Conformal Transformation Technique Research Articles

Some mathematical assumptions for accurate transformation parameters between WGS84 and Nord Sahara geodetic systems

Abstract The transformation of coordinates between different geodetic datums has been a common practice within the geospatial profession. Relating different geodetic datums mostly involves the use of conformal transformation techniques, which could produce results that are not very often satisfactory for certain geodetic, surveying, and mapping purposes. This has been attributed to the inability of the conformal models to resolve the lack (non-exhaustivity) and the heterogeneity of double points existing within the Algerian local geodetic networks. Indeed, the Algerian geodetic network is divided into two main zones completely different in accuracy and points exhaustivity. The North is better and well defined in precision; on the other hand, the South is poorly defined because of the lack of heterogeneity of the points. Mathematical algebraic methods give closed-form solutions to geodetic transformation problems, requiring a high-level computer programming background. In everyday usage, the closed-form solutions are much simpler and have a higher precision than earlier procedures. Thus, it can be predicted that these new solutions will find their place in practice. The present work deals with an important theoretical problem of geodesy: we are looking for a mathematical dependency between two spatial systems using common pairs of points whose coordinates are given in both systems. In geodesy and photogrammetry, the most often used procedure to move from one coordinate system to another is the 3D, seven-parameter similarity transformation (Bursa–Wolf, Molodensky–Badekas, and Helmert).

Analytical Solution of the Interference between Elliptical Inclusion and Screw Dislocation in One-Dimensional Hexagonal Piezoelectric Quasicrystal

This study examines the interference problem between screw dislocation and elliptical inclusion in one-dimensional hexagonal piezoelectric quasicrystals. The general solutions are obtained using the complex variable function method and the conformal transformation technique. When the screw dislocation is located outside or inside the elliptical inclusion, the perturbation method and Laurent series expansion are employed to derive explicit analytical expressions for the complex potentials in the elliptical inclusion and the matrix, respectively. Considering four types of far-field force and electric loading conditions, analytical solutions for various specific cases are obtained by using matrix operations. Expressions for the phonon field stress, phason field stress, and electric displacement are given for special cases, including the absence of a dislocation, the presence of an elliptical hole, and the interference between a screw dislocation and circular inclusion, as well as the case of a circular hole. The design and analysis of quasicrystal inclusion structures can benefit from the results of this work.

Complex function method for the scattering of harmonic plane waves by an arbitrary-shaped cavity in an unsaturated medium

General formulations of the complex function method for the scattering of harmonic waves by an arbitrary-shaped cavity in the unsaturated medium are constructed. In light of the momentum and the mass balance equations, the wave fields of the unsaturated medium with unknown coefficients are solved by Fredlund’s theory and Helmholtz’s decomposition. Complex-valued expressions for the displacements and the total stresses of the solid skeleton, the pore pressures, and the seepages of the unsaturated medium are obtained by the conformal transformation technique to map the arbitrary cavity onto a unit circle in the complex plane. The boundary value problem results in a series of infinite algebraic equations with unknown coefficients by considering the stress, displacement, and permeable conditions along the cavity. A parametric study for the scattering of the incident P1 waves around a rectangular cavity in an unsaturated medium is performed to investigate the dynamic stress concentrations, displacements, and pore pressures. Numerical results show that the saturation degree of the unsaturated medium has a considerable influence on the dynamic response of the cavity by the incident P1 waves. With the increase of the saturation degree, the dynamic stress concentration around the rectangular cavity decreases, and the equivalent pore pressure increases. For obliquely incident P1 waves, the shielding effect of the rectangular cavity is obvious.

Scattering of SH Waves by a Semi-cylindrical Bump in an Inhomogeneous Half-space

The wave propagation in an inhomogeneous half space with a semi-cylindrical surface bump is investigated by the means of complex function method. With the origin of the coordinate system as the center, the density of the medium changes radially. To solve the Helmholtz equation with variable coefficients caused by the inhomogeneous of the medium, a conformal transformation technique based on the theory of complex variable functions is adopted. Then, by adjusting the inhomogeneous parameters and comparing with the published analytical results, the correctness of the method in this paper is verified. Finally, the displacement amplitudes of typical observation points on the surface and inside are given, and the effects of incident wave angle, wave number and inhomogeneous parameters on wave energy distribution are analyzed.

Acoustic levitation of axisymmetric Mie objects above a transducer array by engineering the acoustic radiation force and torque.

