Parabolic Boundaries Research Articles

Scalar-potential mapping of the steady-state magnetosheath model

Abstract. The steady-state magnetosheath model has various applications for studying the plasma and magnetic field profile around the planetary magnetospheres. In particular, the magnetosheath model is analytically obtained by solving the Laplace equation for parabolic boundaries (bow shock and magnetopause). We address the question, how can we utilize the magnetosheath model by transforming into a more general, empirical, non-parabolic magnetosheath geometry? To achieve the goal, we develop the scalar-potential mapping method which provides a semi-analytic estimate of steady-state flow velocity and magnetic field in the empirical magnetosheath domain. The method makes use of a coordinate transformation from the empirical magnetosheath domain into the parabolic magnetosheath domain and evaluates a set of variables (shell variable and connector variable) to utilize the solutions of the Laplace equation obtained for the parabolic magnetosheath domain. Our model uses two invariants of transformation: the zenith angle within the magnetosheath with respect to the direction to the Sun and the ratio of the distance to the magnetopause and the thickness of magnetosheath along the magnetopause-normal direction. The use of magnetopause-normal direction makes a marked difference from the earlier model construction using the radial direction as reference. The plasma flow and magnetic field can be determined as a function of the upstream condition (flow velocity or magnetic field) in a wide range of zenith angles. The scalar-potential mapping method is computationally inexpensive, using analytic expressions as much as possible, and is applicable to various planetary magnetosheath domains.

Parabolic mirrors collimating and focusing fluxes of thermal phonons

Manipulating heat fluxes at the nanoscale has become increasingly important in modern microelectronics. However, many methods of heat manipulations rely on complex nanofabrication. Here, we propose simple designs for collimation and focusing of thermal phonons based on parabolic mirrors that require no nanofabrication. We perform Monte Carlo simulations of a ballistic phonon transport in silicon membranes with parabolic boundaries. Our simulations demonstrate that parabolic surfaces can act as parabolic mirrors for phonons, thus collimating or focusing phonon fluxes in semiconductors. Such parabolic mirrors can create a directional flux of thermal phonons emitted from a nanoscale hot spot or focus a collimated phonon flux into a hot spot. These devices open new possibilities in the thermal management of low-temperature systems, such as quantum circuits or cryogenic particle detectors.

Convergence of shock waves between conical and parabolic boundaries

Convergence of shock waves, generated by underwater electrical explosions of cylindrical wire arrays, between either parabolic or conical bounding walls is investigated. A high-current pulse with a peak of ∼550 kA and rise time of ∼300 ns was applied for the wire array explosion. Strong self-emission from an optical fiber placed at the origin of the implosion was used for estimating the time of flight of the shock wave. 2D hydrodynamic simulations coupled with the equations of state of water and copper showed that the pressure obtained in the vicinity of the implosion is ∼7 times higher in the case of parabolic walls. However, comparison with a spherical wire array explosion showed that the pressure in the implosion vicinity in that case is higher than the pressure in the current experiment with parabolic bounding walls because of strong shock wave reflections from the walls. It is shown that this drawback of the bounding walls can be significantly minimized by optimization of the wire array geometry.

Upper bound analysis of forging penetration in a radial forging process

A theoretical model based on upper bound approach is proposed to study forging penetration in a radial forging process. Using the transformations for both the geometric coordinate system and the velocity field, the parabolic discontinuity boundaries in axisymmetric problem are mapped into straight lines in the transformed field. As a result, the present model is simplified and its implementation is analogous to the rigid block upper bound approach in plane strain problem. In order to quantify the penetration of plastic deformation, forging penetration depth is introduced and defined as the radial distance from the outside surface of the workpiece to the intersection point of the assumed plastic and rigid regions. This model is verified by comparing the predicted forging load with published experimental data, and by comparing the predicted forging penetration depth with that of the finite element simulation. Finally the effects of process parameters such as the radial reduction rate and the inlet angle of hammer on the forging penetration are investigated.

Shock wave convergence in water with parabolic wall boundaries

The convergence of shock waves in water, where the cross section of the boundaries between which the shock wave propagates is either straight or parabolic, was studied. The shock wave was generated by underwater electrical explosions of planar Cu wire arrays using a high-current generator with a peak output current of ∼45 kA and rise time of ∼80 ns. The boundaries of the walls between which the shock wave propagates were symmetric along the z axis, which is defined by the direction of the exploding wires. It was shown that with walls having a parabolic cross section, the shock waves converge faster and the pressure in the vicinity of the line of convergence, calculated by two-dimensional hydrodynamic simulations coupled with the equations of state of water and copper, is also larger.

