Structures having triply periodic minimal surfaces and methods of manufacture and uses thereof

Abstract

Provided are structures having triply periodic minimal surfaces (TPMS), as either full TPMS structures or fractional TPMS structures, wherein all surfaces of the TPMS unit cell are accessible or are substantially visible from the top and bottom view projections. Also provided are methods of manufacturing said TPMS structures, and industrial applications of said TPMS structures.

Description

STRUCTURES HAVING TRIPLY PERIODIC MINIMAL SURFACES AND METHODS OF MANUFACTURE AND USES THEREOFTECHNICAL FIELD

[0001] The present invention relates to structures comprised of triply periodic minimal surfaces (TPMS). More specifically, the present invention relates to TPMS structures wherein all surfaces of the TPMS unit cell are accessible or are substantially visible from the top and bottom view projections. This includes fractional TPMS structures, which are TPMS structures comprised of fractional TPMS unit cells. The present invention also discloses methods of manufacturing said TMPS structures, and industrial applications and uses for said TPMS structures.BACKGROUND

[0002] Triply Periodic Minimal Surfaces (TPMS) are a class of mathematical surfaces that repeat periodically in three dimensions (such as on their x, y, and z axis) and exhibit minimal surface area for a given boundary condition. There are numerous types of TPMS, each defined by a unique periodic equation. As a TPMS just defines a surface, it lacks any wall thickness. When such surfaces are given volume in a 3D space the result is termed a TPMS structure. Depending on the way such surface is given volume a sheet TPMS structures or skeletal TPMS structures can be formed.

[0003] TPMS structures are of significant interest across various technical fields due to their unique geometric properties. TPMS structures are characterized by their continuous, smooth surfaces that divide space into interpenetrating networks. These features make TPMS structures ideal for creating materials with exceptional mechanical and transport properties.

[0004] Different types of TPMS periodic equation can also be combined to form hybrid TPMS structures, where two or more TPMS periodic equations are mathematicallymerged, alternated, or blended while retaining triply periodic behavior. This mathematical versatility allows the customization of TPMS structures for specific applications and manufacturing constraints.

[0005] Mathematically, as TPMS structures are defined by complex equations that describe their surfaces, these equations can be modified to configure the structure in various ways. For example, TPMS structures can be manipulated by stretching, compressing, or twisting their lattice structures, while still maintaining their periodic minimal nature.

[0006] Due to the triply periodic nature of TPMS, a TPMS structure can be composed of many repeating TPMS unit cells within them. For example, a TPMS structure may be composed of tens, hundreds, or even thousands of unit cells arranged in the x, y, and z directions (e.g., 100x1000x10 unit cells), forming a continuous TPMS surface that preserves the characteristics of the repeating unit cell throughout the structure. This repeating nature of TPMS structures inherently enables scalable structures.

[0007] One advantage of TPMS structures lies in their mechanical strength. Due to their inherent geometric configurations, TPMS structures can achieve high strength-to- weight ratios, making them suitable for load-bearing applications. This property aids in industries such as aerospace, automotive, and construction, where materials are preferably both lightweight and robust. The mechanical strength of TPMS structures stems from their efficient distribution of stress and strain, which reduces material fatigue and enhances durability.

[0008] The interconnected nature of TPMS structures also facilitates efficient fluid transport with lower pressure drop. This property makes TPMS structures ideal for use in heat exchangers, filtration systems, and biomedical devices. In heat exchangers, the efficient fluid flow through the TPMS structure enhances heat transfer while minimizing energy losses. Similarly, in filtration systems, the interconnected pathways allow for effective separation of pollutants with minimal resistance to fluid flow. In biomedical applications, TPMS structures can be used to design scaffoldsfor tissue engineering, where efficient nutrient transport is recommended for cell growth and tissue regeneration.

[0009] Despite their numerous advantages, the manufacturing of TPMS structures poses significant challenges. These challenges are due to the complex, interconnected geometries of TPMS structures. Many of these complex, interconnected geometries of TPMS structures are not visible nor accessible from the outer surface of the TPMS structure. Therefore, common mass manufacturing methods, such as machining, molding, casting, or forming are commonly unable to create a TPMS structure.

[0010] TPMS structures are commonly produced using 3D printing techniques, which build up the TPMS structure using a large number of discrete layers. 3D printing enables the formation of interior voids that are not visible or not accessible from the outer surface of the TPMS structure. Examples of 3D printing techniques include selective laser sintering (SLS), stereolithography (SLA), and fused deposition modeling (FDM) can be used to fabricate TPMS structures.

[0011] However, 3D printing techniques have numerous disadvantages limiting their applicability. While 3D printing can offer accurate reproductions of TPMS structures, it is often limited in scalability and speed. Producing large quantities of TPMS structures using 3D printing can be time-consuming and costly. The layer-by- layer construction process is inherently slower than other manufacturing methods, making it challenging to meet the demands of mass production. Additionally, the range of materials that can be used with 3D printing is limited, restricting the application scope of TPMS structures.

[0012] 3D printing techniques such as SLS and SLA also face challenges related to material properties and post-processing. For instance, parts produced using SLS may require additional finishing processes to achieve the desired surface smoothness. This would not be possible if the surface is no longer accessible from the outer surface of the TPMS structure. Similarly, SLA parts may need curing and support removal, adding to the overall production time and cost. These post-processing steps can also introduce variability and potential defects in the final product. The complex nature ofthe geometry of the TPMS structure can also make post-processing difficult, as surfaces may not be accessible to post-processing tools. This can make postprocessing by way of machining or finishing difficult or impossible.

[0013] The challenges associated with current common 3D printing techniques add time, increase expense, and limit options for the production of TPMS structures. This, in turn, limits the industrial applicability of TPMS structures thereof, despite their advantages. For example, there has been limited to no material use of TPMS structures in a variety of mechanical, thermal, chemical, and carbon capture systems where the structural benefit of TPMS structures would provide functional advantages.

[0014] Thus, there is a need for alternative TPMS structures, and alternative methods of manufacturing TPMS structures that are rapid, accurate, cost-effective, conducive to post-processing, and suitable for a broader range of materials, thereby enabling the benefits of TPMS structures to be used in a broader array of industrial applications.SUMMARY

[0015] It has been discovered that a wide variety of TPMS assemblies can be effectively and efficiently manufactured by fractioning the TPMS structure into fractional TPMS subunits. Said fractional TPMS subunits enable all interior voids to be accessible from the outer surface(s), enabling a wide range of manufacturing techniques to be employed, including those techniques not suitable for manufacturing a TPMS structure. This in turn allows for the creation of TPMS structures using materials that are otherwise incompatible with the current manufacturing of TPMS structures, or allow for most cost and time effective manufacturing processes. Fractional TPMS subunits can then be assembled into TPMS structures, and further into large TPMS structure assemblies, to enable industrial applications that have to-date been unable to utilize the benefits of TPMS structures.

[0016] In one aspect, a fractional triply periodic minimal surface (TPMS) structure comprises at least one fractional TPMS unit cell, wherein the at least one fractional TPMS unit cell: has a TPMS surface defined by a periodic equation, the TPMS surface comprises one full period of the periodic equation in at least one of the three x, y, or z dimensions, and the TPMS surface comprises less than one full period of the periodic equation in at least one of the three x, y, or z dimensions, wherein all surfaces of said fractional TPMS structure are accessible from at least one plane parallel to a top face and a bottom face of said at least one fractional TPMS unit cell.

[0017] In a further aspect of the invention, each of said at least one fractional TPMS unit cell of the fractional TPMS structure is substantially one of: a Diamond, a Gyroid, a Primitive, a Neovius, an F-RD, an IWP, a CLP, an Octo, a K, a C(D), a C(G), a C(I2- Y), a C(S), a C(Y), a C(+-Y), a Diamond', a Gyroid', a Gyroid", a Lidinoid, a P+C(P), a Qstar, an S, a Slotted P, a Split P, a Y, or a Y' fractional TPMS unit cell.

[0018] In a further aspect, the fractional TPMS structure comprises a plurality of fractional TPMS unit cells arranged into a repeating array along two of the three x, y, or z dimensions.

[0019] In a further aspect, the fractional TPMS structure comprises at least one of: metals, non-metals, composites, or biomaterials.

[0020] In a further aspect, the less than one full period of the periodic equation in at least one of the three x, y, or z dimensions of the fractional TPMS unit cell of the fractional TPMS structure is a half, third, quarter, fifth, sixth, eighth, or sixteenth of the one full period of the period equation.

[0021] In a further aspect, the less than one full period of the periodic equation in at least one of the three x, y, or z dimensions of the fractional TPMS unit cell of the fractional TPMS structure is a 45 degree rotated fraction, being 1 / ^2 or a half, third, quarter, fifth, sixth, eighth, or sixteenth fraction of 1 / ^2.

[0022] In a further aspect, the fractional TPMS structure according to any one of claims 1 to 7, wherein the faces of said fractional TPMS structure are substantially visible fromat least one plane parallel to a top face and a bottom face of said at least one fractional TPMS unit cell.

[0023] In a further aspect, the start of less than one full period of the periodic equation in at least one of the three x, y, or z dimensions of the fractional TPMS unit cell of the fractional TPMS structure defines a planar surface, or defines a non-planar surface.

[0024] In a further aspect, the fractional TPMS unit cell is a fractional skeletal TPMS unit cell or a fractional sheet TPMS unit cell.

[0025] An alternative embodiment of the invention provides a method of manufacturing a fractional TPMS structure in accordance with the invention, wherein the fractional TPMS structure is manufactured using one or more of a machining, casting, molding, forming, or template manufacturing process.

[0026] In a further aspect, the output of the manufacturing process is a fractional TPMS structure composed of a plurality of fractional TPMS unit cells in two of the x, y, and z directions, but only one fractional TPMS unit cell in the remaining x, y, or z direction.

[0027] In a further aspect, a first and second fractional TPMS structure are manufactured, and wherein the first and second fractional TPMS structure are stacked on one another by translating, rotating, or flipping the second fractional TPMS structure and adjoining it to the first fractional TPMS structure such that the TPMS surfaces maintain continuity.

[0028] In a further aspect, a plurality of fractional TPMS structures are manufactured, and said plurality of fractional TPMS structures are tessellated in one or more of the x, y, and z, dimension.

[0029] In a further aspect, the manufacturing process further comprises an undercutting of the surfaces of the fractional TPMS unit cell, optionally by use of undercutting end mills, lollipop cutters, t-slot cutters, keyway cutters, and dovetail cutters.

[0030] In a further aspect, material is added to the any accessible but not visible surface of the fractional TPMS unit cell, so that all surfaces of the fractional TPMS structure are visible from at least one plane parallel to a top face and a bottom face of said at least one fractional TPMS unit cell.

[0031] A further alternative embodiment of the invention provides the use of a fractional TPMS structure to manufacture according to the invention, wherein the fractional TPMS structure is used to produce a TPMS templating structure from a stacked, or tessellated fractional TPMS structure, and wherein the TPMS templating structure is used to shape a structure through casting, coating, plating, deposition, infiltration, electroforming, molding, or similar templating processes.

[0032] A further alternative embodiment of the invention provides the use of a fractional TPMS structure according to the invention, wherein the fractional TPMS structure is configured for use as: a load bearing structure or structural support; a storage or holding tank; a heater; a battery; a catalyst or catalyst support; a floor plate, optionally further comprising a thermally insulating material; a concrete formwork, or internal geometry within a formwork; a floating structure; decorative or architectural materials; a heat exchanger or heat exchanges core; a thermocoupler; a component in a thermal energy storage system; an adsorbent; an adsorbent component in a carbon capture apparatus; an adsorbent component in a portable oxygen concentrator or oxygen / medical air purification device.

[0033] A further alternative embodiment provides a triply periodic minimal surface (TPMS) structure comprising at least one TPMS unit cell, wherein the at least one TPMS unit cell is a full 1:1:1 skeletal Primitive TPMS unit cell, and wherein the at least one TPMS unit cell: has a TPMS surface defined by a periodic equation; the TPMS surface comprises one full period of the periodic equation in the x, y, or z dimensions; and one of the one full period of the period equation starts at the predetermined period 402.BRIEF DESCRIPTION OF THE DRAWINGS

[0034] The present invention is described herein by reference to the following figures, in which identical reference numerals refer to identical elements and in which:Figures 1A to 1Z are graphic representations of various expressions defining the surface of a full 1:1:1 TPMS unit cell;Figure 2A is a side view showing predetermined periods in the full 1:1:1 skeletal Diamond unit cell;Figure 2B is a perspective view showing two like fractional 1: 1:0.5 skeletal Diamond unit cells starting from predetermined period 202;Figure 2C is a perspective view showing two like rotated fractional 1: 1:0.3536 skeletal Diamond unit cells starting from predetermined period 203;Figure 2D is a side view showing predetermined periods in the full 1:1:1 sheet Diamond unit cell;Figure 2E is a perspective view showing two like rotated fractional 1: 1:0.3536 sheet Diamond unit cells starting from predetermined period 243;Figure 3A is a side view showing predetermined periods in the full 1:1:1 skeletal Gyroid unit cell;Figure 3B is a perspective view showing two fractional 1: 1:0.25 skeletal Gyroid unit cells starting from predetermined period 303;Figure 3C is a perspective view showing two rotated fractional 1: 1:0.3536 skeletal Gyroid unit cells starting from predetermined period 304;Figure 3D is a side view showing predetermined periods in the full 1:1:1 sheet Gyroid unit cell;Figure 3E is a perspective view showing two rotated fractional 1: 1:0.3536 sheet Gyroid unit cells starting from predetermined period 342;Figure 3F is a top view showing the rotated fractional 1: 1:0.3536 sheet Gyroid unit cells starting from predetermined period 342;Figure 4A is a side view showing predetermined periods in the full 1:1:1 skeletal Primitive unit cell;Figure 4B is a perspective view showing two like fractional 1: 1:0.5 skeletal Primitive unit cells starting from predetermined period 402;Figure 4C is a perspective view showing a full 1:1:1 skeletal Primitive unit cell starting from predetermined period 402;Figure 4D is a side view showing predetermined periods in the full 1:1:1 sheet Primitive unit cell;Figure 4E is a perspective view showing two like fractional 1: 1:0.5 sheet Primitive unit cells starting from predetermined period 442;Figure 5A is a side view showing a predetermined period in the full 1:1:1 skeletal Neovius unit cell;Figure 5B is a perspective view showing two like fractional 1: 1:0.5 skeletal Neovius unit cells starting from the predetermined period;Figure 5C is a side view showing a predetermined period in the full 1:1:1 sheet Neovius unit cell;Figure 5D is a perspective view showing two like fractional 1: 1:0.5 sheet Neovius unit cells starting from the predetermined period;Figure 6A is a side view showing a predetermined period in the positive full 1:1:1 skeletal F-RD unit cell;Figure 6B is a perspective view showing two like positive fractional 1: 1:0.5 skeletal F-RD unit cells starting from the predetermined period;Figure 6C is a side view showing predetermined periods in the negative full 1:1:1 skeletal F-RD unit cell;Figure 6D is a perspective view showing two like negative fractional 1: 1:0.25 skeletal F-RD unit cells starting from predetermined period 623;Figure 6E is a side view showing predetermined periods in the full 1:1:1 sheet F-RD unit cell;Figure 6F is a perspective view showing two like fractional 1: 1:0.25 sheet F-RD unit cells starting from predetermined period 643;Figure 7A is a side view showing predetermined periods in the positive full 1:1:1 skeletal IWP unit cell;Figure 7B is a perspective view showing two like positive rotated fractional 1:1:0.7071 skeletal IWP unit cells starting from predetermined period 703;Figure 7C is a side view showing a predetermined period in the negative full 1:1:1 skeletal IWP unit cell;Figure 7D is a perspective view showing two like negative fractional 1: 1:0.5 skeletal IWP unit cells starting from predetermined period 722;Figure 7E is a side view showing a predetermined period in the full 1:1:1 sheet IWP unit cell;Figure 7F is a perspective view showing two like rotated fractional 1: 1:0.3536 sheet IWP unit cells starting from predetermined period 742;Figure 7G is a perspective view showing predetermined period 702 in the positive full 1:1:1 skeletal IWP unit cell;Figure 7H is a perspective view showing a fractional 1: 1:0.5 skeletal IWP unit cell starting from predetermined period 702;Figure 8A is a side view showing predetermined periods in the full 1:1:1 skeletal CLP unit cell;Figure 8B is a perspective view showing two like fractional 1: 1:0.5 skeletal CLP unit cells starting from predetermined period 802;Figure 8C is a perspective view showing two like rotated fractional 1: 1:0.7071 skeletal CLP unit cells starting from predetermined period 803;Figure 8D is a side view showing predetermined periods in the full 1:1:1 sheet CLP unit cell;Figure 8E is a perspective view showing two like rotated fractional 1: 1:0.3536 sheet CLP unit cells starting from predetermined period 843;Figure 9 is a perspective view showing a blank of raw material being machined to produce a positive fractional 1: 1:0.5 IWP unit cell;Figure 10 shows a molding process used to create a fractional skeletal Diamond TPMS structure (fractional 1:1:0.5 skeletal Diamond unit cell);Figure 11 shows a perspective view of a half of a mold for a fractional skeletal Diamond structure 1100 (array of fractional 6:6:0.5 skeletal Diamond unit cells with a solid fraction of 0.5);Figure 12 shows a perspective view of a process for producing two fractional TPMS structures (arrays of fractional 5:5:0.5 skeletal Diamond unit cells with a solid fraction of 0.5) which are then stacked to create a larger TPMS structure;Figure 13 shows a sheet TPMS structure (a modified fractional 1:1:0.25 sheet Diamond unit cell) is created by forming or molding a blank material;Figure 14A shows the process of a skeletal TPMS templating structure being converted into an open-channel sheet TPMS structure by sequential plating or coating;Figure 14B shows the process of a skeletal TPMS templating structure (two stacked layers of fractional 1: 1:0.5 skeletal Diamond unit cells are shown) being plated or coated to produce a sealed sheet TPMS structure with enclosed internal channels;Figure 14C shows a process of a TPMS templating structure being used to produce a second skeletal TPMS structure through casting, infiltration, molding, or similar templating processes;Figure 15 shows a stacked TPMS structure created from fractional TPMS structures by translating and stacking;Figure 16 show a process of assembling and creating a joined TPMS structure using alternating layers of fractional TPMS structures and meltable sheet material;Figure 17 shows a TPMS plate heat exchanger 1701 comprising a heat-exchanger core 1708 made from a sheet TPMS structure;Figure 18 shows a perspective exploded view 1801 and a perspective sectional view 1810 of a TPMS thermosyphon-integrated fin structure 1802 on a heat source 1804;Figure 19A shows a TPMS vessel 1901 containing an internal TPMS structure 1903;Figure 19B shows a wing-shaped TPMS vessel with an internal TPMS structure, with 1905 and without 1906 the external wall present;Figure 20A shows a perspective view of a massive TPMS TES system 2001 incorporating a TPMS-ESM 2008, and a perspective sectional view of the TPMS-TES 2002;Figure 20B shows an arrangement of TPMS-ESM including large-unit-cell TPMS-ESM segments 2011, small-unit-cell TPMS-ESM segments 2013, and medium-unitcell TPMS-ESM segments 2012;Figure 21 shows a containerized TPMS-TES 2101, in which the storage vessel is housed within a transportable enclosure;Figure 22A shows a perspective view of a fractional TPMS structure assembly forming two beds within a carbon capture system;Figure 22B shows a perspective view of a fractional TPMS unit cell and a stretched fractional TPMS unit cell.Figure 23 shows a sectional perspective view of a reduced-mass slab floor plate 2300 constructed using a fractional TPMS structure assembly as a TPMS templating structure.Figure 24 is a sectional perspective view of an insulated concrete form (ICF) wall assembly incorporating a fractional TPMS structure assembly.DETAILED DESCRIPTION

