Approximate Formula Research Articles

Characterization and analysis of attenuation anisotropy in viscoelastic vertical transverse isotropic media

AbstractDue to the intrinsic attenuation of the earth, the study of wave propagation characteristics, considering seismic attenuation plays an important role in high‐precision reservoir prediction. Therefore, we investigate the propagation and reflection characteristics of seismic waves in viscoelastic vertical transverse isotropic media in the complex frequency domain. Specifically, we analyse the response characteristics of velocity, propagation vector and attenuation vector with respect to viscosity media with different attenuation intensities. Furthermore, based on the quasi‐Zoeppritz equation, the variation of reflection coefficient amplitude with offset at different attenuation angles and different attenuation intensities is studied. We also compare the trends in the amplitude variation of reflection coefficients with offset in elastic isotropic, elastic anisotropic and viscoelastic anisotropic media. Due to the complexity of the exact reflection coefficient expression, we first propose the approximate expression of the attenuation–anisotropy parameters and then derive the approximate expression of the reflection coefficient. The numerical simulation results show that the approximate expression of the reflection coefficient is still accurate, even in media with strong anisotropy. Finally, the accuracy evaluations of the reflection coefficient formulas using four typical theory models demonstrate that the approximate reflection coefficient formulas are highly accurate in both weak and strong anisotropic media.

Artificial Neural Network-based robust technique for period prediction of Ottoman minarets in Türkiye

Minarets are crucial as symbolic and indispensable elements in mosques across religious communities worldwide. Beyond architectural culture, hazardous conditions have shaped the minaret structural formations in Islamic geography. Among these, masonry minarets built during the Ottoman Empire faced significant vulnerability to lateral forces due to their slender design, brittle materials, and distinctive shapes. After many earthquakes experienced in the 20th and 21st centuries, the collapse of numerous minarets highlights the urgent need to preserve the remaining structures in high seismicity regions of the historical Ottoman lands, moreover, identifying and fortifying the most vulnerable ones takes precedence. This study proposes an innovative approach, departing from previous research that solely focused on the correlation between geometrical and material properties such as height, cross-section area, moment of inertia, Young’s modulus, and material density using approximate formulas to predict the fundamental periods. Instead, it introduces a novel technique based on Artificial Neural Networks (ANNs) to predict the first three periods of Ottoman masonry minarets. Accurate measurements from 18 minarets located in Bursa City, Türkiye, were meticulously collected under ambient conditions, serving as the foundation for the neural network's comprehensive output database. The distinct parameters (geometrical and material properties) of these minarets form the input dataset. To mitigate the influence of random measurement noise, an effective averaging scheme was implemented. As a result, the proposed ANN technique demonstrates its robustness and holds great promise for practical applications, as it enables accurate estimation of the desired modal information for both the 18 minarets used to train the networks and the remaining three minarets, achieving high levels of accuracy.

Active Kriging-based conjugate first-order reliability method for highly efficient structural reliability analysis using resample strategy

Efficient structural reliability analysis method is crucial to solving reliability analysis of complex structural problems. High-computational cost and low-failure probability problems greatly limit the efficiency in structural reliability analysis problems, causing the safety and reliability of the structure to be questioned. In this work, a highly efficient structural reliability analysis method coupling active Kriging algorithm with conjugate first order reliability method (AK-CFORM) is proposed. Specifically, the resample strategy is considered to reduce the number of samples evaluated in each active learning process; the uniform sampling is used to better balance global and local optimal problems; the conjugate map is used to improve the robustness of analytical first order reliability method; and the approximate numerical differential formula is proposed to solve the problems of non-convergence when solving the gradient of the Kriging surrogate model. Finally, three numerical cases and four engineering cases are used to illustrate the effectiveness and robustness of the proposed method. The results show that the proposed AK-CFORM has greater advantages in the number of calling system response and surrogate model with robust and accurate performance.

Two Approximation Formulas for Gamma Function with Monotonic Remainders

In this paper, two new approximation formulas with monotonic remainders for the gamma function have been presented. Also, we present some numerical comparisons between our new approximation formulas and some known ones, which demonstrate the superiority of our results.

