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Minisurf – A minimal surface generator for finite element modeling and additive manufacturing

      Highlights

      • MiniSurf is an efficient CAD file generator for triply periodic minimal surfaces.
      • MiniSurf creates .slt files for additive manufacturing and input meshes for finite element modeling.
      • MiniSurf allows user-defined minimal surfaces.
      • MiniSurf has a simple and sleek GUI enabling user-controlled patterning and mesh density.

      Abstract

      Triply periodic minimal surfaces (TPMSs) have long been studied by mathematicians but have recently garnered significant interest from the engineering community as ideal topologies for shell-based architected materials with both mechanical and functional applications. Here, we present a TPMS generator, MiniSurf. It combines surface visualization and CAD file generation (for both finite element modeling and additive manufacturing) within one single GUI. MiniSurf presently can generate 19 built-in and one user-defined triply periodic minimal surfaces based on their level-set surface approximations. Users can fully control the periodicity and precision of the generated surfaces. We show that MiniSurf can potentially be a very useful tool in designing and fabricating architected materials.

      Keywords

      Code metadata
      Tabled 1
      Current Code versionv1.0
      Permanent link to code/repository used for this code versionhttps://github.com/SoftwareImpacts/SIMPAC-2020-28
      Permanent link to reproducible capsulehttps://codeocean.com/capsule/1851964/tree/v1
      Legal Software LicenseMIT
      Code versioning system usedNone
      Software code languages, tools, and services usedMatlab
      Compilation requirements, operating environments, & dependencies
      If available Link to developer documentation/manual
      Support email for questions[email protected] and [email protected]
      Software metadata
      Tabled 1
      Current software versionv1.0
      Permanent link to executables of this versionhttps://github.com/mengtinh/MiniSurf
      Permanent link to Reproducible Capsulehttps://codeocean.com/capsule/1851964/tree/v1
      Legal Software LicenseMIT
      Computing platform / Operating SystemMicrosoft Windows
      Installation requirements & dependenciesMatlab Runtime
      If available Link to user manual — if formally published include a reference to the publication in the reference listhttps://github.com/mengtinh/MiniSurf/blob/master/User%20Manual.pdf
      Support email for questions[email protected] and [email protected]

