3D symplectic expansion for piezoelectric media

The symplectic series expansion method is extended to three‐dimensional problem for transversely isotropic piezoelectric media. The governing equations are first derived in Hamiltonian form, and symplectic eigensolutions are directly obtained through analytical method. All solutions of the problem are reduced to finding eigenvalues and eigensolutions. The classical St Venant solutions are described by zero‐eigensolutions, and the localized solutions are depicted by non‐zero‐eigensolutions. Symplectic relationships of the ortho‐normalization are used, end conditions are rewritten by eigensolutions, and a numerical scheme is formed analytically. Some numerical examples are given. Copyright © 2007 John Wiley & Sons, Ltd.     

Matched interface and boundary (MIB) for the implementation of boundary conditions in high‐order central finite differences

High‐order central finite difference schemes encounter great difficulties in implementing complex boundary conditions. This paper introduces the matched interface and boundary (MIB) method as a novel boundary scheme to treat various general boundary conditions in arbitrarily high‐order central finite difference schemes. To attain arbitrarily high order, the MIB method accurately extends the solution beyond the boundary by repeatedly enforcing only the original set of boundary conditions. The proposed approach is extensively validated via boundary value problems, initial‐boundary value problems, eigenvalue problems, and high‐order differential equations. Successful implementations are given to not only Dirichlet, Neumann, and Robin boundary conditions, but also more general ones, such as multiple boundary conditions in high‐order differential equations and time‐dependent boundary conditions in evolution equations. Detailed stability analysis of the MIB method is carried out. The MIB method is shown to be able to deliver high‐order accuracy, while maintaining the same or similar stability conditions of the standard high‐order central difference approximations. The application of the proposed MIB method to the boundary treatment of other non‐standard high‐order methods is also considered. Copyright © 2008 John Wiley & Sons, Ltd.     

弹性地基上矩形薄板自由振动的精确解

根据分离变量法得到了Winkler弹性地基上矩形薄板自由振动问题的精确解, 分析了地基模量对频率的影响, 其中边界条件为CCCC、SCCC和SSCC情况的精确解过去被认为是难以得到的。在分离变量方法中, 不需要事先人为的选取满足某一组对边边界条件的挠度函数, 而是直接利用控制方程本征根给出振型函数通解的一般解析形式, 再利用边界条件得到振型函数系数和频率方程的精确形式。数值结果与有限元结果及文献结果吻合较好, 验证了该文方法和结果的的正确性。     

Solution of laplace equation by the method of separation of variables

Abstract

The Laplace equation, which is used to describe the problem of two‐dimensional heat conduction with appropriate boundary conditions at steady state, is solved in this work by applying the method of separation of variables. The primary objective of this work involves discussing the effects of the constant value of the separation of variables (p) and the sequential order of substituting boundary conditions on the solution. Without appropriately arranging the sequential order of substituting the boundary conditions, the solution for non‐zero constant values of separation of variables (p) can not be obtained. For a zero value for the constant of the separation of variables, the solution obtained is trivial or does not exist. Solutions in different forms are obtained by using different values for the constant of the separation of variables (p) and for the sequential orders of substituting the boundary conditions.     

Numerical determination of the fundamental eigenvalue for the Laplace operator on a spherical domain

Summary Methods for obtaining approximate solutions for the fundamental eigenvalue of the Laplace-Beltrami operator (i.e., the membrane eigenvalue problem for the vibration equation) on the unit spherical surface are developed. Two types of spherical surface domains are considered: (1) the interior of a spherical triangle, and (2) the exterior of a great circle arc extending for less than radians (a spherical surface with a slit). In both cases, zero boundary conditions are imposed. In order to solve the resulting second-order elliptic partial differential equations in two independent variables, a finite difference approximation is employed. The fundamental eigenvalue is approximated by iteration utilizing the power method and point successive overrelaxation. Some numerical results are given and compared, in certain special cases, with analytical solutions to the eigenvalue problem. The significance of the numerical eigenvalue results is discussed in terms of the singularities in the solution of three-dimensional boundary-value problems near a polyhedral corner of the domain.     

