Planar Case Research Articles

Flexible and conformable broadband metamaterial absorber with wide-angle and polarization stability for radar application

In this work, a type of flexible, broadband electromagnetic microwave absorber is designed, fabricated and experimentally characterized. The absorber is composed of lumped resistors loaded frequency selective surface which is mounted on flexible substrate using silicone rubber and in turn backed by copper film. The simulated results show that an effective absorption (over 90%) bandwidth spans from 7.6 to 18.3 GHz, which covers both X (8–12 GHz) and Ku (12–18 GHz) bands, namely a 82.6% fraction bandwidth. And the bandwidth performs a good absorption response by varying the incident angle up to 60° for both TE and TM polarization. Moreover, the flexibility of the substrate enables the absorber conformably to bend and attach to cylinders of various radius without breakdown of the absorber. The designed structure has been fabricated and measured for both planar and conformable cases, and absorption responses show a good agreement of the broadband absorption feature with the simulated ones. This work has demonstrated specifically that proposed structure provides polarization-insensitive, wide-angle, flexible and conformable wideband absorption, which extends the absorber’s application to practical radar cross section reductions for radars and warships.

The Effect of Drying around Power Cables on the Vadose Zone Temperature

Core Ideas We show the impact of soil hydraulic properties on its critical temperature. Ambient conditions have a larger impact in some fields than local soil parameters. We introduce a parameter for optimizing cable backfill based on hydraulic properties. The effects of moisture migration away from power cables depend strongly in most soils on whether the soil has reached the so‐called critical conditions (e.g., critical temperature) for a dry‐out. However, research on how to estimate this critical temperature has been limited. We solved numerically a differential equation for the planar case of steady‐state coupled heat and mass transfer in porous media, using the van Genuchten–Mualem model for the hydraulic properties of the medium. The critical isotherm is assessed by the temperature at which the steepest change occurs in the moisture content with temperature. We considered the impact of the properties of the surrounding medium and of the ambient temperature and moisture content on the dry‐out temperature. We found that seasonal variations in ambient conditions in a field in Italy have a greater impact on the critical temperature than the variations in soil properties in that field (under the assumption of a nearly uniform moisture content with depth). However, our analysis shows that, under certain conditions, a change in the van Genuchten parameter n and in a specific combination of parameters that we call a dynamic parameter can have as large an effect on the critical temperature. This suggests a direction in the optimization of backfill materials.

On the global solvability of the coupled kinetic-fluid system for flocking with large initial data

We study the dynamics of infinitely many Cucker–Smale (CS) flocking particles under the interplay of random communication and compressible fluids in planar wave case. For the dynamics of an ensemble of flocking particles, we use the kinetic Cucker–Smale–Fokker–Planck (CS–FP) equation with a degenerate diffusion, whereas for the fluid component, we use the compressible Navier–Stokes (NS) equations. These two subsystems are coupled via the drag force. For this coupled model, we present a global existence of classical solutions for arbitrarily large initial data which may contain vacuum.

Annular rearrangements, incompressible axi-symmetric whirls and $$L^1$$-local minimisers of the distortion energy

In this paper we consider a variational problem consisting of an energy functional defined by the integral, $$\begin{aligned} \mathbb {F}[u,\mathbf{X}] = \frac{1}{2}\int _{\mathbf{X}} \frac{|\nabla u|^2}{|u|^2} \,dx, \end{aligned}$$and an associated mapping space, here, the space of incompressible Sobolev mappings of the symmetric annular domain in the Euclidean n-space \(\mathbf{X}= \lbrace x \in {\mathbb {R}}^n{:}\,a<|x|<b \rbrace \): $$\begin{aligned} \mathcal {A}_{\phi }(\mathbf{X}) = \bigg \lbrace u \in W^{1,2}(\mathbf{X},{\mathbb {R}}^n){:}\,\det \nabla u = 1 \,\, a.e. \text { and } u |_{_{\partial \mathbf{X}}} \equiv x \bigg \rbrace . \end{aligned}$$The goal is then twofold. Firstly to establish and highlight an unexpected difference in symmetries of the critical points and local minimisers of \({\mathbb {F}}\) over \({\mathcal {A}}_\phi (\mathbf{X})\) in the two special cases \(n=2\) and \(n=3\). More specifically, that when \(n=3\), despite the inherent rotational symmetry in the problem, there are NO non-trivial rotationally symmetric critical points of \({\mathbb {F}}\) over \({\mathcal {A}}_\phi (\mathbf{X})\), whereas in sharp contrast, when \(n=2\), not only that there is an infinitude of rotationally symmetric critical points of the energy but also there is an infinitude of local minimisers of \({\mathbb {F}}\) over \({\mathcal {A}}_\phi (\mathbf{X})\) with respect to the \(L^1\)-metric. At the heart of this analysis is an investigation into the rich homotopy structure of the space of self-mappings of annuli. The second aim is to introduce and implement a novel symmetrisation technique in the planar case \(n=2\) for Sobolev mappings u in \(\mathcal {A}_\phi (\mathbf{X})\) that lowers the energy whilst keeping the homotopy class of u invariant. We finally generalise and extend some of these results to higher dimensions, in particular, we show that only in even dimensions do we have an infinitude of non-trivial rotationally symmetric critical points.

