WO2022212472A1

WO2022212472A1 - Systems and methods for constructing lattice objects for additive manufacturing

Info

Publication number: WO2022212472A1
Application number: PCT/US2022/022503
Authority: WO
Inventors: Ruiqi Chen; Hardik KABARIA
Original assignee: Carbon, Inc.
Priority date: 2021-04-01
Filing date: 2022-03-30
Publication date: 2022-10-06

Abstract

A method for constructing a three-dimensional (3D) object is described. The object is comprised of repeating interconnected unit cells of a selected primary lattice. The method is carried out by: (a) providing a tetrahedral mesh representing the 3D object; (b) selecting a primary lattice unit cell, the primary lattice unit cell including a subsection consisting of a tetrahedral unit cell (preferably a unit cell with T _d symmetry); and then (c) filling said tetrahedral mesh with the tetrahedral unit cell to thereby construct a 3D object comprising repeating interconnected unit cells of the selected primary lattice. In some embodiments the method further includes the step of (d) additively manufacturing the 3D object.

Description

SYSTEMS AND METHODS FOR CONSTRUCTING LATTICE OBJECTS FOR ADDITIVE MANUFACTURING

Field

[0001] This concerns additive manufacturing, and particularly concerns filling three- dimensional shapes with lattice unit cells, such as surface lattice and strut lattice unit cells, for additive manufacturing thereof.

Background

[0002] Three-dimensional objects comprised of surface lattice unit cells suitable for additive manufacturing are beginning to emerge (see, e.g. Boyce, Wang and Lau, US Pat. App. Pub. No. US2012/0196100; Ryan, US Patent No. 9,440,216). Such objects are conventionally generated by mapping cubic unit cells of the surface lattice into hexahedral meshes representing the 3D object. While suitable for simple, uniform, 3D shapes, such methods cannot be easily automated for additive manufacturing of more complex 3D shapes (for example, due the difficulty of generating conformal hexahedral meshes). Tetrahedral mesh representations of 3D objects are, on the other hand, widely used for generating 3D objects, but techniques for filling tetrahedral meshes with surface lattices have not been available.

[0003] Guo, Liu and Yu, Tetrahedron-based porous scaffold design for 3D printing,

Designs 3, 16 (2019) describe a method of generating porous lattices from tetrahedral minimal surfaces, but do not describe isolating tetrahedral subsections from surface lattice unit cells, and do not provide assurance that the cells will properly fit when filled into a mesh.

Summary

[0004] A method for constructing a three-dimensional (3D) object is described. The object is comprised of repeating interconnected unit cells of a selected primary lattice. The method is carried out by: (a) providing a tetrahedral mesh representing the 3D object; (b) selecting a primary lattice unit cell, the primary lattice unit cell including a subsection consisting of a tetrahedral unit cell (preferably a unit cell with Td symmetry); and then (c) filling said tetrahedral mesh with the tetrahedral unit cell to thereby construct a 3D object comprising repeating interconnected unit cells of the selected primary lattice. In some embodiments the method further includes the step of (d) additively manufacturing the 3D object.

[0005] In some embodiments, the primary lattice unit cell comprises a strut lattice unit cell.

[0006] In some embodiments, the primary lattice unit cell comprises a surface lattice unit cell.

[0007] In some embodiments, the primary lattice unit cell is defined by an equation

(i.e., is an implicit lattice unit cell) or by a CAD file.

[0008] In some embodiments, the tetrahedral mesh comprises a conformal tetrahedral mesh.

[0009] In some embodiments, the subsection consisting of a tetrahedral unit cell consists of a I^’ tetrahedrally symmetric unit cell.

[0010] In some embodiments, the primary lattice unit cell is hexahedral ( e.g ., cubic).

[0011] In some embodiments, the selecting step (b) is carried out by selected a primary lattice unit cell from a set of primary lattice unit cells and a corresponding independent tetrahedral unit cells, the corresponding independent tetrahedral unit cell consisting of a tetrahedral subsection of said primary lattice unit cell, with the filling step (c) carried out with the corresponding independent tetrahedral unit cell.

[0012] In some embodiments, the selecting step (b) is carried out by: (i) inputting a primary lattice unit cell, (ii) identifying a presence or absence of a subsection consisting of a tetrahedral unit cell within the primary lattice unit cell, and (in) if present then generating a corresponding independent tetrahedral unit cell from the subsection, with the filling step (c) carried out with said corresponding independent tetrahedral unit cell; and (iv) optionally, adding the primary lattice unit cell to a database of primary lattice unit cells known to contain a subsection consisting of a tetrahedral unit cell.

[0013] In some embodiments, the object comprises a shock absorber, cushion, or pad

(e.g., a wearable shock absorbing cushion such as a helmet liner, garment liner, body pad or body support; an automotive or aerospace body panel or impact absorber; a saddle or seat such as a bicycle saddle, a footwear innersole, midsole, or orthotic, etc.).

[0014] In some embodiments, the method includes the step of: (d) additively manufacturing the 3D object.

[0015] In some embodiments, the additively manufacturing step is carried out by selective laser sintering, selective laser melting, electron beam melting, fused deposition modeling, stereolithography (e.g, continuous liquid interface production), material jetting, or multijet modeling. [0016] In some embodiments, the 3D object is comprised of, consists of, or consists essentially of a polymer (including polymer blends), metal, ceramic, or composite thereof. [0017] In some embodiments, the 3D object is comprised of, consists of, or consists essentially of the reaction products of a dual cure polymer resin.