[0035] The following definitions are used herein to describe the invention:• “TPMS” refers to a triply periodic minimal surface (or surfaces), as defined by periodic equations.• A “full TPMS unit cell” refers to a TPMS which is only composed of one complete period of the TPMS periodic equation, expressed as 1:1:1 (1 full period in the x axis: 1 full period in the y axis: 1 full period in the z axis). The full TPMS unit cell can be repeatedly tiled using only translational operations to form a TPMS structure without discontinuity).• A “full TPMS structure” refers to a physical structure that is comprised of one or more types of repeating full TPMS unit cells.• A “fractional TPMS unit cell” refers to a TPMS which is only composed of a fraction of one complete period of the TPMS periodic equation. For example, a fractional TPMS unit cell which comprises a full period of the periodic equation in the x and y axis, but only half a half period in the z axis, is expressed as 1:1:05• A “fractional TPMS structure” refers to a physical structure that is comprised of one or more types of repeating fractional TPMS unit cells.

[0036] It has been discovered that the limiting factor to the broader use of full TPMS structures is the limitation on suitable manufacturing techniques and materials. Specifically, due to the presence of internal surfaces, full TPMS structures can be manufactured using 3D printing techniques, as the interior surfaces are otherwise inaccessible from a top and bottom view of the full TPMS unit cell and therefore cannot be created through milling, casting, machining, forming, molding, or other common manufacturing methods.

[0037] It has further been discovered that these limitations can be overcome by manufacturing fractional TPMS structures. Unlike full TPMS structures, a fractional TPMS structure can allow for access to all internal surfaces of the fractional TPMS unit cell for manufacturing and processing. This enables the use of broader manufacturing techniques, and a broader array of materials, than is possible for full TPMS structures.Mathematical equations defining full TPMS unit cells

[0038] As referenced herein, a full TPMS unit cell is defined within periods of 1: 1: 1 (x:y:z).The surface of a full TPMS unit cell can be defined by a wide range of periodic equations. However, it should be understood that a full TPMS unit cell can have various dimensions, and that changing the dimensions of the unit cell will stretch or compress the full TPMS unit cell along the axis with the altered dimension. For example, a full 1:1:1 TPMS unit cell having the dimensions of 10,0.5,1 cm will be stretched by a factor of 10 along the x-axis and compressed by half along the y-axis.

[0039] Further, the periodic equations can be used to name the type of full TPMS unit cell.

[0040] A “Diamond” full TPMS unit cell is a full TPMS unit cell wherein the coordinates x,y,z of each point on its surface are defined by Expression 1, which is graphically represented in Figure 1A:sin(x)sin(y)sin(z) + sin(x')cos(y')cos(z') + cos(x')sin(y')cos(z')+ cos(x cos( sin(zExpression 1

[0041] A “Gyroid” full TPMS unit cell is a full TPMS unit cell wherein the coordinates x,y,z of each point on its surface are defined by Expression 2, which is graphically represented in Figure IB:sin x^cosy) + sin(y)cos(z) + sin z)cos x)Expression 2

[0042] A “Primitive” full TPMS unit cell is a full TPMS unit cell wherein the coordinates x,y,z of each point on its surface are defined by Expression 3, which is graphically represented in Figure 1C:cos(x) + cos(y) + cos(z)Expression 3

[0043] A “Neovius” full TPMS unit cell is a full TPMS unit cell wherein the coordinates x,y,z of each point on its surface are defined by Expression 4, which is graphically represented in Figure ID:3(cos(x) + cos(y) + cos(z)) + 4cos(x)cos(y)cos(z)Expression 4

[0044] An “F-RD” full TPMS unit cell is a full TPMS unit cell wherein the coordinates x,y,z of each point on its surface are defined by Expression 5, which is graphically represented in Figure IE:4cos(x)cos(y)cos(z) — cos(2x)cos(2y) — cos(2y)cos(2z)— cos(2x)cos(2z)Expression 5

[0045] An “IWP” full TPMS unit cell is a full TPMS unit cell wherein the coordinates x,y,z of each point on its surface are defined by Expression 6, which is graphically represented in Figure IF:2(cos(x)cos(y) + cos(y)cos(z) + cos(z)cos(x)) — cos(2x) + cos(2y)+ cos(2z))Expression 6

[0046] A “CLP” full TPMS unit cell is a full TPMS unit cell wherein the coordinates x,y,z of each point on its surface are defined by Expression 7, which is graphically represented in Figure 1G:sin(z siny) — 0.4sin(x)cos(z)cos(y)Expression 7

[0047] An “Octo” full TPMS unit cell is a full TPMS unit cell wherein the coordinates x,y,z of each point on its surface are defined by Expression 8, which is graphically represented in Figure 1H:0.6(cos(x)cos(y) + cos(y)cos(z) + cos(z)cos(x)) — 0.4(cos(x) + cos(y)+ cos(z))Expression 8

[0048] A “K” full TPMS unit cell is a full TPMS unit cell wherein the coordinates x,y,z of each point on its surface are defined by Expression 9, which is graphically represented in Figure II:0.3(cos(x) + cos(y) + cos(z)) + 0.3(cos(x)cos(y) + cos(y)cos(z) + cos(z)cos(x)) — 0.4(cos(2x) + cos(2y) + cos(2z))Expression 9

[0049] A “C(D)” full TPMS unit cell is a full TPMS unit cell wherein the coordinates x,y,z of each point on its surface are defined by Expression 10, which is graphically represented in Figure 1J:cos(3x + y)cos(z) — sin(3x — y)'sin(z)' + cos(x + 3y)cos(z) + sin(x— 3y)sin(z) + cos(x — y)cos(3z) — sin(x + y)stn(3z)Expression 10

[0050] A “C(G)” full TPMS unit cell is a full TPMS unit cell wherein the coordinates x,y,z of each point on its surface are defined by Expression 11, which is graphically represented in Figure IK:3(sin(x)cos(y) + sin(y)cos(z) + cos(x)sin(zy) + 2(sin(3x)cos(y) + cos(x)sin(3z) + sin(3y)cos(zy) — 2(sin(x)cos(3y)+ cos(3x}sin(z)' + sin(y)'cos(3zy)'Expression 11

[0051] A “C(I2-Y)” full TPMS unit cell is a full TPMS unit cell wherein the coordinates x,y,z of each point on its surface are defined by Expression 12, which is graphically represented in Figure IL:2(sin(2x)cos(y)sin(z) + sin(x)sin(2y)cos(z) + cos(x)sin(y)sin(2z))+ cos(2x)cos(2y) + cos(2y)cos(2z) + cos(2x)cos(2z)Expression 12

[0052] A “C(S)” full TPMS unit cell is a full TPMS unit cell wherein the coordinates x,y,z of each point on its surface are defined by Expression 13, which is graphically represented in Figure IM:cos(2x) + cos(2y) + cos(2z) + 2(sin(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)Expression 13

[0053] A “C(Y)” full TPMS unit cell is a full TPMS unit cell wherein the coordinates x,y,z of each point on its surface are defined by Expression 14, which is graphically represented in Figure IN:— 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)Expression 14

[0054] A “C(+-Y)” full TPMS unit cell is a full TPMS unit cell wherein the coordinates x,y,z of each point on its surface are defined by Expression 15, which is graphically represented in Figure IO:—2cos(x)cos(y)cos(z) + sin(2x)sin(y) + sin(2y)sin(z) + sin(x)sin(2z)Expression 15

[0055] A “Diamond ” full TPMS unit cell is a full TPMS unit cell wherein the coordinates x,y,z of each point on its surface are defined by Expression 16, which is graphically represented in Figure IP:0.5(cos(x)cos(y)cos(z) + cos x)'sin y)'sin z)' + sin x)cos(y)sin z)+ sin(x)sin(y)cos(z)) — 0.5(sin(2x)sin(2y)+ sin(2y)sin(2z) + sin(2z)sin(2x))Expression 16

[0056] A “Gyroid'” full TPMS unit cell is a full TPMS unit cell wherein the coordinates x,y,z of each point on its surface are defined by Expression 17, which is graphically represented in Figure IQ:sin(2x)cos(y)sin(z) + sin(x)sin(2y)cos(z) + cos(x)sin(y)sin(2z)Expression 17

[0057] A “Gyroid"” full TPMS unit cell is a full TPMS unit cell wherein the coordinates x,y,z of each point on its surface are defined by Expression 18, which is graphically represented in Figure 1R:5(sin(2x)cos(y)sin(z) + sin(x)sin(2y)cos(z) + cos(x)sin(y)sin(2z))+ cos(2x)cos(2y) + cos(2y)cos(2z) + cos(2x)cos(2z) Expression 18

[0058] A “Lidinoid” full TPMS unit cell is a full TPMS unit cell wherein the coordinates x,y,z of each point on its surface are defined by Expression 19, which is graphically represented in Figure IS:(sin(2x)cos(y)sin(z) + sin(2y)cos(z)sin(x) + sin(2z)cos(x)sin(y))— (cos(2x)cos(2y) + cos(2y)cos(2z) + cos(2z)cos(2x))Expression 19

[0059] A “P+C(P)” full TPMS unit cell is a full TPMS unit cell wherein the coordinates x,y,z of each point on its surface are defined by Expression 20, which is graphically represented in Figure IT:0.3cos(x)cos(y)cos(z) + 0.2(cos(x) + cos(y) + cos(z))+ 0. lcos(2x)cos(2y)cos(2z) + 0. l(cos(2x) + cos(2y) + cos(2z)) + 0.05(cos(3x) + cos(3y) + cos(3z))+ 0. l(cos(x)cos(y) + cos(y)cos(z) + cos(z)cos(x))Expression 20

[0060] A “Qstar” full TPMS unit cell is a full TPMS unit cell wherein the coordinates x,y,z of each point on its surface are defined by Expression 21, which is graphically represented in Figure 1U:(cos(x) — 2cos(y))cos(z) — V3 sin(z)(cos(x — y) — cos(x)) + cos(x — y)cos(z)Expression 21

[0061] An “S” full TPMS unit cell is a full TPMS unit cell wherein the coordinates x,y,z of each point on its surface are defined by Expression 22, which is graphically represented in Figure IV:cos(2x)sin(y)cos(z) + cos(x)cos(2y)sin(z) + sin(x)cos(y)cos(2z)Expression 22

[0062] A “Slotted P” full TPMS unit cell is a full TPMS unit cell wherein the coordinates x,y,z of each point on its surface are defined by Expression 23, which is graphically represented in Figure 1W:—2(cos(x)cos(y) + cos(y)cos(z) + cos(z)cos(x)) — 2(cos(2x) + cos(2y)+ cos(2z)) + cos(2x)cos(y) + cos(2y)cos(z)+ cos(2z)cos(x) — (cos(x)cos(2y) + cos(y)cos(2z)+ cos(z)cos(2x))Expression 23

[0063] A “Split P” full TPMS unit cell is a full TPMS unit cell wherein the coordinates x,y,z of each point on its surface are defined by Expression 24, which is graphically represented in Figure IX:1.1(sin(2x)cos(y)sin(z) + sin(2y)cos(z)sin(x) + sin(2z)cos(x)sin(y))— 0.2(cos(2x)cos(2y) + cos(2y)cos(2z) + cos(2z)cos(2x)) — 0.4(cos(2x) + cos(2y) + cos(2z))Expression 24

[0064] A “Y” full TPMS unit cell is a full TPMS unit cell wherein the coordinates x,y,z of each point on its surface are defined by Expression 25, which is graphically represented in Figure 1Y: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)Expression 25

[0065] A “Y'” full TPMS unit cell is a TPMS unit cell wherein the coordinates x,y,z of each point on its surface are defined by Expression 26, which is graphically represented in Figure 1Z:2cos(x)cos(y)cos(z) + sin(2x)sin(y) + sin(2y)sin(z) + sin(x)sin(2z)Expression 26

[0066] Other TPMS geometries that the TPMS unit cell can substantially comprise includes Neovius Versions (i.e., N14, N26, N38), Complementary P Surface Family, Batwing Family, The Starfish Family, Disphenoid Families, P3a surface of Lord and Mackay, Hybrids (combinations of surfaces), Schwarz1H & CLP Surfaces, Schoen's IWP(r), F-RD(r), Hybrid- 1[P, F-RD], GW, O, C-TO, Manta Surface (19, 35, & 51), or Complementary D, RII, RIII, 1-6, 1-8, & 1-9 Surfaces.

[0067] The type of fractional TPMS unit cell can also be classified by reference to the above periodic equations. Where the full TPMS unit cell would comprise a full period of the equation in each direction (e.g., 1: 1: 1), a fractional TPMS unit cell will contain at least one direction that does not contain the full period of the equation, but only a fraction thereof. The specific fractional period is defined within the fractional unit cell (e.g., 1:1:0.5).

[0068] When intending to specify the ‘type’ of full TPMS unit cell or fractional TPMS unit cell when describing the invention, the identified ‘type’ of unit cell is described, along with its unit cell dimensions, after identifying if the unit cell is “full” or“fractional”. For example, a given unit cell could be defined as a “full 1:1:1 skeletal Diamond TPMS unit cell” or “fractional 1: 1:0.5 skeletal Diamond TPMS unit cell”.Fractional TPMS unit cells and fractional TPMS structures

[0069] As described herein, it has been discovered that the advantageous properties of full TPMS structures can be utilized across broader embodiments, including through the use of additional materials and manufacturing techniques, through the use of fractional TPMS structures.

[0070] Fractional TPMS structures have significant manufacturing advantages over the prior art by allowing manufacturing techniques to be used that were previously deemed impossible for full TPMS structures. In a full TPMS structure, for example, the complexity of the TPMS shape leaves surfaces inside the TPMS unit cell that cannot be seen from projections from the top view and the bottom view, as well as not being accessible to traditional manufacturing techniques through the top and bottom views. This hidden geometry complicates the manufacturing process, as not substantially all surfaces are accessible for mass production techniques, limiting the manufacturing options available and likewise the materials that can be used to create the full TPMS structure.

[0071] Given the wide range of fractional TPMS structures that can be created, not all will have a configuration that lacks any inaccessible inner surfaces. However, as used herein the following discussion of fractional TPMS structures in accordance with the present invention, any such fractional TPMS structure must further have all surfaces of the fractional TPMS unit cell accessible ( / .<., have machinable undercuts) or substantially visible (i.e., have no undercuts) from the top and bottom view projections. Where all surfaces are accessible and / or substantially visible from the top and bottom view projections, this visibility allows for easier and more efficient manufacturing using common mass production techniques. This in turn enables theuse a broader array of materials, including those that would otherwise be incompatible with 3D printing.

[0072] In one embodiment, a fractional TPMS structure comprises a fractional unit cell that has a half of one complete period of the TPMS periodic equation (e.g., 1: 1:0.5) where all surfaces are accessible and / or substantially visible from the top and bottom view projections. These fractional TPMS unit cell shapes may be like images of their second half. This symmetry allows a fractional TPMS unit cell to be produced and stacked to produce a structure that resemble a full TPMS unit cell, though doing so may require at least one of: flipping, rotating, or phase shifting before being stacked as discussed herein.

