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MSLattice: A free software for generating uniform and graded lattices based on triply periodic minimal surfaces

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Nature‐inspired materials based on triply periodic minimal surfaces (TPMS) are very attractive in many engineering disciplines because of their topology‐driven properties. However, their adoption across different research and engineering fields is limited by the complexity of their design process. In this work, we present MSLattice, a software that allows users to design uniform, and functionally grade lattices and surfaces based on TPMS using two approaches, namely, the sheet networks and solid networks. The software allows users to control the type of TPMS topology, relative density, cell size, relative density grading, cell size grading, and hybridization between lattices. These features make MSLattice a complete design platform for users in different engineering disciplines, especially in applications that employ additive manufacturing (3D printing) and computational modeling. We demonstrate the capability of the software using several examples.
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SPECIAL ISSUE ARTICLE
MSLattice: A free software for generating uniform and
graded lattices based on triply periodic minimal surfaces
Oraib Al-Ketan
1
| Rashid K. Abu Al-Rub
2,3,4
1
Core Technology Platform, New York
University Abu Dhabi, Abu Dhabi, UAE
2
Digital and Additive Manufacturing
Center, Khalifa University of Science and
Technology, Abu Dhabi, UAE
3
Mechanical Engineering Department,
Khalifa University of Science and
Technology, Abu Dhabi, UAE
4
Aerosapce Engineering Department,
Khalifa University of Science and
Technology, Abu Dhabi, UAE
Correspondence
Oraib Al-Ketan, Core Technology
Platform, New York University Abu
Dhabi, P.O. Box 129188, Abu Dhabi, UAE.
Email: oraib.alketan@nyu.edu,
oraib.alketan@gmail.com
Abstract
Nature-inspired materials based on triply periodic minimal surfaces (TPMS)
are very attractive in many engineering disciplines because of their topology-
driven properties. However, their adoption across different research and engi-
neering fields is limited by the complexity of their design process. In this work,
we present MSLattice, a software that allows users to design uniform, and
functionally grade lattices and surfaces based on TPMS using two approaches,
namely, the sheet networks and solid networks. The software allows users to
control the type of TPMS topology, relative density, cell size, relative density
grading, cell size grading, and hybridization between lattices. These features
make MSLattice a complete design platform for users in different engineering
disciplines, especially in applications that employ additive manufacturing
(3D printing) and computational modeling. We demonstrate the capability of
the software using several examples.
KEYWORDS
additive manufacturing, architected materials, cellular materials, GUI, MSLattice, triply periodic
minimal surfaces
1|INTRODUCTION
In nature, many biological systems such as bone, wood, and marine sponge have a microstructure that consists of inter-
penetrating solid and void phases and exhibit a desirable combination of strength and lightweight necessary for sup-
port, mobility, and protection.
13
Such materials influenced and promoted the emergence of the cellular materials field
of study, which is concerned with synthesizing lightweight, energy absorbing, multifunctional, and strong materials,
composites, and structures. Such materials are obtained by deliberately introducing voids in the bulk of solid material
and can be classified based on their structure into stochastic (foams) or periodic (lattices) unit cells. Also, based on the
unit cell's topology, cellular material can be classified into open-cell and closed-cell foams/lattices.
4
The mechanical
and physical properties of cellular materials depend on their topology. This structureproperty relationship suggests
that the mechanical and physical properties of cellular materials depend not only on the chemical composition of base
materials but also on the geometric features of the unit cell or cells making the cellular material.
In recent years, advances in additive manufacturing (3D printing) facilitated the fabrication of cellular materials
and reduced the limitations in fabrication associated with structural complexity. Synchronously, different design tools
evolve to facilitate the design of complex structures and allow users to realize complex functional designs much easier.
One design approach that attracted lots of interest is based on the mathematical representation of the cellular materials,
in particular, the design of lattices based on triply periodic minimal surfaces (TPMSs). Because of their fascinating
topologies and related geometrical features such as smooth shells, large surface area to volume ratio, and
Received: 21 August 2020 Revised: 20 September 2020 Accepted: 21 September 2020
DOI: 10.1002/mdp2.205
Mat Design Process Comm. 2020;e205. wileyonlinelibrary.com/journal/mdp2 © 2020 John Wiley & Sons, Ltd. 1of10
https://doi.org/10.1002/mdp2.205
interpenetrating void networks, TPMS-based lattices have been proposed in recent years for a wide range of engineering
applications such as scaffolds for tissue engineering and body implants,
59
interpenetrating phase composites with tun-
able mechanical properties,
1019
functionally graded structural lattices,
20
thermal management devices such as heat
sinks
21
and heat exchangers,
22
soft robotics,
19
catalytic substrates,
23
feed spacers,
2427
moving bed biofilm reactors for
wastewater treatment,
28
static mixers for chemical processes,
29
and lightweight structures for mechanical
components.