[0018] Additively manufactured 3D objects produced by methods as described herein are also disclosed.

[0019] The foregoing and other objects and aspects of the present invention are explained in greater detail in the drawings herein and the specification set forth below. The disclosures of all United States patent references cited herein are to be incorporated herein by- reference.

Brief Description of the Drawings

[0020] FIG. 1A is a flow chart illustrating a first embodiment of the systems and methods described herein.

[0021] FIG. IB is a flow chart illustrating a second embodiment of the systems and methods described herein.

[0022] FIG. 2 schematically illustrates a non-limiting embodiment of a system and apparatus for carrying out a method as described herein.

[0023] FIGS. 3A-3C illustrate images depicting an achiral tetrahedral symmetry T_d, including four 3 -fold rotational symmetry axes (FIG. 3A), three 4-fold rotoinversion symmetry axes (which are also 2-fold rotational symmetry axes) (FIG. 3B), and six mirror symmetry planes (FIG. 3C) according to some embodiments.

[0024] FIG. 4A illustrates Chiral tetrahedron with clockwise hole pattern.

[0025] FIG. 4B illustrates Chiral tetrahedron with counterclockwise hole pattern.

[0026] FIG. 4CAchiral tetrahedron according to some embodiments.

[0027] FIGS. 5A-5D illustrate a collection of implicit surfaces that approximate triply periodic minimal surfaces, including Schwarz primitive (Pm3m, no. 221) (FIG. 5A), gyroid (I4_C32, no. 214)(FIG. 5B), diamond (Fd3m, no. 227) (FIG. 5C), CLP (P4₂/mcm, no. 132) (FIG. 5D), and Fisher-Koch S (Ia3d, no. 230) (FIG. 5E) according to some embodiments.

[0028] FIG. 6 illustrates a regular tetrahedron with labeled vertices according to some embodiments.

[0029] FIGS. 7A-7B illustrate unit cells inlcuding an FRD cubic unit cell with regular tetrahedron overlaid in red (FIG. 7A), and an extracted tetrahedron unit cell (FIG.

7B) according to some embodiments.

[0030] FIGS. 8A-F illustrate a process to generate piecewise implicit surface lattice structure, showing surface mesh of the design space (FIG. 8A), tetrahedron mesh of the design space (FIG. 8B), tetrahedron unit cell based on skeletal diamond cubic cell (space group 227) (FIG. 8C), tetrahedron unit cell based on FRD cubic cell (space group 225) (FIG. 8D), mesh populated with skeletal-diamond-derived unit cells (FIG. 8E), and mesh populated with FRD-derived unit cells (FIG. 8F) according to some embodiments.

[0031] FIGS. 9A-9D illustrate a process to generate piecewise surface lattice structure, showing a CAD-drawn, hexahedral (cubic) unit cell (FIG. 9A); a tetrahedral unit cell with Td symmetry isolated from the hexahedral unit cell, as bounded within the tetrahedral shaded border (FIG. 9B); surface mesh of the design space (FIG. 9C), and the mesh populated with the unit cell of FIG. 9B, along with a portion thereof magnified (FIG. 9D) according to some embodiments.

Detailed Description of Illustrative Embodiments

[0032] The present invention is now described more fully hereinafter with reference to the accompanying drawings, in which embodiments of the invention are shown. This invention may, however, be embodied in many different forms and should not be construed as limited to the embodiments set forth herein; rather these embodiments are provided so that this disclosure will be thorough and complete and will fully convey the scope of the invention to those skilled in the art.

[0033] The disclosures of all United States patent references cited herein are incorporated herein by reference in their entirety.

[0034] As used herein, the term "and/or" includes any and all possible combinations of one or more of the associated listed items, as well as the lack of combinations when interpreted in the alternative ("or").

[0035] “Lattice unit cell” as used herein is intended to include both strut lattice unit cells, and surface lattice unit cells. Lattice unit cells may be implicit lattice unit cells (that is, defined by an equation) or can be generated by other means, such as manually by a computer- assisted design (CAD) program. Lattice unit cells as described herein consist of a single interconnected object (to insure manufacturability thereof) and not cells that consist of separate, unconnected, parts.

[0036] “Surface lattice unit cell” as used herein, includes, but is not limited to, triply periodic surface lattice unit cells, ( e.g ., F-RD or gyroid unit cells), all of which which can be, but need not be, minimal surface lattice unit cells.

[0037] Methods. As noted above, a method for constructing a three-dimensional (3D) object is described. The object is comprised of repeating interconnected unit cells of a selected primary lattice, and may optionally include additional features, such as one or more additional, different, lattices, skins or other solid portions, or the like. The method is schematically illustrated in FIGS. 1A-1B, and generally includes the steps of: (a) providing a tetrahedral mesh representing the 3D object (11, 21); (b) selecting a primary lattice unit cell (12), the primary lattice unit cell including a subsection consisting of a tetrahedral unit cell; and then (c) filling said tetrahedral mesh with the tetrahedral unit cell (14, 24) to thereby construct a 3D object comprising repeating interconnected unit cells of the selected primary lattice.

[0038] In some embodiments, the primary lattice unit cell comprises a strut lattice unit cell. In other embodiments, the primary lattice unit cell comprises a surface lattice unit cell.

[0039] In some embodiments, the said primary lattice unit cell is defined by an equation ( i.e ., is an implicit lattice unit cell); in other embodiments, the primary lattice unit cell is defined by a CAD file.

[0040] In some embodiments, the tetrahedral mesh comprises a conformal tetrahedral mesh.

