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Deep Learning for Topology Optimization of Triply Periodic Minimal Surface Based Gyroid-Like Structures



Triply periodic minimal surfaces (TPMS) are non-intersecting complex geometrical surfaces that can be used in unit cell design of cellular structures. TPMS possess attractive properties like large surface area to volume ratio and mathematically controlled geometry which find them applications in catalytic converters, cocontinuous composites, thermal and permeability management, to name a few. The advent of additive manufacturing eased the manufacture of these structures which were previously challenging with traditional methods of manufacturing. Design of TPMS unit-cell based materials involves topology optimization to achieve the desired physical properties depending on the specific application of the structure. Topology optimization, in turn, involves the objective function evaluations for each iteration till converging to an optimal design and this may pose a computational burden when the function evaluations are time consuming finite element or computational fluid simulations. This can be alleviated by employing machine learning based methods for the optimization process. Deep learning using convolutional neural networks (CNN) have effectively been used for prediction of optimal topologies required for desired properties thus eliminating any objective function evaluations. In this paper, we explore the use of 3D CNN models for topology optimization of a TPMS based unit cell. The Solid Isotropic Material Penalization density method in topology optimization is employed on energy based homogenized unit cell properties. The unit cell that is obtained satisfying a desired mechanical property along with their topology parameters is then learnt to build a CNN model which can then be used to predict the optimal unit cell design for any topology parameters. The class of TPMS used in this work is Gyroids. The CNN model is tested for errors in prediction using mean square error metric and dice coefficient of the 2D slices of unit cell. The results indicate that the model can predict the ground truth accurately with few data points. This showed a promising approach in the area of TPMS based unit cell design using CNN.


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Gibson, L. J., and M. F. Ashby. 1999. Cellular Solids: Structure and Properties. Cambridge

University Press.

Yu, X., J. Zhou, H. Liang, Z. Jiang, and L. Wu. 2018. “Mechanical Metamaterials Associated with

Stiffness, Rigidity and Compressibility: A Brief Review,” Progress in Material Science, 94:114-

Aboudi, J. 1991. Mechanics of Composite Materials: A Unified Micromechanical Approach.

Elsevier Science.

Al-Ketan, O., and R. K. Abu Al-Rub. 2019. “Multifunctional Mechanical Metamaterials Based on

Triply Periodic Minimal Surface Lattices: A review,” Advanced Engineering Materials, 21(10).

Torquato, S., and A. Donev. 2004. “Minimal Surfaces and Multifunctionality”, Proc. R. Soc. Lond.

A., 460:1849–1856

Maskery, I., A. O. Aremu, L. Parry, R. D. Wildman, C. J. Tuck, and I. A. Ashcroft. 2018.

“Effective Design and Simulation of Surface-based Lattice Structures Featuring Volume Fraction

and Cell Type Grading,” Materials and Design, 155:220-232.

Abueidda, D. W., Rashid K. Abu Al-Rub, Ahmed S. Dalaq, Dong-Wook Lee, Kamran A. Khan,

Iwona Jasiuk. 2016. “Effective conductivities and elastic moduli of novel foams with triply periodic

minimal surfaces,” Mechanics of Materials 95:102-115.

Alya Alhammadi, Oraib Al-Ketan, Kamran A.Khan, Mohamed Ali, Reza Rowshan, Rashid K.Abu

Al-Rub. 2020. “Microstructural characterization and thermomechanical behavior of additively

manufactured AlSi10Mg sheet cellular materials”, Materials Science and Engineering: A 791,

Kamran A. Khan and Rashid K. Abu Al-Rub. 2018. “Modeling Time and Frequency Domain

Viscoelastic Behavior of Architectured Foams”, Journal of Engineering Mechanics 144(6).

Yang, E., M. Leary, B. Lozanovski, D. Downing, M. Mazur, A. Sarker, A. M. Khorasani, A. Jones,

T. Maconachie, S. Bateman, M. Easton, M. Qian, P. Choong, and M. Brandt. 2019. “Effect of

Geometry on the Mechanical Properties of Ti-6Al-4V Gyroid Structures Fabricated via SLM: A

Numerical Study,” Materials and Design, 184:108-165.