20,3040
TPMS can be represented using level-set approximation equations, which are a set of trigonometric
functions that describe an iso-surface evaluated at an iso-value. Although several studies presented the mathematical
models and examples of the generated lattices, the actual implementation is often not presented. The complexity of
implementing the mathematical representation of the surfaces to create 3D models is a limitation that hinders a wider
adaption of TPMS in deferent fields.
In this work, we recall the mathematical representation of TPMS-based materials and introduce a design tool that
allows scientists, researchers, and engineers to generate uniform and functionally graded 3D printable models and sur-
face meshes of TPMS-based lattices using two approaches, namely, the sheet networks and solid networks. This design
package called MSLattice is freely available for the scientific and engineering community. The tool allows precise con-
trol of the type of TPMS topology, relative density (i.e., the ratio of lattice's density with respect to the density of the
constituent material or, equivalently, the solid volume fraction), cell size, relative density grading, cell size grading, and
hybridization between two different lattices. Furthermore, the software includes flexibility in allowing the user to
design his own TPMS lattice based on level-set approximations.
2|METHODOLOGY
When a surface is characterized by having a mean curvature of zero at any point, it is referred to as a minimal surface.
41
Also, when this surface is infinite and periodic in 3D, it is referred to as a TPMS. Several mathematical approaches were
presented to describe the nodal coordinates that designate a minimal surface.
4244
However, the simplest and most used
method is the level-set approximation approach.
2.1 |Minimal surfaces
Level-set equations are a set of trigonometric functions that combinedly satisfies the equality ϕ(x,y,z)=c. Here, the
function ϕ(x,y,z) is an iso-surface evaluated at an iso-value c. Examples of level-set equations that are most used in liter-
ature are provided below where corresponding surfaces are shown in Figure 1A.
Schoen-Gyroid sin Xcos Y+ sin Ycos Z+ sin Zcos X=cð1Þ
Schwarz-Diamond cos Xcos Ycos Zsin Xsin Ysin Z=cð2Þ
FIGURE 1 A, Unit cell examples of triply periodic minimal surfaces (TPMS); B, strategies to create lattices from a minimal surface
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Schwarz Primitive cos X+ cos Y+ cos Z=cð3Þ
SchoenIWP 2 cos Xcos Y+ cos Ycos Z+ cos Zcos XðÞ
cos 2X+ cos 2Y+ cos 2ZðÞ=cð4Þ
Neovius 3 cos X+ cos Y+ cos ZðÞ+ 4 cos Xcos Ycos ZðÞ=cð5Þ
Fischer Koch S cos 2Xsin Ycos Z+ cos Xcos 2Ysin Z
+ sin Xcos Ycos 2Z=cð6Þ
Schoen-FRD 4 cos Xcos Ycos ZðÞ
cos 2 Xcos 2Y+ cos 2Ycos 2Z+ cos 2Zcos 2XðÞ=cð7Þ
PMY !2 cos Xcos Ycos Z+ sin 2Xsin Y+ sin Xsin 2Z+ sin 2Ysin Z=cð8Þ
In these equations, X=2απx,Y=2βπy,Z=2πγz,α,β, and γare constants related to the unit cell size in the x,y,
and zdirections, respectively.
When the level-set equation is evaluated at a c= 0, the iso-surfaces split the space into subdomains of equal vol-
umes. These subdomains can be controlled through the iso-value constant such that the volumes can be expanded or
contracted by offsetting from the zero value in the normal direction or the opposite direction.
2.2 |Uniform relative density
Creating a TPMS lattice material based on these zero-thickness surfaces is possible following two approaches. First is by
considering one of the volumes divided by the minimal surface as the solid domain and the other as the void domain.
