LSFD method for accurate vibration modes and modal stress-resultants of freely vibrating plates that model VLFS

The Ritz method and some finite element formulations fail to furnish accurate modal stress-resultants for vibrating plates with free edges, even though the natural frequencies and mode shapes are accurately obtained. For example, by using the Ritz method, it was found that the modal twisting moments and shear forces violate the natural boundary conditions and that they contain erroneous “oscillations”. This paper presents the least squares finite difference (LSFD) method for solving the freely vibrating plate problem. It will be shown herein that the modal stress-resultants obtained by the LSFD method satisfy the natural boundary conditions at the free edges without any oscillations.     

Pricing American options using a space-time adaptive finite difference method

American options are priced numerically using a space- and time-adaptive finite difference method. The generalized Black–Scholes operator is discretized on a Cartesian structured but non-equidistant grid in space. The space- and time-discretizations are adjusted such that a predefined tolerance level on the local discretization error is met. An operator splitting technique is used to separately handle the early exercise constraint and the solution of linear systems of equations from the finite difference discretization of the linear complementarity problem. In numerical experiments three variants of the adaptive time-stepping algorithm with and without local time-stepping are compared.     

Free vibration of laminated plates using a finite strip method based on a higher-order plate theory

The finite strip method based on the higher-order plate theory is developed for determining the natural frequencies of laminated plates. This method can accurately predict the through thickness effect of transverse shear deformation. Furthermore, only a few degress of freedom are required in the finite strip method. Some numerical results for various span-to-thickness ratios, material properties and stack sequences are presented for illustrative purposes. The present model provides a better way to obtain more accurate natural frequency results.     

The Galerkin element method applied to the vibration of rectangular damped sandwich plates

The Galerkin element method (GEM), which combines Galerkin orthogonal functions with the traditional finite element formulation, has previously been applied successfully to the vibration analysis of damped sandwich beams, and an improved iteration method was developed for its eigen solution. In the current paper, this promising method is extended to the vibration of damped sandwich plates. A quite different model is formulated which has both nodal coordinates and edge coordinates, while in the case of beams, there are only nodal coordinates. Displacement compatibility over the interfaces between the damping layer and the elastic layers is taken account of in order to ensure a conforming element and thereby guarantee good accuracy. The seed matrix method is proposed for simplifying the building of the element mass, stiffness and damping matrices. Numerical examples show that the application of the GEM to sandwich plate structures is computationally very efficient, while providing accurate estimates of natural frequencies and modal damping over a wide frequency range.     

Free vibration analysis of laminated composite plates based on FSDT using one-dimensional IRBFN method

This paper presents a new effective radial basis function (RBF) collocation technique for the free vibration analysis of laminated composite plates using the first order shear deformation theory (FSDT). The plates, which can be rectangular or non-rectangular, are simply discretised by means of Cartesian grids. Instead of using conventional differentiated RBF networks, one-dimensional integrated RBF networks (1D-IRBFN) are employed on grid lines to approximate the field variables. A number of examples concerning various thickness-to-span ratios, material properties and boundary conditions are considered. Results obtained are compared with the exact solutions and numerical results by other techniques in the literature to investigate the performance of the proposed method.     

平面薄膜自由振动的有限元分析

构造6节点三角形单元,适合于平面薄膜自由振动的有限元分析.文中采用面积坐标,给出单元的形函数,根据哈密顿原理建立薄膜自由振动方程,推导其单元刚度矩阵和单元质量矩阵.3个典型算例表明,6节点三角形单元的计算结果比ANSYS三角形单元更接近理论解,具有更高的精度.     

Free vibration analysis of orthotropic plates by the initial value method

A modified method of the classical initial value technique is presented for the free vibration analysis of rectangular orthotropic plates. The method employs finite difference operators in one direction and uses integration in the other direction. The solutions are superimposed to satisfy the boundary conditions. A detailed discussion of the method is given. A comparative study is presented of the initial value method and other numerical methods. Three examples that show the accuracy of the method are presented.     

