Unified formulation of numerical methods. I. Green's formulas for operators in discontinuous fields

Applying the method of weighted residuals and then interpreting the resulting equations by means of Green's formulas for discontinuous functions, a direct method of analysis is developed. The scheme includes finite differences, finite elements, and boundary methods. This is the first of a sequence of articles in which the methodology is presented. A fundamental ingredient of the procedure are general Green's formulas for operators defined in discontinuous fields. They are developed in this first article.     

Unified approach to numerical methods,part II: Finite elements,boundary methods,and their coupling

This is the second in a series of three papers devoted to the presentation of a direct procedure of analysis of numerical methods for partial differential equations. The procedure consists of applying the method of weighted residuals and then interpreting the resulting equations by means of Green's formulas for discontinuous functions. Here, the general Green's formulas for operators defined in discontinuous fields developed in the first article, are applied to formulate the method of weighted residuals for arbitrary linear operators. Finite elements, boundary methods, and general procedures for coupling finite elements and boundary methods are discussed.     

General theory of domain decomposition: Indirect methods

According to a general theory of domain decomposition methods (DDM), recently proposed by Herrera, DDM may be classified into two broad categories: direct and indirect (or Trefftz‐Herrera methods). This article is devoted to formulate systematically indirect methods and apply them to differential equations in several dimensions. They have interest since they subsume some of the best‐known formulations of domain decomposition methods, such as those based on the application of Steklov‐Poincaré operators. Trefftz‐Herrera approach is based on a special kind of Green's formulas applicable to discontinuous functions, and one of their essential features is the use of weighting functions which yield information, about the sought solution, at the internal boundary of the domain decomposition exclusively. A special class of Sobolev spaces is introduced in which boundary value problems with prescribed jumps at the internal boundary are formulated. Green's formulas applicable in such Sobolev spaces, which contain discontinuous functions, are established and from them the general framework for indirect methods is derived. Guidelines for the construction of the special kind of test functions are then supplied and, as an illustration, the method is applied to elliptic problems in several dimensions. A nonstandard method of collocation is derived in this manner, which possesses significant advantages over more standard procedures. © 2002 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 18: 296–322, 2002; Published online in Wiley InterScience (www.interscience.wiley.com). DOI 10.1002/num.10008     

Theory of differential equations in discontinuous piecewise‐defined functions

A truly general and systematic theory of finite element methods (FEM) should be formulated using, as trial and test functions, piecewise‐defined functions that can be fully discontinuous across the internal boundary, which separates the elements from each other. Some of the most relevant work addressing such formulations is contained in the literature on discontinuous Galerkin (dG) methods and on Trefftz methods. However, the formulations of partial differential equations in discontinuous functions used in both of those fields are indirect approaches, which are based on the use of Lagrange multipliers and mixed methods, in the case of dG methods, and the frame, in the case of Trefftz method. This article addresses this problem from a different point of view and proposes a theory, formulated in discontinuous piecewise‐defined functions, which is direct and systematic, and furthermore it avoids the use of Lagrange multipliers or a frame, while mixed methods are incorporated as particular cases of more general results implied by the theory. When boundary value problems are formulated in discontinuous functions, well‐posed problems are boundary value problems with prescribed jumps (BVPJ), in which the boundary conditions are complemented by suitable jump conditions to be satisfied across the internal boundary of the domain‐partition. One result that is presented in this article shows that for elliptic equations of order 2m, with m ≥ 1, the problem of establishing conditions for existence of solution for the BVPJ reduces to that of the “standard boundary value problem,” without jumps, which has been extensively studied. Actually, this result is an illustration of a more general one that shows that the same happens for any differential equation, or system of such equations that is linear, independently of its type and with possibly discontinuous coefficients. This generality is achieved by means of an algebraic framework previously developed by the author and his collaborators. A fundamental ingredient of this algebraic formulation is a kind of Green's formulas that simplify many problems (some times referred to as Green‐Herrera formulas). An important practical implication of our approach is worth mentioning: “avoiding the introduction of the Lagrange multipliers, or the ‘frame’ in the case of Trefftz‐methods, significantly reduces the number of degrees of freedom to be dealt with.” © 2006 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2007     

On spherical spline interpolation and approximation

Spherical spline functions are introduced by use of Green's surface functions with respect to the (Laplace-)Beltrami operator of the (unit) sphere. Natural (spherical) spline functions are used to interpolate data discretely given on the sphere. A method is presented that allows the smoothing of irregularities in measured values or experimental data. Extensions of Peano's theorem and Sard's theory of best approximation to the spherical case are given by integral formulas. Schoenberg's theorem is transcribed into spherical nomenclature.     

Minimization principle and generalized Fourier series for discontinuous Sturm-Liouville systems in direct sum spaces

By modifing the Green''s function method we study certain spectral aspects of discontinuous Sturm-Liouville problems with interior singularities. Firstly, we define four eigen-solutions and construct the Green''s function in terms of them. Based on the Green''s function we establish the uniform convergeness of generalized Fourier series as eigenfunction expansion in the direct sum of Lebesgue spaces $L_2$ where the usual inner product replaced by new inner product. Finally, we extend and generalize such important spectral properties as Parseval equation, Rayleigh quotient and Rayleigh-Ritz formula (minimization principle) for the considered problem.     