This is done by considering the volume bounded by the minimal surface such that ϕ(x,y,z)>cor ϕ(x,y,z)<cto create
a solid-network lattice. Second is by offsetting the minimal surface along its normal direction and against the normal
direction to create a double surface by solving cϕ(x,y,z)c. The spatial nodes bounded by these two double
surfaces construct the solid domain that represents a thickened shell/sheet-based lattice derived from the minimal
surface. We refer to this lattice as a sheet-network lattice (see Figure 1B).
2.3 |Functional grading of TPMS-based metamaterials
2.3.1 |Relative density grading
Relative density refers to the density of the resulting lattice divided by the density of the base material it is made
of. Relative density is also equivalent to solid volume fraction that represents the solid volume of the lattice with respect
to the volume of the space it occupies. The relative density of sheet- and solid-network lattices derived from TPMS can
be graded by varying the value of the level-set constant cspatially in the Cartesian space depending on a certain func-
tion or tabulated data such that
45
Solid-Networks ϕ>cx,y,zðÞorϕ<cx,y,zðÞ ð9Þ
Sheet-networks cx,y,zðÞ<ϕ<+cx,y,zðÞ ð10Þ
For example, a linear grading is obtained by describing the iso-value as a linear function along one of the Cartesian
coordinates such that c=Ax +Bwhere Aand Bare constants. Similarly, any function can be used to control the rela-
tive density variation of the lattice structure (Figure 2A).
AL-KETAN AND ABU AL-RUB 3of10
2.3.2 |Unit cell size grading
The ability to mathematically control the topology of the lattice allows not only to perform spatial grading of the rela-
tive density but also to change the cell size in a certain direction. This permits varying the surface area and pore size at
a constant relative density. The mathematical procedure to achieve cell size grading is detailed by Liu et al.
46
and
described in detail in the Supporting Information (see Figure 2B).
2.3.3 |Cell type grading (multimorphology)
A multimorphology TPMS lattice can be obtained by transitioning between two or more minimal surfaces.
36,47,48
For this purpose, a weighing function can be used to assign different TPMS topologies to different spaces in the
hybrid lattice. Mathematically, the simple case of a hybrid lattice composed of two cell types can be described
as follows:
ϕMultimorphology=γϕSurface1+1γðÞϕSurface2ð11Þ
where ϕ
Multi morphology
is the multimorphology lattice made of ϕ
Surface1
and ϕ
Surface2
and γis a spatial weighting
function with a value between 0 and 1. γcan be described for example by a sigmoid function such that
47
FIGURE 2 Examples of functionally
graded and hybrid TPMS lattices
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γx,y,zðÞ=1
1+ekG x,y,zðÞ ð12Þ
where G(x,y,z) is the spatial coordinate set that describes the shape of transition between the different regions and
kdefines the width of that transition. The function G(x,y,z) can be any function in x,y, and z. For example, if the func-
tion G(x,y,z)=x, then the transition is simply taking place along the xaxis (Figure 2C). If the function G(x,y,z)=x
2
+y
2
t
2
, where tis a constant, then it defines a circular transition (Figure 2D). Note that if γis constant such that
0<γ< 1, then a new hybrid lattice is obtained without grading.
In a more general approach for hybridization of cell types, the design space can be split into subdomains using con-
trol points such that different TPMS structures can be assigned to different subdomains. Mathematically, the level-set
equation can then be described in the form of a weighted sum of the different subdomains:
ϕMulti-Morphology xðÞ=X
n
i=1
wixðÞϕixðÞ ð13Þ
where the weight functions w
i
(x) are defined by
49
wixðÞ=1 + expðkxxi2

P
n
j=1
1 + expðkxxj2

ð14Þ
where points x
i
lay in the ith subdomain, the ith substructure ϕ
i
is assigned to the ith subdomain, nis the number
of control points, and xdenotes the 3D spatial coordinates (x,y,z). Figure 2E shows an example of a three-lattice hybrid
structure obtained using Equations 13 and 14 (see also section 3 in the Supporting Information).
3|MSLATTICE DEVELOPMENT AND DEMONSTRATION
The mathematical equations presented in Section 2 have been implemented in Matlab and compiled into a
simple-to-use graphical user interface (GUI). We refer to this GUI as MSLattice. MSLattice is a software package written
and compiled into a standalone executable package. It runs on Matlab runtime library. The standalone package does
not require Matlab to be installed. If the runtime library is not available on the user's computer, it will be downloaded
during the installation of the MSLattice software. The installation wizards for Windows and Linux are provided in the
Supporting Information and in the permanent GitHub repository https://github.com/MSLattice. The GUI consists of
four main tabs (see Figure 3), namely, (1) Uniform TPMS lattices, (2) Functional TPMS grading, (3) Implicit functions,
and (4) A simple STL viewer. The Functional TPMS grading tab contains two subtabs, namely, (a) Relative density grad-
ing tab and (b) Cell size grading tab. The Help menu provides a detailed description and documentation of each Tab.
