Conformal Transformation Method Research Articles

Turbomachine criteria for similarity of natural size proportionality

Introduction. It is possible to give rise to synergy as a result of science-intensive industries combination with innovative eco-technologies for subsoil use only by developing a brand new approach to nature-like auxiliary technologies. Insufficient adaptability of turbomachines that ensure industrial safety increases the production cost of the mining and oil and gas complexes of the Russian Federation by more than 15%, reducing its competitiveness. Research methodology. Based on the hypothesis of the hydrodynamic analogy of the mechanisms of deceleration of the flow around the airfoil and the formation of its profile resistance, Karman's theory of attached and free vortices, the Zhukovsky-Chaplygin-Kutta hypothesis, the method of conformal transformations, the theory of similarity, the method of singular points by Chaplygin S. A., the criteria for the similarity of natural proportionality are obtained, that is, for the hydrodynamic similarity of the mechanism of energy interaction between the blades of the turbomachine impeller and the wing of a bird. Results. It has been proved that the dominant control over the nature-like proportionality of the aerodynamics of turbomachines is the ratio between the speed and flow acceleration circulation around the airfoil. It has been established that the coefficients of the airfoil resistance, lift and aerodynamic quality of the airfoil cascade are hydrodynamic analogs of the coefficients of the circulation of the velocity and acceleration of the flow and their ratio. Conclusions. It has been experimentally confirmed that the use of the proposed criterion of natural proportionality in the design of turbomachines increases their coefficient of aerodynamic adaptability by more than 2 times, increasing the area of economical operation by 83%.

The Influence of Geometry on the Loss of Curved Slot Waveguide

In curved slot waveguide devices, there are often waveguide bending situations. Under certain conditions, the smaller the waveguide bending geometry, the more compact device structure; the smaller the bending geometry, the greater the loss caused by waveguide bending. Based on the study of the loss of a curved slot waveguide, this paper combines the geometric dimensions with it to study the influence of the loss. A new method for calculating the loss coefficient caused by the bending of the waveguide: the effective refractive index method with fixed interference correction. Through this method, the relationship between the bending loss coefficient and the width, height and radius of curvature of the dielectric waveguide is analyzed and calculated. Compare the calculation result with the result calculated by the pattern analysis method. Finally, the results were discussed and analyzed. The conformal transformation method and the normalization method of the geometrical size to the bending loss of the waveguide are analyzed, the relationship between the bending geometrical size of the curved narrow slot waveguide and the bending loss is obtained, and the empirical formula for the bending loss is established. From the formula, four Curves of waveguide bending size and bending loss of a common material. Experimental research results show that with the improvement of bending waveguide loss theory and coupling mechanism, low-loss transmission can be achieved under small bending dimensions, so that the integration of overall geometrical dimensions is the development trend of bending.

Exact Solutions for Interaction of Parallel Screw Dislocations with a Wedge Crack in One-Dimensional Hexagonal Quasicrystal with Piezoelectric Effects

The purpose of this paper is to consider the interaction between many parallel dislocations and a wedge-shaped crack and their collective response to the external applied generalized stress in one-dimensional hexagonal piezoelectric quasicrystal, employing the complex variable function theory and the conformal transformation method; the problem for the crack is reduced to the solution of singular integral equations, which can be further reduced to solving Riemann–Hilbert boundary value problems. The analytical solutions of the generalized stress field are obtained. The dislocations are subjected to the phonon field line force, phason field line force, and line charge at the core. The positions of the dislocations are arbitrary, but the dislocation distribution is additive. The dislocation is not only subjected to the external stress and the internal stress generated by the crack, but also to the force exerted on it by other dislocations. The closed-form solutions are obtained for field intensity factors and the image force on a screw dislocation in the presence of a wedge-shaped crack and a collection of other dislocations. Numerical examples are provided to show the effects of wedge angle, dislocation position, dislocation distribution containing symmetric configurations and dislocation quantities on the field intensity factors, energy release rate, and image force acting on the dislocation. The principal new physical results obtained here are (1) the phonon stress, phason stress, and electric displacement singularity occur at the crack tip and dislocations cores, (2) the increasing number of dislocations always accelerates the crack propagation, (3) the effect of wedge angle on crack propagation is related to the distribution of dislocations, and (4) the results of the image force on the dislocation indicate that the dislocations can either be attracted or rejected and reach stable positions eventually.