[0073] As only a one-half fractional unit cell (i.e., a 1: 1:0.5, whereby two fractional unit cells may be assembled to resemble a full unit cell) needs to be produced, and all internal surfaces of the fractional unit cell are accessible from the top and bottom views, this allows the manufacturing to be more efficient, and expands the available materials and manufacturing techniques.

[0074] A fractional unit cell that results in like shapes is a preferred embodiment of the invention, as only a single fractional TPMS structure needs to be manufactured that, with minimal adjustment, may be assembled to produce many unique individual cells that replicate the full TPMS structure.

[0075] Figure 2A is a side view showing predetermined periods in the full 1:1:1 skeletal Diamond TPMS unit cell. Figure 2B is a perspective view showing two like fractional 1:1:0.5 skeletal Diamond TPMS unit cells starting from the predetermined period 202. The fractional 1:1:0.5 skeletal Diamond TPMS unit cell 210, if flipped and rotated 180 degrees, would appear like half unit fractional TPMS unit cell 211 Figure 2C is a perspective view showing two like rotated fractional 1: 1:0.3536 skeletal Diamond TPMS unit cells starting from the predetermined period 203. The rotated fractional 1:1:0.3536 skeletal Diamond TPMS unit cell 212, if flipped and rotated 180 degrees, would appear like rotated fractional TPMS unit cell 213.

[0076] In another embodiment, the fractional TPMS unit cell is rotated 45 degrees and fractionated to create a rotated fractional 1: 1:0.3536 TPMS unit cell (0.3536 refer to half the fraction 1 / ^2) where all surfaces are accessible and / or substantially visible from the top and bottom view projections. These rotated fractional TPMS unit cell shapes may also be like images of their second rotated fraction. This rotation may have advantages in allowing the rotated fractional TPMS unit cell to have more solid fractions whose surfaces are substantially visible from the top and bottom view projections in comparison to a non-rotated fractional TPMS unit cell.

[0077] Figures 2A and 2B show the side and perspective views of a skeletal Diamond TPMS unit cell with a solid fraction of 0.5, meaning that only 50% of the volume of the unit cell is occupied by material. 201 shows a full 1:1:1 skeletal Diamond TPMS unit cell whose void volume would be a mirror image of 201. Because the void volume is a mirror image of the solid volume, a negative of the solid volume can be used to manufacture the solid volume. However, as the void volume of the full TPMS unit cell is not readily accessible from the exterior surfaces, the ability to manufacture a mirror image is limited. It has been discovered that this limitation is overcome by utilizing a fractional TPMS unit cell, thereby having all internal surfaces accessible from the top and bottom view perspectives of the fractional TPMS unit cell.

[0078] The ratio of solid fraction and void fraction within a fractional TPMS unit cell will always total 1. For example, for solid fractions of 0.5, the void fraction will be 0.5. For solid fractions of 0.25, the void fraction would be 0.75. Therefore, to produce a positive 0.25 solid fraction TPMS unit cell, a negative of 0.75 solid fraction would be needed.

[0079] Figures 2D and 2E show the side and perspective views of a sheet Diamond TPMS unit cell with a solid fraction of 0.25. Figure 2D is a side view showing predetermined periods 242 & 243 in the full 1:1:1 sheet Diamond unit cell. Figure 2E is a perspective view showing two like rotated fractional 1: 1:0.3536 sheet Diamond TPMS unit cells starting from the predetermined period 243. The rotatedfractional 1:1:0.3536 sheet Diamond TPMS unit cell 250, if flipped and rotated 180 degrees, would appear like rotated fractional TPMS unit cell 251.

[0080] As seen from the side view of Figure 2A, to create two like fractional 1: 1:0.5skeletal Diamond TPMS unit cells, the predetermined period 202 of the period equation can start at any of the x periods (highlighted with line with arrows 202 indicating the fractionated can be moved in the directions of the arrows). Or to create two like rotated fractional 1:1:0.3536 skeletal Diamond TPMS unit cells (0.3536 = 1 / (2 / 2)), the predetermined period 203 of the period equation can start at any 1 / (2 / 2) fractional period from the predetermined period 203. As TPMS shapes are triply periodic, the predetermined period can also be in the y or the z direction (not highlighted in Figures 2A, 2D, 3A, 3D, 4A, 4D, 5A, 5C, 6A, 6C, 6E, 7A, 7D, 7F, 8A, or 8D). For example, fractioning horizontally along 202, two like fractional 1:1:0.5 skeletal Diamond TPMS unit cells 410, 411 are created, which have all surfaces substantially visible from the top and bottom view projections that were not substantially visible from the full 1:1:1 skeletal Diamond unit cell.

[0081] In another embodiment, the fractional TPMS unit cell is smaller than 1: 1:0.5 (also referred to herein as a 1 / 2 fractional TPMS unit cell). This smaller fractional TPMS unit cell comprises on one axis, 1 / 3, 1 / 4, 1 / 5, 1 / 6, 1 / 8, 1 / 16, or l / 32nd of the full period of the periodic equation, and the resulting fractional TPMS unit cells have all surfaces accessible or substantially visible from projections from the top and bottom view. These fractional shapes are less likely to be like shapes, requiring more and more fractional shapes to be created to resemble the full TPMS unit cell. This would also require more layers to be stacked to resemble the full TPMS unit cell. For certain types of TPMS, fractionating smaller than 1 / 2 fractions may be beneficial. For example, smaller fractional Y, Y’, Split P, C(G), C(S), Lidinoid, and C(D) 1: 1:0.125 TPMS unit cells may have advantages in manufacturing than larger unit cells for certain types of manufacturing. For example, fractional C(I2-Y), C(Y), C(+- Y), Diamond’, Gyroid, Gyroid’, Gyroid”, Lidinoid, and S 1:1:0.25 TPMS unit cell (i.e., 1 / 4 fractional TPMS unit cells), may have advantages in manufacturing compared to larger unit cells for certain types of manufacturing.

[0082] In another embodiment, the fractional TPMS unit cell can be rotated and has a fractional period of a rotation ratio between 1 to 1 / ^2, where all surfaces are accessible and / or substantially visible from the top and bottom view projections. This rotation ratio would be a function of the ratio of the angle of rotation of the TPMS unit cell. For example, for a rotation of 45 degrees, the rotation ratio would be 1 / ^2. For certain rotated fractional TPMS unit cells, rotating the unit cell may be beneficial. For example, a 45-degree rotation of Primitive, IWP, or CLP rotated fractional TPMS unit cells may increase the number of surfaces substantially visible from the top and bottom projections compared to full or half unrotated TPMS unit cells. In certain embodiments, this increased visibility may enable the use of larger fractional TPMS unit cells (e.g, 0.7071 in place of 0.5 fractional TPMS unit cells) or permit a broader range of solid fractions, depending on the specific TPMS periodic equation and application.

[0083] In a further embodiment, the rotated fractional TPMS unit cell has a period smaller than the rotation ratio (1 to 1 / ^2), where all surfaces are accessible and / or substantially visible from the top and bottom view projections. The smaller rotated TPMS unit cell can correspond to 1 / 2, 1 / 3, 1 / 4, 1 / 5, 1 / 6, 1 / 8, 1 / 16, or l / 32ndof the rotation ratio. For certain TPMS periodic equations and / or solid fractions, using these smaller rotated fractional TPMS unit cells may increase the number of surfaces substantially visible from the top and bottom projections relative to the larger rotated fractional TPMS unit cell. These smaller rotated ratio fractional TPMS unit cells may also have a larger solid fraction while also having an increase the number of surfaces substantially visible from the top and bottom projections relative to the non rotated fractional TPMS unit cell. For example, the 45 degree rotated fractional 1:1:0.3536 sheet Diamond TPMS unit cell from predetermined period 243 has a larger fraction than fractional 1: 1:0.25 sheet Diamond unit cell from predetermined period 242 (0.3536 is greater than 0.25).

[0084] Fractional TPMS unit cells smaller than 1: 1:0.5 may have an increased number of surfaces that are accessible or substantially visible from the top and bottom view projections that were not accessible nor substantially visible from the full or halffractional TPMS unit cells. Figure 3A shows a side view of predetermined periods in the full 1:1:1 skeletal Gyroid TPMS unit cell. Figure 3B is a perspective view showing two like fractional 1: 1:0.25 skeletal Gyroid TPMS unit cells starting from the predetermined period 303. For example, the fractional 1: 1:0.5 skeletal Gyroid TPMS unit cell still has a significant area of surfaces that are not substantially visible from projections from the top and bottom view. However, by starting at predetermined period 303, the fractional 1:1:0.25 skeletal Gyroid TPMS unit cell 310 and 311 have more surfaces that are substantially visible from the top and bottom view projections. This would simplify certain types of manufacturing techniques. To create the full 1:1:1 skeletal Gyroid TPMS unit cell from fractional 1: 1:0.25 skeletal Gyroid TPMS unit cell 310 and 311, two-unit cells 310 and 311 may be produced, and then stacked to produce the full 1:1:1 skeletal Gyroid TPMS unit cell.

[0085] Rotated fractional TPMS unit cells may allow all surfaces to be substantially visible from the top and bottom view projections that were not substantially visible from the non-rotated fractional TPMS unit cells. Figure 3C is a perspective view showing two like rotated fractional 1:1:0.3536 skeletal Gyroid TPMS unit cells starting from the predetermined period 304. For example, the fractional 1:1:0.25 skeletal Gyroid TPMS unit cell 310 still has a small amount of surface area that are not substantially visible from projections from the top and bottom view. However, by rotating and starting at predetermined period 304, the rotated fractional 1: 1:0.3536 skeletal Gyroid TPMS unit cell 312 and 313 have all surfaces substantially visible from the top and bottom view projections. This would simplify certain types of manufacturing techniques as there would be no undercuts present. To create larger repeating 1: 1: 1.414 skeletal Gyroid TPMS unit cell from like rotated fractional 1: 1:0.3536 skeletal Gyroid TPMS unit cell 312 & 313, four rotated fractional 1:1:0.3536 skeletal Gyroid TPMS unit cell 312 are manufactured, with two translated and flipped 312 to appear as 313. All unit cells may then be stacked to produce a larger repeating 1: 1: 1.414 skeletal Gyroid TPMS unit cell.

[0086] In another embodiment, the fractional TPMS unit cell can have material added to any hidden surfaces or surfaces that are not accessible, and / or material removed from thetop and / or bottom view projections. This material addition and / or removal can allow all surfaces to be accessible and / or substantially visible from the top and bottom view projections while still maintaining the desirable properties of the TPMS structure. The location of the material addition or removal can be determined through the limitations of the manufacturing technique combined with a top and bottom view projections onto the surface which would show the areas not substantially visible from the top and bottom view projections. The location and amount of the material addition or removal can also be a function of the type of TPMS periodic equation, skeletal or sheet, and solid fraction.

[0087] Figures 3D, 3E, and 3F shows the side, perspective, and top views of a full and fractional sheet Gyroid TPMS unit cell with a solid fraction of 0.25. Figure 3D is a side view showing predetermined periods 342 & 343 in the full 1:1:1 sheet Gyroid TPMS unit cell. Figure 3E is a perspective view showing two like fractional 1: 1: 0.25 sheet Gyroid TPMS unit cells starting from the predetermined period 342.The fractional 1:1:0.25 sheet Gyroid TPMS unit cell 350, if flipped and rotated 180 degrees, would appear like fractional TPMS unit cell 351. Figure 3F is a top view of the fractional 1: 1:0.25 sheet Gyroid TPMS unit cell 350, showing areas 359 which are not substantially visible from the top and bottom view projections. By adding material to the hidden surfaces to area 359, and / or removing material from the top and / or bottom view projections to area 359 would allow all surfaces to be substantially visible from the top and bottom view projections while still maintaining the desirable properties of the TPMS gyroid geometry.

[0088] From the side view of the full 1:1:1 skeletal Primitive TPMS unit cell 401 shown in Figure 4A, predetermined periods are highlighted with 402 & 403. Predetermined period 403 can create two like rotated fractional 1: 1:0.707 skeletal Primitive TPMS unit cells. Predetermined period 402 can create two like fractional 1:1:0.5 skeletal Primitive TPMS unit cells 410 and 411 shown in Figure 4B. These fractional 1:1:0.5 skeletal Primitive TPMS unit cells 410 and 411 provide all surfaces of the unit cell as substantially visible from one projection. For example, when projecting from the topview onto fractional 1:1:0.5 skeletal Primitive TPMS unit cell 410, all surfaces of the fractional 1:1:0.5 skeletal Primitive TPMS unit cell 410 are substantially visible.

[0089] It has surprisingly been discovered that the benefits of fractional 1: 1:0.5 skeletal Primitive TPMS unit cell 410 can alternatively be obtained through a full 1:1:1 skeletal Primitive TPMS unit cell provided that one of the three x,y,z dimensions starts at predetermined period 402. Because all surfaces are substantially visible from one projection of the fractional 1:1:0.5 skeletal Primitive TPMS unit cell, this allows all the surfaces of a full 1:1:1 skeletal Primitive TPMS unit cell that starts from predetermined period 402 to be seen from the top and bottom view projections 409. Figure 4C shows a perspective view of a full 1:1:1 skeletal Primitive TPMS unit cell starting from predetermined period 402. This appears to be an exception to the common limitations of a full TPMS structures.

[0090] As such, in a further embodiment, the benefits of the invention can be obtained by way of a full 1:1:1 skeletal Primitive TPMS structure, provided that one of the three x,y,z dimensions the full 1:1:1 skeletal Primitive TPMS unit cell starts at predetermined period 402.

[0091] Figure 4D shows a side view of predetermined period 442 & 443 in the full 1:1:1 sheet Primitive TPMS unit cell. Predetermined period 443 can create two like rotated fractional 1:1:0.707 sheet Primitive TPMS unit cells. Predetermined period 442 can create two like fractional 1: 1:0.5 sheet Primitive TPMS unit cells 450 and 451 shown in Figure 4E.

[0092] Figure 5A shows a side view of a predetermined period in the full 1:1:1 skeletal Neovius TPMS unit cell 501. From the side view shown in Figure 5A, a predetermined period 502 is shown to create two like fractional 1: 1:0.5 skeletal Neovius TPMS unit cells 510 & 511 shown in Figure 5B. These fractional 1:1:0.5 skeletal Neovius TPMS unit cells reveal all surfaces of the unit cell as substantially visible from the top and bottom view projections. Figure 5B is a perspective view showing two like fractional 1: 1:0.5 skeletal Neovius TPMS unit cells starting from the predetermined period 502.

[0093] Figure 5C shows a side view of a predetermined period in the full 1:1:1 sheet Neovius TPMS unit cell 541. From the side view shown in Figure 5C, a predetermined period 542 is shown to create two like fractional 1: 1:0.5 sheet Neovius TPMS unit cells 550 & 551 shown in Figure 5D. These fractional 1:1:0.5 sheet Neovius TPMS unit cells reveal all surfaces of the unit cell as substantially visible from the top and bottom projections.

[0094] Figures 6A to 6D show perspective and side views of full and fractional skeletal F- RD TPMS unit cells with void fraction of 0.5. A positive full 1:1:1 skeletal F-RD TPMS unit cell 601 is shown. The void volume of the F-RD TPMS unit cell (negative full 1:1:1 F-RD TPMS unit cell 621) is not the same as the solid volume of F-RD (positive full 1: 1: 1 F-RD TPMS unit cell 601).

[0095] From the side view of the positive full 1:1:1 skeletal F-RD TPMS unit cell 601shown in Figure 6A, a predetermined period 602 is shown to create two like positive fractional 1:1:0.5 skeletal F-RD TPMS unit cells 610 and 611 shown in Figure 6B.These positive fractional 1:1:0.5 skeletal F-RD TPMS unit cells 610 and 611 reveal all surfaces of the unit cell as substantially visible from the top and bottom projections.

[0096] From the side view of the negative full 1:1:1 skeletal F-RD TPMS unit cell 621 shown in Figure 6C, a predetermined period 623 is shown to create two like negative fractional 1:1:0.25 skeletal F-RD TPMS unit cells 630 and 631 shown in Figure 6D. These negative fractional 1:1:0.25 skeletal F-RD TPMS unit cells reveal substantially all surfaces of the unit cell from the top and bottom projections.

[0097] Figures 6E and 6F show perspective and side views of full and fractional sheet F- RD TPMS unit cells with solid fraction of 0.25. From the side view of the full 1:1:1 sheet F-RD TPMS unit cell 641 shown in Figure 6E, a predetermined period 643 is shown to create two like fractional 1: 1:0.25 sheet F-RD TPMS unit cells 650 & 651 shown in Figure 6F. These fractional 1: 1:0.25 sheet F-RD TPMS unit cells have a small amount of surface area that are not visible from projections from the top and bottom view but are accessible for machining.

[0098] Figures 7A to 7H show perspective and side views of the TPMS unit cell called IWP with a solid fraction of 0.5 and 0.25 for skeletal and sheet, respectively. The positive full 1:1:1 skeletal IWP TPMS unit cell 701 is unlike its negative full 1:1:1 skeletal IWP TPMS unit cell 721.

