Series Expansion Method Research Articles

Particulate flow of a viscous fluid driven by a torsionally oscillating disk

Abstract A novel hydrodynamic model is used to study oscillatory flow of a particle-laden fluid driven by a torsionally oscillating disk. The present model is featured by a modified form of Navier-Stokes equations which is conceptually different than existing related models and enjoys relatively concise mathematical formulation. Explicit expressions are derived for the radial, azimuthal and axial velocities using the method of power series expansions of small amplitude parameter, and the derived solutions reduce to Rosenblat's earlier results for a clear fluid driven by a torsionally oscillating disk in the absence of suspended particles. The implications of the present results to dusty gases are discussed in detail with particular interest in the effects of suspended particles on the decay indexes and wavelengths of the induced oscillatory velocity fields. The results obtained in this work can be used to quantify the effects of suspended particles on particulate flow driven by a torsionally oscillating disk.

Robust topology optimization of multi-material structures with overhang angle constraints using the material field series-expansion method

This paper presents a novel topology optimization method to enhance the manufacturability and practicality of multi-material structures. Traditional topology optimization techniques excel at generating high-performance designs, but the performance of these designs is often compromised due to the load perturbations encountered in real-world applications or post-processing for additive manufacturing (AM). This work addresses this challenge by incorporating three key features directly into the optimization process. First, the method integrates overhang angle constraints, ensuring designs are inherently manufacturable with AM. Second, it utilizes the material field series-expansion method for a significant reduction in design variables while simultaneously providing a clear and efficient description of material distribution within the optimized designs. Finally, to enhance design robustness under real-world scenarios with uncertain loads, the method employs Monte Carlo simulation, which is decoupled from the optimization process to reduce the computational cost. Numerical examples demonstrate the efficiency and effectiveness of the proposed method in generating robust designs of multi-material structures well-suited for AM. The proposed framework offers a promising pathway for the realization of complex, high-performance composite structures.

Laminar boundary layers in three‐dimensional MHD natural convective flow past a semi‐infinite vertical plate with variable sinusoidal suction

AbstractThe primary goal of this study is to investigate how the diffusion‐thermo and thermo‐diffusion affect a three‐dimensional hydromagnetic natural convective flow of a radiating fluid past a uniformly moving isothermal and isosolutal semi‐infinite vertical plate subjected to a sinusoidal suction velocity. The asymptotic series expansion method is adopted to solve the equations governing the flow model and to find the analytical solutions. The effect of variable suction on the flow and transport characteristics is mainly highlighted in the study. Both the diffusion‐thermo and thermo‐diffusion effects improve the fluid velocity.

Response of semicircular canyons and movable cylindrical cavities to SH waves in anisotropic half-space geology

The development of tunnels or the laying of underground pipelines are essential engineering projects in modern society, and in canyon tunnels and underground pipeline projects, the surface motion and cavity edge motion have been topics of concern in ground vibration problems. We investigate the wave scattering problem in an elastic half-space anisotropic medium containing a semicircular canyon and a subsurface movable cylindrical cavity by using the wave function expansion method, the complex function method, and the mirror method. By deriving the governing equation and transforming it into the standard form of the Helmholtz equation satisfying the zero-stress boundary condition, we solve the corresponding displacement functions. Introducing a position correction coefficient, the scattered wavefield in a half-space anisotropic medium is constructed by the mirror method, which improves the problem of scattered wave source singularity in anisotropic half-space media. Then, combining the free boundary conditions with a Fourier series expansion method, we solve the unknown coefficients in the equations. The correctness of the method is verified by degenerating it to a classical analytic solution. Finally, using frequency- and time-domain analysis, we investigate the effects of the relevant parameters on the surface motion [Formula: see text], the dynamic stress concentration factor, and the displacement amplitude [Formula: see text]. The results indicate that rock anisotropy and the presence of semicircular canyons have a significant effect on the dynamic response of subsurface structures. This not only provides a theoretical basis for practical unlined tunnels or pipeline projects but also can provide a basis for seismic design of underground structures.

Efficient valuation of variable annuities under regime-switching jump diffusion models with surrender risk and mortality risk

We present an efficient valuation approach for guaranteed minimum accumulation benefits (GMABs), guaranteed minimum death benefits (GMDBs), and surrender benefits (SBs) embedded in variable annuity (VA) contracts in a regime-switching jump diffusion model. We incorporate into the contract the risks of mortality and surrender, with these events generally monitored discretely over the life of the policy. Using a combination of the continuous-time Markov chain (CTMC) approximation and the Fourier cosine series expansion (COS) method, we determine that the valuation problem can be resolved within a regime-switching jump diffusion framework. Extensive numerical experiments showcase the efficiency of the proposed method, which proves to be more advantageous when compared to existing approaches like Monte Carlo (MC) simulation. The thorough analysis explores how model parameters affect the valuation outcomes.

