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# 2’D Non-Orthogonal Spline Wavelets and Schneider’s Level- dependent Scheme for 3’D Boundary Elements Method

M.Hooshmand
Candidate of PhD, Civil and Structural Department, University Of Tehran-Iran
K.Bargi
Professor, Civil and Structural Department, University of Tehran-Iran

Key Words: matrix compression scheme, wavelet BEM, 2’D Non-orthogonal wavelet

ABSTRACT: 2’D Non-orthogonal spline wavelets are used as basis function in boundary elements
method(BEM).This kind of wavelet has compact supports and closed-formed expression that have been
proposed by authors. Besides one can choose the vanishing moments of the wavelets Independently of the order
of B-splines. The adaptive meshing for boundary elements makes it possible to reduce the degree of freedom
(DOF) required for a specified accuracy. Sparse coefficient matrices are obtained by truncating the small
elements a priori. The level-dependent schemes enable us to reduce the matrix entries. In the present paper we
investigate the matrix truncation using Schneider’s level-dependent algorithm. The level-dependent truncation
schemes select the truncated entries by comparing the predetermined threshold with the distance between the
supports of two basis function. Through numerical examples, the efficiency of compression scheme together with
Non-orthogonal surface wavelet is investigated.

1 Introduction
The main advantage of application of Boundary Element Method (BEM) is reduction of computational work.
Using BEM , we can decrease the computational space by one dimension,So the amount of computations will be
decreased greatly. Aplication of wavelet basis function is very suitable for truncation of the matrix
coeficients.Using wavelets as basis function enables us to utilize two important properties of wavelets. These are
the local support and vanishing moment.
The vanishing moments imply the orthogonality of the wavelet basis to all polynomials of certain degree or less.
The wavelet basis is originally developed for image processing, and hence the orthonormal wavelets have an
advantage which it can easily perform decomposition and reconstruction of signals or images. The orthogonality
of wavelets plays an important role in image processing and signal analysis ,whereas it is not always essential
property in boundary element analysis. In application of wavelets to BEM , the local support and vanishing
moments of wavelets are essential for reduction of computational cost. By this property the leading order of
integrals including wavelets shifts from the 0th-order term to the nth-order one where n is the order of
wanishing moments. As a result, the integrals have small values if their kernel functions are smooth. Since many
coefficients of boundary element equation are given by such integrals, we can obtain a sparse coefficient matrix
by truncation of these small entries.
The enhancement of wavelet BEM can be accomplished by evolving the wavelet basis. The orthonormal
wavelets have been constructed by Haar(1910),Stromberg(1982),Meyer(1986),Lemarie(1988),Battle(1987)
,Daubechies(1988) on real line,and Von Petersdorff et al.(1997) on a surface. In wavelet BEM , the orthonormal
wavelets are mainly employed as basis functions for descretization of a Boundary Integral Equation (BIE), in
early stage of the research.
Streinberg and Leviatan (1993) and Sabetfakhri and Kateki(1994) have attempted to use the Battle-Lemarie
wavelets in Galerkin BEM for2D scattering problems.Wang(1995,1997) has proposed boundary element analysis
using Daubechies orthogonal wavelets. Steinberg et al. and Sabetfakhri et al. have attempted to employ the
Battle-lemarie wavelet. Wang(1995,1997) has proposed the boundary element analysis using the Daubechies
wavelet. These wavelets are classified in orthonormal wavelets. Although the orthonormal wavelets were
employed in many works , they are disadvantages in application to BEM.
The orthonormal wavelets have infinite supports and they are not closed-form except Haar wavelet that is a
compactly supported wavelet with a closed-form.Goswami et al. have employed a semi-orthogonal wavelet, in

1

which bases in the same subspace do not have orthogonality.Kazuhisa Abe et al.(2003-2005) have proposed a
compact supported wavelet with arbitary vanishing moments. They developed a non-ortogonal wavelet that is
suitable for boundary element analysis.
In the present paper we express the non-orthogonal B-spline surface wavelet that was proposed by authors and
investigate the matrix compression scheme. The matrix compression for the proposed wavelet BEM enables us
to generate a sparse matrix. The matrix truncation scheme is applied using the Schneider level-dependent
method together with Non-orthogonal surface wavelet that was proposed by authors.