带化学反应的边界层流动问题中一类弱奇异Volterra积分方程的近似解

研究带化学表面反应的边界层流动问题导出的一类弱奇异Volterra积分方程的近似解。以一些化学反应的阶数为例求出解在零点的分数阶级数展开式及其■有理逼近。通过将发散积分解释为Hadamard有限部分积分,并借助数值积分方法导出解在无穷远点的带高阶对数项的渐近展开式。实际计算表明,给出的解在零点和无穷远点展开式的联合使用可以在整个半无限区间上高效地求解这类带化学表面反应的边界层流动问题。     

An application of the symplectic system in two-dimensional viscoelasticity

This paper redescribes fundamental problem of the two-dimensional viscoelasticity in symplectic system. With the aid of the symplectic character and integral transformation, solutions of duality equations are obtained, or Saint-Venant solutions of extension and bend and local solutions of boundary effects. Thus the original problem is reduced to finding zero eigenvalue eigensolutions and non-zero eigenvalue eigensolutions. Meanwhile, adjoint relationships of the symplectic orthogonality in the Laplace domain are generalized to in the time domain. After obtaining fundamental eigensolutions, the problem can be discussed in the eigensolution space of the time domain without the need of the Laplace transformation and inverse one. As its application, a direct method is shown and some examples are discussed, which reveal relations between the creep or relaxation and eigensolutions. The symplectic method and numerical method provide an idea for other researching as well.     

ON THE INADMISSIBILITY OF SEPARATION OF VARIABLES SOLUTIONS IN LINEAR ANISOTROPIC VISCOELASTICITY

SUMMARY

The field and constitutive equations as well as the boundary conditions of linear anisotropic viscoelasticity are examined for possible solutions which are separable into products of time and spatial functions. It is shown that under no circumstances other than for constant Poisson's ratios are such separation of variables solutions admissible, even though under these or some additional restrictive conditions they can exist in isotropic viscoelasticity.     

正交压电复合材料层板各类边界的解析解

压电复合材料层板由压电片与纤维层叠合而成。基于等效单层理论的位移场和电势场,针对正交铺层压电复合材料层板柱面弯曲问题,建立了力电耦合平衡方程,获得了一般边界的解析解。解析解由特解和通解两部分组成,特解对应于简支边界条件,通解由其他各类边界条件确定。平衡方程的变量仅4个,且不随层数变化。如采用相应的位移和电势分布函数,可以得到一阶理论、高阶理论、指数型理论等多种理论的解析解。算例中给出了各种边界条件下位移、应力和电势的解,讨论了各种理论的精度,观察到了固支边界的应力奇异现象。     

Separation and the compressible boundary layer

Summary The behavior of the compressible boundary layer equations close to a point of zero skin-friction is studied using a perturbation technique of Kaplun. The behavior of the skin friction is reduced to a study of a nonlinear integral equation with an Abel kernel. For a cold wall this yields the singular behavior described by Buckmaster [2] from a different point of view. For a hot wall the behavior is apparently regular when the heat transfer is non zero, in agreement with Stewartson [11]. There is no evidence that the boundary layer breaks down anywhere than at a point of zero skin friction and the implications of this for the similarity solutions of Cohen and Reshotko are discussed.     

Non‐reflecting artificial boundaries for transient scalar wave propagation in a two‐dimensional infinite homogeneous layer

This paper presents an exact non‐reflecting boundary condition for dealing with transient scalar wave propagation problems in a two‐dimensional infinite homogeneous layer. In order to model the complicated geometry and material properties in the near field, two vertical artificial boundaries are considered in the infinite layer so as to truncate the infinite domain into a finite domain. This treatment requires the appropriate boundary conditions, which are often referred to as the artificial boundary conditions, to be applied on the truncated boundaries. Since the infinite extension direction is different for these two truncated vertical boundaries, namely one extends toward x →∞ and another extends toward x→‐ ∞, the non‐reflecting boundary condition needs to be derived on these two boundaries. Applying the variable separation method to the wave equation results in a reduction in spatial variables by one. The reduced wave equation, which is a time‐dependent partial differential equation with only one spatial variable, can be further changed into a linear first‐order ordinary differential equation by using both the operator splitting method and the modal radiation function concept simultaneously. As a result, the non‐reflecting artificial boundary condition can be obtained by solving the ordinary differential equation whose stability is ensured. Some numerical examples have demonstrated that the non‐reflecting boundary condition is of high accuracy in dealing with scalar wave propagation problems in infinite and semi‐infinite media. Copyright © 2003 John Wiley & Sons, Ltd.     