A family of stadium-like billiards with parabolic boundaries under scaling analysis

Some chaotic properties of a family of stadium-like billiards with parabolic focusing components, which is described by a two-dimensional nonlinear area-preserving map, are studied. Critical values of billiard geometric parameters corresponding to a sudden change of the maximal Lyapunov exponent are found. It is shown that the maximal Lyapunov exponent obtained for chaotic orbits of this family is scaling invariant with respect to the control parameters describing the geometry of the billiard. We also show that this behavior is observed for a generic one-parameter family of mapping with the nonlinearity given by a tangent function.

Hartree-Fock approximation of bipolaron state in quantum dots and wires

The bipolaronic ground state of two electrons in a spherical quantum dot or a quantum wire with parabolic boundaries is studied in the strong electron-phonon coupling regime. We introduce a variational wave function that can conveniently conform to represent alternative ground state configurations of the two electrons, namely, the bipolaronic bound state, the state of two individual polarons, and two nearby interacting polarons confined by the external potential. In the bipolaron state the electrons are found to be separated by a finite distance about a polaron size. We present the formation and stability criteria of bipolaronic phase in confined media. It is shown that the quantum dot confinement extends the domain of stability of the bipolaronic bound state of two electrons as compared to the bulk geometry, whereas the quantum wire geometry aggravates the formation of stable bipolarons.

3P2b-6 Imaging of surface acoustic wave reflection and focusing using circular, ellipsoidal and parabolic boundaries(Poster Session)

3P2b-6 Imaging of surface acoustic wave reflection and focusing using circular, ellipsoidal and parabolic boundaries(Poster Session)

Novel Current Error Space Phasor Based Hysteresis Controller Using Parabolic Bands for Control of Switching Frequency Variations

A current error space phasor based simple hysteresis controller is proposed in this paper to control the switching frequency variation in two-level pulsewidth-modulation (PWM) inverter-fed induction motor (IM) drives. A parabolic boundary for the current error space phasor is suggested for the first time to obtain the switching frequency spectrum for output voltage with hysteresis controller similar to the constant switching frequency voltage-controlled space vector PWM-based IM drive. A novel concept of online variation of this parabolic boundary, which depends on the operating speed of motor, is presented. A generalized technique that determines the set of unique parabolic boundaries for a two-level inverter feeding any given induction motor is described. The sector change logic is self-adaptive and is capable of taking the drive up to the six-step mode if needed. Steady-state and transient performance of proposed controller is experimentally verified on a 3.7-kW IM drive in the entire speed range. Close resemblance of the simulation and experimental results is shown

Earth Dam with Toe Drain on an Impervious Base

The required height of a toe drain for a homogeneous earth dam on an impervious base has been determined considering reservoir water level, capillary rise for the embankment soil, free board, top width of the earth dam, embankment slopes, and tail-water position, such that the surface of seepage does not develop on the downstream sloping face of the earth dam and capillary saturation above phreatic line is contained well within the downstream sloping face. Using Kozeny’s analytic function, exact solution to the unconfined flow through an earth dam having parabolic equipotential boundaries on either side has been obtained. For straight toe drain face, and for various positions of tailwater, approximate toe drain heights and heights of surface of seepage have been determined using the Kozeny’s function and the method of fragments. It has been found that for an earth dam with 1∕2 upstream slope, −1∕3 downstream slope, no tailwater, and 2 m capillary rise, capillary saturation is contained within the earth da...

“Gradual cut-off” method of the sequential procedure in the case of a Rayleigh distribution

The effectiveness of optimum statistical sequential criteria for signal detection on the background of noise is studied when the signal and noise have a Rayleigh distribution. A sequential criterion with parabolic boundaries of zones that is sometimes more economic is suggested. The properties of criteria are compared by a mathematical simulation method.

Inelastic Gravitational Billiards

We present an experimental investigation of gravitational billiards where the particle undergoes inelastic collisions with its boundary. The motion is mapped for an inelastic particle contained within parabolic, wedge, and hyperbolic boundaries. For the parabola, stable orbits are found and the wedge demonstrates a characteristic instability for its vertex angle. In the instance of the hyperbola, there are several features of the dynamics similar to the parabola at low driving and the wedge for higher driving. However, the low driving case for a hyperbola can only be completely understood by considering inelasticity effects predicted by a numerical simulation and the observation that the velocity dependent inelasticity allows the particle to sample several nearby trajectories for fixed driving.

Chaotic dynamics and orbit stability in the parabolic oval billiard.

Chaotic properties of the one-parameter family of oval billiards with parabolic boundaries are investigated. Classical dynamics of such billiard is mixed and depends sensitively on the value of the shape parameter. Deviation matrices of some low period orbits are analyzed. Special attention is paid to the stability of orbits bouncing at the singular joining points of the parabolic arcs, where the boundary curvature is discontinuous. The existence of such orbits is connected with the segmentation of the phase space into two or more chaotic components. The obtained results are illustrated by numerical calculations of the Poincaré sections and compared with the properties of the elliptical stadium billiards.

