On modified Green functions in exterior problems for the Helmholtz equation

1982 ◽  
Vol 383 (1785) ◽  
pp. 313-332 ◽  
Keyword(s):  
Integral Equation ◽  
Green Function ◽  
Helmholtz Equation ◽  
Least Squares ◽  
Present Note ◽  
Boundary Integral ◽  
Green Functions ◽  
Exterior Problems ◽  
The Green Function ◽  
Definition Of

The question of non-uniqueness in boundary integral equation formu­lations of exterior problems for the Helmholtz equation has recently been resolved with the use of additional radiating multipoles in the definition of the Green function. The present note shows how this modification may be included in a rigorous formalism and presents an explicit choice of co­efficients of the added terms that is optimal in the sense of minimizing the least-squares difference between the modified and exact Green functions.

scholarly journals Superalgebraically convergent smoothly windowed lattice sums for doubly periodic Green functions in three-dimensional space

2016 ◽  
Vol 472 (2191) ◽  
pp. 20160255 ◽  
Author(s):  
Oscar P. Bruno ◽  
Stephen P. Shipman ◽  
Catalin Turc ◽  
Stephanos Venakides
Keyword(s):  
Green Function ◽  
Dimensional Space ◽  
Three Dimensional ◽  
Integral Equation Method ◽  
Boundary Integral ◽  
Green Functions ◽  
Lattice Sums ◽  
Periodic Arrays ◽  
The Green Function ◽  
Three Dimensional Space

This work, part I in a two-part series, presents: (i) a simple and highly efficient algorithm for evaluation of quasi-periodic Green functions, as well as (ii) an associated boundary-integral equation method for the numerical solution of problems of scattering of waves by doubly periodic arrays of scatterers in three-dimensional space. Except for certain ‘Wood frequencies’ at which the quasi-periodic Green function ceases to exist, the proposed approach, which is based on smooth windowing functions, gives rise to tapered lattice sums which converge superalgebraically fast to the Green function—that is, faster than any power of the number of terms used. This is in sharp contrast to the extremely slow convergence exhibited by the lattice sums in the absence of smooth windowing. (The Wood-frequency problem is treated in part II.) This paper establishes rigorously the superalgebraic convergence of the windowed lattice sums. A variety of numerical results demonstrate the practical efficiency of the proposed approach.

New Higher Order Numeric Quadratures for Regular or Singular Functions on an Interval: Applications for the Helmholtz Integral Equation

1999 ◽  
Author(s):  
Philippe Helluy ◽  
Sylvain Maire ◽  
Patrice Ravel
Keyword(s):  
Integral Equation ◽  
Green Function ◽  
Integral Equations ◽  
Integration Method ◽  
Boundary Integral Equations ◽  
Higher Order ◽  
Boundary Integral ◽  
Numerical Resolution ◽  
Singular Functions ◽  
The Green Function

Abstract A high order integration method is presented for regular or singular integrands over an integral. This method appears to be very useful to compute the integrals of the green function in the numerical resolution of boundary integral equations.

ON RECURSIVE DEFINITION OF THE GREEN FUNCTION OF FINITE DIFFERENCE AND DIFFERENTIAL OPERATORS

1991 ◽  
Vol 05 (03) ◽  
pp. 201-209 ◽  
Author(s):  
V.I. KABANOVICH ◽  
A.N. ERMILOV ◽  
A.M. KURBATOV
Keyword(s):  
Green Function ◽  
Finite Difference ◽  
Differential Operators ◽  
Rational Functions ◽  
Difference Operators ◽  
Green Functions ◽  
Recursive Definition ◽  
Constant Coefficients ◽  
The Green Function ◽  
Definition Of

The possibility of recursive definition of the Green function of finite difference operators with constant coefficients is considered. New equations for the Green functions are found. In particular cases a representation for the Green functions in the form of a series of rational functions is obtained. Generalization for the case of differential operators is carried out. The general problems of the Green functions recursive definition are formulated.

scholarly journals An integral equation–based numerical method for the forced heat equation on complex domains

2020 ◽  
Vol 46 (5) ◽  
Author(s):  
Fredrik Fryklund ◽  
Mary Catherine A. Kropinski ◽  
Anna-Karin Tornberg
Keyword(s):  
Integral Equation ◽  
Heat Equation ◽  
Helmholtz Equation ◽  
Singular Integrals ◽  
Boundary Integral ◽  
Boundary Integral Method ◽  
Elliptic Pdes ◽  
Time Step ◽  
Modified Helmholtz Equation ◽  
High Regularity

Abstract Integral equation–based numerical methods are directly applicable to homogeneous elliptic PDEs and offer the ability to solve these with high accuracy and speed on complex domains. In this paper, such a method is extended to the heat equation with inhomogeneous source terms. First, the heat equation is discretised in time, then in each time step we solve a sequence of so-called modified Helmholtz equations with a parameter depending on the time step size. The modified Helmholtz equation is then split into two: a homogeneous part solved with a boundary integral method and a particular part, where the solution is obtained by evaluating a volume potential over the inhomogeneous source term over a simple domain. In this work, we introduce two components which are critical for the success of this approach: a method to efficiently compute a high-regularity extension of a function outside the domain where it is defined, and a special quadrature method to accurately evaluate singular and nearly singular integrals in the integral formulation of the modified Helmholtz equation for all time step sizes.

