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Figure 16: Convergence of a shape optimization on pattern “(EL E2)(E1,E4)(E2,E4).” Left: optimization starting point. Right: optimized shape.

Figure 16 Convergence of a shape optimization on pattern “(EL E2)(E1,E4)(E2,E4).” Left: optimization starting point. Right: optimized shape.

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Figure 1: Six basic elastic textures are used to obtain a large range of homogenized isotropic material properties. A 3 x 3 x 1 tiling of each pattern is shown, along with rendered (left) and fabricated (right) cell geometry below. The naming convention is explained in Section 4.
Figure 1: Six basic elastic textures are used to obtain a large range of homogenized isotropic material properties. A 3 x 3 x 1 tiling of each pattern is shown, along with rendered (left) and fabricated (right) cell geometry below. The naming convention is explained in Section 4.
Figure 2: Overview of elastic texture generation and use. In this section, we describe our overall approach, visualized in Fig- ure 2, and a specific set of patterns that we have obtained.
Figure 2: Overview of elastic texture generation and use. In this section, we describe our overall approach, visualized in Fig- ure 2, and a specific set of patterns that we have obtained.
Figure 3: Region of the (Ev) space covered by the selected set of patterns. Each topology’s coverage is shown in a different color. We do not claim that the proposed family is in any sense optimal. It is most likely possible to extend the coverage or to cover the same domain with fewer topologies. However, the presented set is already quite useful for controlling material properties, as the examples of Section 7 show.
Figure 3: Region of the (Ev) space covered by the selected set of patterns. Each topology’s coverage is shown in a different color. We do not claim that the proposed family is in any sense optimal. It is most likely possible to extend the coverage or to cover the same domain with fewer topologies. However, the presented set is already quite useful for controlling material properties, as the examples of Section 7 show.
Figure 5: Compression test results for eight patterns with varying ment measurements, we computed the X, Y, and Z strains and their ratios. SE eR MERE ES ER Re OMe ERE Jee SEB ee Peet tee PEs Pee yer ee: homogenized Young’s moduli (6 x 6 x 2 tiling of 5mm cells). (a) Slopes extracted from the measured force vs. displacement curves along with a best-fit line through the origin. (b) Moduli extracted from simulated compression tests, with and without mod- eling compression plate friction. Without friction, the simulated test agrees with homogenization perfectly, but friction introduces error.
Figure 5: Compression test results for eight patterns with varying ment measurements, we computed the X, Y, and Z strains and their ratios. SE eR MERE ES ER Re OMe ERE Jee SEB ee Peet tee PEs Pee yer ee: homogenized Young’s moduli (6 x 6 x 2 tiling of 5mm cells). (a) Slopes extracted from the measured force vs. displacement curves along with a best-fit line through the origin. (b) Moduli extracted from simulated compression tests, with and without mod- eling compression plate friction. Without friction, the simulated test agrees with homogenization perfectly, but friction introduces error.
Figure 4: Samples of the (E,v) space reached by patterns with topology “(E1,E2)(E1,E4)(E2,E4).”
Figure 4: Samples of the (E,v) space reached by patterns with topology “(E1,E2)(E1,E4)(E2,E4).”
Figure 7: We extracted a 45° rotated rectangular block from a reg- ularly tiled 10mm cell microstructure to test Young’s modulus in non-axis aligned directions. Figure 6: Poisson’s ratios measured from 3 X 3 x 1 printed tilings of 10mm cells vs. homogenized properties. The v = —0.67 sample, outside our family’s range, violates isotropy and printability con- straints (we added support structure manually for this experiment).
Figure 7: We extracted a 45° rotated rectangular block from a reg- ularly tiled 10mm cell microstructure to test Young’s modulus in non-axis aligned directions. Figure 6: Poisson’s ratios measured from 3 X 3 x 1 printed tilings of 10mm cells vs. homogenized properties. The v = —0.67 sample, outside our family’s range, violates isotropy and printability con- straints (we added support structure manually for this experiment).