Multi-Objective Stochastic Paint Optimizer for Solving Dynamic Economic Emission Dispatch with Transmission Loss Prediction Using Random Forest Machine Learning Model

Dynamic economic emission dispatch problems are complex optimization tasks in power systems that aim to simultaneously minimize both fuel costs and pollutant emissions while satisfying various system constraints. Traditional methods often involve solving intricate nonlinear load flow equations or employing approximate loss formulas to account for transmission losses. These methods can be computationally expensive and may not accurately represent the actual transmission losses, affecting the overall optimization results. To address these limitations, this study proposes a novel approach that integrates transmission loss prediction into the dynamic economic emission dispatch (DEED) problem. A Random Forest machine learning model was offline-trained to predict transmission losses accurately, eliminating the need for repeated calculations during each iteration of the optimization process. This significantly reduced the computational burden of the algorithm and improved its efficiency. The proposed method utilizes a powerful multi-objective stochastic paint optimizer to solve the highly constrained and complex dynamic economic emission dispatch problem integrated with random forest-based loss prediction. A fuzzy membership-based approach was employed to determine the best compromise Pareto-optimal solution. The proposed algorithm integrated with loss prediction was validated on widely used five and ten-unit power systems with B-loss coefficients. The results obtained using the proposed algorithm were compared with seventeen algorithms available in the literature, demonstrating that the multi-objective stochastic paint optimizer (MOSPO) outperforms most existing algorithms. Notably, for the Institute of Electrical and Electronics Engineers (IEEE) thirty bus system, the proposed algorithm achieves yearly fuel cost savings of USD 37,339.5 and USD 3423.7 compared to the existing group search optimizer algorithm with multiple producers (GSOMP) and multi-objective multi-verse optimization (MOMVO) algorithms.

Polarization of disk electrodes in high-conductivity electrolyte solutions.

We investigate the polarization of disk electrodes immersed in an electrolyte solution and subjected to a small external AC voltage over a wide range of frequencies. A mathematical model is developed based on the Debye-Falkenhagen approximation to the coupled Poisson-Nernst-Planck equations. Analytical techniques are used for predicting the spatial distribution of the electric potential and the complex impedance of the system. Scales for impedance and frequency are identified, which lead to a self-similar behavior for a range of frequencies. Experiments are conducted with gold electrodes of sizes in the range 100-350 μm immersed in a high-conductivity KCl solution over five orders of magnitude in frequency. A collapse of data on impedance magnitude and phase angle onto universal curves is observed with scalings motivated by the mathematical model. A direct comparison with the approximate analytical formula for impedance is made without any fitting parameters, and a good agreement is found for the range of frequencies where the analytical model is valid.

A Novel Low-Frequency Lateral Wave Approximation Formula of Submerged Magnetic Dipoles Near Sea Surface

A Novel Low-Frequency Lateral Wave Approximation Formula of Submerged Magnetic Dipoles Near Sea Surface

Best sum-throughput evaluation of cooperative downlink transmission nonorthogonal multiple access system

In cooperative simultaneous wireless information and power transfer (SWIPT) nonorthogonal multiple access (NOMA) downlink situations, the current research investigates the total throughput of users in center and edge of cell. We focus on creating ways to solve these problems because the fair transmission rate of users located in cell edge and outage performance are significant hurdles at NOMA schemes. To enhance the functionality of cell-edge users, we examine a two-user NOMA scheme whereby the cell-center user functions as a SWIPT relay using power splitting (PS) with a multiple-input single-output. We calculated the probability of an outage for both center and edge cell users, using closed-form approximation formulas and evaluate the system efficacy. The usability of cell edge users is maximized by downlink transmission NOMA (CDT-NOMA) employing a SWIPT relay that employs PS. The suggested approach calculates the ideal value of the PS coefficient to optimize the sum throughput. Compared to the noncooperative and single-input single-output NOMA systems, the best SWIPT-NOMA system provides the cell-edge user with a significant throughput gain. Applying SWIPT-based relaying transmission has no impact on the framework’s overall throughput.

Second quantization-based symmetry-adapted perturbation theory: Generalizing exchange beyond single electron pair approximation.

