B-Spline approximation and fast wavelet transform for an ... - UTC

Engineering Analysis with Boundary Elements 26 (2002) 1±13

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B-Spline approximation and fast wavelet transform for an ef®cient evaluation of particular solutions for Poisson's equation E. Perrey-Debain* ,1, H.G. ter Morsche Department of Mathematics and Computing Science, Eindhoven University of Technology, P.O. Box 513, 5600 MB Eindhoven, The Netherlands Received 9 May 2001; revised 6 July 2001; accepted 18 July 2001

Abstract Based on the idea of the Dual Reciprocity Method, a numerical method has been devised to interpolate the source term of the Poisson's equation by using B-spline approximation and then use them to approximate particular solutions. The advantage of such a construction is the possibility of using the Fast Wavelet Transform to minimize the number of coef®cients in the wavelet expansion. Practical concerns in the implementation of the method are discussed. Three numerical examples are presented to validate the approach. q 2001 Elsevier Science Ltd. All rights reserved. Keywords: Poisson's equation; Particular solutions; Dual reciprocity method; Newton potential; B-Spline; Wavelets

1. Introduction The numerical solution of the Poisson's equation is an important problem in numerical analysis with many practical rami®cations in the applied sciences. There are direct applications in incompressible ¯uid mechanics, heat transfer, electrostatics, to name but a few. Moreover, linear partial differential equations with varying coef®cients [1] and non-linear problems [2] can be sometimes solved by iterative procedures where each iterative step involves solving the Poisson's equation. The importance of a reliable and computationally ef®cient solution to this equation is dif®cult to overstate. Although the Finite Difference Method (FDM) and the Finite Element Method (FEM) can provide excellent approximation, they usually fail to represent correctly complicated domains and boundary conditions. Furthermore, both methods require solving large systems of equations if an accurate solution is sought. The Boundary Element Method (BEM) gains its main advantage over these last methods by reducing the dimension of the problem: only the boundary discretization is required and the number of unknowns is signi®cantly reduced. However, * Corresponding author. Tel.: 144-191-374-7094; fax: 144-191-3747253. E-mail address: [email protected] (E. Perrey-Debain). 1 Present address: School of Engineering, University of Durham, South Road, Durham, Co. Durham, DH1 3LE, UK.

the presence of the inhomogeneous term (or source term) in the Poisson's equation makes the BEM less attractive because the integral formulation involves a domain integral whose evaluation may consume the majority of the computation time. In the last two decades, a number of alternatives to direct integration of the domain integral have been proposed, among them can be noted the important paper of Atkinson [3] and the pioneering work of Nardini and Brebbia [4] on the dual reciprocity method (DRM) in 1982. Surveys on this topic were given by Golberg and Chen in 1994 [5] and Golberg in 1995 [6]. Since the introduction of the DRM, there has been increasing interest in the use of Radial Basis Functions (RBFs) in the BEM community. The main reasons are (i) RBFs often provide excellent interpolants for multidimensional data; (ii) for any rotationally symmetric operator (Laplace, Biharmonic, Helmholtz, etc.), the particular solution is found by solving an ordinary differential equation and an analytical form is then often available; (iii) problems for which the domain integral includes the problems variables can be handled; (iv) the numerical implementation is relatively easy. Nevertheless, to our point of view, two major facts are still restricting the DRM. The ®rst is the uncertainties associated with the placement of the interpolation points. The second is that the non-compact support of the RBFs makes the interpolation matrix dense, time consuming to invert and sometimes highly ill-conditioned, especially for a large number of interpolation points [7]. Recently, however, Chen et al. [8] have begun to explore

0955-7997/02/$ - see front matter q 2001 Elsevier Science Ltd. All rights reserved. PII: S 0955-799 7(01)00080-7

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the use of compactly supported radial basis functions to alleviate this last dif®culty. Their results showed signi®cant improvements (interpolation matrix density of 20% with remarkable accuracy) but it is not clear how to properly de®ne the scaling parameter that controls the support of the RBF. As discussed by the authors, the interpolation matrix is diagonal when the scaling parameter is suf®ciently small. In this case the source term is approximated as a linear combination of sharp spikes and the quality of approximation is poor. This last statement is only true because the number of collocation points usually used in the DRM is rather small (few hundreds at most). Consider now the same problem with few dozen of thousands of collocation points; the approximation will be then of good quality even for `pathological' functions presenting some discontinuities. The drawback is that the evaluation of the particular solution will be too much time consuming making this idea unrealistic. In this paper, we show that, by replacing the RBF by BSpline, it is then possible to use a ®lter process called the Fast Wavelet Transform (FWT). This process allows to reduce drastically the degree of freedom while preserving a good quality of approximation even for `pathological' functions. The Section 2 contains a brief introduction to the method of particular solutions. In Section 3, we present the theoretical ingredients needed for our numerical applications: wavelet theory and Fast Wavelet Transform. The numerical evaluation of the associated particular solutions is carried out in Section 4. As there is no analytical form for these particular solutions, we show how to store them in Section 5. Three numerical examples of increasing complexity are given in Section 6 to illustrate the effectiveness of our proposed method. We also compare our results with previous methods. 2. Method of particular solutions Let V , R 2 be an open bounded simply connected set with boundary G . We consider solving the boundary value problem Du…x† ˆ f …x†;

