Conformal Transformation Method Research Articles

Scattering of SH waves by elliptical cavity and type-III crack in deep anisotropic geology

Most of the underground rock formations are anisotropic, and the study of the dynamic response of anisotropic geology containing internal defects is necessary for the development of underground tunnels or caverns. In this paper, the wave function expansion method, the complex function method, and the Green's function method are used to solve the problem of scattering of SH waves in anisotropic rock in the presence of elliptical cavity and type-III crack. Boundary conditions for solving the problem are given using the conformal transformation method, and crack in the medium is constructed using the "crack cutting" technique. The scattered wave field generated by the crack is constructed based on the Green function method, and the displacement and stress fields in the model are obtained by the wave field superposition principle in the case of SH wave incidence. The dynamic stress concentration factor (DSCF) on the boundary of the elliptical cavity and the dynamic stress intensity factor (DSIF) at the crack tip are derived. The correctness of the method is verified by degenerating it to a classical analytic solution. Finally, the effects of the anisotropic strength of the rock mass, the dimensions of the cavern, and the incidence angle and wave number of the SH wave on the DSCF and DSIF are analyzed by numerical calculations in both the frequency and time domains. The results show that the anisotropy of the rock mass, the shape of the cavern and the defects of the surrounding medium have a great influence on the dynamic response of the underground structures, which should be given enough attention in the development and design of deep underground caverns or tunnels.

Simulation of the Wave Turbulence of a Liquid Surface Using the Dynamic Conformal Transformation Method

The dynamic conformal transformation method has been generalized for the first time to numerically simulate the capillary wave turbulence of a liquid surface in the plane symmetric anisotropic geometry. The model is strongly nonlinear and involves effects of surface tension, as well as energy dissipation and pumping. Simulation results have shown that the system of nonlinear capillary waves can pass to the quasistationary chaotic motion regime (wave turbulence). The calculated exponents of spectra do not coincide with those for the classical Zakharov–Filonenko spectrum for isotropic capillary turbulence but are in good agreement with the estimate obtained under the assumption of the dominant effect of five-wave resonant interactions.

Analytical study of steady state seepage in a circular tunnel considering the outer boundary of the grouting ring as a non-constant head boundary

The influence of groundwater seepage on tunnel is not negligible, so it is very important to calculate the distribution of pore water pressure and the seepage volume accurately. The current analytical studies of seepage field in tunnels with grouting ring assume the head outside the grouting ring to be constant, which is accurate when there is a large difference between the permeability of the grouting ring and the soil body, but less accurate when there is a small difference in permeability. Accordingly, this paper combines a new conformal transformation method and the separated variable method to overcome the current problem of not being able to obtain an analytical solution after assuming the head at the outer boundary of the grouting circle as a non-constant head. And according to the obtained analytical solution and the existing analytical solutions and numerical simulation results for comparison, comparison results show that: when 1 ≤ kr/kg ≤ 100, the absolute value of the maximum relative error between the calculation results of the external pore water pressure of the grouting circle of the CVM solution and those of the numerical solution is more than 1.5 times of that of the analytical solution of this paper, and the maximum value is nearly 3 times of that of the analytical solution of this paper. Therefore, the analytical solution obtained in this paper by assuming that the outer boundary of the grout ring is a non-constant head is more accurate and has some applicability in the case where the permeability of the soil and the grout ring do not differ much. Finally, extensive parametric analyses of the permeability and thickness of the grouting ring and the depth of the tunnel are also performed to demonstrate the capability of the proposed analytical solution. In addition, the proposed analytical solution is much less computationally demanding compared to numerical software, but the accuracy is comparable.

Construction of rose flower-like NiCo-LDH electrode derived from bimetallic MOF for highly sensitive electrochemical sensing of hydrazine in food samples.

