Noncanonical Shape Research Articles

Inverse Equivalent-Source Techniques for Lossless and Passive Metasurfaces With Noncanonical Shapes and Geometries

Inverse Equivalent-Source Techniques for Lossless and Passive Metasurfaces With Noncanonical Shapes and Geometries

Temperature Field Around Nearly Spherical Cavities in Uniform Heat Flow

Abstract The problem of determining temperature distribution near a spherical cavity in an otherwise infinite medium which is under uniform heat flow is a classical problem of linear heat conduction theory. Extensive reviews of relevant work can be found in [1-5]. It is clear from these studies is that what has been achieved in this regard is largely related to cavities with highly canonical shapes, mostly spherical. Solutions of similar problems for cavities with non-canonical shapes are rare. In the present article, we consider the case of a cavity in an infinite solid in a uniform heat flow, whose shape deviates slightly from a perfectly spherical shape (hereafter called nearly spherical cavity). To the first order in the small parameter characterizing the boundary perturbation, we are able to derive closed-form expressions for the temperature field around the nearly spherical cavity for sufficiently smooth boundary perturbations that are arbitrary functions of the azimuthal and polar angles.

Surface parameterization complex geometry

Among thin-walled structures, including building structures and constructions, shells of complex geometry are effective in their rigidity and strength characteristics, which are also distinguished by architectural harmony. For a wider application of shells of complex geometry, it is necessary to reliably assess their stress-strain state. In this case, an integral part of the calculation is the parametrization stage of the median surface of shells of complex geometry. There are shells of complex geometry of canonical and non-canonical forms. For shells of non-canonical shape, the median surface cannot be defined by analytical formulas. At the same time, difficulties arise at the stage of specifying (parameterizing) the shape of the median surface. The task becomes more complicated when the shell fragment has a complex contour and one or more surface points have fixed coordinates. For building structures, this is, for example, the presence of additional internal supports. Information about the spline version of the FEM is presented. Some well-known parametrization methods are noted. The approach of parametrization of a minimal surface of a complex shape bounded by four curved contours and a given (fixed) coordinate of one inner point of the surface is considered. An algorithm for constructing a spatial network, as well as determining coordinates, metric tensor components and Christoffel symbols necessary for solving parametrization problems in the spline version of the finite element method is described.

Dielectric, magnetic and electromagnetic shielding properties of Poly-(3,4ethylenedioxythiophene)-maghnite associated with different fillers with any non-canonical shape

A new inverse measuring technique that corrects non-desired sample displacements and can handle any kind of sample shapes is presented. It is applied to the estimation of permittivity and permeability of Poly-(3,4ethylenedioxythiophene)-maghnite (PEDOT-Mag) associated with different additives such as iron, copper, and hydrogen. This electromagnetic characterization has been performed over the 9 to 11 GHz frequency range by comparing the simulated and measured scattering parameters of a partially filled WR-90 waveguide. Additionally, the reflection, absorption multiple reflection losses and the shielding effectiveness of these materials have been calculated. Results show higher values of dielectric constant and loss factor for PEDOT-Mag-copper, than those compounds associated with hydrogen or iron. As expected, the highest permeability values are achieved by PEDOT-Mag-iron. The composite PEDOT-Mag-copper exhibits higher attenuation constant, absorption loss, multiple reflection loss and shielding efficiency values than composites associated with iron or hydrogen. PEDOT-Mag-hydrogen, however, shows the highest reflection loss values.

The calculation of the thermal stressed state of multilayer platesofa non-canonical shape

A method for solving the problem of stationary thermoelasticity of multilayer plates of non-canonical shape in plan is proposed, which is based on the method of immersing a complex domain into the domain of a canonical form. A non-canonical plate with arbitrary boundary conditions "immerses" into a canonical plate. To ensure that the specified boundary conditions are met, additional compensating loads are added to the auxiliary structure, distributed along the contour of the initial structure. The intensities of compensating loads are determined from a system of integral equations. Deformation of the layers of the plates are described within the framework of the first-order theory, taking into account the transverse shear strains in each layer. The field of temperature loads is obtained by solving the non-stationary problem of thermal conductivity of a multilayer plate. The temperature stresses in a five-layer plate when heated by a film heat source are investigated.