Measurement of Water Holdup in Oil-Water Two-Phase Flows Using Coplanar Microstrip Transmission Lines Sensor

This paper presents the research of water holdup detection in oil-water two-phase flows and sensor design based on coplanar microstrip transmission line (also known as coplanar waveguide, CPW). The relationship between the CPW structure/material parameters and electromagnetic wave propagation characteristics was analyzed using conformal transformation method. Adopting double-sided and S-shaped wiring sheet structure, the dynamic range of detection is improved while the structure of the sensor is miniaturized. Therefore, multiple sensors can be arranged on the cross-section of the oil well to form a detection array, which can provide differential information of water holdup on the cross-section of horizontal wells or highly deviated wells and improve the spatial resolution of the measurement. Numerical simulations and experimental results show that the phase shifts of the signal detected by sensors are approximately linearly proportional to the water holdup of the measured fluid, the resolution is up to 3% in the whole range of water holdup from 0% to 100%. This method makes up for the disadvantages of capacitance method and conductance method, which are only effective for low water holdup and high holdup scenarios respectively.

The DMCM-Based Analytical Calculation of Radial Vibration Force in a Nonsalient Pole Synchronous Generator in No-Load Conditions

In this paper, the distributed magnetic circuit method (DMCM) is used to calculate the magnetic circuit of a nonsalient pole synchronous generator and deal with the saturation effect of the iron-core material. The conformal transformation method is adopted for calculating the air-gap relative permeance to determine the slot effects of both the stator and the rotor and obtain the spatial and temporal distribution of air-gap flux density and radial vibration force wave. The results of the analytical calculation are very consistent with those of the finite-element method (FEM) and the time (8.47 s) spent on the calculation by the DMCM is much less than the time by the FEM (40 min 45 s). Through measuring the no-load back electromotive force waveform of the generator and the vibration acceleration spectrum of the generator feet, the DMCM is proved to be correct and effective, which will provide the theoretical basis for the accurate design of a magnetic circuit of the generator and for the quantitative analysis of its electromagnetic vibration.

Analytical Solution for the Electric Field Inside Dynamically Harmonized FT-ICR Cell.

Dynamically harmonized FT-ICR cell has a saddle-like hyperbolic field distribution inside when averaged over a cyclotron trajectory around the axis of the cell. Such a field distribution makes the motion along the magnetic field independent of the motion in the x,y-plane, as well as the cyclotron motion independent of the magnetron motion and prevents any disintegration of excited coherent ion clouds, which is ruining the resolution in the other types of FT-ICR cells providing by this ideal phasing of single-m/z ion clouds in the entire volume of the cell. FT-ICR instruments with such a cell show resolutions of more than ten million at m/z 1000 at relatively small magnetic fields like 7 Tesla in quadrupole detection mode, what is not reachable by any other type of modern mass spectrometers. We have found that for such ion traps, it is possible to find the analytical solution in the working volume of the trap without any averaging. The potential distribution for the almost whole volume of such a cell can be presented in the form ϕ(x, y, z) = αz2 + f2D(x, y), where f2D(x, y) is the solution of 2D Poisson equation, which could be found by the method of conformal transformation. This solution is applicable in the practical case and can serve as a base for an analytical theory of signal detection using such cells and as a standard for solutions obtained by numerical simulations of the cell field.

A Transformation-Domain Image Reconstruction Method for Open Electrical Impedance Tomography Based on Conformal Mapping

Open electrical impedance tomography (OEIT) is a nondestructive testing technique imaging the objects within the infinite observation domain through the electrodes fixed at one side of this domain. A novel image reconstruction method is proposed based on the conformal transformation method. The proposed method solves the EIT problems, including both the forward and inverse problems, in the conformal transform domain, instead of original Euclidean domain. The numerical and experimental results show that the proposed method is with higher computational accuracy for the forward and inverse problems comparing with the traditional methods of truncating the infinite domain to a small closed region. The quantitative analyses show that the proposed method can improve the reconstructed accuracy of OEIT, especially for the conductivity changes far from the electrodes.