The Uniform TPMS lattices tab allows the user to plot and export a number of uniform lattices and surfaces by input-
ting the needed parameters such as cell type (solid networks or sheet networks), cell topology, relative density, cell size,
sample dimensions, and mesh density per unit cell (see Figure 3). The mesh density per unit cell means that the user spec-
ifies the grid points for a single unit cell and the software will recalculate the grid points for the full structure. As such, the
user will not need to guess the optimal grid points. The user can also choose to generate cuboid, cylindrical, or spherical
samples. Upon clicking on plot, the new lattice will be displayed on the plot on the right-hand side of the interface and will
activate the Save button. STL files can then be exported for the purpose of 3D printing or numerical simulations.
The Relative density grading Tap under Functional TPMS grading Tab allows the user to generate lattices with line-
arly graded lattices such that the relative density changes along the zdirection. Apart from the above-mentioned
parameters in Uniform TPMS grading, the user also specifies the start and end relative density in order to control the
grading (see Figure S3).
The Cell size grading Tap under Functional TPMS grading allows the user to linearly vary the cell size of the sample
in the zdirection. Here, the user will specify the cell type, cell geometry, relative density, sample length and height,
AL-KETAN AND ABU AL-RUB 5of10
initial and final required cell sizes, and finally, the mesh density. Similar to the previous tabs, once the lattice is plotted,
the Save button will be activated and the user can export the STL file (see Figure S4).
Implicit functions tab can be viewed as the user's gate to generate user-defined level-set functions. In this tab, the
user can define the function, the iso-value, the number of cells in the x,y, and zdirections and the lattice type
(i.e., solid networks or sheet networks). The software will generate the plot and return the relative density of the plotted
structure. For example, other level-set equations that are not part of the built-in TPMS structures can be plotted and
exported. An example is presented in Figure 4A (see also Figure S5 for numerical values). This tab can also be used to
generate functionally graded lattices by controlling the relative density through controlling the iso-value. The user can
define the grading function, and it does not necessarily be a linear function. For example, a sample graded in the xand
ydirections using exponential grading is shown in Figure 4B (see also Figure S5 for numerical values). Also, the user
can do hybridization of different lattices such as the hybridization between Gyroid, Diamond, and Primitive lattices as
shown in Figure 2E. The detailed mathematical implementation for this three-lattice hybrid structure is detailed in
section 3 of the Supporting Information for readers to examine. It worth mentioning that the Implicit function tab is
not limited to TPMS structures as it can be used to plot any implicit function that defines a surface of certain topology.
For example, plotting the arbitrary function x×y×z=cwith sheet-network category yields the nonperiodic structure
shown in Figure 4C. This structure resembles a plate-like simple cubic lattice.
50
4|3D PRINTING AND COMPUTATIONAL ANALYSIS
TPMS-based lattices are self-supporting structures that do not require any support materials or structures during 3D
printing. STL files generated using MSLattice can be uploaded to any 3D printing system and fabricated directly. Exam-
ples of metallic 3D printed samples generated using this software are presented in Figure 5. These samples have been
FIGURE 3 MSLattice GUI showing the different tabs and the interface of the Uniform TPMS lattices
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printed using the EOS M280 powder bed fusion system available at the core technology platform at the New York
University in Abu Dhabi such that 316L powder and the standard EOS parameters have been used.
For the purpose of numerical simulations, the user can convert the STL file to a CAD file using the freely available
software FreeCAD (https://www.freecadweb.org/). In the Supporting Information, we detail the steps of converting an
STL file into a STEP file that can be used in computational modeling using, for example, finite element method or com-
putational fluid dynamics packages.
5|IMPACT OVERVIEW
By making MSLattice freely available for the research and engineering community, it is expected to have a wider adap-
tion of TPMS-based materials in different fields of engineering. Other software packages that can generate TPMS-based
materials have been made available.