[0041] In some embodiments, the primary lattice unit cell is hexahedral (e.g., cubic).

[0042] In preferred embodiments, the subsection consisting of a tetrahedral unit cell consists of a I^’ tetrahedrally symmetric unit cell. This serves to ensure that the subsection will fit well into the tetrahedral mesh when the object is populated with the lattice, as discussed further below.

[0043] In the embodiment of the method set forth in FIG. 1A, the selecting step (b) is carried out by selecting a primary lattice unit cell from a set of primary lattice unit cells and a corresponding independent tetrahedral unit cells (13). The corresponding independent tetrahedral unit cell consists of a tetrahedral subsection of said primary lattice unit cell, with the filling step (c) carried out with the corresponding independent tetrahedral unit cell (preferably a Td symmetric unit cell, as noted above and below).

[0044] In some embodiments, a 3d object filed with the selected lattice unit cell is additively manufactured (15)

[0045] In the alternate embodiment of the method set forth in FIG. IB, the selecting step (b) is carried out by: (i) inputting a primary lattice unit cell (22), (ii) identifying the presence or absence of a subsection consisting of a tetrahedral unit cell within the primary lattice unit cell cell (again preferably a Td symmetric unit cell, as noted above and below)

(23), and (in) if present then generating a corresponding independent tetrahedral unit cell from the subsection (23’), with said filling step (c) carried out with said corresponding independent tetrahedral unit cell (24). Since new and useful primary lattice unit cells can be found this way, the method may optionally include the step of (iv) adding the primary lattice unit cell to a database of primary lattice unit cells known to contain a subsection consisting of a tetrahedral unit cell (26). In some embodiments, a 3D objected filled with the selected lattice unit cell is additively manufacture (25).

[0046] Additive manufacturing. Techniques for additive manufacturing are known.

Suitable techniques include, but are not limited to, such as selective laser sintering (SLS)

(e.g, of metal powders, ceramic powders, etc.), selective laser melting (SLM) (e.g, of polymer powders), electron beam melting (EBM) (e.g, of metal powders), fused deposition modeling (FDM), stereolithography (SLA), material jetting including three-dimensional printing (3DP) and multijet modeling (MJM) (MJM including Multi- Jet Fusion such as available from Hewlett Packard), and others. See, e.g, H. Bikas et ak, Additive manufacturing methods and modelling approaches: a critical review, Int. J. Adv. Manuf. Technol. 83, 389-405 (2016).

[0047] Methods and apparatus for stereolithography are known and described in, for example, U.S. Patent No. 5,236,637 to Hull, US Patent Nos. 5,391,072 and 5,529,473 to Lawton, U.S. Patent No. 7,438,846 to John, US Patent No. 7,892,474 to Shkolnik, U.S.

Patent No. 8,110,135 to El-Siblani, U.S. Patent Application Publication No. 2013/0292862 to Joyce, and US Patent Application Publication No. 2013/0295212 to Chen et al. The disclosures of these patents and applications are incorporated by reference herein in their entirety.

[0048] In some embodiments, the additive manufacturing step is carried out by one of the family of methods sometimes referred to as as continuous liquid interface production (CLIP). CLIP is known and described in, for example, US Patent Nos. 9,211,678; 9,205,601; 9,216,546; and others; in J. Tumbleston et ak, Continuous liquid interface production of 3D Objects, Science 347, 1349-1352 (2015); and in R. Janusziewcz et ak, Layerless fabrication with continuous liquid interface production, Proc. Natl. Acad. Sci. USA 113, 11703-11708 (October 18, 2016). Other examples of methods and apparatus for carrying out particular embodiments of CLIP include, but are not limited to: Batchelder et ak, US Patent Application Pub. No. US 2017/0129169 (May 11, 2017); Sun and Lichkus, US Patent Application Pub. No. US 2016/0288376 (Oct. 6, 2016); Willis et al., US Patent Application Pub. No. US 2015/0360419 (Dec. 17, 2015); Lin et al., US Patent Application Pub. No. US 2015/0331402 (Nov. 19, 2015); D. Castanon, S Patent Application Pub. No. US 2017/0129167 (May 11,

2017). B. Feller, US Pat App. Pub. No. US 2018/0243976 (published Aug 30, 2018); M. Panzer and J. Tumbleston, US Pat App Pub. No. US 2018/0126630 (published May 10,

2018); K. Willis and B. Adzima, US Pat App Pub. No. US 2018/0290374 (Oct. 11, 2018) L. Robeson et al., PCT Patent Pub. No. WO 2015/164234 (see also US Patent Nos. 10,259,171 and 10,434,706); and C. Mirkin et al., PCT Patent Pub. No. WO 2017/210298 (see also US Pat. App. US 2019/0160733).

[0049] While any suitable resin can be used, in some embodiments, dual cure resins are preferred. Such resins are known and described in, for example, US Patent Nos. 9,676,963, 9,453,142 and 9,598,606 to Rolland et al. Particular examples of suitable dual cure resins include, but are not limited to, Carbon Inc. medical polyurethane, elastomeric polyurethane, rigid polyurethane, flexible polyurethane, cyanate ester, epoxy, and silicone dual cure resins, all available from Carbon, Inc., 1089 Mills Way, Redwood City, California 94063 USA.

