Numerical analysis of the one-dimensional heat equation subject to a boundary integral specification

In this research a numerical technique is developed for the one-dimensional heat equation that combines classical and integral boundary conditions. New matrix formulation techniques with arbitrary polynomial bases are proposed for the numerical/analytical solution of this kind of partial differential equation. We give a simple and efficient algorithm based on an iterative process for numerical solution of the method.     

The attenuation of the higher-order cross-section modes in a duct with a thin porous layer

A numerical method for sound propagation of higher-order cross-sectional modes in a duct of arbitrary cross-section and boundary conditions with nonzero, complex acoustic admittance has been considered. This method assumes that the cross-section of the duct is uniform and that the duct is of a considerable length so that the longitudinal modes can be neglected. The problem is reduced to a two-dimensional (2D) finite element (FE) solution, from which a set of cross-sectional eigen-values and eigen-functions are determined. This result is used to obtain the modal frequencies, velocities and the attenuation coefficients. The 2D FE solution is then extended to three-dimensional via the normal mode decomposition technique. The numerical solution is validated against experimental data for sound propagation in a pipe with inner walls partially covered by coarse sand or granulated rubber. The values of the eigen-frequencies calculated from the proposed numerical model are validated against those predicted by the standard analytical solution for both a circular and rectangular pipe with rigid walls. It is shown that the considered numerical method is useful for predicting the sound pressure distribution, attenuation, and eigen-frequencies in a duct with acoustically nonrigid boundary conditions. The purpose of this work is to pave the way for the development of an efficient inverse problem solution for the remote characterization of the acoustic boundary conditions in natural and artificial waveguides.     

Free vibration analysis of continuous orthotropic bridge decks

While studies of the free vibration problem of single span bridge slabs have been undertaken by a number of authors, literature on continuous span orthotropic bridge slabs is rather scarce. Furthermore, general continuous bridge deck problems have been dealt with by approximate methods only for specific types of boundary conditions. In this paper an attempt is made to formulate a general analytical solution which would be applicable to all types of boundary conditions. The solution developed is discussed here with special reference to “bridge type” boundary conditions. The analysis is based on the ordinary theory of thin plates and is formulated for linearly elastic materials with isotropic or orthotropic properties. A Levy-type series solution is employed and the problem of free vibration analysis of continuous isotropic and orthotropic bridge slabs is solved by using the principle of superposition. The solution is tested for convergence by varying the number of terms in the solution and the convergence is found to be excellent. Results obtained for continuous isotropic bridge decks are compared with published solutions and close agreements are found. For orthotropic bridge decks a similar comparison was not possible because of a lack of published results in the technical literature.     

A Chebyshev–Lagrangian method for acoustic analysis of a rectangular cavity with arbitrary impedance walls

A general Chebyshev–Lagrangian method is proposed to obtain the analytical solution for a rectangular acoustic cavity with arbitrary impedance boundary conditions. The originality of the present paper is the successful attempt of applying orthogonal polynomials, such as Chebyshev polynomials of the first kind, to the analysis of a rectangular sound field with general wall impedance. The sound pressure is uniformly expressed as triplicate Chebyshev polynomial series which is independent in each direction. The Chebyshev polynomial series solution is obtained using the Rayleigh–Ritz procedure after considering the influence of boundary impedance on the cavity as the work done by the impedance surfaces in the Lagrangian function. The accuracy and reliability of the proposed method are validated against the analytical solutions and some numerical results available in the literature. Excellent orthogonality and complete properties of the Chebyshev polynomials ensure the rapid convergence, numerical stability, high accuracy of the current solution. The simplicity and low computational cost of the present approach make it preferable to obtain the results of complex models even in the relative high frequency range by choosing enough truncated terms in the sound pressure expression. Numerous cases with various uniform or non-uniform impedance boundary conditions are analyzed numerically and some of the results can be used as benchmark. It is shown that the impedance boundary condition can effectively influence or modify the acoustic characteristics and response of a cavity.     

The R-function method for the free vibration analysis of thin orthotropic plates of arbitrary shape

Free flexural vibrations of homogeneous, thin, orthotropic plates of an arbitrary shape with mixed boundary conditions are studied using the R-function method. The proposed method is based on the use of the R-function theory and variational methods. In contrast to the widely used methods of the network type (finite differences, finite element, and boundary element methods), in the R-function method all the geometric information given in the boundary value problem statement is represented in an analytical form. This allows one to seek a solution in a form of some formulas called a solution structure. These solution structures contain some indefinite functional components that can be determined by using any variational method. A method of constructing the solution structures satisfying the required mixed boundary conditions for eigenvalue plate bending problems is described. Numerical examples for the vibration analysis of orthotropic plates of complex geometry with mixed boundary conditions for illustrating the aforementioned R-function method and comparison against the other methods are made to demonstrate its merits.     