      1. Introduction

      For decades, scientists and engineers have been striving to design and fabricate new multiphase materials with controlled phase topologies – often termed “architected materials” or “metamaterials” – with unprecedented and tunable combinations of properties; architected cellular materials, where one phase is void, are the most notable examples. In terms of mechanical behavior, significant efforts have focused on designing architected materials that are stiff, strong and tough at very low density, by optimizing the topology of the material phases. Traditionally, topologies have largely been limited to beam-based structures, such as honeycombs in 2D [
      • Wadley H.N.
      Multifunctional periodic cellular metals.
      ,
      • Zhang Y.H.
      • Qiu X.M.
      • Fang D.N.
      Mechanical properties of two novel planar lattice structures.
      ,
      • Asadpoure A.
      • Valdevit L.
      Topology optimization of lightweight periodic lattices under simultaneous compressive and shear stiffness constraints.
      ,
      • Russell B.P.
      • Deshpande V.S.
      • Wadley H.N.G.
      Quasistatic deformation and failure modes of composite square honeycombs.
      ,
      • Dharmasena K.P.
      • Wadley H.N.G.
      • Xue Z.
      • Hutchinson J.W.
      Mechanical response of metallic honeycomb sandwich panel structures to high-intensity dynamic loading.
      ,
      • Christodoulou I.
      • Tan P.J.
      Crack initiation and fracture toughness of random Voronoi honeycombs.
      ,
      • Hsieh M.-T.
      • Deshpande V.S.
      • Valdevit L.
      A versatile numerical approach for calculating the fracture toughness and R-curves of cellular materials.
      ] and octet lattices in 3D [
      • Deshpande V.S.
      • Fleck N.A.
      • Ashby M.F.
      Effective properties of the octet-truss lattice material.
      ,
      • O’Masta M.R.
      • Dong L.
      • St-Pierre L.
      • Wadley H.N.G.
      • Deshpande V.S.
      The fracture toughness of octet-truss lattices.
      ,
      • Dong L.
      • Deshpande V.
      • Wadley H.
      Mechanical response of ti-6al-4v octet-truss lattice structures.
      ,
      • Wendy Gu X.
      • Greer J.R.
      Ultra-strong architected Cu meso-lattices.
      ,
      • Bagheri A.
      • Buj-Corral I.
      • Ferrer M.
      • Pastor M.M.
      • Roure F.
      Determination of the elasticity modulus of 3D printed octet-truss structures for use in porous prosthesis implants.
      ,
      • Zheng X.Y.
      • Lee H.
      • Weisgraber T.H.
      • Shusteff M.
      • Deotte J.
      • Duoss E.B.
      • Kuntz J.D.
      • Biener M.M.
      • Ge Q.
      • Jackson J.A.
      • Kucheyev S.O.
      • Fang N.X.
      • Spadaccini C.M.
      Ultralight, ultrastiff mechanical metamaterials.
      ,
      • Meza L.R.
      • Das S.
      • Greer J.R.
      Strong, lightweight, and recoverable three-dimensional.
      ]. More recently, interest has shifted to shell-based topologies with minimal surface characteristics, such as triply periodic minimal surfaces (TPMS) [
      • Maskery I.
      • Sturm L.
      • Aremu A.O.
      • Panesar A.
      • Williams C.B.
      • Tuck C.J.
      • Wildman R.D.
      • Ashcroft I.A.
      • Hague R.J.M.
      Insights into the mechanical properties of several triply periodic minimal surface lattice structures made by polymer additive manufacturing.
      ,
      • Al-ketan O.
      • Rezgui R.
      • Rowshan R.
      • Du H.
      • Fang N.X.
      Microarchitected stretching-dominated mechanical metamaterials with minimal surface.
      ,
      • Helou M.
      • Kara S.
      • Design R.
      Analysis and manufacturing of lattice structures: An overview.
      ,
      • Maskery I.
      • Aboulkhair N.T.
      • Aremu A.O.
