Abstract:
A unit cell for a lattice structure includes eight unit trusses disposed at vertices of the unit cell. A single unit truss is disposed at a centroid of the unit cell. Each of the nine unit trusses includes fourteen struts. Lattice structures are commonly used to connect various loads within a volume of space. Most such structures, however, have a rigid definition for their topology, and are unable to conform to shape or load directions. Additionally, conventional lattice structures are homogeneous, having dimensions and properties that are consistent throughout. These constraints, generally imposed for ease of manufacturing and assembly, prevent the development of highly robust and efficient structures, and limit the potential for multi-functional applications.

Description:
CROSS-REFERENCE TO RELATED APPLICATIONS 
       [0001]    This application is a non-provisional of and claims priority to U.S. Provisional Application No. 61/851,751, filed on Mar. 13, 2013 and U.S. Provisional Application No. 61/851,776, filed on Mar. 13, 2013. The entire contents of both applications are incorporated herein by reference. 
     
    
     BACKGROUND OF THE INVENTION 
       [0002]    Lattice structures are commonly used to connect various loads within a volume of space. Most such structures, however, have a rigid definition for their topology, and are unable to conform to shape or load directions. Additionally, conventional lattice structures are homogeneous, having dimensions and properties that are consistent throughout. These constraints, generally imposed for ease of manufacturing and assembly, prevent the development of highly robust and efficient structures, and limit the potential for multi-functional applications. 
       SUMMARY OF THE INVENTION 
       [0003]    The present invention relates to lattice structures, and in particular to unit trusses for building lattice structures. 
         [0004]    In accordance with one construction of the invention, a unit cell for a lattice structure includes eight unit trusses disposed at vertices of the unit cell, and a single unit truss positioned within the unit cell, wherein each of the nine unit trusses includes fourteen struts. 
         [0005]    In accordance with another construction of the invention, a unit truss for a lattice structure includes a junction, and fourteen struts coupled to the junction, six of the struts being mutually orthogonal, and eight of the inner struts oriented diagonally relative to each of the six mutually orthogonal struts. 
         [0006]    In accordance with yet another construction of the invention, a lattice structure includes a unit cell having a plurality of struts that absorb loads selected from a group consisting of tensile loads, compressive loads, and shear loads, and a dual enclosing the unit cell, the dual represented by intersections between a rectangular prism, the unit cell, and octahedra. 
         [0007]    Other aspects of the invention will become apparent by consideration of the detailed description and accompanying drawings. 
     
    
     
       BRIEF DESCRIPTION OF THE DRAWINGS 
         [0008]      FIG. 1  is a perspective view of a hexahedral unit cell according to one construction that includes nine unit trusses. 
           [0009]      FIG. 2  is a schematic representation of the hexahedral unit cell. 
           [0010]      FIG. 3  is a perspective view of one of the unit trusses. 
           [0011]      FIGS. 4 and 5  are perspective views of constructions of lattice structures with unit cells that are modified to be non-rectangular. 
           [0012]      FIGS. 6-8  are perspective views of the unit trusses being assembled together. 
           [0013]      FIGS. 9-11  are perspective views of the unit trusses, illustrating different materials and sizes for struts within each of the unit trusses. 
           [0014]      FIGS. 12-14  are perspective views of a lattice structure according to one construction, wherein at least one of the struts is made of an electrically-insulating material. 
           [0015]      FIG. 15  is a perspective view of a cubic unit cell according to one construction. 
           [0016]      FIG. 16  is a perspective view of a lattice structure according to one construction that incorporates a plurality of the cubic unit cells. 
           [0017]      FIG. 17  is a diagram illustrating tensile and compressive loads on the lattice structure of  FIG. 16 . 
           [0018]      FIGS. 18 and 19  are perspective views of principal stress and shear planes, and struts of one of the unit trusses aligned with the principal stress and shear planes. 
           [0019]      FIG. 20  is a perspective view of a cube unit cell according to one construction, filled with material. 
           [0020]      FIG. 21  is a perspective view of a supercube unit cell according to one construction, filled with material. 
           [0021]      FIG. 22  is a perspective view of the cubic unit cell of  FIG. 15 , filled with material. 
           [0022]      FIG. 23  is a perspective view of an octet unit cell according to one construction, filled with material. 
           [0023]      FIG. 24  is a perspective view of an ultracube unit cell according to one construction, filled with material. 