Fabrication of monolithic diamond photodetector with microlenses.

A monolithic diamond photodetector with microlenses is fabricated by etching microlens arrays (MLAs) on single crystal diamond surface and patterning tungsten electrode strips on the edge of these arrays. Firstly, compact MLAs are etched on half of diamond sample surface by thermal reflow method. Secondly, via magnetron sputtering technique, two sets of interdigitated tungsten electrodes are patterned on the sample surface, one set is on the edge of MLAs, the other set is on the planar area. The optoelectronic performances of photodetectors have been investigated and indicated that the photocurrent of microlens photodetector increases by 74.8 percent at 10 V under 220 nm UV light illumination by comparing with that in planar case. Simulations of photodetectors' electrical and optical properties have been carried out, illustrating an improvement of charge collection ability and light absorption efficiency in microlens case. Furthermore, the present device structure can be extended to other semiconductor photodetectors.

An accurate singularity-free and locking-free formulation of a three-dimensional shear-deformable beam using Euler parameters

An accurate singularity-free and locking-free formulation of a three-dimensional shear-deformable beam with large deformations and large rotations is developed. The position of the centroid line of the beam is integrated from its slope that is related to the rotation of a corresponding cross-section and stretch and shear strains. The rotation is parametrized by Euler parameters, which can naturally avoid the singularity problem that Euler angles or the rotation vector can have. While stretch and shear strains are interpolated using polynomial functions, Euler parameters are interpolated using a C1-continuous interpolation function to guarantee that curvatures of the centroid line of the beam are continuous at nodes of its elements. Governing equations of the beam are obtained using Lagrange’s equations for systems with constraints, and several examples are simulated to show the performance of the current formulation. Results show that the current formulation does not suffer from shear locking and Poisson locking. Results from the current formulation for a planar static case are compared with its exact solutions, and they are in excellent agreement, which verifies accuracy of the current formulation. Results from the current formulation for some other cases are also in excellent agreement with those from commercial software ABAQUS and ADAMS.

Quadrature Domains for the Bergman Space in Several Complex Variables

We make use of the Bergman kernel function to study quadrature domains whose quadrature identities hold for \(L^2\) holomorphic functions of several complex variables. We generalize some mapping properties of planar quadrature domains and point out some differences from the planar case. We then show that every smooth bounded convex domain in \({\mathbb {C}}^n\) is biholomorphic to a quadrature domain. Finally, the possibility of continuous deformations within the class of planar quadrature domains is examined.

Dual conformal symmetry, integration-by-parts reduction, differential equations, and the nonplanar sector

We show that dual conformal symmetry, mainly studied in planar $\mathcal N = 4$ super-Yang-Mills theory, has interesting consequences for Feynman integrals in nonsupersymmetric theories such as QCD, including the nonplanar sector. A simple observation is that dual conformal transformations preserve unitarity cut conditions for any planar integrals, including those without dual conformal symmetry. Such transformations generate differential equations without raised propagator powers, often with the right hand side of the system proportional to the dimensional regularization parameter $\epsilon$. A nontrivial subgroup of dual conformal transformations, which leaves all external momenta invariant, generates integration-by-parts relations without raised propagator powers, reproducing, in a simpler form, previous results from computational algebraic geometry for several examples with up to two loops and five legs. By opening up the two-loop three- and four-point nonplanar diagrams into planar ones, we find a nonplanar analog of dual conformal symmetry. As for the planar case this is used to generate integration-by-parts relations and differential equations. This implies that the symmetry is tied to the analytic properties of the nonplanar sector of the two-loop four-point amplitude of $\mathcal N = 4$ super-Yang-Mills theory.