Transducer arrays are a versatile tool for the contactless manipulation of spherical Rayleigh objects. Here we propose an analytical model for stable levitation of axisymmetric Mie objects through directly engineering the desired radiation force and torque. Acoustic contributions from multiple transducers are superimposed through the translation addition theorem, and the nonspherical objects are mapped into a sphere using the conformal transformation technique so that the scattered field can be asymptotically obtained. Then we give the acoustic radiation force and torque applied to a rigid nonspherical Mie object, which can be reconstructed as a series of quasiexplicit functions of the transducer (amplitude and phase) parameters. Through specifying the desired radiation force and torque exerted on the objects, a system of nonlinear equationsis produced, which could be iteratively solved to retrieve appropriate transducer parameters that stabilize the object in an equilibrium position. Practically, we demonstrate several examples of stable levitation of a sphere, a spheroid, and a disk with an averaged radius of a=7mm (size parameter of ka≈5.18) above a transducer array. The absolute acoustic pressure field surrounding the objects simulated by the finite-element method is illustrated to verify the trapping results. The developed analytical model provides an alternative approach to retrieve the transducer parameters for levitating macroscopic nonspherical rigid objects, which may help design the systematic dynamical manipulation of Mie particles.

Scattering of harmonic P1 and SV waves by a shallow lined circular tunnel in a poroelastic half-plane

An analytical solution for the scattering of harmonic P1 and SV waves in a poroelastic half-plane with a shallow lined tunnel is obtained using the plane complex theory in elastodynamics. In light of the wave function expansions, the wave fields of the poroelastic medium and the liner with unknown coefficients are obtained based on Biot's theory and Helmholtz decomposition. Complex-valued expressions of the effective stresses, the fluid stress, and the displacements of the poroelastic medium and the liner are expressed by the complex variable function method and the conformal transformation technique. With the boundary conditions and the continuity of the medium-liner interface, the boundary value problem results in a series of algebraic equations. The unknown coefficients in the infinite set of algebraic equations can be solved numerically by truncating the series number. A parametric study for the incident SV waves is performed to investigate dynamic stress concentrations and fluid stress of the medium and the liner. Numerical results show that the embedment depth of the tunnel, the incident angle of the excitations, and the porosity of the medium have considerable influence on the dynamic responses of the medium and the liner. The shielding effect of the tunnel on the incident SV waves is obvious. For the big embedment depth of the tunnel, the scattered waves contribute little to the displacements and dynamic stress concentration of the medium and the liner. For a high porosity close to the critical value, the response of the medium-liner system to the incident waves is great.

Plasmonic modes in nanowire dimers: A study based on the hydrodynamic Drude model including nonlocal and nonlinear effects

A combined analytical and numerical study of the modes in two distinct plasmonic nanowire systems is presented. The computations are based on a Discontinuous Galerkin Time-Domain approach and a fully nonlinear and nonlocal hydrodynamic Drude model for the metal is utilized. In the linear regime, these computations demonstrate the strong influence of nonlocality on the field distributions as well as on the scattering and absorption spectra. Based on these results, second-harmonic generation efficiencies are computed over a frequency range that covers all relevant modes of the linear spectra. In order to interprete the physical mechanisms that lead to corresponding field distributions, the associated linear quasi-electrostatic problem is solved analytically via conformal transformation techniques. This provides an intuitive classification of the linear excitations of the systems that is then applied to the full Maxwell case. Based on this classification, group theory facilitates the determination of the selection rules for the efficient excitation of modes, both in the linear and the nonlinear regime. This leads to significantly enhanced second-harmonic generation via judiciously exploiting the system symmetries. These results regarding the mode structure and second-harmonic generation are of direct relevance to other nano-antenna systems.

Conformal transformation route to gravity’s rainbow

Conformal transformation as a mathematical tool has been used in many areas of gravitational physics. In this paper, we would consider the gravity's rainbow, in which the metric could be treated as a conformal rescaling of the original metric. By using the conformal transformation technique, we get a specific form of modified Newton's constant and cosmological constant in gravity's rainbow, which implies that the total vacuum energy is dependent on probe energy. Moreover, the result shows that the Einstein gravity's rainbow could be described by an energy-dependent $f(E,\tilde R)$ gravity. At last, we study the $f(R)$ gravity, when the gravity's rainbow is considered, it can also be described as another energy-dependent $\tilde f(E,\tilde R)$ gravity.