           [0024]      FIG. 25  is a perspective view of a lattice structure according to one construction with an outer cube-based layer and an inner ultracube-based layer. 
           [0025]      FIG. 26  is a perspective view of an axial load member for the lattice structure of  FIG. 25 . 
           [0026]      FIGS. 27 and 28  are perspective views of a lattice structure according to one construction that is a combination of cube unit cells and ultracube unit cells. 
           [0027]      FIGS. 29-38  are perspective views of constructions of unit cells with associated duals. 
           [0028]      FIG. 39  is a perspective view of a lattice structure according to one construction with hollow struts. 
           [0029]      FIG. 40  is a perspective view of a lattice structure according to one construction with struts that have hydrofoil or airfoil geometry. 
           [0030]      FIGS. 41 and 42  are schematic representations of stress ellipses for cross-sections of a strut. 
           [0031]      FIG. 43  is a perspective view of a lattice structure according to one construction that includes conical struts. 
           [0032]      FIG. 44  is a perspective view of a lattice structure according to one construction that includes sinusoidal struts. 
           [0033]      FIG. 45  is a perspective view of a dual that includes faces made of different material. 
           [0034]      FIG. 46  is a perspective view of a multifunctional thermal-management lattice structure according to one construction. 
           [0035]      FIGS. 47-49  are perspective views of constructions of lattice structures that include custom composite gradients. 
       
    
    
     DETAILED DESCRIPTION 
       [0036]    Before any embodiments of the invention are explained in detail, it is to be understood that the invention is not limited in its application to the details of construction and the arrangement of components set forth in the following description or illustrated in the following drawings. The invention is capable of other embodiments and of being practiced or of being carried out in various ways. 
         [0037]    With reference to  FIGS. 1 and 2 , a hexahedral unit cell  10  includes nine unit trusses  14 , eight of the unit trusses  14  being disposed at vertices  18  of the hexahedral unit cell  10  and one of the unit trusses  14  being positioned within the hexahedral unit cell  10 . In one construction, one of the unit trusses  14  is disposed at a centroid  22  of hexahedral unit cell  10 . In the illustrated construction, each unit truss  14  has exactly the same geometry and orientation within the hexahedral unit cell  10 . As illustrated in  FIG. 2 , the illustrated hexahedral unit cell  10  resembles a body-centered cubic (BCC) crystal structure, with the vertices  18  forming six faces  26 , the faces  26  defining planar, rectangular surfaces. The term unit truss  14 , as referred to herein, is a unit lattice that may be of any size or scale. 
         [0038]    With reference to  FIG. 3 , each unit truss  14  has a maximum of fourteen struts  30  that are connection elements used to couple one or more of the trusses  14  together, as well as to absorb one or more loads (e.g., tensile, compressive, and shear). Depending on boundary conditions of the unit truss  14  (e.g., anticipated loads), some of the struts  30  in one or more of the unit trusses  14  may not be used (e.g., are removed) within each unit truss  14 , to save material. Additionally, in some constructions, the faces  26  of the hexahedral unit cell  10  may not be arranged as planar, rectangular (or other parallelogram) surfaces like that in  FIG. 2 . For example,  FIG. 4  illustrates a two-dimensional portional cross-section of a load-bearing lattice structure  34  utilizing multiple hexahedral unit cells  10  coupled together, where some of the struts  30  have been removed.  FIG. 5  illustrates a three-dimensional image of a cubic-like lattice structure  38  of multiple hexahedral unit cells  10  where some of the struts  30  are oriented in an irregular pattern, such that faces in the lattice structure  38  are non-planar, and non-rectangular (in contrast to the faces  26  in  FIG. 2 ). 
         [0039]    With continued reference to  FIG. 3 , in the illustrated construction, six of the fourteen struts  30  of the unit truss  14  are mutually orthogonal struts  42  and eight of the fourteen struts  30  are diagonally-oriented struts  46  oriented diagonally relative to each of the six mutually orthogonal struts  42 . As illustrated in  FIG. 3 , each of the eight diagonally-oriented struts  46  extends between three of the mutually orthogonal struts  42 . In some construction fillets are disposed between the struts  30 . 
