More Like this

1. Investigation of the Homogenized Anisotropy for the Triply Periodic Minimal Surface Lattice under Varying Applied Stress Orientation

   Wei, Chongyi; Smith, DE (September 2024, 2024 Solid Freeform Fabrication Conference)

   The advance of additive manufacturing makes it possible to design spatially varying lattice structures with complex geometric configurations. The homogenized elastic properties of these periodic lattice structures are known to deviate significantly from isotropic behavior where orthotropic material symmetry is often assumed. This paper addresses the need for a robust homogenization method for evaluating anisotropy of periodic lattice structures including an understanding of how the elastic properties transform under rotation. Here, periodic boundary conditions are applied on two-material representative volume element (RVE) finite element models to evaluate the complete homogenized stiffness tensor. A constrained multi-output regression approach is proposed to evaluate the elasticity tensor components under any assumed material symmetry model. This approach is applied to various lattice structures including scaffold and surface-based Triply Periodic Minimal Surface (TPMS). Our approach is used to assess the accuracy of rotation for assumed anisotropic and orthotropic homogenized material models over a range of lattice structures. 

   more » « less
2. A Heat Method for Generalized Signed Distance

   https://doi.org/10.1145/3658220  

   Feng, Nicole; Crane, Keenan (July 2024, ACM Transactions on Graphics)

   We introduce a method for approximating the signed distance function (SDF) of geometry corrupted by holes, noise, or self-intersections. The method implicitly defines a completed version of the shape, rather than explicitly repairing the given input. Our starting point is a modified version of theheat methodfor geodesic distance, which diffuses normal vectors rather than a scalar distribution. This formulation provides robustness akin togeneralized winding numbers (GWN), but provides distance function rather than just an inside/outside classification. Our formulation also offers several features not common to classic distance algorithms, such as the ability to simultaneously fit multiple level sets, a notion of distance for geometry that does not topologically bound any region, and the ability to mix and match signed and unsigned distance. The method can be applied in any dimension and to any spatial discretization, including triangle meshes, tet meshes, point clouds, polygonal meshes, voxelized surfaces, and regular grids. We evaluate the method on several challenging examples, implementing normal offsets and other morphological operations directly on imperfect curve and surface data. In many cases we also obtain an inside/outside classification dramatically more robust than the one obtained provided by GWN. 

   more » « less
3. Quasiperiodic Composites: Multiscale Reiterated Homogenization

   https://doi.org/10.1109/MetaMaterials.2019.8900926  

   Cherkaev, E.; Guenneau, S.; Hutridurga, H.; Wellander, N. (September 2019, 2019 Thirteenth International Congress on Artificial Materials for Novel Wave Phenomena (Metamaterials))

   With recent technological advances, quasiperiodic and aperiodic materials present a novel class of metamaterials that possess very unusual, extraordinary properties such as superconductivity, unusual mechanical properties and diffraction patterns, extremely low thermal conductivity, etc. As all these properties critically depend on the microgeometry of the media, the methods that allow characterizing the effective properties of such materials are of paramount importance. In this paper, we analyze the effective properties of a class of multiscale composites consisting of periodic and quasiperiodic phases appearing at different scales. We derive homogenized equations for the effective behavior of the composite and discover a variety of new effects which could have interesting applications in the control of wave and diffusion phenomena. 

   more » « less
4. Spatially quasi-periodic bifurcations from periodic traveling water waves and a method for detecting bifurcations using signed singular values

   https://doi.org/10.1016/j.jcp.2023.111954  

   Wilkening, Jon; Zhao, Xinyu (April 2023, Journal of Computational Physics)

   We present a method of detecting bifurcations by locating zeros of a signed version of the smallest singular value of the Jacobian. This enables the use of quadratically convergent root-bracketing techniques or Chebyshev interpolation to locate bifurcation points. Only positive singular values have to be computed, though the method relies on the existence of an analytic or smooth singular value decomposition (SVD). The sign of the determinant of the Jacobian, computed as part of the bidiagonal reduction in the SVD algorithm, eliminates slope discontinuities at the zeros of the smallest singular value. We use the method to search for spatially quasi-periodic traveling water waves that bifurcate from large-amplitude periodic waves. The water wave equations are formulated in a conformal mapping framework to facilitate the computation of the quasi-periodic Dirichlet-Neumann operator. We find examples of pure gravity waves with zero surface tension and overhanging gravity-capillary waves. In both cases, the waves have two spatial quasi-periods whose ratio is irrational. We follow the secondary branches via numerical continuation beyond the realm of linearization about solutions on the primary branch to obtain traveling water waves that extend over the real line with no two crests or troughs of exactly the same shape. The pure gravity wave problem is of relevance to ocean waves, where capillary effects can be neglected. Such waves can only exist through secondary bifurcation as they do not persist to zero amplitude. The gravity-capillary wave problem demonstrates the effectiveness of using the signed smallest singular value as a test function for multi-parameter bifurcation problems. This test function becomes mesh independent once the mesh is fine enough. 

   more » « less
5. Generative Design of Multi-Material Hierarchical Structures via Concurrent Topology Optimization and Conformal Geometry Method

   https://doi.org/10.1115/DETC2019-97617  

   Jiang, Long; Chen, Shikui; Gu, Xianfeng David (August 2019, ASME IDETC/CIE 2019)

   Abstract Topology optimization has been proved to be an automatic, efficient and powerful tool for structural designs. In recent years, the focus of structural topology optimization has evolved from mono-scale, single material structural designs to hierarchical multimaterial structural designs. In this research, the multi-material structural design is carried out in a concurrent parametric level set framework so that the structural topologies in the macroscale and the corresponding material properties in mesoscale can be optimized simultaneously. The constructed cardinal basis function (CBF) is utilized to parameterize the level set function. With CBF, the upper and lower bounds of the design variables can be identified explicitly, compared with the trial and error approach when the radial basis function (RBF) is used. In the macroscale, the ‘color’ level set is employed to model the multiple material phases, where different materials are represented using combined level set functions like mixing colors from primary colors. At the end of this optimization, the optimal material properties for different constructing materials will be identified. By using those optimal values as targets, a second structural topology optimization is carried out to determine the exact mesoscale metamaterial structural layout. In both the macroscale and the mesoscale structural topology optimization, an energy functional is utilized to regularize the level set function to be a distance-regularized level set function, where the level set function is maintained as a signed distance function along the design boundary and kept flat elsewhere. The signed distance slopes can ensure a steady and accurate material property interpolation from the level set model to the physical model. The flat surfaces can make it easier for the level set function to penetrate its zero level to create new holes. After obtaining both the macroscale structural layouts and the mesoscale metamaterial layouts, the hierarchical multimaterial structure is finalized via a local-shape-preserving conformal mapping to preserve the designed material properties. Unlike the conventional conformal mapping using the Ricci flow method where only four control points are utilized, in this research, a multi-control-point conformal mapping is utilized to be more flexible and adaptive in handling complex geometries. The conformally mapped multi-material hierarchical structure models can be directly used for additive manufacturing, concluding the entire process of designing, mapping, and manufacturing. 

   more » « less