      • Tuck C.J.
      • Ashcroft I.A.
      Compressive failure modes and energy absorption in additively manufactured double gyroid lattices.
      ,
      • Bai L.
      • Gong C.
      • Chen X.
      • Sun Y.
      • Zhang J.
      • Cai L.
      • Zhu S.
      • Xie S.Q.
      Additive manufacturing of customized metallic orthopedic implants: Materials, structures, and surface modifications.
      ,
      • Ataee A.
      • Li Y.
      • Brandt M.
      • Wen C.
      Ultrahigh-strength titanium gyroid scaffolds manufactured by selective laser melting (SLM) for bone implant applications.
      ] and isotropic stochastic spinodal minimal surfaces [
      • Hsieh M.-T.
      • Endo B.
      • Zhang Y.
      • Bauer J.
      • Valdevit L.
      The mechanical response of cellular materials with spinodal topologies.
      ,
      • Guell Izard A.
      • Bauer J.
      • Crook C.
      • Turlo V.
      • Valdevit L.
      Ultrahigh energy absorption multifunctional spinodal nanoarchitectures.
      ,
      • Kochmann D.M.
      • Hopkins J.B.
      • Valdevit L.
      Multiscale modeling and optimization of the mechanics of hierarchical metamaterials.
      ]; while more challenging to fabricate, these topologies are devoid of nodes and other stress intensification regions, which results in improved strength and toughness [
      • Hsieh M.-T.
      • Endo B.
      • Zhang Y.
      • Bauer J.
      • Valdevit L.
      The mechanical response of cellular materials with spinodal topologies.
      ,
      • Schwaiger R.
      • Meza L.R.
      • Li X.
      The extreme mechanics of micro- and nanoarchitected materials.
      ,
      • Han S.C.
      • Lee J.W.
      • Kang K.
      A new type of low density material: Shellular.
      ,
      • Garcia A.E.
      • Wang C.S.
      • Sanderson R.N.
      • McDevitt K.M.
      • Zhang Y.
      • Valdevit L.
      • Mumm D.R.
      • Mohraz A.
      • Ragan R.
      Scalable synthesis of gyroid-inspired freestanding three-dimensional graphene architectures.
      ] as well as efficient fluid transport at low pressure drops [
      • McDevitt K.M.
      • Thorson T.J.
      • Botvinick E.L.
      • Mumm D.R.
      • Mohraz A.
      Microstructural characteristics of bijel-templated porous materials.
      ,
      • Mohraz A.
      • Thorson T.J.
      Post-processing bijels for applications.
      ,
      • Thorson T.J.
      • Botvinick E.L.
      • Mohraz A.
      Composite bijel-templated hydrogels for cell delivery.
      ]. Many studies of these minimal surface topologies have been motivated by the development of superior additive manufacturing (AM) technologies that enable their fabrication, and generally employ finite element modeling (FEM) for calculation of their mechanical and functional response; as a consequence, there is an increasing need for quick and accurate generation of computer-aided design (CAD) files for periodic cellular materials based on TPMS topologies, to be employed both for numerical analysis and additive manufacturing.
      Figure thumbnail gr1
      Fig. 1Display of MiniSurf GUI: Control panel is on the left and visualization panel is on the right.
      Figure thumbnail gr2
      Fig. 2Display of (a) single unit cell and (b) 4 × 2 × 2 unit cells of Schwarz P surface.
      In this article, we present an efficient software application called “MiniSurf”, which combines surface visualization and CAD file generation (for both FEM and AM) within one single graphical user interface (GUI). We briefly describe and illustrate the main software features. In addition, we highlight the impact of this package on current and potential applications in the field of architected materials design. Finally, we discuss the software limitations and future improvements.

