Wavelet-based method for stability analysis of vibration control systems with multiple delays

A new method based on wavelets is proposed for stability analysis of vibration control system with multiple delays. In this method, solutions of the initial-value problem of time delayed vibration control systems are approximated by wavelets. Applying wavelet collocation method, the initial-value problem of the time delayed vibration control systems is transformed to a mapping system. The stability of the original system depends on the maximal modulus of eigenvalues of the mapping matrix, and the numerical solution of the initial-value problem are obtained by solving the mapping system. Numerical examples show that the method can locate stable and unstable regions in parameter planes and produces accurate numerical solutions of initial-value problems of time delayed systems.     

Multiscale modeling of beam and plates using customized second-generation wavelets

We have designed bicubic Hermite-type finite-element wavelets that decouple the multiresolution stiffness matrix obtained from the discretization of the biharmonic equation. The scale decoupling basis makes the stiffness matrix block diagonal and hence eliminates the coupling between scales. The scale-decoupled system leads to an incremental procedure for systematic enrichment of the solution without the need for costly remeshing of the whole domain and recalculation of the solution. The solution is obtained by injection of finer-scale wavelets at locations with high detail coefficients. We conducted some numerical experiments to demonstrate the customized wavelet-based finite-element method for the problem of bending of Euler’s beam and Kirchhoff’s plates; we also demonstrate the role of wavelets in resolving localized phenomena.     

A Kansa-type MFS scheme for two-dimensional time fractional diffusion equations

In this paper, an efficient Kansa-type method of fundamental solutions (MFS-K) is extended to the solution of two-dimensional time fractional sub-diffusion equations. To solve initial boundary value problems for these equations, the time dependence is removed by time differencing, which converts the original problems into a sequence of boundary value problems for inhomogeneous Helmholtz-type equations. The solution of this type of elliptic boundary value problems can be approximated by fundamental solutions of the Helmholtz operator with different test frequencies. Numerical results are presented for several examples with regular and irregular geometries. The numerical verification shows that the proposed numerical scheme is accurate and computationally efficient for solving two-dimensional fractional sub-diffusion equations.     

A Legendre spectral element method for the Stefan problem

In this paper we present a Legendre spectral element method for solution of multi-dimensional unsteady change-of-phase Stefan problems. The spectral element method is a high-order (p-type) finite element technique, in which the computational domain is broken up into general (curved) quadrilateral macroelements, and the solution, data and geometry are expanded within each element in terms of tensor-product Lagrangian interpolants. The discrete equations are generated by a Galerkin formulation followed by Gauss–Lobatto Legendre quadrature, for which it is shown that exponential convergence to smooth solutions is obtained as the polynomial order of fixed elements is increased. The spectral element equations are inverted by conjugate gradient iteration, in which the matrix-vector products are calculated efficiently using tensor-product sum-factorization. To solve the Stefan problem numerically, the heat equations in the liquid and solid phases are transformed to fixed domains applying an interface-local time-dependent immobilization transformation technique. The modified heat equations are discretized using finite differences in time, resulting at each time step in a Helmholtz equation in space that is solved using Legendre spectral element elliptic discretizations. The new interface position is then computed using a variationally consistent flux treatment along the phase boundary, and the solution is projected upon the corresponding updated mesh. The rapid convergence rate and stability of the method are discussed, and numerical results are presented for a one-dimensional Stefan problem using both a semi-implicit and a fully implicit time-stepping scheme. Finally, a two-dimensional Stefan problem with a complex phase boundary is solved using the semi-implicit scheme.     

Analytical extraction of leaky modes in circular slab waveguides with arbitrary refractive index profile

Circular slab waveguides are conformally transformed into straight inhomogeneous waveguides, whereupon electromagnetic fields in the core are expanded in terms of Legendre polynomial basis functions. Thereafter, different analytical expression of electromagnetic fields in the cladding region, viz. Wentzel-Kramers-Brillouin solution, modified Airy function expansion, and the exact field solution for circular waveguides, i.e., Hankel function of complex order, are each matched to the polynomial expansion of the transverse electric field within the guide. This field matching process renders different boundary conditions to be satisfied by the set of orthogonal Legendre polynomial basis functions. In this fashion, the governing wave equation is converted into an algebraic and easy to solve eigenvalue problem, which is associated with a matrix whose elements are analytically given. Various numerical examples are presented and the accuracy of each of the abovementioned different boundary conditions is assessed. Furthermore, the computational efficiency and the convergence rate of the proposed method with increasing number of basis functions are briefly discussed.     