Exceptional times for percolation under exclusion dynamics

We analyse in this paper a conservative analogue of the celebrated model of dynamical percolation introduced by H\aggstr\om, Peres and Steif in [HPS97]. It is simply defined as follows: start with an initial percolation configuration $\omega(t=0)$. Let this configuration evolve in time according to a simple exclusion process with symmetric kernel $K(x,y)$. We start with a general investigation (following [HPS97]) of this dynamical process $t \mapsto \omega_K(t)$ which we call $K$-exclusion dynamical percolation. We then proceed with a detailed analysis of the planar case at the critical point (both for the triangular grid and the square lattice $Z^2$) where we consider the power-law kernels $K^\alpha$ \[ K^{\alpha}(x,y) \propto \frac 1 {\|x-y\|_2^{2+\alpha}} \, . \] We prove that if $\alpha > 0$ is chosen small enough, there exist exceptional times $t$ for which an infinite cluster appears in $\omega_{K^{\alpha}}(t)$. (On the triangular grid, we prove that it holds for all $\alpha < \alpha_0 = \frac {217}{816}$.) The existence of such exceptional times for standard i.i.d. dynamical percolation (where sites evolve according to independent Poisson point processes) goes back to the work by Schramm-Steif in [SS10]. In order to handle such a $K$-exclusion dynamics, we push further the spectral analysis of exclusion noise sensitivity which had been initiated in [BGS13]. (The latter paper can be viewed as a conservative analogue of the seminal paper by Benjamini-Kalai-Schramm [BKS99] on i.i.d. noise sensitivity.) The case of a nearest-neighbour simple exclusion process, corresponding to the limiting case $\alpha = +\infty$, is left widely open.

Singularities at the contact point of two kissing Neumann balls

We investigate eigenfunctions of the Neumann Laplacian in a bounded domain Ω⊂Rd, where a cuspidal singularity is caused by a cavity consisting of two touching balls, or discs in the planar case. We prove that the eigenfunctions with all of their derivatives are bounded in Ω‾, if the dimension d equals 2, but in dimension d≥3 their gradients have a strong singularity O(|x−O|−α), α∈(0,2−2] at the point of tangency O. Our study is based on dimension reduction and other asymptotic procedures, as well as the Kondratiev theory applied to the limit differential equation in the punctured hyperplane Rd−1∖O. We also discuss other shapes producing thinning gaps between touching cavities.

An investigation on the effect of pitchwise endwall design in a turbine cascade at different incidence angles

This paper describes the effects of non-axisymmetric endwall profiling on the aerodynamic performance of a linear turbine cascade at different incidence angles. The sinusoidal profiling is carried out with constant profile curvature along the mean streamline path. Three different profiles, with varying hump to dip height, are analyzed numerically and the performances are compared with the planar profile. Reynolds Averaged Navier Stokes (RANS) equations are solved in their conservative form using Finite Volume Method with SST turbulence model. The calculated results indicate that the profiled endwall minimizes the lateral movement of weaker boundary layer fluid from the hub-pressure side corner. In comparison with planar case, the flow deviations are largely contained with endwall profiling but closer to the endwall it enhances the overturning and secondary flow kinetic energy. The reduction in loss coefficient is estimated to be 1.3%, 8.7% and 38% for incidence angles of −10°, nominal and +15° respectively. The sinusoidal profiling has brought down the pitch averaged flow deviation and secondary flow kinetic energy at nominal and positive incidence angles but the impact is insignificant at negative incidence. Profiling minimizes the rolling up of the passage vortex and makes the passage vortex to migrate closer to the endwall. This flow modification brings down the losses in the core flow but enhances the losses near the endwall.