This paper presents a general second-quantized form of a permutation operator interchanging n pairs of electrons between interacting subsystems in the framework of the symmetry-adapted perturbation theory (SAPT). We detail the procedure for constructing this operator through the consecutive multiplication of single-pair permutation operators. This generalized form of the permutation operator has enabled the derivation of universal formulas for S2n approximations of the exchange energies in the first and second order of the interaction operator. We present expressions for corrections of S4 approximations and assess its efficacy on a selection of systems anticipated to exhibit a slowly converging overlap expansion. Additionally, we outline a method to sum the overlap expansion series to infinity in second-quantization, up to the second order in V. This new approach offers an alternative to the existing formalism based on density-matrix formulations. When combined with a symbolic algebra program for automated derivations, it paves the way for advancements in SAPT theory, particularly for intricate wavefunction theories.

New Accurate Approximation Formula for Gamma Function

In this paper, a new approximation formula for the gamma function and some of its symmetric inequalities are established. We prove the superiority of our results over Yang and Tian’s approximation formula for the gamma function of order v−9.

A Note on a Fractional Extension of the Lotka–Volterra Model Using the Rabotnov Exponential Kernel

In this article, we study the fractional form of a well-known dynamical system from mathematical biology, namely, the Lotka–Volterra model. This mathematical model describes the dynamics of a predator and prey, and we consider here the fractional form using the Rabotnov fractional-exponential (RFE) kernel. In this work, we derive an approximate formula of the fractional derivative of a power function ζp in terms of the RFE kernel. Next, by using the spectral collocation method (SCM) based on the shifted Vieta–Lucas polynomials (VLPs), the fractional differential system is reduced to a set of algebraic equations. We provide a theoretical convergence analysis for the numerical approach, and the accuracy is verified by evaluating the residual error function through some concrete examples. The results are then contrasted with those derived using the fourth-order Runge-Kutta (RK4) method.

Differential Equations for Approximate Solutions of Painlevé Equations: Application to the Algebraic Solutions of the Painlevé-III $({\rm D}_7)$ Equation

It is well known that the Painlevé equations can formally degenerate to autonomous differential equations with elliptic function solutions in suitable scaling limits. A way to make this degeneration rigorous is to apply Deift-Zhou steepest-descent techniques to a Riemann-Hilbert representation of a family of solutions. This method leads to an explicit approximation formula in terms of theta functions and related algebro-geometric ingredients that is difficult to directly link to the expected limiting differential equation. However, the approximation arises from an outer parametrix that satisfies relatively simple conditions. By applying a method that we learned from Alexander Its, it is possible to use these simple conditions to directly obtain the limiting differential equation, bypassing the details of the algebro-geometric solution of the outer parametrix problem. In this paper, we illustrate the use of this method to relate an approximation of the algebraic solutions of the Painlevé-III (D$_7$) equation valid in the part of the complex plane where the poles and zeros of the solutions asymptotically reside to a form of the Weierstraß equation.

Effective reduced models from delay differential equations: Bifurcations, tipping solution paths, and ENSO variability

Conceptual delay models have played a key role in the analysis and understanding of El Niño-Southern Oscillation (ENSO) variability. Based on such delay models, we propose in this work a novel scenario for the fabric of ENSO variability resulting from the subtle interplay between stochastic disturbances and nonlinear invariant sets emerging from bifurcations of the unperturbed dynamics.To identify these invariant sets we adopt an approach combining Galerkin–Koornwinder (GK) approximations of delay differential equations and center-unstable manifold reduction techniques. In that respect, GK approximation formulas are reviewed and synthesized, as well as analytic approximation formulas of center-unstable manifolds. The reduced systems derived thereof enable us to conduct a thorough analysis of the bifurcations arising in a standard delay model of ENSO. We identify thereby a saddle–node bifurcation of periodic orbits co-existing with a subcritical Hopf bifurcation, and a homoclinic bifurcation for this model. We show furthermore that the computation of unstable periodic orbits (UPOs) unfolding through these bifurcations is considerably simplified from the reduced systems.These dynamical insights enable us in turn to design a stochastic model whose solutions – as the delay parameter drifts slowly through its critical values – produce a wealth of temporal patterns resembling ENSO events and exhibiting also decadal variability. Our analysis dissects the origin of this variability and shows how it is tied to certain transition paths between invariant sets of the unperturbed dynamics (for ENSO’s interannual variability) or simply due to the presence of UPOs close to the homoclinic orbit (for decadal variability). In short, this study points out the role of solution paths evolving through tipping “points” beyond equilibria, as possible mechanisms organizing the variability of certain climate phenomena.