It is necessary to efficient detection hydrazine in food. Exploring highly sensitive, low-cost and fast response electrochemical hydrazine sensing methods has been a challenge in this field. In this paper, a conformal transformation method is used to prepare rose flower-like NiCo-LDH derivating from the bimetallic NiCo-MOFs, and the N2H4 sensing platform with a large electrocatalytic area, high conductivity and good stability was constructed. Based on the synergy between Ni and Co and the remarkable catalytic activity of the rough 3D flower-like structure, the N2H4 sensor has a linear response in the concentration range of 0.001-1mmol/L and 1-7mmol/L, with a sensitivity of 5342 μA L mmol-1 cm-2 and 2965 μA L mmol-1 cm-2 (S/N=3), respectively, and low limit of detection of 0.043μmol/L. This study opens a new door for the successful application of electrochemical sensors to detect N2H4 in real food samples.

Fully Nonlinear Evolution of Free-Surface Waves with Constant Vorticity under Horizontal Electric Fields

This work presents the results of a direct numerical simulation of the nonlinear free surface evolution of a finite-depth fluid with a linear shear flow under the action of horizontal electric fields. The method of time-dependent conformal transformation for the description of the combined effects of the electric fields and constant vorticity is generalized for the first time. The simulation results show that strong shear flow co-directed in the wave propagation direction leads to the formation of large-amplitude surface waves, and, for some limiting vorticity value, a wave breaking process with the formation of an air bubble in the liquid is possible. The oppositely directed shear flow can cause the retrograde motion of a surface wave (wave propagation in the opposite direction to the linear wave speed). The simulations conducted taking into account the electro-hydrodynamic effects demonstrate that a high enough external horizontal electric field suppresses these strongly nonlinear processes, and the surface waves tend to preserve their shape.

Study on the risk of seepage field of Qiantang River underground space excavated in water-rich rheological rock area

Abstract With the rapid development of underwater space construction, the theoretical research is at the initial stage. At present, the theoretical research methods mainly include mirror image method and conformal transformation method. At present, almost all the research objects in analytical research are homogeneous soil layers. However, in practical engineering, soil layers are often distributed in layers. In this study, the seepage field of layered soil layers is analyzed, and the analytical derivation method of seepage field in heterogeneous media based on traditional conformal transformation method is given, which is verified by numerical simulation. The hydraulic head loss of the medium with better permeability in shallow layer is very small, and the streamline is vertically distributed. In the weak interlayer, the hydraulic head loss is huge, and the streamline bends greatly, showing a strong horizontal distribution and converging towards the center. It can provide theoretical basis for the design of water and soil pressure in practical engineering.

Subsection and superposition method for reservoir formation damage evaluation of complex-structure wells

Kinds of complex-structure wells can effectively improve production, which are widely used. However, in the process of drilling and completion, complex-structure wells with long drilling cycle and large exposed area of reservoir can lead to the fact that reservoir near wellbore is more vulnerable to the working fluid invasion, resulting in more serious formation damage. In order to quantitatively describe the reservoir formation damage in the construction of complex-structure well, taking the inclined well section as the research object, the coordinate transformation method and conformal transformation method are given according to the flow characteristics of reservoir near wellbore in anisotropic reservoir. Then the local skin factor in orthogonal plane of wellbore is deduced. Considering the uneven distribution of local skin factor along the wellbore, the oscillation decreasing model and empirical equation model of damage zone radius distribution along the wellbore direction are established and then the total skin factor model of the whole well is superimposed to realize the reservoir damage evaluation of complex-structure wells. Combining the skin factor model with the production model, the production of complex-structure wells can be predicted more accurately. The two field application cases show that the accuracy of the model can be more than 90%, which can also fully reflect the invasion characteristics of drilling and completion fluid in any well section of complex-structure wells in anisotropic reservoir, so as to further provide guidance for the scientific establishment of reservoir production system.