A second generation wavelet based finite elements on triangulations

In this paper we have developed a second generation wavelet based finite element method for solving elliptic PDEs on two dimensional triangulations using customized operator dependent wavelets. The wavelets derived from a Courant element are tailored in the second generation framework to decouple some elliptic PDE operators. Starting from a primitive hierarchical basis the wavelets are lifted (enhanced) to achieve local scale-orthogonality with respect to the operator of the PDE. The lifted wavelets are used in a Galerkin type discretization of the PDE which result in a block diagonal, sparse multiscale stiffness matrix. The blocks corresponding to different resolutions are completely decoupled, which makes the implementation of new wavelet finite element very simple and efficient. The solution is enriched adaptively and incrementally using finer scale wavelets. The new procedure completely eliminates wastage of resources associated with classical finite element refinement. Finally some numerical experiments are conducted to analyze the performance of this method.     

APPLICATION OF WAVELETS ON THE INTERVAL TO NUMERICAL ANALYSIS OF INTEGRAL EQUATIONS IN ELECTROMAGNETIC SCATTERING PROBLEMS

The wavelet expansions on the interval are employed for solving the problems of the electromagnetic (EM) scattering from two-dimensional (2-D) conducting objects. The arbitrary configurations of scatterers are modeled using the boundary element method (BEM). By using the wavelets on the interval as basis and test functions, a sparse matrix equation is generated from the integral equation under study. The resulted sparse matrix equation allows the use of sparse matrix solvers or multi-level iterations for rapid solution. The utilization of wavelets on the interval circumvents the difficulties in the application of the wavelets on the real line to finite interval problems, and has no periodicity constraint to the unknown function that is usually imposed by periodic wavelets. Numerical examples are provided and compared with the previously published data or other methods. © 1997 by John Wiley & Sons, Ltd.     

Meshless and wavelets based complex quadrature of highly oscillatory integrals and the integrals with stationary points

In this paper, we formulate and employ efficient and accurate methods for numerical evaluation of highly oscillatory integrals and integrals having stationary points. Two new approaches using radial basis function (RBF) and wavelets are discussed. The first approach is related to meshless method (MM) which is based on multiquadric (MQ) RBF, and is specially designed for integrands having oscillatory character. This approach stems from the Levin's method. In this procedure, the solution is obtained by solving the corresponding ODE or PDE instead of finding a numerical solution of the integration problem. In situations when the integrand has stationary points, MM fails to deliver. We opt for quadrature rules based on Haar wavelets and hybrid functions. The proposed methods are tested on a number of benchmark tests considered in available literature. The performance of the new methods is compared with the existing methods. Better accuracy of the proposed methods is reported for a variety of problems.     

二维 Volterra-Fredholm 积分方程数值算法研究及收敛性分析

针对一类二维非线性Volterra-Fredholm积分方程,提出利用二维Block-Pulse函数为基函数进行数值求解。首先,引入Block-Pulse函数的定义及基函数的向量表示形式;其次,根据二维Block-Pulse函数的不相交性和正交性推导了基向量的积分算子矩阵和乘积算子矩阵;然后,基于该算子矩阵将待求问题转化为一系列向量的乘积形式,利用配点法离散未知变量获得原问题的数值解;最后,通过两个具体的数值算例对所提算法的可行性和收敛性进行了验证。     

Study on the elastic‐plastic buckling strength of plate structures

Abstract

Shifted Legendre polynomials are applied to solve the state equations of linear system. The computation procedure is greatly simplified by introducing the operational matrix for the integration of shifted Legendre vectors whose elements are shifted Legendre polynomials. The key of the method is that the state and forcing functions are expressed in terms of a series of shifted Legendre polynomials with expansion coefficients. Ordinary differential equations of state system are transformed into a series of algebraic equations of the shifted Legendre expansion coefficients and then are solved by employing the technique of matrix inverse. The methods of the computational algorithms are also investigated in order to simplify the calculation procedure and make the calculation convergent.     

