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Title
Abstract
Figures
Introduction
Related Work
Overview and Main Results
From Patterns to Material Properties
Conclusions
References
All Topics
Computer Science
Information Systems

Elastic textures for additive fabrication

Profile image of Nico PietroniNico Pietroni

2015, ACM Transactions on Graphics

https://doi.org/10.1145/2766937
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12 pages

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Abstract

Figure 1: Six basic elastic textures are used to obtain a large range of homogenized isotropic material properties. A 3 × 3 × 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.

Figures (21)
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 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.
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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Additive manufacturing (AM) processes are a types of methods that fabricate parts by adding elements and segments of a designed feedstock material. These materials can range from polymeric and plastic to metallic and ceramic. Based on different needs, a specific method can be implemented. Different methods utilize different deposition techniques. Some of them melt the materials and some change the materials into semisolid form. Different heating sources such as laser and resistance heaters van be used to change material states. The most common processes are Stereolithography (SLA), Liquid thermal polymerization (LTP), Fused deposition modeling (FDM), Ballistic particle manufacturing (BPM), Selective laser melting (SLM), Laser engineered net shaping (LENS) and Binder jet printing (BJP).

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Design and fabrication of materials with desired deformation behavior
Hyunho Lee

ACM Transactions on Graphics, 2010

: Two examples of real and replicated objects. Thanks to our data-driven process, we are able to measure, simulate, and obtain material combinations of non-linear base materials that match a desired deformation behavior. We can then print those objects with multimaterial 3D printers using two materials (blue and black) with varying internal microstructure.

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Materials design for the anisotropic linear elastic properties of textured cubic crystal aggregates using zeroth-, first- and second-order bounds
Mauricio Lobos

International Journal of Mechanics and Materials in Design, 2014

For polycrystals made of cubic materials like copper, aluminum, iron and other metals and ceramics, the macroscopic elastic behavior can be bounded using minimum energy principles. Böhlke and Lobos (Acta Mater. 67:324-334, 2014) have shown that not only the Voigt and the Reuss bound but also the Hashin-Shtrikman bounds can be represented explicitly depending on the texture in form of the fourth-order texture coefficient. Considering the inequalities due to these bounds, the texture can be enclosed independently of the specific cubic material parameters. This implies domains for the texture parameters. Materials design is defined as the identification of materials and microstructures such that the effective constitutive properties correspond best to a prescribed properties profile. The design space is proposed to be constituted by the material design space and microstructure design space, delivering a total of twelve scalar design variables in the present model for linear elasticity of cubic crystal aggregates. Based on analytical results, materials design is established as an algorithm following Adams et al. (Microstructure Sensitive Design for Performance Optimization, 2013). In the present work, the scheme consists of four steps: (i) material selection, (ii) homogenization scheme, (iii) properties closure, and (iv) microstructure optimization. As an example, Young's modulus of a polycrystal is designed with respect to four prescribed directions for a macroscopical orthotropic sample symmetry. For the orthotropic texture domain, a mathematically equivalent parametrization is derived in order to facilitate the constrained numerical optimizations. Keywords Cubic crystal aggregates Á Materials design Á Hashin-Shtrikman bounds Á Tensorial texture coefficients M. Lobos Á T. Böhlke (&) Chair for Continuum Mechanics,

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    Ella Moore

    Proceedings of the 2nd ACM Symposium on Computational Fabrication

    This paper presents an exploration of computer-controlled embroidery design and stitching with the goal of making purposeful, precision changes to material properties of a base elastic textile. Two techniques are proposed for adding stitches through designed microstructural cells and stitch-level planning. For the latter, a novel path planning algorithm is proposed to serialize stitches that is similar to a greedy solution for the travelling salesman problem with a set of domain-specific constraints that dictate edge cost. We show the efficacy of the concept through a set of simple design examples that undergo mechanical load testing and discuss the value of the technique in future applications using computational design. CCS CONCEPTS • Computing methodologies → Graphics systems and interfaces; Computer graphics;

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    State of the Art on Stylized Fabrication
    Nico Pietroni

    Computer Graphics Forum

    Digital fabrication devices are powerful tools for creating tangible reproductions of 3D digital models. Most available printing technologies aim at producing an accurate copy of a tridimensional shape. However, fabrication technologies can also be used to create a stylistic representation of a digital shape. We refer to this class of methods as "stylized fabrication methods." These methods abstract geometric and physical features of a given shape to create an unconventional representation, to produce an optical illusion, or to devise a particular interaction with the fabricated model. In this state-of-the-art report, we classify and overview this broad and emerging class of approaches and also propose possible directions for future research.

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    Globally Approximate Gaussian Processes for Big Data With Application to Data-Driven Metamaterials Design
    Yu-Chin Chan

    Journal of Mechanical Design

    We introduce a novel method for Gaussian process (GP) modeling of massive datasets called globally approximate Gaussian process (GAGP). Unlike most large-scale supervised learners such as neural networks and trees, GAGP is easy to fit and can interpret the model behavior, making it particularly useful in engineering design with big data. The key idea of GAGP is to build a collection of independent GPs that use the same hyperparameters but randomly distribute the entire training dataset among themselves. This is based on our observation that the GP hyperparameter approximations change negligibly as the size of the training data exceeds a certain level, which can be estimated systematically. For inference, the predictions from all GPs in the collection are pooled, allowing the entire training dataset to be efficiently exploited for prediction. Through analytical examples, we demonstrate that GAGP achieves very high predictive power matching (and in some cases exceeding) that of state-...

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    METASET: Exploring Shape and Property Spaces for Data-Driven Metamaterials Design
    Yu-Chin Chan

    Journal of Mechanical Design

    Data-driven design of mechanical metamaterials is an increasingly popular method to combat costly physical simulations and immense, often intractable, geometrical design spaces. Using a precomputed dataset of unit cells, a multiscale structure can be quickly filled via combinatorial search algorithms, and machine learning models can be trained to accelerate the process. However, the dependence on data induces a unique challenge: an imbalanced dataset containing more of certain shapes or physical properties can be detrimental to the efficacy of data-driven approaches. In answer, we posit that a smaller yet diverse set of unit cells leads to scalable search and unbiased learning. To select such subsets, we propose METASET, a methodology that (1) uses similarity metrics and positive semi-definite kernels to jointly measure the closeness of unit cells in both shape and property spaces and (2) incorporates Determinantal Point Processes for efficient subset selection. Moreover, METASET al...

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    Developable Metamaterials: Mass-fabricable Metamaterials by Laser-Cutting Elastic Structures
    alexandra ion

    Proceedings of the 2021 CHI Conference on Human Factors in Computing Systems, 2021

    Figure 1: We propose a novel material design that can be laser cut with user-defined compliance. We show how such fast to produce materials allow users to create custom materials, such as (a) stuffing for toys with custom softness, (b) lamp shades with variable transparency, (c) damping materials for packaging, or (d) be used for static support (e.g., formwork for architecture).

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