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.     

Unified formulation of numerical methods

A summary of the unified theory of numerical methods recently published by the author is presented [5–8]. In the theory, a direct method of analysis is developed applying the method of weighted residuals and then interpreting the resulting equations by means of Green's formulas for discontinuous functions. The scheme includes finite differences, finite elements and boundary methods. A fundamental ingredient of the procedure are general Green's formulas for operators defined in discontinuous fields.     

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     

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.     

Green's function-based integral approaches to nonlinear transient boundary-value problems (II)

Two Green's function-based numerical formulations are used to solve the time-dependent nonlinear heat conduction (diffusion) equation. These formulations, which are an extension of the first paper, utilize two fundamental solutions and the Green's second identity to achieve integral replications of the governing partial differential equation. The integral equations thus derived are discretized in space and time and aggregated in a finite element sense to give a system of nonlinear discrete equations that are solved by the Newton–Raphson algorithm. The mathematical simplicity of the Green's function of the first formulation facilitates its numerical implementation. The performance of the formulations is assessed by comparing their results with available numerical and analytical solutions. In all cases satisfactory and physically realistic results are obtained.     

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)     

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.     

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)     

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.     

Analysis of arbitrarily shaped planar cracks in two-dimensional hexagonal quasicrystals with thermal effects. Part II: Numerical solutions

The extended displacement discontinuity (EDD) boundary element method is developed to analyze an arbitrarily shaped planar crack in two-dimensional (2D) hexagonal quasicrystals (QCs) with thermal effects. The EDDs include the phonon and phason displacement discontinuities and the temperature discontinuity on the crack face. Green's functions for uniformly distributed EDDs over triangular and rectangular elements for 2D hexagonal QCs are derived. Employing the proposed EDD boundary element method, a rectangular crack is analyzed to verify the Green's functions by discretizing the crack with rectangular and triangular elements. Furthermore, the elliptical crack problem for 2D hexagonal QCs is investigated. Normal, tangential, and thermal loads are applied on the crack face, and the numerical results are presented graphically.     

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.     

Approximate Separability of the Green's Function of the Helmholtz Equation in the High Frequency Limit

The minimum number of terms that are needed in a separable approximation for a Green's function reveals the intrinsic complexity of the solution space of the underlying differential equation. It also has implications for whether low‐rank structures exist in the linear system after numerical discretization. The Green's function for a coercive elliptic differential operator in divergence form was shown to be highly separable [2], and efficient numerical algorithms exploiting low‐rank structures of the discretized systems were developed. In this work, a new approach to study the approximate separability of the Green's function of the Helmholtz equation in the high‐frequency limit is developed. We show (1) lower bounds based on an explicit characterization of the correlation between two Green's functions and a tight dimension estimate for the best linear subspace to approximate a set of decorrelated Green's functions, (2) upper bounds based on constructing specific separable approximations, and (3) sharpness of these bounds for a few case studies of practical interest. © 2018 Wiley Periodicals, Inc.     

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.     

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.     

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)     

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.     

A Novel Time‐Domain BIEM for Wave Propagation Analysis in Anisotropic Solids With Cracks

Analysis of elastic wave propagation in anisotropic solids with cracks is of particular interest to quantitative non‐destructive testing and fracture mechanics. For this purpose, a novel time‐domain boundary integral equation method (BIEM) is presented in this paper. A finite crack in an unbounded elastic solid of general anisotropy subjected to transient elastic wave loading is considered. Two‐dimensional plane strain or plane stress condition is assumed. The initial‐boundary value problem is formulated as a set of hypersingular time‐domain traction boundary integral equations (BIEs) with the crack‐opening‐displacements (CODs) as unknown quantities. A time‐stepping scheme is developed for solving the hypersingular time‐domain BIEs. The scheme uses the convolution quadrature formula of Lubich [1] for temporal convolution and a Galerkin method for spatial discretization of the BIEs. An important feature of the present time‐domain BIEM is that it uses the Laplace‐domain instead of the more complicated time‐domain Green's functions. Fourier integral representations of Laplace‐domain Green's functions are applied. No special technique is needed in the present time‐domain BIEM for evaluating hypersingular integrals.     

Green's function of two-dimensional time-fractional diffusion equation using addition formula of Wright function

In this paper, using the Mellin transform of Wright function we derive an addition formula for the Wright function. In some special cases, addition formulas for the Hermite, Bessel and Mittag-Leffler functions are also given and the Green's function of two-dimensional time-fractional diffusion equation is presented in the whole plane.