CALCULATION OF THE IMPROPER INTEGRALS FOR FOURIER BOUNDARY ELEMENT METHOD

Abstract. The traditional Boundary Element Method (BEM) is a collection of numerical techniques for solving some partial differential equations. The classical BEM produces a fully populated coefficients matrix. With Galerkin Boundary Element Method (GBEM) is possible to produce a symmetric coefficients matrix. The Fourier BEM is a more general numerical approach. To calculate the final matrix coefficients it is necessary to find the improper integrals. The article presents the method for calculation of such integrals.


Introduction
Basic integral equation for the Boundary Element Method (BEM) is constructed by the convolution with the fundamental solution [2,3,6].  The basic principles of the traditional BEM are presented for the paradigmatic example of the n-dimensional stationary heat conduction described by [2]: where: ∆ -Laplace operator, u -the unknown quantity, f -the known volume sources in Ω.
To obtain a well posed problem, half of the boundary data (either u on Γ u or t on Γ t ) should be defined by boundary conditions, i.e. Ω ∂ = Γ ∪ Γ t u .

The distribution theory
Distributions are objects which generalize functions [2,8]. They extend the concept of derivative to all locally integral functions and are used to formulate generalized solutions of partial differential equations. They are important in physics and engineering where many non-continuous problems naturally lead to differential equations whose solutions or initial conditions are distributions, such as the Dirac delta distribution.
The basic idea is to identify functions with abstract linear functionals on a space of well-behaved test functions.
For example, let: u: R → R, be a locally integrable, and φ: R → R ,be a smooth (infinitely differentiable) function with compact support (i.e., identically zero outside of some bounded set). The function φ is the test function and: This is a real number which depends on φ.
The differentiation of generalized functions is defined as: Because of the definition of the test function φ, distributions are infinitely differentiable. Jumps and singularities can be differentiated [2].
By using a larger space of test functions, it is possible to define the tempered distributions, useful for the Fourier transformation in generality. All tempered distributions have a Fourier transformation, but not all distributions have one [2].
The invariance of the scalar product concerning the Fourier transformation is called Parseval's identity: It is possible to define the Fourier transformation of tempered distributions. These include all the integrable functions, as well as well-behaved functions of polynomial growth and distributions of compact support, and have the added advantage that the Fourier transformation of any tempered distribution is again a tempered distribution.

The special distributions
The n-dimensional Dirac distribution: is defined by: . 0 x all for 0 ) ( The Dirac distribution is the identity object concerning convolution: and its Fourier transformation is: (10) The Heaviside distribution is obtained by the integration of the Dirac distribution: (11) In the literature, there are several definitions for the value at x=0. For the linear distribution it is determined by: For the multidimensional Heaviside distribution, the cutoff distribution for a domain n R ∈ Ω is defined: which can be expressed by: with a function ) The integration of a distribution u over the domain Ω can be described by: The main advantage of the theory of distribution is that it re-establishes differentiation as the simple procedure and all quantities are differentiable even if they exhibit singularities and jumps [2].

Fourier BEM
To obtain the Fourier transformation of the Boundary Integral Equations (BIE), all quantities have to be extended from Ω to R n . It can be done by defining a cutoff distribution χ [2]. All quantities are multiplied by χ and finally transformed into Fourier space. Mathematically this extension and transformation is justified only in the frame of the theory of distributions [2,8].
The main advantage of the distributional BIE is that the integrals extend formally over the entire R n and therefore the Fourier transformation can be applied to these integral equation.
For the definition of the trial functions it is needed to define a cutoff distribution [2] for a rectangular element: The trial functions are obtained by multiplying with the translation vector b i and the dilation matrix a i . Finally the unknown and the known quantities on the boundaries are approximated by: The n-dimension Fourier transformation: The basics of Fourier BEM are two known theorems of the Fourier transformation.
The theorem of Parseval states the invariance of energy or work with respect to the dimensional Fourier transformation: The convolution theorem links the convolution in the original space to a simple multiplication in the transformed space: In the notation: these two theorems may be described as: (28) The Fourier BEM method analysed by [2] is especially of interest for cases where the fundamental solution is not known.
The transformation of the cutoff distribution 0 χ is: • for reference element in R 2 : • for reference element in R 3 : For straight elements and for arbitrary polynomial trial functions p 0 (x), the transformed expressions are analytically known in R 2 and R 3 [2].
The discretized Fourier BIE lead to an algebraic system identical to that obtained in the original space (Galerkin BEM [7]), where the matrices are computed in the transformed space and:

Numerical example
The problem of the numerical integration for Fourier BEM formulation is presented for the boundary integral equations limited to constant elements and 2D space. As the test example, the Dirichlet problem of the Poisson equation is considered [2].
The Dirichlet problem for Poisson equation: is solved in a quadratic two-dimensional domain ] The interior is subjected to stationary heat source f. The boundary Ω ∂ is divided into 8 elements (Fig. 2).

Fig. 2. Quadratic domain Ω with 8 boundary elements and constant trial function
The fundamental solution and its transformation for the Laplacian ∆ are [2]: Taking into account the fact that u=0 at the boundaries, the general system of BIE can be reduced to: For the R 2 elements the cuttoff distribution definition is described by eq.13 .
and its Fourier transformation is described by: In our example, for constant elements, the trial function and its Fourier transformation is: (38) Additionally, the Fourier transformation for dilation and translation operators is described as (eq. 19): The trial functions should be defined for 16 constant elements. For every element, the coefficients for Heaviside and Dirac distribution should be modified to receive the value of the product ) ( equal to one inside the element and equal to zero outside. From the definition, the Dirac distribution is equal to one only for x=0, and Heaviside distribution is equal to one for x>0.
and similar for other The equation system in Fourier space is: where: Computer implementation of Fourier BEM requires the same skills as classical BEM [1,3,6,7]. Integration with respect to unknowns in the Fourier approach is equivalent to the integral: where: i, j -number of elements,

The integral determination
After dividing the infinite area into four subareas (Fig. 3) we have: To calculate the integrals (46) the Gaussian quadrature with the -1 and 1 integration limits was used. To do that every subarea was transformed to a local coordinate system using the transformations T 1 , T 2 and T 3 (Fig. 4).

Results
For 8 elements the matrix H coefficients were calculated symbolically [4] and numerically. Table 1 presents the value of the final solution (the solutions for 8 elements discretization are the same). For the numerical calculation the 80 integration points were used [5]. The numerical calculation of the integrals (46) is very complicated. The calculation based on the proposed algorithm allows us to achieve the 1.13% accuracy.

Conclusion
Numerical method for determination of improper integrals occurring in FBEM allowed to determine the results with an error of 1.13%. The Fourier BEM method is more difficult than the standard BEM method but is specially of interest for cases where the fundamental solution is not known.