         [0040]    With continued reference to  FIG. 3 , the illustrated unit truss  14  further includes a junction  50  coupled to ends of each of the struts  30 . In the illustrated construction, the junction  50  is a sphere (e.g., a hollow sphere) that minimizes stress concentrations within the unit truss  14 , but in other constructions the junction  50  has other suitable geometries that minimize stress concentrations. In some constructions, the volume of the junction  50  is scaled depending on applied loads. In some constructions, the junction  50  includes chamfered and filleted gussets or webs (as seen in  FIGS. 29-31  and  34 - 38 )  52  between at least two of the struts  30 , and/or is made of nested components. The webs  52  may extend between the two struts  30  at a portion of the length of the struts or along the entire length of the struts. The webs  52  may be solid or may include holes or occlusions or may be solid at one portion and have holes at another portion. The holes may be in a particular pattern or randomly distributed on the web  52 . 
         [0041]    With reference to  FIGS. 6-8 , the unit trusses  14  are coupled together by coupling one or more of the struts  30  of one unit truss  14  with one or more of the struts  30  of another unit truss  14 . As illustrated in  FIG. 6 , for example, the vertices  18  of the hexahedral unit cell or cells  10  are coupled together with at least three (and up to eight) of the mutually orthogonal struts  42  from each of the unit trusses  14 . With reference to  FIG. 7 , the unit truss  14  at the centroid  22  is coupled to the unit trusses  14  at the vertices  18  with the diagonally-oriented struts  46  from each of the unit trusses  14  at the vertices  18  and the unit truss  14  at the centroid  22 . With reference to  FIG. 8 , the orthogonally-oriented struts  42  (up to six) of the unit truss  14  at the centroid  22  couple the unit truss  14  at the centroid  22  with other unit trusses  14  at centroids of adjacent hexahedral unit cells  10 . 
         [0042]    Within each unit truss  14 , the individual struts  30  absorb one or more loads (e.g., tensile, compressive, and shear loads). In some constructions, the unit trusses  14  and struts  30  are oriented specifically with directions of force at each location throughout a lattice structure. With reference to  FIGS. 6-8 , in some constructions, the struts  30  are all made of the same material. With reference to  FIGS. 9-11 , in other constructions, some of the struts  30  (e.g., the mutually orthogonal struts  42 ) are made of a first material or composite material, and other struts (e.g., the diagonally-oriented struts  46 ) are made of a different material or composite material. In some constructions, the materials of the struts  30  differ from unit truss  14  to unit truss  14 . With continued reference to  FIGS. 9-11 , in some constructions, the individual struts  30  of each unit truss  14  are sized independently according to anticipated forces applied to the struts  30 . For example, as illustrated in  FIG. 10 , the mutually orthogonal struts  34  may have a diameter larger than that of the diagonally-oriented struts  38 . 
         [0043]    When load conditions and/or fabrication constraints demand, the mutually orthogonal struts  42  from the unit truss  14  at the centroid  22  are removed, leaving only those between the vertices  18 . In these cases, the diagonally-oriented struts  46  connecting the unit truss  14  at the centroid  22  to the vertices  18  are “de-coupled” such that the diagonally-oriented struts  46  at the vertices  46  rotate about one orthogonal axis to lie between two of the mutually orthogonal struts  42 . In this way, the diagonally-oriented struts  46  are coupled to those two directions. 
         [0044]    With reference to  FIGS. 1 and 2 , the unit truss  14 , and any lattice structure that is made of one or more of the unit trusses  14  (or hexahedral unit cells  10 ), includes at least one void region  54  between the struts  30  that in some constructions is occupied by particulates, foam, aerogels, electrolytes, or other material, allowing for additional functionality (e.g., pressurized-fluid storage, heat dissipation, or acoustic damping) of the unit truss  14 . 
         [0045]    The apparent density at each unit truss  14  is a function of the size and composition of each strut  30 . Therefore, the apparent density at each point within any particular lattice structure that includes one or more of the unit trusses  14  is a function of the load at that point. This relation can be exploited to generate lattice structures that represent variable-density output of structural-optimization routines. 
         [0046]    In some constructions, a lattice structure is defined by two “intertwined” orthogonal lattices, interconnected along the diagonally-oriented struts  46  of the unit trusses  14  to form a composite lattice configuration. In some of these composite lattice configurations, each of the orthogonal lattices separately handles a different load condition. For example, one of the orthogonal lattices is comprised of a material better suited for tensile loads, while the other is comprised of material better suited for compressive loads (e.g., one with larger diameter struts  30  like those illustrated in  FIGS. 10 and 11 ). In some constructions of the composite lattice configuration the struts  30  of each orthogonal lattice structure are scaled independently, to a minimum of zero cross-sectional area, to minimize mass (effective density) within the composite lattice structure, and the diagonally-oriented struts  46  are used to carry the difference, (i.e., shear) in both tension and compression. 
