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Research Papers

Optimizing Topology and Gradient Orthotropic Material Properties Under Multiple Loads

[+] Author and Article Information
Anthony Garland

Department of General Engineering,
Clemson University,
Clemson, SC 29634
e-mail: apg@clemson.edu

Georges Fadel

Department of Mechanical Engineering,
Clemson University,
Clemson, SC 29634
e-mail: fgeorge@clemson.edu

Manuscript received September 19, 2017; final manuscript received September 26, 2018; published online February 6, 2019. Assoc. Editor: Rahul Rai.

J. Comput. Inf. Sci. Eng 19(2), 021007 (Feb 04, 2019) (12 pages) Paper No: JCISE-17-1188; doi: 10.1115/1.4041744 History: Received September 19, 2017; Revised September 26, 2018

The goal of this research is to optimize an object's macroscopic topology and localized gradient material properties (GMPs) subject to multiple loading conditions simultaneously. The gradient material of each macroscopic cell is modeled as an orthotropic material where the elastic moduli in two local orthogonal directions we call x and y can change. Furthermore, the direction of the local coordinate system can be rotated to align with the loading conditions on each cell. This orthotropic material is similar to a fiber-reinforced material where the number of fibers in the local x and y-directions can change for each cell, and the directions can as well be rotated. Repeating cellular unit cells, which form a mesostructure, can also achieve these customized orthotropic material properties. Homogenization theory allows calculating the macroscopic averaged bulk properties of these cellular materials. By combining topology optimization with gradient material optimization and fiber orientation optimization, the proposed algorithm significantly decreases the objective, which is to minimize the strain energy of the object subject to multiple loading conditions. Additive manufacturing (AM) techniques enable the fabrication of these designs by selectively placing reinforcing fibers or by printing different mesostructures in each region of the design. This work shows a comparison of simple topology optimization, topology optimization with isotropic gradient materials, and topology optimization with orthotropic gradient materials. Finally, a trade-off experiment shows how different optimization parameters, which affect the range of gradient materials used in the design, have an impact on the final objective value of the design. The algorithm presented in this paper offers new insight into how to best take advantage of new AM capabilities to print objects with gradient customizable material properties.