Imposing Dirichlet boundary conditions with Nitsche's method and spline‐based finite elements

A key challenge while employing non‐interpolatory basis functions in finite‐element methods is the robust imposition of Dirichlet boundary conditions. The current work studies the weak enforcement of such conditions for B‐spline basis functions, with application to both second‐ and fourth‐order problems. This is achieved using concepts borrowed from Nitsche's method, which is a stabilized method for imposing constraints on surfaces. Conditions for the stability of the system of equations are derived for each class of problem. Stability parameters in the Nitsche weak form are then evaluated by solving a local generalized eigenvalue problem at the Dirichlet boundary. The approach is designed to work equally well when the grid used to build the splines conforms to the physical boundary of interest as well as to the more general case when it does not. Through several numerical examples, the approach is shown to yield optimal rates of convergence. Copyright © 2010 John Wiley & Sons, Ltd.     

Perfectly matched layers for transient elastodynamics of unbounded domains

One approach to the numerical solution of a wave equation on an unbounded domain uses a bounded domain surrounded by an absorbing boundary or layer that absorbs waves propagating outward from the bounded domain. A perfectly matched layer (PML) is an unphysical absorbing layer model for linear wave equations that absorbs, almost perfectly, outgoing waves of all non‐tangential angles‐of‐incidence and of all non‐zero frequencies. In a recent work [Computer Methods in Applied Mechanics and Engineering 2003; 192: 1337–1375], the authors presented, inter alia, time‐harmonic governing equations of PMLs for anti‐plane and for plane‐strain motion of (visco‐) elastic media. This paper presents (a) corresponding time‐domain, displacement‐based governing equations of these PMLs and (b) displacement‐based finite element implementations of these equations, suitable for direct transient analysis. The finite element implementation of the anti‐plane PML is found to be symmetric, whereas that of the plane‐strain PML is not. Numerical results are presented for the anti‐plane motion of a semi‐infinite layer on a rigid base, and for the classical soil–structure interaction problems of a rigid strip‐footing on (i) a half‐plane, (ii) a layer on a half‐plane, and (iii) a layer on a rigid base. These results demonstrate the high accuracy achievable by PML models even with small bounded domains. Copyright © 2004 John Wiley & Sons, Ltd.     

A new methodology to solve non‐linear equation systems using genetic algorithms. Application to combined cyclegas turbine simulation

This paper shows a methodology to sort out the equations of a non‐linear system in order to solve it by the fixed‐point method. The arrangement of the equations is established by a genetic algorithm that deals with a population of possible resolution processes of the system. The method is specially useful in the following situations: first, when the system is very non‐linear and has many variables (where the Newton–Raphson method does not work properly); second, when the number of equations and variables may be altered because the equation system may change in each simulation and, therefore, more than one only solution process is needed if the fixed‐point process is employed. As an example, the methodology has been applied to solve the equation system that models the behaviour of a heat recovery steam generator of a combined cycle power plant at full load and part load conditions. Copyright © 2005 John Wiley & Sons, Ltd.     

Direct computation of critical equilibrium states for spatial beams and frames

In this paper, explicit formulas for second order derivatives of the residual vector with respect to the state variables for a geometrically exact 3D beam element based on the Reissner's model are presented. These derivatives are required when a direct non‐linear stability eigenvalue problem is solved by the Newton's method. If the external load is parametrized by a single parameter, such an eigenvalue problem consists of solving the critical state variables, the eigenmode, and the critical load parameter from the equation system consisting of the equilibrium equations, the criticality condition, and some auxiliary conditions depending on the type of a critical point. Copyright ©2011 John Wiley & Sons, Ltd.     