Multifocus Image Reconstruction and Fusion for 3-D Flow Visualization Using an Optimized Four-Plane ECT Sensor

3D image reconstruction of permittivity distribution is very challenging for electrical capacitance tomography (ECT) because of the limited number of capacitance measurements and the non-uniformly distributed sensitivity, especially for axial distribution. In this paper, a 3D image reconstruction method based on fuzzy adaptive Kalman filter (FAKaF) and multi-focus image fusion is used to improve the 3D image quality. A four-plane ECT sensor is constructed and optimized based on the conformal transformation method, whose sensitivity in the imaging volume is more uniform in the axial direction. Then a FAKaF is established to obtain a series of initial reconstructed images with higher image quality in the central region, which can be seen as a sequence of multi-focus 3D images. Finally, an image registration and fusion method based on cross correlation and spatial frequency is used to extract and fuse the high-quality parts of each image in the sequence to further improve the quality of the reconstructed images. Simulations and experiments were carried out to evaluate the performance of the proposed method. The results show that compared with the widely-used methods, initial images reconstructed using the proposed FAKaF-based method possess higher image quality. Through image fusion, higher correlation coefficients between reference distributions and fused images can be obtained, proving the effectiveness of the proposed method.

Comparison of terrain corrections based on the point source and line source DC methods

In the direct current (DC) exploration method, topographic relief distorts the apparent resistivity curve. To eliminate effects of terrain fluctuations, two undulating terrains comprising valleys and ridges were investigated in the present study. An unstructured triangular mesh method in which the wave number k and its coefficient g were obtained using the integral method and the point and line source surveys were conducted using comsol multiphysics. Current sources were evaluated using two-dimensional (2-D) finite element forward modeling, whereas terrain correction was performed using both the comparison and conformal transformation methods. The results reveal comparable theoretical curves for the line and point sources, but quantitative characteristics of the curves differ. The comparison method is suitable for both curves, whereas the conformal transformation method is only applicable to the line source. Even though electric fields associated with the line and point sources differ, the comparison method that is based on the electrical cross-section curve of the line source and the electric profile curve of the point source remains effective.

Broadband achromatic flexural wave Mikaelian lens for high resolution focusing

Conformal transformation method (CTM) has been extensively applied to control propagation of electromagnetic waves and acoustics waves due to the form-invariant property of Maxwell equations and acoustic equations. However, CTM’s application in elastic waves is rarely reported due to the governing equation of elastic waves do not have form invariant property. In this paper, through igniting evanescent waves at the interface of conformally mapped Mikaelian lens with hyperbolic secant refractive index profile, CTM is successfully used to achieve highly efficient (above 75%) broadband (30−80 kHz) achromatic high-resolution flexural wave focusing in thin plate with full width at half maximum (FWHM) around 0.2λ. The proposed Mikaelian lens is designed by linking refractive index with the thickness in plates. Simulated results agree well with theoretical prediction. This high performance for flexural wave focusing could be used for energy harvesting and medical imaging.

Mathematical model of wave diffraction for symmetrically arranged breakwaters

Wave diffraction is a typical problem that is encountered in breakwater construction because knowledge of diffraction behavior has important application in the design and location of breakwaters. With the rapid development of harbor, more and more unconventional layout forms of breakwaters are often encountered in actual engineering. Therefore, this paper introduces a mathematical model of regular waves diffracted by symmetrically arranged rigid and thin breakwaters with a gap, using the conformal transformation method and Green's function method; an appropriate numerical method is described to obtain the approximate analytical solutions. Wave diffraction coefficients were calculated for cases involving different incident angles of the waves, different gap widths, and the included angles between two breakwaters, and the diffraction coefficient diagrams were plotted. The results of the present mathematical model were compared with the those of previous analytical solutions. It was found that the variation trends of the present results are consistent with those of previous studies, which proves the correctness of the model. The influence of the included angle between two breakwaters on the distribution of diffraction coefficients was also analyzed. The mathematical model is unity of the Penny and Price's [“Part I. The diffraction theory of sea waves and the shelter afforded by breakwaters,” Philos. Trans. R. Soc. London 244, 236–253 (1952)] and Kirby et al.'s (1994) [“Parabolic approximations for water waves in conformal coordinate systems,” Coastal Eng. 23, 185–213 (1994)] models; it effectively overcomes the insufficient accuracy of the superposition method for cases involving oblique incidence or gap widths of less than one wavelength. Furthermore, Lamb's [Hydrodynamics, 6th ed. (Cambridge University Press, 1932), p. 305] solution is reproduced when the ratio of the gap width to the wavelength is very small.

