Asymmetric Green's functions for a functionally graded transversely isotropic tri-material

This paper considers the elastic analysis of a functionally graded transversely isotropic tri-material solid under the arbitrary distribution of applied static loads. Using two displacement potential functions, for three-dimensional point-load and patch-load configurations, Green's functions for displacement and stress components are generated in the form of infinite line-integrals. These solutions are shown to be analytically reducible to the special cases of exponentially graded bi-material, exponentially graded half-space and homogeneous tri-material Green's functions. It also encompasses a functionally graded finite layer on a rigid base with surface loading with two cases of interfacial conditions, rigid-bonded and rigid-frictionless. Finally, for the special case of a functionally graded layer sandwiched between two homogeneous layers, using several numerical displays, the effect of material inhomogeneity on the responses is studied and the accuracy of numerical scheme is verified.     

3D dynamic Green’s functions in a multilayered poroelastic half-space

The complete 3D dynamic Green’s functions in the multilayered poroelastic media are presented in this study. A method of potentials in cylindrical coordinate system is applied first to decouple the Biot’s wave equations into four scalar Helmholtz equations, and then, general solutions to 3D wave propagation problems are obtained. After that, a three vector base and the propagator matrix method are introduced to treat 3D wave propagation problems in the stratified poroelastic half-space disturbed by buried sources. It is known that the original propagator algorithm has the loss-of-precision problem when the waves become evanescent. At present, an orthogonalization procedure is inserted into the matrix propagation loop to avoid the numerical difficulty of the original propagator algorithm. At last, the validity of the present approach for accurate and efficient calculating 3D dynamic Green’s functions of a multilayered poroelastic half-space is confirmed by comparing the numerical results with the known exact analytical solutions of a uniform poroelastic half-space.     

Variational iteration method for elastodynamic Green’s functions

In this study, a new application of the variational iteration method is presented. We use this method to obtain approximations to 3D Green’s function for the dynamic system of anisotropic elasticity. The numerical results obtained from convolution of Green’s function and data of the Cauchy problem are compared with the exact solution for cubic media.     

On boundary-volume element discretization of inhomogenous elastodynamic problems

This paper presents an integral equation formulation and its discretization scheme for the elastodynamic problem in which the material properties are prescribed as arbitrary, continuous and differentiable functions of the spatial coordinates. The formulation is made by using the Green's function for the corresponding problem in homogenous elasticity. From a weighted residual statement of the problem, the governing differential equation is transformed into a set of the integral equations in the inner domain as well as on the boundary. These integral equations are discretized by introducing a finite number of the boundary-volume-time elements, and the solution for the system of linear equations thus obtained is discussed.     

On the Green’s function for the Helmholtz operator in an impedance circular cylindrical waveguide

This paper addresses the problem of finding a series representation for the Green’s function of the Helmholtz operator in an infinite circular cylindrical waveguide with impedance boundary condition. Resorting to the Fourier transform, complex analysis techniques and the limiting absorption principle (when the undamped case is analyzed), a detailed deduction of the Green’s function is performed, generalizing the results available in the literature for the case of a complex impedance parameter. Procedures to obtain numerical values of the Green’s function are also developed in this article.     

3D modeling of crack growth in electro-magneto-thermo-elastic coupled viscoplastic multiphase composites

This work presents a time-domain hypersingular integral equation (TD-HIE) method for modeling 3D crack growth in electro-magneto-thermo-elastic coupled viscoplastic multiphase composites (EMTE-CVP-MCs) under extended incremental loads rate through intricate theoretical analysis and numerical simulations. Using Green’s functions, the extended general incremental displacement rate solutions are obtained by time-domain boundary element method. Three-dimensional arbitrary crack growth problem in EMTE-CVP-MCs is reduced to solving a set of TD-HIEs coupled with boundary integral equations, in which the unknown functions are the extended incremental displacement discontinuities gradient. Then, the behavior of the extended incremental displacement discontinuities gradient around the crack front terminating at the interface is analyzed by the time-domain main-part analysis method of TD-HIE. Also, analytical solutions of the extended singular incremental stresses gradient and extended incremental integral near the crack fronts in EMTE-CVP-MCs are provided. In addition, a numerical method of the TD-HIE for a 3D crack subjected to extended incremental loads rate is put forward with the extended incremental displacement discontinuities gradient approximated by the product of time-domain basic density functions and polynomials. Finally, examples are presented to demonstrate the application of the proposed method.     