         [0047]    In some constructions, the composite lattice structure allows for the intertwining of mutually reactive materials, where one, or both may carry a load. In these constructions the diagonally-oriented struts  46  (non-reactive to either of the other two materials, or protected from reacting) hold the materials apart until a separator or barrier material is dissolved, melted, destroyed or otherwise removed. Alternatively, a catalyst, or third reactive material, is introduced, e.g., as, or carried by, a fluid, to initiate a reaction. 
         [0048]    In some constructions, the diagonally-oriented struts  46  are made of an electrically-insulating material, allowing, for example, for each of the orthogonal lattices in the composite lattice structure to carry differing voltage potentials. With reference to  FIGS. 12 and 13 , for example, multi-material lattices  56  are illustrated, where at least one of the struts  30  is made of an electrically-insulating material, and where struts of different textures and sizes are used. 
         [0049]    With reference to  FIG. 14 , in some constructions, the lattice structure  56  includes anodic struts  58  and cathodic struts  62  (in the illustrated construction both diagonally-oriented struts  46 ), and wherein the void region  54  is filled with electrolyte for a load-bearing power source. 
         [0050]    With reference to  FIG. 15 , a cubic unit cell  66  includes one of the unit trusses  14  described above, wherein each of the fourteen struts  30  is an inner strut. The cubic unit cell  66  also includes twelve outer struts  70  that form edges of the cubic unit cell  66  and enclose the fourteen inner struts  30  in a cube-like manner. 
         [0051]    In the illustrated construction, the twelve outer struts  70  carry either tensile loads or compressive loads, the six mutually orthogonal struts  42  carry the opposite loads, and the eight diagonally-oriented struts  46  carry resultant shear loads. 
         [0052]    In some constructions each of the inner and outer struts  30 ,  70  is made of the same material. In some constructions each of mutually orthogonal struts  42  is made of a first material, each of the diagonally-oriented struts  46  is made of a second material, and each of the outer struts  70  is made of a third material, the first, second, and third materials each being different. Other constructions include different combinations of materials, sizes, and dimensions for the inner and outer struts  30 ,  70  than that illustrated. 
         [0053]    With reference to  FIG. 16 , a composite lattice structure  74  is illustrated that includes a plurality of the cubic unit cells  66 . The struts  30  within the composite lattice structure  74  vary in size so as to absorb two loads of one type (e.g., tensile) and one load of another type (e.g., compressive) (as illustrated in the diagram in  FIG. 17 ). The remaining loads are zero, so the remaining struts are scaled accordingly (shown as lines in  FIG. 16  representing zero diameter for illustrative purposes). 
         [0054]    The unit trusses  14 , hexahedral unit cells  10 , and cubic unit cells  66  are scale independent, and in some constructions are hierarchical. For example, a structure may be built with members having a lattice structure that includes one or more unit trusses  14 , hexahedral unit cells  10 , and/or cubic unit cells  66 . Additionally, in some constructions, a hexahedral unit cell  10 , cubic unit cell  66 , or other lattice structure made of unit trusses  14  at one scale can occupy one octant of a unit cell (e.g., a hexahedral unit cell  10  or a cubic unit cell  66 ) of another scale. 
         [0055]    With reference to  FIGS. 18 and 19 , each of struts  30  within the unit truss  14  is aligned along the intersection of one or more principal stress planes  78  and principal shear planes  82 , such that the principal stress and shear planes  78 ,  82  define a “structural skeleton” along these planes. In particular, each of the mutually orthogonal struts  42  is aligned along the intersection of two or more of the principal stress planes  78 , and each of the diagonally-oriented struts  46  is aligned along the intersection of two or more of the principal shear planes  82 . Depending on loads, scale, material, and fabrication constraints, and as described above, in some constructions, the unit truss  14  does not include one or more of the struts  30 . For example, and with continued reference to  FIG. 18 , in some constructions, the unit truss  14  degenerates into only four tetralattice struts  30  (i.e., those with larger diameter illustrated in  FIG. 18 , one of which is not visible behind the planes  78 ,  82 ). 