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References

Sobczak, J. J. , and Drenchev, L. , 2013, “ Metallic Functionally Graded Materials: A Specific Class of Advanced Composites,” J. Mater. Sci. Technol., 29(4), pp. 297–316. [CrossRef]
Birman, V. , and Byrd, L. W. , 2007, “ Modeling and Analysis of Functionally Graded Materials and Structures,” Appl. Mech. Rev., 60(5), pp. 195–216. [CrossRef]
Griffith, M. L. , Harwell, L. D. , Romero, J. T. , Schlienger, E. , Atwood, C. L. , and Smugeresky, J. E. , 1997, “ Multi-Material Processing by LENS,” Solid Freeform Fabrication Symposium, Austin, TX, Aug. 11–13, p. 387. http://citeseerx.ist.psu.edu/viewdoc/download?doi=10.1.1.476.3973&rep=rep1&type=pdf
Tammas-Williams, S. , and Todd, I. , 2016, “ Design for Additive Manufacturing With Site-Specific Properties in Metals and Alloys,” Scr. Mater., 135, pp. 105–110.
Guedes, J. M. , and Kikuchi, N. , 1990, “ Preprocessing and Postprocessing for Materials Based on the Homogenization Method With Adaptive Finite Element Methods,” Comput. Methods Appl. Mech. Eng., 83(2), pp. 143–198. [CrossRef]
Shankar, P. , Fazelpour, M. , and Summers, J. D. , 2015, “ Comparative Study of Optimization Techniques in Sizing Mesostructures for Use in Nonpneumatic Tires,” ASME J. Comput. Inf. Sci. Eng., 15(4), p. 041009. [CrossRef]
Thompson, M. K. , Moroni, G. , Vaneker, T. , Fadel, G. , Campbell, R. I. , Gibson, I. , Bernard, A. , Schulz, J. , Graf, P. , Ahuja, B. , and Martina, F. , 2016, “ Design for Additive Manufacturing: Trends, Opportunities, Considerations, and Constraints,” CIRP Ann. Manuf. Technol., 65(2), pp. 737–760. [CrossRef]
Garland, A. , and Fadel, G. , 2015, “ Design and Manufacturing Functionally Gradient Material Objects With an Off the Shelf Three-Dimensional Printer: Challenges and Solutions,” ASME J. Mech. Des., 137(11), p. 111709. [CrossRef]
Garland, A. , and Fadel, G. , 2016, “ Multi-Objective Optimal Design of Functionally Gradient Materials,” ASME Paper No. DETC2016-59298.
Matsuzaki, R. , Ueda, M. , Namiki, M. , Jeong, T.-K. , Asahara, H. , Horiguchi, K. , Nakamura, T. , Todoroki, A. , and Hirano, Y. , 2016, “ Three-Dimensional Printing of Continuous-Fiber Composites by In-Nozzle Impregnation,” Sci. Rep., 6, p. 23058. [CrossRef] [PubMed]
Huang, Y. , Leu, M. C. , Mazumder, J. , and Donmez, A. , 2014, “ Additive Manufacturing: Current State, Future Potential, Gaps and Needs, and Recommendations,” ASME J. Manuf. Sci. Eng., 137, p. 014001.
Sigmund, O. , and Maute, K. , 2013, “ Topology Optimization Approaches,” Struct. Multidiscip. Optim., 48(6), pp. 1031–1055. [CrossRef]
Tang, Y. , Hascoet, J.-Y. , and Zhao, Y. F. , 2014, “ Integration of Topological and Functional Optimization in Design for Additive Manufacturing,” ASME Paper No. ESDA2014-20381.
Tang, Y. , and Zhao, Y. F. , 2016, “ A Survey of the Design Methods for Additive Manufacturing to Improve Functional Performance,” Rapid Prototyping J., 22(3), pp. 569–590. [CrossRef]
Rosen, D. W. , 2014, “ Multiscale, Heterogeneous Computer Aided Design Representation for Metal Alloy Microstructures,” ASME J. Comput. Inf. Sci. Eng., 14(4), p. 41003. [CrossRef]
Muller, P. , Hascoet, J.-Y. , and Mognol, P. , 2014, “ Toolpaths for Additive Manufacturing of Functionally Graded Materials (FGM) Parts,” Rapid Prototyping J., 20(6), pp. 511–522. [CrossRef]
Comotti, C. , Regazzoni, D. , Rizzi, C. , and Vitali, A. , 2017, “ Additive Manufacturing to Advance Functional Design: An Application in the Medical Field,” ASME J. Comput. Inf. Sci. Eng., 17(3), p. 031006. [CrossRef]
Eschenauer, H. A. , and Olhoff, N. , 2001, “ Topology Optimization of Continuum Structures: A Review*,” ASME Appl. Mech. Rev., 54(4), pp. 331–390. [CrossRef]
Wang, M. , 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. [CrossRef]
van Dijk, N. P. , Maute, K. , Langelaar, M. , and Van Keulen, F. , 2013, “ Level-Set Methods for Structural Topology Optimization: A Review,” Struct. Multidiscip. Optim., 48(3), pp. 437–472. [CrossRef]
Bendsoe, M. P. , 1995, Optimization of Structural Topology, Shape, and Material, Springer-Verlag, Berlin, Chap. 1.
Sigmund, O. , 2001, “ A 99 Line Topology Optimization Code Written in Matlab,” Struct. Multidiscip. Optim., 21(2), pp. 120–127. [CrossRef]
Garland, A. , Mocko, G. , and Fadel, G. , 2014, “ Challenges in Designing and Manufacturing Fully Optimized Functional Gradient Material Objects,” ASME Paper No. DETC2014-34544.
Hughes, T. J. R. , Cottrell, J. A. , and Bazilevs, Y. , 2005, “ Isogeometric Analysis: CAD, Finite Elements, NURBS, Exact Geometry and Mesh Refinement,” Comput. Methods Appl. Mech. Eng., 194(39–41), pp. 4135–4195. [CrossRef]
Morvan, S. , 2001, “ MMa-Rep, A Representation for Multimaterial Solids,” Clemson University, Clemson, SC.
Kou, X. Y. , and Tan, S. T. , 2007, “ Heterogeneous Object Modeling: A Review,” Comput.-Aided Des., 39(4), pp. 284–301. [CrossRef]