Determination of pressure properties of superconducting systems based on characteristic ratios 2\u0394(0)/Tc and \u0394C(Tc)/CN(Tc)

A simplified analytical model of the effect of high pressure on the critical temperature and other thermodynamic properties of superconducting systems is developed using the general conformal transformation method and group-theoretical arguments. Relationships between the characteristic ratios { {mathcal R} }_{1}equiv 2{rm{Delta }}(0)/{T}_{{rm{c}}} and { {mathcal R} }_{2}equiv {rm{Delta }}C({T}_{{rm{c}}})/{C}_{{rm{N}}}({T}_{{rm{c}}}) and the stability of the superconducting state is discussed. Including a single two-parameter fluctuation in the density of states, placed away from the Fermi level, stable solutions determined by { {mathcal R} }_{1} are found. It is shown that the critical temperature Tc(p), as a function of high external pressure, can be predicted from experimental data, based on the values of the two characteristic ratios, the critical temperature, and a pressure coefficient measured at zero pressure. The model can be applied to s-wave low-temperature and high-temperature superconductors, as well as to some novel superconducting systems of the new generation. The problem of emergence of superconductivity under high pressure is explained as well. The discussion is illustrated by using experimental data for superconducting elements available in the literature. A criterion for compatibility of experimental data is formulated, allowing one to identify incompatible measurement data for superconducting systems for which the maximum or the minimum critical temperature is achieved under high pressure.

Conformally Mapped Multifunctional Acoustic Metamaterial Lens for Spectral Sound Guiding and Talbot Effect.

We demonstrate a conformally mapped multifunctional acoustic metamaterial Mikaelian lens. Mikaelian lens is a gradient medium with a hyperbolic secant refractive index profile that can realize functions like beam self-focusing. Unlike the conventional design approaches, with a conformal transformation method, only isotropic material parameters with gradient refractive index profiles are required for the construction of such lens. To realize desired gradient index distribution, we carefully design a new type of cross-channel-shaped acoustic metamaterial, whose refractive index can be effectively modulated by simply changing the slit opening size. The distinct capabilities of the metamaterial Mikaelian lens in manipulating acoustic waves are experimentally verified with the fabricated sample. Simultaneous sound guiding and Talbot effects, which normally require respective geometrical and wave acoustic approximations, are observed in simulations and experiments. Furthermore, those effects of shaping acoustic wave propagations were validated within a relatively wide frequency range. Our study reveals how the conformal transformation method can help to bridge the ray acoustics with wave acoustics. It offers opportunities to the development of novel multifunctional acoustic devices for various applications, such as sound and particle manipulations.

Simulation of the electromagnetic wall response during Vertical Displacement Events (VDE) in ITER tokamak

The key basis for tokamak plasma disruption modeling is to understand how currents flow to the plasma facing surfaces during plasma disruption events. In ITER tokamak, the occurrence of a limited number of major disruptions will definitively damage the chamber with no possibility to restore the device. In the current exchange plasma-wall-plasma, according to the Helmholtz decomposition theorem, our surface current density in the conducting shell - the unknown of our problem - being a vector field twice continuously differentiable in 3D, has been splited into two components: an irrotational (curl-free) vector field and a solenoidal (divergence-free) vector field. Developing a weak formulation form and minimizing the correspondent energy functionals in a Finite Element approach, we have obtained the space and time distribution of the surface currents. We verified successfully our numerical simulation with an analytical solution with pure homogeneous Neumann B.C. and satisfying the necessary existence condition. By considering the iron core presence in JET tokamak, we have split the magnetization currents - the unknowns in some integral equations - into two components, the first producing a magnetic field in the iron region only and the second producing a magnetic field in the vacuum, obtaining thus a better evaluation of the influence of the iron core on the plasma equilibrium. To reduce the influence of the singularities appearing during the surface currents determination in multiply connected domains (L-shaped domains) we have used a conformal transformation method.

Characteristics optimization in single and dual coupled silicon-on-insulator ring (disk) photonic biosensors

Silicon-on-insulator (SOI) evanescent field whispering gallery mode (WGM) ring (disk) sensors are promising components for biological sensing applications. We study the optimum design parameters that result in the highest sensitivity (S) and figure of merit (FOM) simultaneously, for single and dual coupled SOI WGM ring (disk) based sensors. These devices are respectively consisting of one and two silicon ring (disk) resonator coupled to a straight waveguide. Periodic behavior of coupling parameters in biological solutions, are modelled by analytical conformal transformation method and coupled mode theory. These methods are verified by experimental results. We demonstrate optimum design parameters at twelve different configurations of single SOI ring (disk) based sensors. Based on the results, the calculated quality factor of second TE WGM resonance at these resonators in water solutions is Qt ≥ 25000. This gives S and FOM parameters to be 193 (134) nm/RIU and 3112 (2161) /RIU, for single ring (disk) resonator sensor. We also obtain an ultra-improved intrinsic limit of detection (ILOD=7.32 × 10−8 RIU) biosensor based on two coupled ring resonators. The ILOD value, shows that our structure performs two and three orders of magnitude better than respectively two and three SOI cascade ring resonators based sensors, reported lately.

