Fourier Series Expansion Method Research Articles

Photothermoelastic interaction in a two-dimensional semiconductor with nonlocal stress theory

The present contribution enlightens a viable simulation which describes the photothermoelastic interactions for a two-dimensional thermoelastic transversely isotropic thick semiconducting plate due to the presence of a varying heat source. The upper surface of the semiconductor is traction-free, subjected to prescribed surface temperature and prescribed carrier density flux while the lower surface of the plate rests on a rigid foundation and is thermally insulated. In this new model, the conventional Fourier law has been modified assimilating the memory-dependent derivative within a slipping interval in association with the nonlocal stress theory. On utilizing Laplace and Fourier transformation mechanism, the governing equations for displacement, carrier density and temperature have been obtained in the Laplace–Fourier transform domain. In order to have the solutions in the space-time domain, the inversion of the double transform have been carried out numerically based on the method of Fourier series expansion. From the graphical representations corresponding to the numerical results, the significance of effective parameters such as nonlocality parameter, delay-time parameter is discussed. Noteworthy differences in the results have been demonstrated for different kernel functions. Also, the superiority of a nonlinear kernel function and its effectiveness is also reported in the study.

A study of a nonlinear vibration isolator supported on an imperfect boundary plate

The research on the nonlinear vibration isolation of continuum systems always focuses on the perfect boundary conditions. In this work, nonlinear isolation of transverse vibrations of the arbitrary boundary rectangular plate was investigated. A novel analytical approach was proposed. The rigid body dynamics and high-order energy Fourier series expansion method were utilized to establish a mathematical model of the arbitrary boundary nonlinear continuum system. This method involves the simultaneous Fourier series expansion in temporal and spatial domains. The dynamic behaviors of an arbitrary boundary rectangular plate with nonlinear vibration isolators can be described through a harmonic balance analysis together with arc-length continuation. The convergence and stability for the frequency response functions were checked via the Lyapunov stability theory. The numerical results supported the analytical solutions. Both the analytical and numerical results demonstrated that nonlinear broadband vibration isolation is sensitive to the elasticity of the arbitrary boundary flexible foundation. Furthermore, increasing damping can effectively suppress the transmission of low-order vibrations and control the divergence of higher-order vibrations. Finally, an experiment rig was designed to validate the nonlinear isolation of the transverse vibrations of an arbitrary boundary plate.

Valuing equity-linked guaranteed minimum death benefits with European-style Asian payoffs under a regime switching jump-diffusion model

This paper presents a novel and efficient approach to pricing equity-linked guaranteed minimum death benefits (GMDB) with European-style geometric Asian and arithmetic Asian payoffs. Our method assumes that the underlying asset process follows a regime-switching Lévy model, which captures the key features of market dynamics in the continuous transition of the economy. To derive the approximate value of GMDB products, we employ the complex Fourier series (CFS) expansion method. Our error analysis demonstrates that this approach exhibits an exponential convergence rate. In our numerical experiments, we compare the CFS approach to other Fourier transform methods and Monte Carlo simulation. The results show that our method outperforms the other approaches in terms of both efficiency and accuracy. This paper contributes to the literature on pricing equity-linked GMDB products by proposing a novel and efficient approach based on regime-switching Lévy models and complex Fourier series expansion. The implications of our results may have significant practical implications for the insurance industry and financial markets.

Waveform Selection Based on Discrete Prolate Spheroidal Sequences for Near-Optimal SNRs for Photoacoustic Applications

Waveform engineering is an important topic in imaging and detection systems. Waveform design for the optimal Signal-to-Noise Ratio (SNR) under energy and duration constraints can be modelled as an eigenproblem of a Fredholm integral equation of the second kind. SNR gains can be achieved using this approach. However, calculating the waveform for optimal SNR requires precise knowledge of the functional form of the absorber, as well as solving a Fredholm integral eigenproblem which can be difficult. In this paper, we address both those difficulties by proposing a Fourier series expansion method to convert the integral eigenproblem to a small matrix eigenproblem which is both easy to compute and gives a heuristic view of the effects of different absorber kernels on the eigenproblem. Another important result of this paper is to provide an alternate waveform, the Discrete Prolate Spheroidal Sequences (DPSS), as the input waveform to obtain near optimal SNR that does not require the exact form of the absorber to be known apriori.