Excess Pore Water Pressure and Ground Consolidation Settlement Caused by Grouting of Shield Tunnelling

This study investigates the effect of shield tunnel grouting on excess pore pressure distributions around tunnel and ground consolidation settlement. First, it presents half-plane excess pore pressure around an unlined tunnel caused by grouting pressure based on the complex variable theory and the conformal mapping method. Second, the analytical solution for the soil consolidation is derived using the conformal transformation method and the consolidation theory of Terzaghi–Rendulic. The drainage conditions of the tunnel lining are idealized into three cases: fully drained, partially drained, and impermeable along the tunnel circumference. The impacts of the burial depth of the tunnel, the permeability coefficient of the tunnel lining and the surrounding soil, the Poisson’s ratio of the soil, the lateral Earth pressure coefficient of the soil, and the shear modulus are investigated.

Optimization of Numerical Methods for Transforming UTM Plane Coordinates to Lambert Plane Coordinates

The rapid transformation from UTM (Universal Transverse Mecator) projection to Lambert projection helps to realize timely merging, inversion, and analysis of high-frequency partitioned remote sensing images. In this study, the transformation error and the efficiency of the linear rule approximation method, the improved linear rule approximation method, the hyperbolic transformation method, and the conformal transformation method were compared in transforming the coordinates of sample points on WGS84 (The World Geodetic System 1984)-UTM zonal projections to WGS84-Lambert projection coordinates. The effect of the grid aspect ratio on the coordinate transformation error of the conformal transformation method was examined. In addition, the conformal transformation method-based error spatial pattern of the sample points was analyzed. The results show that the conformal transformation method can better balance error and efficiency than other numerical methods. The error of the conformal transformation method is less affected by grid size. The maximum x-error is less than 0.36 m and the maximum y-error is less than 1.22 m when the grid size reaches 300 km × 300 km. The x- and y-error values decrease when square grids are used; namely, setting the grid aspect ratio close to 1 helps to weaken the effect of increasing grid area on the error. The dispersion of the error distribution and the maximum error of sample points both decrease relative to their minimum distance to the grid edge and stabilize at a minimum distance equal to 70 km. This study can support the rapid integration of massive remote sensing data over large areas.

Dynamic stress analysis of scattering by a circular cavity in a radially inhomogeneous unbounded space under SH waves

Abstract The density of a radially inhomogeneous unbounded space is derived as a function form. Harmonic dynamics stress of the radially inhomogeneous medium with a circular cavity is investigated by the complex variable function method. The governing equation under incident SH waves in the radially inhomogeneous unbounded medium is expressed as a Helmholtz equation with a variable coefficient. It is equivalently transformed into a standard Helmholtz equation by the conformal transformation method. Then, the stress fields in the radially inhomogeneous medium can be proposed. The results indicate that the changes in density parameter of the medium and wave number further affect the dynamic stress concentration factor around the circular cavity.