Figure 8: (a) The tetrahedral cube decomposition used to generate 3D patterns; (b) The 15 nodes defined on a tetrahedron together with their degrees of freedom. Consider the group of symmetries of a cube O;,, which includes reflections about three symmetry planes orthogonal to the X, Y, Z axes and the six planes orthogonal to the bisector of each pair of axes. By partitioning the cube according to these symmetry planes, we obtain 48 equal tetrahedra as in Figure 8a. On maps a single one of these tetrahedra to any other, so it is sufficient to define the nodes and edges of the pattern graph—as well as their offsets and thicknesses—on a single tetrahedron.
Figure 8: (a) The tetrahedral cube decomposition used to generate 3D patterns; (b) The 15 nodes defined on a tetrahedron together with their degrees of freedom. Consider the group of symmetries of a cube O;,, which includes reflections about three symmetry planes orthogonal to the X, Y, Z axes and the six planes orthogonal to the bisector of each pair of axes. By partitioning the cube according to these symmetry planes, we obtain 48 equal tetrahedra as in Figure 8a. On maps a single one of these tetrahedra to any other, so it is sufficient to define the nodes and edges of the pattern graph—as well as their offsets and thicknesses—on a single tetrahedron.
Figure 9: Symmetry orbits are colored with yellow, red and green. Left: vertex symmetry orbits. Right: edge symmetry orbits. We generate the different topological configurations by chang- ing the connectivity between 15 nodes on a tetrahedron (see Figure 8a): vertex nodes {Vo, Vi, V2, V3}, edge nodes {Eo, Fi, E2, Es, Es, Es}, faces nodes {Fo, Fi, Fo, F3}, and a single internal node, Jo. Configurations are named by their graphs’ edge sets (see labels in Figure 1). Each node is constrained to stay on its respective simplex to preserve the topology, so vertex nodes are fixed, edge nodes have a single offset, and so on (Figure 8b). Figure 9 shows an example topology colored by its node and edge orbits with respect to symmetry group On, and Figure 10 demon- strates the effects of the node offset and edge thickness parameters.
Figure 9: Symmetry orbits are colored with yellow, red and green. Left: vertex symmetry orbits. Right: edge symmetry orbits. We generate the different topological configurations by chang- ing the connectivity between 15 nodes on a tetrahedron (see Figure 8a): vertex nodes {Vo, Vi, V2, V3}, edge nodes {Eo, Fi, E2, Es, Es, Es}, faces nodes {Fo, Fi, Fo, F3}, and a single internal node, Jo. Configurations are named by their graphs’ edge sets (see labels in Figure 1). Each node is constrained to stay on its respective simplex to preserve the topology, so vertex nodes are fixed, edge nodes have a single offset, and so on (Figure 8b). Figure 9 shows an example topology colored by its node and edge orbits with respect to symmetry group On, and Figure 10 demon- strates the effects of the node offset and edge thickness parameters.
Figure 12: 2D examples of the printability detection algorithm. Vertices with supporting nodes are marked (green), then a breadth- first search extends the supported vertex front to horizontal neigh- bors. The remaining unmarked nodes are unsupported (red). Two cases are shown: unprintable (top) and printable (bottom). Printability can be tested by a simple algorithm: we first mark as supported all nodes with a supporting node (considering periodic- ity). Then we propagate the front of supported nodes to neighbors at the same level. When this breadth first search terminates, the pattern is printable if and only if all nodes are marked as supported. The procedure is illustrated in Figure 12. We also note that this constraint can be expressed algebraically as a set of inequality con- straints on the offset variables, which can be enforced by an opti- mization solver.