The free vibration of skew plates using the hierarchical finite element method

The hierarchical finite element method is used to determine the natural frequencies and modes of flat, isotropic skew plates. A number of such plates with different boundary conditions—including free edges and point supports—are considered in this paper. The dependence of frequency on skew angle, aspect ratio and Poisson's ratio is investigated, though succinctness prohibits a complete study exploring the full interrelation of these parameters. Extensive results are presented in diagrammatic, graphical, and tabular format; these are shown to be in very good agreement with the work of other investigators, and should prove a valuable source of data for use by engineers and scientists.     

基于修正变分法开孔板和加强板结构的自由振动分析

为研究船舶开孔板和加强板结构的振动特性,用1阶剪切变形板理论描述各向同性板的位移场,并采用修正变分原理和区域分解方法建立板的离散动力学模型.每一块子域板的位移和转角分量通过第一类切比雪夫正交多项式展开.针对加强板模型,将该方法获得的结果与已经发表的文献和有限元商用软件计算结果进行对比,验证该方法的收敛性和正确性.基于修正变分法探讨多种开孔和加强板模型的自由振动特性,充分说明该数理模型和半解析方法是一种适合处理复杂板结构问题的数值工具.     

Numerical solution of potential flow equations with a predictor-corrector finite difference method

We develop a numerical solution algorithm of the nonlinear potential flow equations with the nonlinear free surface boundary condition.A finite difference method with a predictor-corrector method is applied to solve the nonlinear potential flow equations in a two-dimensional (2D) tank.The irregular tank is mapped onto a fixed square domain with rectangular cells through a proper mapping function.A staggered mesh system is adopted in a 2D tank to capture the wave elevation of the transient fluid.The finite difference method with a predictor-corrector scheme is applied to discretize the nonlinear dynamic boundary condition and nonlinear kinematic boundary condition.We present the numerical results of wave elevations from small to large amplitude waves with free oscillation motion,and the numerical solutions of wave elevation with horizontal excited motion.The beating period and the nonlinear phenomenon are very clear.The numerical solutions agree well with the analytical solutions and previously published results.     

A finite difference variational method for bending of plates

The method of analysis for bending of plates presented in this paper combines a finite difference scheme for the plate strain components and a variational derivation of the equations of motion or equilibrium. The plate strain components are expressed in terms of discrete nodal displacements with the aid of the two dimensional Taylor expansion. Consequently, the virtual work, or the first variation of the strain energy, in an area element is found as a function of the nodal displacements. The derivation of the element forces or the element stiffness matrices and the assembly of the equations of motion or equilibrium follows closely the steps of the finite element method.     

A generalized finite difference method using Coatmèlec lattices

Generalized finite difference methods require that a properly posed set of nodes exists around each node in the mesh, so that the solution for the corresponding multivariate interpolation problem be unique. In this paper we first show that the construction of these meshes can be computerized using a relatively simple algorithm based on the concept of a Coatmèlec lattice. Then, we present a generalized finite difference method which provides a numerical solution of a partial differential equation over an arbitrary domain, using the generated meshes. The accuracy and mesh adaptivity of the method is evaluated using elliptical equations in several domains.     

Computing the least quartile difference estimator in the plane

A common problem in linear regression is that largely aberrant values can strongly influence the results. The least quartile difference (LQD) regression estimator is highly robust, since it can resist up to almost 50% largely deviant data values without becoming extremely biased. Additionally, it shows good behavior on Gaussian data—in contrast to many other robust regression methods. However, the LQD is not widely used yet due to the high computational effort needed when using common algorithms. It is shown that it is possible to compute the LQD estimator for n bivariate data points in expected running time O(n²logn) or deterministic running time . Additionally, two easy to implement algorithms with slightly inferior time bounds are presented. All of these algorithms are also applicable to least quantile of squares and least median of squares regression through the origin, improving the known time bounds to expected time O(nlogn) and deterministic time . The proposed algorithms improve on known results of existing LQD algorithms and hence increase the practical relevance of the LQD estimator.     

Free vibration of laminated plates using a high-order element

Free vibration analysis of laminated composite plates using the finite element method has been presented. A high-order quadratic isoparametric element has been employed in the analysis. Both the eight-node serendipity and the nine-node Lagrangian shape functions have been used and their performances have been compared. Various schemes for the generation of the mass matrix have been discussed and a comparative study of these schemes has been presented. The results from the present method have been compared with the closed form solutions and experimental observations of the previous investigators.     

