Generation of a Quadrilateral Mesh based on NURBS for Gyroids ‎of Variable Thickness and Porosity

Document Type : Research Paper


1 School of Engineering and Sciences, Tecnológico de Monterrey. Av. General Ramón Corona 2514, Col. Nuevo México, Zapopan, Jalisco, México, CP 45138‎

2 School of Engineering and Sciences, Tecnológico de Monterrey. Calle del Puente 222, Col. Ejidos de Huipulco, Tlalpan, Ciudad de México, México, CP 14380‎


The Gyroid is a periodic minimal surface explored in different applications, such as architecture and nanotechnology. The general topology is suitable for the construction of porous structures. This paper presents a non-iterative, novel methodology for the generation of a NURBS-based Gyroid volume. The Gyroid fundamental patch is defined with the Weierstrass parameterization. Furthermore, the geometry is manipulated to generate a structured mesh, allowing better element quality for FEM and IGA simulations. The re-parametrization is carried out by a Least-Squares approximation with a parametric NURBS surface, enabling a better definition of the mid-surface normals for the generation of the complete Gyroid volume. Different cases of variational thickness and porosity are presented to validate the versatility of our method.


Main Subjects

Publisher’s Note Shahid Chamran University of Ahvaz remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

[1] Karcher, H., Polthier, K., Construction of Triply Periodic Minimal Surfaces, Philosophical Transactions: Mathematical, Physical and Engineering Sciences, 354(1715), 1996, 2077-2104. 
[2] Han, L., Che, S., An Overview of Materials with Triply Periodic Minimal Surfaces and Related Geometry: From Biological Structures to Self-Assembled Systems, Advanced Materials, 30(17), 2018, 1705708.
[3] Emmer, M., Minimal Surfaces and Architecture: New Forms, Nexus Network Journal, 15(2), 2013, 227-239.
[4] Sychov, M.M., Lebedev, L.A., Dyachenko, S.V., Nefedova, L.A., Mechanical properties of energy-absorbing structures with triply periodic minimal surface topology, Acta Astronautica, 150, 2018, 81-84.
[5] Thomas, N., Sreedhar, N., Al-Ketan, O., Rowshan, R., Al-Rub, R.K., Arafat, H., 3D printed triply periodic minimal surfaces as spacers for enhanced heat and mass transfer in membrane distillation, Desalination, 443, 2018, 256-271.
[6] Park, J.H., Lee, J.C., Unusually high ratio of shear modulus to Young’s modulus in a nano-structured gyroid metamaterial, Scientific Reports, 7(10533), 2017. 
[7] Wei, D., Scherer, M.R., Bower, C., Andrew, P., Ryhanen, T., Steiner, U., A nanostructured electrochromic supercapacitor, Nano Letters, 12(4), 2012, 1857-62.
[8] Qin, Z., Jung, G.S., Kang, M.J., Buehler, M.J., The mechanics and design of a lightweight three-dimensional graphene assembly, Science Advances, 3(1), 2017, 1601536.
[9] Yang, Y., Wang, G., Liang, H., Gao, C., Peng, S., Shen, L., Shuai, C., Additive manufacturing of bone scaffolds, International Journal of Bioprinting, 5(1), 2019, 148.
[10] Bobber, F.S.L, Zadpoor, A.A., Effects of bone substitute architecture and surface properties on cell response, angiogenesis, and structure of new bone, Journal of Materials Chemistry B, 5(31), 2017, 6175-6192.
[11] Melchels, F.P.W., Bertoldi, K., Gabbrielli, R., Velders, A.H., Feijen, J., Grijpma, D.W., Mathematically defined tissue engineering scaffold architectures prepared by stereolithography, Biomaterials, 31(27), 2010, 6909-6916.
[12] Callens, S.J.P, Uyttendaele, R.J.C, Fratila-Apachitei, L.E., Zadpoor, A.A., Substrate curvature as a cue to guide spatiotemporal cell and tissue organization, Biomaterials, 23, 2020, 119739.
[13] Fantini, M., Curto, M., Crescenzio, F., TPMS for interactive modelling of trabecular scaffolds for Bone Tissue Engineering, Advances on Mechanics, Design Engineering and Manufacturing, Springer, 2016.
[14] Walker, J.M., Bodamer, E., Kleinfehn, A., Luo, Y., Becker, M., Dean, D., Design and mechanical characterization of solid and highly porous 3D printed poly(propylene fumarate) scaffolds, Progress in Additive Manufacturing, 2, 2017, 99-108.
[15] Al-Ketan, O., Al-Rub, R.K.A., MSLattice: A free software for generating uniform and graded lattices based on triply periodic minimal surfaces, Material Design & Processing Communications, 2020, e205.
[16] Shi, J., Yang, J., Zhu, L., Li, L., Li, Z., Wang, X., A Porous Scaffold Design Method for Bone Tissue Engineering Using Triply Periodic Minimal Surfaces, IEEE Access, 6, 2018, 1015-1022.