A Green’s function formulation of nonlocal finite-difference schemes for reaction-diffusion equations

Reaction-diffusion equations are commonly used in different science and engineering fields to describe spatial patterns arising from the interaction of chemical or biochemical reactions and diffusive transport mechanisms. The aim of this work is to show that a Green’s function formulation of reaction-diffusion PDEs is a suitable framework to derive FD schemes incorporating both O(h²) accuracy and nonlocal approximations in the whole domain (including boundary nodes). By doing so, the approach departs from a Green’s function formulation of the boundary-value problem to pose an approximation problem based on a domain decomposition. Within each subdomain, the corresponding integral equation is forced to have zero residual at given grid points. Different FD schemes are obtained depending on the numerical scheme used for computing the Green’s integral over each subdomain. Dirichlet and Neumann boundary conditions are considered, showing that the FD scheme based on the Green’s function formulation incorporates, in a natural way, the effects of boundary nodes in the discretization approximation.     

A Green’s function model for the analysis of laser heating of materials

An analytical model based on Green’s function method is developed to analyze the temperature distribution and heated regions in a material irradiated by a high-energy laser beam. The model is multi-dimensional, transient and incorporates different types of beam characteristics and boundary conditions. The multi-dimensional integration formulas in the Green’s function solution equation are evaluated using an adaptive numerical integration algorithm. A parametric study is conducted to show the effect of various laser beam parameters and material properties on the laser heating process.     

The Broadwell model with a subsonic boundary

In this paper, we focus on the time-asymptotic behavior of an initial boundary value problem (IBVP) for the Broadwell model with a subsonic physical boundary. By using the Green’s function for the initial problem established in [C.-Y. Lan, H.-E. Lin, S.-H. Yu, The Green’s functions for the Broadwell model in half space problem, Netw. Heterog. Media 1 (1) (2006)] and the weighted energy estimates, we construct the Green’s function for IBVP and show that the solution converges pointwise to the equilibrium state when the perturbations are sufficiently small.     

The space fractional diffusion equation with Feller’s operator

This paper investigates the space fractional diffusion equation with fractional Feller’s operator. The Green’s function is obtained by using Fourier transform, and the analytical solutions of some space fractional diffusion equations with initial (or initial and boundary) condition are obtained in terms of Green’s function. In addition, numerical simulations are discussed. The results indicate that the effect range of skewness parameter θ has more effect on probability density than that of parameter α. The results also explain the property of the skewness and long tail in the asymmetry diffusion process.     

轴对称弹性动力学问题的重构核插值法

重构核插值法是近年来提出的一种新型无网格方法.该方法的形函数具有点插值性和高阶光滑性,不仅能够直接施加本质边界条件,而且能保证较高的计算精度.为了更有效地求解三维轴对称弹性动力学问题,对重构核插值法(reproducing kernel interpolation method, RKIM)应用于此类问题进行了研究,并发展了相应的数值模拟方法.由于几何形状和边界条件的轴对称性,计算时只需要横截面上离散节点的信息,因而前处理变得简单.采用Newmark-β法进行了时域积分.数值算例表明,轴对称弹性动力学分析的重构核插值法既有无网格方法的优势,又有较高的计算精度.     

New closed-form Green function and integral formula for a thermoelastic quadrant

In this study a new Green’s function and a new Green-type integral formula for a boundary value problem (BVP) in thermoelastostatics for a quadrant are derived in closed form. On the boundary semi-straight-lines twice mixed homogeneous mechanical boundary conditions (one boundary semi-straight-line is free of loadings and normal displacements and tangential stresses are prescribed on the other one) are prescribed. The thermoelastic displacements are subject by a heat source applied in the inner points of the quadrant and by mixed non-homogeneous boundary heat conditions (on one boundary semi-straight-line the temperature is prescribed and the heat flux is given on the other one). When thermoelastic Green’s function is derived the thermoelastic displacements are generated by an inner unit point heat source, described by δ Dirac’s function. All results are obtained in elementary functions that are formulated in a special theorem. A closed-form solution for a particular BVP of thermoelastostatics for a quadrant also is included. Using the proposed approach it is possible to extend the obtained for quadrant results to any other canonical Cartesian domain.     

2D Green’s functions for semi-infinite orthotropic thermoelastic plane

Based on the 2D general solutions of orthotropic thermoelastic material, the Green’s function for a steady point heat source in the interior of semi-infinite orthotropic thermoelastic plane is constructed by three newly introduced harmonic functions. All components of coupled field in semi-infinite thermoelastic plane are expressed in terms of elementary functions. Numerical results are given graphically by contours.     