[0099] From the side view of the positive full 1:1:1 skeletal IWP TPMS unit cell 701 as shown in Figure 7A, predetermined periods 702 and 703 are shown. Predetermined period 703 can create two like positive rotated fractional 1:1:0.7071 skeletal IWP TPMS unit cells 712 & 713 as shown in Figures 7B. These positive rotated fractional 1:1:0.7071 skeletal IWP TPMS unit cell 712 & 713 reveals all surfaces of the unit cell as substantially visible from the top and bottom view projections. At certain higher solid fractions, some surfaces of positive rotated fractional 1:1:0.7071 skeletal IWP TPMS unit cell 712 remain hidden from the top and bottom view projections. In these cases, it is advantageous to use a smaller rotated fractional period equal to half of the rotation ratio (1 / ^2), such as 17(2^2). This reduced fraction increases surface visibility from top and bottom projections, enabling manufacturable configurations for higher-solid-fraction skeletal IWP geometries.

[0100] From the side view of the negative full 1:1:1 skeletal IWP TPMS unit cell 721 as shown in Figure 7C, a predetermined period 722 is shown to create two like negative fractional 1: 1:0.5 IWP TPMS unit cells 730 and 731 shown in Figure 7D.These negative fractional 1:1:0.5 IWP TPMS unit cells reveals all surfaces of the unit cell as substantially visible from the top and bottom projections.

[0101] From the side view of the full 1:1:1 sheet IWP TPMS unit cell 741 as shown in Figure 7E, a predetermined period 742 is shown. Predetermined period 742 can be used to create two like fractional 1: 1:0.3536 sheet IWP TPMS unit cells 750 and 751, as shown in Figure 7F. At this division, the surface of fractional 1:1:0.3536 sheet IWP TPMS unit cell 750 is accessible for manufacturing, but not all surfaces are fully visible when viewed from the top and bottom projections.

[0102] In another embodiment, the predetermined period does not need to originate from a planar surface and can instead be defined along non-planar surface such as a wavy orcurved surface based on the TPMS geometry. This wavy predetermined period may follow a surface generated from the implicit equation of the TPMS, rather than a flat plane, allowing the fractional division to align with natural surface contours of the unit cell. As shown in Figure 7G, predetermined period 702 is illustrated on the positive full 1:1:1 skeletal IWP TPMS unit cell 701. Starting from this wavy predetermined period, a positive fractional 1:1:0.5 skeletal IWP TPMS unit cell 710 can be created as shown in Figure 7H. This positive fractional 1:1:0.5 skeletal IWP TPMS unit cell 710 is fully visible from both the top and bottom view projections, producing two like positive fractional 1: 1:0.5 skeletal IWP TPMS unit cells when cut at predetermined period 702. Two such fractional 1:1:0.5 skeletal TPMS IWP unit cells may be translated, rotated, or flipped and then stacked together to reconstruct the larger 1:1:1 skeletal IWP unit cell.

[0103] By defining the wavy predetermined period along a curved surface rather than a planar one, a larger fractional TPMS segment may be produced. In some unit cells, defining the predetermined period along a wavy surface may enable the creation of larger fractional TPMS unit cells, rather than requiring smaller fractional divisions, which may reduce the total number of segments needed to reconstruct a full TPMS unit cell (for example, enabling two 1: 1:0.5 fractions instead of requiring four 1:1:0.25 fractions). This wavy predetermined period also provides manufacturing advantages, as the complementary wave geometry enables self-seating during stacking, improving alignment between layers. The increased interfacial area created by the wavy surface further enhances layer adhesion during joining, while the curved geometry increases interlayer shear strength relative to flat planar mating surfaces. Although Figures 7G & 7H illustrate this embodiment using IWP, this approach is applicable to any type of TPMS.

[0104] Figures 8A to 8E show perspective and side views of the CLP TPMS unit cell for skeletal and sheet configurations at solid fractions of 0.5 and 0.25, respectively. From the side view of the full 1:1:1 skeletal CLP TPMS unit cell 801 shown in Figure 8A, a predetermined period 802 is shown. Predetermined period 802 can be used to create two like fractional 1: 1:0.5 skeletal CLP TPMS unit cells 810 and 811,as illustrated in Figure 8B. These fractional 1:1:0.5 skeletal CLP unit cells 810 and 811 reveal all surfaces of the unit cell as substantially visible from the top and bottom projections at solid fractions below 0.5; however, at higher solid fractions, some surfaces may remain hidden. Figure 8C shows that starting from predetermined period 803, two rotated fractional 1:1:0.7071 skeletal CLP TPMS unit cells 812 and 813 can be produced. In this rotated configuration, all surfaces are substantially visible from the top and bottom view projections at all solid fractions, in contrast to the fractional 1: 1:0.5 skeletal CLP unit cells 810 and 811.

[0105] Figures 8D and 8E illustrate the sheet CLP configuration. Figure 8D shows a side view of the full 1:1:1 sheet CLP TPMS unit cell 841 with predetermined periods 842 & 843. Starting at predetermined period 842 creates two like fractional 1: 1:0.25 sheet CLP TPMS unit cells, while Figure 8E shows that starting from predetermined period 843, two rotated fractional 1: 1:0.3536 sheet CLP TPMS unit cells 850 and 851 can be created.Methods of Manufacturing

[0106] A fractional TPMS structure comprises one or more fractional TPMS unit cells.Within the fractional TPMS structure, the fractional TPMS unit cells may be rotated, modified, arranged as an array, or combined in any configuration described herein. The dimensions of the fractional TPMS unit cell are application dependent, as are the dimensions of the fractional TPMS structure. For example, the overall size of the fractional TPMS structure can be adjusted by adjusting the dimension of the fractional TPMS unit cell used, the number of fractional TPMS unit cells within the fractional TPMS structure, and combinations thereof.

[0107] The fractional TPMS structure can be created using a variety of means of shaping a raw material. To produce a larger structure, comprised of more than one fractional TPMS structure, a plurality of fractional TPMS structure can be manufactured and then be stacked to produce a stacked TPMS structure. The stacked TPMS structureof such an embodiment comprises various layers of fractional TPMS structures. The fractional TPMS structure layers of the stacked fractional TPMS structure can be joined together, or finished. To produce an even larger structure, the stacked fractional TPMS structure can be tessellated to produce a tessellated fractional TPMS structure. The fractional TPMS structure, stacked fractional TPMS structure, or the tessellated fractional TPMS structure can be used as a template to produce a templated fractional TPMS structure.

[0108] It should be understood that the raw material can comprise one or multiple types of materials. Such materials can comprise metals, non-metals, composites, or biomaterials. Metals can comprise, for example, aluminium, iron, steel, titanium, silicon, and copper. Non-metals can comprise, for example, plastics, elastomers, carbons, silicas, aluminas, ceramics (e.g., zeolites), gels, cement, or glass. Composites can comprise combinations of metals, non-metals, or biomaterials, as well as metal matrix composites, ceramic matrix composites, concretes, polymer matrix composites, carbon fibers, fiberglass, and para-aramids (e.g., Kevlar™). Biomaterials include, but are not limited to, biocompatible polymers, culinary materials, and natural materials such as biogels, wood, lignin, and cellulose.

[0109] The raw material can be formed into the fractional TPMS structure using a variety of means of shaping, including but not limited to machining, casting, molding, forming, or a combination thereof. Machining can comprise, for example, 3, 4, and 5-axis CNC, router tables, 5-axis lathes, turning centers, electrical discharge machining, laser cutting, and laser engraving. Casting can comprise, for example, and casting, die casting, investment casting, gravity casting, lost foam casting, permanent mold casting, plaster casting, and vacuum casting. Molding can comprise, for example, injection molding, blow molding, extrusion molding, rotational molding, thermoforming, and powder metallurgy. Forming can comprise, for example, forging, bending, stamping, shearing, extrusion, and vacuum forming.

[0110] In one embodiment, an array of fractional TPMS unit cells is formed as a fractional TPMS structure, where the array comprises multiple fractional TPMS unit cellspattemed periodically along two axes. For example, an array may correspond to a period such as 10:10:0.5, representing ten repeated periods of the periodic equation along two axes, and one-half of the periodic equation along the third axis. This can also be expressed as an array of 1: 1:0.5 fractional TPMS unit cells arranged with ten in each of the x and y direction, but only one in the z direction. Producing an array in this manner allows multiple fractional TPMS unit cells to be formed simultaneously, enabling efficient scaling to larger component sizes rather than forming unit cells individually. This embodiment may reduce manufacturing time, improve geometric repeatability, and allow batch-scale fabrication.

[0111] In another embodiment, larger structures can be created from a plurality of fractional TPMS structures by stacking individually manufactured fractional TPMS structures to create a stacked fractional TPMS structure. Stacking fractional TPMS structures increases the overall dimension of the structure along the axis in which the fractional period occurs. For example, 20 fractional TPMS structures with a fractional 10:10:0.5 TPMS unit cell, may be produced using various techniques. These fractional TPMS structures are then methodically stacked to create a larger stacked fractional TPMS structure at a dimension of 10:10:10 TPMS unit cells. Importantly, such an embodiment differs from a full TPMS structure, notwithstanding the end structure comprising a complete period of the periodic equation in the z dimension, as the stacked fractional TPMS structure is composed of the individually manufactured fractional TPMS structures.

[0112] In another embodiment, the plurality of the fractional TPMS structures in the stacked fractional TPMS structure are joined together using a means of joining. The means of joining of fractional TPMS structures into larger fractional TPMS structures, of any desired size, can be done by various means including but not limited to welding, brazing, soldering, adhesive joining, fastening, fusing, or curing. Using this approach, stacked fractional TPMS structure can be created at scales, while ensuring the stacked fractional TPMS structure maintains high structural integrity and continuity.

[0113] In another embodiment, larger structures can be created from a plurality of stacked fractional TPMS structures by tessellating them to create a tessellated fractional TPMS structure. Tessellated fractional TPMS structures increases the overall dimension of the structure along all three axis. For example, 50 stacked fractional TPMS structures at a dimension of stacked 10:10:10 TPMS unit cells, may be produced using various techniques. These stacked fractional TPMS structures are then tessellated to create a larger tessellated fractional TPMS structure at a dimension of 20:50:50 TPMS unit cells. Because TPMS are triply periodic, their surfaces and networks align when repeated along one or more axes, enabling direct geometric continuation without mismatch at boundaries.

[0114] In a further embodiment, tessellation may be performed using overlapping stacked fractional TPMS structures, where adjacent stacks partially overlap at their boundary region rather than meeting at a single planar interface. This overlapping configuration increases interfacial contact area, improves load transfer between tessellated segments, and may enhance structural rigidity and resistance to shear or delamination within the tessellated fractional TPMS structure. Overlapping may be implemented along any axis of tessellation, depending on the fractional periods of the stacked structures. In another embodiment, tessellated fractional TPMS structures are joined using a means of joining. In further embodiments, a first tessellated fractional TPMS structure may itself be tessellated again to create even larger volumetric or surface-scale tessellated fractional TPMS structures. The resulting TPMS structures retain the periodic geometry and functional characteristics of the underlying pattern, enabling scaling to component-, panel-, or system-level structures without loss of continuity, while maintaining ease of manufacturing.

[0115] In one embodiment, machining methods including but not limited to 3, 4, and 5-axis CNC, router tables, 5-axis lathes, turning centers, electrical discharge machining, laser cutting, and laser engraving, can be used to produce a fractional TPMS structure where all TPMS surfaces are accessible to the CNC machine from a plane substantially parallel to at least one of the top and bottom of the unit cell.

[0116] For example, a fractional skeletal IWP TPMS structure (comprising a positive fractional 1:1:0.5 skeletal IWP TPMS unit cell) is manufactured using a 3-axis CNC machining process beginning from a wavy predetermined period 702 as shown in Figure 9. First, a blank of raw material is secured in the CNC machine 901. The blank may comprise a machinable material, such as aluminum. A roughing cut is performed to remove the majority of accessible material corresponding to the void volume of the fractional skeletal IWP structure. A contouring cut may follow to refine the surface. A final surfacing pass produces a smooth fractional skeletal IWP structure profile on the first face 902. The half machined blank is then inverted 180° about the x-axis, re-fixtured, and machined from the opposite side. The roughing, contouring, and surfacing sequence is repeated until the fractional skeletal IWP TPMS structure 903 is fully formed.

[0117] In another embodiment, a laser-based fabrication process is used to produce a fractional TPMS structure, either continuously or in batches. It should be understood that laser-based fabrication process may include laser engravers, laser cutters, excimer lasers, or photolithography machines. The process begins by feeding a sheet-roll of raw material into a laser-based fabrication process. The laser selectively removes material from the top surface, bottom surface, or both surfaces of the sheet of raw material to generate the TPMS curvature pattern required for the fractional TPMS structure. Laser parameters such as power, scanning speed, wavelength, and dwell time may be selected based on material and desired depth of cut.

[0118] In embodiments where the fractional TPMS structure includes void passages extending through the sheet thickness, the laser may operate in through-cutting mode to form perforations or continuous voids. The patterned sheet can then be advanced automatically to fabricate the fractional TPMS structure continuously along the roll, after which it may be cut to length to form a plurality of fractional TPMS structures. These structures may subsequently be stacked, joined, or further processed to form larger TPMS structures suitable for the desired application, such as in thermal, mechanical, or chemical systems.

[0119] In another embodiment, casting methods such as sand casting, die casting, investment casting, gravity casting, lost foam casting, permanent mold casting, plaster casting, and vacuum casting can be used to shape a fractional TPMS structure. In such an embodiment, all surfaces of the fractional TPMS structure should be accessible via a plane parallel to the top and bottom surfaces.

[0120] In another embodiment, molding methods such as injection molding, blow molding, extrusion molding, rotational molding, thermoforming, and powder metallurgy can be used to shape a fractional TPMS structure. In such an embodiment, all surfaces of the fractional TPMS structure should be accessible via a plane parallel to the top and bottom surfaces.

[0121] In another embodiment, forming processes are used to create a fractional TPMS structure. Forming process may include stamping, punching, press forming, deep drawing, roll forming, hydroforming, incremental sheet forming, superplastic forming, thermoforming of polymer sheets, or any combination of forming processes. In such an embodiment, all surfaces of the fractional TPMS structure should be accessible via a plane parallel to the top and bottom surfaces. These methods deform a sheet, foil, or ductile workpiece so that the material conforms to a TPMS curvature pattern. In some embodiments, forming is performed in stages to gradually introduce curvature and prevent wrinkling or tearing. Forming processes may be applied individually or in combination, for example roll-forming followed by stamping or hydroforming for final shape refinement.

[0122] In another embodiment, a mold may be fabricated to produce fractional TPMS structures. This mold can be used for casting, molding, or forming. The mold may comprise a top mold half and a bottom mold half, each manufactured as a negative representation of the desired fractional TPMS unit cell. The mold halves may be produced by machining, casting, additive manufacturing, EDM, or other suitable fabrication processes. When aligned, the top and bottom mold halves define a cavity corresponding to a TPMS surface, skeletal form, or sheet form, enabling material to be shaped into the corresponding geometry.

[0123] Figure 10 shows an example of the molding process for producing a fractional skeletal Diamond TPMS structure (comprising a fractional 1:1:0.5 skeletal Diamond unit cell with a solid fraction of 0.5) from a raw material substrate. In the first step 1010, a substrate sheet 1001 is placed between a top mold half 1002 and a bottom mold half 1003, where both mold halves are shaped as negatives of the desired fractional skeletal Diamond TPMS structure. In the second step 1011, mold halves are then pressed together, deforming the substrate sheet 1001 into the mold cavity to form a fractional skeletal Diamond TPMS structure 1004. In the third step 1012 after pressing, the mold halves are separated and the formed fractional skeletal Diamond TPMS structure 1004 is removed.

[0124] It should be understood that the top half mold 1002 and the bottom half mold 1003have a structure capable of being used with other manufacturing methods. For example, the void space between the top half mold 1002 and the bottom half mold 1003 can be filled with molten material in a die casting process or filled with liquid plastic in injection molding.

[0125] Figure 11 shows a perspective view of a half of a mold for a fractional skeletal Diamond structure (comprising an array of fractional 6:6:0.5 skeletal Diamond unit cells with a solid fraction of 0.5) 1100. When this half-mold is paired with a corresponding opposing half-mold, the two molds together define the full cavity geometry required to produce an array of fractional 6:6:0.5 skeletal Diamond unit cells.

[0126] Figure 12 shows a perspective view of a process for producing two fractional TPMS structures (comprising an array of fractional 5:5:0.5 skeletal Diamond unit cells with a solid fraction of 0.5) which are then stacked to create a stacked fractional TPMS structure. First in 1210, a raw material first substrate sheet 1201 is provided and the top half mold 1202 and a bottom half mold 1203 are opened. Then in 1211, the first substrate sheet 1201 is then placed between a top half mold 1202 and a bottom half mold 1203. Then in 1212, the top half mold 1202 and bottom half mold 1203 are then closed around the first substrate sheet 1201 to produce a first fractional TPMSstructure 1204. Then in 1213, first fractional TPMS structure 1204 is then removed from the top half mold 1202 and bottom half mold 1203, and a raw material second substrate sheet 1205 is provided. Lastly in 1214, as with the first substrate sheet 1201, the top half mold 1202 and bottom half mold 1203 are closed around the second substrate sheet 1205 to produce a second fractional TPMS structure 1206. The second fractional TPMS structure 1206 is then removed from the top half mold 1202 and bottom half mold 1203. The first fractional TPMS structure 1204 and the second fractional TPMS structure 1206 may then be stacked to form a stacked fractional TPMS structure.