Jackson, T. R. , 2000, “ Analysis of Functionally Graded Material Object Representation Methods,” Ph.D. thesis, Massachusetts Institute of Technology, Cambridge, MA. https://dspace.mit.edu/handle/1721.1/9032
Gupta, V. , and Tandon, P. , 2017, “ Heterogeneous Composition Adaptation With Material Convolution Control Features,” ASME J. Comput. Inf. Sci. Eng., 17(2), p. 021008. [CrossRef]
Taheri, A. H. , Hassani, B. , and Moghaddam, N. Z. , 2014, “ Thermo-Elastic Optimization of Material Distribution of Functionally Graded Structures by an Isogeometrical Approach,” Int. J. Solids Struct., 51(2), pp. 416–429. [CrossRef]
Huang, J. , Fadel, G. M. , Blouin, V. Y. , and Grujicic, M. , 2002, “ Bi-Objective Optimization Design of Functionally Gradient Materials,” Mater. Des., 23(7), pp. 657–666. [CrossRef]
Kou, X. Y. , and Tan, S. T. , 2005, “ A Hierarchical Representation for Heterogeneous Object Modeling,” Comput.-Aided Des., 37(3), pp. 307–319. [CrossRef]
Kou, X. Y. , Parks, G. T. , and Tan, S. T. , 2012, “ Optimal Design of Functionally Graded Materials Using a Procedural Model and Particle Swarm Optimization,” Comput.-Aided Des., 44(4), pp. 300–310. [CrossRef]
Hu, Y. , Fadel, G. M. , Blouin, V. Y. , and White, D. R. , 2006, “ Optimal Design for Additive Manufacturing of Heterogeneous Objects Using Ultrasonic Consolidation,” Virtual Phys. Prototyping, 1(1), pp. 53–62. [CrossRef]
Cho, J. R. , and Ha, D. Y. , 2002, “ Optimal Tailoring of 2D Volume-Fraction Distributions for Heat-Resisting Functionally Graded Materials Using FDM,” Comput. Methods Appl. Mech. Eng., 191(29–30), pp. 3195–3211. [CrossRef]
Cho, J. R. , and Shin, S. W. , 2004, “ Material Composition Optimization for Heat-Resisting FGMs by Artificial Neural Network,” Compos. Part A Appl. Sci. Manuf., 35(5), pp. 585–594. [CrossRef]
Coelho, P. G. , Guedes, J. M. , and Rodrigues, H. C. , 2015, “ Multiscale Topology Optimization of Bi-Material Laminated Composite Structures,” Compos. Struct., 132, pp. 495–505. [CrossRef]
Stegmann, J. , and Lund, E. , 2005, “ Discrete Material Optimization of General Composite Shell Structures,” Int. J. Numer. Methods Eng., 62(14), pp. 2009–2027. [CrossRef]
Henrichsen, S. , Lindgaard, E. , and Lund, E. , 2015, “ Free Material Stiffness Design of Laminated Composite Structures Using Commercial Finite Element Analysis Codes,” Struct. Multidiscip. Optim., 51(5), pp. 1097–1111. [CrossRef]
Liu, J. , Duke, K. , and Ma, Y. , 2015, “ Computer-Aided Design–Computer-Aided Engineering Associative Feature-Based Heterogeneous Object Modeling,” Adv. Mech. Eng., 7(12), epub.
Lipton, R. , and Stuebner, M. , 2007, “ Optimal Design of Composite Structures for Strength and Stiffness: An Inverse Homogenization Approach,” Struct. Multidiscip. Optim., 33(4–5), pp. 351–362. [CrossRef]
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. [CrossRef]
Garland, A. , and Fadel, G. , 2015, “ Manufacturing Functionally Gradient Material Objects With an Off the Shelf 3D Printer: Challenges and Solutions,” ASME Paper No. DETC2015-47841.
Garland, A. , 2017, “ garland3/clemsonPhD: JCISE Paper Snapshot,” accessed Nov. 24, 2018, https://doi.org/10.5281/zenodo.148127
Rodrigues, H. , Guedes, J. M. , and Bendsoe, M. P. , 2002, “ Hierarchical Optimization of Material and Structure,” Struct. Multidiscip. Optim., 24(1), pp. 1–10. [CrossRef]
Coelho, P. G. , Fernandes, P. R. , Guedes, J. M. , and Rodrigues, H. C. , 2008, “ A Hierarchical Model for Concurrent Material and Topology Optimisation of Three-Dimensional Structures,” Struct. Multidiscip. Optim., 35(2), pp. 107–115. [CrossRef]
Xia, Q. , and Wang, M. Y. , 2008, “ Simultaneous Optimization of the Material Properties and the Topology of Functionally Graded Structures,” Comput.-Aided Des., 40(6), pp. 660–675. [CrossRef]
Dunning, P. D. , Brampton, C. J. , and Kim, H. A. , 2015, “ Simultaneous Optimisation of Structural Topology and Material Grading Using Level Set Method,” Mater. Sci. Technol., 31(8), pp. 884–894. [CrossRef]
Sigmund, O. , 1995, “ Tailoring Materials With Prescribed Elastic Properties,” Mech. Mater., 20(4), pp. 351–368. [CrossRef]
Zuo, Z. H. , Huang, X. , Yang, X. , Rong, J. H. , and Xie, Y. M. , 2013, “ Comparing Optimal Material Microstructures With Optimal Periodic Structures,” Comput. Mater. Sci., 69, pp. 137–147. [CrossRef]
Huang, X. , Radman, A. , and Xie, Y. M. , 2011, “ Topological Design of Microstructures of Cellular Materials for Maximum Bulk or Shear Modulus,” Comput. Mater. Sci., 50(6), pp. 1861–1870. [CrossRef]
Van Der Klift, F. , Koga, Y. , Todoroki, A. , Ueda, M. , Hirano, Y. , and Matsuzaki, R. , 2015, “ 3D Printing of Continuous Carbon Fibre Reinforced Thermo-Plastic (CFRTP) Tensile Test Specimens,” Open J. Compos. Mater., 6(1), pp. 18–27. https://www.scirp.org/journal/PaperInformation.aspx?PaperID=62614