Hybrid Laplace transform-finite element method to a generalized electromagneto-thermoelastic problem

A finite element method based on the Laplace transform technique is developed for a two-dimensional problem in electromagneto-thermoelasticity. The problem is in the context of the following generalized thermoelasticity theories: Lord–Shulman’s, Green–Lindsay’s, the Chandrasekharaiah–Tzou, as well as the dynamic coupled theory. The Laplace transform method is applied to the time domain and the resulting equations are discretized using the finite element method. The inversion process is carried out using a numerical method based on a Fourier series expansions. Numerical results compared with those given in literature prove the good performance of the used method. It is demonstrated that the Chandrasekharaiah–Tzou theory can be considered as an extension of Lord–Shulman’s, and the generalized heat conduction mechanism is completely different from the classical Fourier’s in essence.     

Analytical modelling for a three-dimensional hydrofoil with winglets operating beneath a free surface

The current study focuses on establishing a theoretical lifting surface model for predicting the hydrodynamic loads acting on the three-dimensional hydrofoil with winglets, which is considerably influenced by the proximity to the free surface through finding the three-dimensional Green’s function for the planar and vertical horseshoe vortices operating below a free surface. The hydrofoil surface is decomposed into a finite number of elements along the span direction and the chord directions, each of which can then be represented by a horseshoe vortex. The linearized free surface boundary condition is applied to analyze the influence of the free surface on the hydrofoil as well as the winglets. The thickness problem is considered using the source distribution among the hydrofoil and winglets surfaces and the analytical Green’s function that satisfies the linearized free surface boundary condition is used. As a sample application, numerical examples were conducted to show the performance of the hydrodynamic characteristics for the hydrofoil with winglets as a function of the Froude number. It was concluded that there are significant efficiency benefits from using winglets inside the free surface proximity effect. These results are substantiated by the comparison with the available published data.     

On a Weyl-Titchmarsh theory for discrete symplectic systems on a half line

Recently, Bohner and Sun [9] introduced basic elements of a Weyl-Titchmarsh theory into the study of discrete symplectic systems. We extend this development through the introduction of Weyl-Titchmarsh functions together with a preliminary study of their properties. A limit point criterion is described and characterized. Green’s function for the half-line is introduced as a limit of such functions in the regular case and half-line solutions obtained are seen to satisfy λ-dependent boundary conditions at infinity.     

Two-dimensional steady-state general solution for isotropic thermoelastic materials with applications. II. Green’s function for two-phase infinite plane

Green’s function for isotropic thermoelastic two-phase infinite plane under a line heat source is established in this paper. By virtue of the fourth compact general solutions in Part I which is expressed in three harmonic functions, six new suitable harmonic functions with undetermined constants are constructed for the two semi-infinite planes of the two-phase infinite plane, respectively. The corresponding thermoelastic field can be obtained by substituting these harmonic functions into the general solution, and the undetermined constants can be determined by compatibility conditions and the equilibrium conditions. Numerical results are given graphically by contours.     

Ostrowski type methods for solving systems of nonlinear equations

Four generalized algorithms builded up from Ostrowski’s method for solving systems of nonlinear equations are written and analyzed. A development of an inverse first-order divided difference operator for functions of several variables is presented, as well as a direct computation of the local order of convergence for these variants of Ostrowski’s method. Furthermore, a sequence that approximates the order of convergence is generated for the examples and it confirms in a numerical way that the order of the methods is well deduced.     

A study on the method of fundamental solutions using an image concept

In this paper, both analytical and semi-analytical solutions for Green’s functions are obtained by using the image method which can be seen as a special case of method of fundamental solutions (MFS). The image method is employed to solve the Green’s function for the annular, eccentric and half-plane Laplace problems. In addition, an analytical solution is derived for the fixed-free annular case. For the half-plane problem with a circular hole and an eccentric annulus, semi-analytical solutions are both obtained by using the image concept after determining the strengths of two frozen image points and a free constant by matching boundary conditions. It is found that two frozen images terminated at the two focuses in the bipolar coordinates for the problems with two circular boundaries. A boundary value problem of an eccentric annulus without sources is also considered. Error distribution is plotted after comparing with the analytical solution derived by Lebedev et al. using the bipolar coordinates. The optimal locations for the source distribution in the MFS are also examined by using the image concept. It is observed that we should locate singularities on the two focuses to obtain better results in the MFS. Besides, whether the free constant is required or not in the MFS is also studied. The results are compared well with the analytical solutions.     

Conduction and convection phenomena through a slab with thermal heterogeneities

This paper addresses the computation of the three-dimensional transient heat transfer through a layered solid and/or fluid formation containing irregular inclusions. The use of appropriate Green’s functions for a flat layer formation in a boundary element method formulation avoids the discretization of the layer interface boundaries.