[0127] It should be understood that this process can be repeated as many times as necessary to produce a suitably large stacked fractional TPMS structure. Further, the first fractional TPMS structure 1204 and the second fractional TPMS structure 1206 can be joined using any suitable means and along any suitable face.

[0128] Figure 13 illustrates an embodiment in which a fractional sheet TPMS structure (comprising a modified fractional 1: 1:0.25 sheet Diamond TPMS unit cell) is created by forming or molding a raw material. A raw material blank sheet 1303 is placed between an upper punch 1302 and a lower punch 1304, where the punch surfaces contain the negative shape of the TPMS geometry. In the initial state 1301 the blank sheet 1303 is flat. As pressing begins 1306, the blank sheet 1301 gradually forced into the TPMS features of the punch. Continued deformation results in the blank sheet 1301 conforms into the fractional sheet TPMS structure 1305 at the bottom of the punch 1307. The completed fractional TPMS structure is then removed from the punches 1308. This method enables rapid fabrication of fractional TPMS structures using metals, polymers, or ductile composite laminates, and may be performed at ambient or elevated temperature depending on material formability.

[0129] Figure 15 illustrates an embodiment of creating a stacked fractional TPMS structure from fractional TPMS structures by translating and stacking. In Figure 15, a misaligned stack 1501 is shown from a side view, where multiple identical fractional TPMS structures (six fractional 2:2:0.25 sheet Diamond unit cells) are layeredwithout positional adjustment, resulting in discontinuity between repeating features. Alignment is achieved by translating every second fractional layer by a quarter period, producing an aligned arrangement 1502. When the layers are stacked in this offset configuration, a stacked fractional TPMS structure 1503 is formed. Cut lines 1504 indicate the region of excess material that may be removed to generate straight external boundaries while retaining internal TPMS surface continuity. This removal of excess material can happen to the fractional TPMS structure prior to stacking, or after the stacked fractional TPMS structure is created. It should be understood that the translating period size depends on the type of TPMS, as well as if it is skeletal or sheet. In a further embodiment, different translations, rotations, or flips depending on sheet or skeletal, or different fractional TPMS unit cells to achieve seamless periodic continuity when stacked.

[0130] If all surfaces are not substantially visible but are accessible from a plane parallel to the top and bottom of the fractional TPMS structure, advanced tooling and techniques can be used. In one embodiment, machining with undercutting tools such as undercutting end mills, lollipop cutters, t-slot cutters, keyway cutters, and dovetail cutters can shape fractional TPMS structures with undercuts. In another embodiment, a 5-axis CNC machine or more can also be used to produce these structures without undercutting tools by machining from multiple viewpoints.

[0131] In another embodiment, casting or molding can be utilized to shape fractional TPMS structures with undercuts using specialized undercutting tooling. This tooling includes collapsible cores, lifter and sliding mechanisms, bump-offs for flexible materials, and inserts that are held and released by the mold. This allows for the shaping of undercut geometries that are not achievable through conventional molding or casting methods. For example, collapsible cores can retract to release the undercut features once the material has solidified, while lifter and sliding mechanisms enable the movement of mold components to clear undercut areas. Bump-offs leverage the flexibility of certain materials to deform temporarily, allowing the part to be removed from the mold without damage. Additionally, insertscan be strategically placed within the mold to form undercut features and then removed post-molding.

[0132] Further, two or more manufacturing techniques can be combined when creating a fractional TPMS structure. For example, a fractional TPMS structure can be roughly formed using die-casting and subsequently refined using a CNC machine.

[0133] Similarly, a first portion of a fractional TPMS structure can be formed using a first manufacturing technique while a second portion of the same fractional TPMS structure can be formed using a second manufacturing technique. In some embodiments, the use of different manufacturing techniques does not impact the ability of the fractional TPMS structures created to be stacked together to form larger tessellated fractional TPMS structures of desired size for their intended use.

[0134] It should be understood that various other combinations of techniques are possible (e.g., the use of three or more methods or the use of alternative techniques). It should also be understood that the choice of manufacturing technique, and choice of material, will depend upon the desired application of the fractional TPMS structures.

[0135] Figure 16 illustrates an embodiment for stacking and joining a stacked fractional TPMS structure from a plurality of fractional TPMS structures, using alternating layers of fractional TPMS structures and meltable sheet material. In 1601, an example layup is shown where multiple fractional TPMS structures 1605 (comprising of seven fractional 1:2:0.25 sheet Diamond TPMS unit cells) are arranged with meltable sheets 1604 positioned between them in alternating layers. In 1602, the layers are stacked to form a multilayer assembly suitable for joining. In 1603, the assembly is heated such that the meltable sheets soften, flow, or wet the surface of the fractional TPMS structures, forming a continuous bonded and joined stacked fractional TPMS structure after cooling. This joining method enables the creation of mechanically unified stacked fractional TPMS structures with enhanced surface connectivity and improved heat or load transfer paths between fractional TPMS unit cells.

[0136] In one embodiment, the meltable sheets 1604 comprise copper and the fractional TPMS structures 1605 comprise stainless-steel fractional sheet TPMS structures. The stacked assembly is heated in a brazing furnace such that the copper wets and bonds to the stainless-steel surfaces, producing a joined stacked fractional TPMS structure with continuous metallurgical interfaces between all fractional TPMS unit cells.Templating Manufacturing

[0137] In another embodiment, a templated TPMS structure is created from a stacked, or tessellated fractional TPMS structure, and functions as a TPMS templating structure used to shape a structure through casting, coating, plating, deposition, infiltration, electroforming, molding, or similar templating processes. The TPMS templating structure may be used as (i) a sacrificial template that is fully removed after forming the templated TPMS structure, (ii) a permanent internal component that remains embedded within the final templated TPMS structure, or (iii) a partial remaining template where only selected material is removed, leaving the remainder. The TPMS templating structure can be a skeletal, sheet, or a combination of the two.

[0138] In one embodiment, a sheet TPMS templating structure can have one or many layers of templated TPMS material on it forming a thickened sheet geometry. In a further embodiment, removal of the TPMS templating structure can leave a third interpenetrating network in the templated TPMS structure. Depending on the layering of the thickened sheet geometry, greater than three interpenetrating networks can be created by including additional templated materials on layers.

[0139] In another embodiment, one or both of the void networks of a sheet TPMS templating structure can be filled with one or two templated TPMS materials. The templated TPMS material would form a skeletal geometry in network of the sheet TPMS templating structure. If both void networks are filled and the sheet TPMS templating structure is removed, a void network in the shape of the sheet TPMS templating structure would be left.

[0140] In another embodiment, a skeletal TPMS templating structure can have one or many layers of material on it forming a sheet templated TPMS structure. In a further embodiment, the void network of the skeletal TPMS templating structure with the sheet templated TPMS structure can be filled with the same or a second templated TPMS materials.

[0141] In another embodiment, a skeletal TPMS templating structure void network can be filled with a material to form a templated TPMS structure.

[0142] The TPMS templating structure may be fabricated from raw materials including, but not limited to: metals, non-metals, composites, or biomaterials. In further embodiment, the template material is polystyrene, polyurethane foam, wax, paraffin, lost-foam polymer, zinc, aluminum, magnesium, salt-based forms, cellulose, cardboard, wood fiber, ceramic greens, starch-based matrices, or other removable materials. In embodiments requiring coating or plating, the template may be coated with metals (e.g., copper, nickel, silver, chromium), ceramics, carbon, conductive paints, electroless deposition layers, CVD films, or combinations thereof.

[0143] The TPMS template structure removal may be performed by melting, burning-out, dissolving, reacting away, vaporizing, leaching, evaporating, etching, or chemical dissolution, depending on material compatibility. The template may be removed fully to produce a templated TPMS structure (e.g., skeletal or sheet), partially removed to form hybrid internal geometries, or retained completely to provide advantages such as insulation, stiffening, acoustic damping, buoyancy, chemical potential, or thermal regulation.

[0144] Figure 14A illustrates an embodiment in which a skeletal TPMS templating structure is converted into an open-channel sheet templated TPMS structure by sequential plating or coating. The skeletal TPMS templating structure 1401 (for example zinc) is prepared from stacking or joining fractional TPMS structures (two stacked layers of rotated fractional 1:1.414:0.7071 skeletal CLP TPMS unit cells are shown). A first plating or coating layer 1402 is applied, followed by optional additional coating layers 1403, 1404 to increase wall thickness, alter surfacechemistry, or tailor final material composition (e.g., nickel — copper — nickel stack). Selected external regions 1408 are subsequently removed using techniques such as machining, etching, grinding, laser ablation, or chemical stripping as shown in 1405-1406, exposing the skeletal TPMS templating structure. The templating material may then be removed using techniques such as by melting, dissolution, or chemical reaction (e.g., melting out zinc, dissolving with mild acid), leaving a sheet templated TPMS structure 1407 formed by the deposited plating layers.

[0145] Figure 14B illustrates a related embodiment where a skeletal TPMS templating structure (two stacked layers of fractional 1:1:0.5 skeletal Diamond TPMS unit cells are shown) is plated or coated to produce a sealed sheet templated TPMS structure with enclosed internal channels. A skeletal TPMS templating structure!410 which is shown comprising two stacked layers of fractional 1:1:0.5 skeletal Diamond TPMS unit cells, is plated or coated with one or more layers 1411. Only selected access regions 1414 are removed to create openings which allows for the removal of the skeletal TPMS templating structure 1412. After the template is removed, the remaining structure forms a closed-wall sheet templated TPMS structure 1413, with sealed internal voids for one half of the sheet network. After removal of the templating material, the final sheet templated TPMS structure retains internal fluid pathways while maintaining continuous exterior surfaces. This templated TPMS structure has benefits and uses in thermosyphons, heat-pipes, fluid routing fins, catalytic reactors, CO2 adsorbers, or other systems requiring internal cavities.

[0146] Figure 14C illustrates an embodiment where a skeletal TPMS templating structure is used to produce a skeletal templated TPMS structure through casting, infiltration, molding, or similar templating processes. First, multiple fractional TPMS structures (shown as eight fractional 4:4:0.5 skeletal Primitive TPMS unit cells 1425) are produced and arranged 1420 to be layered to produce a continuous stacked TPMS structure. These layers are stacked and / or joined 1421 to form the skeletal TPMS templating structure 1426 that is larger than just one fractional TPMS structure. Then the void network within the skeletal TPMS templating structure 1426 is filled or infiltrated 1422 with a castable or moldable material, forming the skeletal templatedTPMS structure 1427 which is the negative of the skeletal TPMS templating structure. A sectional view 1423 shows how the skeletal templated TPMS structure 1427 occupies the negative fraction defined by the fractional TPMS unit cell of the skeletal TPMS templating structure 1426. If it is desirable, the skeletal TPMS templating structure 1426 may be removed 1424, leaving behind the skeletal templated TPMS structure 1427.

[0147] In one embodiment of the process shown in Figure 14C, the TPMS templating structure 1426 is fabricated from a TPMS templating material such as polystyrene foam or polyurethane foam, and the castable material comprises concrete, cementitious grout, or geopolymer slurry. The castable material is poured or injected into the void network of the template 1426 and allowed to cure. In one embodiment, the TPMS templating structure 1426 is fully removed following curing, producing the templated TPMS structure 1427 made of the castable material. Full removal may be desirable where maximized void volume, fluid flow pathways, drainage channels, weight reduction, or integration with reinforcement are required. Removal may be achieved through solvent dissolution (e.g., acetone for polystyrene foam), thermal burnout, melting, or mechanical extraction, depending on material compatibility. In another embodiment, the TPMS templating structure 1426 is retained within the cured concrete, forming a composite structure of the templated TPMS structure and the TPMS templating structure. Retention may be beneficial where the TPMS templating structure provides functional benefits to the intended application of the composite structure, such as thermal insulation, acoustic damping, buoyancy, improved energy absorption during impact, reduced concrete volume, reduced selfweight, or simplified construction without stripping forms.Applications of Fractional TPMS Structures

[0148] The fractional TPMS structures described herein may be applied across a range of mechanical, thermal, chemical, biological, acoustic, optical, and energy-related systems, or a combination described herein. These fractional TPMS structures canexist in a fractional TPMS structure assembly. The fractional TPMS structures assembly comprises a fractional TPMS structure, a stacked fractional TPMS structure, a tessellated fractional TPMS structure, a templated TPMS structure, a TPMS templating structure, or any combination listed herein. The following embodiments illustrate representative use cases enabled by fractional TPMS structure assemblies and manufacturing methods described in this disclosure.Mechanical Applications

[0149] In mechanical applications, fractional TPMS structure assemblies may provide enhanced structural performance, reduced weight, vibration isolation, and improved stiffness-to-mass ratio compared to conventional bulk or lattice materials. Their periodic architecture distributes loads throughout the volume, enabling predictable deformation modes and energy absorption under mechanical stress.

[0150] In one embodiment, a fractional TPMS structure assembly is integrated as a loadbearing core in a TPMS sandwich panel, where two sheets or plates are bonded, brazed, fused, glued, or otherwise joined to the fractional TPMS structure assembly. Such panels may be used in aerospace, space, automotive, civil, marine, or architectural applications where high stiffness and low mass are desirable. This innovation allows engineering applications to use TPMS sandwich panels which have can have mechanical advantages over honeycomb sandwich panels for potentially lesser cost and use a larger selection of materials.

[0151] In another embodiment, fractional TPMS structure assemblies may be stacked or tiled to form thicker mechanical members, beams, panels, or modular blocks.Stacked arrays may interlock or be bonded to form larger assemblies, enabling graded stiffness regions, impact-absorbing layers, or vibration-controlled structures tailored to load requirements.

[0152] In another embodiment, a fractional TPMS structure assembly is used as an internal structural component within a vessel or tank for transporting or storing fluids such as fuel, oxidizer, water, chemicals, or cryogenic liquids. The fractional TPMS structureassembly may act as both a mechanical reinforcement member and a sloshmitigation structure, replacing or reducing the need for internal baffles or dampers used in conventional tank designs. This may be advantageous in applications including, but not limited to, rocket propellant tanks, ship and marine tankers, road and rail tanker cars, aviation fuel tanks, and pressurized storage tanks, where fluid motion must be controlled during acceleration, maneuvering, or partial fill states.

[0153] In one variation, the fractional TPMS structure assembly consists of a sheet TPMS unit cells (e.g., diamond, CLP, gyroid, primitive) arranged to divide the interior volume into two or more compartments. Fuel may be withdrawn from one compartment before the other, providing a more consistent mass distribution over time and reducing shifting center-of-gravity effects — particularly valuable in aerospace or aircraft fuel management. In one embodiment, the fractional sheet TPMS structures assembly may be fabricated using the manufacturing methods described in Figure 13, Figure 14A, or Figure 14B allowing lightweight structures with very low solid fraction to maximize usable tank volume while minimizing added mass.

[0154] An advantage of the fractional TPMS structure assembly compared to traditional baffles is that the fractional TPMS structure assembly provides a continuous loadbearing network rather than isolated plates, which may significantly increase buckling resistance and compressive stability, especially in thin-wall tanks used in rockets that are structurally stable only when pressurized. The fractional TPMS structure assembly may therefore improve tank rigidity when internal pressure decreases, reducing collapse risk during low-pressure phases such as coast or vent cycles.

[0155] Figure 19A illustrates an embodiment of a vessel 1901 comprising an internal fractional TPMS structure assembly 1903. The internal fractional TPMS structure assembly 1903 is shown as a diamond sheet geometry, though other types of TPMS may be used depending on mechanical or fluid-dynamic requirements. A sectional view of the vessel 1902 demonstrates the internal fractional TPMS structureassembly 1903 bonded to or mechanically interfacing with an outer tank shell 1904, forming a reinforced vessel that limits sloshing and distributes load throughout the interior of the vessel 1901.

[0156] Figure 19B illustrates another embodiment of a wing-shaped vessel with an internal fractional TPMS structure assembly, depicted with 1905 and without 1906 the external wall present for clarity. In this embodiment, the fractional TPMS structure assembly may contribute to increased flexural strength or tailored flexibility within the wing profile while simultaneously functioning as a fuel tank or fluid containment structure. Optionally, fuel may be drained from compartments formed by the fractional TPMS structure assembly in a controlled sequence, or structural thickness and solid fraction may be varied along the wing span to achieve desired stiffness or weight distribution at specific locations.

[0157] In civil and infrastructure applications, a fractional TPMS structure assembly may be used as a formwork or internal geometry within concrete, cementitious grout, geopolymer, or similar cast materials. When cast material is cast around or through a fractional TPMS structure assembly acting as a template — as demonstrated in the manufacturing embodiment of Figure 14C — the cured material forms a negative imprint of the fractional TPMS structure assembly, producing a lightweight structural member with internal fractional TPMS unit cells. Depending on design requirements, the fractional TPMS structure assembly may be retained permanently to provide insulation, void creation, acoustic damping, buoyancy, or weight reduction, or may be removed post-curing. Integration of TPMS into concrete elements can reduce material consumption, lower structural mass, enable reinforcement routing, improve thermal performance, maintain drainage pathways, and enable new structural forms not feasible with conventional formwork. The following embodiments illustrate specific uses of fractional TPMS structures assembly in civil and infrastructure applications.