      2. Description and features

      Minisurf is a software package that runs on Matlab Runtime (a freely accessible Matlab compiler) for visualization and generation of triply periodic minimal surface CAD files (with .inp extension for FEM through Simulia Abaqus and/or .stl extension for AM). The software package has a sleek and simple GUI consisting of two panels: a control panel (left) and a visualization panel (right), as shown in Fig. 1. The control panel allows users to select from the built-in library of minimal surfaces, as well as to type in the custom level-set equation of any desired surface. To facilitate generation of periodic architected materials, users can adjust the number of unit cells Ni, with i=x,y,z, along the x, y and z-directions, to produce specimens of different aspect ratios and number of unit cells, as shown in Fig. 2(a) and (b). In addition, the precision of the generated surfaces, governed by number of composing facets, can be fine-tuned by changing the number of mesh grid points Pi (with i=x,y,z) along the x, y, and z-directions. The generated minimal surfaces will be shown in the visualization panel in either non-mesh or mesh mode, as illustrated in Fig. 3(a) and (b).
      MiniSurf currently has 19 built-in minimal surfaces. All these minimal surfaces are generated by meshing their implicit level-set approximations fx,y,z=c, where c is a constant and x, y, and z represent the location of Px×Py×Pz grid points in a 3D volume of size Nx×Ny×Nz; equations for all built-in surfaces are reported in Table 1. The meshing is executed via the Matlab built-in function isosurface, that discretizes the minimal surfaces into many triangular facets, thus providing information on the facet-vertex connectivity. Information on the connectivity is then subsequently used to write CAD files in .inp and .stl formats.
      Figure thumbnail gr3
      Fig. 3Display of (a) non-mesh mode and (b) mesh mode of Neovius surface.
      Table 1The level-set surface equations in the form of fx,y,z=c for the 19 built-in triply periodic minimal surfaces. For a single unit cell, x,y, and z are bounded by [0,2π].
      TPMSLevel-set equation for the TPMS fx,y,z=c
      Schwarz P
      • Wohlgemuth M.
      • Yufa N.
      • Hoffman J.
      • Thomas E.L.
      Triply periodic bicontinuous cubic microdomain morphologies by symmetries.
      ,
      • Schoen A.H.
      Infinite Periodic Minimal Surfaces Without Self-Intersections.
      cos(x)+cos(y)+cos(z)=0
      Double Primitive
      • Blanquer S.B.G.
      • Werner M.
      • Hannula M.
      • Sharifi S.
      • Lajoinie G.P.R.
      • Eglin D.
      • Hyttinen J.
      • Poot A.A.
      • Grijpma D.W.
      Surface curvature in triply-periodic minimal surface architectures as a distinct design parameter in preparing advanced tissue engineering scaffolds.
      0.5cos(x)cos(y)+cos(y)cos(z)+cos(z)cos(x)+0.2[cos(2x)+cos(2y)+cos(2z)]=0
      Schwarz D
      • Schoen A.H.
      Infinite Periodic Minimal Surfaces Without Self-Intersections.
      ,
      • Michielsen K.
      • Kole S.
      Photonic band gaps in materials with triply periodic surfaces and related tubular structures.
      sin(x)sin(y)sin(z)+sin(x)cos(y)cos(z)+cos(x)sin(y)cos(z)+cos(x)cos(y)sin(z)=0
      Complementary D
      • Michielsen K.
      • Kole S.
      Photonic band gaps in materials with triply periodic surfaces and related tubular structures.
      cos3x+ycos(z)sin3xysin(z)+cosx+3ycos(z)+sinx3ysin(z)+cosxycos(3z)sinx+ysin(3z)=0
      Double Diamond
      • Wohlgemuth M.
      • Yufa N.
      • Hoffman J.
      • Thomas E.L.
      Triply periodic bicontinuous cubic microdomain morphologies by symmetries.
      0.5sin(x)sin(y)+sin(y)sin(z)+sin(x)sin(z)+0.5cos(x)cos(y)cos(z)=0
      D’
      • Wohlgemuth M.
      • Yufa N.
      • Hoffman J.
      • Thomas E.L.
      Triply periodic bicontinuous cubic microdomain morphologies by symmetries.
      0.5cos(x)cos(y)cos(z)+cos(x)sin(y)sin(z)+sin(x)cos(y)sin(z)+sin(x)sin(y)cos(z)0.5sin(2x)sin(2y)+sin(2y)sin(2z)+sin(2z)sin(2x)=0.2
      Gyroid
      • Wohlgemuth M.
      • Yufa N.
      • Hoffman J.
      • Thomas E.L.
      Triply periodic bicontinuous cubic microdomain morphologies by symmetries.
      ,
      • Schoen A.H.
      Infinite Periodic Minimal Surfaces Without Self-Intersections.
      ,
      • Michielsen K.
      • Kole S.
      Photonic band gaps in materials with triply periodic surfaces and related tubular structures.
      cos(x)sin(y)+cos(y)sin(z)+cos(z)sin(x)=0
