Functions Of Complex Variables Research Articles

Theoretical Analysis of Stress Intensity Factor for Two Asymmetric Cracks Emanating from Water Conveyance Tunnel

The water conveyance tunnel has been widely applied in tunnel engineering, which may be subjected to compressive stress and shear stress from rock and faults, respectively. The internal pressure caused by water in the tunnel will make the appearance of cracks and propagation, resulting in disasters. In order to study the effects of the cracks and compressive stress on the cracked tunnel model, a plane model containing a circular hole with two unequal cracks under the compression loads, shear stress and internal pressure was established in this study. Complex variable function and conformal mapping function were implemented to theoretically derive the stress intensity factors (SIFs) at the crack tip in the rock mass, the numerical simulation was conducted to validate the theoretical solutions. The results indicate that the numerical results using the ABAQUS code were in good agreement with the theoretical solutions. The effects of compressive stress, shear stress, internal pressure, crack length and crack inclination angle on model II SIFs were discussed, the initiation angle and initiation pressure of different rock materials were also studied. When the crack inclination angle changes from [Formula: see text] to [Formula: see text], the tunnel is easily destroyed. With the increase of the right crack length, the increasing rate of the SIFs increases quickly first and then linearly, however, the left crack length has a slight effect on SIFs. When the crack inclination angle is [Formula: see text], the crack initiation angle reaches the maximum value.

Analysis of stress intensity factor for a crack emanating from elliptical hole subjected to compressive stress and shear stress

Underground caves often exist in karst areas, which will cause many difficulties in tunnel construction. The underground caves are often subjected to compressive stress, shear stress and internal pressure come from rock, faults and groundwater, respectively. The crack emanating from the cave will propagate under the effect of external forces, leading to the disaster of cave collapse and water gushing. To research the stability of crack emanating from the elliptical cave, a model of a plane containing an elliptical hole with a single crack under the compressive stress, shear stress and internal pressure was established. The theoretical solutions of stress intensity factors (SIFs) for crack tip were derived using complex variable function and conformal mapping function. The theoretical results were compared with the results of numerical simulation to verify the correctness of the theoretical solutions. The effects of crack length, ellipse shape, compressive stress, shear stress, internal pressure and crack inclination angle on SIFs were discussed. We can conclude that when the crack inclination angle is varied from 0° to 30°, the initiation pressure is small, and the cave is easily destroyed by groundwater. The flatter the elliptical hole, the easier the crack propagates. With an increasing of crack length, the value of SIFs at the crack tip increase quickly first and then increase linearly. When the crack inclination angle is 45°, the initiation angle of the crack reaches the maximum value.

Complex function solution for deformation and failure mechanism of inclined coal seam roadway

The stressed environment of the inclined coal seam roadway is complex and changeable, and the damage degree of surrounding rock increases, threatening the safe mining of coal mines. In order to take targeted support measures to control the stability of roadway surrounding rock, it is very important to study the stress and deformation characteristics of roadway surrounding rock in inclined coal seam. Therefore, this paper analyzes the deformation and failure law of inclined coal seam roadway according to the theory of complex variable function. It optimizes the solution process and accuracy of the mapping function coefficient and deduces the analytical solution of surrounding rock stress and deformation inclined coal seam roadway. The deformation and failure mechanism of surrounding rock in inclined coal seam roadway is revealed theoretically, and further use numerical simulation and physical simulation tests for supplementary analysis and verification. The results show that the stress and deformation of roadway surrounding rock in inclined coal seam show obvious asymmetric distribution characteristics. The stress and deformation of roadway surrounding rock on the right side are greater than on the left side. The two sides of the roadway, the right side of the roof and the roof angle of the right side, are the key positions of roadway stress concentration and deformation. According to the variation law of stress and deformation distribution of roadway surrounding rock, roadway cyclic deformation and failure theory is put forward. The numerical simulation and physical simulation tests show that the deformation and failure law of roadway is consistent with the theoretical analysis results, and there are differences in numerical values. The cyclic deformation and failure mechanism of roadway in inclined coal seam is verified, which can provide theoretical guidance for roadway support design.

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.