A finite difference continuation method for computing energy levels of Bose-Einstein condensates

We study a finite difference continuation (FDC) method for computing energy levels and wave functions of Bose-Einstein condensates (BEC), which is governed by the Gross-Pitaevskii equation (GPE). We choose the chemical potential λ as the continuation parameter so that the proposed algorithm can compute all energy levels of the discrete GPE. The GPE is discretized using the second-order finite difference method (FDM), which is treated as a special case of finite element methods (FEM) using the piecewise bilinear and linear interpolatory functions. Thus the mathematical theory of FEM for elliptic eigenvalue problems (EEP) also holds for the Schrödinger eigenvalue problem (SEP) associated with the GPE. This guarantees the existence of discrete numerical solutions for the ground-state as well as excited-states of the SEP in the variational form. We also study superconvergence of FDM for solution derivatives of parameter-dependent problems (PDP). It is proved that the superconvergence O(ht) in the discrete H¹ norm is achieved, where t=2 and t=1.5 for rectangular and polygonal domains, respectively, and h is the maximal boundary length of difference grids. Moreover, the FDC algorithm can be implemented very efficiently using a simplified two-grid scheme for computing energy levels of the BEC. Numerical results are reported for the ground-state of two-coupled NLS defined in a large square domain, and in particular, for the second-excited state solutions of the 2D BEC in a periodic potential.     

Nonlinear forced vibration of damped plates by an asymptotic numerical method

This work deals with damped nonlinear forced vibrations of thin elastic rectangular plates subjected to harmonic excitation by an asymptotic numerical method. Using the harmonic balance method and Hamilton’s principle, the governing equation is converted into a static formulation. A mixed formulation is used to transform the problem from cubic nonlinearity to quadratic one sequence. Displacement, stress and frequency are represented by power series with respect to a path parameter. Equating the like powers of this parameter, the nonlinear governing equation is transformed into a sequence of linear problems with the same stiffness matrix. Through a single matrix inversion, a considerable number of terms of the perturbation series can easily be computed with a limited computation time. The starting point, corresponding to a regular solution, is obtained by the Newton–Raphson method. In order to increase the step length, Padé approximants are used. Numerical tests are presented and compared with numerical and analytical results in the literature, for different boundary conditions, excitations and damping coefficients.     

Nonlinear analysis of skew plates using the finite element method

In this paper finite element analysis of the large deflection behaviour of skew plates has been done. A high precision conforming triangular plate bending element has been used. The central deflection, bending and membrane stresses have been reported for simply supported and clamped rhombic plates. The variations of these quantities have been studied for different skew angles.     

Full‐wave analysis of circular guiding structures using the finite difference frequency domain method

In this article, a general full‐wave two dimensional finite difference frequency domain (2D‐FDFD) method is presented that could be used to analyze general circular multi‐layered multi‐conductor guiding structures. The FDFD method is mainly used to get the dispersion curves for these structures. The results which are obtained using the FDFD equations come through solving an eigen‐value problem, where the obtained eigen‐values and eigen‐vectors are used to produce the propagation constants, distribution of the fields and the characteristic impedances for these structures. Several examples ranging from simple coaxial lines to coupled circular microstrip lines are presented. The FDFD results are compared with those obtained through other analytical and numerical techniques. © 2010 Wiley Periodicals, Inc. Int J RF and Microwave CAE, 2010.     

Preconditioned variational methods on rotated finite difference discretisation

Due to their rapid convergence properties, recent focus on iterative methods in the solution of linear system has seen a flourish on the use of gradient techniques which are primarily based on global minimisation of the residual vectors. In this paper, we conduct an experimental study to investigate the performance of several preconditioned gradient or variational techniques to solve a system arising from the so-called rotated (skewed) finite difference discretisation in the solution of elliptic partial differential equations (PDEs). The preconditioned iterative methods consist of variational accelerators, namely the steepest descent and conjugate gradient methods, applied to a special matrix ‘splitting’ preconditioned system. Several numerical results are presented and discussed.