The variational principle for probabilistic measure and Hashin–Shtrikman bounds

The paper is a review of Hashin–Shtrikman type bounds for effective moduli of conductivity and elasticity of polycrystals and composites written from the perspective of the variational principle for probabilistic measure. The results for such bounds are rederived in probabilistic terms. Remarkably, in probabilistic terms the Hashin–Shtrikman approach gets especially simple form. Besides, a clear distinction arises between the basic assumption, the choice of the trial field, and the simplifying assumptions, like geometrical isotropy, physical isotropy, texture isotropy, etc. We filled out several gaps. First, we derive an integral equation to be solved to get the bounds when the simplifying assumptions do not hold. Second, we extend the bounds for polycrystals with the cubic symmetry of crystallites to all thermodynamically possible crystallites; previously such bounds were found for crystallites with special elastic properties. One practical outcome considered is the derivation of approximate formulae for the temperature dependence of effective elastic moduli. Third, for crystallites with non-cubic symmetries, we formulated algebraic variational problems to be solved numerically to obtain the bounds, and solved these problems for several materials.

Common mode far field radiation mechanisms and prediction of a buck converter with cables based on common mode voltage mutation method

AbstractIn the field of electromagnetic compatibility (EMC) research, the knowledge of the formation mechanism of the far field electromagnetic radiation and its prediction is essential for the electromagnetic noise suppression design of a switched mode power supply. This paper investigates the equivalent driving sources of the common mode (CM) current of a Buck converter with cables. Based on the concept of CM voltage mutation, this paper proposes two mechanism models of the same‐direction CM voltage driving mode and the opposite‐direction CM voltage driving mode and derives the approximate analytical formulas for the far field intensity estimation of the horizontal cables. To evaluate the predictive effectiveness of the CM far field radiation model, a 30 MHz very high frequency (VHF) Buck prototype with shielded cables is built and tested, and a shielding shell and low impedance ground conductor suppression designs are then verified. The CM electromagnetic radiation mechanisms of a non‐isolated power converter are demonstrated here. The proposed electromagnetic radiation models, approximate analytical formulas, and improvement design can help to advance the CM radiation analysis and optimization design.

Parameterization of anisotropic media by A-parameters

AbstractMost common parameterization of anisotropic media is by twenty one independent elements aijkl of the density-normalized stiffness tensor or by twenty one independent elements Aαβ of the density-normalized matrix of elastic parameters in the Voigt notation. These parameters are commonly of significantly different sizes, are dimensional, in (km/s)2, often appear in combinations. We are offering an alternative parameterization by twenty one A-parameters (anisotropic parameters), which removes the mentioned disadvantages and possesses some additional useful properties. For example, axes or planes of coordinate systems, in which A-parameters are defined, need not be related to symmetry axes or planes of the considered anisotropy symmetry as required in other similar parameterizations. In combination with the first-order weak-anisotropy approximation, in which anisotropy is considered as the first-order perturbation of reference isotropy, parameterization by A-parameters yields insight into the role of individual A-parameters in the wave propagation problems. For example, it turns out that in the first-order weak-anisotropy approximation, P- and S-wave velocities are each controlled by fifteen A-parameters. A set of six of them appears only in the expression for P-wave velocity, a set of other six A-parameters appears only in S-waves velocity expressions. Remaining set of nine A-parameters is common for both waves. We present transformation of A-parameters, analogue to Bond transformation, and useful formulae for the weak-anisotropy approximation for anisotropy of any symmetry and arbitrary tilt.