(Almost) everything is a Dicke model - Mapping non-superradiant correlated light-matter systems to the exactly solvable Dicke model

We investigate classes of interacting quantum spin systems in a single-mode cavity with a Dicke coupling, as a paradigmatic example of strongly correlated light-matter systems. Coming from the limit of weak light-matter couplings and large number of matter entities, we map the relevant low-energy sector of a broad class of models in the non-superradiant phases onto the exactly solvable Dicke model. We apply the outcomes to the Dicke-Ising model as a paradigmatic example [Phys. Rev. Res. 2, 023131 (2020), Sci. Rep. 4, 4083 (2014)], in agreement with results obtained by mean-field theory [Sci. Rep. 4, 4083 (2014)]. We further accompany and verify our findings with finite-size calculations, using exact diagonalization and the series expansion method pcst^{++}++ [Phys. Rev. A 108, 013323 (2023)]

3-D Current Density and Magnetic Field of 3-D MR Scanner Gradient Coil

The topic of this paper is to describe the 3-D current density in the windings of a 3-D coil, which fills the volume between two coaxial cylinders at a precisely defined distance from each other, and which serves to generate a magnetic field gradient in the center of the cylinder axis. The 3-D current density is considered an unknown input quantity, which is calculated from the known gradient magnetic field output. It is an inverse problem in mathematics, where the direct problems are the calculation of unknown output quantities based on known input quantities. Fourier series expansion methods in the context of cylindrical coordinates were used to describe the 3-D current density. In that case, Bessel functions are used as development components. The current densities, at each point in space, were lined up to represent current lines. Each power line is associated with a coil winding through which a current of a certain strength flows. After that, the principle of discretization of coil windings was applied. Each winding is divided into a large number of elementary segments that were considered as current elements, which create, based on Bio-Savar's law, an elementary magnetic field. In this way, the total, continuous magnetic field is broken into many elementary components, which come from different current elements. An important result of this process is that each current element can be controlled independently by a current source. This means that the output magnetic field of the gradient can be controlled by current sources, which are the input sizes, and this is what is at the core of the topic of this paper.

The influence of gravity modulation on double-diffusive convection of Jeffrey fluids in an isotropic porous media

This article explores the double-diffusive convection of Jeffrey fluids under the gravity modulation. The primary objective is to investigate the impact of time-periodic gravitational variations on Darcy-Brinkman convection of Jeffrey fluids within a porous layer. A weak nonlinear analysis is employed, utilizing a power series expansion method, where disturbances are expressed as power series. Nusselt and Sherwood numbers are employed to measure heat and mass transport, respectively. The study analyzes the influence of various parameters, including the Jeffrey parameter, Darcy number, Vadasz number, Lewis number, Rayleigh-Darcy number, and solutal Rayleigh-Darcy number, on heat and mass transfer. Graphical representations of the results illustrate that an increase in the Jeffrey parameter, modulation amplitude, Lewis number, Vadasz number, and solutal Rayleigh-Darcy number enhances heat and mass transport, thereby promoting instability. Conversely, an increase in the Darcy number and modulation frequency leads to a decrease in heat transfer and mass transport. Comparisons between unmodulated and modulated cases highlight that the modulated case has a more significant effect on stability compared to the unmodulated case. This underscores the crucial role of external parameters, such as gravity modulation, in controlling heat and mass transfer within the system.

POLYNOMIAL SOLUTIONS FOR THE KOLMOGOROV–WIENER PREDICTION OF MODELED SMOOTHED HEAVY-TAIL PROCESS

Nowadays telecommunication traffic in systems with data packet transfer is considered as a heavy-tail random process. In a couple of rather simple models traffic is considered to be stationary one. In our recent papers we generated modeled heavy-tail data, which is based on the smoothing of the fractional Gaussian noise. In particular, the applicability if the continuous Kolmogorov–Wiener filter to the prediction of such data was investigated, the corresponding Wiener–Hopf integral equation was solved on the basis of the truncated Walsh function expansion. However, a question occurs – may another truncated orthogonal function expansion be applied to the problem under consideration? So, the corresponding investigation may be an actual question. In our recent papers we investigated theoretical fundamentals of the Kolmogorov–Wiener filter construction for different models, in particular, on the basis of the truncated polynomial expansion method and on the basis of the truncated trigonometric Fourier series expansion method. In this paper we restrict ourselves to the investigation of the applicability of the truncated polynomial expansion method to the problem under consideration, the corresponding method is based on the Сhebyshev polynomials of the first kind. The applicability of another polynomial or trigonometric expansions to the problem under consideration may be discussed in other papers. The aim of the work is to investigate the applicability of the Galerkin method based on the Chebyshev polynomials of the first kind to the Kolmogorov–Wiener prediction of smoothed heavy-tail data. The methodology consists in the solving of the Wiener–Hopf integral equation on the basis of the truncated polynomial expansion method which is based on the Chebyshev polynomials of the first kind. The scientific novelty consists in the proof of the fact that the Galerkin method based on the Chebyshev polynomials of the first kind may be applied to the Kolmogorov–Wiener prediction of smoothed heavy-tail data. The conclusions are as follows. The truncated polynomial expansion method based on the Chebyshev polynomials of the first kind may give reliable results in the framework of the Kolmogorov–Wiener prediction of smoothed heavy-tail data.