         [0056]    With reference to  FIGS. 20-24 , the intersection of the principal stress and shear planes  78 ,  82  define various types of unit cells, including a cube unit cell  86  ( FIG. 20 ), a supercube unit cell  90  ( FIG. 21 ), a modified supercube unit cell (i.e., the cubic unit cell  66  described above) ( FIG. 22 ), an octet unit cell  98  ( FIG. 23 ), and an ultracube unit cell  102  ( FIG. 24 ). Each of these unit cells  86 ,  90 ,  66 ,  98 , and  102  is formed at the intersection between four or five unit trusses  14 , and in the illustrated construction, each of these unit cells  86 ,  90 ,  66 ,  98 ,  102  includes a void region  54  filled with an inner material  106  (e.g., a solid matrix, fluid, particulates, aerogels, etc.). 
         [0057]    The “structural skeletons” defined by the struts  30  also include any transformation of these geometries, such as scaling, shearing, and bending, through which the defining planes  78 ,  82  may become surfaces, and their intersections may become curves. For example,  FIGS. 25 and 26  illustrate a conceptual, lattice-based, multifunctional, composite, pressure-vessel section  110  ( FIG. 25 ) and an axial-load member  114  ( FIG. 26 ) for use, for example, within the pressure-vessel section  110  ( FIG. 25 ) with geometric transformations applied to achieve a conformal lattice. In one construction,  FIG. 25  illustrates a cube-based layer  118 , an inner ultracube-based layer  122 , and a solid inner liner  126 . In some constructions the axial-load member  114  is a model of a low-mass, buckle-resistant, multi-function structure that carries load along and about its axis. As a hierarchical structure, this axial-load member  114  may be one strut in a larger unit cell (e.g., one strut in one of the ultracube unit cells  102  in the pressure-vessel structure  110 . 
         [0058]    In contrast to traditional composites, where composites are layered down in layers (i.e., a “layup” process) typically following a part&#39;s contour, the “layup” of lattice structures that employ unit trusses  14  has 360° of freedom about all three axes. In particular, the lattice structures follow loads, not necessarily a pre-defined part-volume geometry. For some geometries, such as pressure-vessels, the resultant orientations may be similar (e.g., as seen in  FIG. 25 ). However, lattice structures may vary from one unit lattice to the next, allowing for much more complex geometries (e.g., as seen in  FIGS. 4 and 5 ). 
         [0059]    With reference to  FIGS. 27 and 28 , in some constructions, a lattice structure  130  includes a combination of cube unit cells  86  and ultracube unit cells  102 . The layers of the lattice  130  can be offset, creating a body-centered-cubic-like connectivity. For example, and with continued reference to  FIG. 27 , the second layer of ultracube unit cells  102  from the bottom is shifted by one-half unit cell size in both directions in the plane of that layer. This connectivity includes a tetralattice geometry and an inverted tetralattice that may be removed for structural efficiency (reduced mass for similar performance), creating a rhombic dodecahedron geometry. With reference to  FIG. 28 , the connectivity may be altered between unit lattices, from offset-ultracube or tetralattice to modified supercube. 
         [0060]    With reference to  FIGS. 29-38 , in some constructions a lattice structure includes a unit cell (e.g. hexahedral unit cell  10 , cubic unit cell  66 , a cube unit cell  86 , supercube unit cell  90 , an octet unit cell  98 , or an ultracube unit cell  102 ) that is enclosed by a corresponding dual  134 . 
         [0061]    The unit cell and its dual  134  can be represented by intersections between a rectangular prism, the unit cell, and octahedral (dependent on size, position and rotation of octahedron relative to the rectangular prism). In some constructions, the unit cell and its dual  134  can be represented by truncated octahedron with orthogonal octahedral, each subdivided into four tetrahedral about principle axes. The strut count of the truncated octahedron can be reduced from three orthogonal rings to two, or just one, for further mass reduction and compliance. In some constructions the unit cell and its dual  134  can be represented by rhombic dodecahedron with octahedron and tetrahedral (octet). The strut count of the rhombic dodecahedron can be reduced to the tetralattice for further mass reduction and increased compliance. 
         [0062]    With reference to  FIGS. 29-31 , planar web extrusions from each of the dual&#39;s faces generate struts  138  in the dual  134 . For example,  FIG. 29  illustrates a cube unit cell  86  enclosed within its dual  134 , another cube.  FIG. 30  illustrates a cubic unit cell  66  enclosed within its dual  134 , a truncated octahedron.  FIG. 31  illustrates an octet unit cell  98 , composed of web extrusions in one principal stress and all six principal shear-stress planes, within its dual  134 , a rhombic dodecahedron. 