Performance evaluation of artificial neural networks for planimetric coordinate transformation—a case study, Ghana

Two national horizontal geodetic datums, namely, the Accra and Leigon datum, have been the only available datum used in Ghana. These two datums are non-geocentric and were established based on astro-geodetic observations. Relating these different geodetic datums mostly involves the use of conformal transformation techniques which could produce results that are not very often satisfactory for certain geodetic, surveying and mapping purposes. This has been ascribed to the incapability of the conformal models to absorb more of the heterogeneous and local character of deformations existing within the local geodetic networks. Presently, application of new approaches such as artificial neural network (ANN) is highly recommended. Whereas the ANN has been gaining much popularity to solving coordinate transformation-related problems in recent times, the existing researches carried out in Ghana have shown that only three-dimensional conformal transformation methods have been utilized. To the best of our knowledge, plane coordinate transformation between the two local geodetic datums in Ghana has not been investigated. In this paper, an attempt has been made to explore the plane coordinate transformation performance of two different ANN approaches (backpropagation neural network (BPNN) and radial basis function neural network (RBFNN)) compared with two different traditional techniques (six- and four-parameter models) in the Ghana national geodetic reference network. The results revealed that transforming plane coordinates from Leigon to Accra datum, the RBFNN was better than the BPNN and traditional techniques. Transforming from Accra to Leigon datum, both the BPNN and RBFNN produced closely related results and were better than the traditional methods. Therefore, this study will create the opportunity for Ghana to recognize the significance and strength of the ANN technology in solving coordinate transformation problems.

Modelling and analysis of the dynamic behavior of inhomogeneous continuum containing a circular inclusion

Based on the complex function theory, an universal approach of solving the dynamic stress concentration around a circular inclusion in two-dimensional (2D) inhomogeneous medium is presented. The inhomogeneous Helmholtz equation with variable coefficient is converted to the standard Helmholtz equation by using the general conformal transformation technique analytically. An inhomogeneity with the density varying as a smooth function of two spatial coordinates and the constant elastic modulus is established to verify the accuracy of analytical results. As a typical example, an exponential variation of the density is introduced for analyzing the dynamic stress concentration factor (DSCF) around the inclusion. Numerical results show the efficiency of the method and the effects of the medium inhomogeneity, reference wave number of the background medium, the reference wave number and shear modulus ratios between the background medium and inclusion.

A facile method to realize perfectly matched layers for elastic waves

In perfectly matched layer (PML) technique, an artificial layer is introduced in the simulation of wave propagation as a boundary condition which absorbs all incident waves without any reflection. Such a layer is generally thought to be unrealizable due to its complicated material formulation. In this paper, on the basis of transformation elastodynamics and complex coordinate transformation, a novel method is proposed to design PMLs for elastic waves. By applying the conformal transformation technique, the proposed PML is formulated in terms of conventional constitutive parameters and then can be easily realized by functionally graded viscoelastic materials. We perform numerical simulations to validate the material realization and performance of this PML.

NMC and the Fine-Tuning Problem on the Brane

We propose a new solution to the fine-tuning problem related to coupling constantλof the potential. We study a quartic potential of the formλϕ4in the framework of the Randall-Sundrum type II braneworld model in the presence of a Higgs field which interacts nonminimally with gravity via a possible interaction term of the form-(ξ/2)ϕ2R. Using the conformal transformation techniques, the slow-roll parameters in high energy limit are reformulated in the case of a nonminimally coupled scalar field. We show that, for some value of a coupling parameterξand brane tensionT, we can eliminate the fine-tuning problem. Finally, we present graphically the solutions of several values of the free parameters of the model.

Radiated Electromagnetic Immunity Analysis of Flex Cable with Ground Plane Using Transmission Line Equations

This paper presents the analysis of radiated immunity of flex cable with ground plane using transmission line equations. The conformal mapping transformation technique and the integral equations are employed to extract the equivalent RLCG parameters for canonical types and complicated types of structures, respectively. The external electromagnetic field sources are modeled as equivalent voltage and current sources in the transmission line equations. After the equivalent circuit is obtained, the induced voltage is computed via the SPICE solver. Validation with case studies shows that the approach is capable of efficiently modeling the radiated electromagnetic interference to evaluate the immunity level of devices under test.

Sloshing characteristics in half-full horizontal elliptical tanks with vertical baffles

The effects of surface-piercing or bottom-mounted vertical baffles on two-dimensional liquid sloshing characteristics in a half-full non-deformable horizontal cylindrical container of elliptical cross section is investigated. The problem solution is achieved by employing the linear potential theory in conjunction with the successive conformal transformation technique, leading to matrix eigenvalue problems on simple (rectangular and semi-infinite strip) regions. Plots of the resonant eigen-frequencies as a function of the baffle extension at selected container aspect ratios are presented and discussed for the three lowest antisymmetric and symmetric transverse oscillation modes. Also, the effects of vertical baffles on the hydrodynamic pressure mode shapes and sloshing flows are examined through appropriate 2D images. The surface-piercing vertical baffle is demonstrated to be an effective tool in reducing the antisymmetric sloshing frequencies, especially for lower aspect ratio tanks and higher modes. On the other hand, the bottom-mounted baffle is shown to have a great influence on the higher antisymmetric slosh modes only when its tip approaches the liquid free surface. Limiting cases are considered and good agreements with available analytic and numerical solutions as well as experimental data are obtained.