Investigation of the soil-lining interaction at a tunnel entrance during freeze-thaw cycling: Model testing and semi-empirical modeling

Periodic freeze-thaw cycling can significantly reduce the bearing capacity of mountain tunnel linings. In this study, a model test is carried out to research the lining stress characteristics and the freezing depth of tunnel entrances in cold regions. Then, specific consideration is given to the freezing and thawing depths, the frost heaving pressure, and the internal forces around the lining. Generally, the results show that the occurrence times of the freezing and thawing depths at different positions are similar, but the various trends are different. Moreover, the frost heaving pressure at point 4 (sidewall) has the earliest occurrence time, the fastest rate of increase, and the most prolonged increase duration. Under the influence of the frost heaving pressure, the maximum axial force position shifts from the knee to the spandrel, and the sidewall is in tension. Furthermore, the periodic change in the frost heaving pressure may induce a permanent deformation at the tunnel entrance lining, especially at the position between the knee and the sidewall. Finally, based on the conformal transformation and complex variable method, a semi-empirical model is developed to predict the distribution of internal forces around the non-circular shallow tunnel subjected to frost heaving. Overall, this study helps to estimate the soil-lining interaction at a tunnel entrance during freeze-thaw cycling.

An approximation method combined with conformal transformation for the fast measurement of blade section curves

To realize high-precision machining of free-form blade surfaces, it is necessary to measure and reconstruct the blade morphology several times. Sampling measurements and data compression through free-curve approximation technology are of great benefit in shortening the processing time. A novel B-spline approximation method is proposed in this paper, which can determine a few dominant points based on curvature characteristics to approximate dense and noisy measured points with high precision. We adopt a unique adaptive knot placement strategy combined with a curvature dependent evaluation parameter and a conformal transformation method. This strategy is error-bounded in the initial knot placement, which greatly reduces the time consumed by subsequent iterative insertions. The proposed method shows advantages in approximation accuracy, compression rate, and especially in computational efficiency, in comparison with four traditional knot placement methods. Experiments were carried out and the results show that the dominant point selected by this method can replace the fine measurement results, and the compression rate of the measurement points can reach more than 90%. This method is suitable for the sampling measurement and compression of the data of blades, aircraft wings, and other similar free-form surfaces, and is especially suitable for solving the inefficiency problem caused by repeated measurements made in the process of iterative machining of aero-engine blades.

THE OPTIMIZATION OF THE SHAPE AND SIZE OF THE INJECTION CONTACTS OF THE INTEGRATED P-I-N-STRUCTURES ON THE BASE OF USING THE CONFORMAL MAPPING METHOD

Context. P-i-n-diodes are widely used in a microwave technology to control the electromagnetic field. The field is controlled by the formation of an electron-hole plasma in the region of an intrinsic semiconductor (i-region) under the influence of a control current. The development of control devices on p-i-n-diodes has led to the emergence of integral p-i-n-structures of various types, the characteristics of which (for example, switching speed, switched power level, etc.) exceed the similar characteristics of volume diodes. The properties of p-i-n-structures are determined by a number of processes: the diffusion-drift charge transfer process, the recombination-generation, thermal, injection, and the so on. Obviously, these processes should be taken into account (are displayed) in the mathematical model of the computer-aided design system for control devices of a microwave systems. Integrated process accounting leads to the formulation of complex tasks. One of them is the task of optimizing the shape, geometric dimensions and placement of the injected contacts (an active region).
 Objective. The goal of the work is the development of a mathematical model and the corresponding software of the process of a microwave waves interaction with electron-hole plasma in an active region of the surface-oriented integral p-i-n-structures with ribbon-type freeform contacts to optimize an active region shape and its geometric dimensions.
 Method. The main idea of the developed algorithm is to use the conformal mapping method to bring the physical domain of the problem to canonical form, followed by solving internal boundary value problems in this area for the ambipolar diffusion equation and the wave equation using numerical-analytical methods (the finite difference method; partial domains method using projection boundary conditions similar to the Galerkin method). The optimization algorithm is based on a phased solution of the following problems (the shape and geometric dimensions of the active region are specified at each stage): a computational grid of nodes for the physical regions of the problem is being found, in an active region the carriers concentration distribution is being determined and the energy transmitted coefficient in the system under study is being calculated, which is used in the proposed optimization functional. The extreme values of the functional are found by the uniform search method.
 Results. The proposed mathematical model and the corresponding algorithm for optimizing the shape and geometric dimensions of the active region (i-region) of integrated surface-oriented p-i-n-structures expands the tool base for the design of semiconductor circuits of microwave frequencies (for example, similar to CST MICROWAVE STUDIO).
 Conclusions. An algorithm has been developed to optimize the shape and geometrical dimensions of the active region of integrated surface-oriented p-i-n-structures with in-depth contacts intended for switching millimeter-wave electromagnetic signals. The universality of the algorithm is ensured by applying the method of conformal transformations of spatial domains. The example of the application of the proposed algorithm to search for the optimal sizes of wedge-shaped (in cross-section) contacts of silicon structures is considered.

