Complex Fourier Series Expansion Research Articles

Analysis of the thermo-hydro-mechanical coupled dynamic response of sediments under eccentric driving of deep-sea mining vehicle

This study establishes a coupled thermo-hydro-mechanical dynamic model(THMD) for saturated porous sediments under the eccentric motion of a deep-sea mining vehicle, considering the characteristics of deep-sea sediments. We applied the method of complex Fourier series expansion to simplify the coupled equations into ordinary differential equations and obtained the complex Fourier series expansion forms of various physical quantities of sediments during the eccentric movement of the mining vehicle. The undetermined constants are determined based on boundary conditions, yielding a series representation of the solution for THMD of deep-sea sediments. Subsequently, the solution was utilized to investigate the influence of mining vehicle parameters on the dynamic response of deep-sea sediments. This study demonstrates that the vertical displacement, vertical stress, and excess pore water pressure of sediments increase as the speed of the mining vehicle increases. Furthermore, the vertical displacement, vertical stress, and excess pore water pressure increase with the increment of load amplitude, and the differences in the curves at different eccentricities gradually become larger as the load amplitude increases. The proposed model can be applied to geotechnical engineering problems in saturated porous foundations, providing a necessary theoretical basis and analytical approach.

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.

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.

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.

Pricing equity-linked death benefits by complex Fourier series expansion in a regime-switching jump diffusion model

In this paper, we consider the valuation problem of equity-linked annuity with guaranteed minimum death benefit (GMDB) by complex Fourier series method under regime-switching jump diffusion models. We show that the price formulas can be expressed by some discounted density functions associated with the residual lifetime random variable. Fourier transforms for discounted density functions are derived, and complex Fourier series expansion method is applied to recover the discounted density functions. Some explicit formulas for computing the GMDB price are given. Finally, we give some numerical results to show effectiveness of our method.

Complex Fourier series expansion for the liquid-solid phase transition in PCM layers: transient and steady state periodic regimes

Semi-analytical solutions to the classical two phase Stefan problem are proposed. Time dependent solutions to the one-dimensional liquid-solid phase transition in a PCM wallboard subjected to isothermal and periodic Dirichlet boundary conditions are obtained. Transient and steady state solutions are found in finite size systems, and the semi-analytical solutions are validated through the asymptotic time limit behaviour of the phase transition. In this work, complex Fourier methods are proposed to find the solutions in the transient and steady state periodic regimes. Semi-analytical solutions based on the heat balance integral method (HBIM) are used to verify the consistency of the proposed method. The Fourier method can be pictured as a generalization of the phasors based method recently introduced by other authors. The proposed method incorporates a complete set of complex functions, which allows finding the transient and steady state response of the system. Finally, solutions for the time dependent interface position, liquid and solid temperature distributions and the thermal energy penetrating through the PCM wallboard, are shown. The solutions from the proposed method are found to be consistent when compared to the semi-analytical solutions estimated through the HBIM.

Nonparametric estimation of ruin probability by complex Fourier series expansion in the compound Poisson model

In this paper, we consider the estimation of ruin probability in the classical compound Poisson model. We first show that the ruin probability can be expressed by the complex Fourier series expansion. Then using a random sample on claim number and individual claim sizes we construct a nonparametric estimator of the ruin probability. Error analysis of the new estimator is made when the sample size is large, and simulation results are also provided to show the efficiency of our method when the sample size is finite.

Effect of porous layer on the Faraday instability in viscous liquid

A linear stability analysis of a viscous liquid on a vertically oscillating porous plane is performed for infinitesimal disturbances of arbitrary wavenumbers. A time-dependent boundary value problem is derived and solved based on the Floquet theory along with the complex Fourier series expansion. Numerical results show that the Faraday instability is dominated by the subharmonic solution at high forcing frequency, but it responds harmonically at low forcing frequency. The unstable regions corresponding to both subharmonic and harmonic solutions enhance with the increasing value of permeability and yields a destabilizing effect on the Faraday instability. Further, the presence of porous layer makes faster the transition process from subharmonic instability to harmonic instability in the wavenumber regime. In addition, the first harmonic solution shrinks gradually and becomes an unstable island, and ultimately disappears from the neutral curve if the porous layer thickness is increased. In contrast, the first and second subharmonic solutions coalesce, and the onset of Faraday instability is dominated by the subharmonic solution. In a special case, the study of Faraday instability of a viscous liquid on a porous substrate can be replaced by a study of Faraday instability of a viscous liquid on a slippery substrate when the permeability of the porous substrate is very low. Further, the Faraday instability can be destabilized by introducing a slip effect at the bottom plane.

Hedging and pricing early-exercise options with complex fourier series expansion

We introduce a new numerical method called the complex Fourier series (CFS) method proposed by Chan (2017) to price options with an early-exercise feature—American, Bermudan and discretely monitored barrier options—under exponential Lévy asset dynamics. This new method allows us to quickly and accurately compute the values of early-exercise options and their Greeks. We also provide an error analysis to demonstrate that, in many cases, we can achieve an exponential convergence rate in the pricing method as long as we choose the correct truncated computational interval. Our numerical analysis indicates that the CFS method is computationally more comparable or favourable than the methods currently available. Finally, the superiority of the CFS method is illustrated with real financial data by considering Standard & Poor’s depositary receipts (SPDR) exchange-traded fund (ETF) on the S&P 500® index options, which are American options traded from November 2017 to February 2018 and from 30 January 2019 to 21 June 2019.