[0158] In a further embodiment, rebar or conduit tubing is placed along curved surfaces of fractional TPMS structures during assembly, simplifying alignment and reducinginstallation labor by using the configuration of the fractional TPMS unit cell as a routing and positioning guide. The installation process may include placing a first fractional TPMS structure, positioning rebar or conduit tubing along the exposed curved TPMS pathways, stacking a second fractional TPMS structure on top of the first fractional TPMS structure, and repeating the process layer-by-layer until a fractional TPMS structure assembly achieves the desired size. The rebar or conduit may rest directly on the fractional TPMS unit cell surfaces during stacking, reducing misalignment errors, decreasing the number of reinforcement ties typically required to hold rebar grids in place prior to pouring, and allowing the creation of predefined conduit pathways without drilling or coring after curing. Fractional TPMS structures that provide continuous surface channels across arrays of fractional unit cells may be particularly suitable for this purpose, including but not limited to CLP, Primitive, Diamond, and IWP fractional TPMS unit cells, as their repeating curvature allows reinforcement or conduit to naturally follow consistent paths through the slab.

[0159] In another embodiment, a fractional TPMS structure assembly is used in floor plate construction, where cast material is poured around and over the fractional TPMS structure assembly acting as a template, to form a reduced-mass slab while preserving mechanical strength. In a further embodiment, the solid fraction of the fractional TPMS structure assembly is varied spatially. For example, having higher solid fractions of the concrete near support columns and lower solid fraction in midspan regions. This configuration reduces concrete material consumption and slab mass while maintaining load-bearing performance.

[0160] In another embodiment, a fractional TPMS structure assembly fabricated from a thermally insulating material is incorporated into concrete walls, facade panels, or building envelopes to reduce thermal bridging. The internal TPMS voids interrupt direct conduction paths, which may improve thermal insulation and reduce heat loss while maintaining structural continuity. In another embodiment, a fractional TPMS structure assembly fabricated from an acoustic insulating material is incorporated into concrete walls, facade panels, or building envelopes to reduce acoustic transmission or reflection. In this case, the distributed void network within thefractional TPMS structure assembly with the acoustic insulating material may scatter, absorb, or diffuse sound waves, reducing transmitted acoustic energy compared to solid concrete of equivalent thickness. In some cases, for both thermal or acoustic insulation, the fractional TPMS structure assembly can cause a reduction of concrete volume within the wall assembly, which may lower material usage and overall weight, as well as manufacturing costs.

[0161] Figure 23 is a sectional perspective view of a reduced-mass slab floor plate 2300 constructed using a fractional TPMS structure assembly as a TPMS templating structure. The templating structure is assembled by first placing a first fractional TPMS structure 2301 (shown as an array of fractional Primitive TPMS unit cells). The continuous curved channels of this fractional TPMS structure enable rebar 2321 to be positioned, using the TPMS geometry as an integrated reinforcement-routing guide. A second fractional TPMS structure 2302 is then stacked over the first.Additional fractional TPMS structures (for example, 2303, 2304, 2305; a topmost layer not shown) may be stacked sequentially along with rebar until the fractional TPMS structure assembly reaches the desired slab depth.

[0162] In the illustrated embodiment, the solid fraction of the fractional TPMS unit cells varies across the assembly. At the slab perimeter, the fractional TPMS structure has a 50% solid fraction, while nearer the center of the plate, the fractional TPMS structure has a 75% solid fraction 2306 are used. This variation may be smooth or discrete from the load bearing wall to the center of the floor plate or with varying values for the solid fractions. This allows the designer to increase material density and stiffness in high-load or column-supported regions while reducing mass in mid-span areas. Once the fractional TPMS structure assembly is complete, cast material is poured around and over it, forming the casted templated TPMS structure. In regions templated by 50%-solid-fraction fractional TPMS structures, the final slab exhibits a 50% void fraction 2312; in regions templated by 75%-solid-fraction structures, the casted region exhibits a 25% void fraction 2311.

[0163] In another embodiment, the fractional TPMS structure assembly is positioned between insulated concrete form (ICF) panels such that insulation spans continuously from one wall face to the other. This configuration may eliminate the need for conventional ICF plastic web ties that normally bridge between foam walls, and may provide a built-in routing mechanism for rebar placement prior to concrete pouring. In a further embodiment, the fractional TPMS structure includes curved or wavy predetermined periods that mechanically interlock with the opposing ICF panels. The wavy surfaces of the fractional TPMS structure may increase contact area relative to traditional peg-style interlocks, improving form alignment during assembly and providing structural engagement under load. In a further embodiment, fractional TPMS structures with high compressive load capacity, such as fractional CLP TPMS structures oriented with structural pathways aligned to the load direction, or fractional Primitive TPMS structures, may be particularly suited for this application.

[0164] Figure 24 illustrates a sectional perspective view of an insulated concrete form (ICF) wall assembly incorporating a fractional TPMS structure assembly for improved insulation continuity, and rebar routing. Three full ICF panels 2401, 2402, and 2403 are shown, with a fractional TPMS structure assembly 2413 positioned between opposing foam panels 2411 and 2412. The fractional TPMS structure assembly fills the interior cavity normally occupied by discrete plastic web ties, providing a continuous TPMS-based geometry across the wall thickness.

[0165] During assembly, horizontal reinforcement 2421 (e.g., rebar) may be placed along the curved TPMS surfaces, which act as natural positioning guides. Vertical reinforcement 2422 may be inserted through predefined openings in the fractional TPMS structure assembly either prior to stacking the ICF panels or after partial assembly, depending on construction sequencing. Once concrete is poured around the fractional TPMS structure assembly, the resulting wall incorporates continuous insulation, integrated reinforcement routing, and a structurally engaged TPMS- derived internal geometry.

[0166] In one embodiment, a fractional TPMS structure assembly is incorporated directly into one or both concrete wythes of a concrete sandwich panel. In this configuration, the fractional TPMS structure assembly functions as an embedded insulation element, a templating form, a reinforcement-routing guide, and / or a decorative geometry within the concrete face. Rebar may be placed within the fractional TPMS structure layers prior to casting and may optionally be pretensioned or posttensioned. Concrete is then poured around the fractional TPMS structure assembly, forming an integral wall section with reduced concrete volume and improved thermal resistance compared to a solid slab.

[0167] In another embodiment, the fractional TPMS structure assembly is positioned within the core region between two concrete wythes of a concrete sandwich panel. The fractional TPMS structure assembly forms continuous channels through the panel thickness, which may be used to (i) place shear connectors during manufacturing, (ii) allow the cured concrete to act as its own shear transfer mechanism, or (iii) enable casting of both wythes from one side of the form. This approach simplifies manufacturing, and preserves mechanical connectivity between wythes without conventional plastic or metallic ties.

[0168] In another embodiment, the fractional CLP TPMS structure assembly aligned with the primary load direction, or the fractional Primitive TPMS structure assembly, may be preferred for vertical load-bearing wall assemblies due to their compressive performance and efficient stress distribution. Alternative fractional TPMS structure assembly may be selected based on shear transfer, ductility, insulation, drainage, or architectural considerations.

[0169] In another embodiment, a fractional TPMS structure assembly formed from low- density or buoyant materials — such as styrofoam, polyurethane foam, or similar lightweight foams — is incorporated into floating civil structures including bridge pontoons, marine platforms, floating walkways, or buoyant foundations. The fractional TPMS structure assembly is placed within a mold or formwork, and reinforcement members (e.g., rebar, steel cable, carbon-fiber rods, or tensionmembers) may be incorporated prior to casting. A lightweight or marine-grade concrete mixture is then poured around the fractional TPMS structure assembly and cured to form a composite concrete shell, while the embedded fractional TPMS structure assembly maintains flotation, reduces overall density, and provides internal stiffening and impact tolerance. Use of fractional TPMS structure assemblies may allow large floating modules to be cast as single monolithic elements, reducing fabrication complexity compared to traditional pontoon construction or laminated foam-core assemblies.

[0170] In a further embodiment, the interior of the floating body comprises a high-solid- fraction fractional TPMS structure assembly to maximize internal buoyancy, while the outer surface comprises a cast concrete or fiber-reinforced composite shell for abrasion resistance, hydrodynamic stability, and environmental durability.

[0171] In another embodiment, fractional TPMS structure assemblies are used for the production of shoreline protection blocks, seawall elements, or breakwater modules. The fractional TPMS structure assembly may be completely or partially removed post-curing to create a negative fractional TPMS structure assembly, or remain embedded as a core material depending on material, buoyancy, or drainage requirements. The cast structure may include reinforcement, fiber additives, or corrosion-resistant materials for marine durability.

[0172] In a variation of this embodiment, the fractional TPMS structure assembly is removed after curing to produce a hollow TPMS breakwater unit. The internal TPMS void network allows waves to enter the structure and dissipate energy internally, reducing peak loading and storm surge force transfer.

[0173] In another variation of this embodiment, the fractional TPMS structure assembly comprises a porous or water-wicking material and is retained inside the concrete to form a porous fractional TPMS structure assembly in the form of a shoreline protection unit. A porous or water-wicking material — such as compressed organic fiber board, biodegradable natural fibers, wood pulp fiberboard, or open-cell mineral wool — may be used to enable controlled water drainage. This configuration reducesdownstream pooling, allows slow water permeation similar to geotextile fabrics, and mitigates soil erosion behind the structure.

[0174] In another embodiment, fractional TPMS structure assemblies are used as decorative or architectural bricks, tiles, facade modules, roadway units, or walkway pavers. The resulting product provides a number of possible benefits obtained through the use of fractional TPMS structure assemblies. For example, the periodic TPMS geometry may provide visually distinct surface patterns, including due to light passing through the TPMS channels to create aesthetically striking visual and illumination effects. Such structures may be used for architectural facades, landscape installations, or artistic lighting features. When used as roadway or pedestrian paving blocks, the TPMS geometry may also allow water to drain through the structure, reducing pooling and slip hazards while lowering material volume and weight relative to solid bricks. The interior voids of the TPMS form may remain open or be partially filled with aggregate, soil, vegetation, permeable media, or lighting elements, enabling functional integration for stormwater infiltration, green-infrastructure design, acoustic diffusion, or embedded utility routing within the same decorative element.

[0175] In another embodiment, a fractional TPMS structure assembly is used to cast roadway units, driveway slabs, or walkway pavers in which the fractional TPMS structure assembly remains embedded after casting. The fractional TPMS structure assembly may be fabricated from a low-density TPMS templating material such as expanded polystyrene, polyurethane foam, cork-based composites, or other lightweight TPMS materials that reduce total concrete volume and overall structural mass. In another example, the fractional TPMS structure assembly may be fabricated from water-wicking or permeable TPMS templating materials, allowing water to infiltrate and drain through the finished paver to reduce pooling, promote freezethaw durability, and improve surface safety. Retention of the fractional TPMS structure assembly after casting provides several functional benefits. The internal TPMS void geometry maintains compressive load-carrying capability while reducing the amount of concrete required, thereby lowering material use without sacrificing structural performance. The TPMS channels also provide predefined routingpathways for heating pipes, resistive heating elements, or hydronic snow-melt systems, enabling precise placement and improved thermal efficiency by directing heat toward the surface rather than the underlying ground. In a further embodiment, the TPMS geometry provides stable, repeatable support locations for rebar, steel mesh, or other tensile reinforcement members, simplifying construction and reducing installation variability. The embedded fractional TPMS structure assembly may also contribute to thermal insulation, reduced weight, and improved drainage in the completed roadway or paving element.Thermal Applications

[0176] In one embodiment, a fractional TPMS structure assembly is used as a heat exchanger core to transfer heat to one fluid or between two fluids. The interconnected network pathways of the TPMS geometry may allow a fluid to flow through one set of channels, allowing heat to transfer from the solid fractional TPMS structure assembly to the fluid. For fractional sheet TPMS structures, there are two separate networks. For fractional skeletal TPMS structures, there is one network. The large internal surface area of the fractional TPMS structure assemblies may increase thermal exchange efficiency. Depending on the design, the fractional TPMS structure assembly may be used for solid-gas, liquid-liquid, gas-gas, gas-liquid, or phase-change heat exchange.

[0177] In a further embodiment, a sheet type fractional TPMS structure assembly is used in the heat-exchanger core that separates and transfers heat between two fluid flows. A first fluid flows through a first TPMS network, while a second fluid flows through a second TPMS network, with the two networks thermally coupled through the sheet type fractional TPMS structure assembly walls. Heat is transferred from the first fluid, through the wall, and into the second fluid. The flow configuration may be counter-current or co-current depending on thermal design requirements. Adjusting the fractional sheet TPMS solid fraction, channel topology, or wall thickness may be used to tune pressure drop, heat transfer area, and overall thermal performance.

[0178] Figure 17 illustrates an embodiment of a sheet type fractional TPMS structure assembly configured as a plate heat exchanger 1701 comprising a heat-exchanger core 1708. The plate heat exchanger 1701 includes a fluid one inlet 1702, fluid one outlet 1712, fluid two inlet 1713, fluid two outlet 1703, a fluid one inlet distribution plate 1706, fluid one outlet distribution plate 1710, fluid two inlet distribution plate 1709, fluid two outlet distribution plate 1705, a lower cover 1711, an upper cover 1704, an outer shell 1707, and a heat-exchanger core 1708 positioned between the distribution plates. In Figure 17, the sheet type fractional TPMS structure assembly is shown as a fractional diamond sheet TPMS unit cell having approximately 10% solid fraction, however other types of TPMS, with varying shapes or solid fraction, may also be used. Types of TPMS structures that provide more uniform hydraulic pathways may be particularly preferred for even flow distribution through the exchanger.

[0179] The solid fraction of the sheet type fractional TPMS structure assembly may be varied to adjust heat transfer performance, allowable pressure differential, and mechanical rigidity. The heat-exchanger core 1708 may be fabricated from thermally conductive and / or corrosion-resistant materials including, but not limited to, copper, brass, stainless steel, or aluminum. In one embodiment, multiple fractional TPMS structures are produced, stacked, and joined using a thermally conductive material such as a braze alloy, solder, conductive adhesive, or diffusion-bonded interface, to create a continuous heat-transfer pathway between layers of the fractional TPMS structure of the heat-exchanger core 1708. This joining approach is similar to the layered assembly and metallurgical joining embodiment illustrated in Figure 16.

[0180] In another embodiment, layers of fractional walled sheet TPMS structures are stacked alternately with solder / brazing sheets with the ends being stacked with distribution plates, upper cover, and lower cover, after which the entire assembly is placed in a heat-treating equipment to create internal sealed flow passages. As used herein, a fractional walled sheet TPMS structure refers to a fractional sheet TPMS structure enclosed or bordered by an outer wall or perimeter frame such that the TPMS geometry is contained within a defined boundary for stacking and sealing. Inone example, the fractional walled sheet TPMS structures, covers, and distribution plates are formed from stainless steel, stacked with copper sheets in-between them, and then joined using copper brazing to create a high-temperature, corrosionresistant, leak-proof fractional TPMS structure plate heat exchanger.

[0181] In Figure 17, during operation, fluid one flows within the one network, while fluid two flows through the second network which was directed using the distribution plates, enabling thermal exchange across the walls. The curved TPMS geometry may disrupt boundary layers, generate controlled mixing, and increase convective transfer area, resulting in higher heat flux compared to planar or pin-fin heat exchangers. The configuration shown in Figure 17 is arranged as a counterflow exchanger, with crossed inlet / outlet routing to maintain a high temperature gradient between fluids. Distribution plates may be designed to ensure uniform channel loading, minimize bypass regions, and prevent maldistribution.

[0182] In another embodiment, a fractional TPMS structure assembly acts as a TPMS air fin for convective heat removal into air. The TPMS air fin geometry generates controlled flow disruption, reduces boundary layer formation, and increases turbulence at moderate Reynolds numbers relative to planar or pin-fin geometries. The high surface-area-to-volume ratio of the fractional TPMS structure assembly enables greater heat transfer per unit mass, and the smooth continuous curvature reduces stagnant thermal zones. Solid fraction, wall thickness, orientation of the layers of the fractional TPMS structure, and type TPMS may be selected to tune airflow resistance and heat transfer performance. In a further embodiment, the TPMS air-fin comprises a sheet-type TPMS that presents a large continuous curved surface to airflow, increasing convective heat transfer efficiency. Fractional sheet TPMS structure assemblies provide extensive uninterrupted surface area, uniform fluid contact, and enhanced heat rejection capability, making them well suited for use as thermal fins or extended heat-exchange surfaces. In some embodiments, the TPMS air-fin is manufactured using the coating approach of Figure 14A to produce a fractional sheet TPMS structure assembly from a skeletal template. In some embodiments, the TPMS air-fin is manufactured using the press-formed method ofFigure 13 to produce layers of fractional sheet TPMS structures, which may then be stacked or joined to form the TPMS air-fin.