Increasing surface active Co2+ sites of MOF-derived Co3O4 for enhanced supercapacitive performance via NaBH4 reduction

Dodecahedral NaBH4-reduced Co3O4 (R-Co3O4) was successfully prepared via a conformal transformation method. Such facile procedure leads to abundance of electrochemically active sites of Co2+ ions. The as-prepared R-Co3O4 affords the material an improved specific capacitance of 329 F/g, which is triply that of the pristine Co3O4. In addition, an enhanced cycling stability is also achieved for the R-Co3O4. Thus, we develop a novel but simple route for large-scale industrial production of electrochemically active materials via NaBH4 reduction.

Hydrodynamic pressure and its effects on embedded tube tunnel in high permeable stratum

When dynamic loadings act on the embedded structure, vibration will be generated and transmitted to the surrounding ground causing dynamic deformation and stress of structure. A semi-analytical solution for hydrodynamic pressure of circular cavity in high permeable stratum using the conformal transformation method is proposed and the amplitude of hydrodynamic water pressure about the frequency is investigated under the harmonic vibration loadings. Furthermore, based on Kirchhoff - love assumption, considering the dynamic interaction of soil and tunnel structure, by means of numerical examples, the dynamic characteristics of a single embedded tunnel was analysed due to the hydrodynamic pressure, the calculation results show that, the vibration response of the tunnel structure is under the similar tendency for different water depth, but affected variously; the impedances of soil can significantly reduce the deformation and internal force of tunnel to a certain extent.

Fast analytical modelling of an SOI micro-ring resonator for bio-sensing application

In this work, we present a closed-form model of silicon-on-insulator (SOI) micro-ring resonators, based on the conformal transformation method. We compute the propagation characteristics of the whispering gallery modes at resonance wavelengths. The results of the proposed model are compared to numerical simulations based on the finite element method (FEM). The proposed method allows comparable accuracy to that of FEM simulations, with a significant reduction in computer resource requirements (time, speed, memory). To further validate the proposed modelling, we also compare the modelling results to experimental data. Finally, we model a ring resonator for biosensing applications. We compute the bulk refractive index sensitivity (S), the figure of merit (FOM) and the intrinsic limit of detection (ILOD). In the geometry under study, our proposed model predicts an ideal ILOD of refractive index units, which is 0.24 smaller than what has been reported in SOI optimized structure. These results validate the proposed modelling technique for the fast design of optical biosensors where both the figure of merit and the geometries can be optimized.

Simple analytical model of the effect of high pressure on the critical temperature and other thermodynamic properties of superconductors

Within the general conformal transformation method a simplified analytical model is proposed to study the effect of external hydrostatic pressure on low- and high-temperature superconducting systems. A single fluctuation in the density of states, placed away from the Fermi level, as well as external pressure are included in the model to derive equations for the superconducting gap, free energy difference, and specific heat difference. The zero- and sub-critical temperature limits are discussed by the method of successive approximations. The critical temperature is found as a function of high external pressure. It is shown that there are four universal types of the response of the system, in terms of dependence of the critical temperature on increasing external pressure. Some effects, which should be possible to be observed experimentally in s-wave superconductors, the cuprates (i.e. high-Tc superconductors) and other superconducting materials of the new generation such as two-gap superconductors, are revealed and discussed. An equation for the ratio {{boldsymbol{ {mathcal R} }}}_{{bf{1}}} ≡ 2Δ(0)/Tc, as a function of the introduced parameters, is derived and solved numerically. Analysis of other thermodynamic quantities and the characteristic ratio {{boldsymbol{ {mathcal R} }}}_{{bf{2}}} ≡ ΔC(Tc)/CN(Tc) is performed numerically, and mutual relations between the discussed quantities are investigated. The simple analytical model presented in the paper may turn out to be helpful in searching for novel superconducting components with higher critical temperatures induced by pressure effects.

Stabilizing effect of the power law inflation on isentropic relativistic fluids

This paper is concerned with the stabilizing effect of the power law inflation on relativistic Eulerian fluids. We prove the global stability of the background solutions to the relativistic fluids including the isothermal gases and generalized Chaplygin gases by the method of conformal transformation when the initial data is a small perturbation to the background solution. We also prove the blowup phenomena of the relativistic Euler fluids including the isothermal gases and polytropic gases when the initial data satisfies suitable assumptions.