Über die Gaußsche Methode zur angenäherten Berechnung von Integralen

The purpose of this paper is the application of Green's theory and Green-Lagrange integral formulas relative to Legendre's differential operator to obtain integral expressions of remainder terms in Gaussian mechanical quadratures.     

Stress and free vibration analysis of functionally graded beams using static Green's functions

In this paper static Green's functions for functionally graded Euler-Bernoulli and Timoshenko beams are presented. All material properties are arbitrary functions along the beam thickness direction. The closed-form solutions of static Green's functions are derived from a fourth-order partial differential equation presented in [2]. In combination with Betti's reciprocal theorem the Green's functions are applied to calculate internal forces and stress analysis of functionally graded beams (FGBs) under static loadings. For symmetrical material properties along the beam thickness direction and symmetric cross-sections, the resulting stress distributions are also symmetric. For unsymmetrical material properties the neutral axis and the center of gravity axis are located at different positions. Free vibrations of functionally graded Timoshenko beams are also analyzed [3]. Analytical solutions of eigenfunctions and eigenfrequencies in closed-forms are obtained based on reference [2]. Alternatively it is also possible to use static Green's functions and Fredholm's integral equations to obtain approximate eigenfunctions and eigenfrequencies by an iterative procedure as shown in [1]. Applying the Sensitivity Analysis with Green's Functions (SAGF) [1] to derive closed-form analytical solutions of functionally graded beams, it is possible to modify the derived static Green's functions and include terms taking cracks into account, which are modeled by translational or rotational springs. Furthermore the SAGF approach in combination with the superposition principle can be used to take stiffness jumps into account and to extend static Green's functions of simple beams to that of discontinuous beams by adding new supports. (© 2011 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Existence result and discontinuous finite element discretization for a plane stresses Hencky problem

We hereafter propose and analyse a discontinuous finite element method for a plane stress Hencky problem. For that purpose we begin by proving an existence result for the continuous problem. A kind of Green's formula between and other intermediate results that may be of independent interest are presented and established separately. Then we formulate the discretized problem, give an existence result for it and prove a result of weak convergence of a subsequence of discrete solutions to a solution of the continuous problem.     

A Boundary Integral Method for Compressible Stokes Flow

A recently developed type of biochip employs ultrasonic surface acoustic waves (SAWs) as a microscale pumping and mixing mechanism for fluids. The driving force for fluid flow is an effect of nonlinear acoustics known as acoustic streaming. We recently studied a two–scale numerical model to describe this effect, which was discretized using classical finite element methods. The micro–scale part of the model describes the propagation of damped acoustic waves. Since the used equations are linear and homogeneous, it is natural to look toward a boundary integral method and attempt a coupling with the FEM scheme still employed in the macro–scale model part. One main ingredient for this approach, namely explicit formulas for free–space Green's functions describing damped acoustics, appear to be novel. We will describe some details of the new scheme, which shows a promising gain of efficiency compared to using FEM for damped acoustics. (© 2008 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

On the structure of linear semigroups

Green's Lemma [1, Lemma 2.2] is one of the most important theorems in the theory of semigroups. The main purpose of this note is to establish a generalized Green's Lemma and a generalized Clifford and Miller's Theorem [1, p. 59] in linear semigroups. A generalized Green's Lemma describes the behavior of certain mappings between two distinct D-classes.     

An enhanced finite difference time domain method for two dimensional Maxwell's equations

An enhanced finite-difference time-domain (FDTD) algorithm is built to solve the transverse electric two-dimensional Maxwell's equations with inhomogeneous dielectric media where the electric fields are discontinuous across the dielectric interface. The new algorithm is derived based upon the integral version of the Maxwell's equations as well as the relationship between the electric fields across the interface. To resolve the instability issue of Yee's scheme (staircasing) caused by discontinuous permittivity across the interface, our algorithm revises the permittivities and makes some corrections to the scheme for the cells around the interface. It is also an improvement over the contour-path effective permittivity algorithm by including some extra terms in the formulas. The scheme is validated in solving the scattering of a dielectric cylinder with exact solution from Mie theory and is then compared with the above contour-path method, the usual staircasing and the volume-average method. The numerical results demonstrate that the new algorithm has achieved significant improvement in accuracy over other methods. Furthermore, the algorithm has a simple structure and can be merged into current FDTD software packages easily. The C++ source code for this paper is provided as supporting information for public access.     