Large deflection of clamped circular plate and accuracy of its approximate analytical solutions

A different set of governing equations on the large deflection of plates are derived by the principle of virtual work(PVW), which also leads to a different set of boundary conditions. Boundary conditions play an important role in determining the computation accuracy of the large deflection of plates. Our boundary conditions are shown to be more appropriate by analyzing their difference with the previous ones. The accuracy of approximate analytical solutions is important to the bulge/blister tests and the application of various sensors with the plate structure. Different approximate analytical solutions are presented and their accuracies are evaluated by comparing them with the numerical results. The error sources are also analyzed. A new approximate analytical solution is proposed and shown to have a better approximation. The approximate analytical solution offers a much simpler and more direct framework to study the plate-membrane transition behavior of deflection as compared with the previous approaches of complex numerical integration.     

Boundary conditions for the bending of a piezoelectric beam

For beam bending in transversely isotropic piezoelectric media, the reciprocal theorem and the general solution of piezoelasticity are applied in a novel way to obtain the appropriate stress and mixed boundary conditions accurate to all orders for the beam of general edge geometry and loadings. By generalizing the method developed by Gregory and Wan, a set of necessary conditions on the edge-data for the existence of a rapidly decaying solution is established. The prescribed edge-data of the beam must satisfy these conditions in order that they could generate a decaying state within the beam. When stress and mixed conditions are imposed on the beam edge, these decaying state conditions for the case of bending deformation of piezoelectric beam are derived explicitly. They are then used for the correct formulation of boundary conditions for the beam theory solution (or the interior solution). Besides, an analytical solution of elastic beam is formulated to verify validity of our boundary conditions. For the stress data, our boundary conditions coincide with those obtained in conventional forms of beam theories. More importantly, the appropriate boundary conditions with two sets of mixed edge-data are obtained for the first time.     

Natural frequencies of a pre-twisted blade in a centrifugal force field

In this paper, starting with the thin shell theory, the governing partial differential equation of motion for the transverse deflection of a rotating pre-twisted plate is derived. Strain-displacement relationships include the effect of warping of the cross-section due to twist-bend coupling effect introduced as a result of pre-twist in the plate of non-circular (rectangular) cross-section. Then the equation of motion, thus derived, is used to formulate the free vibration of a typical turbo-machinery cantilevered airfoil blade by considering it as a plate of an equivalent rectangular cross-section subjected to a quasi-static load due to centrifugal force field. The analytical derivation considers both the stress-stiffening as well as stress-softening effects of the centrifugal forces on the spinning airfoil. The partial differential equation governing the flexural motion of the plate is transformed into a matrix-eigenvalue form using a Rayleigh-Ritz technique. The plate deformations are represented by a set of ‘admissible’ sinusoidal trial functions, which fully satisfy all the clamped-end constrains as well as the free-edge boundary conditions. The results of the analytical model exhibit an excellent agreement with the previously published test data both for thin and thick plate geometries and even in highly twisted configurations. The results of the eigenvalue solution are presented in a non-dimensional form for plates of varying aspect ratios and different amounts of pre-twist in the plate. The numerical results are directly applicable in determining the static and running frequencies of typical blades used in turbo-machinery.     

超短脉冲径向偏振光束的解析解及时空耦合效应

从超短脉冲光束的傍轴传输方程出发, 运用傅里叶变换和相关数学算符的对易性, 得到了超短脉冲径向偏振光束的解析表达式.该解可适用于任意脉冲驱动的径向偏振光束. 基于该解析表达式并结合具体例子,讨论了超短脉冲径向偏振光束在自由空间中的传输性质. 结果表明, 在传输过程中时空耦合主要体现在光束边沿的脉冲延迟. 这一效应导致了脉冲不同时间位置处横向光强分布随传输的变化, 以及脉冲前后沿关于束腰的不对称性. 本文的方法同样适用于得到超短脉冲方位角偏振光束的解析解和传输性质.     

