Series Expansion Method Research Articles

Thermal effect on vibration and buckling analysis of thin isotropic/orthotropic rectangular plates with crack defects

The present research is concerned with the vibration analysis of thin isotropic and orthotropic rectangular plates with crack defects under thermal environmental conditions. In the literature, there are only few studies reported in this direction. Based on the classical plate theory, the governing equations of the isotropic and orthotropic cracked rectangular plates can be derived, in which a surface crack located at the plate center is formulated based on a line-spring model. Since the dynamic behavior of structural elements is significantly affected by thermal effects, a thermal buckling analysis for isotropic and orthotropic plates is also conducted. A uniform heating load on the cracked rectangular plates is considered and the critical buckling temperature of the plates with or without cracks is investigated. The discrete singular convolution (DSC) method is then applied to formulate the eigenvalue equations for the cracked rectangular plates under various thermal conditions. The DSC technique is an ingenious method in stability and dynamic analysis of plates, not only it is a flexible local method to handle complex geometries and boundary conditions, but also it performs as a global approach with a high degree of accuracy. To go beyond the limitation of the DSC method, the use of Taylor’s series expansion method is incorporated for the treatment of free boundary conditions. In addition, this is the first attempt to explore its application on the analysis of cracked rectangular plates under thermal effects. In this work, the vibration of isotropic and orthotropic cracked rectangular plates with various combinations of boundary conditions is studied. A special restrained manner of simply supported conditions that are permissible for in-plane movements is also analyzed. The obtained solutions herein are compared with the existing results to verify the accuracy and reliability. Besides, accurate first-known solutions are also presented.

Interaction between two nanoscale elliptical holes with surface tension

In this paper, the plane problem of two elliptical nanoscale holes with surface tension is investigated. Firstly, the basic equations are given via the complex variable methods. Then, the stress boundary condition caused by surface tension is derived through the integral-form Gurtin–Murdoch model. The problem is finally solved by the conformal mapping along with the series expansion methods. The results show that the stress field decreases as the two holes become further away from each other. When the distance between the two holes is more than three times the sum of their sizes, the interaction between the two holes can be neglected. In addition, the stress field is greatly influenced by the orientation, aspect ratio and size of the holes. The positions of the maximum hoop stress are also discussed. When the two elliptical holes are put close horizontally, the hoop stress around one hole usually obtain its maximum at the endpoint close to the other hole. However, if one elliptical hole is not horizontal, the hoop stress around it will no longer attain its maximum at the endpoints. Another exception is that when one elliptical hole becomes larger, the hoop stress around the smaller hole would tend to achieve a local minimum at the endpoint close to the larger hole.

Phase transition of the frustrated antiferromagntic J1-J2-J3 spin-1/2 Heisenberg model on a simple cubic lattice

We have comprehensively investigated the frustrated J1-J2-J3 Heisenberg model on a simple cubic lattice. This model allows three regimes of magnetic order, viz., (π, π, π), (0, π, π) and (0, 0, π,), denoted as AF1, AF2, and AF3, respectively. The effects of the interplay of neighboring couplings on the model are studied in the entire temperature range. The zero temperature magnetic properties of this model are discussed utilizing the linear spin wave (LSW) theory, nonlinear spin wave (NLSW) theory, and Green’s function (GF) method. The zero temperature phase diagrams evaluated by the LSW and NLSW methods are illustrated, and are observed to exhibit different parameter boundaries. In certain regions and along the parameter boundaries, the possible phase transformations driven by the parameters are discussed. The results obtained using the LSW, NLSW, and GF methods are compared with those obtained using the series expansion (SE) method, and are observed to be in good agreement when the value of J2 is not close to the parameter boundaries. The ground state energies obtained using the LSW and NLSW methods are close to that obtained using the SE method. At finite temperatures, only the GF method is employed to evaluate the magnetic properties, and the calculated phase diagram is observed to be identical to the classical phase diagram. The results indicate that at the parameter boundaries, a temperature-driven first-order phase transition between AF1 and AF2 may occur along the boundary line. Along the AF1-AF3 and AF2-AF3 boundary lines, AF3 is less stable than AF1 and AF2. Our calculated critical temperature agrees with that obtained using Monte Carlo simulations and pseudofermion functional renormalization group scheme.

A hierarchy of dynamic equations for solid isotropic micropolar circular cylinders

This work considers the derivation procedure and evaluation of dynamic equations for isotropic micropolar circular cylinders by adopting a power series expansion method in the radial coordinate. Variationally consistent equations of motion together with pertinent sets of boundary conditions are expressed in a systematic fashion up to arbitrary order. The numerical results cover eigenfrequencies, mode shapes and field distributions over cross sections for axisymmetric and flexural motion adopting different sets of end boundary conditions for equations of different truncation orders of the present method. The results illustrate that the present approach may render benchmark solutions provided that higher order equations are used, and act as accurate approximate engineering solution for lower order equations.