Analytical method for a tunnel with periodically jointed lining in the poroelastic soil under seismic waves

In this study, a longitudinal equivalent periodic shell (LEPS) model and corresponding continuous joint model are proposed for the periodically jointed lining (PJL) tunnel. Based on the LEPS model for the PJL tunnel, an analytical method for the PJL tunnel under seismic waves is established. The surrounding soil of the PJL tunnel is assumed to be the poroelastic medium, and described by the Biot's theory. The lining of the PJL tunnel is assumed to be composed of a series of lining rings linked periodically by ring joints along the longitudinal direction, which are simplified as a LEPS and described by the cylindrical shell theory. Due to the periodicity of the LEPS lining, the tunnel-soil system is treated as a periodic structure in this study. Based on the periodicity condition for the tunnel-soil system as well as the wave function expansion method, the representation for the scattering wave field in the soil is established. By using the aforementioned cylindrical shell theory and Fourier series expansion method, the Fourier space equations of motion and convolution type constitutive relations for the LEPS lining are derived. By using the lining-soil continuity conditions, the Fourier space equations of motion and convolution type constitutive relations for the LEPS lining as well as the representation for the wave field in the soil, the coupled equations for the tunnel-soil system are established, with which the Fourier components of the displacements of the LEPS lining and wave function coefficients for the scattering waves in soil can be solved for. Numerical results indicate that the presence of the ring joint in the LEPS lining of the tunnel will affect the distribution of the internal forces in the lining considerably, and it will also enhance the maximum internal forces occurring in the lining.

Pricing equity-linked guaranteed minimum death benefits with surrender risk by complex Fourier series expansion method

This paper considers the pricing problem of an equity-linked guaranteed minimum death benefit product with an expiration time of T, under the assumption of allowing surrender. To model the dynamics of the underlying asset process, we employ several jump diffusion models with regime switching. As many exotic financial products exhibit path dependency, Monte Carlo simulation methods have commonly been used for their valuation. In this paper, we propose a novel and efficient Fourier-based numerical method that utilizes complex Fourier series expansion to value such products. Through both error analysis and numerical experiments, we demonstrate that our proposed method is more effective than the Monte Carlo method. Finally, we conduct a sensitivity analysis on the contract parameters. Our findings have significant implications for the pricing of complex financial products since our proposed method can be readily applied to a variety of derivative products with path dependency.

Accurate Modeling and Optimization of Electromagnetic Forces in an Ironless Halbach-Type Permanent Magnet Synchronous Linear Motor

In order to solve the electromagnetic force optimization problem of a high-power-density ironless Halbach-type permanent magnet synchronous linear motor, this paper adopts an electromagnetic force optimization method based on magnetic field analysis, electromagnetic force modeling, and genetic algorithm optimization: Firstly, the magnetic field of the Halbach permanent magnet array is solved by the combination of the equivalent magnetization strength method and the pseudo-periodic method, which takes into account the influence of the edge effect of the secondary magnetic field, and the magnetic field of the primary winding is solved by Fourier series expansion method. Secondly, the Maxwell tensor method is used to establish the functional relationship between the electromagnetic thrust and the main structural parameters of the unilateral motor. Finally, based on the parameter sensitivity analysis of the optimized variables and the response surface calculation, the optimal combination of the optimized variables to meet the optimization objective is found by a genetic algorithm. This method of the accurate modeling and optimization of an electromagnetic force can accurately calculate the motor air gap magnetic field and electromagnetic thrust, and the optimization speed is fast, which can greatly save time. The optimization results show that, under the premise of constant input power, the unilateral average thrust of the motor is increased by 18.75%, the peak value of thrust fluctuation is decreased by 30.27%, and the results match well with the finite element results, which verifies the correctness of the optimization results of the electromagnetic force and the reasonableness of the optimization method.

Pricing ratchet equity index annuity with mortality risk by complex Fourier series method

In this paper, we consider the pricing of the ratchet equity-indexed annuity with the incorporation of mortality risk by an efficient and accurate method under the general time-changed Lévy processes. The pricing of the contract is a two-dimensional problem: we apply the complex Fourier series method to the log return dimension of the asset and high-order quadrature approximation to the other dimensions. Depending on the subordinating dynamic of the volatility, we use the time-changed Lévy process to model stochastic volatility via the random time change in the Lévy process, which compensates for limitations in the common Lévy process. By combining the recursive algorithm and complex Fourier series method, we obtain the approximate price of the contract. Error analysis proves that the total error can achieve an exponential convergence rate, and numerical examples demonstrate the accuracy and efficiency of the complex Fourier series expansion method.