Design of shaped piezoelectric modal sensor for beam with arbitrary boundary conditions by using Adomian decomposition method

The theory for designing distributed piezoelectric modal sensors is well established for beam structures. However, the current modal sensor theory is limited in scope in that it can only be applied in the case of classical boundary conditions (i.e., either clamped, free, simply supported or sliding). In this paper a solution to the problem of finding the shape of piezoelectric modal sensors for a beam with arbitrary boundary conditions is proposed, using the Adomian decomposition method (ADM). A general expression for designing the shape of a piezoelectric modal sensor is presented, in which the output signal of the designed sensor is proportional to the response of the target mode. Other modes are filtered out. The modal sensor shape is expressed as a function of the second spatial derivative of the structural mode shape function. Based on the ADM and employing some simple mathematical operations, the closed-form series solution of the second spatial derivative of the mode shapes can be determined. Then the shapes of the designed modal sensors are obtained. Finally, some numerical examples are given to demonstrate the feasibility of the proposed modal sensors. It is shown that, for classical boundary conditions, the shapes of the modal sensors based on the ADM agree well with analytical and numerical results given in the literature. For general boundary conditions it is found that the shape of the modal sensors is influenced by the number of modes of interest because the second spatial derivatives of the mode shapes are not orthogonal to one another. The modal sensors for general boundary conditions can be considered as modal filters within a limited frequency band.     

基于局部加密纯无网格法非线性Cahn-Hilliard方程的模拟

为数值求解描述不同物质间相位分离现象的高阶非线性Cahn-Hilliard(C-H)方程,发展了一种基于局部加密纯无网格有限点集法(local refinement finite pointset method,LR-FPM).其构造过程为:1)将C-H方程中四阶导数降阶为两个二阶导数,连续应用基于Taylor展开和加权最小二乘法的FPM离散空间导数;2)对区域进行局部加密和采用五次样条核函数以提高数值精度;3)局部线性方程组求解中准确施加含高阶导数Neumann边值条件.随后,运用LR-FPM求解有解析解的一维/二维C-H方程,分析粒子均匀分布/非均匀分布以及局部粒子加密情况的误差和收敛阶,展示了LR-FPM较网格类算法在非均匀布点情况下的优点.最后,采用LR-FPM对无解析解的一维/二维C-H方程进行了数值预测,并与有限差分结果相比较.数值结果表明,LR-FPM方法具有较高的数值精度和收敛阶,比有限差分法更易数值实现,能够准确展现不同类型材料间相位分离非线性扩散现象随时间的演化过程.     

Analytical and numerical results for vibration analysis of multi-stepped and multi-damaged circular arches

This paper investigates the in-plane linear dynamic behaviour of multi-stepped and multi-damaged circular arches under different boundary conditions. Cracked cross-sections are modelled as massless elastic rotational hinges. In damaged configuration, cracks can be located both at the interface between two adjacent portions as well as inside the portion itself. For each arch portion bounded by two cracks, the differential equations of motion have been established considering axial extension, transverse shear effects and rotatory inertia. The equilibrium equations of arch portions are combined in the coupled fundamental system in terms of radial displacement, tangential displacement and rotation. Analytical and numerical solutions for multi-stepped arches, in undamaged as well as in damaged configurations, are proposed. The analytical solution is based on the Euler characteristic exponent procedure involving the roots of characteristic polynomials, while the numerical method is focused on the Generalized Differential Quadrature (GDQ) method and the Generalized Differential Quadrature Element (GDQE) technique. Numerical results are shown in terms of the first 10 analytical and numerical frequencies of multi-stepped and multi-damaged arches with different boundary conditions. Finally, convergence and stability characteristics of the GDQE procedure are investigated. The convergence rate of the natural frequencies is shown to be very fast and the stability of the numerical procedure is very good.     

Numerical-experimental hybrid method for stress separation in digital gradient sensing method

A numerical-experimental hybrid method for the stress separation in the digital gradient sensing (DGS) method is proposed in this study. In the proposed hybrid method, boundary conditions for a local finite element model, that is, nodal force along boundaries are inversely determined from experimental values obtained by the digital gradient sensing method. The hybrid method follows two stages. In stage 1, the DGS method measures the Cartesian stress gradient components directly and, subsequently, the sum in Cartesian stresses at all interesting points on the surface; stress sum are used to compute the unknown boundary conditions for the local model. In stage 2, the individual stress components are calculated by the direct finite element method using the computed boundary conditions from stage 1. The effectiveness is demonstrated by applying the proposed method to a stress concentration problem involving concentrated load acting on an edge of a large planar sheet. The individual stress components thus determined are summed and compared with analytical stress sum, confirming the effectiveness and accuracy of the proposed technique.     

Integral transform method in radiative transfer

A method based on integral Fourier transform technique is proposed for obtaining the radiation intensity in a nonconservative, isotropically scattering, finite slab with a general internal isotropic source and with general transparent boundary conditions. The theory of the method is first presented and it is shown to provide an exact analytical solution for the problem. Numerical results for various physical situations are then reported and compared with previous other results wherever possible. The case of the universal functions Θ(τ) and Θ_g(τ) is also considered.     