Weighted Measurement Fusion Particle Filter for Nonlinear Systems with Correlated Noises.

The multi-sensor information fusion particle filter (PF) has been put forward for nonlinear systems with correlated noises. The proposed algorithm uses the Taylor series expansion method, which makes the nonlinear measurement functions have a linear relationship by the intermediary function. A weighted measurement fusion PF (WMF-PF) was put forward for systems with correlated noises by applying the full rank decomposition and the weighted least square theory. Compared with the augmented optimal centralized fusion particle filter (CF-PF), it could greatly reduce the amount of calculation. Moreover, it showed asymptotic optimality as the Taylor series expansion increased. The simulation examples illustrate the effectiveness and correctness of the proposed algorithm.

A Nanoscale Hole of Arbitrary Shape with Surface Elasticity

In this paper, we study the plane deformation of an arbitrarily-shaped nanoscale hole subjected to a uniform mechanical load at the infinity. Stress boundary condition along the hole surface is constructed by incorporating the deformation-dependent surface elasticity. The solution is given via the complex variable techniques, including the conformal mapping and series expansion methods. To verify the present formulation, we compare our results for the elliptical holes with those from the literature and observe good agreement between the two sets of results. Then we present numerical examples for four shapes of holes (ellipse, square, pentagon and triangle) with varying sizes to investigate the stress field with surface elasticity. The first aspect we are interested in is what factors can affect the stress field. Our results show that for nanoscale holes with surface elasticity, the stress field can be greatly influenced by the hole size and the hole shape. Specifically, when the hole size decreases, the hoop stress usually decreases, while the normal and shear stresses always increase. However, the hole size hardly influences the stress distribution pattern, which is, in fact, determined by the hole shape. In addition, the hole shape strongly influences the stress magnitude. Another aspect concerned is the positions of the maximum stresses. The results show that for the most cases, the stresses obtain their maximums at a certain corner of the shapes. However, for some shapes, there might be certain stresses that attain their maximums nearby, not at, the corner of the shapes.

Estimating the Gerber–Shiu function in the perturbed compound Poisson model by Laguerre series expansion

In this paper, we study the statistical estimation of the Gerber–Shiu function in the compound Poisson risk model perturbed by diffusion. This problem has been solved in [32] by the Fourier–Sinc series expansion method. Different from [32], we use the Laguerre series to expand the Gerber–Shiu function and propose a relevant estimator. The estimator is easily computed and has fast convergence rate. Various simulation studies are presented to confirm that the estimator performs well when the sample size is finite.

A complex-variable virtual crack extension finite element method for elastic-plastic fracture mechanics

The virtual crack extension (VCE) approach for computing the energy release rate using a complex-variable finite element method (ZFEM) is extended to nonlinear materials undergoing plastic deformation. The method consists of performing a numerical derivative of the potential energy with respect to a crack extension via the complex Taylor series expansion method (CTSE). ZFEM does not depend on the existence of a strain energy density and does not require contour or domain integrals. As a result, ZFEM overcomes the contour path dependency and monotonic loading limitations of the J-integral in elastic-plastic fracture problems. The nonlinear fracture analysis capability is demonstrated using material models based on the deformation and incremental theory of plasticity and using a loading/unloading cycle. Under monotonic loading, the ZFEM method remains robust at all levels of plasticity and its results are of the same accuracy as those provided by the ASTM standards and the saturated J-integral value. Moreover, when unloading the elasto-plastic material, the J-integral contours provide erroneous energy release rate values, whereas ZFEM results are in excellent agreement with those obtained by finite differencing.

A series expansion method aided design of current mode second generation current conveyor based active control circuit

Current mode circuits are receiving considerable interest from engineering community due to their unique characteristics that are not found in their voltage mode counterparts. This paper presents a series method based design of a current conveyor active control circuit. An algorithm based on the principle of the series expansion of the pulse transfer function is employed for obtaining the transfer function of the controller. The controller is subsequently realized using the second generation current conveyors. The proposed methodology is used to design a CCII controller for a practical system, which is subsequently simulated using the AD844 IC’s in the NI’s Multisim simulation tool. Subsequently, the controller was designed and the comparisons are drawn between the experimental and simulation results in order to validate the approach.

Hydrogen-like donor in a core-shell nanowire under electric and magnetic fields

The electric and magnetic field effects on the energy spectrum, the charge distribution and the electric dipole moment induced by the electron released from an off-axis donor confined inside the thin shell layer is studied by using the double Fourier series expansion method. The density of states exhibits strongly marked Aharonov-Bohm oscillations induced by the magnetic field, while the electric field modifies the charge distribution. The average electron-donor distance increases slowly as the electric field increases from zero, but then it rises sharply as the electric field exceeds a threshold value forming a giant permanent dipole moment.