Anti-plane dynamics of right-angle rare earth giant magnetostrictive medium with cylindrical conducting inclusion

In this paper, the anti-plane dynamics of rare earth giant magnetostrictive material with a cylindrical magnetically conducting inclusion in right-angle domain under the incidence of SH waves are studied. The governing equation and constitutive equation of magneto-elastic coupling are established. The outward scattered wave and internal standing wave caused by the inclusion are given by combining the complex function method, the wave function expansion method and mirror method. At the same time, the magnetic potential caused by wave is given by solving the Laplace equation. On the basis of boundary conditions, the Fourier series expansion method and effective truncated series are introduced to establish and solve infinite linear algebraic integral equations. Finally, based on numerical calculation, the analysis of dynamic stress concentration factor and magnetic field intensity concentration factor under different parameter combinations is given. The purpose of this paper is to provide an effective analytical method for the research of fracture dynamics of giant magnetostrictive materials under multi-magnetic field coupling.

Application of the Fourier Series Expansion Method for the Inversion of Gravity Gradients using Gravity Anomalies

Accurate and highly precise gravity gradient data are an important component of, for example, gravity field modeling, seabed topography inversion, and resource exploration. However, high-precision gravity gradient data are difficult to obtain. To address this difficulty, this work introduces the Fourier series expansion method to the modeling of gravity gradient fields. Based on gravity anomalies, the analytic expressions of the gravity gradient tensors have been deduced, which provides a new mathematical method for obtaining gravity gradient data. The expression’s derivation and verification processes are as follows. First, these analytic expressions for inverting the gravity gradient based on gravity anomaly data are derived according to the Laplace equation, the boundary value conditions of spherical approximation, and the Fourier series expansion method. Then, global 1’ × 1’ gravity field data provided by UCSD are used to verify the accuracy of these formulas. Finally, the results are analyzed. The experimental results show that the results obtained based on this inversion formula can sufficiently show the details of gravity gradient changes. The formulas derived in this paper have good computational efficiency in the inversion of regional gravity gradients and provide a new mathematical method for gravity gradient data acquisition.

Development of Converter Output Voltage Phase-Shift Keying Methods as Applied to a Seven-Phase Motor

Phase-shift keying (modulation) of the output voltage of a three-phase converter connected to a symmetrical three-phase winding is one of the stages of the motor vector control technology. This process has well been studied and lies in a practical application field. One of the supposed lines in the development of modern electric traction systems is to construct an electric drive based on a multiphase motor (with the number of phases more than three). In implementing a multiphase motor vector control technology, new opportunities appear, in particular, a new phase-shift keying method. The article considers the voltage phase-shift keying process as applied to a seven-phase winding with the aim to develop seven-phase converter control algorithms implementing the output voltage phase-shift keying, which is possible due to the features of a seven-phase symmetrical motor winding. In the study, the vector analysis techniques and Fourier series expansion methods were used. Experimental studies confirming the obtained theoretical results have been carried out. The scientific novelty of the work lies in that a new converter phase output voltage phase-shift keying method as applied to a seven-phase symmetrical winding and a new converter control algorithm implementing this method have been proposed. The results can be useful in implementing the vector control of a multiphase motor.

A unified approach for dynamic characteristic analysis of a standing-wave thermoacoustic engine with general impedance ends

As promising devices for energy conversion, thermoacoustic engines (TAEs) are attracting more and more research attention with regard to renewable thermal energy application. For a TAE, the oscillation frequency is a critical parameter in its design and operation, which can only be estimated currently through an often tedious process using a shooting method or a numerical iteration method, and an analysis model with classical boundary conditions. In this context, this study seeks to establish a unified model for dynamic analysis of a one-dimensional (1D) standing-wave TAE with general impedance ends. First, the differential equations governing the thermoacoustic behavior of the TAE and its impedance boundary conditions are simultaneously solved using the Galerkin method and an improved Fourier series expansion method. Then, the characteristic distributions of oscillation frequency, sound pressure, and volume flow rate are obtained by solving a standard eigenvalue problem. Finally, the effects of engine length, stack length and position, and impedance ends on the dynamic behavior of the TAE’s thermoacoustic system are investigated and analyzed. The results show that engine length has a significantly negative correlation with the modal frequencies in different working gases, and stack position and stack length have a slight effect on the oscillation frequency of the TAE. The arbitrary variations in impedance boundary conditions can be simulated from the open end to the rigid wall and all intermediate cases. The proposed model can efficiently predict changes in dynamic parameters of 1-D TAEs with impedance ends in a unified pattern and can serve as a reliable analysis tool for further research on TAEs coupled with various energy harvesters.