Figures

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Fig. 1

Bridge macrostructure with varying mesostructure

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Fig. 2

Increasing the range of material properties allowed in the design decreases the objective function

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Fig. 3

Evaluation of the objective function value of an element shown as the solid black line. The vertical dotted line shows the actual minimum. The shaded region shows a local minimum region trap to avoid.

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Fig. 4

Equivalent states of the design variables are shown

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Fig. 5

Optimization flow chart

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Fig. 6

The top left image shows the discretization of the domain. In the remaining images, the six arrows represent six different loading cases where the arrow shows the location of the load. The bottom left image shows four loading conditions where the load is applied on the top of the structure and distributed over one quarter of the columns. The left images show two sheer cases where the load is distributed over the entire top node row and the load direction is to the left or right. The arrows do not represent point loads, but instead show the region where a distributed load is applied. The bottom of the design domain is fixed in the X and Y directions.

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Fig. 7

Output of the algorithm for the bridge design

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Fig. 8

“Vol Target” is ρtarget, “Actual Vol” is the current average ρ in the domain, “E avg” is 1/2Etarget∑e=1N(Exx+Eyyρ/2), “E target” is 1/2EtargetEtarget, and “Elast Obj” is objective J. The x-axis is optimizer calls, so three points represent one complete loop from Fig. 5. The objective is normalized by dividing by the greatest objective value.

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Fig. 9

Comparison of topology optimization techniques. Left: topology, Exx, Eyy, and rotation; center: topology and isotropic gradient materials; right: topology only.

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Fig. 10

Output design of the load on the left and clamped on the right. The output is represented using Eqs. (32) and (33).

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Fig. 11

Optimization metrics plot for the design in Fig. 10. “Vol Target” is ρtarget, “Actual Vol” is the current average ρ in the domain, “E avg” is 1/2Etarget∑e=1N(Exx+Eyyρ/2), “E target” is 1/2EtargetEtarget, and “Elast Obj” is objective J. The x-axis is optimizer calls, so three points represent one complete loop from Fig. 5. The objective is normalized by dividing by the greatest objective value.

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Fig. 12

The bar plot shows the relationship between the objective and ρtarget. Three of the final designs are also shown.

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