A boundary method of Trefftz type for PDEs with scattered data

This paper presents the application of a Trefftz type method for partial differential equations (PDEs) of the elliptic type with inhomogeneous term given by a set of scattered data. The method of particular solutions is used. Basis functions of a new type were introduced to approximate the scattered data. Using these basis functions, we get the approximation in the form of series over some orthogonal system of eigenfunctions. The particular case of the trigonometric eigenfunctions is considered. The corresponding approximation of the inhomogeneous term allows to get a particular solution for PDEs with constant coefficients or for the systems of such PDEs easily. We test our basis functions on recovering well-known Franke's and PEAKS functions given by scattered data. We also present results of solution Helnholtz PDE, PDE with differential operator of 4th order and system of PDEs arising in shell deflection problems. A comparison of the numerical solutions with analytic solutions is performed for all the problems.     

A one-stage meshless method for nonhomogeneous Cauchy problems of elliptic partial differential equations with variable coefficients

A one-stage meshless method is devised for solving Cauchy boundary value problems of elliptic partial differential equations (PDEs) with variable coefficients. The main idea is to approximate an unknown solution using a linear combination of fundamental solutions and radial basis functions. Compared with the two-stage method of particular solution, the proposed method can deal with more general elliptic PDEs with variable coefficients. Several numerical results in both two- and three-dimensional space show that our proposed method is accurate and effective.     

Faber series method for plane problems of an arbitrarily shaped inclusion

This paper presents the theoretical and numerical results for the plane problem of an arbitrarily shaped inclusion in an infinite isotropic matrix based on the Faber series method. The key of the method is to express the complex potentials in the arbitrary inclusion in the form of Faber series with unknown coefficients and then substitute them directly into the boundary conditions on the interface. These conditions lead to a set of linear equations containing all the unknown coefficients. Through solving these linear equations, one can obtain the complex potentials both inside the inclusion and in the matrix. Then, numerical results are presented and graphically shown for the cases of an elliptic, square, and triangle inclusions, respectively. It is found that as the stiffness of the inclusion increases, the hoop stress decreases at the rim of the inclusion, while the radial and shear stresses increase. Especially, it is also found that the stresses show the nature of intense fluctuations near the corners of the triangle inclusion, since the inclusion in this case is similar to a wedge.     

On the anisotropic piezoelastic contact problem for an elliptical punch

Summary This paper firstly conducts a systematic investigation of the problem of a rigid punch indenting an anisotropic piezoelectric half-space. The Fourier transform method is employed to the mixed boundary value problem. Using the principle of linear superposition, the resulting transformed (algebraic) equations, whose right-hand sides contain both pressure and electric displacement terms, can be solved by superposing the solutions of two sets of algebraic equations, one containing pressure and another containing electric displacement. For an arbitrarily shaped punch, two governing equations are derived, which can be solved numerically. In the case of transversely isotropic piezoelectric media, the two governing equations are corresponding with that given by others using potential theory. Particularly, when the punch has elliptic cross-section, and the pressure and electric displacement are given by some certain forms of polynomial functions, then the displacement and electric potential are prescribed by polynomial functions in the contact area. The parameters contained in it satisfy a set of linear algebraic equations, whose coefficients involve contour integrals. The problem of indentation by a smooth flat punch is examined for special orthotropic piezoelectric media, and some results obtained can be degenerated to the case of transversely isotropic piezoelectric media.     

Boundary value problems in thin shallow shells of arbitrary plan form

Summary This paper is concerned with a general method of analysis of boundary-value problems in thin shallow shells of arbitrary plan form. Two specific shell configurations are considered. General solutions to the governing partial differential equations are obtained in complex form, containing a sufficient number of arbitrary elements to satisfy the four boundary conditions permitted by classical thin shell theory. An algorithm for the determination of these arbitrary elements from a general form of boundary condition is presented. The method of solution is based on I.N.Vekua's theory of elliptic partial differential equations.Part 1 of the paper is devoted to shallow spherical shells. An example calculation is given for a circular planform shell, for which a closed form of solution may be obtained. The computed results show close agreement with the exact solution.Part 2 of the paper deals with shallow circular cylindrical shells, including the calculation of a shell the planform of which is a square with rounded corners. Graphs of deflection and stress function are given.     