[0050] Systems and apparatus. A non-limiting example of an apparatus for carrying out a non-limiting embodiment of the methods described herein is schematically illustrated in FIG. 2. Such an apparatus includes a user interface 3 for inputting instructions (such as selection of an object to be produced, and selection of features to be added to the object), a controller 4, and a stereolithography apparatus 5 such as described above. An optional washer (not shown) can be included in the system if desired, or a separate washer can be utilized. Similarly, for dual cure resins, an oven (not shown) can be included in the system, although operated separate oven can also be utilized.

[0051] Connections between components of the system can be by any suitable configuration, including wired and/or wireless connections. The components may also communicate over one or more networks, including any conventional, public and/or private, real and/or virtual, wired and/or wireless network, including the Internet.

[0052] The controller 4 may be of any suitable type, such as a general-purpose computer. Typically, the controller will include at least one processor 4a, a volatile (or “working”) memory 4b, such as random-access memory, and at least one non-volatile or persistent memory 4c, such as a hard drive or a flash drive. The controller 4 may use hardware, software implemented with hardware, firmware, tangible computer-readable storage media having instructions stored thereon, and/or a combination thereof, and may be implemented in one or more computer systems or other processing systems. The controller 4 may also utilize a virtual instance of a computer. As such, the devices and methods described herein may be embodied in any combination of hardware and software that may all generally be referred to herein as a "circuit," "module," "component," and/or "system." Furthermore, aspects of the present invention may take the form of a computer program product embodied in one or more computer readable media having computer readable program code embodied thereon.

[0053] Any combination of one or more computer readable media may be utilized.

Example embodiments of the present inventive concepts may be embodied in various devices, apparatuses, and/or methods. For example, example embodiments of the present inventive concepts may be embodied in hardware and/or in software (including firmware, resident software, micro-code, etc.). Furthermore, example embodiments of the present inventive concepts may take the form of a computer program product comprising a non- transitory computer-usable or computer-readable storage medium having computer-usable or computer-readable program code embodied in the medium for use by or in connection with an instruction execution system. In the context of this document, a computer-usable or computer-readable medium may be any non-transient medium for use by or in connection with the instruction execution system, apparatus, or device.

[0054] The computer-usable or computer-readable medium may be, for example but not limited to, any non-transient computer readable medium, includingan electronic, magnetic, or semiconductor system, apparatus, or device. More specific examples (a nonexhaustive list) of the computer-readable medium would include the following: a portable computer diskette, a random access memory (RAM), a read-only memory (ROM), an erasable programmable read-only memory (EPROM or Flash memory), and a portable compact disc read-only memory (CD-ROM).

[0055] Example embodiments of the present inventive concepts are described herein with reference to flowchart and/or block diagram illustrations. It will be understood that each block of the flowchart and/or block diagram illustrations, and combinations of blocks in the flowchart and/or block diagram illustrations, may be implemented by computer program instructions and/or hardware operations. These computer program instructions may be provided to a processor of a general purpose computer, a special purpose computer, or other programmable data processing apparatus to produce a machine, such that the instructions, which execute via the processor of the computer or other programmable data processing apparatus, create means and/or circuits for implementing the functions specified in the flowchart and/or block diagram block or blocks.

[0056] These computer program instructions may also be stored in a computer usable or computer-readable memory that may direct a computer or other programmable data processing apparatus to function in a particular manner, such that the instructions stored in the computer usable or computer-readable memory produce an article of manufacture including instructions that implement the functions specified in the flowchart and/or block diagram block or blocks.

[0057] The computer program instructions may also be loaded onto a computer or other programmable data processing apparatus to cause a series of operational steps to be performed on the computer or other programmable apparatus to produce a computer implemented process such that the instructions that execute on the computer or other programmable apparatus provide steps for implementing the functions specified in the flowchart and/or block diagram block or blocks.

[0058] The at least one processor 4a of the controller 4 may be configured to execute computer program code for carrying out operations for aspects of the present invention, which computer program code may be written in any combination of one or more programming languages, including an object oriented programming language such as Java, Scala, Smalltalk, Eiffel, JADE, Emerald, C++, C#, VB.NET, or the like, conventional procedural programming languages, such as the "C" programming language, Visual Basic, Fortran 2003, COBOL 2002, PHP, ABAP, dynamic programming languages such as Python, PERL, Ruby, and Groovy, or other programming languages.

[0059] The at least one processor 4a may be, or may include, one or more programmable general purpose or special-purpose microprocessors, digital signal processors (DSPs), programmable controllers, application specific integrated circuits (ASICs), programmable logic devices (PLDs), field-programmable gate arrays (FPGAs), trusted platform modules (TPMs), or a combination of such or similar devices, which may be collocated or distributed across one or more data networks.

[0060] Connections between internal components of the controller 4 are shown only in part and connections between internal components of the controller 4 and external components are not shown for clarity, but are provided by additional components known in the art, such as busses, input/output boards, communication adapters, network adapters, etc. The connections between the internal components of the controller 4, therefore, may include, for example, a system bus, a Peripheral Component Interconnect (PCI) bus or PCI-Express bus, a HyperTransport or industry standard architecture (ISA) bus, a small computer system interface (SCSI) bus, a universal serial bus (USB), IIC (I2C) bus, an Advanced Technology Attachment (ATA) bus, a Serial ATA (SATA) bus, and/or an Institute of Electrical and Electronics Engineers (IEEE) standard 1394 bus, also called "Firewire."