Generalized Gauge Transformation with PT$PT$‐Symmetric Non‐Unitary Operator and Classical Correspondence of Non‐Hermitian Hamiltonian for a Periodically Driven System

AbstractThis paper demonstrates that the parity‐time ()‐symmetric non‐Hermitian Hamiltonian for a periodically driven system can be generated from a kernel Hamiltonian by a generalized gauge transformation. The kernel Hamiltonian is Hermitian and static, while the time‐dependent transformation operator has to be symmetric and non‐unitary in general. Biorthogonal sets of eigenstates appear necessarily as a consequence of the non‐Hermitian Hamiltonian. The wave functions and associated non‐adiabatic Berry phase for the nth eigenstate are obtained analytically. The classical version of the non‐Hermitian Hamiltonian becomes a complex function of canonical variables and time. The corresponding kernel Hamiltonian is derived with symmetric canonical‐variable transfer in the classical gauge transformation. Moreover, with the change of position‐momentum to angle‐action variables it is revealed that the non‐adiabatic Hannay's angle and Berry phase satisfy precisely the quantum‐classical correspondence, .

Slice holomorphic functions in the unit ball: boundedness of $L$-index in a direction and related properties

Let $\mathbf{b}\in\mathbb{C}^n\setminus\{\mathbf{0}\}$ be a fixed direction. We consider slice holomorphic functions of several complex variables in the unit ball, i.e. we study functions which are analytic in intersection of every slice $\{z^0+t\mathbf{b}: t\in\mathbb{C}\}$ with the unit ball $\mathbb{B}^n=\{z\in\mathbb{C}^: \ |z|:=\sqrt{|z|_1^2+\ldots+|z_n|^2}<1\}$ for any $z^0\in\mathbb{B}^n$. For this class of functions we consider the concept of boundedness of $L$-index in the direction $\mathbf{b},$ where $\mathbf{L}: \mathbb{B}^n\to\mathbb{R}_+$ is a positive continuous function such that $L(z)>\frac{\beta|\mathbf{b}|}{1-|z|}$ and $\beta>1$ is some constant.For functions from this class we deduce analog of Hayman's Theorem. It is criterion useful in applications todifferential equations. We introduce a concept of function having bounded value $L$-distribution in direction forthe slice holomorphic functions in the unit ball. It is proved that slice holomorphic function in the unit ball has bounded value $L$-distribution in a direction if and only if its directional derivative has bounded $L$-index in the same direction. Other propositions concern existence theorems. We show that for any slice holomorphic function $F$ with bounded multiplicities of zeros on any slice in the fixed direction there exists such a positive continuous function $L$that the function $F$ has bounded $L$-index in the direction.

Optimal thermal design of anisotropic plates with arbitrary cutouts using genetic algorithm

Anisotropic plates in different applications may have geometric defects such as openings and cracks. The presence of the opening disturbs the heat flow, which creates significant thermal stress around the opening. When the heat flux is high enough, these extreme stresses can lead to structural failure. This article aims to obtain the optimal parameters for achieving the minimum value of the normalized stress near the cutout's boundary in perforated anisotropic plates utilizing the genetic algorithm. Optimization parameters include the curvature of opening's corners, orientation angle of opening, fibers angle, heat flux angle, and opening's elongation. The plate is under heat flux, and the opening's border is thermally insulated. The stress distribution around the opening is calculated using Lekhnitskii's complex variable method and complex potential functions. The genetic algorithm is then implemented to find the optimal values for design parameters. The results show that by selecting the optimal parameters related to the anisotropic material and the opening's geometry, the stress intensity factor of the perforated anisotropic plates is remarkably reduced. Furthermore, this optimization algorithm can be extended to find the optimized parameters and achieve the optimal designs in anisotropic and isotropic perforated plates under thermal loadings.

Non-symmetric indentation of an elastic half-plane

In this work, the problems of wave propagation in a half-space due to the indentation by a rigid wedge at a constant speed and by a parabolic punch at a constant acceleration have been considered separately. The elastodynamics problems of non-symmetric indentation over a contact region expanding at a constant speed and constant acceleration have been solved using the method of homogeneous functions. Following Cherepanov and Cherepanov et al., the general solution of the problems has been derived in terms of an analytic function of complex variables. The expressions for the stress component under the contact region and the torque over the contact region have been derived. Numerical results of the particular cases of the Problems I and III and of the Problems II and IV have been presented in the form of graphs. This work and its applications are expected to be helpful in the study of indentation-related problems of solid mechanics.