On the propagation of electromagnetic waves through parabolic cylindrical chiroguides with small flare angles

AbstractWith the use of Maxwell's field equations, an analytical investigation of the propagation of electromagnetic (EM) waves through a three‐layer parabolic cylindrical chirowaveguide is presented; different regions of the guide have different values of chirality admittance. The analytical approaches to derive the dispersion relations and the eigenvalue equations for cutoff are discussed under the assumption of small flare angles of the parabolic cylindrical boundaries. However, more emphasis is given on the modal field cutoffs. It is observed that the bunching tendency of field cutoffs, which is seen in the cases of parabolic cylindrical dielectric guides, is greatly affected in this case, which is attributed to the presence of chirality in the medium. © 2002 Wiley Periodicals, Inc. Microwave Opt Technol Lett 33: 414–419, 2002; Published online in Wiley InterScience (www.interscience.wiley.com). DOI 10.1002/mop.10338

Rationalization of Sequential Monitoring of Reliability

A discussion is given of the problem of increasing the efficiency of monitoring the quality of industrial products for which the moments of occurrence of failures are distributed in accordance with a binomial law or a Weibull law with unknown scale parameters. A mathematical modeling method is used to perform a comparison of the properties of different criteria and a sequential criterion with parabolic boundaries of the acceptance and rejection zones is proposed which is most economic in a number of cases.

Fundamental Systems of Numerical Schemes for Linear Convection-Diffusion Equations and their Relationship to Accuracy

A new approach towards the assessment and derivation of numerical methods for convection dominated problems is presented, based on the comparison of the fundamental systems of the continuous and discrete operators. In two or more space dimensions, the dimension of the fundamental system is infinite, and may be identified with a ball. This set is referred to as the true fundamental locus. The fundamental system for a numerical scheme also forms a locus. As a first application, it is shown that a necessary condition for the uniform convergence of a numerical scheme is that the discrete locus should contain the true locus, and it is then shown it is impossible to satisfy this condition with a finite stencil. This shows that results of Shishkin concerning non-uniform convergence at parabolic boundaries are also generic for outflow boundaries. It is shown that the distance between the loci is related to the accuracy of the schemes provided that the loci are sufficiently close. However, if the loci depart markedly, then the situation is rather more complicated. Under suitable conditions, we develop an explicit numerical lower bound on the attainable relative error in terms of the coefficients in the stencil characterising the scheme and the loci.

How parabolic free boundaries approximate hyperbolic fronts

A rather complete study of the existence and qualitative behaviour of the boundaries of the support of solutions of the Cauchy problem for nonlinear first-order and second-order scalar conservation laws is presented. Among other properties, it is shown that, under appropriate assumptions, parabolic interfaces converge to hyperbolic ones in the vanishing viscosity limit.

Electromagnetic waves at parabolic boundaries

We first discuss the scattering of an electromagnetic plane wave at a perfectly conducting parabolic cylinder when the incident wave travels toward the edge of the cylinder and parallel to its axis. We get in terms of Weber functions an exact solution which is simple enough to make possible a comparison with approximate solutions developed to tackle diffraction at rounded wedges. Then, we consider electromagnetic waves capable of propagating inside the parabolic cylinder (eigenmodes). We prove that these eigenmodes correspond to the zeros of Weber functions which are real on the real axis. For the fundamental modes these Weber functions reduce to the Bessel functions J±¼ and we discuss the zeros of these functions.

Ordering in Ising models of parabolic shape

The features of order in narrow systems are studied for isotropic square lattice Ising models with general parabolic boundaries. Using Monte Carlo methods, magnetization profiles are calculated which agree very well with the conformal results and scale properly with the geometrical length parameter. The variation of the tip magnetization with temperature is obtained and the predicted stretched exponential form in the critical region is confirmed.

Analysis of the guidance of electromagnetic waves by a deformed planar waveguide with parabolic cylindrical boundaries

An analytical study of a waveguide with two parabolic cylindrical surfaces separating the guiding region of refractive index n1 from two cladding regions of common refractive index n2 with n1≳n2 is presented. The cutoff conditions for modes are derived as equations containing trigonometric functions. The modes show a bunching tendency instead of well-defined discreteness, and several mode bunches occur because the distance of separation between the two interfaces does not remain constant but continues to increase as one moves away from the region near the vertices of the parabolas. Unlike the planar waveguide, a nonzero cutoff is deduced which may be attributed to the curvature of the boundaries, and the flare.