[17] Feng, J., Fu, J., Shang, C., Lin, Z., Li, B., Porous scaffold design by solid T-splines and triply periodic minimal surfaces, Computer Methods in Applied Mechanics and Engineering, 336, 2018, 333-352.
[18] Brakke, K., The Surface Evolver and the Stability of Liquid Surfaces, Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences, 354(1715), 1996, 2143-2157.
[19] Otoguro, Y., Mochizuki, H., Takizawa, K., Tezduyar, T.E., Space–Time Variational Multiscale Isogeometric Analysis of a tsunami-shelter vertical-axis wind turbine, Computational Mechanics, 66(6), 2020, 1443-1460.
[20] Costa, G., Montemurro, M., Pailhès, J., Minimum length scale control in a NURBS-based SIMP method, Computer Methods in Applied Mechanics and Engineering, 354, 2019, 963-989.
[21] Costa, G., Montemurro, M., Pailhès, J., Perry, N., Maximum length scale requirement in a topology optimization method based on NURBS hyper-surfaces. CIRP Annals Manufacturing Technology, 68, 2019, 153-156.
[22] Roiné, T., Montemurro, M., Pailhès, J., Stress-based topology optimization through non-uniform rational basis spline hyper-surfaces, Mechanics of Advanced Materials and Structures, 2021, 1-29.
[23] Montemurro, M., Bertolino, G., Roiné, T., A general multi-scale topology optimisation method for lightweight lattice structures obtained through additive manufacturing technology, Composite Structures, 258, 2021, 113360.
[24] Montemurro, M., Costa, G., Eigen-frequencies and harmonic responses in topology optimization: A CAD-compatible algorithm, Engineering Structures, 214, 2020, 110602.
[25] Montemurro, M., Refai, K., A Topology Optimization Method Based on Non-Uniform Rational Basis Spline Hyper-Surfaces for Heat Conduction Problems, Symmetry, 13(5), 2021, 888.
[26] Leyendecker, S., Penner, J., Biomechanical simulations with dynamic muscle paths on NURBS surfaces, PAAM, 19(1), 2019, 201900230.
[27] Audoux, Y., Montemurro, M., Pailhès, J., Non-Uniform Rational Basis Spline hyper-surfaces for metamodelling, Computer Methods in Applied Mechanics and Engineering, 364(4), 2020, 112918. 
[28] Audoux, Y., Montemurro, M., Pailhès, J., A Metamodel Based on Non-Uniform Rational Basis Spline Hyper-Surfaces for Optimisation of Composite Structures, Composite Structures, 247, 2020, 112439.
[29] Moutsanidis, G., Li, W., Bazilevs, Y., Reduced quadrature for FEM, IGA and meshfree methods, Computer Methods in Applied Mechanics and Engineering, 373, 2021, 113521.
[30] Voruganti, H.K., Gondeagaon, S., Comparative study of isogeometric analysis with finite element analysis, Energy Procedia, 2016, 8.
[31] Fiordilino, G.A., Izzi, M.I., Montemurro, M., A general isogeometric polar approach for the optimisation of variable stiffness composites: Application to eigenvalue buckling problems, Mechanics of Materials, 153, 2021, 103574.
[32] Gandy, P., Klinowski, J., Exact computation of the triply periodic G (‘Gyroid’) minimal surface, Chemical Physics Letters, 321(5-6), 2000, 363-371.
[33] Cottrell, J., Hughes, T., Bazilevs, Y., Isogeometric Analysis. Toward Integration of CAD and FEA, John Wiley & Sons Inc, United States, 2009.
[34] Floater, M., Parametrization and smooth approximation of surface triangulations, Computer Aided Geometric Design, 14(3), 1997, 231-250. 
[35] Costa, G., Montemurro, M., Pailhès, J., A General Optimization Strategy for Curve Fitting in the Non-Uniform Rational Basis Spline Framework, Journal of Optimization Theory and Applications, 176, 2018, 225-251. 
[36] Bertolino, G., Montemurro, M., Perry, N., Pourroy, F., An Efficient Hybrid Optimization Strategy for Surface Reconstruction, Computer Graphics Forum, 40(6), 2021, 215-241.
[37] Chen, X.D., Ma, W., Paul, J.C., Cubic B-spline curve approximation by curve unclamping, Computer-Aided Design, 42(6), 2010, 523-534.
[38] Vartziotis, D., Wipper, J., Papadrakakis, M., Improving mesh quality and finite element solution accuracy by GETMe smoothing in solving the Poisson equation, Finite Elements in Analysis and Design, 66, 2013, 36-52.
[39] Gandy, P., Cvijovic, D., Mackay, A., Klinowski, J., Exact computation of the triply periodic D (‘diamond’) minimal surface, Chemical Physics Letters, 314(5-6), 1999, 543-551.
[40] Gandy, P., Klinowski, J., Exact computation of the triply periodic Schwarz P minimal surface, Chemical Physics Letters, 322(6), 2000, 579-586.
[41] Cvijovic, D., Klinowski, J. The T and CLP families of triply periodic minimal surfaces. Part 1. Derivation of parametric equations, Journal de Physique I., 1992, 137-147.