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1. Level-Set-Based Shape & Topology Optimization of Thermal Cloaks

   Xu, Xiaoqiang; Chen, Shikui (January 2022, ASME Design Engineering Technical Conferences)

   Thermal metamaterials are gaining increasing popularity, especially for heat flux manipulation purposes. However, due to the high anisotropy of the structures resulting from the transformation thermotics or scattering cancellation methods, researchers are resorting to topology optimization as an alternative to find the optimal distribution of constituent bulk materials to realize a specific thermal function. This paper proposes to design a thermal cloak using the level-set-based shape and topology optimization. The thermal cloak design is considered in the context of pure heat conduction. The cloaking effect is achieved by reproducing the reference temperature field through the optimal distribution of two thermally conductive materials. The structural boundary is evolved by solving the Hamilton-Jacobi equation. The feasibility and validity of the proposed method to design thermal meta-devices with cloaking functionality are demonstrated through two numerical examples. The optimized structures have clear boundaries between constituent materials and do not exhibit thermal anisotropy, making it easier for physical realization. The first example deals with a circular cloaking region as a benchmark design. The robustness of the proposed method against various cloaking regions is illustrated by the second example concerning a human-shaped cloaking area. This work can inspire a broader exploration of the thermal meta-device in the heat flux manipulation regime. 

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2. Active thermal cloaking and mimicking

   https://doi.org/10.1098/rspa.2020.0941  

   Cassier, Maxence; DeGiovanni, Trent; Guenneau, Sébastien; Guevara Vasquez, Fernando (May 2021, Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences)

   null (Ed.)

   We present an active cloaking method for the parabolic heat (and mass or light diffusion) equation that can hide both objects and sources. By active, we mean that it relies on designing monopole and dipole heat source distributions on the boundary of the region to be cloaked. The same technique can be used to make a source or an object look like a different one to an observer outside the cloaked region, from the perspective of thermal measurements. Our results assume a homogeneous isotropic bulk medium and require knowledge of the source to cloak or mimic, but are in most cases independent of the object to cloak. 

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3. A Framework for Simulation of Multiple Elastic Scattering in Two Dimensions

   https://doi.org/10.1137/18M1232814  

   Lai, J.; Li, P. (January 2020, SIAM journal on scientific computing)

   Consider the elastic scattering of a time-harmonic wave by multiple well-separated rigid particles with smooth boundaries in two dimensions. Instead of using the complex Green's tensor of the elastic wave equation, we utilize the Helmholtz decomposition to convert the boundary value problem of the elastic wave equation into a coupled boundary value problem of the Helmholtz equation. Based on single, double, and combined layer potentials with the simpler Green's function of the Helmholtz equation, we present three different boundary integral equations for the coupled boundary value problem. The well-posedness of the new integral equations is established. Computationally, a scattering matrix based method is proposed to evaluate the elastic wave for arbitrarily shaped particles. The method uses the local expansion for the incident wave and the multipole expansion for the scattered wave. The linear system of algebraic equations is solved by GMRES with fast multipole method (FMM) acceleration. Numerical results show that the method is fast and highly accurate for solving elastic scattering problems with multiple particles. 

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4. A High-Order Nyström-Based Scheme Explicitly Enforcing Surface Density Continuity for the Electric Field Integral Equation

   https://doi.org/10.1109/LAWP.2024.3402749  

   Hu, Jin; Sideris, Constantine (January 2024, IEEE Antennas and Wireless Propagation Letters)

   This paper introduces an efficient approach for solving the Electric Field Integral Equation (EFIE) with highorder accuracy by explicitly enforcing the continuity of the impressed current densities across boundaries of the surface patch discretization. The integral operator involved is discretized via a Nystrom-collocation approach based on Chebyshev polynomial expansion within each patch and a closed quadrature rule is utilized such that the discretization points inside one patch coincide with those inside another patch on the shared boundary of those two patches. The continuity enforcement is achieved by constructing a mapping from those coninciding points to a vector containing unique discretization points used in the GMRES iterative solver. The proposed approach is applied to the scattering of several different geometries including a sphere, a cube, a NURBS model imported from CAD software, and a dipole structure and results are compared with the Magnetic Field Integral Equation (MFIE) and the EFIE without enforcing continuity to illustrate the effectiveness of the approach. 

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5. Incompressible active phases at an interface. Part 1. Formulation and axisymmetric odd flows

   https://doi.org/10.1017/jfm.2022.856  

   Jia, Leroy L.; Irvine, William T.M.; Shelley, Michael J. (November 2022, Journal of Fluid Mechanics)

   Inspired by the recent realization of a two-dimensional (2-D) chiral fluid as an active monolayer droplet moving atop a 3-D Stokesian fluid, we formulate mathematically its free-boundary dynamics. The surface droplet is described as a general 2-D linear, incompressible and isotropic fluid, having a viscous shear stress, an active chiral driving stress and a Hall stress allowed by the lack of time-reversal symmetry. The droplet interacts with itself through its driven internal mechanics and by driving flows in the underlying 3-D Stokes phase. We pose the dynamics as the solution to a singular integral–differential equation, over the droplet surface, using the mapping from surface stress to surface velocity for the 3-D Stokes equations. Specializing to the case of axisymmetric droplets, exact representations for the chiral surface flow are given in terms of solutions to a singular integral equation, solved using both analytical and numerical techniques. For a disc-shaped monolayer, we additionally employ a semi-analytical solution that hinges on an orthogonal basis of Bessel functions and allows for efficient computation of the monolayer velocity field, which ranges from a nearly solid-body rotation to a unidirectional edge current, depending on the subphase depth and the Saffman–Delbrück length. Except in the near-wall limit, these solutions have divergent surface shear stresses at droplet boundaries, a signature of systems with codimension-one domains embedded in a 3-D medium. We further investigate the effect of a Hall viscosity, which couples radial and transverse surface velocity components, on the dynamics of a closing cavity. Hall stresses are seen to drive inward radial motion, even in the absence of edge tension. 

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