Bound state solutions to the Schrödinger equation for some diatomic molecules

The bound state solutions to the radial Schrodinger equation are obtained in three-dimensional space using the series expansion method within the framework of a general interaction potential. The energy eigenvalues of the pseudoharmonic and Kratzer potentials are given as special cases. The obtained analytical results are applied to several diatomic molecules, i.e. $$\mathrm {N}_2, \mathrm {CO}, \mathrm {NO}$$ and $$\mathrm {CH}$$ . In order to check the accuracy of the present method, a comparison is made with similar results obtained in the literature by using other techniques.

An improved method for fuzzy–interval uncertainty analysis and its application in brake instability study

Most of the existing methods of brake squeal instability analysis are merely available to handle single type of uncertain case. In this study, an improved unified method is developed for uncertainty quantification, which is capable of handling two types of fuzzy–interval cases. In the first fuzzy–interval case, uncertain parameters of engineering structures are assumed as either fuzzy variables or interval variables, which exist in structures simultaneously and independently. In the second fuzzy–interval case, all uncertain parameters are represented by interval variables, but their lower and upper bounds just can be expressed as fuzzy variables instead of deterministic values. In the proposed method, fuzzy–boundary interval variables are introduced to handle fuzzy–interval uncertainties, and based on which an improved response analysis model is established. In the improved model, the fuzzy–boundary interval variables are firstly converted into interval-boundary variables by level-cut technique. Then by temporarily neglecting boundary uncertainties, the initial interval responses can be approximated via conducting once Taylor series expansion and subinterval analysis. Next, Taylor series expansion and central difference method are combined twice to deal with boundary uncertainties, and the interval responses of the structures with interval-boundary variables are yielded. Finally, the fuzzy–interval responses of the structures are derived on the basis of interval union operation and fuzzy decomposition theorem. The improved method is subsequently extended to quantify the uncertainties in brake squeal instability analysis involving two types of fuzzy–interval uncertainties. The effectiveness of the proposed method on tackling fuzzy–interval problems is demonstrated by numerical examples.

Ocean Surface Current Extraction Scheme With High-Frequency Distributed Hybrid Sky-Surface Wave Radar System

The high-frequency hybrid sky-surface wave radar (HF HSSWR) has recently been used to monitor large-area sea states. However, most of the HF HSSWR detection methods are based on the assumption of a no-tilt and constant height ionospheric model, and the influences caused by uneven electron density are ignored. This paper proposes a new surface current inversion scheme for the HF distributed HSSWR system, which considers the unknown ionospheric state as a black box and extracts the key parameters to compute the surface current based on a scattering model. The computational formula of the component of the current vector is explored using spatial scattering theory instead of an approximate bistatic model. In addition, the Fourier series expansion method is applied to the HF data to extract the real first-order Bragg frequency. Subsequently, the grazing angle and the bistatic angle can be found by inversion using the first-order Bragg frequency formula after searching out the common scattering patch of two receiving stations. Simultaneously, the coordinate registration of the currents can also be determined. The feasibility and effectiveness of this new algorithm are verified with field experimental results by comparing the current vectors derived from HSSWR and traditional HF SWR. The RMS differences of the magnitude and direction of the current vectors within the core common area of the two detection systems are about 10.2 cm/s and 9.5°, respectively.

Computational method for a fractional model of the helium burning network

Stellar cores may be considered as a nuclear reactor that play important role in injecting new synthesized elements in the interstellar medium. Helium burning is an important stage that contribute to the synthesis of key elements such as carbon, through the triple- α process, and oxygen. In the present paper, we introduce a computational method for the fractional model of the nuclear helium burning in stellar cores. The system of fractional differential equations is solved simultaneously using series expansion method. The calculations are performed in the sense of modified Riemann–Liouville fractional derivative. Analytic expressions are obtained for the abundance of each element as a function of time. Comparing the abundances calculated at the fractional parameter α=1, which represents the integer solution, with the numerical solution revealed a good agreement with maximum error ɛ=0.003. The product abundances are calculated at α=0.25,0.5,0.75 to declare the effects of changing the fractional parameters on the calculations.

Numerical study on the manoeuvring of a container ship in regular waves

ABSTRACTShips operate in seaways, but ship manoeuvrability is usually studied in calm water conditions, thus ignoring the effects of wave on the ship hydrodynamic behaviour. So, the assessment of surface ship manoeuvrability in wave environment is more realistic and accurate than its estimation in calm water condition. In the present study, captive dynamic ship model tests are numerically simulated in a computational fluid dynamics environment in regular head sea waves and the hydrodynamic derivatives are derived from the estimated force/moment time series using the Fourier series expansion method. These derivatives and wave excitation forces are fitted in the manoeuvring equations of motion and are solved to simulate the ship standard manoeuvres in head sea waves. Parameters of these definitive manoeuvres in wave condition are compared with those in still-water condition.