Uniform surface‐to‐line integral reduction of physical optics for curved surfaces by modified edge representation with higher‐order correction

AbstractThe modified edge representation is one of the equivalent edge currents approximation methods for calculating the physical optics surface radiation integrals in diffraction analysis. The Stokes' theorem is used in the derivation of the modified edge representation from the physical optics for the planar scatterer case, which implies that the surface integral is rigorously reduced into the line integral of the modified edge representation equivalent edge currents, defined in terms of the local shape of the edge. On the contrary, for curved surfaces, the results of radiation integrals depend upon the global shape of the scatterer. The physical optics surface integral consists of two components, from the inner stationary phase point and the edge. The modified edge representation is defined independently from the orientation of the actual edge, and therefore, it could be available not only at the edge but also at the arbitrary points on the scatterer except the stationary phase point where the modified edge representation equivalent edge currents becomes infinite. If stationary phase point exists inside the illuminated region, the physical optics surface integration is reduced into two kinds of the modified edge representation line integrations, along the edge and infinitesimally small integration around the inner stationary phase point, the former and the latter give the diffraction and reflection components, respectively. The accuracy of the latter has been discussed for the curved surfaces and published. This paper focuses on the errors of the former and discusses its correction. It has been numerically observed that the modified edge representation works well for the physical optics diffraction in flat and concave surfaces; errors appear especially for the observer near the reflection shadow boundary if the frequency is low for the convex scatterer. This paper gives the explicit expression of the higher‐order correction for the modified edge representation.

Ground states and high energy solutions of the planar Schrödinger–Poisson system

In this paper, we are concerned with the Schrödinger–Poisson system0.1{−Δu+u+ϕu=|u|p−2uin Rd,Δϕ=u2in Rd.Due to its relevance in physics, the system has been extensively studied and is quite well understood in the case . In contrast, much less information is available in the planar case which is the focus of the present paper. It has been observed by Cingolani S and Weth T (2016 On the planar Schrödinger–Poisson system Ann. Inst. Henri Poincare 33 169–97) that the variational structure of (0.1) differs substantially in the case and leads to a richer structure of the set of solutions. However, the variational approach of Cingolani S and Weth T (2016 On the planar Schrödinger–Poisson system Ann. Inst. Henri Poincare 33 169–97) is restricted to the case which excludes some physically relevant exponents. In the present paper, we remove this unpleasant restriction and explore the more complicated underlying functional geometry in the case with a different variational approach.

A case of multivariate Birkhoff interpolation using high order derivatives

We consider a specific scheme of multivariate Birkhoff polynomial interpolation. Our samples are derivatives of various orders kj at fixed points vj along fixed straight lines through vj in directions uj, under the following assumption: the total number of sampled derivatives of order k,k=0,1,… is equal to the dimension of the space homogeneous polynomials of degree k. We show that this scheme is regular for general directions. Specifically this scheme is regular independent of the position of the interpolation nodes. In the planar case, we show that this scheme is regular for distinct directions.Next we prove a “Birkhoff-Remez” inequality for our sampling scheme extended to larger sampling sets. It bounds the norm of the interpolation polynomial through the norm of the samples, in terms of the geometry of the sampling set.

High-Order-Harmonic Generation from a Relativistic Circularly Polarized Laser Interacting with Over-Dense Plasma Grating**Supported by the National Natural Science Foundation of China under Grant Nos 11375265, 11475259 and 11675264, the National Basic Research Program of China under Grant No 2013CBA01504, and the Science Challenge Project under Grant No JCKY2016212A505.

The simple surface current model is extended to study the generation of high-order harmonics for a relativistic circularly polarized laser pulse interacting with a plasma grating surface. Both exact relativistic electron dynamics and optical interference of surface periodic structure are considered. It is found that high order harmonics in the specular direction are obviously suppressed whereas the harmonics of the grating periodicity are strongly enhanced and folded into small solid angles with respect to the surface direction. The conversion efficiency of certain harmonics is five orders of magnitude higher than that of the planar target cases. It provides an effective approach to generate a coherent radiation within the so-called ‘water window’ while maintaining high conversion efficiency and narrow angle spread.