Optimum design of fiber angle and hole orientation of an orthotropic plate

With the goal of decreasing the stress concentration along the hole boundary in an orthotropic plate under inequi-biaxial loadings, an optimum design of the fiber angle and hole orientation is presented. The maximum absolute tangential stress along the hole boundary is taken as the objective function, and the fiber orientation angle and the hole orientation angle are considered as design variables. The conformal transformation method of a complex function and the Differential Evolution (DE) algorithm are used. Two non-circular shapes, ellipse and hexagon are taken as examples to analyze the problem. Based on the results, we can conclude that the major axis of elliptical holes should be designed in the direction of the maximum external loading for a perforated structure in an orthotropic plate. However, the principal direction that has the larger Young’s modulus should be inclined to the direction of the minimum loading, especially for a significantly orthotropic plate.

Formation of Regions with High Energy and Pressure Gradients at the Free Surface of Liquid Dielectric in a Tangential Electric Field

Nonlinear dynamics of the free surface of an ideal incompressible non-conducting fluid with high dielectric constant subjected by strong horizontal electric field is simulated on the base of the method of conformal transformations. It is demonstrated that interaction of counter-propagating waves leads to formation of regions with steep wave front at the fluid surface; angles of the boundary inclination tend to {\pi}/2, and the curvature of surface extremely increases. A significant concentration of the energy of the system occurs at these points. From the physical point of view, the appearance of these singularities corresponds to formation of regions at the fluid surface where pressure exerted by electric field undergoes a discontinuity and dynamical pressure increases almost an order of magnitude.

Analytical assessment of some characteristic ratios for s-wave superconductors

We evaluate some thermodynamic quantities and characteristic ratios that describe low- and high-temperature s-wave superconducting systems. Based on a set of fundamental equations derived within the conformal transformation method, a simple model is proposed and studied analytically. After including a one-parameter class of fluctuations in the density of states, the mathematical structure of the s-wave superconducting gap, the free energy difference, and the specific heat difference is found and discussed in an analytic manner. Both the zero-temperature limit T = 0 and the subcritical temperature range T ≲ Tc are discussed using the method of successive approximations. The equation for the ratio R1, relating the zero-temperature energy gap and the critical temperature, is formulated and solved numerically for various values of the model parameter. Other thermodynamic quantities are analyzed, including a characteristic ratio R2, quantifying the dynamics of the specific heat jump at the critical temperature. It is shown that the obtained model results coincide with experimental data for low-Tc superconductors. The prospect of application of the presented model in studies of high-Tc superconductors and other superconducting systems of the new generation is also discussed.

Comparison of algorithms for converting 3D objects into rolls, using a spiral coordinate system

ABSTRACTThe definition of a point’s location in volume by Cartesian, spherical and cylindrical coordinate systems requires three dimensions (x, y, z), (r, θ, ϕ) and (ρ, ϕ, z) accordingly. It is obvious that superfluity, only two dimensions and the constant (a point’s height or plane thickness) are enough because, when the plane is transformed into a roll, the third dimension appears. It is easy to convert a three-dimensional space to a two-dimensional one if you know the thickness of the spatial layer and the equation of equivalence between 3D and 2D spaces. This work discusses several advantages of conformal transformation methods to convert different volume objects into a ribbon, based on its symmetry and the spiral coordinate system for roll powder sintering (RPS) [Shulunov, V.R., 2014. A high performance, high precision, low cost rapid prototyping and manufacturing technology. AUSMT Copyright ©. International Journal of Automation Smart Technology, 4 (3). doi:10.5875/ausmt.v4i3.718, Shulunov, V.R., 2015a. A roll powder sintering additive manufacturing technology. Applied Mechanics and Materials, 789–790, 1212–1216. © Trans Tech Publications, Switzerland. doi:10.4028/www.scientific.net/AMM.789-790.1212, Shulunov, V.R., 2015b. Several advantages of the ultra high-precision additive manufacturing technology, © Springer-Verlag London. International Journal of Advance Manufacturing Technology. doi:10.1007/s00170-015-7533-0] additive manufacturing technology. RPS requires slicing a 3D object with Archimedes spiral scanning. The proposed methods can be used to transform 3D objects into 2D objects.