Finite amplitude spherically symmetric wave propagation in a compressible hyperelastic solid

Summary Some aspects of the wave propagation, resulting from the spherically symmetric expansion of a thick walled hyperelastic shell and the limiting case of expansion of a cavity in an unbounded medium, are investigated. It is assumed that the shell is isotropic and uniform in the natural reference state and its strain energy function is a particular compressible generalization of that for the neo-Hookean solid. The response of a compressible shell, due to a spatially uniform time dependent application of internal pressure, is compared with that for the neo-Hookean shell taken as a limiting case of the compressible shell. This is also done for an unbounded medium.A finite difference method which uses the relation along one of the families of characteristics is used to obtain numerical results. In order to implement this method the governing equations are expressed as a system of first order partial differential equations in conservation form.With 9 Figures     

The embedded discontinuous Galerkin method: application to linear shell problems

A new discontinuous Galerkin method for elliptic problems which is capable of rendering the same set of unknowns in the final system of equations as for the continuous displacement‐based Galerkin method is presented. Those equations are obtained by the assembly of element matrices whose structure in particular cases is also identical to that of the continuous displacement approach. This makes the present formulation easily implementable within the existing commercial computer codes. The proposed approach is named the embedded discontinuous Galerkin method. It is applicable to any system of linear partial differential equations but it is presented here in the context of linear elasticity. An application of the method to linear shell problems is then outlined and numerical results are presented. Copyright © 2006 John Wiley & Sons, Ltd.     

A regularization method for the approximate particular solution of nonhomogeneous Cauchy problems of elliptic partial differential equations with variable coefficients

Radial basis functions (RBFs) have proved to be very flexible in representing functions. Based on the idea of the analog equation method and radial basis functions, in this paper, ill-posed Cauchy problems of elliptic partial differential equations (PDEs) with variable coefficients are considered for the first time using the method of approximate particular solutions (MAPS). We show that, using the Tikhonov regularization, the MAPS results an effective and accurate numerical algorithm for elliptic PDEs and irregular solution domains. Comparing the proposed MAPS with Kansa's method, numerical results show that the proposed MAPS is effective, accurate and stable to solve the ill-posed Cauchy problems.     

On the use of radial basis functions in the solution of elliptic boundary value problems

A numerical technique for solving elliptic boundary value problems is developed. The method expands the solution in terms of radial basis functions and chooses expansion coefficients such that the governing equations and boundary conditions are approximately satisfied. The method is demonstrated through its application to some non-trivial problems involving the Laplace and biharmonic equations.     

Fundamental solutions for second order linear elliptic partial differential equations

This paper is concerned with outlining some fundamental solutions and Green's functions for a system of second order linear elliptic partial differential equations in two independent variables. The fundamental solution and a number of Green's functions are given in relatively elementary closed form for some cases when the coefficients in the equations are constant. When the coefficients are variable the fundamental solution is obtained for some particular classes of equations.     

The buckling of an orthotropic composite truncated conical shell with continuously varying thickness subject to a time dependent external pressure

In this study, the buckling of an orthotropic composite truncated conical shell with continuously varying thickness, subject to a uniform external pressure which is a power function of time, has been considered. At first, the fundamental relations and the Donnell type stability equations of an orthotropic composite truncated conical shell, subject to an external pressure, have been obtained. Then, employing Galerkin method, those equations have been reduced of time dependent differential equation with variable coefficients. Finally, applying the variational method of Ritz method type, the critical static and dynamic loads, the corresponding wave numbers and the dynamic factor have been found analytically. Using those results, the effects of the variations of the power in the thickness expression, the semi-vertex angle, the power of time in the external pressure expression and the ratio of the Young's moduli on the critical parameters are studied numerically, for the case when the thickness of the conical shell varies as a power and exponential function. It is observed, from the computations carried out, that these factors have appreciable effects on the critical parameters of the problem in the heading.     