Hedging and Pricing Early-Exercise Options With Complex Fourier Series Expansion

Hedging and Pricing Early-Exercise Options With Complex Fourier Series Expansion

Influence of the Slot and Pole Number Combination on the Eddy Current Loss of Permanent Magnet Machines

In this study, the influence of the combination of slot number and pole number of permanent magnet machines on the iron loss is mathematically investigated in order to obtain guidelines that can help reduce iron loss; such reduction can facilitate miniaturization and improve the efficiency. By simplifying the machine electromagnetic model, the iron loss can be evaluated for any combination of slot number and pole number. The distribution of the magnetomotive force on the stator teeth and its complex Fourier series expansion are obtained so that coordinate transform can be easily done and the rotor iron loss can thereby be calculated. A core factor is defined on the basis of the calculated iron losses, and a core factor map of slot and pole number combinations is prepared.

Linear elastic constitutive relation for multiphase porous media using microstructure superposition: Dry media

A constitutive relation for multiphase porous media containing a phase-change constituent is developed based on a periodic representative unit cell, microstructure superposition technique, and homogenization scheme. Multiphase porous mediae, consisting of different solid phases, fluid phases, and phase-change constituents, play an important role in variety engineering applications. A microstructure superposition method, stemmed from linear superposition theory, is devised such that the constitutive relation of the media can be obtained in an analytical form and the derivation is straightforward and tractable. The original microstructure of the media is firstly divided into several simple microstructure configurations. The microscopic stresses and strains are obtained for each individual microstructure configuration based on its own equilibrium and equivalent equations, and complex Fourier series expansion of strains. The microscopic strains are then superposed together to obtain the constitutive relation of the multiphase porous media using homogenization scheme. The derived constitutive relation can be applied to study the mechanical behavior of porous media consisting of two solid phases with one being able to change phase between solid and liquid and one liquid phase. As a particular application of the derived constitutive relation, the mechanical behavior of porous media with a single solid phase under dry condition was studied and the results were consistent with other models and agreed well with available experimental data.

257 BEMによる網目状構造物の効率的な腐食解析(OS08-2 境界要素法とその周辺技術(2))(OS08 境界要素法とその周辺技術)

An efficient numerical method is proposed for three-dimensional corrosion analysis of structures having net geometory, such as rebar in concrete, fence with sea-side structure. The knowledge of physical quantities (potential and current density) on the surface of corroding materials is very important in corrosion monitoring and corrosion protection. Boundary element method is suitable for corrosion analysis, because it doesn't require discretization of internal domain. Enormous number of elements is necessary to discretize the structure having net geometory. To reduce the number of boundary elements, our method is based on complex Fourier series expansion of boundary quantities in circumferential and axial direction of net structures. In order to demonstrate the usefulness of the method, Example problem of rebar in concrete with cathodic protection system are presented.

パイプ形状を有する非軸対称3次元腐食問題の効率的な境界要素解析法

An efficient numerical method is proposed for non-axisymmetric three-dimensional corrosion problems involving axisymmetric structures. The knowledge of physical quantities (potential and current density) on the surface of corroding materials is very important in corrosion monitoring and corrosion protection. Boundary element method is suitable for corrosion analysis, because it doesn't require discretization of internal elements. Although geometry of a structure is axisymmetric, enormous number of elements is necessary to discretize the realstructure, since surface physical quantities of the axisymmetric structure are affected by otherstructures and electrodes. To reduce the boundary elements, our method is based on complex Fourier series expansion of boundary quantities in circumferential direction of axisymmetricstructures. When axisymmetric structures depart from other structures and electrodes, this method has the advantage of decreasing the number of series, because surface physical quantities are more axisymmetric. In order to demonstrate the usefulness of the method, two example problems of buried pipelines in soil and a sea-water pump for cathodic protection system are presented.

ON THE USE OF FFT ALGORITHM FOR THE CIRCUMFERENTIAL CO-ORDINATE IN BOUNDARY ELEMENT FORMULATION OF AXISYMMETRIC PROBLEMS

A new numerical method is proposed for the boundary element analysis of axisymmetric bodies. The method is based on complex Fourier series expansion of boundary quantities in circumferential direction, which reduces the boundary element equation to an integral equation in (r–z) plane involving the Fourier coefficients of boundary quantities, where r and z are the co-ordinates of the (r, θ, z) cylindrical co-ordinate system. The kernels appearing in these integral equations can be computed effectively by discrete Fourier transform formulas together with the fast Fourier transform (FFT) algorithm, and the integral equations in (r–z) plane can be solved by Gaussian quadrature, which establishes the Fourier coefficients associated with boundary quantities. The Fourier transform solution can then be inverted into (r, θ, z) space by using again discrete Fourier transform formulas together with FFT algorithm. In the study, first we present the formulation of the proposed method which is outlined above. Then, the method is assessed by using three sample problems. A good agreement is observed in the comparisons of the predictions of the method with those available in the literature. It is further found that the proposed method provides considerable saving in computer time compared to existing methods of literature. © 1997 by John Wiley & Sons, Ltd.

State of stress in a plat circular ring with a crack: PMM vol. 39, n≗ 6, 1975, pp. 1118–1128

The stress distribution in a circular isotropie ring with a crack on part of the concentric circle is investigated. A system of functional equations governing the coefficients of the complex Fourier series expansion of the stresses acting on the circle on which the crack is located is obtained. The solution of the mentioned system of equations is obtained by using a factorization method, which permitted reduction of the initial system of equations to two coupled infinite systems of algebraic equations. The possibility of using the method of truncation to solve these systems is proved. The singularity originating in the neighborhood of ends of the crack in the formulas governing the stresses is isolated. Stress intensity coefficients for the effect of a uniform load on the external contour are presented.