Introduction to wavelets in engineering

The aim of this paper is to provide an introduction to the subject of wavelet analysis for engineering applications. The paper selects from the recent mathematical literature on wavelets the results necessary to develop wavelet-based numerical algorithms. In particular, we provide extensive details of the derivation of Mallat's transform and Daubechies' wavelet coefficients, since these are fundamental to gaining an insight into the properties of wavelets. The potential benefits of using wavelets are highlighted by presenting results of our research in one- and two-dimensional data analysis and in wavelet solutions of partial differential equations.     

A boundary element analysis of metal matrix composite materials

The numerical modelling of metal matrix composites is an important part of the research now being conducted on these materials. Due to the numerical complexity of a fully three-dimensional analysis, two-dimensional approximations are normally used with finite element methods. While these analyses are informative, they cannot treat complex particle shapes or examine three-dimensional effects in the composite. The use of boundary element methods in place of the more widely used finite element methods significantly reduces the computing power necessary to obtain a solution to a given problem, making it possible to simulate fully three-dimensional geometries. In the present paper a two-dimensional form of the BEM is applied to the study of metal matrix composite materials, and its performance compared with that of similar FEM stadies. We also compare the predicted composite properties with existing and new experimental results. We conclude that the BEM is an effective tool for the analysis of this class of problems.     

An inverse method for determining elastic material properties and a material interface

A numerical procedure which integrates optimization, finite element analysis and automatic finite element mesh generation is developed for solving a two-dimensional inverse/parameter estimation problem in solid mechanics. The problem consists of determining the location and size of a circular inclusion in a finite matrix and the elastic material properties of the inclusion and the matrix. Traction and displacement boundary conditions sufficient for solving a direct problem are applied to the boundary of the domain. In addition, displacements are measured at discrete points on the part of the boundary where the tractions are prescribed. The inverse problem is solved using a modified Levenberg-Marquardt method to match the measured displacements to a finite element model solution which depends on the unknown parameters. Numerical experiments are presented to show how different factors in the problem and the solution procedure influence the accuracy of the estimated parameters.     

Numerical integration of multi-dimensional highly oscillatory,gentle oscillatory and non-oscillatory integrands based on wavelets and radial basis functions

In this paper Haar wavelets (HWs), hybrid functions (HFs) and radial basis functions (RBFs) are used for the numerical solution of multi-dimensional mild, highly oscillatory and non-oscillatory integrals. Part of this study is extension of our earlier work [9], [47] to multi-dimensional oscillatory and non-oscillatory integrals. Second part of the study is focused on coupling Levin's approach [30] with meshless methods. In first part of the paper, application of the numerical algorithms based on HWs and HFs [9] is extended to integrals having a varying oscillatory and non-oscillatory integrands defined on circular and rectangular domains. In second part of the paper, we propose a meshless method based on multiquadric (MQ) RBF for highly oscillatory multi-dimensional integrals. The first approach is directly related to numerical quadrature with wavelets basis. Like classical numerical quadrature, this approach does not need any intermediate numerical technique. The second approach based on meshless method of Levin's type converts numerical integration problem to a partial differential equation (PDE) and subsequently finding numerical solution of the PDE by a meshless method. The computational algorithms thus derived are tested on a number of benchmark kernel functions having varying oscillatory character or integrands with critical points at the origin. The novel methods are compared with the existing methods as well. Accuracy of the methods is measured in terms of absolute and relative errors. The new methods are simple, more efficient and numerically stable. Theoretical and numerical convergence analysis of the HWs and HFs is also given.     