51,52
However, these packages are very limited in their functionalities and often
require the use of third-party software to thicken the generated surfaces of zero thickness. They are also limited in their
ability to create solid-network-type structures. The ability to do functional grading is also limited even with the use of a
third-party software for thickening. On the other hand, other commercially available tools can be very expensive.
MSLattice is a much-advanced software with the capability of generating surface meshes, uniform lattices, and graded
lattices. Also, through its implicit function window, the software allows the user to generate any TPMS-based lattice
provided that the level-set approximation equation is available. In addition, multimorphology lattices can be obtained
through the implicit function window. For this reason, the authors have provided examples in the help menu of the
implicit functions tab to support the user.
FIGURE 4 Examples of
structures that can be generated
using the implicit functions tab
FIGURE 5 Examples of 3D printed metallic TPMS-based lattices generated using MSLattice
AL-KETAN AND ABU AL-RUB 7of10
6|LIMITATIONS
Currently, the software is limited to not being able to lattice an arbitrary CAD shape for the purpose of light-weighting.
This is currently achieved by importing the generated unit cell to a third part package such as Magics (Materialise,
Belgium) to lattice the structure. Moreover, the software is limited in its ability to grade a structure based on a grayscale
image resulting from topology-optimization. In the future, the authors are planning to incorporate such functionalities
into this software.
7|CONCLUSIONS AND FUTURE WORK
In this short paper, we present MSLattice, a standalone software written in Matlab for the generation of 3D printable lat-
tices based on TPMS structures. The software allows the user to generate uniform and functionally graded lattices with
sheet- or solid-network configurations for a wide range of TPMS topologies. The software also allows the user to export
the STL file for the purpose of 3D printing and numerical simulations. In the future, the authors aim to expand this
package to allow for lattice-based topology-optimization, as well as the ability to lattice any arbitrary CAD geometry.
ACKNOWLEDGEMENT
The first author would like to acknowledge the help of Juan Esteban Villegas Delgado in certain aspects of coding. The
metal 3D printing was performed done using the system available in the Core Technology Platforms at New York Uni-
versity Abu Dhabi.
ORCID
Oraib Al-Ketan https://orcid.org/0000-0003-2736-3779
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SUPPORTING INFORMATION
Additional supporting information may be found online in the Supporting Information section at the end of this article.
How to cite this article: Al-Ketan O, Abu Al-Rub RK. MSLattice: A free software for generating uniform and
graded lattices based on triply periodic minimal surfaces. Mat Design Process Comm. 2020;e205. https://doi.org/
10.1002/mdp2.205
10 of 10 AL-KETAN AND ABU AL-RUB

Supplementary resource (1)

... Designing lattice structures involved a combination of computational modeling and experimental validation. Five distinct lattice structures were designed in this study using the parametric open-source modelling software (MSLattice) [14], Rhino 3D (Rhinoceros 3D, Version 6 SR22, Robert McNeel & Associates, Seattle, WA, USA), and nTopology (nTop Edu, Release 3.26, nTop Inc., New York, NY, USA): gyroid, diamond, Schwarz P, Split P, and honeycomb. Table 1 represents the equations of the Triply Periodic Minimal Structures (TPMS) [14][15][16]. ...
... Five distinct lattice structures were designed in this study using the parametric open-source modelling software (MSLattice) [14], Rhino 3D (Rhinoceros 3D, Version 6 SR22, Robert McNeel & Associates, Seattle, WA, USA), and nTopology (nTop Edu, Release 3.26, nTop Inc., New York, NY, USA): gyroid, diamond, Schwarz P, Split P, and honeycomb. Table 1 represents the equations of the Triply Periodic Minimal Structures (TPMS) [14][15][16]. These structures were chosen to represent a range of geometries with varying mechanical properties and potential applications in footwear components. ...
... Split P, and honeycomb. Table 1 represents the equations of the Triply Periodic Minimal Structures (TPMS) [14][15][16]. These structures were chosen to represent a range of geometries with varying mechanical properties and potential applications in footwear components. ...