      G’
      • Wohlgemuth M.
      • Yufa N.
      • Hoffman J.
      • Thomas E.L.
      Triply periodic bicontinuous cubic microdomain morphologies by symmetries.
      sin(2x)cos(y)sin(z)+sin(2y)cos(z)sin(x)+sin(2z)cos(x)sin(y)=0.32
      Double gyroid
      • Blanquer S.B.G.
      • Werner M.
      • Hannula M.
      • Sharifi S.
      • Lajoinie G.P.R.
      • Eglin D.
      • Hyttinen J.
      • Poot A.A.
      • Grijpma D.W.
      Surface curvature in triply-periodic minimal surface architectures as a distinct design parameter in preparing advanced tissue engineering scaffolds.
      2.75sin(2x)sin(z)cos(y)+sin(2y)sin(x)cos(z)+sin(2z)sin(y)cos(x)cos(2x)cos(2y)+cos(2y)cos(2z)+cos(2z)cos(2x)=0.95
      Karcher K
      • Wohlgemuth M.
      • Yufa N.
      • Hoffman J.
      • Thomas E.L.
      Triply periodic bicontinuous cubic microdomain morphologies by symmetries.
      0.3cos(x)+cos(y)+cos(z)+0.3cos(x)cos(y)+cos(y)cos(z)+cos(z)cos(x)0.4cos(2x)+cos(2y)+cos(2z)=0.2
      O, CT-O
      • Wohlgemuth M.
      • Yufa N.
      • Hoffman J.
      • Thomas E.L.
      Triply periodic bicontinuous cubic microdomain morphologies by symmetries.
      0.6cos(x)cos(y)+cos(y)cos(z)+cos(z)cos(x)0.4cos(x)+cos(y)+cos(z)=0.25
      Lidinoid
      • Michielsen K.
      • Kole S.
      Photonic band gaps in materials with triply periodic surfaces and related tubular structures.
      ,
      • Lidin S.
      • Larsson S.
      Bonnet transformation of infinite periodic minimal surfaces with hexagonal symmetry.
      0.5sin(2x)cos(y)sin(z)+sin(2y)cos(z)sin(x)+sin(2z)cos(x)sin(y)0.5cos(2x)cos(2y)+cos(2y)cos(2z)+cos(2z)cos(2x)=0.15
      Neovius
      • Wohlgemuth M.
      • Yufa N.
      • Hoffman J.
      • Thomas E.L.
      Triply periodic bicontinuous cubic microdomain morphologies by symmetries.
      ,
      • Schoen A.H.
      Infinite Periodic Minimal Surfaces Without Self-Intersections.
      3cos(x)+cos(y)+cos(z)+4cos(x)cos(y)cos(z)+cos(z)cos(x)=0
      I-WP
      • Schoen A.H.
      Infinite Periodic Minimal Surfaces Without Self-Intersections.
      ,
      • Michielsen K.
      • Kole S.
      Photonic band gaps in materials with triply periodic surfaces and related tubular structures.
      2cos(x)cos(y)+cos(y)cos(z)+cos(z)cos(x)cos(2x)+cos(2y)+cos(2z)=0
      Fisher–Koch S
      • Blanquer S.B.G.
      • Werner M.
      • Hannula M.
      • Sharifi S.
      • Lajoinie G.P.R.
      • Eglin D.
      • Hyttinen J.
      • Poot A.A.
      • Grijpma D.W.
      Surface curvature in triply-periodic minimal surface architectures as a distinct design parameter in preparing advanced tissue engineering scaffolds.
      ,
      • Michielsen K.
      • Kole S.
      Photonic band gaps in materials with triply periodic surfaces and related tubular structures.
      cos(2x)sin(y)cos(z)+cos(x)cos(2y)sin(z)+sin(x)cos(y)cos(2z)=0
      Fisher–Koch C(S)
      • Michielsen K.
      • Kole S.
      Photonic band gaps in materials with triply periodic surfaces and related tubular structures.
      cos(2x)+cos(2y)+cos(2z)+2sin(3x)sin(2y)cos(z)+cos(x)sin(3y)sin(2z)+sin(2x)cos(y)sin(3z)+2sin(2x)cos(3y)sin(z)+sin(x)sin(2y)cos(3z)+cos(3x)sin(y)sin(2z)=0
      Fisher–Koch Y
      • Michielsen K.
      • Kole S.
      Photonic band gaps in materials with triply periodic surfaces and related tubular structures.
      cos(x)cos(y)cos(z)+sin(x)sin(y)sin(z)+sin(2x)sin(y)+sin(2y)sin(z)+sin(x)sin(2z)+sin(2x)cos(z)+cos(x)sin(2y)+cos(y)sin(2z)=0
      Fisher–Koch C(Y)
      • Michielsen K.
      • Kole S.
      Photonic band gaps in materials with triply periodic surfaces and related tubular structures.
      sin(x)sin(y)sin(z)+sin(2x)sin(y)+sin(2y)sin(z)+sin(x)sin(2z)cos(x)cos(y)cos(z)+sin(2x)cos(z)+cos(x)sin(2y)+cos(y)sin(2z)=0
      F-RD
      • Wohlgemuth M.
      • Yufa N.
      • Hoffman J.
      • Thomas E.L.
      Triply periodic bicontinuous cubic microdomain morphologies by symmetries.
      ,
      • Blanquer S.B.G.
      • Werner M.
      • Hannula M.
      • Sharifi S.
      • Lajoinie G.P.R.
      • Eglin D.
      • Hyttinen J.
      • Poot A.A.
      • Grijpma D.W.
      Surface curvature in triply-periodic minimal surface architectures as a distinct design parameter in preparing advanced tissue engineering scaffolds.
      ,
      • Michielsen K.
      • Kole S.
      Photonic band gaps in materials with triply periodic surfaces and related tubular structures.
      4cos(x)cos(y)cos(z)cos(2x)cos(2y)+cos(2x)cos(2z)+cos(2y)cos(2z)=0