         [0063]      FIG. 32  illustrates a packed rhombic dodecahedra, which is a geometric dual  134  of the octet unit cell  98  in  FIG. 31 , with tetralattice generated along half of their edges, and  FIG. 33  illustrates an octet unit cell  98  within a tetralattice dual  134 . These are further examples of the degenerative potential of the lattice structures (i.e., from rhombic dodecahedron to tetralattice, by removing four struts). 
         [0064]      FIGS. 34 and 35  illustrate a truncated-octahedron dual  134 . The strut counts of the truncated-octahedron dual  134  (and rhombic-dodecahedron dual  134 ) can be reduced from three orthogonal rings (in the planes of the octahedron edges, connecting the vertices) to two, or just one, for further mass reduction and compliance. In particular,  FIG. 34  illustrates one orthogonal ring of connections having been removed from the truncated-octahedron dual  134 .  FIG. 35  illustrates two rings of connection having been removed, as well as faces that the two rings were supporting. The struts  138  of the truncated-octahedron dual  134  are generated from planar web extrusions, resulting in straight connections. These could follow the profile of the conic section that connects the vertices in that plane, or their mirror images about those edges. 
         [0065]    With reference to  FIGS. 36-38 , in some constructions the unit cell (e.g. hexahedral unit cell  10 , cubic unit cell  66 , cube unit cell  86 , supercube unit cell  90 , octet unit cell  98 , or ultracube unit cell  102 ) is coupled to its dual  134  with ligaments  142 , to provide for minimal addition of mass. For example,  FIG. 36  illustrates a cube unit cell  86  and its dual  134  coupled together with ligaments  142 ,  FIG. 37  illustrates a cubic unit cell  66  and its dual  134  coupled together with ligaments  142 , and  FIG. 38  illustrates an octet unit cell  98  and its dual  134  coupled together with ligaments  142 . 
         [0066]    The ligaments  142  couple the central junctions  50  of the unit cells to nodes  146  of the dual  134 . In some constructions the ligaments  142  are made of the same material as the unit cell or dual  134 . In other constructions the ligaments  142  are made of different material. 
         [0067]    When there is no shear, i.e. pure hydrostatic loading within the unit cell, the diagonally-oriented struts  46  in the shear planes  82  can be removed and since there is only compression or tension, only one of the remaining intertwined cubic structures may be required, also negating the ligaments  142  there between. Under pure shear the “hydrostatic” struts (i.e., the mutually orthogonal struts  42 ) can be removed, as well as the ligaments  142  if they are not also the shear struts. 
         [0068]    As described above, in some constructions the struts  30  are generated along the intersections of the principal stress and principal shear planes  78 ,  82 . In other constructions the struts  30  are generated along bisectors between two such intersections in a plane, or about either (e.g., a spiral). The struts  30  may be of any cross-sectional type, including hollow. For example, and with reference to  FIGS. 39 and 40 , in some constructions a supercube lattice  150  includes a plurality of hollow hydrostatic-load struts (e.g., mutually orthogonal struts  42 ) enveloped within hollow shear-load struts (e.g., diagonally-oriented struts  46 ). The junctions of the shear-load struts are shown here with stress-minimizing bulb geometry. The separation of the two hollow structures provides separate fluid-flow paths for heat exchange, in addition the potential for external fluid through-flow. 
         [0069]    With reference to  FIG. 40 , in some constructions the supercube lattice  150  is modified and degenerated into an octet. The struts  30  illustrated in  FIG. 40  have hydrofoil (or airfoil) geometry (solid or hollow) for increased surface area and reduced drag for heat transfer to or from a flowing fluid. 