Dynamic response of a half-space with a depression covered by viscous liquid under SH waves

This study examines the problem of the displacement amplitude at the interface of an elastic half-space with a semicircular depression covered by a viscous liquid under the action of shear horizontal (SH) waves. First, the half-space incident wave field is constructed. Simultaneously, the scattering displacement field of the problem is obtained using the large circle and complex variable function methods. Thereafter, based on the boundary conditions at the fluid-solid interface, an infinite linear algebraic equation is established to obtain the unknown coefficients of the wave function. Finally, the changes in the interface displacement amplitude caused by SH wave scattering are obtained. Numerical examples reveal that the thickness, dynamic viscous coefficient, density of the viscous liquid, and incident wave frequency all affect the displacement amplitude at the interface. This paper also presents valuable conclusions that are expected to help guide the study of engineering structures with recessed half-spaces covered by viscous liquids.

Exploration of new analytical correlations as an alternative to the Kramers-Kronig transforms for the assessment of impedance spectroscopy data

• Alternative method to detect errors in EIS spectra. • Overcome of some practical issues for assessment of simple spectra. • Elucidation of analytical correlations between magnitude and phase. New analytical correlations between impedance magnitude and phase are developed and presented. These correlations are used for the assessment of impedance spectroscopy data. As correlations are obtained, they can be calculated point to point at each frequency of the spectra with no need to establish explicit premises related to stability, linearity or finite limits for their derivation. Their use is simpler than that based on the direct evaluation of the Kramers-Kroning transforms or alternative methods. An example of their derivation from basic equations involving complex numbers is described. They were obtained by looking for an explanation of the empirical fact that the slope of the Bode-Magnitude diagram vs frequency resembles the shape of the corresponding Bode-Phase diagram. Unlike the Kramers-Kronig transform, these new correlations can also be directly applied to test spectra with highly predominant capacitive or inductive responses or even incomplete loops. Their analytical expressions are exact for simple ideal elements and circuits, including the Warburg diffusion element. These correlations were tested with simulated spectra altered with known random error distributions, and they were able to detect the random distribution and the error level for simple spectra when the errors were > 0.5%. For more sophisticated spectra, the minimum error that can be detected without interference of the deviations due to the proper algorithm are around 2.5%. No exact correlation was found for a constant phase element (CPE), but a close empirical approximation was obtained for a single CPE. It is necessary to improve this approximation when a CPE is combined with some other elements because of detected deviations. In addition, it was found that in a first approach the algorithm does not detect non-linear behavior on a simple experimental circuit, but the test gave insights of how to detect it. On the contrary, it is well suited for detection of system stability issues. Examples of their application to actual experimental spectra, high capacitive or with several loops, are presented and the limitations discussed. More work and exhaustive tests are necessary to improve the method and stablish its real limitations, but its use for assessment of simple spectra at certain error level can already be done. Given that the derivation of the new correlations is only based on its complex number nature, this study concludes that the magnitude and phase of any complex number or 2D vector are not independent, and it is expected that similar correlations can be applied to any frequency-dependent complex variable, vector or transfer function, such as those used in optics or rheology.

Thermal fracture resistance of functionally graded thermal barrier coatings with systems of multiple cracks. Application of rule of mixtures

The problem of thermal fracture of functionally graded thermal barrier coatings (FGCs) on a homogeneous substrate under the influence of thermo-mechanical loadings is studied in the presented work. It is assumed that FGCs contain a pre-existing system of multiple cracks, edge and/or internal. The method of analytical functions of complex variables and singular integral equations, as well as a mechanical quadrature method for solving singular integral equations were used. The functional graded material properties (including the fracture toughness) are modelled by the formulas based on the rule of mixtures (RoM) with a power-law coefficient λ as the grading parameter. The following fracture characteristics are calculated: stress intensity factors, critical stresses and fracture angles. A series of computational experiments is carried out for FGC/H structures with typical systems of multiple cracks in the FGC at different grading parameters. As illustrative examples, systems of edge cracks and edge cracks with internal cracks are investigated. The results show that the grading parameter λ has a dominant effect on the fracture characteristics. The presented model allows to correlate the material parameters, geometrical parameters and the main fracture characteristics, which can be the basis for the solution for an optimization problem, e.g. to find the grading parameter λ and the crack arrangements that increase the fracture toughness.