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Diabetic foot complications pose significant health risks, necessitating innovative approaches in orthotic design. This study explores the potential of additive manufacturing in producing functional footwear components with lattice-based structures for diabetic foot orthoses. Five distinct lattice structures (gyroid, diamond, Schwarz P, Split P, and honeycomb) were designed and fabricated using stereolithography (SLA) with varying strand thicknesses and resin types. Mechanical testing revealed that the Schwarz P lattice exhibited superior compressive strength, particularly when fabricated with flexible resin. Porosity analysis demonstrated significant variations across structures, with the gyroid showing the most pronounced changes with increasing mesh thickness. Real-time pressure distribution mapping, achieved through integrated force-sensitive resistors and Arduino- based data acquisition, enabled the visualization of pressure hotspots across the insole. The correlation between lattice properties and pressure distribution was established, allowing for tailored designs that effectively alleviated high-pressure areas. This study demonstrates the feasibility of creating highly personalized orthotic solutions for diabetic patients using additive manufacturing, offering a promising approach to reducing the plantar pressure in foot and may contribute to improved outcomes in diabetic foot care.
... Since the bone structure has structural and material heterogeneity, the major focus of our work is to imitate it. There are five methods of modelling functionally graded TPMS structure (Al-Ketan and Al-Rub [149]), i.e., (i) relative density grading of TPMS lattice, (ii) cell size grading of TPMS lattices, (iii) two lattices with linear transition, (iv) two lattices with circular transition, and (v) three lattices hybridization. For TPMS structures to meet the requisite structures, they are functionally graded. ...
... This software is not easily accessible to research groups, small industries, or individuals due to licensing and pricing issues (nTopology [195]). A TPMS lattice can be designed for integration into basic geometries (cuboids, cylinders, and spheres) using opensource software such as MS Lattice, whose code is written in MATLAB (Natick, Massachusetts, USA: The MathWorks, Inc.) (Al-Ketan and Al-Rub [149]). Region TPMS is an additional open-source program that allows you to adjust the structure's resolution to improve the TPMS structures' surface finish (Karakoç [196]). ...
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... Рис. 2. Этапы подготовки геометрии: а -¼ элементарной ячейки в декартовой системе координат; б -¼ элементарной ячейки в цилиндрической системе координат; в -регуляризация фасетной геометрии; г -построение конструкции из элементарных ячеек На основе программ для автоматической генерации stl-файлов основных типов TPMS [2,10] была разработана программа для построения фасетной геометрии трубчатых элементов. В силу симметрии структур ИВП и примитивов для создания геометрии всей конструкции достаточно построить ¼ часть элементарной ячейки. ...
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. Results of finite element modeling of tubular structural elements consisting of triply periodic minimal surfaces under axial tensile and torsion are presented in this paper. The influence of the size and structure type with equal volume of elementary cell on stress state and stiffness of tubular element is shown. Ключевые слова: трижды периодические минимальные поверхности, трубчатый элемент, конечно-элементный анализ, напряжённое состояние, примитивы Шварца, поверхности ИВП Шоэна. Введение. Возможности многоматериальной печати, предоставляемые аддитивным произ-водством, способны обеспечить достаточную точность при изготовлении сложных пористых структур [4; 12]. За счёт настраиваемой геометрии механические метаматериалы, тип которых включает структуру, основанную на трижды периодических минимальных поверхностях (triply periodic minimal surface, TPMS), оказываются более предпочтительными, чем традиционные сото-вые или пористые структуры, благодаря высоким удельным показателям механических характери-стик. TPMS представляют собой поверхности, состоящие из точек со средней кривизной, равной нулю, и проявляющие пространственную симметрию и периодичность. Гладкие поверхности та-ких структур могут быть выражены математическими функциями, изменением параметров кото-рых можно контролировать пористость и площадь поверхности [8]. В последние годы на основе TPMS были разработаны различные конструкции для контроля вибрации [19], поглощения энергии [14] и звука [17], включая многослойные пластины [1], пены, соты, решётки [15]. Они применяются также для улучшения характеристик теплообменников [7; 18], мембран [5], костных имплантов [20]. Несмотря на то что инструменты проектирования и анализа TPMS достаточно хорошо разработаны, некоторые аспекты их механического поведения
... A triply periodic minimal surface (TPMS) is characterized by a topological structure with a mean curvature of zero throughout its surface [12,13]. Compared with traditional porous structures, TPMS structures offer smoothness, complete interconnectivity, and a natural porosity that mitigate the issues of stress concentration and a lack of adjustability associated with conventional designs [14]. ...
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... The isometric and frontal views of the pin-fin designs considered in this work -TPMS-IWP, Hybrid A, and Hybrid B -are represented in Fig. 4, along with the conventional square pin. The main IWP lattice was created using MSLattice software [29]. The overall dimensions of the square pin are 75 × 75 × 150 μm. ...
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