x[V

…1†

with boundary conditions B i …u†uG i ˆ 0;

i ˆ 1; 2; ¼; m

…2†

where G i is a partition of G and B iˆ1;¼;m are the usual boundary operators (Dirichlet, Neumann and Robin type). To solve Eq. (1), it is convenient to ®rst reduce it to an equivalent homogeneous equation, since the resulting boundary integral reformulation will then be free of domain integrals. To do this, let up be a particular solution of Eq. (1), i.e. a solution which does not necessarily satisfy the given boundary conditions, and let u ˆ v 1 up : Then by linearity, v satis®es Dv…x† ˆ 0;

x[V

…3†

with boundary conditions derived from that of u and up. Once up and 2u p =2n are known, Eq. (3) can be solved by a standard BEM [9]. The key issue is how to determine the particular solution up and its normal derivative. In Section 3 we brie¯y give some background relative to the wavelets theory with special regard for the B-spline wavelets basis for approximating f. 3. Biorthogonal B-spline wavelets 3.1. Multi-resolution analysis on R2 The concept of multi-resolution analysis plays a central role in the context of classical wavelets. Although it is not the place for a complete introduction, we shall recall the basic setting of wavelet analysis on R2 as far as it is needed for our purposes. For the reader who is not familiar with wavelets theory, mathematical concepts such as Riesz basis, biorthogonality or dual pairs can be found in Appendix A. For a lot more details, we refer to Chui [10], Dahmen [11] and the overview of Jawerth and Sweldens [12]. To facilitate the discussion in the sequel of this paper, we denote a point in R2 by x ˆ …x1 ; x2 † and we return to the usual notation x ˆ x in one dimension. Furthermore, we denote a couple of integer in Z2 by k ˆ …k1 ; k2 † and l ˆ …l1 ; l2 †: All the other variables involved have the usual meaning. Let H be the Hilbert space L2 …R2 † of square integrable bivariate functions. A multi-resolution sequence V ˆ {Vj }j[Z consists of nested closed subspaces Vj , H whose union is dense in H, i.e. 0 1 [ Vj , Vj11 ; closH @ Vj A ˆ H: …4† j[Z

A scaling function f [ V0 ; with a non-vanishing integral, exists such that the collection {f…x 2 k†uk [ Z2 } is a Riesz basis of V0. Since f [ V0 , V1 ; a sequence (ak) exists such that the scaling function satis®es X f…x† ˆ 2 ak f…2x 2 k†: …5† k

This functional equation is often called the re®nement equation. It can be shown that the collection of functions

fj;k …x† ˆ 2j f…2j x 2 k† is also a Riesz basis of Vj. We will use Wj to denote a space complementing Vj in Vj11 ; i.e. a space that satis®es Vj11 ˆ Vj % Wj

…6†

where the symbol % stands for direct sum. In other words, each element of Vj11 can be written, in a unique way, as the sum of an element of Wj and an element of Vj. The spaces Wj is sometime called the detail space because it contains the `detail' information needed to go from an approximation at

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resolution j to an approximation at resolution j 1 1: Now, e for j ®xed, the wavelet functions {c j;k ue ˆ 1; 2; 3; k [ Z2 } constitute Riesz basis of the detail space Wj. They are obtained by dilatations and integer translations of a ®nite number of mother functions (`mother wavelets'). In fact, one has

books such as Ref. [10]. In signal processing this idea is known as subband ®ltering. It consists of applying a low~ and a band-pass …B† ~ ®lter followed by a downpass …A† sampling (i.e. retaining only the even index sample).

e c j;k …x† ˆ 2j c e …2j x 2 k†:

One of the basic methods for constructing wavelets involves the use of cardinal B-spline functions (or simply B-splines). These are probably the simplest functions with small supports that are most ef®cient for numerical implementation. Moreover, they give rise to an important class of dual pairs where both scaling functions have compact support. Because bivariate B-splines as the wavelets stemming from them are simply tensor products of the univariate B-splines and univariate wavelets, we ®rst deal with the onedimensional setting. For simplicity we have chosen a B-spline of order 2 (i.e. the well-known hat function) as the univariate scaling function w