[0061] The user interface 3 may be of any suitable type. The user interface 3 may include a display and/or one or more user input devices. The display may be accessible to the at least one processor 4a via the connections between the system components. The display may provide graphical user interfaces for receiving input, displaying intermediate operation/data, and/or exporting output of the methods described herein. The display may include, but is not limited to, a monitor, a touch screen device, etc., including combinations thereof. The input device may include, but is not limited to, a mouse, keyboard, camera, etc., including combinations thereof. The input device may be accessible to the at least one processor 4a via the connections between the system components. The user interface 3 may interface with and/or be operated by computer readable software code instructions resident in the volatile memory 4b that are executed by the processor 4a.

[0062] Products. Examples of objects that can be produced by the methods described herein include, but are not limited to, shock absorbers, cushions, or pads ( e.g a wearable shock absorbing cushion such as a helmet liner, garment liner, body pad or body support; an automotive or aerospace body panel or impact absorber; a saddle or seat such as a bicycle saddle; a footwear innersole, midsole, or orthotic, etc.). Such objects can be comprised of, consist of, or consist essentially of a polymer (including polymer blends), metal, ceramic, or composites thereof, depending on the particular additive manufacturing technique employed. In some embodiments (typically where the object is produced by bottom-up or top-down stereolithography) the object can be comprised of, consist of, or consist essentially of the reaction products of a dual cure polymer resin (including but not limited to those noted above).

[0063] EXAMPLES AND DETAILED DESCRIPTION

[0064] Tetrahedron-based unit cells are generated by first constructing an implicit surface cubic unit cell using a 3D Fourier series. The terms of the Fourier series are carefully chosen such that the cubic unit cell possesses achiral tetrahedral symmetry. The tetrahedron- based unit cell is then isolated by taking the intersection of a properly oriented regular tetrahedron with the implicit surface. To generate the lattice structure, a tetrahedron mesh of the input domain is created using an appropriate algorithm (described later). Finally, the tetrahedral unit cell is mapped to each tetrahedron in the mesh through an affine transformation, creating a piecewise implicit surface lattice structure. A brief overview of tetrahedral symmetry, space groups, and implicit surfaces is provided before proceeding to the lattice structure generation procedure.

[0065] 1.1 Symmetry considerations

[0066] A regular tetrahedron with no markings or coloring possesses achiral (or full) tetrahedral symmetry, denoted as T_d. The symmetry group T_d is comprised of the identity, four 3-fold rotational symmetry axes, three 2-fold rotational symmetry axes (which are also 4-fold rotoinversion symmetry axes), and six mirror planes, yielding a symmetry order of 24. The rotational axes, rotoinversion axes, and mirror planes of a regular tetrahedron are depicted in FIG. 3.

[0067] On the other hand, a tetrahedron with four different environments at its corners possesses chiral tetrahedral symmetry, denoted as T. The symmetry group T is a subgroup of T_d and is comprised of only the rotational symmetries of T_d, yielding a symmetry order of 12 (i.e. no reflections or rotoinversions). Two chiral tetrahedrons and an achiral tetrahedron are depicted in FIG. 4. If two tetrahedrons of the type shown in FIG. 4A or FIG. 4B are arranged such that their faces join together, the holes on the faces would be partially covered up due to the chirality. However, if two achiral tetrahedrons of the type shown in FIG. 4C are arranged in the same way, the holes would not be covered up. Because this work is interested in joining together tetrahedron-based unit cells to create a lattice structure, the unit cells should have of T_d symmetry to allow faces to join without obscuring features.

[0068] To find tetrahedron-based unit cells with T_d symmetry, this works looks at a related family of periodic space filling structures based on a cubic unit cell. All periodic structures (cubic or otherwise) can be categorized into space groups. A space group consists of the set of transformations, or symmetry operations, that, when applied to an object that belongs to that space group, produces the same object. Unlike the point groups T and T_d (which keep a center point fixed in space), space groups also include additional symmetry operations such as translations, glide planes, and screw axes. For periodic structures, there are 230 distinct crystallographic space groups. Each space group is assigned a number and the set of transformations for each space group are tabulated in the International Tables for Crystallography (ITC) [Hahn, 1983] Out of the 230 space groups, there are a handful that possess T_d symmetry: 215 (P43m), 216 (F43m), 217 (/43m), 221 (Pm3m), 224 ( Pn3m ), 225 (Fm3m), 227 (Fd3m), and 229 (/m3m). These space groups are all part of the cubic crystal system. By using a cubic unit cell from one of these space groups as the starting point, a tetrahedron-based unit cell with T_d symmetry can be derived from the cubic unit cell. While theoretically any cubic unit cell from one of these space groups can be used, this work is interested in cubic unit cells generated by implicit equations in the form of a 3D Fourier series, as these have been shown to produce lightweight sheet-like and skeletal-like lattice structures.

[0069] 1.2 Periodic implicit surfaces constructed using 3D Fourier series.

[0070] An implicit surface is a surface in 3D defined by the equation /(x, y, z) = 0.

All points that satisfy the equation are located on the surface. Some examples of implicit surfaces include /(x, y, z) = x² + y² + z² — r² (representing an origin-centered sphere with radius r) and /(x,y,z) = cos(x) + cos(y) + cos (z) (an approximation to the Schwarz primitive minimal surface). Implicit surfaces partition space into interior and exterior regions depending on the sign of the function / (x, y, z) at a given point in space. This work uses the convention that interior points take on negative values. Implicit surfaces can also be defined in non-Euclidean space such as hyperbolic space, but this work only considers them in Euclidean space.