ГЕОМЕТРИЧЕСКИЕ СВОЙСТВА ИНТЕГРАЛЬНОГО ОПЕРАТОРА БЕРНАЦКОГО

In the geometric theory of complex variable functions, the study of mapping of classes of regular functions using various operators has now become an independent trend. The connection f(z)∈S o⇔ g(z) = zf'(z) ∈ S * of the classes S o and S * of convex and star-shaped functions can be considered as mapping using the differential operator G[f](x) = zf'(z) of class S o to class S * , that is, G: S o → S * or G(S o ) = S * . The impetus for studying this range of issues was M. Bernatsky's assumption that the inverse operator G –1 [f](x), which translates S * → S o and thereby “improves” the properties of functions, maps the entire class S of single-leaf functions into itself. At present, a number of articles have been published which study the various integral operators. In particular, they establish sets of values of indicators included in these operators where operators map class S or its subclasses to themselves or to other subclasses. This paper determines the values of the Bernatsky parameter included in the generalized integral operator, at which this operator transforms a subclass of star-shaped functions allocated by the condition a < Re zf'(z)/ f(z) < b (0 < a < 1 < b), in the class K(γ) of functions, almost convex in order γ. The results of the article summarize or reinforce previously known effects.

Conformal projections of a tri-axial ellipsoid based on isometric coordinates: history, methodology, and examples

Abstract The paper presents a review of the conformal projections of a tri-axial ellipsoid and the methodology of creating these projections with the use of isometric coordinates. The concept is very simple and has been known for a long time; if isometric coordinates are introduced on the surface of the original and on the plane of the image, then any analytical function of the complex variable, i.e. a function that has a continuous derivative, creates a conformal projection. The introduction presents the history of conformal projections. Then, existing projections are presented, including the Bugayevskiy projection and several projections developed by the author that apply selected functions of the complex variable. Scripts were prepared in the Octave software with the use of the presented methodology. Programming in Octave offers a possibility of a simple implementation of complex variable functions, which is also briefly discussed in the paper. The developed scripts were then used to perform calculations and to draw cartographic grids and distortion isolines in the selected conformal projections. The test object was the tri-axial ellipsoid that represents Phobos.

Mean-squared approximation of some classes of complex variable functions by Fourier series in the weighted Bergman space 𝐵2,𝛾

Mean-squared approximation of some classes of complex variable functions by Fourier series in the weighted Bergman space 𝐵2,𝛾

Study on the penetration of elliptical cross-section projectiles into concrete targets: theory and experiment

To better understand the penetration mechanism of the elliptical cross-section projectile (ECSP) into semi-infinite concrete target, penetration experiments using three types of ECSPs with different shape ratios (1, 1.25 and 1.61) and with striking velocities ranged from 550 m/s to 1050 m/s were conducted. Penetration depths, penetration trajectory and mass erosion rates of the projectile were obtained after the experiments. The experiment results show that the penetration performance and ballistic stability of the ECSP are equivalent to those of the circular cross-section projectile (CCSP). Based on the theory of complex variable function and conformal transformation, a semi-analytical model which can calculate the cavity boundary stress distribution of elliptical section cavity controlled by the displacement boundary condition was established and the model was validated by comparing the model degenerate solution with Kirsch problem results. Theoretical calculation results show that the radial stress of elliptical section cavity increases progressively from the minor axis to the major axis. In addition, a formula combining with the semi-analytical theoretical model and the local interaction theory was developed. The predicted penetration depths were compared with 30 groups of experiment data with different projectile parameters and striking velocities and coincide quite well with the corresponding experimental data. Finally, the influence of shape ratio and caliber-radius-head (CRH) on the penetration performance of projectile and the application prospect of ECSPs on hypersonic weapon platform were studied.