Figure 12: 2D examples of the printability detection algorithm. Vertices with supporting nodes are marked (green), then a breadth- first search extends the supported vertex front to horizontal neigh- bors. The remaining unmarked nodes are unsupported (red). Two cases are shown: unprintable (top) and printable (bottom). Printability can be tested by a simple algorithm: we first mark as supported all nodes with a supporting node (considering periodic- ity). Then we propagate the front of supported nodes to neighbors at the same level. When this breadth first search terminates, the pattern is printable if and only if all nodes are marked as supported. The procedure is illustrated in Figure 12. We also note that this constraint can be expressed algebraically as a set of inequality con- straints on the offset variables, which can be enforced by an opti- mization solver.
Figure 10: The results of varying the thickness (top) and offset (bottom) parameters of a particular pattern topology.
Figure 10: The results of varying the thickness (top) and offset (bottom) parameters of a particular pattern topology.
Figure 11: Two pattern topologies from each of three different families, shown with the families’ interfaces (nodes on the cube cell faces)
Figure 11: Two pattern topologies from each of three different families, shown with the families’ interfaces (nodes on the cube cell faces)
Figure 13: (Schematic) Periodic tiling of a domain Q with base cell Y having geometry w and length scale e. Defining the homogenized tensor. Consider heterogeneous object Q&° filled with a periodic microstructure, as shown schematically in Figure 13. Parameter € determines the size of cell Y relative to the object 0° and permits asymptotic analysis as € — 0.
Figure 13: (Schematic) Periodic tiling of a domain Q with base cell Y having geometry w and length scale e. Defining the homogenized tensor. Consider heterogeneous object Q&° filled with a periodic microstructure, as shown schematically in Figure 13. Parameter € determines the size of cell Y relative to the object 0° and permits asymptotic analysis as € — 0.
Figure 14: Deformation of an object with varying material proper- ties per voxel, and the same object with the material in each voxel replaced with the corresponding pattern. The deformed objects are colored by max Stress. Convergence rate. Remarkably, we have observed that the homog- enized coefficients remain accurate when the microstructure varies across cells (with few or no repetitions). In our experiments, the de- formation behavior of even very coarse tilings closely matches the homogenized behavior (Figure 14), and we expect similar agree- ment in general for smoothly varying loads.
Figure 14: Deformation of an object with varying material proper- ties per voxel, and the same object with the material in each voxel replaced with the corresponding pattern. The deformed objects are colored by max Stress. Convergence rate. Remarkably, we have observed that the homog- enized coefficients remain accurate when the microstructure varies across cells (with few or no repetitions). In our experiments, the de- formation behavior of even very coarse tilings closely matches the homogenized behavior (Figure 14), and we expect similar agree- ment in general for smoothly varying loads.
Figure 15: Left: a shape derivative, visualized as a steepest ascent normal velocity field for objective (8). Right: the shape velocity induced by one of the pattern’s thickness parameters.
Figure 15: Left: a shape derivative, visualized as a steepest ascent normal velocity field for objective (8). Right: the shape velocity induced by one of the pattern’s thickness parameters.
Figure 17: The path in (E, v) space traversed by the optimiza- tion of pattern “(E1,E2)(E1,E4)(E2,E4)” shown in Figure 16. The brown points are intermediate anisotropic microstructures.
Figure 17: The path in (E, v) space traversed by the optimiza- tion of pattern “(E1,E2)(E1,E4)(E2,E4)” shown in Figure 16. The brown points are intermediate anisotropic microstructures.
Figure 18: Examples of objects with painted material properties. All are fabricated with 5mm cells.
Figure 18: Examples of objects with painted material properties. All are fabricated with 5mm cells.
References Figure 19: Convergence of material optimization.
References Figure 19: Convergence of material optimization.
Figure 21: Compression of an anisotropic sample along the X, Y, and Z directions.
Figure 21: Compression of an anisotropic sample along the X, Y, and Z directions.
Figure 20: Examples of objects with optimized material properties. All are fabricated with 5mm cells.
Figure 20: Examples of objects with optimized material properties. All are fabricated with 5mm cells.
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