A dual reciprocity boundary element formulation using the fractional step method for the incompressible Navier–Stokes equations

This paper presents a dual reciprocity boundary element solution method for the unsteady Navier–Stokes equations in two-dimensional incompressible flow, where a fractional step algorithm is utilized for the time advancement. A fully explicit, second-order, Adams–Bashforth scheme is used for the nonlinear convective terms. We performed numerical tests for two examples: the Taylor–Green vortex and the lid-driven square cavity flow for Reynolds numbers up to 400. The results in the former case are compared to the analytical solution, and in the latter to numerical results available in the literature. Overall the agreement is excellent demonstrating the applicability and accuracy of the fractional step, dual reciprocity boundary element solution formulations to the Navier–Stokes equations for incompressible flows.     

Direct numerical simulation of pipe flow using a solenoidal spectral method

In this study, a numerical method based on solenoidal basis functions, for the simulation of incompressible flow through a circular–cylindrical pipe, is presented. The solenoidal bases utilized in the study are formulated using the Legendre polynomials. Legendre polynomials are favorable, both for the form of the basis functions and for the inner product integrals arising from the Galerkin-type projection used. The projection is performed onto the dual solenoidal bases, eliminating the pressure variable, simplifying the numerical approach to the problem. The success of the scheme in calculating turbulence statistics and its energy conserving properties is investigated. The generated numerical method is also tested by simulating the effect of drag reduction due to spanwise wall oscillations.     

An alternative method for degenerate scale problems in boundary element methods for the two-dimensional Laplace equation

The boundary integral equation approach has been shown to suffer a nonunique solution when the geometry is equal to a degenerate scale for a potential problem. In this paper, the degenerate scale problem in boundary element method for the two-dimensional Laplace equation is analytically studied in the continuous system by using degenerate kernels and Fourier series instead of using discrete system using circulants [Engng Anal. Bound. Elem. 25 (2001) 819]. For circular and multiply-connected domain problems, the rank-deficiency problem of the degenerate scale is solved by using the combined Helmholtz exterior integral equation formulation (CHEEF) concept. An additional constraint by collocating a point outside the domain is added to promote the rank of influence matrix. Two examples are shown to demonstrate the numerical instability using the singular integral equation for circular and annular domain problems. The CHEEF concept is successfully applied to overcome the degenerate scale and the error is suppressed in the numerical experiment.     

Potential discontinuity simulations with the numerical Green’s function in steady and transient problems

This paper applies the numerical Green’s function (NGF) boundary element formulation (BEM) first in standard form to solve the Laplace equation and then, coupled to the operational quadrature method (OQM), to solve time domain problems (TD-BEM). Both involve the analysis of potential discontinuities in the respective scalar model simulation. The implementation of the associated Green’s function acting as the fundamental solution is advantageous since element discretization of actual discontinuity surfaces are no longer required. In the OQM the convolution integral is substituted by a quadrature formula, whose weights are computed using the fundamental solution in the Laplace domain, producing the direct solution to the problem in the time domain. Applications of the NGF to problems involving the Laplace equation and its transient counterpart are presented for two-dimensional potential flow examples, confirming that the formulation is stable and accurate.     

Simultaneous inversion for a diffusion coefficient and a spatially dependent source term in the SFADE

This paper deals with an inverse problem of determining a diffusion coefficient and a spatially dependent source term simultaneously in one-dimensional (1-D) space fractional advection–diffusion equation with final observations using the optimal perturbation regularization algorithm. An implicit finite difference scheme for solving the forward problem is set forth, and a fine estimation to the spectrum radius of the coefficient matrix of the difference scheme is given with which unconditional stability and convergence are proved. The simultaneous inversion problem is transformed to a minimization problem, and existence of solution to the minimum problem is proved by continuity of the input–output mapping. The optimal perturbation algorithm is introduced to solve the inverse problem, and numerical inversions are performed with the source function taking on different forms and the diffusion coefficient taking on different values, respectively. The inversion solutions give good approximations to the exact solutions demonstrating that the optimal perturbation algorithm with the Sigmoid-type regularization parameter is efficient for the simultaneous inversion problem in the space fractional diffusion equation.