
, and Hague
, R.
, 2011
, “Topology Optimization for Additive Manufacturing
,” Proceedings of the Solid Freeform Fabrication Symposium
, Austin, TX
, Aug. 8–10
, pp. 348
–362
.25.
Gaynor
, A. T.
, Meisel
, N. A.
, Williams
, C. B.
, and Guest
, J. K.
, 2014
, “Topology Optimization for Additive Manufacturing: Considering Maximum Overhang Constraint
,” 15th AIAA/ISSMO Multidisciplinary Analysis and Optimization Conference
, Atlanta, GA
, June 16–20
, pp. 20
–36
.26.
Gaynor
, A. T.
, and Guest
, J. K.
, 2016
, “Topology Optimization Considering Overhang Constraints: Eliminating Sacrificial Support Material in Additive Manufacturing Through Design
,” Struct. Multidiscipl. Optim.
, 54
(5
), pp. 1157
–1172
. 27.
Langelaar
, M.
, 2016
, “An Additive Manufacturing Filter for Topology Optimization of Print-Ready Designs
,” Struct. Multidiscipl. Optim.
, 55
, pp. 871
–883
. 28.
Langelaar
, M.
, 2016
, “Topology Optimization of 3D Self-Supporting Structures for Additive Manufacturing
,” Addit. Manuf.
, 12
, pp. 60
–70
.29.
Qian
, X.
, 2017
, “Undercut and Overhang Angle Control in Topology Optimization: A Density Gradient Based Integral Approach
,” Int. J. Numer. Meth. Eng.
, 111
(3
), pp. 247
–272
. 30.
Wang
, C.
, Qian
, X.
, Gerstler
, W. D.
, and Shubrooks
, J.
, 2019
, “Boundary Slope Control in Topology Optimization for Additive Manufacturing: for Self-Support and Surface Roughness
,” ASME J. Manuf. Sci. Eng.
, 141
(9
), p. 091001
. 31.
Wang
, C.
, and Qian
, X.
, 2020
, “Simultaneous Optimization of Build Orientation and Topology for Additive Manufacturing
,” Addit. Manuf.
, 34
, p. 101246
.32.
Zhang
, K.
, Cheng
, G.
, and Xu
, L.
, 2019
, “Topology Optimization Considering Overhang Constraint in Additive Manufacturing
,” Comput. Struct.
, 212
, pp. 86
–100
.33.
Zhang
, K.
, and Cheng
, G.
, 2020
, “Three-Dimensional High Resolution Topology Optimization Considering Additive Manufacturing Constraints
,” Addit. Manuf.
, 35
, p. 101224
.34.
Wang
, C.
, 2022
, “Simultaneous Optimization of Build Orientation and Topology for Self-Supported Enclosed Voids in Additive Manufacturing
,” Comput. Meth. Appl. Mech. Eng.
, 388
, p. 114227
. 35.
Langelaar
, M.
, 2018
, “Combined Optimization of Part Topology, Support Structure Layout and Build Orientation for Additive Manufacturing
,” Struct. Multidiscipl. Optim.
, 57
(5
), pp. 1985
–2004
. 36.
Mezzadri
, F.
, Bouriakov
, V.
, and Qian
, X.
, 2018
, “Topology Optimization of Self-Supporting Support Structures for Additive Manufacturing
,” Addit. Manuf.
, 21
, pp. 666
–682
.37.
Kuo
, Y.
, Cheng
, C.
, Lin
, Y.
, and San
, C.
, 2018
, “Support Structure Design in Additive Manufacturing Based on Topology Optimization
,” Struct. Multidiscipl. Optim.
, 57
(1
), pp. 183
–195
. 38.
Wang
, C.
, and Qian
, X.
, 2023
, “Simultaneous Optimization of Part and Support for Heat Dissipation in Additive Manufacturing
,” Struct. Multidiscip. Optim.
, 66
(1
), p. 3
. 39.
Wang
, C.
, and Qian
, X.
, 2020
, “Simultaneous Optimization of Part and Support for Heat Dissipation in Additive Manufacturing
,” 40th Computers and Information in Engineering Conference (CIE) of International Design Engineering Technical Conferences and Computers and Information in Engineering Conference
, Virtual, Online
, Aug. 17–19
, Vol. 9, p. V009T09A018.40.
Allaire
, G.
, and Bogosel
, B.
, 2018
, “Optimizing Supports for Additive Manufacturing
,” Struct. Multidiscipl. Optim.
, 58
(6
), pp. 2493
–2515
. 41.
Cheng
, L.
, Liang
, X.
, Bai
, J.
, Chen
, Q.
, Lemon
, J.
, and To
, A.
, 2019
, “On Utilizing Topology Optimization to Design Support Structure to Prevent Residual Stress Induced Build Failure in Laser Powder Bed Metal Additive Manufacturing
,” Addit. Manuf.
, 27
, pp. 290
–304
.42.
Zhou
, M.
, Liu
, Y.
, and Lin
, Z.
, 2019
, “Topology Optimization of Thermal Conductive Support Structures for Laser Additive Manufacturing
,” Comput. Meth. Appl. Mech. Eng.
, 353
, pp. 24
–43
. 43.
Jiang
, J.
, Xu
, X.
, and Stringer
, J.