[0071] Periodic implicit surfaces can be constructed as a sum of trigonometric functions, representing a 3D Fourier series. The general form is given by

(1) where

and h , k, l are positive integers, L_x, L_y, and L_z are lengths of the unit cell along the Cartesian x, y, and z directions, and c_hki are complex- valued weighting constants. Note that while Eq. (1) is in general complex-valued, either the real or imaginary part of Eq. (1) can be used to define the implicit surface. In addition, for cubic unit cells, L_x = L_y = L_z = L, so without loss of generality, L can be set to unity.

[0072] To generate cubic unit cells of a particular space group, every symmetry operation from the chosen space group is applied to f_hk and the resulting transformed f_hki equations are summed up [Wohlgemuth, 2001] For example, consider the cubic space group 215 (P43m) with 24 symmetry operations (x,y,z), (x,y,z), (x,y,z), ..., (y,x,z). Applying all 24 operations to f_hki and summing the terms yields

[0073] Eq. (3) can be used as the starting point to generate a cubic unit cell within space group 215 by choosing non-negative integer values of h, k , and l. In addition, linear combinations of Eq. (3) with complex- valued weights c_hki will also produce a cubic unit cell within space group 215. Thus, the implicit surface equation for a cubic unit cell within space group 215 is given by

where either the real or imaginary part of Eq. (4) can be used to define the implicit surface. Analogous equations can be developed for any other space group, including non-cubic space groups, with appropriate choices of L_x , L_y , and L_z (for certain lower-numbered space groups, skew coordinates may be necessary).

[0074] The implicit surface may also be modified by a constant offset parameter t, yielding a skeletal lattice [Al-Ketan, 2018]: f skeletal ^~ foriginal t (5) where f _original refers to the implicit surface equation before offsetting, and t is the offset amount. In general, t may be positive or negative. To create a sheet-like implicit surface, an implicit equation can be offset in both directions by some constant t:

[0075] Offsetting using these two approaches does not change the space group that the original implicit surface belongs to. The offset parameter t is not exactly a thickness parameter, but changing it does cause various measures of thickness (e.g. minimum wall thickness, maximum wall thickness, etc.) to change. However, even though t is constant, the change in thickness is likely not spatially uniform.

[0076] Implicit surfaces generated using this method can be used to model many well-known geometries, such as approximations to triply periodic implicit surfaces (TPMS). Several TPMS approximations are depicted in FIG. 5 along with their corresponding space groups. The research conducted by Wohlgemuth and associates [Wohlgemuth, 2001] used this method as a way to discover several new TPMS approximations, but the present work is not concerned with whether a particular implicit surface is also a minimal surface approximation. Rather, any implicit surface from the aforementioned space groups can be used to generate a tetrahedron-based unit cell, as long as the resulting tetrahedron unit cell is composed of a single connected component (for manufacturing reasons).

[0077] 1.3 Generating the piecewise lattice structure

[0078] 1.3.1 Tetrahedron-based unit cell

[0079] The tetrahedron-based unit cell with T_d symmetry is created by taking the intersection of the cubic unit cell (from one of the aforementioned space groups) with an appropriately oriented regular tetrahedron. The intersection operation can be done explicitly or implicitly. For the explicit intersection, the implicit surface of the cubic unit cell first needs to be triangulated using some appropriate method (e.g. marching cubes [Lorensen, 1987] or marching triangles [Hartmann, 1998]), then a boolean intersection operation can be performed on the explicit triangle meshes using an appropriate mesh processing package such as CGAL [Loriot, 2020] For the implicit intersection, the regular tetrahedron must first be expressed as an implicit surface. This can be accomplished by considering the regular tetrahedron with vertices p₄, p₂, p₃, and p₄ (in the order depicted in FIG. 6) as the intersection of four halfspaces, each of which is tangent to a face of the tetrahedron:

where x = ( x,y , z) and the Intersection function can be accomplished using an R-function [Pasko, 1995] such as Maximum:

Intersection (f_l f₂, f₃, /₄) = (L, f₂, h, U) (8) or a smooth approximation such as LogSumExp:

where k is a smoothing factor. The tetrahedron unit cell can then be isolated by performing another Intersection operation, this time between f tetrahedron ^and the implicit surface of the cubic unit cell. Finally, the implicit surface of the tetrahedron unit cell is triangulated using marching cubes or a related scheme.

[0080] The location and orientation of the regular tetrahedron depends on the particular space group of the cubic unit cell. For all space groups of interest, candidate tetrahedron barycenter coordinates c are tabulated in Table 1 assuming Cartesian axis- aligned cubic unit cells with unity edge length. Without loss of generality, only barycenters with coordinates all in the range [0,1) are listed. In addition, several barycenters may yield the same tetrahedron unit cell due to the symmetry of the cubic cell; in this case, only one barycenter is listed. For each barycenter point, there are two possible tetrahedrons that may be used to generate the tetrahedron unit cell because the regular tetrahedron is self dual. The first one has vertices defined by:

where m is some positive value. If the desired tetrahedron edge length is given by a, then setting m = — will yield a tetrahedron with the edge length a. The dual tetrahedron has vertices defined by:

Table 1: Location of tetrahedron barycenters for various cubic space groups.

[0081] The tetrahedral unit cell should be composed of a single connected component for fabrication reasons. The number of connected components can be found by considering the triangulated surface mesh of the tetrahedral unit cell as a graph and searching for connected nodes [Hopcraft, 1973]

[0082] An example of a tetrahedron unit cell generated from the FRD implicit surface

(space group Fm3m , no. 225) is shown in FIG. 7. The regular tetrahedron has a barycenter of (0.25,0.25,0.25) with m = 0.25. The resulting tetrahedron unit cell bears strong resemblance to the “TIS P Surface” in the work by Guo and associates [Guo, 2019] In fact, closer inspection of their work reveals that the equation used to create the “TIS P Surface” actually also creates a cubic unit cell that strongly resembles the FRD implicit surface.