Surface Plasmons and Singularities

We apply the conformal transformation technique to study systematically a variety of singular plasmonic structures, including two-dimensional sharp edges, rough surfaces, and nanocrescents. These structures are shown to exhibit two distinct features. First, different from a planar metallic surface, the surface plasmon excitations on the examined structures have a lower bound cutoff at a finite frequency; second, the electric field diverges below a critical frequency even when metallic losses are considered. For rough surfaces and open-crescent nanostructures, the influence of the structure shapes on the absorbance characteristics is also discussed. Our analysis gives a unique insight into the light capture process on singular structures and holds the promise of detection of single molecules and greatly enhanced nonlinear effects.

Potential flow solution for a free surface flow past a sudden slope change

An analytical model was developed to calculate the pressure distribution in a free surface flow past a sudden change in channel slope. A conformal transformation technique was used to solve the problem analytically in a way that there is no need for trial and error to find the location of the free surface. Two methods were employed for this simulation: flow at a corner and free streamline theory. It was shown that free streamline theory is more accurate. Experiments were conducted to verify the ability of the analytical model to calculate the pressure distribution in a channel with a sudden change in slope. Slope changes of 6.22°, 10°, and 15° were tested with various flow discharges. The analytical model predictions of pressure distribution along the channel bed and with depth agreed well with the experimental measurements. Pressures up to 25 times the hydrostatic pressure were experimentally measured near the point of sudden change in slope. These pressures were reproduced by the model. The analytical model predictions of the water surface profile over a ramp in a prototype spillway were compared with those of a numerical model. The comparison showed a good agreement.Key words: pressure distribution, free surface flow, analytical model, chute spillway, aerator ramp, potential flow.

Groundwater flow in anisotropic aquifers reduced to the isotropic case

AbstractBy applying the technique of conformal transformation of the metric, widely used in general relativity and cosmology, the analysis of groundwater flow in an anisotropic aquifer (in which the hydraulic conductivity is a tensor) is reduced to the case of an isotropic one. This reduction is achieved for a particular conformal factor of the transformation. Examples of application of this technique are given. Copyright © 2004 John Wiley & Sons, Ltd.

Capacitor Cilíndrico Excêntrico

Consideramos o capacitor cilíndrico com a placa interna fora de centro e calculamos a capacitância correspondente usando a técnica da transformação conforme. Observamos que a capacitância pode ser ampliada e a quantidade de material para construir o capacitor pode ser reduzida com a geometria excêntrica.Este resultado pode ser usado como tutorial de uma aula experimental para medir o efeito da excentricidade na capacitância e na construção do capacitor cilíndrico.

On the apparent permeability of a porous layer backed by a perforated plate

Two-dimensional slow viscous flow from a fluid reservoir, through a porous layer and then through a perforated plate is studied assuming Stokes flow in the fluid reservoir and Darcy flow within the porous medium. It is first shown that the coupled Stokes/Darcy problem can be reduced to a Darcy problem when the various length scales are constrained such that Darcy's law is appropriate to describe flow in the porous layer in the vicinity of the perforations of the plate. The apparent permeability of the porous layer is studied as a function of the (uniform) thickness of the layer, and as a function of the size and spacing of the performations in the plate. The apparent permeability is shown to be significantly lower than the intrinsic permeability of the porous layer when the layer is sufficiently thin. Closed-form expressions for the apparent permeability are derived using conformal transformation techniques. We then present a model of particle deposition onto the perforated plate. The growth of the porous layer resulting from the deposition is studied, as is the evolution of its apparent permeability.

Transverse demagnetizing factors of long rectangular bars: I. Analytical expressions for extreme values of susceptibility

The demagnetizing problem should be studied numerically in nonellipsoids with nonzero susceptibility χ, except for a few limiting cases where analytical treatment turns out to be practical. In this work, for an infinitely long bar with rectangular cross section 2a×2b and χ=∞ and −1 in a uniform transverse applied field Ha along dimension a or b, analytical expressions for the surface pole or current distributions and the fluxmetric and magnetometric demagnetizing factors Nf,m are derived using a technique of conformal transformation. From the new as well as the existing formulas, Nf,m(χ=∞,0,−1) are evaluated, plotted, and tabulated as functions of a/b.