Dielectric Resonator Antenna Geometry-Dependent Performance Tradeoffs

A flexible shape generation technique (Surface Contour Shape Generation) is combined with multiobjective evolutionary optimization algorithms in order to synthesize geometrically-diverse dielectric resonator antennas and explore the impact geometry has on relevant performance metrics. It is shown that noncanonical shapes can improve gain and bandwidth when compared to conventional dielectric resonator antenna geometries. Previously described free-form techniques (such as the superformula) have demonstrated improved performance over canonical shapes, but are difficult to integrate with inverse-design. In contrast, the proposed shape generation technique is coupled with multiobjective optimization in order to illustrate the solution space tradeoffs between the various design goals. This technical approach is compatible with various feed designs, antenna performance parameters, and 3D printing techniques.

Quantum theory of radiative decay rate and frequency shift of surface plasmon modes

In this paper we study, in the time domain, the interaction between localized surface plasmons and photons in arbitrarily shaped metal nanoparticles, by using the Hopfield approach to quantize the plasmon modes, where the electron oscillations are represented by a harmonic matter field linearly coupled to the electromagnetic radiation. The plasmon - photon coupling gives rise to dressed plasmon modes. We have found that the radiation does not induce a significant coupling among the different quasi-electrostatic plasmon modes for particles of size up to the plasma wavelength, but causes a frequency shift and an exponential decay in time of the modes. By solving the equations governing the expectation values of the plasmon creation and annihilation operators, we obtain a new closed-form full-wave expression for the decay rate and for the frequency shift of the plasmon modes. It is non-perturbative and it only depends on the surface charge distribution of the quasi-electrostatic plasmon modes. We validate the expression against the Mie theory for a nano-sphere of radius comparable to the plasma wavelength. Eventually, we investigate the decay rate and the frequency shift of the plasmon modes in isolated and interacting nanoparticle of non-canonical shape, as their size increases up to the plasma wavelength.

On the spreading of non-canonical thermals from direct numerical simulations

We present results from direct numerical simulations on laminar and turbulent non-canonical thermals with an initial rectangular density distribution at a Reynolds number of Re = 500 and Re = 5000, respectively. We find the non-canonical shape to induce strong azimuthal variations in the thermal for both the laminar and turbulent cases. These include noticeable differences in downward and horizontal propagation speeds as well as differences in the strength of the vortex tube. These differences persist over a significant period of time and help generate a cross-flow component that is otherwise not present in canonical cases. The cross-flow component is in the opposite direction to that observed in gravity currents with the same initial density distribution. This is counterintuitive seeing that both flows are solely driven by buoyancy. By extracting the three-dimensional streamlines, we find the descending vortex tube to force the dense fluid to follow a helical path.

Bending Vibrations of Bimorph Piezoceramic Plates of Noncanonical Shape

The superposition method is used to develop an effective method for analytical solution of the problem of the harmonic bending vibrations of parallelogram-shaped bimorph piezoceramic plates. The reduction of infinite series allows deriving a finite-dimensional system of algebraic equations by satisfying given boundary conditions by minimizing the standard deviation and using the collocation method.

Interface integral technique for the thermoelasticity of random structure matrix composites

We consider linear thermoelastic composite media, which consist of a homogeneous matrix containing a statistically homogeneous random set of aligned homogeneous heterogeneities of non-canonical (i.e. non-ellipsoidal) shape. The representations of the effective properties (effective moduli, thermal expansion, and stored energy) are expressed through the statistical averages of the interface polarization tensors introduced apparently for the first time. The properties of the interface polarization tensors are described. The new general integral equations connecting the stress and strain fields in the point being considered with the stress and strain fields in the surrounding points are obtained for the random fields of heterogeneities. The method is based on a recently developed centering procedure where the notion of a perturbator is introduced in terms of boundary interface integrals estimated by the method of fundamental solution for a single inclusion inside the infinite matrix. This enables us to reconsider basic concepts of micromechanics such as effective field hypothesis, quasi-crystalline approximation, and the hypothesis of “ellipsoidal symmetry.” Effective properties (such as effective moduli, thermal expansion, and stored energy) as well as the first statistical moments of stresses in the phases are estimated for statistically homogeneous composites with the general case of the inclusion shape. The results of this reconsideration are quantitatively estimated for some modeled statistically homogeneous composites reinforced by aligned homogeneous heterogeneities of non-canonical shape. The explicit new representations of the effective thermoelastic properties and stress concentration factor are expressed through some building blocks ( perturbators) described by numerical solutions for one heterogeneity inside the infinite medium subjected to the homogeneous remote loading. Some new effects are detected that are impossible in the framework of a classical background of micromechanics.