Linear analytical model for the seismic response of a straight‐jointed shield tunnel

AbstractIn this study, the segment joint and ring joint of the lining of the straight‐jointed tunnel are simplified as an equivalent open cylindrical shell with a small central angle and an equivalent closed cylindrical shell with a small width, and the lining of the straight‐jointed tunnel is thus simplified as an equivalent continuous periodic shell (ECPS) described by the cylindrical shell theory. Since the ECPS lining is periodic along the longitudinal direction, the tunnel‐soil system can thus be treated as a periodic system, which is referred to as the periodic tunnel‐soil system (PTSS) in this study. Based on the proposed ECPS lining model, an analytical method for the tunnel with the ECPS lining (ECPS tunnel) under seismic waves is established in this study. By employing the periodicity condition for the PTSS as well as the wave function expansion method, the representation for the wave field in the soil is established. With the aforementioned cylindrical shell theory and Fourier series expansion method, the convolution type constitutive relation for the ECPS lining and the Fourier space equations of motion are derived. By using the soil‐lining continuity condition and aforementioned formulations for the ECPS lining and soil, the coupled Fourier space equations of motion for the PTSS are established, with which the response of the ECPS lining and scattered waves in the soil can be determined.

Efficient strategy for topology optimization of stochastic viscoelastic damping structures

In this study, the dynamic topology optimization (TO) of stochastic viscoelastic damping structures (VDSs) is performed for the first time. However, the expensive computation cost seriously hinders the TO process. To address this problem, an efficient strategy is proposed. On the one hand, in the stochastic structural response analysis, a fully adaptive method based on direct probability integral method is presented to determine the number and locations of samples. Meanwhile, to improve the computational efficiency of structural response for each sample, a piecewise model-order reduction method based on Krylov subspace is adopted to generate the orthonormal basis and project the original large-scale system onto a low-order system. On the other hand, to overcome the optimization challenge arising from large number of design variables in the density-based topology method, a material-field series-expansion method is employed to describe the topology layout and significantly reduce the number of design variables. Moreover, the sensitivity of the optimization model is derived by the adjoint method and the method of moving asymptotes (MMA) is used to efficiently update the design variables. Some numerical examples comprehensively demonstrate the effectiveness and efficiency of the proposed strategy. The results indicate that the uncertainty of structural parameters greatly affect both the topology layout of VDS and vibration damping capacity.

Efficient valuation of guaranteed minimum accumulation benefits in regime switching jump diffusion models with lapse risk

We present a streamlined valuation method for the guaranteed minimum accumulation benefits incorporated within variable annuity contracts. At each contract renewal date, the insurer updates the policyholder's account value to the higher of the guaranteed value and the equity-linked investment value. In addition, we introduce lapse risks into the variable annuity contract, modeling the lapse decision under the assumption of stochastic intensity. Utilizing a combination of continuous-time Markov chain approximation and the Fourier cosine series expansion method, we derive closed-form valuation formulas under regime switching jump diffusion models. Numerical simulations showcase the precision and effectiveness of the proposed approach.

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.

Random dynamic response analysis of an asphalt concrete core wall dam on deep overburden with double random factors

The spatial variability of material parameters and the randomness of ground motion are factors that influence the seismic response of a dam. To obtain more reliable seismic response results, it is crucial to study the influence of various uncertain factors on the response to earthquakes. In this paper, the K-L series expansion method and LH sampling technique are combined to characterize the spatial correlation of material parameters using a Gaussian autocorrelation function. This allows for the simulation of random fields of material parameters. By improved the Clough-Penzien power spectrum model and incorporating the concept of a random function, it becomes possible to generate nonstationary random ground motion that accurately reflects the site conditions. Taking an asphalt concrete core dam on deep overburden as an example, a random dynamic response analysis was conducted, considering various random factors. Statistical analysis and distribution testing were conducted on the horizontal acceleration and vertical permanent deformation responses. The process of the evolution of the probability density of the horizontal acceleration at the top of the dam was determined using the generalized probability density evolution method. This method helps to reveal the influence of various random factors on the response of the dam body. The research results show that among the spatial variability of material parameters and the randomness of ground motions, the randomness of ground motions is the main influencing factor of the random response results of the dam body. The mean response of two random factors is significantly larger than that of a single random factor. When considering dual random factors simultaneously, the complexity of the random factors increases, resulting in statistical properties that differ from those of single random factors. Therefore, to obtain more reliable random seismic response results, it is necessary to consider both the spatial variability of material parameters and the randomness of ground motions.