         [0070]    With reference to  FIGS. 41 and 42 , the cross-sections of the struts  30  in the lattice structures described herein, regardless of material, can also be of any shape, optimizable for desired functionality, and can change along the length of the struts  30 . Generally, the minimum cross-sectional area for each strut is set so that: 
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         [0071]    The cube unit cell  86 , having no shear struts (i.e. diagonally-oriented struts  46 ), can have the cross-sections of its mutually orthogonal struts  42  scaled proportionally to the average stress ellipse for that unit cell, while maintaining its required minimal cross-sectional area. For example, and as illustrated in  FIG. 41  (which illustrates an example stress ellipse, shown in the X-Y, Y-Z, and X-Z planes, as well as a depiction of the proportional scaling that could be applied to the mutually orthogonal strut  42  of each) and  FIG. 42  (which illustrates examples of strut cross-section stress-ellipse-proportional scaling, for both a cross and conic section type), the Y-direction strut  42  is normal to the X-Z plane, so the cross-section of the Y-direction strut  42  can be scaled to match the proportionality between the X-direction and Z-direction stresses. This improves shear-loading capacity. 
         [0072]    With reference to  FIGS. 43 and 44 , in some constructions the struts  30  are not straight. For example, as illustrated in  FIG. 43 , in some constructions some of the struts  30  are conical. As illustrated in  FIG. 44 , in some constructions the struts  30  are sinusoidal. The conical and sinusoidal struts  30  contain all cell nodes in one plane and can replace straight struts  30  for stiffness or compliance modification. 
         [0073]    In some constructions the struts  30  are multiple entities bundled like wire, and their separation varies along the length of the strut  30  or ligament  142 . In some constructions the struts  30  bend around other struts  30  or ligaments  142  instead of intersecting with them. In some constructions the struts  30  are generated as a web extruded from a plane (e.g., like struts  138  described above). In some constructions the struts  30  are enclosed with a shell or shrink-wrap geometry, and/or have edges and corners that are filleted or chamfered. 
         [0074]    With reference to  FIG. 45 , in some constructions the duals  134  include multiple faces  154  (e.g., six faces of a truncated-octahedron dual  134  as illustrated in  FIG. 45 ) that each have a different material, thickness, and/or profile, and wherein the struts  138  are made of yet another material different than the material for each of the faces  154 . 
         [0075]    In some constructions, a lattice structure (e.g., one which includes the unit truss  14 , hexahedral unit cell  10 , cubic unit cell  66 , cube unit cell  86 , supercube unit cell  90 , octet unit cell  98 , and/or ultracube unit cell  102 ) includes protrusions and/or intrusions on internal or external surfaces of the lattice structure for increased surface area. The protrusions and/or intrusions provide heat transfer, electrochemical reactions, and biological cell growth. 
         [0076]    In some constructions a lattice structure includes metal plating or other conformal coatings. Proportions of mixed materials in fabrication can lead to a gradient between stiffness and compliance within the lattice structure. 
         [0077]    In some constructions, an octahedral lattice structure is subdivided into tetrahedral lattice structures. In some constructions a rectangular prism is subdivided into smaller rectangular prisms, each of which is further subdivided. This subdivision continues until limits of fabrications are met, at both ends of the structure&#39;s scale. These prisms are then used for generating a structural skeleton at their respective size scales, resulting in a fractal lattice structure. 
         [0078]    As the minimal strut dimensions approach fabrication limits during assembly of a lattice structure, struts  30  can be removed, allowing for the remainder to be scaled up while maintaining low total mass. The minimal form is the tetralattice. An offset ultracube unit cell  102  degenerates into the tetralattice, oriented with principle stress planes  78 . A cubic unit cell  66  degenerates into a tetralattice rotated with one principle stress plane  78  and two shear stress planes  82 . As noted above with regards to  FIGS. 34 and 35 , a truncated octahedron can be reduced from three orthogonal rings, to two or one. 
         [0079]    In some constructions, a lattice structure includes multiple different types of unit cells or modified unit cells (e.g. hexahedral unit cell  10 , cubic unit cell  66 , cube unit cell  86 , supercube unit cell  90 , octet unit cell  98 , and ultracube unit cell  102 ) and their duals  134 . The lattice structures can be aligned with potential fields (e.g., pressure, temperature, voltage, magnetism and gravity). In some constructions, two struts  30  lie in, or are tangent to, an isosurface (surface of constant magnitude through a potential field), while a third strut  30  is normal to the isolevel at that point, or tangent to that normal at that point. 
         [0080]    With reference to  FIG. 46 , a multifunctional thermal-management lattice structure  158  is illustrated that is formed using the cube unit cells  86  to safely integrate a power source into a medical device. In addition to sinking heat, the lattice structure  158  is intended to handle axial loading and moderate torsion. 