Three-dimensional piezoelasticity solution for dynamics of cross-ply cylindrical shells integrated with piezoelectric fiber reinforced composite actuators and sensors

A benchmark three-dimensional (3D) exact piezoelasticity solution is presented for free vibration and steady state forced response of simply supported smart cross-ply circular cylindrical shells of revolutions and panels integrated with surface-bonded or embedded monolithic piezoelectric or piezoelectric fiber reinforced composite (PFRC) layers. The effective properties of PFRC laminas for the 3D case are obtained based on a fully coupled iso-field model. The governing partial differential equations are reduced to ordinary differential equations in the thickness coordinate by expanding all entities for each layer in double Fourier series in span coordinates, which identically satisfy the boundary conditions at the simply-supported ends. These equations with variable coefficients are solved using the modified Frobenius method, wherein the solution is constructed as a product of an exponential function and a power series. The unknown constants of the general solution are finally obtained by employing the transfer matrix method across the layers. Results for natural frequencies and the forced response are presented for single layer piezoelectric and multilayered hybrid composite and sandwich shells of revolution and shell panels integrated with monolithic piezoelectric and PFRC actuator/sensor layers. The present benchmark solution would help assess 2D shell theories for dynamic response of hybrid cylindrical shells.     

On an analytical-numerical procedure for the analysis of cylindrical shells with arbitrarily oriented cracks

An analytical-numerical procedure for obtaining stress intensity factor solutions for an arbitrarily oriented crack in a long, thin circular cylindrical shell is presented. The method of analysis involves obtaining a series solution to the governing shell equation in terms of Mathieu and modified Mathieu functions by the method of separation of variables and satisfying the crack surface boundary conditions numerically using collocation. The solution is then transformed from elliptic coordinates to polar coordinates with crack tip as the origin through a Taylor series expansion and membrane and bending stress intensity factors are computed. Numerical results are presented and discussed for the pressure loading case.     

Transfer matrix method for the solution of multiple elliptic layers with different elastic properties. Part I: infinite matrix case

This paper provides the transfer matrix method for the solution of multiple elliptic layers with different elastic properties. In the study, the medium is composed of an elliptic inclusion and many confocal elliptic layers. The conformal mapping and the continuation of analytic functions are used. In the mapping plane, the complex potentials in the inclusion and the individual layer are expressed in the form of Laurent series. The correct form of the complex potentials in the inclusion is addressed. From the continuation condition for traction and displacement along the interface, the relation between the coefficients in the Laurent series of complex potentials for two adjacent layers can be evaluated. This relation is expressed in a matrix form, and it is called the transfer matrix. Using the transfer matrixes successively and the traction condition at remote place, the problem is finally solved. Numerical examples for two cases, the two-phase case and the three-phase case, are presented.     

The method of approximate fundamental solutions (MAFS) for elliptic equations of general type with variable coefficients

The paper presents a meshless method for solving elliptic equations of general type with variable coefficients. It is based on the use of the delta-shaped functions and the method of approximate fundamental solutions first suggested for solving equations with constant coefficients. The method assumes that the solution domain is embedded in a square and the initial equation is extended onto the square with the help of the CICE −(Chebyshev interpolation + C-expansion) approximation scheme. As a result the coefficients of the equation are approximated by the truncated Fourier series over some orthogonal system in the square. The approximate fundamental solutions (AFSs) satisfy L[u]=I(x), where I(x) is the delta shaped function in the form of the truncated Fourier series. Thus, the AFSs due to the special form of the operator can be obtained in the similar form of truncated series. The next part of the MAFS follows the general scheme of the MFS. The numerical examples are presented and the results are compared with the analytical solutions. The comparison shows that the method presented provides a very high precision in solution of two-dimensional elliptic equations of general type with different boundary conditions (Dirichlet, Neumann, mixed) in arbitrary domains.     