Robust topology optimization of multi-material structures based on the material-field series-expansion method

The performance of composite structures can be dramatically improved through the synergy between different constituent materials. However, the topology optimization of multi-material structures poses a heavy computational burden, which becomes more significant when load uncertainty is considered. This article proposes an efficient method to address the problem of robust topology optimization of multi-material structures. Employing the property that the material-field series-expansion method can significantly reduce the number of design variables, the proposed method improves the computational efficiency of the robust optimization problem. Meanwhile, the modification of the original multi-material interpolation model improves the quality of the optimized designs. Monte Carlo simulation is used to treat the load uncertainty, but it is decoupled from the optimization process based on linear elasticity theory to reduce computation cost. The effectiveness of the proposed method is demonstrated by numerical examples, which show that robust design results can be efficiently generated by this method.

Computing N-dimensional polytrope via power series

Abstract Polytropic equations (Lane–Emden [LE] equations) are valuable because they offer a simple explanation for a star’s interior structure, interstellar matter, molecular clouds, and even spiral arms that can be calculated and used to estimate various physical parameters. Many analytical and numerical methods are used to solve the polytropic LE equation. The series expansion method played an essential role in many areas of science and has found application in many branches of physical science. This work uses the series expansion method to examine N-dimensional polytropes (i.e., slab, cylinder, and sphere). To solve LE-type equations, a computational method based on accelerated series expansion (ASE) is applied. We calculate several models for the N-dimensional polytropes. The numerical results show good agreement between the ASE and numerical and analytical models of the N-dimensional polytropes.

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.

Soret and Dufour effects on two dimensional steady MHD flow past a semi‐infinite vertical porous plate in the presence of thermal radiation and radiation heat absorption with constant heat and mass flux

AbstractThe problem of a two‐dimensional steady, MHD convective, incompressible, viscous flow past a uniformly moving semi‐infinite vertical porous plate embedded in a porous medium in the presence of radiation heat absorption, thermal radiation, thermal diffusion, and diffusion thermo effects with constant heat and mass flux are studied. In the fluid region, a transversely oriented, homogeneous magnetic field of intensity is applied. The non‐dimensional governing equations are solved analytically adopting asymptotic series expansion method. The impacts of several physical quantities including Grashof number, Schmidt number, Prandtl number, Soret number, radiation absorption, radiation parameter, magnetic field within the flow domain and transport characteristic have been explored graphically. It has been found that fluid velocity and skin friction fall with enhancing the values of magnetic parameter. Further, temperature and concentration fields rise with an increase in the Soret effect.

Harmonic wave scattered by an inclusion in an elastic plane: The complete Gurtin-Murdoch model

We consider the propagation of a harmonic elastic wave in a composite inclusion–matrix structure subjected to plane deformation. The interface between the inclusion and matrix is described by the complete Gurtin-Murdoch model with non-vanishing interface tension and interface stretching rigidity. We consider an inclusion of general shape and formulate the corresponding boundary value problem for the wave functions in the inclusion and matrix when a harmonic compressional or shear wave is incident on the edge of the matrix. The problem is then solved by series expansion methods for the case of a circular inclusion embedded in an infinite matrix. The series solution obtained is validated by checking its convergence and via comparisons with existing static and dynamic solutions for certain reduced cases. Numerical examples are presented for the case of a small-sized circular hole embedded in a soft matrix demonstrating the influence of surface tension on the incident wave-induced dynamic stress concentration in the matrix. We find that the presence of surface tension relieves the peak stress around the circular hole when the frequency of the incident wave is below a certain critical value, while it tends to intensify the peak stress for a high-frequency incident wave.

On Computing General Root Algorithms Based on Binomial Expansion, Bernoulli’s Method of Continuous Compound Interest, Series Expansion and Other Modification Methods

Most of the Pth root algorithms exhibit high latency or small convergence rates when iteratively computing the roots. Here we present a slew of algorithms based on series expansions of binomial form, and exponential terms that show low latency or small computational cost for finding the Pth root. The latency decreases if the family of series taken are truncated at higher terms. We show that Babylonian method converges quadratically while the binomial series show an increase in convergence rate from quadratic, cubic and higher order O(t^N) convergence rate as the series is sequentially truncated at higher order terms.