Useful Physical Images and Algorithms for Vector Dyadic Green's Functions [Wireless Corner

This article gathers useful, simple algorithms and their physical interpretations for field solutions from incremental sources in three-dimensional (3-D) spatial, two-dimensional (2-D) spectral, and one-dimensional (1-D) spectral domains. The interpretations of the 1-D spectral Green's functions are visualized in space as fields from current sheets, tubes, and shells for the planar, circular cylindrical, and spherical cases, respectively. A joint algorithm is presented for solving the multilayer case for all three cases. Similarly, field problems involving cylindrical objects or bodies of revolution (BOR) are structured into spectrums of 2-D spatial solutions from line sources and ring sources, respectively. The formulations and physical images are pedagogical and open up for new creative ways of teaching electromagnetic (EM) field theory as well as structuring numerical algorithms for field solutions that take known symmetries into account. It is also shown that the 3-D spatial Green's functions can be approximated to improve physical interpretation by omitting higher-order 1/r terms when r &#x0003E; 2&#x03BB;.

On the isotopy problem for quasiconformal mappings

The question of the isotopy of a quasiconformal mapping and its special aspects in dimension greater than 2 are considered. It is shown that an arbitrary quasiconformal mapping of a ball has an isotopy to the identity map such that the coefficient of quasiconformality (dilatation) of the mapping varies continuously and monotonically. In contrast to the planar case, in dimension higher than 2 such an isotopy is not possible in an arbitrary domain. Examples showing specific features of the multidimensional case are given. In particular, they show that even when such an isotopy exists, it is not always possible to perform an isotopy so that the coefficient of quasiconformality approaches 1 monotonically at each point in the source domain.

On a bifurcation problem arising in cholesteric liquid crystal theory

In a cholesteric liquid crystal the director field $n(x,y,z)$ tends to form a right-angle helicoid around a twist axis in order to minimize the internal energy; however, a fixed alignment of the director field at the boundary (strong anchoring) can give rise to distorted configurations of the director field, as oblique helicoid, in order to save energy. The transition to this distorted configurations depend on the boundary conditions and on the geometry of the liquid crystal, and it is known as Freedericksz transition (without external fields). We consider the classical situation of a thin layer between two glass sheet assuming the Oseen-Frank model for the energy, and that the director field depend only on the direction $z$ orthogonal to the layer; then we focus on two kinds of boundary conditions: the planar case and the orthogonal case. In the first, we impose that $n(0)=(1,0,0)$, $n(d)=(\cos\alpha,\sin\alpha,0)$ (where $z=0$ and $z=d>0$ are, respectively, the bottom and the top of the layer), and search for the couples $(d,\alpha)$ such that oblique helicoid appear. In the case $K_1>0$, $K_2=K_3=1$ for the elastic constants of the Oseen-Frank energy, we completely characterize these couples. In the second case it is a classical result that oblique helicoid bifurcates from the trivial solution $n(z)=(0,0,1)$ for suitable values of $d$; then we study the exact number of these nontrivial solutions and their stability.

On a Liouville-type theorem for the Ginzburg–Landau system

We prove that entire, complex valued solutions to the Ginzburg–Landau system with positive real and imaginary parts are constant in any spatial dimension. This property was shown very recently only in the planar case.

Genealogy and stability of periodic orbit families around uniformly rotating asteroids

Resonance orbits around a uniformly rotating asteroid are studied from the approach of periodic orbits in this work. Three periodic families (denoted as I, II, and III in the paper) are fundamental in organizing the resonance families. For the planar case: (1) Genealogy and stability of Families I, II and the prograde resonance families are studied. For extremely irregular asteroids, family genealogy close to the asteroid is greatly distorted from that of the two body-problem (2BP), indicating that it is inappropriate to treat the orbital motions as perturbed Keplerian orbits. (2) Genealogy and stability of Family III are also studied. Stability of this family may be destroyed by the secular resonance between the orbital ascending node’s precession and the asteroid’s rotation. For the spatial case: (1) Genealogy of the near circular three-dimensional periodic families are studied. The genealogy may be broken apart by families of eccentric frozen orbits whose argument of perigee is “frozen” in space. (2) The joint effects between the secular resonance and the orbital resonances may cause instability to three-dimensional orbital motion with orbit inclinations close to the critical values. Applying the general methodology to a case study — the asteroid Eros and also considering higher order non-spherical terms, some extraordinary orbits are found, such as the ones with orbital plane co-rotating with the asteroid, and the stable frozen orbits with argument of perigee librating around values different from 0°, 90°, 180°, 270°.