Effects of geometric toy design on parent–child interactions and spatial language

Geometric forms have formal definitions. While knowing shape names is considered important for school-readiness, many children do not understand the defining features of shapes until well into elementary school (Satlow & Newcombe, 1998). One reason is likely that they do not encounter enough variety in the shapes they see (Resnick, Verdine, Golinkoff, & Hirsh-Pasek, 2016). The present study observed 60 parents and their 3-year-old children during play with geometric toys, exploring how spatial language varied with the nature of the shape-toy set (canonical shapes versus a mix of canonical and unusual or less-canonical variants) and whether geometric shapes were presented as tangible, traditional toys or shown on a touchscreen tablet app. Although children in the app condition heard more shape names than the other conditions due to the language produced by the app itself, children used more overall words and more spatial language with tangible toys that included varied shapes. In addition, parents used more shape names with sons than with daughters and tended to adjust their use of spatial language more in response to varied shape sets with boys, although these findings need replication to evaluate generality. These data suggest that including non-canonical shapes in tangible shape toys may provide a low-cost, high-impact way of refining adult-child interactions that might facilitate children’s early geometric knowledge.

Effective field hypothesis in Hashin–Shtrikman bounds estimations for effective moduli of composites with noncanonical shape of inclusions

One consider linear elastic composite materials (CMs), which consist of a homogeneous matrix containing a statistically homogeneous random set of identical aligned inhomogeneous inclusions of non-canonical (i.e. nonellipsoidal) shape. Estimates of the Hashin-Shtrikman type are developed by extremizing of the classical variational functional involving a general integral equation (GIE) connecting the strain fields in the point being considered with the strain fields in the surrounding points. An improvement of the approach proposed is based on the use of a piece wise constant approximation of the trial effective field rather than the trial polarisation tensor that, in its turn, allows one to take into account any inhomogeneity of the strain distributions inside the inclusions. Due to a sensitivity of the proposed approach to the inhomogeneity of these strain distributions, a final Hashin-Shtrikman bounds representation demonstrates more significant dependence on the shape of noncanonical inclusions than the corresponding classical bounds (that is shown by the concrete numerical examples).

Method of fundamental solutions in micromechanics of elastic random structure composites

One considers linearly elastic composite media, which consist of a homogeneous matrix containing a statistically homogeneous random set of aligned homogeneous heterogeneities of non-canonical (i.e. nonellipsoidal) shape. The new general integral equations connecting the stress and strain fields in the point being considered with the stress and strain fields in the surrounding points are obtained for the random fields of heterogeneities. The method is based on a recently developed centering procedure where the notion of a perturbator is introduced in terms of boundary interface integrals that makes it possible to reconsider basic concepts of micromechanics such as effective field hypothesis, quasi-crystalline approximation, and the hypothesis of ellipsoidal symmetry. The results of this reconsideration are quantitatively estimated for some modeled composite reinforced by aligned homogeneous heterogeneities of non canonical shape. Some new effects are detected that are impossible in the framework of a classical background of micromechanics.

К оценке критического давления для замкнутой, строго выпуклой оболочки неканонической формы

К оценке критического давления для замкнутой, строго выпуклой оболочки неканонической формы

Estimation of effective elastic moduli of random structure composites by the method of fundamental solutions

One considers linearly elastic composite media, which consist of a homogeneous matrix containing a statistically homogeneous random set of aligned homogeneous heterogeneities of non-canonical shape. Effective elastic moduli as well as the first statistical moments of stresses in the phases are estimated through the averaged boundary integrals over the inclusion boundaries. The modified popular micromechanical models are based on the numerical solution for one inhomogeneity inside the infinite matrix loaded by remote homogeneous effective field. This solution is obtained by a meshfree method based on fundamental solutions basis functions for a transmission problem in linear elasticity. The problem here addressed, consists in computing the displacement and traction fields of an elastic object, which has piecewise constant Lamé coefficients, from a given displacement (or stress) field on the infinity. The main properties of the method are analyzed and illustrated with several numerical simulations in 2D domains.