Multiquadric based RBF-HFD approximation formulas and convergence properties

RBF-FD formulas have received special attention due to their superior accuracy, well-conditioning, and ease of implementation in solving partial differential equations. To further improve the accuracy of RBF-FD formulas, without increasing the stencil size RBF-HFD formulas are proposed in the literature. In this article, we obtain leading/higher order analytical expressions for weights and local truncation errors associated with various such formulas using the multiquadric radial function. We also study the convergence properties of these formulas. Symbolic computation in Mathematica is employed for analytically solving linear systems for the unknown weights in the RBF-HFD formulas. Symmetry properties of central difference approximations help reduce the number of unknown weights and thereby the size of the linear system. We compute the analytical expressions of weights and local truncation errors for first and second-derivative approximation formulas till tenth order; and for two-dimensional Laplacian operator approximation formulas till sixth order. The obtained formulas are free from ill-conditioning. It is observed that the RBF-HFD formula weights converge to corresponding order classical compact FD scheme weights when the shape parameter (ϵ) tends to zero. The local truncation error has global minimum for an optimal value of the shape parameter. Further, the optimal shape parameter is independent of nodal distance but depends on choice of test function and its derivative values at a reference node.

Optimal power flow and computer‐based analysis of a simultaneous AC–DC power transmission system

AbstractElectricity demand is constantly increasing with the growing population, and the process of establishing new transmission lines has become complicated due to the rise in industrialization and urbanization. One solution to meeting this increased demand is to improve the power transfer capability by increasing the transmission voltage. However, the existing transmission lines may not be able to handle the increased load if they are pushed to their thermal limit. This paper presents a mathematical analysis of a simultaneous AC–DC transmission system transmitting power at different values of parameters. The computational analysis is carried out using MATLAB, and the circuit model simulation is validated by using power systems computer aided design (PSCAD/EMTDC). The transmitted power is evaluated by varying the transmission angle and transmission length for a fixed voltage, and then by varying the transmission angle and voltage for a fixed transmission length. Both these cases are analyzed using approximate and actual formulas. The results of the computation and simulation show close conformity.

Second-Order Approximate Reflection Coefficient of Thin Interbeds with Vertical Fractures

The horizontal fractures in the strata will close in the compaction effect of overlying strata, while the vertical cracks are widely developed, which can be equivalent to HTI (transverse isotropy with a horizontal axis of symmetry) medium. When an S-wave propagates into HTI media, the shear wave will divide into two types of waves: a fast S-wave and slow S-wave. When the strata of HTI are thin and overlapping, called the thin interbeds model, the wave field exhibits complex primary reflections, converted waves, and multiples. We introduce a new second-order approximation of the total reflection coefficient, with the incidence angle lower than the critical angle in thin-interbed HTI media using a recursive algorithm. We verify the effectiveness of the second-order approximation by analyzing the energy of multiples. Comparing the second-order approximate solution that degenerates the HTI medium into isotropic and Kennett’s exact solution, we find that our solution has an accuracy of over 99.9% in any azimuth, with the incidence angle lower than the critical angle under P-wave incidence. However, our solution of the SP wave field is suitable for incidence azimuth angles between 0–75° and 120–180°, with the lowest accuracy occurring at an incidence angle of 25° and a relative error of 6.4%. The approximate solution in the SS wave field has the same applicable range as the SP wave, with the maximum error of 6.3% occurring at the incident angle of 1°. This new second-order approximate formula for the total reflection coefficient of thin interbeds composed of HTI helps us to understand the reflection characteristics of complex thin interbeds. It also lays a theoretical foundation for the development of AVO (Amplitude Versus Offset) analysis and inversion techniques for lithological and stratigraphic oil and gas reservoirs.

Fractional Boundary Layer Flow: Lie Symmetry Analysis and Numerical Solution

In this paper, we present a fractional version of the Sakiadis flow described by a nonlinear two-point fractional boundary value problem on a semi-infinite interval, in terms of the Caputo derivative. We derive the fractional Sakiadis model by substituting, in the classical Prandtl boundary layer equations, the second derivative with a fractional-order derivative by the Caputo operator. By using the Lie symmetry analysis, we reduce the fractional partial differential equations to a fractional ordinary differential equation, and, then, a finite difference method on quasi-uniform grids, with a suitable variation of the classical L1 approximation formula for the Caputo fractional derivative, is proposed. Finally, highly accurate numerical solutions are reported.