Green's functions for forced vibration analysis of bending-torsion coupled Timoshenko beam

A method based on Green's functions is proposed for the analysis of the steady-state dynamic response of bending-torsion coupled Timoshenko beam subjected to distributed and/or concentrated loadings. Damping effects on the bending and torsional directions are taken into account in the vibration equations. The elastic boundary conditions with bending-torsion coupling and damping effects are derived and the classical boundary conditions can be obtained by setting the values of specific stiffness parameters of the artificial springs. The Laplace transform technology is employed to work out the Green's functions for the beam with arbitrary boundary conditions. The Green's functions are obtained for the beam subject to external lateral force and external torque, respectively. Coupling effects between bending and torsional vibrations of the beam can be studied conveniently through these analytical Green's functions. The direct expressions of the steady-state responses with various loadings are obtained by using the superposition principle. The present Green's functions for the Timoshenko beam can be reduced to those for Euler–Bernoulli beam by setting the values of shear rigidity and rotational inertia. In order to demonstrate the validity of the Green's functions proposed, results obtained for special cases are given for a comparison with those given in the literature and they agree with each other exactly. The influences of external loading frequency and eccentricity on Green's functions of bending-torsion coupled Timoshenko beam are investigated in terms of the numerical results for both simply supported and cantilever beams. Moreover, the symmetric property of the Green's functions and the damping effects on the amplitude of Green's functions of the beam are discussed particularly.     

A method for calculating the long-range behavior of the spin correlator in a two-dimensionalising model

A method is developed for studying the long-range behavior of the spin correlator in a two-dimensional Ising model, based on an approximate solving of the equation for the resolvent of a Toeplitz matrix whose determinant is a correlator. In the scaling domain the answer is expressed in terms of the Green's functions of certain singular equations. The bounds obtained for the norm of the matrix-resolvent yield the possibility of a rigorous justification of the asymptotic formulas.     

Static and dynamic analysis of cracked or weakened structures

Stiffness modifications in engineering structures, for example due to damage and cracking, will inevitably also lead to changes in deformations, internal forces, natural frequencies and mode shapes of the structures. In this paper, an efficient and simple method for sensitivity analysis of cracked or weakened structures under time-harmonic loading is presented. The method is based on a comparison between the strain energy and the kinetic energy of an uncracked structure and that of a cracked structure in conjunction with the application of exact or approximate Green's functions as described in [3] for the static case. The present analysis enables the prediction of any changes in the displacements and stresses and has a lower computational effort as compared to available classical methods, because only the damaged region has to be re-considered in the method. Green's functions are taken as a basis of the approach, which have the ability to weight the influence of the stiffness modifications in a region of a structure and show how sensitive other regions respond to the stiffness modifications. Based on linear elastic fracture mechanics, cracked or damaged regions are approximated by spring models in the analytical solution of some simple beam problems, while cracked finite elements are used for complicated cases where analytical solutions cannot be obtained. Sensitivity analysis with Green's functions (SAGF) approach is applied to static and dynamic analysis of cracked and weakened structures, which consist of homogeneous materials or fiber reinforced composites like reinforced concretes. (© 2010 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Wavelet compression of integral operators arising from boudary-domain integral method

The boundary-domain integral method uses Green's functions to write integral representations of partial differential equations. Since Green's functions are non-local, the systems of linear equations arising from the discretization of integral representations are fully populated. Several algorithms have been proposed, which yield a data-sparse approximation of these systems, such as the fast multipole method, adaptive cross approximation and among others, wavelet compression. In the framework of solving the Navier-Stokes equations in velocity-vorticity form one may utilize the boundary-domain integral method for the solution of the kinematics equation to calculate the boundary vorticity values. Since the kinematics equation is a Poisson type equation, usually its integral representation is written with the Green's function for the Laplace operator. In this work, we introduce a false time into the equation and parabolize its nature. Thus, a time-dependent Green's function may be used. This provides a new parameter, the time step, which can be set to control the Green's function. The time-dependent Green's function is a global function, but by carefully choosing the time step, its behaviour is almost local. This makes it a good candidate for wavelet compression, yielding much better compression ratios at a given accuracy than when using the Green's function for the Laplace operator. However, as false time is introduced, several time steps must be solved in order to reach a converged solution. (© 2014 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Chung–Yau Invariants and Graphs with Symmetric Hitting Times

The Chung–Yau graph invariants were originated from Chung–Yau's work on discrete Green's function. We show how they could be used to derive new explicit formulas and estimates for hitting times of random walks. We also apply them to study graphs with symmetric hitting times.     

The Green polynomials via vertex operators

An iterative formula for the Green polynomial is given using the vertex operator realization of the Hall-Littlewood function. Based on this, (1) a general combinatorial formula of the Green polynomial is given; (2) several compact formulas are given for Green's polynomials associated with upper partitions of length ≤3 and the diagonal lengths ≤3; (3) a Murnaghan-Nakayama type formula for the Green polynomial is obtained; and (4) an iterative formula is derived for the bitrace of the finite general linear group G and the Iwahori-Hecke algebra of type A on the permutation module of G by its Borel subgroup.     

Full-Discrete Finite Element Method for Stochastic Hyperbolic Equation

This paper is concerned with the finite element method for the stochastic wave equation and the stochastic elastic equation driven by space-time white noise. For simplicity, we rewrite the two types of stochastic hyperbolic equations into a unified form. We convert the stochastic hyperbolic equation into a regularized equation by discretizing the white noise and then consider the full-discrete finite element method for the regularized equation. We derive the modeling error by using "Green's method" and the finite element approximation error by using the error estimates of the deterministic equation. Some numerical examples are presented to verify the theoretical results.