Semi-Analytical Calculation of No-Load Radial and Tangential Electromagnetic Force Waves of a Non-Salient Pole Synchronous Generator

For a marine electrical machine, its high power density and large electromagnetic (EM) load tend to bring high saturation to the magnetic circuit. When the non-linearity of ferromagnetic material is not taken into consideration, large errors will appear in the calculation of air-gap flux density and EM force wave. Considering the radial and tangential EM force waves that cause the stator core to vibrate and deform are the primary sources of EM vibration of a non-salient pole synchronous generator (NSPSG), this paper introduces a semi-analytical method of calculating the radial and tangential EM force waves, with the non-linearity of ferromagnetic material taken into account. The method includes the distributed magnetic circuit method (DMCM), the separation variable method (SVM) and the complex variable conformal transformation method (CVCTM). First of all, the radial air-gap flux density is obtained by using the DMCM to calculate the no-load magnetic circuit of the NSPSG and dealing with the non-linearity of ferromagnetic material by <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">B-H</i> curve. Then, the tangential air-gap flux density is calculated by the SVM and the complex relative air-gap permeance is calculated by the CVCTM to determine the effect of slotting. Finally, the radial and tangential EM force waves are obtained by Maxwell stress tensor method. The proposed semi-analytical method is verified through measuring vibration acceleration of the NSPSG.

The characteristics of unconventional tight oil reservoir and its modification technology and productivity prediction

The zonal productivity prediction model of volume fractured horizontal wells in tight reservoirs is established in this study based on the physical model of volume fractured horizontal wells to study the near-well seepage law of volume fractured horizontal wells in unconventional tight reservoirs. The conformal transformation and iteration method are used to solve the problem. Compared with Fan Zifei's and Ning Zhengfu's method, the optimization of fractured horizontal wells' production parameters is studied. The effects of starting pressure gradient and fracture cluster number on horizontal wells' productivity with volume fracturing are analyzed. Multivariate regression analysis is used to rank the importance of influencing factors. The results show that the starting pressure gradient has the most significant impact on capacity. The application example of volume fractured horizontal wells in Lucaogou tight reservoir shows that the zonal productivity model established in this study can be used to predict the initial productivity of volume fractured horizontal wells. Thus, through theoretical analysis and numerical simulation, the productivity prediction model of horizontal wells with volume fracturing has certain guiding significance for the effective development of tight reservoirs.

Dynamic response of a cylindrical cavity to SH wave in inhomogeneous continuum with modulus varying two-dimensionally

Due to the inhomogeneity of shear modulus, the constitutive form of the medium becomes more complicated, which increases the difficulty of solving the analytical solution. In this paper, a new form of shear modulus is derived. Meanwhile, the wave number in the medium is also variable. By using the method of complex function and conformal transformation, the problem of dynamic stress concentration around a cylindrical cavity in an infinite inhomogeneous medium is solved. Combined with an auxiliary displacement function and a pair of new mapping functions, in the solving process, the analytical expressions of displacement field and stress field are given. The validity of this method is verified by comparing with the published results. Through numerical results, the distribution of dynamic stress concentration factor around a cylindrical cavity is analyzed under different inhomogeneous parameters and reference wave numbers.