         [0081]    In some constructions, the thermal conductivity of a lattice structure is optimized to match that of thermoelectric generators, maximizing power conversion. A void region between the unit cell and its dual  134  can be filled with a phase-change material for latent-heat storage, such as that desired for solar water heaters, with one lattice connected to a source (solar heater), and the other to a sink (“hot water” pipe). 
         [0082]    In some constructions, a lattice structure optimizes material, strut geometry and cell size for manipulation and optimal attenuation of acoustic (fluid pressures) waves, for example via reflection and interference (“sonic crystal”) or via viscous damping of fluid oscillatory flow. The lattice structure can route pressure waves through the three-dimensional structure. 
         [0083]    In some constructions, a lattice structure optimizes material, strut geometry and cell size for manipulation of electromagnetic radiation transmission, through filtering, reflection and refraction, for applications such as routing, collimation and lensing, including concentration and diffusion. 
         [0084]    In some constructions, a lattice structure optimizes material, strut geometry and cell size for manipulation of magnetic fields and flux. 
         [0085]    In some constructions, a lattice structure includes custom composite gradients (e.g., solid to foam, and stiff to compliant). For example, and with reference to  FIG. 47 , a lattice structure  162  includes a transition between solid cube unit cells  86  (disposed at the bottom) and foam cube unit cells  86  (disposed at the top), connected by ligaments  142 . With reference to  FIG. 48 , a lattice structure  166  includes a transition between solid cubic unit cells  66  and foam truncated octahedron duals  134 , connected by ligaments  142 . With reference to  FIG. 49 , a lattice structure  170  includes a transition between solid octet unit cells  98  and foam rhombic-docecahedron duals  134 , connected by ligaments  142 . In some constructions a lattice changes relative density (volume fraction) of either unit lattice for gradient density too. In some constructions select facts of the foam cells may be closed, and/or multiple foam cell types are used. 
         [0086]    In some constructions, cell unit size, strut dimensions, and material selection are set such that the applied load will drastically deform the lattice (including failure) for impact absorption (strain energy converted to heat, rather than stored). The modifications may be made globally or locally within the full structure, and may be a gradient or multiple gradients. 
         [0087]    In some constructions, cell unit size, strut dimensions, and material selection are set such that the applied load elastically deforms the lattice (without plastic deformation) at a requisite strain to achieve a target storage capacity of strain energy. 
         [0088]    In some constructions, cell unit size, strut dimensions, and material selection are set for filtration of particulates from fluids, and separation of mixed fluids that have different viscosities, including routing of the fluids through the lattice structure. A volume fraction (ratio of fluid to solid structure) can vary throughout the structure for variable filterability. Fractal-generated structures provide more pores in specific regions for a given volume fraction. Free struts  30 , for example those that do not mate with a strut  30  of an adjacent unit truss  14 , may be removed if a one-unit-cell length extension does not result in a mate. In fractals, the extension may be up to one cell length of a cell one level up in the hierarchy. 
         [0089]    The lattice structures described herein advantageously reduce weight. For example, the high inter-connectivity of the unit trusses  14  minimizes oversizing, and the unit trusses  14  are custom-sized to handle loads in multiple directions (e.g., with safety factors included). 
         [0090]    The unit trusses  14  can be manufactured with up to three separate materials (e.g., one for tensile loads, one for compressive loads, and one for shear loads). 
         [0091]    Bearing surfaces are achieved with functionally-gradient lattices where the composition of the unit trusses  14  changes through the component, based on functional requirements. The custom inter-connectedness of intertwined lattices minimizes weight. 
         [0092]    In some constructions, the unit trusses  14  are made of a more compliant material for energy absorption. In some constructions the diagonally-oriented struts  46  are made of an elastomer, whose stiffness may vary throughout the structure. 
         [0093]    The high surface-to-volume ratio of internal channels allows for effective heat transfer between the lattice structure and any fluid(s) within the channels. In some constructions, the unit trusses  14  are also be hollow to allow fluid to flow internal to the lattice structure. The custom inter-connectedness of intertwined lattices allows the lattice (functioning as a heat exchanger) to more effectively bear mechanical loads. 
         [0094]    In some constructions, the lattice structure includes phononic band gaps. For example, the unit trusses  14  re spaced for noise filtering. These band gaps may be “stackable.” 
         [0095]    Although the invention has been described in detail with reference to certain preferred embodiments, variations and modifications exist within the scope and spirit of one or more independent aspects of the invention as described.