A Series Solution for SH Wave Scattering by Irregularly Shaped Surface Topographies

ABSTRACT This paper proposes a series solution for studying the ground motion of twin-peaks hill, double triangular hills, and hill-canyon composite topography. The propagation medium of SH wave is linearly elastic, isotropic, and homogeneous. A flexible multi-region-matching technique (MRMT) and complex function method are adopted to construct the wave field expression in each sub-region. To solve the unknown coefficients in the wave field expressions, Fourier series expansion method in the complex domain is adopted. Finally, some typical numerical examples are calculated to analyse the influence of the topography shape parameters, wave number, and the incident angle on ground motion.

Robust adaptive neural networks control for a class of time-delay parabolic systems with nonlinear periodic time-varying parameter

This paper mainly focuses on the stabilization of a class of parabolic systems with time-varying delay, external disturbance and nonlinear periodic time-varying parameter (NPTVP). Firstly, combined with Fourier series expansion (FSE) method and neural networks (NNs) approximation technology, a NNs approximator is designed to approximately describe the uncertain dynamic term with NPTVP. Then, based on adaptive control theory, NNs approximation technology and reparameterization method, two robust adaptive neural network control (RANNC) algorithms are designed to make the parabolic systems with time-varying delay, external disturbance and NPTVP achieve asymptotic stability and meet a prescribed adaptive H∞ performance of disturbance attenuation. Sufficient conditions of the asymptotical stability while satisfying the prescribed adaptive H∞ performance index for the resulting closed-loop systems are also derived. Finally, several simulations are carried out to verify the effectiveness of the designed RANNC algorithms.

Analytical solution for the consolidation of unsaturated vertical drain foundation under depth‐dependent initial conditions

AbstractInsufficient bearing capacity of foundations is hazardous for construction engineering, and improper construction scheme can also lead to unexpected problems. Vertical drains are always installed in the soil foundations to accelerate the consolidation process and thus improve the bearing capacity. According to the consolidation theory proposed by Fredlund, the consolidation of unsaturated vertical drain foundation is investigated with analytical solutions, in which smear effect, drain resistance, and depth‐dependent initial excess pore pressures (i.e., excess pore‐air and pore‐water pressures) are considered. In the derivative process, the decoupling method and Fourier series expansion method are applied. The reliability of the current solutions is verified by comparing the degenerate ones with existing solutions. Sensitivity analysis is conducted to assess the consolidation characteristics of the proposed model against modeling sizes, smear coefficients, drain resistance factors, and distribution coefficients. The analytical model has the advantages of simplicity and easy to implement in practical engineering.

Dynamical analysis of fractional oscillator system with cosine excitation utilizing the average method

In this paper, an analytical and numerical algorithm for solving displacement response of fractional oscillator system with cosine excitation is established. The analytical method is to solve steady‐state response and transient response solutions of the system by means of average method and then the total displacement response solution is the sum of steady‐state solution and transient solution. The numerical method is to use the Grünwald–Letnikov definition of fractional derivatives to discretize the fractional differential term in the system and then to reduce the order of the original system. Then, the approximate response solution of the system under the general periodic excitation is obtained by using the Fourier series expansion method and the superposition principle of linear system. Finally, the effectiveness and feasibility of the analytical technique are verified by numerical simulation, and the influence of fractional order, linear damping coefficient and fractional derivative coefficient on the amplitude of the steady‐state response and the total displacement response of the system is analyzed.