Depolarization of the electromagnetic field scattered by a three-dimensional large-scale irregularity of the lower ionosphere

This paper develops analytical and numerical methods for the solution of three-dimensional problems of radio wave propagation. We consider a three-dimensional vector problem for the electromagnetic field of a vertical electric dipole in a planar Earth-ionosphere waveguide with a largescale local irregularity of negative characteristics at the anisotropic ionospheric boundary. The field components at the boundary surfaces obey the Leontovich boundary conditions. The problem is reduced to a system of two-dimensional integral equations taking into account the overexcitation and depolarization of the field scattered by the irregularity. Using asymptotic (with respect to the parameter kr≫1, where r is the distance from the source or receiver to the nearest point of the irregularity, k=2π/λ, and λ is the radio wavelength) integration over the direction perpendicular to the ray path, we transform this system to one-dimensional integral equations where integration contours represent the geometric contour of the irregularity. The system is numerically solved in the diagonal approximation, combining direct inversion of the Volterra integral operator and subsequent iterations. The proposed numerical algorithm reduces the computer time required for the solution of this problem and is applicable for studying both small-scale and large-scale irregularities. We obtained novel estimates for the field components that are not excited by the source but result entirely from scattering by the sample three-dimensional ionospheric irregularity.     

Electromagnetic plane-wave scattering from a sectorial groove in a perfectly conducting plane

Scattering characteristics of plane waves by a sectorial groove in a perfectly conducting plane are investigated. Both the transverse magnetic (TM) and transverse electric (TE) polarizations of the incident wave are considered. Judicious use of the region-matching technique provides a rigorous series solution to the problem. The analyzed region is separated into two sub-regions by choosing a semi-circular auxiliary boundary. Thefield in each sub-region is expanded as a summationof proper wave functions with unknown coefficients. Enforcing the matching of conditions on the auxiliary boundary and of boundary condition on the circular-arc surface of the groove leads to a linear set of equations and the unknown coefficients are then determined. Numerical results demonstrate the influence of central angles of the sectorial groove on echo width, far-field pattern and near-field distribution. The presented geometry is easily applicable to the design and fabrication of a grating structure for optical switches and tunable filters.      

Free vibration analysis of beams and rectangular plates with free edges by the discrete singular convolution

In this paper, free vibration of beams, annular plates, and rectangular plates with free boundaries are analyzed by using the discrete singular convolution (DSC). A novel method to apply the free boundary conditions is proposed. Detailed derivations are given. To validate the proposed method, eight examples, including the free vibrations of beams, annular plates and rectangular plates with free boundaries are analyzed. Two kernels, the regularized Shannon's kernel and the non-regularized Lagrange's delta sequence kernel, are tested. DSC results are compared with either analytical solutions or/and differential quadrature (DQ) data. It is demonstrated that the proposed method to incorporate the free boundary conditions is simple to use and can yield accurate frequency data for beams with a free end and plates with free edges. Thus, the proposed method for applying the boundary conditions extends the application range of the DSC.     

An exact analytical approach to the free vibration analysis of rectangular plates with mixed boundary conditions

A comprehensive analytical technique is developed for the free vibration analysis of rectangular plates with discontinuities along the boundaries. For illustrative purposes a solution is obtained for plates with edges partially clamped and partially simply supported and plates with edges partially and partially simply supported. A vast array of first mode eigenvalues is provided for these families of plates. Solutions to the equations are obtained by exploiting a mathematical technique described by the author during an earlier publication. It is shown that eigenvalue matrices are easily generated for a wide range of plates with discontinuities in boundary conditions.     

A simple enthalpy-based lattice Boltzmann scheme for complicated thermal systems

To extend the lattice Boltzmann (LB) method to describe the applicable energy systems, the first key step is to build a suitable thermal LB model and corresponding boundary treatments. There are two main shortcomings in the existing related works: either some additional energy source terms are inconvenient to be naturally incorporated or the implementation of non-Dirichlet-type thermal boundary conditions is extremely difficult and sometimes impossible in them for complicated thermal systems, which restrict their applicability to only a few special classes of problems. In order to overcome these drawbacks by a simple way, in this paper a thermal LB model and corresponding boundary treatments are constructed based on the total enthalpy. The specific benefits due to the introduction of the total enthalpy are analyzed and it is found that the numerical results obtained by the present scheme agree well with the analytical solutions and/or the data reported in previous studies.