SERIES SOLUTION OF LAPLACE PROBLEMS

At the ANZIAM conference in Hobart in February 2018, there were several talks on the solution of Laplace problems in multiply connected domains by means of conformal mapping. It appears to be not widely known that such problems can also be solved by the elementary method of series expansions with coefficients determined by least-squares fitting on the boundary. (These are not convergent series; the coefficients depend on the degree of the approximation.) Here we give a tutorial introduction to this method, which converges at an exponential rate if the boundary data are sufficiently well-behaved. The mathematical foundations go back to Runge in 1885 and Walsh in 1929. One of our examples involves an approximate Cantor set with up to 2048 components.

Time fractional third-order variant Boussinesq system: Symmetry analysis, explicit solutions, conservation laws and numerical approximations

The current work provides comprehensive investigation for the time fractional third-order variant Boussinesq system (TFTOBS) with Riemann-Liouville (RL) derivative. Firstly, we obtain point symmetries, similarity variables, similarity transformation and reduce the governing equation to a special system of ordinary differential equation (ODE) of fractional order. The reduced equation is in the Erdelyi-Kober (EK) sense. Secondly, we solve the reduced system of ODE using the power series (PS) expansion method. The convergence analysis for the power series solution is analyzed and investigated. Thirdly, the new conservation theorem and the generalization of the Noether operators are applied to construct nonlocal conservation laws (CLs) for the TFTOBS. Finally, we use residual power series (RPS) to extract numerical approximation for the governing equations. Interesting figures that explain the physical understanding for both the explicit and approximate solutions are also presented.

Scattering of an arbitrary order acoustical Bessel beam by a rigid off-axis spheroid.

In this study, the authors use the partial-wave series expansion method in the spherical coordinate system and provide a theoretical formula for the off-axis acoustical scattering by rigid prolate and oblate spheroids illuminated by an arbitrary-order Bessel beam to explore the physical characteristics of the off-axis far-field scattering (the axis of the Bessel beam is parallel to the axis of the spheroids). The proposed method is verified and several three-dimensional far-field directivity patterns of rigid spheroids illuminated by zeroth- and first-order Bessel beams are depicted to explore the off-axis scattering. A special emphasis is placed on the geometry size of the prolate and oblate spheroids, the half-cone angle of the Bessel beam, the order of the Bessel beam, and the offset position. Compared with the on-axis scattering, the derivation and calculation of the off-axis scattering takes into account the spherical harmonic function using a numerical integral procedure. In addition, no symmetry was observed in the patterns of the far-field acoustic scattering form function for the zeroth- and higher-order Bessel beams with the off-axis illumination. This investigation provides a tool for the exploration of the complex scattering problems of underwater non-spherical particles under the illumination of the Bessel beam.

Free vibration analysis of dowelled rectangular isotropic thin plate on a Modified Vlasov soil type by using discrete singular convolution method

• There is a certain depth of soil beyond which frequency do not vary consistently. • Translational stiffness of Modified Vlasov soil is important in free vibration . • Rotational stiffness of Modified Vlasov soil is negligible in free vibration . • Dynamically activated depth affects frequency values for plates more than for beams. • Proposed method allows same calculation code for all boundary conditions by varying plate edge stiffness. The behaviour of plates resting on elastic foundations has wide interest because it is an important model for many engineering applications. Natural frequencies and corresponding mode shapes are guides to investigating the behaviour of a soil-structure system with time. Therefore, free vibration analysis constitutes a considerable part of research done on this topic. In this study, the effects of the stiffness of a surrounding Modified Vlasov soil type, the soil depth, and the aspect ratio on the free vibration of a thin rectangular dowelled plate on the Modified Vlasov soil type are investigated. We used a discrete singular convolution (DSC) with a Taylor series expansion method to apply the boundary conditions. The two formulas of the Taylor series expansion are used in the DSC to eliminate the fictitious points outside the physical domain. It is noticed that beyond a certain depth, neither the effective depth of the soil nor the, frequency varies consistently. Furthermore, the dynamically activated depth influences the frequency values for plates more than for beams. In addition, the effect of the elastic rotational stiffness of the surrounding soil is found to be negligible compared to that of the elastic vertical translational one. The results obtained can serve as a basis for the dynamic analysis of rigid pavements.

Energy of the amplitude mode in the bicubic antiferromagnet: Series expansion results

Series expansion methods are used to study the quantum critical behavior of the bicubic spin-1/2 antiferromagnet. Excitation energies are computed throughout the Brillouin zone, for both the N\'eel and dimer phases. We compute the energy of the amplitude/Higgs mode and show that it becomes degenerate with the magnon modes at the quantum critical point, as expected on general symmetry grounds.