[0183] In one embodiment, a TPMS air-fin contains an integrated thermosyphon (or heatpipe) internal vapor chamber to create a TPMS thermosyphon-integrated fin structure, where one channel network of the fractional TPMS structure assembly contains the vapor chamber, while the second channel network is open to air, allowing simultaneous vapor condensation on one side and forced / natural convection of air on the other. Figure 18 illustrates a perspective exploded view 1801 and a perspective sectional view 1810 of a TPMS thermosyphon-integrated fin structure 1802 on a heat source 1804. The working system includes: (i) a heat source 1804 (e.g., electronic processor, ASIC, power module), (ii) a thermal interface material 1803 such as thermal paste, thermal pad, metal thermal interface materials, or thermal adhesive, (iii) a TPMS thermosyphon-integrated fin structure 1802 comprising an internal vapor chamber 1806, and an external airflow region for convective cooling 1805. The heat source 1804 transfers heat through the thermal interface material 1803 into the evaporator section of the internal vapor chamber 1806 (the closest part of the heat source 1804 to the TPMS thermosyphon-integrated fin structure 1806), causing working fluid to vaporize and spread through the internal vapor chamber. Vapor then condenses over the large TPMS surface and releases heat into the forced / natural convection of air on the other side.

[0184] In one embodiment, the bottom region of the TPMS thermosyphon-integrated fin structure includes a void volume that allows vapor to freely access the full underside of the internal vapor chamber, enabling uniform vapor spreading and efficient condensation. This TPMS thermosyphon-integrated fin structure increases total condensing area and heat removal efficiency compared to conventional heat pipes attached to fins, by eliminating interfaces between the pipe and the fin.

[0185] In another embodiment, the TPMS thermosyphon-integrated fin structure is manufactured using the process shown in Figure 14B. In this embodiment, one or more layers of arrays of skeletal TPMS structures formed from zinc are firstfabricated. These zinc TPMS layers are stacked to the final full-size of the TPMS thermosyphon-integrated fin structure. This fractional stacked TPMS structure is then coated or plated with copper to create a sheet type templated TPMS structure comprising copper deposited over the skeletal zinc template. The thickness of the copper depends on the application with some embodiments varying between 0.01mm to 1mm thick. Drain or melt-out openings are then formed or exposed in the copper shell, and the zinc is removed by heating, melting, or dissolution, leaving behind a hollow copper sheet templated TPMS structure suitable for use as a TPMS thermosyphon-integrated fin structure.

[0186] In a further embodiment, one or both external surfaces of the copper templated TPMS structure may be plated or coated with nickel to increase corrosion resistance. Nickel deposition may occur after zinc removal, or in another embodiment before or after copper coating or stacking of the zinc layers, depending on desired properties.

[0187] In another class of embodiments, a TPMS energy storage material (TPMS-ESM) comprising a fractional TPMS structure or a templated TPMS structure is used within a thermal energy storage system (TPMS-TES) to store sensible, latent, or thermo-chemical heat. The TPMS-ESM consists of a TPMS geometry, providing continuous 3D network for heat transfer fluids. Compared to monolithic storage blocks, the TPMS-ESM provides improved heat penetration, faster charge / discharge rates, and reduced thermal gradients due to increased internal surface area and distributed flow access. Compared to pellet or packed-bed media, the TPMS-ESM offers significantly lower pressure drop, reduced bed settling / compaction, and improved structural integrity under high temperature operation. The smooth curvature of TPMS geometries may reduce thermal shock stresses and increase mechanical stability during temperature cycling. The TPMS-ESM may be manufactured as individual modular TPMS blocks, a monolithic TPMS cast structure, or combination of the two.

[0188] In another embodiment, heaters, thermocouples, or other instrumentation are routed within the TPMS-ESM by placing them within the interconnected void network ofthe TPMS geometry. The open continuous channels of fractional TPMS unit cells may allow heaters, thermocouples, or other instrumentation to be positioned inside the TPMS ESM. Suitable types of TPMS for heaters, thermocouples, or other instrumentation routing may include Primitive, CLP, IWP, Diamond, Q-Star, or other types of TPMS shapes with continuous through-channels. Heaters, thermocouples, or other instrumentation routed within the TPMS-ESM can make manufacturing of the TPMS TES easier while improving heat-up rates, supports active thermal management, and allows detection of temperature gradients not measurable in pellet or packed bed media.

[0189] The TPMS-ESM may be formed from materials selected according to the desired storage mechanism, including sensible, latent, or thermochemical heat storage. Sensible heat storage materials may include refractory ceramics, high-temperature castables, concrete or aggregate-enhanced concrete, iron-oxide aggregates (e.g., magnetite, hematite), or other solid media with high volumetric heat capacity. Latent heat storage materials may include salts, salt hydrates, metals, eutectics, or phasechange materials retained within or coated onto the TPMS geometry to permit melt / solidification cycling. Thermochemical storage materials may include metal oxides, carbonates, hydroxides, or other reversible reaction media contained within the TPMS network for chemical heat storage. Material selection may consider thermal stability at operating temperature, compatibility with the working fluid, corrosion resistance, thermal conductivity, and compressive strength under TES operating loads.

[0190] In another embodiment, fractional TPMS unit cells of varying dimensions within the TPMS-ESM are arranged in graded layers within a TPMS-TES to tailor pressure drop, flow distribution, heat transfer properties, and charge / discharge uniformity. As illustrated in Figure 20B, an arrangement may include large-unit-cell TPMS-ESM segments 2011, small-unit-cell TPMS-ESM segments 2013, and medium-unit-cell TPMS-ESM segments 2012. Large-unit-cell segments positioned near the fluid inlet and outlet may enable low-resistance entry and exit flow paths. Small-unit-cell segments increased surface-area contact promote uniform flow distribution acrossthe bed. Medium-unit-cell segments act as the bulk TPMS-ESM having better pressure drop than small-unit-cell segments and better heat transfer properties than large-unit-cell segments. In a further embodiment, the fractional TPMS unit cells of varying dimensions within the TPMS-ESM are arranged in graded layers into large, small, medium, small, then large-unit-cell segments as shown in Figure 21. In another further embodiment, the fractional TPMS unit cells of varying dimensions within the TPMS-ESM are arranged in graded layers into large, small, medium, then large-unit-cell segments as shown in Figure 20B. These graded layers configuration may reduce channeling, improve thermal utilization of the storage medium, and permit tuning of TES performance for charging or discharging conditions.

[0191] In another embodiment, fractional TPMS unit cells of varying solid fractions within the TPMS-ESM are arranged in graded layers within a TPMS-TES to tailor pressure drop, flow distribution, heat transfer properties, and charge / discharge uniformity. Low-solid-fraction TPMS-ESM segments behave similarly to large-unit-cell segments by providing low flow resistance and reduced pressure drop, improving access to inlet and outlet regions. High-solid-fraction TPMS-ESM segments behave similarly to small-unit-cell segments by promoting uniform flow distribution across the bed. However, high-solid-fraction segments store more energy per unit volume and may reduce the required footprint of the TPMS-TES, while low-solid-fraction segments may provide improved hydraulic performance. Intermediate-solid-fraction segments may be used as the bulk region to balance energy density and flow characteristics. In a further embodiment, solid-fraction-graded layers are arranged as low — high — medium — high — low or low — high — medium — low, depending on flow direction and thermal cycling requirements. In another embodiment, graded solid-fraction layering is combined with graded unit-cell sizing to achieve dual control of flow distribution and heat-transfer characteristics within the TPMS-TES.

[0192] Figure 20A shows a perspective view of a massive TPMS TES system 2001incorporating a TPMS-ESM 2008. A sectional view of the TPMS-TES 2002 shows six main components: (i) TPMS-ESM 2008, which serve as the heat-storagemedium; (ii) heaters 2003 and 2006, which transfer thermal energy into the TPMS- ESM during charging; (iii) inlet port 2004, through which the heat-transfer fluid is introduced; (iv) outlet port 2005, through which heat transfer fluid exits the system; (v) a sealed vessel 2007, containing the TPMS-ESM and process environment; and (vi) interior insulation 2009, which reduces heat loss to the environment.

[0193] In Figure 20A, the heat transfer fluid enters the inlet port 2004 and is directed to the bottom of the massive TPMS TES. This heat transfer fluid would then be spread using a mechanism such as a large-unit-cell TPMS-ESM segments followed by small-unit-cell TPMS-ESM segments, and allowed to flow through the bulk of the TPMS ESM. By flowing through the TPMS-ESM 2008, heat is transferred from the TPMS-ESM 2008 to the heat transfer fluid, creating a hot heat transfer fluid. This hot heat transfer fluid would then exit at the outlet port 2005.

[0194] Figure 20A illustrates an embodiment of a massive TPMS-TES system in which TPMS-ESM is used as the primary heat storage medium within a large insulated vessel. In this embodiment, the massive TPMS-TES may be assembled from modular TPMS-ESM blocks that are manufactured on or off site as discrete segments or a monolithic TPMS-ESM casted structure. Modular construction of the components may reduce capital cost, shorten installation time, and enable field- scalable systems in remote or industrial locations. The system may be installed above ground, below ground, or partially buried depending on site constraints and insulation strategy. The TPMS-ESM blocks may be arranged such that all heater elements, thermocouples, and internal components are serviceable from a single face of the TES, allowing components to be removed or replaced without full disassembly. In Figure 20A, the access surface is illustrated at the top, although side-access configurations may be used similarly.

[0195] In another embodiment, a partition wall is positioned between the inlet port 2004 and outlet port 2005 to establish a U-shaped flow path within the TPMS-TES, allowing fluid to enter and exit near the same surface. This configuration may simplify construction and reduce plumbing requirements, while still enabling effective chargeand discharge when adequate internal flow distribution is provided. In such an embodiment, segments of small-solid-fraction or small-unit-cell TPMS-ESM may be placed near the turnaround region to enhance lateral distribution and prevent preferential channeling.

[0196] Figure 21 illustrates an embodiment of a containerized TPMS-TES 2101, in which the storage vessel is housed within a transportable enclosure, such as a standard shipping container. The system may comprise an external container 2102, external insulation 2103, internal insulation 2104, a sealed internal vessel 2105, and one or more segments of TPMS-ESM 2106-2108. Heaters 2111 may extend longitudinally between the inlet and outlet faces of the TPMS-ESM to promote uniform charging by distributing heat input through the interior of the storage medium. Caps or ports 2110 may be used for insertion, replacement, or maintenance of heater elements or thermocouples without requiring disassembly of the entire storage system.

[0197] In this embodiment, the containerized TPMS-TES may be delivered as fully modular, installed adjacent to industrial processes, district energy networks, or renewable generation assets, and connected through fluid lines to act as a behind-the- meter heat battery. This approach enables rapid deployment, factory fabrication, and field replacement without requiring large civil infrastructure. A containerized TPMS- TES may serve industrial heat loads directly, operate as long-duration storage for excess renewable power, or function as a dispatchable thermal asset for gridbalancing and demand-response applications.

[0198] Compared to pellet-bed thermal storage, the TPMS-TES provides lower pressure drop, no media settling or compaction, and improved maintainability, reducing operational expenditure. Fractional TPMS structures of this invention may also permit higher effective solid fraction relative to pellet beds, increasing volumetric energy density and reducing footprint for equivalent storage capacity. The continuous curvature and structural continuity of TPMS-ESM may improve mechanical stability, thermal shock tolerance, and cycling durability under repeated charge / discharge conditions. In large-scale systems, this architecture may provide aplatform for grid-integrated demand response, allowing renewable electricity to be converted to heat during surplus generation and dispatched as thermal energy to industrial or di strict- scale users. The combination of high surface area, modular maintainability, and scalable manufacturing may enable economical deployment of long-duration thermal storage for both centralized and microgrid applications.

[0199] In another embodiment, if the type of TPMS shape contains closed or partially closed channels, dedicated instrument channels may be formed during manufacturing of the fractional TPMS structures or during stacking of the fractional TPMS structures to permit placement of heaters, thermocouples, or other instrumentation at desired locations in the final fractional TPMS structure assembly.

[0200] In another embodiment, the TPMS-TES uses internal heaters to charge the TPMS- ESM by resistive, inductive, or radiative heating. Suitable heater elements may include, but are not limited to: Inconel-clad heating rods, stainless-steel sheath heaters, Kanthal or nichrome resistive coils, ceramic insulated heaters, graphite or carbon-based resistive heaters, immersion tubular heaters, induction heating coils, or embedded heat-pipe / thermosyphon elements. Heater selection may depend on operating temperature, corrosive environment, fluid composition, and electrical compatibility. Heaters may be inserted within continuous TPMS channels, routed through dedicated heater conduits formed during fabrication of the fractional TPMS structures or stacking, or embedded permanently within the TPMS-ESM during manufacture.

[0201] In another embodiment, insulation within the TPMS-TES comprises one or multiple layers of thermal insulation materials. Suitable materials may include but are not limited to microporous insulation, fumed-silica panels, vacuum insulation panels, mineral wool, ceramic fiber blanket, refractory insulation board, perlite or aerogel- filled panels, or combinations thereof. Insulation may be arranged as internal insulation surrounding the TPMS-ESM, external insulation around the sealed vessel, or nested multi-layer insulation to meet required thermal retention. The insulationstack may be chosen based on allowable temperature, thermal cycling durability, moisture exposure, or installation configuration (above- or below-ground).

[0202] In another embodiment, temperature sensors, thermocouples, fiber-optic temperature probes, RTDs, or distributed sensing cables are inserted into the TPMS-ESM using the continuous void network of the TPMS geometry. Sensors may monitor temperature gradients during charge and discharge, enabling control algorithms to regulate heater power, flow rate, or storage set-points. In one embodiment, sensor feedback controls heater activation such that heating is increased when measured temperature is below a threshold and decreased when the TPMS-TES is fully charged, preventing over-heating. In another embodiment, multi-zone sensing allows stratified heating, where only selected regions of the TPMS-ESM are heated to maintain thermal layers or to reduce electrical demand during off-peak operation. In a further embodiment, data from distributed temperature sensors informs predictive control or demand-response dispatch strategies for integration with renewable power or district-heat networks.

[0203] In one embodiment, the TPMS-ESM is fabricated using a refractory concrete matrix containing energy-dense iron-ore aggregates such as magnetite or hematite to increase volumetric heat capacity and temperature stability using a templating process similar to the method illustrated in Figure 14C, wherein a TPMS templating structure (e.g., polystyrene or polyurethane foam) defines the negative geometry of the TPMS network. The refractory concrete matrix is poured into the voids of the TPMS template, cured, and optionally heat-treated. The templating material is removed (e.g., solvent dissolution, melting, burnout) to yield a skeletal TPMS-ESM with high thermal mass and high-temperature durability. This embodiment offers enhanced sensible-heat storage density, mechanical stability under thermal cycling, and compatibility with high-temperature TES operation.

[0204] In another embodiment, a closed-wall sheet fractional TPMS structure assembly, such as those produced by the plating or coating processes illustrated in Figures 14A and 14B (for example, using layers 1402, 1403, 1404, or sealed layer 1411), is usedas the containment architecture of a TPMS TES. The closed-wall sheet fractional TPMS structure assembly defines a continuous, sealed network of cavities that may be filled with a phase-change material (PCM) such as zinc, aluminum, molten salts, metallic PCMs, hydrated salts, paraffins, or eutectic alloys. During charging, heat is transferred through the walls to melt the PCM, storing thermal energy as latent heat. During discharge, the PCM solidifies while releasing heat back through the walls. The continuous sheet TPMS geometry increases internal heat-transfer surface area, reduces thermal gradients, and enables faster melting and solidification relative to monolithic PCM modules.

[0205] In another embodiment, a sheet-type fractional TPMS structure assembly with closed walls is configured such that one side of the TPMS wall contains a chemical or thermochemical heat-storage medium, while the opposite side forms a separate flow network for a heat-transfer fluid in the TPMS TES. The walls act as a barrier that allows heat conduction while maintaining chemical isolation between the heat storage medium and the heat-transfer fluid. This configuration enables the storage material to operate under optimal chemical, thermal, or atmospheric conditions (for example, controlled oxygen levels, humidity, pressure, or reactive gas environment) while preventing contamination of the working fluid. Suitable chemi cal -storage media may include reversible redox materials, metal hydrides, carbonate looping materials, ammonia-salt sorbents, or other thermochemical energy carriers requiring environmental separation from the process fluid. The dual-network sheet TPMS architecture allows heat to be supplied or withdrawn efficiently through the sealed wall, supporting continuous cycling and high utilization of the chemical storage medium.Chemical Applications

[0206] In another class of embodiments, a fractional TPMS structure assembly is used as a chemically active or passive contact medium for gas, liquid, or multiphase processing. The continuous, triply-periodic geometry provides large surface-area-to-volume ratio, low pressure drop relative to foams or packed beds, and uniform flow distribution with reduced channeling. These characteristics enable enhanced mass transfer, reaction kinetics, and sorbent utilization for separation, catalysis, chemical conversion, electrochemical processes, scrubbing, and filtration.

[0207] In another embodiment, a fractional TPMS structured adsorbent is configured as an adsorbent, which comprises a fractional TPMS structure assembly, for gas separation, purification, or storage. The fractional TPMS structured adsorbent may be coated, infiltrated, or formed entirely from adsorbent materials such as zeolites, activated carbon, silica gel, alumina, MOFs, amine-functionalized solids, or mixed adsorbent composites. The smooth periodic curvature of TPMS geometry enables high surface-area exposure while maintaining continuous, low-pressure-drop channels, improving mass-transfer rates compared to pelletized or granular beds. The interconnected through-channels reduce intra-particle diffusion resistance, allowing faster adsorption and desorption cycles, improved working capacity, and reduced regeneration energy.