A Boundary Integral Analysis of Lugs Attached to Cylindrical Pressure Vessels

1988 ◽  
Vol 110 (4) ◽  
pp. 355-360 ◽  
Author(s):  
G. N. Brooks
Keyword(s):  
Integral Equation ◽  
Pressure Vessels ◽  
Boundary Integral ◽  
Green Functions ◽  
Integral Analysis ◽  
Simply Supported ◽  
Integral Equation Formulation ◽  
Load Carrying ◽  
Solution Methods ◽  
Rigid Attachment

Integral equations are derived to calculate the stresses and displacements in the neighborhood of a load-carrying rigid attachment in a shallow cylindrical shell. The integral equation formulation is simplified by modifying existing Green functions for the unbounded shell to account for simply supported boundary conditions at the ends of the vessel. The resulting equations are solved numerically. Three forces and three moments applied to the attachment are the loadings considered. Results for circular attachments agree with those found by other authors using different solution methods and with experiments.

Application of the Boundary Element Method to Acoustic Cavity Response and Muffler Analysis

1987 ◽  
Vol 109 (1) ◽  
pp. 15-21 ◽  
Author(s):  
A. F. Seybert ◽  
C. Y. R. Cheng
Keyword(s):  
Integral Equation ◽  
Boundary Element Method ◽  
Boundary Element ◽  
Boundary Integral Equation ◽  
Transmission Loss ◽  
Radiation Condition ◽  
Boundary Integral ◽  
Exterior Problems ◽  
Integral Equation Formulation ◽  
Element Method

This paper is concerned with the application of the Boundary Element Method (BEM) to interior acoustics problems governed by the reduced wave (Helmholtz) differential equation. The development of an integral equation valid at the boundary of the interior region follows a similar formulation for exterior problems, except for interior problems the Sommerfeld radiation condition is not invoked. The boundary integral equation for interior problems does not suffer from the nonuniqueness difficulty associated with the boundary integral equation formulation for exterior problems. The boundary integral equation, once obtained, is solved for a specific geometry using quadratic isoparametric surface elements. A simplification for axisymmetric cavities and boundary conditions permits the solution to be obtained using line elements on the generator of the cavity. The present formulation includes the case where a node may be placed at a position on the boundary where there is not a unique tangent plane (e.g., at an edge or a corner point). The BEM capability is demonstrated for two types of classical interior axisymmetric problems: the acoustic response of a cavity and the transmission loss of a muffler. For the cavity response comparison data are provided by an analytical solution. For the muffler problem the BEM solution is compared to data obtained by a finite element method analysis.

scholarly journals Simplifying the one-loop five graviton amplitude in type IIB string theory

2017 ◽  
Vol 32 (14) ◽  
pp. 1750074 ◽  
Author(s):  
Anirban Basu
Keyword(s):  
Green Function ◽  
String Theory ◽  
Fundamental Domain ◽  
Green Functions ◽  
Considerable Simplification ◽  
The Green Function ◽  
Type Iib String Theory ◽  
The One

We consider the [Formula: see text] and [Formula: see text] terms in the low momentum expansion of the five graviton amplitude in type IIB string theory at one loop. They involve integrals of various modular graph functions over the fundamental domain of [Formula: see text]. Unlike the graphs which arise in the four graviton amplitude or at lower orders in the momentum expansion of the five graviton amplitude where the links are given by scalar Green functions, there are several graphs for the [Formula: see text] and [Formula: see text] terms where each of these two links are given by a derivative of the Green function. Starting with appropriate auxiliary diagrams, we show that these graphs can be expressed in terms of those which do not involve any derivatives. This results in considerable simplification of the amplitude.

scholarly journals Use of Clement’s ODEs for the Speedup of Computation of the Green Function and its Derivatives for Floating or Submerged Bodies in Deep Water

2018 ◽  
Author(s):  
Chunmei Xie ◽  
Aurélien Babarit ◽  
François Rongère ◽  
Alain H. Clément
Keyword(s):  
Green Function ◽  
Deep Water ◽  
Boundary Integral ◽  
Computational Time ◽  
Floating Bodies ◽  
Acceleration Technique ◽  
Singular Kernels ◽  
Classical Boundary ◽  
The Green Function ◽  
Rankine Sources

A new acceleration technique for the computation of first order hydrodynamic coefficients for floating bodies in frequency domain and in deep water is proposed. It is based on the classical boundary element method (BEM) which requires solving a boundary integral equation for distributions of sources and/or dipoles and evaluating integrals of Kelvin’s Green function and its derivatives over panels. The Kelvin’s Green function includes two Rankine sources and a wave term. In present study, for the two Rankine sources, analytical integrations of strongly singular kernels are adopted for the linear density distributions. It is shown that these analytical integrations are more accurate and faster than numerical integrations. The wave term is obtained by solving Clément’s ordinary differential equations (ODEs) [1] and an adaptive numerical quadrature is performed for integrations over the panels. It is shown here that the computational time of the wave term by solving the ODEs is greatly reduced compared to the classical integration method [7].

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