      3. Impact overview

      The idea of using level-set surface equations to approximate TPMSs has been explored extensively in various multidisciplinary research projects for years [
      • Gandy P.J.F.
      • Bardhan S.
      • Mackay A.L.
      • Klinowski J.
      Nodal surface approximations to the P, G, D and I-WP triply periodic minimal surfaces.
      ,
      • Jung Y.
      • Torquato S.
      Fluid permeabilities of triply periodic minimal surfaces.
      ,
      • Torquato S.
      • Donev A.
      Minimal surfaces and multifunctionality.
      ,
      • Dolan J.A.
      • Wilts B.D.
      • Vignolini S.
      • Baumberg J.J.
      • Steiner U.
      • Wilkinson T.D.
      Optical properties of gyroid structured materials: From photonic crystals to metamaterials.
      ,
      • Wilts B.D.
      • Michielsen K.
      • De Raedt H.
      • Stavenga D.G.
      Iridescence and spectral filtering of the gyroid-type photonic crystals in parides sesostris wing scales.
      ,
      • Maldovan M.
      • Urbas A.M.
      • Yufa N.
      • Carter W.C.
      • Thomas E.L.
      Photonic properties of bicontinuous cubic microphases.
      ]; however, to the best of our knowledge, there is no available software package like MiniSurf that includes a nearly complete library of equations for the most interesting TPMSs (with additional ones frequently added) and automatically creates CAD files for both AM and FEM. TPMS shell-based architected materials have remarkable mechanical properties, which makes them superior to classic truss-based lattices in terms of specific strength and toughness [
      • Rajagopalan S.
      • Robb R.A.
      Schwarz meets schwann: Design and fabrication of biomorphic and durataxic tissue engineering scaffolds.
      ,
      • Al-Ketan O.
      • Al-Rub R.K.A.
      • Rowshan R.
      Mechanical properties of a new type of architected interpenetrating phase composite materials.
      ,
      • Bonatti C.
      • Mohr D.
      Mechanical performance of additively-manufactured anisotropic and isotropic smooth shell-lattice materials: Simulations & experiments.
      ]. These studies are recent and the interest of the mechanics community in the structural performance of TPMS-based materials is only expected to grow. MiniSurf will certainly support a number of future projects in this field. As examples, MiniSurf is currently used in two ongoing projects in our research group: (i) Mechanical properties of 3D printed interpenetrating phase composites with shell-based reinforcements [
      • Zhang Y.
      • Hsieh M.-T.
      • Valdevit L.
      Mechanical performance of 3D printed interpenetrating phase composites with spinodal topologies.
      ]. MiniSurf is used to generate CAD files for Schwarz P surface shell-based reinforcements for interpenetrating phase composites. These composites can be readily fabricated by multi-material jetting in VeroWhite (a hard polymeric material for reinforcement) and Agilus (a soft elastomeric material for the matrix) using a Connex 3D printer. The effect of the matrix/reinforcement interpenetration on the mechanical properties of the composites are subsequently investigated both experimentally and numerically (for the numerical studies, MiniSurf-generated meshes are used in finite elements analyses of deformation and damage of the composites). (b) Architected materials designs for long bone implants [

      M.-T. Hsieh, M. Begley, L. Valdevit, Architected implant designs for long bones: The advantage of minimal surface-based topologies, Acta Biomaterialia, in preparation.

      ]. In this effort, we are investigating the performance of minimal surface-based porous materials as implants for long bone repair. Schwarz P CAD files are generated using MiniSurf for the purpose of surface area calculations and finite element modeling. The results are then used to draw comparisons among different topological designs and identify optimal topologies.
      At the same time, we expect MiniSurf to have a broad impact on multidisciplinary studies far beyond the solid mechanics field. The interest of the engineering community in TPMS shell-based materials is documented in several recent studies where TPMS-based architected materials are manufactured and investigated for their multifunctionality, including (1) thermal properties (e.g., thermal conductivity [
      • Abueidda D.W.
      • Abu Al-Rub R.K.
      • Dalaq A.S.
      • Lee D.W.
      • Khan K.A.
      • Jasiuk I.
      Effective conductivities and elastic moduli of novel foams with triply periodic minimal surfaces.
      ,
      • Jung G.S.
      • Yeo J.
      • Tian Z.
      • Qin Z.
      • Buehler M.J.
      Unusually low and density-insensitive thermal conductivity of three-dimensional gyroid graphene.
      ], coefficient of thermal expansion [
      • Abueidda D.W.
      • Dalaq A.S.
      • Abu Al-Rub R.K.
      • Jasiuk I.
      Micromechanical finite element predictions of a reduced coefficient of thermal expansion for 3D periodic architectured interpenetrating phase composites.
      ] and heat exchange [
      • Peng H.
      • Gao F.
      • Hu W.
      Design, modeling and characterization of triply periodic minimal surface heat exchangers with additive manufacturing.
      ,
      • Kim J.
      • Yoo D.-J.
      3D Printed compact heat exchangers with mathematically defined core structures.
      ,
      • Li W.
      • Yu G.
      • Yu Z.
      Bioinspired heat exchangers based on triply periodic minimal surfaces for supercritical CO2 cycles.
      ]), (2) acoustic properties (e.g., sound absorption and acoustic bandgaps [
      • Yang W.
      • An J.
      • Chua C.K.
      • Zhou K.
      Acoustic absorptions of multifunctional polymeric cellular structures based on triply periodic minimal surfaces fabricated by stereolithography.
      ,
      • Abueidda D.W.
      • Jasiuk I.
      • Sobh N.A.
      Acoustic band gaps and elastic stiffness of PMMA cellular solids based on triply periodic minimal surfaces.
      ] and audible coloration [
      • Murphy D.S.H.
      Simulation of acoustic wave propagation in 3-D sonic crystals based on triply periodic minimal surfaces.
      ]) and (3) electrochemical properties (e.g., electrical conductivity [
      • R.K. A.A.-R.
      • D.W. A.
      • A.S. D.
      Thermo-electro-mechanical properties of interpenetrating phase composites with periodic architectured reinforcements.
      ,
      • Ye X.C.
      • Lin X.C.
      • Xiong J.Y.
      • Wu H.H.
      • Zhao G.W.
      • Fang D.
      Electrical properties of 3D printed graphite cellular lattice structures with triply periodic minimal surface architectures.
      ]).