[0208] In a further embodiment, the fractional TPMS structured adsorbent is a TPMS structured zeolite comprises a fractional TPMS structure assembly is manufactured from a zeolite material, and comprises zeolite particles (including but not limited to 13X, 5A, LiX, NaX, ZSM-5, LiLSX, NaMOR, or other combinations of structures listed on the International Zeolite Association Structure Database with counterbalancing metal cations from the periodic table). Zeolites may be incorporated by forming, molding, or casting into fractional TPMS structures or by slurry infiltration followed by drying and sintering, wash-coating, or in-situ crystallization on a fractional TPMS structure assembly that acts as a scaffold.Compared to traditional zeolite beads or pellets, the TPMS structured zeolite reduces channeling, eliminates pellet crushing and attrition, and enables uniform heat transfer during regeneration. Depending on the application, the TPMS structured zeolite can also act as a catalyst.

[0209] In another embodiment, the fractional TPMS structured adsorbent is used for portable oxygen concentrators or oxygen / medical air purification devices, wherein the periodic TPMS network reduces pressure drop for cyclic adsorption processes such as pressure swing adsorption (PSA), vacuum swing adsorption (VS A), or rapid- pressure-swing oxygen generation. The TPMS geometry can shorten cycle time, improve breakthrough performance, reduce sorbent mass, and enable compact devices with lower compressor power.

[0210] In a further embodiment, the TPMS structured adsorbent comprises segmented adsorbent regions, such as: high-capacity zeolite segments for bulk capture, fast- kinetic sorbent segments for polishing, or hydrophobic / hydrophilic gradient segments for humidity tolerance.

[0211] In another embodiment, a TPMS structured catalyst comprises a fractional TPMS structure assembly is configured as a catalyst, and comprises a catalyst support or structured reactor medium. The continuous high-curvature surface of TPMS geometry provides catalyst exposure similar to foams and honeycombs, while reducing stagnant zones and enabling more uniform reactant contact. The open, periodic flow pathways may reduce pressure drop relative to packed catalyst pellets, improving space velocity, reducing blower / compressor power, and allowing higher throughput operation. Catalyst material may be coated, impregnated, wash-coated, infiltrated, or grown in-situ onto or within the TPMS structured catalyst. Active phases may include noble metals (Pt, Pd, Rh, Ru), base metals (Ni, Co, Cu, Fe), oxides, perovskites, MOFs, carbon-based catalysts, or any catalytic formulation suitable to the chemical environment.

[0212] In a further embodiment, the TPMS structured catalyst is used as a clean combustion exhaust catalyst substrate, including for automotive, marine, industrial burner, biomass boiler, or residential furnace emissions. The fractional TPMS structure assembly may be made of or a be support structure for a catalyst oxidation / reduction catalyst for CO, NOx, VOC, or particulate oxidation. The increased surface- accessible area-to-volume ratio may reduce required catalyst loading and lower light-off temperature, enabling more efficient pollutant conversion during cold-start or low-temperature operation. Continuous pathways reduce soot clogging and ash buildup compared to honeycomb monoliths, extending service lifetime.

[0213] In another embodiment, the TPMS structured catalyst is used for ozone reduction, smog abatement, or photocatalytic oxidation. A photocatalyst such as TiO2, ZnO, or doped titania may be applied as a thin coating onto the TPMS surface. UV illumination or visible-light active catalysts may break down ozone and VOCs in indoor environments, HVAC systems, tunnels, cold storage facilities, or aircraft cabin recirculation. The TPMS geometry provides enhanced photon exposure area, increases active site access, and improves mass transfer relative to planar filters. In a further embodiment, the TPMS structured catalyst may be electrically heated or thermally cycled to accelerate regeneration or self-cleaning.

[0214] Catalyst washcoat thickness, porosity, surface roughness, or solid fraction of the TPMS structured catalyst may be tuned to optimize mass-transfer coefficient, residence time, and conversion efficiency. Multi-layer stacked fractional TPMS structures configured as catalytic blocks may be replaced individually, enabling modular maintenance without full reactor shutdown.

[0215] In another embodiment, a fractional TPMS structure assembly comprises a conductive material and is configured as an electrode, flow field, or active current collector within an electrochemical system. Applications include CO2 electrolyzers, water or hydrogen electrolyzers, fuel cells, redox flow batteries, metal-air batteries, or other electrochemical reactors. The open interconnected TPMS pathways increase electrochemically accessible surface area, reduce pressure drop for reactant flow, and enable uniform current and ion distribution across the active region. The fractional TPMS structure may be manufactured from, or subsequently coated with, conductive materials such as carbon, graphite, nickel, copper, stainless steel, titanium, conductive ceramic coatings, or combinations thereof.

[0216] In one embodiment, a fractional TPMS structure assembly is configured as a battery, and comprises an electrochemical cell constructed from a TPMS anode, a TPMSseparator, and a TPMS cathode arranged in sequence as skeletal — sheet — skeletal fractional TPMS structures. The TPMS geometry defines a 3 -dimensional porous network that simultaneously provides current-collector pathways, ion transport channels, and mechanical reinforcement. Unlike planar foil electrodes, this configuration produces a volumetrically structured cell where electrochemical reactions occur throughout a continuous 3D surface.

[0217] In one embodiment, the fractional TPMS structure assembly is configured as a battery, and is manufactured by first producing an anode scaffold fractional skeletal TPMS structure which functions as the anode scaffold. The anode scaffold fractional skeletal TPMS structure is then coated or infiltrated with anode active material using slurry coating, melt infiltration, electrodeposition, vapor deposition, sol-gel infiltration, or powder sintering. A TPMS separator layer is then deposited conformally onto the surface of the anode scaffold fractional skeletal TPMS structure, forming a sheet-type TPMS separator. The separator may be applied via dip-coating, polymer electrolyte casting, ceramic electrolyte deposition, spray coating, electrophoretic deposition, or in situ polymerization. After the separator is established, the remaining interconnected void network is filled or infiltrated with cathode active material, forming a second skeletal TPMS cathode structure geometrically interlocked with the separator. Assembly results in a TPMS anodeseparator-cathode structure.

[0218] This TPMS battery architecture increases electrochemically active surface area, shortens ion-transport pathways, and reduces internal resistance compared to planar electrodes. The interpenetrating TPMS geometry mechanically suppresses dendrite growth and improves cycling durability by distributing mechanical stress through the 3D lattice. The sheet-type separator conforms to the TPMS curvature, preventing electrode contact while permitting ion transport throughout the full volume of the cell. The skeletal cathode and anode may act as self-supporting current collectors, enabling thin or fully removed metallic foils. The fractional TPMS structure assembly configured as a battery may be assembled into modules by stacking ortessellating TPMS anode-separator-cathode structure, optionally allowing roll-to- roll production if fabricated as repeating TPMS anode-separator-cathode structure.

[0219] In one embodiment, a fractional TPMS structure assembly is used as a structured packing contactor for gas-liquid mass transfer, enabling absorption, stripping, carbonation, humidification, dehumidification, or capture of gaseous species. The continuous triply-periodic geometry provides a tortuous but unobstructed flow network, promoting gas-liquid interfacial renewal and suppressing channeling relative to conventional random packings. The high surface-area-to-volume ratio of fractional TPMS structure assemblies increases contact area between phases while maintaining low pressure drop. Fractional TPMS solid fraction, wall thickness, and unit-cell size may be adjusted to tune wettability, residence time, pressure drop, and interfacial area depending on the scrubbing duty. Wetted operation may be countercurrent, cocurrent, or crossflow, with the fractional TPMS structure assembly acting as fixed packing inside towers, columns, membrane-less contactors, or modular cartridge systems.

[0220] In another embodiment, the fractional TPMS structure is fabricated as sheet or skeletal packing blocks which are stacked or tessellated to form tall packed beds within scrubbers or absorber columns. Fractional sheet-type TPMS geometries enable continuous liquid films and uniform wetting patterns, while skeletal geometries allow deeper penetration of gas phase into liquid films, potentially improving mass transfer coefficients.Carbon Capture System

[0221] Referring now to Figure 22 A, as a non -limiting example, such fractional TPMS structure assemblies can form one or more beds within a carbon capture system.

[0222] In an embodiment, the carbon capture system comprises two beds 2201, with each bed comprising a TPMS structure assembly. The first carbon capture bed 2203 is positioned upstream of the second carbon capture bed 2202. That is, when thecarbon capture system is in operation, air passes through an air inlet, then through the first carbon capture bed 2203, then through the second carbon capture bed 2202, and finally through a depleted air exhaust. The first carbon capture bed 2203 may be fused directly to the second carbon capture bed 2202, or a duct may connect the beds. It should be understood that additional carbon capture beds can be added to the carbon capture system to optimize carbon capture.

[0223] Preferably, the fractional TPMS structure assembly is formed of zeolite. However, it should be understood that any suitable absorbent material may be used for the fractional TPMS structure.

[0224] In an embodiment, the first carbon capture bed 2203 comprises 90% of the combined volume of the first carbon capture bed 2203 and the second carbon capture bed 2202.Correspondingly, the second carbon capture bed 2202 comprises 10% of the combined volume. However, it should be understood that various proportions are possible depending on various parameters, such as bed volume, air flow, pressure, and TPMS shape.

[0225] As shown in Figure 22B, the first carbon capture bed has a Diamond unit cell dimension that is stretched along the axis of air flow, thus forming a first-bed fractional 1: 1:0.5 Diamond unit cell 2205. Each fractional TPMS unit cell is a i fraction of a full TPMS unit cell. The individual fractional TPMS unit cells in the first carbon capture bed are arranged in an array to form the first carbon capture bed.

[0226] In one embodiment, the first carbon capture bed has dimensions of 200,180,200 centimeters. Each fractional 1:1:0.5 TPMS unit cells have dimensions of 2,10,1 centimeters. Thus, by stacking 200 arrays of fractional 100:18:0.5 TPMS unit cells would create the dimensions 200,180,200 centimeters.

[0227] As further shown in Figure 14, each fractional TPMS unit cell in the second carbon capture bed has a Diamond geometry and is a 1 / 2 fraction of a full TPMS unit cell (1: 1:0.5), thus forming a second-bed fractional unit cell 2204. In this embodiment, the second carbon capture bed has dimensions of 200,20,200 centimeters. Eachfractional 1:1:0.5 TPMS unit cells have dimensions of 0.5,0.5,0.25 centimeters. Thus, stacking 800 arrays of fractional 400:40:0.5 TPMS unit cells would create the dimensions 200,20,200 centimeters.

[0228] As can be seen, this embodiment uses different dimensions of fractional TPMS cells in the first carbon capture bed and the second carbon capture bed. Advantageously, the geometry of the first carbon capture bed results in the processing of a large amount of carbon dioxide while minimizing pressure drop. Further, the first carbon capture bed is simpler to manufacture due to the large fractional TPMS unit cell sizes and can thus form a larger portion of the overall carbon capture system. In contrast, the geometry of the second carbon capture bed maximizes the capture of any remaining carbon dioxide.

[0229] In one embodiment to manufacture the TPMS beds, a powder metallurgy process is employed using zeolite. First, the raw zeolite ingredients are dried to the desired dryness levels by heating them to 110°C. The desired dryness level varies with the zeolite selected. Once dried, the materials are mixed using a ball mill to ensure even distribution, and the bulk density of the mixture is measured.

[0230] After mixing, the dry ingredients are transferred into a Z-blade mixer, where water is added incrementally until the desired moisture content is reached, as determined by the specific properties of the zeolite material. This mixture is then aged. This aged mixture is then granulated in a granulator into particles with sizes ranging approximately between 10-40 mesh.

[0231] Using a mold designed to match the fractional TPMS structures similar to that highlighted in Figure 12, a desired amount of granulated material is pressed into the TPMS shape to produce a fractional TPMS structure.

[0232] After pressing, the fractional TPMS structure undergoes drying and calcination.Multiple calcined fractional TPMS structures are then stacked to form the first carbon capture bed. The arrays are glued together using small dots of epoxy or calcium silicate (CaSi) to create a continuous and durable joined fractional TPMSstructure of the desired configuration. A similar process is used to create the second carbon capture bed. The first carbon capture bed and the second carbon capture bed are placed next to each other and are optionally joined.

[0233] A person understanding this invention may now conceive of alternative structures and embodiments or variations of the above all of which are intended to fall within the scope of the invention as defined in the claims that follow.

Claims

We claim:

1. A fractional triply periodic minimal surface (TPMS) structure comprising at least one fractional TPMS unit cell, wherein the at least one fractional TPMS unit cell:a. has a TPMS surface defined by a periodic equation,b. the TPMS surface comprises one full period of the periodic equation in at least one of the three x, y, or z dimensions, andc. the TPMS surface comprises less than one full period of the periodic equation in at least one of the three x, y, or z dimensions,wherein all surfaces of said fractional TPMS structure are accessible from at least one plane parallel to a top face and a bottom face of said at least one fractional TPMS unit cell.

2. The fractional TPMS structure according to claim 1, wherein each of said at least one fractional TPMS unit cell is substantially one of: a Diamond, a Gyroid, a Primitive, a Neovius, an F-RD, an IWP, a CLP, an Octo, a K, a C(D), a C(G), a C(I2-Y), a C(S), a C(Y), a C(+-Y), a Diamond', a Gyroid', a Gyroid", a Lidinoid, a P+C(P), a Qstar, an S, a Slotted P, a Split P, a Y, or a Y' fractional TPMS unit cell.

3. The fractional TPMS structure according to any one of claims 1 or 2, comprising a plurality of said fractional TPMS unit cells arranged into a repeating array along two of the three x, y, or z dimensions.

4. The fractional TPMS structure according to any one of claims 1 to 3, wherein said fractional TPMS structure comprises at least one of: metals, non-metals, composites, or biomaterials.

5. The fractional TPMS structure according to any one of claims 1 to 4, wherein the less than one full period of the periodic equation in at least one of the three x, y, or z dimensions is a half, third, quarter, fifth, sixth, eighth, or sixteenth of the one full period of the period equation.

6. The fractional TPMS structure according to any one of claims 1 to 4, wherein the less than one full period of the periodic equation in at least one of the three x, y, or z dimensions is a 45 degree rotated fraction, being 1 / ^2 or a half, third, quarter, fifth, sixth, eighth, or sixteenth fraction of 1 / ^2.

7. The fractional TPMS structure according to any one of claims 1 to 6, wherein the faces of said fractional TPMS structure are substantially visible from at least one plane parallel to a top face and a bottom face of said at least one fractional TPMS unit cell.

8. The fractional TPMS structure according to any one of claims 1 to 7, wherein the start of less than one full period of the periodic equation in at least one of the three x, y, or z dimensions defines a planar surface, or defines a non-planar surface.

9. The fractional TPMS structure according to any one of claims 1 to 8, wherein the fractional TPMS unit cell is a fractional skeletal TPMS unit cell or a fractional sheet TPMS unit cell.

10. A method of manufacturing a fractional TPMS structure as defined by any one of claims 1 to 9, wherein the fractional TPMS structure is manufactured using one or more of a machining, casting, molding, forming, or template manufacturing process.

11. The method of claim 10, wherein the output of the manufacturing process is a fractional TPMS structure composed of a plurality of fractional TPMS unit cells in two of the x, y, and z directions, but only one fractional TPMS unit cell in the remaining x, y, or z direction.

12. The method of any one of claims 10 or 11, wherein a first and second fractional TPMS structure are manufactured, and wherein the first and second fractional TPMS structure are stacked on one another by translating, rotating, or flipping the second fractional TPMS structure and adjoining it to the first fractional TPMS structure such that the TPMS surfaces maintain continuity.

13. The method of any one of claims 10 to 12, wherein a plurality of fractional TPMS structures are manufactured, and said plurality of fractional TPMS structures are tessellated in one or more of the x, y, and z, dimension.

14. The method of any one of claims 10 to 13, wherein the manufacturing process further comprises an undercutting of the surfaces of the fractional TPMS unit cell, optionally by use of undercutting end mills, lollipop cutters, t-slot cutters, key way cutters, and dovetail cutters.

15. The method of any one of claims 10 to 13, wherein material is added to the any accessible but not visible surface of the fractional TPMS unit cell, so that all surfaces of the fractional TPMS structure are visible from at least one plane parallel to a top face and a bottom face of said at least one fractional TPMS unit cell.

16. The use of a fractional TPMS structure to manufacture according to any one of claims 1 to 10, wherein the fractional TPMS structure is used to produce a TPMS templating structure from a stacked, or tessellated fractional TPMS structure, and wherein the TPMS templating structure is used to shape a structure through casting, coating, plating, deposition, infiltration, electroforming, molding, or similar templating processes.

17. The use of a fractional TPMS structure as defined by any one of claims 1 to 10, wherein the fractional TPMS structure is configured for use as: a load bearing structure or structural support; a storage or holding tank; a heater; a battery; a catalyst or catalyst support; a floor plate, optionally further comprising a thermally insulating material; a concrete formwork, or internal geometry within a formwork; a floating structure; decorative or architectural materials; a heat exchanger or heat exchanges core; a component in a thermal energy storage system; an adsorbent; an adsorbent component in a carbon capture apparatus; an adsorbent component in a portable oxygen concentrator or oxygen / medical air purification device.

8. A triply periodic minimal surface (TPMS) structure comprising at least one TPMS unit cell, wherein the at least one TPMS unit cell is a full 1:1:1 skeletal Primitive TPMS unit cell, and wherein the at least one TPMS unit cell:a. has a TPMS surface defined by a periodic equation,b. the TPMS surface comprises one full period of the periodic equation in the x, y, or z dimensions, andc. one of the one full period of the period equation starts at the predetermined period 402.

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