Radiation of a Bunch Crossing a Boundary between a Vacuum and a Cold Magnetized Plasma in a Waveguide

We analyze the electromagnetic field of a small bunch that uniformly moves in a circular waveguide and transverses a boundary between an area filled up with cold magnetized electron plasma and a vacuum area. The magnetic field is supposed to be strong but finite so the perturbation technique can be applied. Two cases are studied in detail: the bunch is flying out of the plasma into a vacuum, and, inversely, the bunch is flying into the plasma out of the vacuum area of waveguide. The investigation of the waveguide mode components is performed analytically with methods of the complex variable function theory. The main peculiarities of the bunch radiation in such situations are revealed.

Dynamic Analysis of Nanosized Arbitrary Shape Inclusion under SH Waves

The scattering of shear waves (SH waves) by nano-scale arbitrary shape inclusion in infinite plane is studied by complex variable function theory. Firstly, the governing equation and the relationships between stress and displacement are given by classical elastic theory. Secondly, the arbitrary shape inclusion in the two-dimensional plane is transformed into a unit circle domain by conformal mapping, the incident wave field and the scattered wave field are presented. Next, the stress and displacement boundary conditions are established by considering surface elasticity theory, The infinite algebraic equations for solving the unknown coefficients of the scattered and standing waves are obtained. Finally, the influence of surface effect, non-dimensional wave number, Shear modulus and hole curvature on the dynamic stress concentration factor are analyzed by some examples, the numerical results show that the surface effect weakens the dynamic stress concentration. With the increase of wave number, the dynamic stress concentration factor (DSCF) decreases. Shear modulus and hole curvature have significant effects on DSCF.

Visual positioning system for marine industrial robot assembly based on complex variable function

Abstract With the deepening of human understanding of marine resources and the acceleration of resource development, marine industrial robots have come to be widely used. Robots are not simply a substitute for manual labour, but a personified electronic and mechanical device that combines human and machine strengths. They have both the ability to react quickly and analyse and judge the state of the environment, and the ability of the machine to work continuously for a long time, with high accuracy and resistance to adverse conditions. In industrial machinery, the gravitational field exists in threedimensional space, but if the distribution of the known field is independent of a coordinate, it is simplified to a twodimensional field, which is called a plane scalar field. There are many two-dimensional field problems in the visual positioning system of assembly system. The exact solution can be obtained by using complex function theory to analyse two-dimensional field in complex plane. In this paper, the application of complex function theory in gravitational field is discussed. In the process of cooperative assembly of marine industrial robots, it is necessary to make the correct choice to improve the cooperative assembly method of marine industrial robots, so that the robot industry in China can undergo development and renewal.

Deformation and stress theory of surrounding rock of shallow circular tunnel based on complex variable function method

Abstract After reviewing many literature foundations, the thesis combines the basic methods of elastic mechanics with mathematical knowledge, sets the bipotential stress potential complex function and analyses the relationship between stress component, strain component and stress potential function, and applies the complex variable function. The expression of the relevant stress component is derived, and the displacement boundary conditions of the surrounding rock of shallow circular tunnel are obtained. Furthermore, the paper applies the basic theory of complex variable function to solve the boundary condition complex variable function for common tunnel sections, and obtains the analytical expression of the surrounding rock stress of shallow circular tunnel. The simulation is carried out by finite element method. The establishment of complex variable function has a good application value in solving the stress of surrounding rock of shallow tunnel.

Using the neural network in clinical systems

The article examines the identification of human bone fractures using convoluted neural networks. The method of recognition of photographs of patients is intended for automated systems of identification and video recording of images. Convolutional neural networks have a number of advantages, such as invariability when reducing or increasing image size, immunity to photo movements and deviations, changes in image perspective, and many other image errors. In addition, convolutional neural networks allow you to combine neurons at a local level in two dimensions, connect photographic elements in any place, and also reduce the total number of weights. The work describes a multi-layer convolutional network. The layers of which it consists are divided into two types: convolutional and sub-selective. Of interest is the use of the principle of weighting in the work. This principle allows you to reduce the number of characteristics of the neural network that can be trained. Network training is based on the rule of minimizing empirical error. This rule is based on the algorithm of inverse error propagation. This algorithm provides an instant calculation of the gradient of a complex function of several variables in case the function itself is predefined. Neural network training is based on probabilistic method. This method leads to more optimal results due to interference in the restructuring of network weights. The work confirms the axiomatics of the applied neural network, its architecture and its learning algorithm.