Al-Ketan, O., Pelanconi, M., Ortona, A., Abu Al-Rub, R.K. 2019. “Additive manufacturing of

architected catalytic ceramic substrates based on triply periodic minimal surfaces,” Journal of the

American Ceramic Society, 102: 6176-6193.

Schoen, A. H. 1970. Infinite periodic minimal surfaces without self-intersections. Nasa Technical

Note D-5541.

Michielsen, K., D. G. Stavenga. 2008. “Gyroid Cuticular Structures in Butterfly Wing Scales:

Biological Photonic Crystals,” J. Roy. Soc. Interface., 5(18):85-94.

Sigmund, O. 1994. “Materials with Prescribed Constitutive Parameters: An Inverse

Homogenization problem,” Int. J. Solids. Struct., 31(17):2313-2329.

Neves, M. M., H. Rodrigues, and J. M. Guedes. 2000. “Optimal design of periodic linear elastic

microstructures,” Comput. Struct., 76: 421-429.

Xia, Li., P. Breitkopf. 2015. “Design of Materials Using Topology Optimization and Energy-based

Homogenization approach in Matlab,” Struct. Multidiscipl. Optim., 52:1229-1241.

Gao, J., H. Li, L. Gao, and M. Xiao. 2018. “Topological Shape Optimization of 3D Microstructured

Materials Using Energy-based Homogenization Method,” Adv. Eng. Softw., 116:89-102.

Gao, J., H. Li, Z. Luo, L. Gao, and P. Li. 2020. “Topology Optimization of Micro-Structured

Materials Featured with the Specific Mechanical Properties,” Int. J. Comput. Methods, 17(3).

Kollmann, H. T., D. W. Abueidda, S. Koric, E. Guleryuz, and N.A. Sobh. 2020. “Deep Learning for

Topology Optimization of 2D metamaterials,” Mater. Des., 196:189-206.

Dong, G., Y. Tang, and Y. F. Zhao. 2019. “A 149 Line Homogenization Code for Three-

Dimensional Cellular Materials Written in MATLAB,” ASME. J. Eng. Mater. Technol., 141(1).

Guedes, J., and N. Kikuchi. 1990. “Preprocessing and Postprocessing for Materials Based on the

Homogenization Method with Adaptive Finite Element Methods,” Comput. Methods Appl. Mech.

Eng., 83(2):143-198.

Xia, Z., C. Zhou, Q. Yong, and X. Wang. 2006. “On Selection of Repeated Unit Cell Model and

Application of Unified Periodic Boundary Conditions in Micro-Mechanical Analysis of

Composites,” Int. J. Solids Struct., 43(2):266-278.

Bendsøe, M. P. 1989. “Optimal Shape Design as a Material Distribution Problem,” Struct. Optim.,


Sigmund, O. 2007. “Morphology-based Black and White Filters for Topology Optimization,”

Struct. Multidisc. Optim., 33: 401–424.

Sigmund, O., and J. Petersson. 1998. “Numerical Instabilities in Topology Optimization: A Survey

on Procedures Dealing with Checkerboards, Mesh-dependencies and Local Minima,” Struct.

Optim., 16:68-75.

Bruns, T. E., and D. A. Tortorelli. 2001. “Topology Optimization of Non-linear Elastic Structures

and Compliant Mechanisms,” Comput. Methods Appl. Mech. Eng.,190: 3443-3459.

Zhang, Z., Q. Liu, and Y. Wang. 2018. “Road Extraction by Deep Residual U-Net,” IEEE. Geosci.

Remote. Sens. Lett., 15(5):749-753.

Zou, K. H., S. K. Warfield, A. Bharatha, C. M. C. Tempany, M. R. Kaus, S. J. Haker, W. M.

Wells, F. A. Jolesz, and R. Kikinis. 2004. “Statistical Validation of Image Segmentation Quality

Based on a Spatial Overlap Index,” Acad. Radiol., 11(2): 178-189.


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