The Shape of Things: The Origin of Young Children’s Knowledge of the Names and Properties of Geometric Forms

How do toddlers learn the names of geometric forms? Previous work suggests that preschoolers have fragmentary knowledge and that defining properties are not understood until well into elementary school. The current study investigated when children first begin to understand shape names and how they apply those labels to unusual instances. We tested 25- and 30-month-old children’s (N = 30 each) understanding of names for canonical shapes (commonly encountered instances, e.g., equilateral triangles), noncanonical shapes (more irregular instances, e.g., scalene triangles), and embedded shapes (shapes within a larger picture, e.g., triangular slices of pizza). At 25 months, children knew very few names, including those for canonical shapes. By 30 months, however, children had acquired more shape names and were beginning to apply them to some of the less typical instances of the shapes. Possible mechanisms driving this initial development of shape knowledge and implications of that development for school readiness are explored.

The Green’s function method in the problem of sound diffraction by an elastic shell of noncanonical shape

Using the Green function method and dynamic elasticity theory, we obtain a solution to the problem of sound diffraction by elastic shells of noncanonical shapes formed by spheroidal, cylindrical, and spherical bodies. The angular scattering characteristics of such compound bodies with different wave sizes are calculated.

Light dynamics in materials with radially inhomogeneous thermal conductivity

We study the properties of bright and vortex solitons in thermal media with nonuniform thermal conductivity and homogeneous refractive index, whereby the local modulation of the thermal conductivity strongly affects the entire refractive index distribution. While regions where the thermal conductivity is increased effectively expel light, self-trapping may occur in the regions with reduced thermal conductivity, even if such regions are located close to the material boundary. As a result, strongly asymmetric self-trapped beams may form inside a ring with reduced thermal conductivity and perform persistent rotary motion. Also, such rings are shown to support stable vortex solitons, which may feature strongly noncanonical shapes.

Iteration method in linear elasticity of random structure composites containing heterogeneities of noncanonical shape

We consider a linearly elastic composite medium, which consists of a homogeneous matrix containing a statistically inhomogeneous random set of heterogeneities of arbitrary shape. The general integral equations connecting the stress and strain fields in the point being considered with the stress and strain fields in the surrounding points are obtained for the random fields of heterogeneities. The method is based on a recently developed centering procedure where the notion of a perturbator is introduced and statistical averages are obtained without any auxiliary assumptions such as, e.g., effective field hypothesis implicitly exploited in the known centering methods. Effective elastic moduli and the first statistical moments of stresses in the heterogeneities are estimated for statistically homogeneous composites with the general case of both the shape and inhomogeneity of the heterogeneities moduli. The explicit new representations of the effective moduli and stress concentration factors are built by the iteration method in the framework of the quasicristallite approximation but without basic hypotheses of classical micromechanics such as both the EFH and “ellipsoidal symmetry” assumption. Numerical results are obtained for some model statistically homogeneous composites reinforced by aligned identical homogeneous heterogeneities of noncanonical shape. Some new effects are detected that are impossible in the framework of a classical background of micromechanics.

Thermoelastic effective properties and stress concentrator factors of composites reinforced by heterogeneities of noncanonical shape

We consider a linearly thermoelastic composite medium, which consists of a homogeneous matrix containing a statistically inhomogeneous random set of heterogeneities of arbitrary shape. The general integral equations connecting the stress and strain fields in the point being considered with the stress and strain fields in the surrounding points are obtained for the random fields of heterogeneities. The method is based on a recently developed centering procedure (Buryachenko, 2010a) where the notion of perturbator is introduced and statistical averages are obtained without any auxiliary assumptions such as, e.g., effective field hypothesis implicitly exploited in the known centering methods. Effective properties (such as compliance and thermal expansion) as well as the first statistical moments of stresses in the phases are estimated for statistically homogeneous composites with the general case of both the shape and inhomogeneity of the thermoelastic heterogeneities properties. The explicit new representations of the effective thermoelastic properties and stress concentration factor are expressed through some building blocks described by numerical solutions for one heterogeneity inside the infinite medium subjected to the homogeneous remote loading. Numerical results are obtained for some model statistically homogeneous composites reinforced by aligned identical homogeneous heterogeneities of noncanonical shape. Some new effects are detected that are impossible in the framework of a classical background of micromechanics.