Three-dimensional theoretical analysis of seepage field in front of shield tunnel face

To evaluate hydraulic head distribution in front of a shield tunnel in a saturated soil layer, theoretical analysis and numerical simulations are carried out in this study. Based on the partial differential equilibrium equation of seepage flow, a three-dimensional (3D) theoretical analytical model of the shield tunnel face and the seepage field in front of it is established using the eigenfunction and the Fourier series expansion methods, and the hydraulic head calculation formula is derived. Combined with engineering cases, the theoretical analysis results and the 3D numerical simulation results are compared and analyzed. The effect of the water pressure of the tunnel face on the hydraulic head distribution is also analyzed. The results of the proposed analytical solution are in agreement with those of the numerical simulation solutions; moreover, the proposed analytical solution requires less time to calculate the seepage hydraulic head than the numerical simulation. The ratio of the initial water table to the diameter (D) of tunnel face has a more significant impact on the hydraulic head distribution at a position 0.5D above the tunnel vault. When the water pressure on the tunnel face is not considered, the values of the hydraulic head are significantly underestimated.

Agglomerated carbon nanotubes effects on buckling behavior of fiber metal laminated nanocomposite shells

In this research, the influence of carbon nanotubes agglomeration on the buckling behavior of multi-phase CNTs/fiber/polymer/metal composite laminates cylindrical shells under hydrostatic pressure was investigated. Governing equations were derived according to the Kirchhoff-Love’s first approximation shell theory and solved by a combination of the Galerkin and Fourier series expansion methods. Equivalent elastic properties of multi-phase CNTs/fiber/polymer/metal laminated (CNTFPML) cylindrical shell were obtained using the Eshelby-Mori-Tanaka approach, considering the dispersion and agglomeration effects. Primarily, CNTs were added to the polymer matrix and afterwards, this new matrix was reinforced by carbon or glass fiber materials. Finally, the composite layer was joined with metal layers and a hybrid shell prepared. The accuracy of the applied method was validated with the finite element method and experimental tests on carbon/epoxy and glass/epoxy composite cylinders under hydrostatic pressure. The results indicate that the CNTs agglomeration, weight and volume fraction of CNTs and type of fiber materials, have a key role on the critical buckling capacity of multi-phase composite shells.

Fourier Series Expansion Based Filter Parametrization for Equivariant Convolutions.

It has been shown that equivariant convolution is very helpful for many types of computer vision tasks. Recently, the 2D filter parametrization technique has played an important role for designing equivariant convolutions, and has achieved success in making use of rotation symmetry of images. However, the current filter parametrization strategy still has its evident drawbacks, where the most critical one lies in the accuracy problem of filter representation. To address this issue, in this paper we explore an ameliorated Fourier series expansion for 2D filters, and propose a new filter parametrization method based on it. The proposed filter parametrization method not only finely represents 2D filters with zero error when the filter is not rotated (similar as the classical Fourier series expansion), but also substantially alleviates the aliasing-effect-caused quality degradation when the filter is rotated (which usually arises in classical Fourier series expansion method). Accordingly, we construct a new equivariant convolution method based on the proposed filter parametrization method, named F-Conv. We prove that the equivariance of the proposed F-Conv is exact in the continuous domain, which becomes approximate only after discretization. Moreover, we provide theoretical error analysis for the case when the equivariance is approximate, showing that the approximation error is related to the mesh size and filter size. Extensive experiments show the superiority of the proposed method. Particularly, we adopt rotation equivariant convolution methods to a typical low-level image processing task, image super-resolution. It can be substantiated that the proposed F-Conv based method evidently outperforms classical convolution based methods. Compared with pervious filter parametrization based methods, the F-Conv performs more accurately on this low-level image processing task, reflecting its intrinsic capability of faithfully preserving rotation symmetries in local image features.

Scattering of SH waves by an isosceles flat-topped hill with multistage slopes

This paper presents a flexible mechanical model, which can be used to study the dynamic response of irregular surfaces with complex shapes under the action of seismic waves. Take flat-topped hill as an example, an isosceles model with complex boundaries is established, and an appropriate multi-region-matching technique (MRMT) is proposed to simplify the complex boundary value problem and solve it analytically. Based on the wave function expansion method and the complex function method, the wave field expressions of each subregion, including the singular shape subregion, are constructed. By introducing multipolar coordinate transformation and combining the continuous conditions at three auxiliary boundaries, the infinite algebraic equations are established. To solve the unknown coefficients in the wave field expressions, the Fourier series expansion method in the complex domain is adopted. Finally, some typical numerical examples are calculated to analyze the influence of the shape parameters of the flat-topped hill, the incident angle and the wave number of seismic waves on surface displacement amplitudes.