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1. 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)

   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.

    

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2. Robust Topology Optimization of Synchronous Reluctance Motors using Cardinal Basis Function Based Level Set Method

   Tian, Jiawei ; Torrey, David ; Luo, Fang ; Longtin, Jon ; Chen ; Shikui ( August 2023 , ASME Design Engineering Technical Conferences)

   Synchronous reluctance motors (SynRMs) have gained considerable attention in the field of electric vehicles as they reduce the need for permanent magnets in the rotor, resulting in less material and manufacturing costs. However, their lower average torque and torque ripple vibrations have been identified as key issues that require resolution. In this study, we present a SynRM design framework employing the cardinal basis functions (CBF)-based parametric level set method. The SynRms design problem is recast as a variational problem constrained by Maxwell's equations which describe the behavior of electric and magnetic fields in the SynRM. A continuum shape sensitivity analysis is carried out using the material derivative and adjoint method. A distance regularization energy function is employed to maintain the level set function as a signed distance function during the optimization. The parametric topology optimization problem is computationally solved using the Method of Moving Asymptotes (MMA). To demonstrate the effectiveness of our approach, we present a numerical example that compares the torque characteristics of the optimal design with those of a reference design. Preliminary results show that the optimized SynRM has a 30.30% increase in average torque, along with a slight increase in torque ripple, compared to the reference model. 

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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. 

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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. 

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5. Perfect reflection by dielectric subwavelength particle arrays: causes, implications, and technology

   https://doi.org/10.1117/12.2578953  

   Magnusson, Robert ; Ko, Yeong H. ; Lee, Kyu J. ; Bootpakdeetam, Pawarat ; Razmjooei, Nasrin ; Simlan, Fairooz A. ; Chen, Ren-Jie ; Buchanan-Vega, Joseph ; Carney, Daniel J. ; Hemmati, Hafez ; et al ( April 2021 , Proc. SPIE, Integrated Optics: Devices, Materials, and Technologies XXV)

   García-Blanco, Sonia M. ; Cheben, Pavel (Ed.)

   Periodic arrays of resonant dielectric nano- or microstructures provide perfect reflection across spectral bands whose extent is controllable by design. At resonance, the array yields this result even in a single subwavelength layer fashioned as a membrane or residing on a substrate. The resonance effect, known as guided-mode resonance, is basic to modulated films that are periodic in one dimension (1D) or in two dimensions (2D). It has been known for 40 years that these remarkable effects arise as incident light couples to leaky Bloch-type waveguide modes that propagate laterally while radiating energy. Perfect reflection by periodic lattices derives from the particle assembly and not from constituent particle resonance. We show that perfect reflection is independent of lattice particle shape in the sense that it arises for all particle shapes. The resonance wavelength of the Bloch-mode-mediated zero-order reflectance is primarily controlled by the period for a given lattice. This is because the period has direct, dominant impact on the homogenized effective-medium refractive index of the lattice that controls the effective mode index experienced by the mode generating the resonance. In recent years, the field of metamaterials has blossomed with a flood of attendant publications. A significant fraction of this output is focused on reflectors with claims that local Fabry-Perot or Mie resonance causes perfect reflection with the leaky Bloch-mode viewpoint ignored. In this paper, we advance key points showing the essentiality of lateral leaky Bloch modes while laying bare the shortcomings of the local mode explanations. The state of attendant technology with related applications is summarized. The take-home message is that it is the assembly of particles that delivers all the important effects including perfect reflection. 

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