[0083] 1.3.2 Testing candidate tetrahedron unit cells for symmetry

[0084] By generating tetrahedron-based unit cells using the methodology outlined in

Section 1.3.1, the unit cells are guaranteed to possess T_d symmetry as long as the generating implicit surface equations are from any of the space groups identified in Table 1. However, tetrahedron-based unit cells may also be created by other means, such as using solid modeling techniques (e.g. B-reps, constructive solid geometry, NURBS) built into computer aided design (CAD) packages. To check whether a candidate unit cell possesses T_d symmetry, the unit cell must be tested against the 24 symmetry operations within the T_d symmetry group. Each symmetry operation is applied to the unit cell one at a time to see if the unit cell has changed shape before and after the operation is applied. If no shape change occurs for all 24 symmetry operations, then the candidate unit cell possesses T_d symmetry. Practically, to detect shape change, numerical metrics such as Hausdorff distance or Jaccard similarity may be used. For example, if the Hausdorff distance of the candidate unit cell before and after a symmetry operation is applied is small (i.e. below some threshold value), then the unit cell shape has not changed.

[0085] 1.3.3 Tetrahedron mesh of input domain

[0086] Once a tetrahedron unit cell has been created, a tetrahedron mesh of the input domain is provided or created. This can be done in a routine manner as there are numerous meshing algorithms available, including Delaunay -based methods [Shewchuk, 1998; Si, 2010; Si, 2015], methods suitable for polygon soups [Hu, 2018; Hu, 2020], and methods utilizing a static universal background mesh [Kabaria, 2017]

[0087] 1.3.4 Mapping tetrahedron unit cell to arbitrary tetrahedron in mesh

[0088] The tetrahedron unit cell is mapped to every tetrahedron in the tetrahedron mesh of the input domain through any suitable technique, such as an affine transformation:

where x_a is a point in the tetrahedron unit cell, x_b is the corresponding mapped point in the mesh tetrahedron, p_4a is the vertex p₄ in the unit tetrahedron, p_4fc is the corresponding vertex in the mesh tetrahedron, and the matrix A is given by

PlX — P4x Plx — P4x Plx — P4x (13)

A = Ply ^~ P4y Ply — P4y Ply ^~ P4y . Plz — P4z Plz — P4z Plz — P4z.

For A_a, the vertices in Eq. (12) refer to those in the tetrahedron unit cell while for A_b, the vertices in Eq. (12) refer to those in the mesh tetrahedron. Every point in the tetrahedron unit cell is mapped using Eq. (12) to a particular tetrahedron in the mesh. The process is repeated for every tetrahedron in the mesh until the entire tetrahedron mesh of the design space is populated with tetrahedron unit cells.

[0089] The generation of implicit surface lattice structures by systems and methods described above is demonstrated in FIGS. 8A-8E. The design space of the structure, as a surface mesh, is shown in FIG. 8 A. A tetrahedral mesh of that design space is generated in accordance with known techniques, as shown in FIG. 8B. A tetrahedral unit cell substructure of the skeletal diamond cubic cell (space group 227) is shown in FIG. 8C, and a tetrahedron unit cell substructure of an FRD cubic cell (space group 225) is shown in FIG. 8D.

[0090] FIG. 8E shows a tetrahedral mesh of FIG. 8B filled with the tetrahedral unit cell of FIG. 7C, yielding a skeletal diamond cubic lattice object conforming to the design space of FIG. 7A.

[0091] FIG. 8F shows a tetrahedral mesh of FIG. 8B filled with the tetrahedral unit cell of FIG. 7D, yielding an FRD lattice object conforming to the design space of FIG. 8A. [0092] While the examples above have been primarily directed to implicit lattice structures, the process described herein is readily adaptable to unit cells that are not defined by an equation, such as a CAD-drawn unit cell. This is illustrated in FIG. 9, where FIG. 9A shows what is clearly a cubic, or hexahedral, and CAD-drawn, unit cell. In this case, a tetrahedral unit cell with Td symmetry is readily isolated from the hexahedral unit cell as shown in FIG. 9B, where the tetrahedral subunit is bounded by the shaded tetrahedron. FIG. 9C shows a surface mesh design space (or “primitive”) for filling, and FIG. 9D shows that mesh filled with the subunit lattice cell.

[0093] The foregoing is illustrative of the present inventive concept and is not to be construed as limiting thereof. Although a few example embodiments have been described, those skilled in the art will readily appreciate that many modifications are possible in the exemplary embodiments without materially departing from the novel teachings of this inventive concept. Accordingly, all such modifications are intended to be included within the scope of this inventive concept as defined in the claims. Therefore, it is to be understood that the foregoing is illustrative of the present inventive concept and is not to be construed as limited to the specific embodiments disclosed, and that modifications to the disclosed embodiments, as well as other embodiments, are intended to be included within the scope of the appended claims.

REFERENCES:

Guo, Y., Liu, K., & Yu, Z. (2019). Tetrahedron-Based Porous Scaffold Design for 3D Printing. Designs, 3(1), 16.

Hahn, T., Shmueli, U., & Arthur, J. W. (Eds.). (1983). International tables for crystallography (Vol. 1, pp. pp-182). Dordrecht: Reidel.