, 2018
, “Support Structures for Additive Manufacturing: A Review
,” J. Manuf. Mater. Process.
, 2
(4
), p. 64
.44.
Bendsøe
, M. P.
, and Sigmund
, O.
, 2003
, Topology Optimization – Theory, Methods and Applications
, 2nd ed., Springer Verlag
, Berlin, Germany
.45.
Zhou
, M.
, and Rozvany
, G. I. N.
, 1991
, “The COC Algorithm, Part II: Topological, Geometrical and Generalized Shape Optimization
,” Comput. Meth. Appl. Mech. Eng.
, 89
(1–3
), pp. 309
–336
. 46.
Wang
, C.
, and Qian
, X.
, 2018
, “Heaviside Projection-Based Aggregation in Stress-Constrained Topology Optimization
,” Int. J. Numer. Meth. Eng.
, 115
(7
), pp. 849
–871
. 47.
Wang
, C.
, and Qian
, X.
, 2020
, “A Density Gradient Approach to Topology Optimization Under Design-Dependent Boundary Loading
,” J. Comput. Phys.
, 411
, p. 109398
. 48.
Haber
, R. B.
, Jog
, C. S.
, and Bendsøe
, M. P.
, 1996
, “A New Approach to Variable-Topology Shape Design Using a Constraint on Perimeter
,” Struct. Optim.
, 11
(1–2
), pp. 1
–12
. 49.
Clausen
, A.
, Aage
, N.
, and Sigmund
, O.
, 2015
, “Topology Optimization of Coated Structures and Material Interface Problems
,” Comput. Meth. Appl. Mech. Eng.
, 290
, pp. 524
–541
. 50.
Clausen
, A.
, Aage
, N.
, and Sigmund
, O.
, 2016
, “Exploiting Additive Manufacturing Infill in Topology Optimization for Improved Buckling Load
,” Engineering
, 2
(2
), pp. 250
–257
. 51.
Clausen
, A.
, Andreassen
, E.
, and Sigmund
, O.
, 2017
, “Topology Optimization of 3D Shell Structures With Porous Infill
,” Acta Mech. Sin.
, 33
, pp. 778
–791
. 52.
Sigmund
, O.
, and Maute
, K.
, 2013
, “Topology Optimization Approaches
,” Struct. Multidiscipl. Optim.
, 48
(6
), pp. 1031
–1055
. 53.
Lazarov
, B. S.
, and Sigmund
, O.
, 2011
, “Filters in Topology Optimization Based on Helmholtz-Type Differential Equations
,” Int. J. Numer. Meth. Eng.
, 86
(6
), pp. 765
–781
. 54.
Xu
, S.
, Cai
, Y.
, and Cheng
, G.
, 2010
, “Volume Preserving Nonlinear Density Filter Based on Heaviside Functions
,” Struct. Multidiscipl. Optim.
, 41
(4
), pp. 495
–505
. 55.
Eslami
, M. R.
, Hetnarski
, R. B.
, Ignaczak
, J.
, Noda
, N.
, Sumi
, N.
, and Tanigawa
, Y.
, 2013
, Theory of Elasticity and Thermal Stresses
, Vol. 197
, Springer
, Dordrecht
.56.
Bendsøe
, M. P.
, 1989
, “Optimal Shape Design as a Material Distribution Problem
,” Struct. Optim.
, 1
(4
), pp. 192
–202
.57.
Stolpe
, M.
, and Svanberg
, K.
, 2001
, “An Alternative Interpolation Scheme for Minimum Compliance Topology Optimization
,” Struct. Multidiscipl. Optim.
, 22
(2
), pp. 116
–124
. 58.
Bruyneel
, M.
, and Duysinx
, P.
, 2005
, “Note on Topology Optimization of Continuum Structures Including Self-Weight
,” Struct. Multidiscipl. Optim.
, 29
(4
), pp. 245
–256
. 59.
Zhang
, W.
, Yang
, J.
, Xu
, Y.
, and Gao
, T.
, 2014
, “Topology Optimization of Thermoelastic Structures: Mean Compliance Minimization or Elastic Strain Energy Minimization
,” Struct. Multidiscipl. Optim.
, 49
(3
), pp. 417
–429
. 60.
Logg
, A.
, Mardal
, K.
, and Wells
, G.
, 2012
, Automated Solution of Differential Equations by the Finite Element Method: The FEniCS Book
, Vol. 84
, Springer Science & Business Media
, Berlin
.61.
Svanberg
, K.
, 1987
, “The Method of Moving Asymptotes – A New Method for Structural Optimization
,” Int. J. Numer. Meth. Eng.
, 24
(2
), pp. 359
–373
. 62.
Holmberg
, E.
, Thore
, C. J.
, and Klarbring
, A.
, 2015
, “Worst-Case Topology Optimization of Self-Weight Loaded Structures Using Semi-Definite Programming
,” Struct. Multidiscip. Optim.
, 52
, pp. 915
–928
. 63.
Kumar
, P.
, 2022
, “Topology Optimization of Stiff Structures Under Self-Weight for Given Volume Using a Smooth Heaviside Function
,” Struct. Multidiscip. Optim.
, 64
(4
), p. 128
. Copyright © 2023 by ASME
You do not currently have access to this content.