Solving high‐order partial differential equations with indirect radial basis function networks

This paper reports a new numerical method based on radial basis function networks (RBFNs) for solving high‐order partial differential equations (PDEs). The variables and their derivatives in the governing equations are represented by integrated RBFNs. The use of integration in constructing neural networks allows the straightforward implementation of multiple boundary conditions and the accurate approximation of high‐order derivatives. The proposed RBFN method is verified successfully through the solution of thin‐plate bending and viscous flow problems which are governed by biharmonic equations. For thermally driven cavity flows, the solutions are obtained up to a high Rayleigh number of 10⁷. Copyright © 2005 John Wiley & Sons, Ltd.     

Exact static element stiffness matrices of non‐symmetric thin‐walled curved beams

An evaluation procedure of exact static stiffness matrices for curved beams with non‐symmetric thin‐walled cross section are rigorously presented for the static analysis. Higher‐order differential equations for a uniform curved beam element are first transformed into a set of the first‐order simultaneous ordinary differential equations by introducing 14 displacement parameters where displacement modes corresponding to zero eigenvalues are suitably taken into account. This numerical technique is then accomplished via a generalized linear eigenvalue problem with non‐symmetric matrices. Next, the displacement functions of displacement parameters are exactly calculated by determining general solutions of simultaneous non‐homogeneous differential equations. Finally an exact stiffness matrix is evaluated using force–deformation relationships. In order to demonstrate the validity and effectiveness of this method, displacements and normal stresses of cantilever thin‐walled curved beams subjected to tip loads are evaluated and compared with those by thin‐walled curved beam elements as well as shell elements. Copyright © 2004 John Wiley & Sons, Ltd.     

Numerical effects in the simulation of Ginzburg–Landau models for superconductivity

Inadequate spatial mesh resolution for simulation of the time‐dependent Ginzburg–Landau equations is shown to give rise to spurious solutions. Phenomenological studies to examine the effect of the physical parameters and boundary conditions in 2D and 3D are presented to illustrate the solution structure and to highlight non‐linear effects related to evolving vortex patterns. We illustrate and explore this issue further by considering a related simplified model in which the number of vortices is equal to the ‘winding number’ that is associated with the applied boundary conditions. Using this model we demonstrate that the solution structure is non‐unique for several values of winding number. Copyright © 2004 John Wiley & Sons, Ltd.     

Free vibration analysis of thin isotropic and anisotropic rectangular plates by the discrete singular convolution algorithm

This paper presents the free vibration analysis of thin isotropic and anisotropic rectangular plates with various boundary conditions by using the discrete singular convolution (DSC) algorithm. Based on Taylor's series expansion, a unique scheme is proposed to handle various boundary conditions, including the simply supported (S), clamped (C) and free (F) edge. To validate the proposed method, the non‐dimensional frequency parameters of isotropic, orthotropic and angle‐ply symmetric laminated rectangular plates with various combinations of boundary conditions are obtained by using the DSC algorithm and compared with the analytical and/or numerical solutions. Comparisons reveal that the proposed method can handle the zero bending moment and zero shear force conditions for the isotropic as well as anisotropic plates. The proposed method provides an alternative way for applying the simply supported boundary conditions in applications of the DSC algorithm to plate structures. This investigation extends the application range of the DSC algorithm to vibration analysis of anisotropic plates with various combinations of boundary conditions. Copyright © 2010 John Wiley & Sons, Ltd.     

A 3D finite element approach for the coupled numerical simulation of electrochemical systems and fluid flow

A comprehensive finite element method for three‐dimensional simulations of stationary and transient electrochemical systems including all multi‐ion transport mechanisms (convection, diffusion and migration) is presented. In addition, non‐linear phenomenological electrode kinetics boundary conditions are accounted for. The governing equations form a set of coupled non‐linear partial differential equations subject to an algebraic constraint due to the electroneutrality condition. The advantage of a convective formulation of the ion‐transport equations with respect to a natural application of homogeneous flux boundary conditions is emphasized. For one of the numerical examples, an analytical solution for the coupled problem is provided, and it is demonstrated that the proposed computational approach is robust and provides accurate results. Copyright © 2011 John Wiley & Sons, Ltd.