Analysis of cylindrical shell panels. A meshless solution

The Meshless Analog Equation Method, a purely meshless method, is applied to the static analysis of cylindrical shell panels. The method is based on the concept of the analog equation of Katsikadelis, which converts the three governing partial differential equations in terms of displacements into three substitute equations, two of second order and one fourth order, under fictitious sources. The fictitious sources are represented by series of radial basis functions of multiquadric type. Thus the substitute equations can be directly integrated. This integration allows the representation of the sought solution by new radial basis functions, which approximate accurately not only the displacements but also their derivatives involved in the governing equations. This permits a strong formulation of the problem. Thus, inserting the approximate solution in the differential equations and in the associated boundary conditions and collocating at a predefined set of mesh-free nodal points, a system of linear equations is obtained, which gives the expansion coefficients of radial basis functions series that represent the solution. The minimization of the total potential of the shell results in the optimal choice of the shape parameter of the radial basis functions. The method is illustrated by analyzing several shell panels. The studied examples demonstrate the efficiency and the accuracy of the presented method.     

A numerical upscaling method for an elliptic equation with heterogeneous tensorial coefficients

The aim of this paper is to develop a new numerical method to determine the effective permeability on a coarse scale level of problems with strongly heterogeneous tensorial coefficients defined on a fine scale. This method stems from a primal–dual mixed formulation and the upscaled permeability is obtained by imposing the continuity of the numerical flux across the lines of discontinuities. Moreover, a finite volume scheme is built to numerically solve elliptic equations with this kind of coefficients. Finally, the method is illustrated by several numerical experiments. Copyright © 2002 John Wiley & Sons, Ltd.     

轴压粘弹性圆柱壳在横向扰动下的混沌行为

基于大挠度薄壳的Donnell-Kármán理论和Kelvin–Voigt粘弹性本构关系,对轴压粘弹性圆柱壳在横向扰动下的混沌行为进行了研究。导出了关于挠度和应力函数的控制方程,借助Galerkin原理将粘弹性圆柱壳的控制方程转化为二阶三次非线性微分动力系统,用Melnikov函数给出了系统发生Smale马蹄型混沌的临界条件。数值计算分析了轴压载荷和粘性阻尼系数对混沌运动的影响。通过分岔图、位移时程曲线、相平面图和Poincaré映射描述了系统的运动行为。研究表明：当轴压载荷与圆柱壳的材料参数满足一定关系时,系统才有可能发生Smale马蹄型混沌;随着轴压载荷的增大,混沌运动区域逐渐减小;随着粘性系数与外阻尼系数比值的增大,混沌运动区域逐渐减小;轴压粘弹性圆柱壳在横向扰动下既会发生定常运动也会发生混沌运动     

Transverse impact of a ball on a sphere with allowance for waves in the target

The transverse impact of a solid projectile on an elastic spherical shell with a pivoting contour support has been studied. Inside the contact zone, the projectile-target interaction is described by a solution of the standard system of equations. Outside the contact zone, the points of the shell are displaced and the shell is deformed due to propagation of a nonstationary wave front. A solution in this region is constructed using ray series with variable coefficients representing jumps of the time derivatives of the unknown functions on the wave surface of strong discontinuity. These coefficients are determined to within arbitrary constants using momentless equations of motion of the shell points. The constants are determined by matching two solutions at the contact zone boundary. Using the obtained analytical expressions and plotted dependences for the contact force and dynamic inflection, it is possible to judge on the influence of the shell structure design on the dynamic characteristics of impact interaction.     

High frequency oscillating viscous flow over elliptic cylinder at incidence

Summary The problem of high frequency oscillating viscous flow over an elliptic cylinder at incidence is investigated. The method of matched inner-outer asymptotic expansion to second order is used to solve the governing equations. The steady and the unsteady modes of flow, related to the present work, are identified and separated. Both steady and unsteady drag and lift coefficients are presented and discussed. The effect of different parameters such as the Strouhal number, Reynolds number, focal length and angle of attack are explored.