      4. Limitations

      Despite being user friendly and freely accessible to all researchers and engineers, MiniSurf has three main limitations:
      • (1)
        Suboptimal mesh
        In general, meshing in Matlab is done through the Delaunay triangulation algorithm [
        • Barber C.B.
        • Dobkin D.P.
        • Huhdanpaa H.
        The quickhull algorithm for convex hulls.
        ,
        • Lee D.T.
        • Schachter B.J.
        Two algorithms for constructing a delaunay triangulation.
        ], which connects a given set of discrete points. Although such algorithm tends to avoid triangular facets with acute angles, meshing of highly curved minimal surfaces – based on the initial user-defined 3D uniform grid points – still results in many triangular facets with bad aspect ratios (thin and long).
      • (2)
        Zero-thickness surface
        MiniSurf generates minimal surfaces composed of many facets without any physical thickness. Postprocessing to thicken these surfaces is often required. Fortunately, many commercial finite element packages (for example, Simulia Abaqus) or additive manufacturing software (for example, Geomagic Design X) have such postprocessing ability.
      • (3)
        Nonparallel computing
        Currently, MiniSurf can only execute calculations with one single-core processor, although it can still efficiently generate highly meshed surfaces (300 × 300 × 300 initial mesh grid points) under one minute.

      5. Conclusion and future improvements

      In this paper, we presented a software package, MiniSurf, that efficiently produces CAD files of shell-based architected materials consisting of periodic arrays of minimal surface unit cells, for additive manufacturing and finite element modeling. The surface description is provided via implicit level-set equations. Currently, the software library has 19 built-in minimal surfaces, but any user-defined level-set surface is also allowed. Despite the limitations discussed in Section 4, we expect the software package to be impactful, given the profound interest in TPMS-based architected materials across a wide range of multidisciplinary fields.
      In the future, we plan to further improve MiniSurf by focusing on its remeshing algorithm, its thickening functionality (triangular facets to triangular prisms), and a parallel computing implementation. Furthermore, we will keep adding new minimal surfaces to our existing library.

      Declaration of Competing Interest

      The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.

      Acknowledgments

      The authors gratefully acknowledge financial support from the Office of Naval Research (program Manager: D. Shifler, Award No. N00014-17-1-2874) and theNASA Early Stage Innovation Program (Award No. 80NSSC18K0259).

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      Linked Article

      • Update (2.0) to MiniSurf—A minimal surface generator for finite element modeling and additive manufacturing
        Software ImpactsVol. 6
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          This is an update to PII: S2665963820300178. We present an updated version v2.0 of our minimal surface generator MiniSurf v1.0, that creates triply periodic minimal surface (TPMS) computer-aided design (CAD) files for both finite element modeling and additive manufacturing. Besides making the GUI more user-friendly, in this new version we significantly improve the mesh quality of the generated CAD files by incorporating a mesh smoothing feature. With this new smoothing feature, MiniSurf v2.0 can now produce high quality CAD files for more accurate finite element modeling and more precise additive manufacturing.
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