Hartmann, E. (1998). A marching method for the triangulation of surfaces. The Visual Computer, 14(3), 95-108.

Hopcroft, J., & Taijan, R. (1973). Algorithm 447: efficient algorithms for graph manipulation. Communications of the ACM, 16(6), 372-378.

Hu, Y., Zhou, Q., Gao, X., Jacobson, A., Zorin, D., & Panozzo, D. (2018). Tetrahedral meshing in the wild. ACM Trans. Graph., 37(4), 60-1.

Hu, Y., Schneider, T., Wang, B., Zorin, D., & Panozzo, D. (2020). Fast tetrahedral meshing in the wild. ACM Transactions on Graphics (TOG), 39(4), 117-1. Kabaria, H., & Lew, A. J. (2017). Universal meshes for smooth surfaces with no boundary in three dimensions. International Journal for Numerical Methods in Engineering, 110(2), 133- 162.

Lorensen, W. E., & Cline, H. E. (1987). Marching cubes: A high resolution 3D surface construction algorithm. ACM siggraph computer graphics, 21(4), 163-169.

Loriot, S., Rouxel-Labbe, M., Toumois, J., & Yaz, I. O. (2020). Polygon Mesh Processing. In CGAL User and Reference Manual. CGAL Editorial Board, 5.1.1 edition, 2020.

Pasko, A., Adzhiev, V., Sourin, A., & Savchenko, V. (1995). Function representation in geometric modeling: concepts, implementation and applications. The visual computer, 11(8), 429-446.

Shewchuk, J. R. (1998, June). Tetrahedral mesh generation by Delaunay refinement. In Proceedings of the fourteenth annual symposium on Computational geometry (pp. 86-95).

Si, H. (2010). Constrained Delaunay tetrahedral mesh generation and refinement. Finite elements in Analysis and Design, 46(1-2), 33-46.

Si, H. (2015). TetGen, a Delaunay-based quality tetrahedral mesh generator. ACM Transactions on Mathematical Software (TOMS), 41(2), 1-36.

Wohlgemuth, M., Yufa, N., Hoffman, J., & Thomas, E. L. (2001). Triply periodic bicontinuous cubic microdomain morphologies by symmetries. Macromolecules, 34(17), 6083-6089.

The foregoing is illustrative of the present invention, and is not to be construed as limiting thereof. The invention is defined by the following claims, with equivalents of the claims to be included therein.

Claims

We claim:

1. A computer-implemented method for constructing a three-dimensional (3D) object, the object comprising repeating interconnected unit cells of a selected primary lattice, the method comprising:

(a) providing a tetrahedral mesh representing the 3D object;

(b) selecting a primary lattice unit cell, the primary lattice unit cell including a subsection consisting of a tetrahedral unit cell; and then

(c) filling said tetrahedral mesh with the tetrahedral unit cell to thereby construct a 3D object comprising repeating interconnected unit cells of the selected primary lattice.

2. The method of claim 1, wherein said primary lattice unit cell comprises a strut lattice unit cell.

3. The method of claim 1, wherein said primary lattice unit cell comprises a surface lattice unit cell.

4. The method of any preceding claim, wherein said primary lattice unit cell is defined by an equation ( i.e is an implicit lattice unit cell) or by a CAD file.

5. The method of any preceding claim, wherein said tetrahedral mesh comprises a conformal tetrahedral mesh.

6. The method of any preceding claim, wherein said subsection consisting of a tetrahedral unit cell consists of a / /tetrahedrally symmetric unit cell.

7. The method of any preceding claim, wherein said primary lattice unit cell is hexahedral ( e.g cubic).

8. The method of any preceding claim, wherein said selecting step (b) is carried out by selecting a primary lattice unit cell from a set of primary lattice unit cells and a corresponding independent tetrahedral unit cells, the corresponding independent tetrahedral unit cell consisting of a tetrahedral subsection of said primary lattice unit cell, with said filling step (c) carried out with said corresponding independent tetrahedral unit cell.

9. The method of claim 1 to 7, wherein said selecting step (b) is carried out by:

(i) inputting a primary lattice unit cell,

(ii) identifying the presence or absence of a subsection consisting of a tetrahedral unit cell within the primary lattice unit cell, and

(in) if present then generating a corresponding independent tetrahedral unit cell from the subsection, with said filling step (c) carried out with said corresponding independent tetrahedral unit cell; and

(iv) optionally, adding said primary lattice unit cell to a database of primary lattice unit cells known to contain a subsection consisting of a tetrahedral unit cell.

10. The method of any preceding claim, wherein said object comprises a shock absorber, cushion, or pad (e.g., a wearable shock absorbing cushion such as a helmet liner, garment liner, body pad or body support; an automotive or aerospace body panel or impact absorber; a saddle or seat such as a bicycle saddle, a footwear innersole, midsole, or orthotic, etc.).

11. The method of any preceding claim, further comprising the step of:

(d) additively manufacturing the 3D object.

12. The method of any preceding claim, wherein said additively manufacturing step is carried out by selective laser sintering, selective laser melting, electron beam melting, fused deposition modeling, stereolithography ( e.g ., continuous liquid interface production), material jetting, or multijet modeling.

13. The method of any preceding claim, wherein said 3D object is comprised of, consists of, or consists essentially of a polymer (including polymer blends), metal, ceramic, or composite thereof.

14. The method of any preceding claim, wherein said 3D object is comprised of, consists of, or consists essentially of the reaction products of a dual cure polymer resin.

15. An additively manufactured 3D object produced by a method of any preceding claim.