Notes on numerical fluid mechanics

A new approach to the numerical solution of weakly singular Volterra integral equations

The boundary element method (BEM) in the context of acoustics or Helmholtz problems is reviewed in this paper. The basis of the BEM is initially developed for Laplace’s equation. The boundary integral equation formulations for the standard interior and exterior acoustic problems are stated and the boundary element methods are derived through collocation. It is shown how interior modal analysis can be carried out via the boundary element method. Further extensions in the BEM in acoustics are also reviewed, including half-space problems and modelling the acoustic field surrounding thin screens. Current research in linking the boundary element method to other methods in order to solve coupled vibro-acoustic and aero-acoustic problems and methods for solving inverse problems via the BEM are surveyed. Applications of the BEM in each area of acoustics are referenced. The computational complexity of the problem is considered and methods for improving its general efficiency are reviewed. The significant maintenance issues of the standard exterior acoustic solution are considered, in particular the weighting parameter in combined formulations such as Burton and Miller’s equation. The commonality of the integral operators across formulations and hence the potential for development of a software library approach is emphasised.

The Boundary Element Method in Acoustics: A Survey

For an integrand with a 1/r vertex singularity, the Duffy transformation from a triangle (pyramid) to a square (cube) provides an accurate and efficient technique to evalu- ate the integral. In this paper, we generalize the Duffy trans- formation to power singularities of the form p(x)/r α , where p is a trivariate polynomial and α> 0 is the strength of the singularity. We use the map (u ,v , w)→ (x, y, z) : x = u β , y = xv, z = xw, and judiciously choose β to accurately estimate the integral. For α = 1, the Duffy transformation (β = 1) is optimal, whereas if α �= 1, we show that there are other values of β that prove to be substantially better. Numerical tests in two and three dimensions are presented that reveal the improved accuracy of the new transforma- tion. Higher-order partition of unity finite element solutions for the Laplace equation with a derivative singularity at a re-entrant corner are presented to demonstrate the benefits of using the generalized Duffy transformation.

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Generalized Duffy transformation for integrating vertex singularities

A new family of systematically constructed near-singularity cancellation transformations is presented, yielding quadrature rules for integrating near-singular kernels over triangular surfaces. This family results from a structured augmentation of the well-known Duffy transformation. The benefits of near-singularity cancellation quadrature are that no analytical integral evaluations are required and applicability in higher-order basis function and curvilinear settings. Six specific transformations are constructed for near-singularities of orders one, two and three. Two of these transformations are found to be equivalent to existing ones. The performance of the new schemes is thoroughly assessed and compared with that of existing schemes. Results for the gradient of the scalar Green function are also presented. For simplicity, static kernel results are shown. The new schemes are competitive with and in some cases superior to the existing schemes considered.

A Family of Augmented Duffy Transformations for Near-Singularity Cancellation Quadrature

To solve one-dimensional linear weakly singular integral equations on bounded intervals, with input functions which may be smooth or not, we propose to introduce first a simple smoothing change of variable, and then to apply classical numerical methods such as product-integration and collocation based on global polynomial approximations. The advantage of this approach is that the order of the methods can be arbitrarily high and that the associated linear systems one has to solve are very well-conditioned.

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High order methods for weakly singular integral equations with nonsmooth input functions

Numerical integration of functions with boundary singularities

In this paper, we propose an efficient strategy to compute nearly singular integrals over planar triangles in R3 arising in boundary element method collocation. The strategy is based on a proper use of various non-linear transformations, which smooth or move away or quite eliminate all the singularities close to the domain of integration. We will deal with near singularities of the form 1/r, 1/r2 and 1/r3, r=∥x−y∥ being the distance between a fixed near observation point x and a generic point y of a triangular element. Extensive numerical tests and comparisons with some already existing methods show that the approach proposed here is highly efficient and competitive. Copyright © 2007 John Wiley & Sons, Ltd.

On the computation of nearly singular integrals in 3D BEM collocation

In this paper we consider the (2D and 3D) exterior problem for the non homogeneous wave equation, with a Dirichlet boundary condition and non homogeneous initial conditions. First we derive two alternative boundary integral equation formulations to solve the problem. Then we propose a numerical approach for the computation of the extra “volume” integrals generated by the initial data. To show the efficiency of this approach, we solve some test problems by applying a second order Lubich discrete convolution quadrature for the discretization of the time integral, coupled with a classical collocation boundary element method. Some conclusions are finally drawn.

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A space–time BIE method for nonhomogeneous exterior wave equation problems. The Dirichlet case

In the last years several authors have used Lubich convolution quadrature formulas to discretize space-time boundary integral equations representing time dependent problems. These rules have the fundamental property of not using explicitly the expression of the kernel of the integral equation they are applied to, which is instead replaced by that of its Laplace transform, usually given by a simple analytic function. In this paper, a review of these rules, which includes their main properties, several new remarks and some conjectures, will be presented when they are applied to the heat and wave space-time boundary integral equation formulations. The construction and behavior of the corresponding coefficients are analyzed and tested numerically. When the quadrature is defined by a BDF method, a new approach for the representation of its coefficients is presented.

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Letizia Scuderi

Papers

High order methods for weakly singular integral equations with nonsmooth input functions

Numerical integration of functions with boundary singularities

On the computation of nearly singular integrals in 3D BEM collocation

A space–time BIE method for nonhomogeneous exterior wave equation problems. The Dirichlet case

Lubich convolution quadratures and their application to problems described by space-time BIEs