Eq. (6) implies that there exist a sequence …bek † such that X c e …x† ˆ 2 bek f…2x 2 k†: …7† k

To show how to pass between to successive resolutions, we need to have access to a suitable biorthogonal wavelet basis. Here, a dual scaling function f~ and dual wavelets c~ e generate a dual multi-resolution of H with subspaces V~ j and W~ j ; such that they form a dual pair and must satisfy the biorthogonality conditions. Since the dual basis de®ne a multi-resolution, the dual functions must satisfy a relation similar to Eqs. (5) and (7) X f~ …x† ˆ 2 a~ k f~ …2x 2 k† …8† k

and

c~ e …x† ˆ 2

X k

b~ ek f~ …2x 2 k†:

…9† e

By construction, wavelets c and their duals all have vanishing integrals. Therefore, by Eqs. (7) and (9), we have the following properties X e X e bk ˆ …10† b~ k ˆ 0: k

k

3.2. Fast wavelet transform Since Vj11 ˆ Vj % Wj ; a function vj11 [ Vj11 can uniquely be written as the sum of a function vj [ Vj and a function wj [ Wj ; i.e. X vj11 …x† ˆ cj11;k fj11;k …x† ˆ vj …x† 1 wj …x† k

ˆ

X l

cj;l fj;l …x† 1

3 X X eˆ1

l

e e dj;l c j;l …x†:

The following relations show how to pass between these two representations (see Ref. [12] for details) X cj;l ˆ …11† a~ k22l cj11;k k

and e dj;l ˆ

X k

b~ ek22l cj11;k :

3.3. B-Spline wavelet on the real line

w…x† ˆ max{0; 1 2 uxu}:

…13†

Its re®nement equation reads p X w…x† ˆ 2 hn w…2x 2 n†

…14†

n

with

p 2 …h21 ; h0 ; h1 † ˆ …1; 2; 1†: 4

…15†

Cohen et al. [13] showed that there is a compactly supported dual scaling function w~ such that w and w~ form a dual pair. The dual function w~ is not unique and in our application, we consider the simplest case (see Table 6.1, page 542 in Ref. [13]) p X w~ …x† ˆ 2 h~ n w~ …2x 2 n† …16† n

with

p 2 …21; 2; 6; 2; 21†: …17† 8 The univariate wavelet c and its dual c~ must form a dual pair as and verify the biorthogonality conditions. It is known that such candidate are given by (see for instance Chapter 4 in Ref. [11]) p X c…x† ˆ 2 …21† n h~12n w…2x 2 n†;

…h~ 22 ; h~21 ; h~ 0 ; h~ 1 ; h~2 † ˆ

n

p X c~ …x† ˆ 2 …21† n h12n w~ …2x 2 n†: n

…12†

When applied recursively, these two formulas de®ne a forward Fast Wavelet Transform (FWT). The inverse formula (not needed here) can be found in standard text-

In Fig. 1, we plot the scaling function w and the wavelet c . We note that the supports are respectively ‰21; 1Š and ‰21; 2Š and the wavelet is symmetric about x ˆ 1=2: Graphs of the corresponding dual functions are given in page 546 of Ref. [13].

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Fig. 1. Scaling function and wavelet.

3.4. The bivariate case The simplest way of generating orthogonal or biorthogonal wavelets on R 2 is via tensor products. Given a dual pair …w; w~ † of univariate scaling functions, the products

f…x† ˆ w…x1 †w…x2 †;

form a dual pair in L2 …R2 †: The three `mother wavelets' are obtained by the products

c 1 …x† ˆ w…x1 †c…x 2 †; c 2 …x† ˆ c…x 1 †w…x2 †; c 3 …x† ˆ c…x 1 †c…x 2 †: The dual functions are de®ned analogously. The corresponding sequences are obtained from the univariate ones in a straightforward fashion: X f…x† ˆ 2 hk1 hk2 w…2x1 2 k1 †w…2x2 2 k2 † k1 ;k2

X k

and

c e …x† ˆ 2

ak f…2x 2 k†

X k

…18†

bek f…2x 2 k†; e ˆ 1; 2; 3:

where b1k ˆ …21†k2 hk1 h~ 12k2 ; b3k ˆ …21†k1 1k2 h~ 12k1 h~ 12k2 :

4. The numerical evaluation of particular solutions 4.1. Interpolation scheme and data compression

f~ …x† ˆ w~ …x1 †w~ …x2 †

ˆ2

The sequences …a~ k † and …b~ ek † for the dual functions are obtained in the same manner.

b2k ˆ …21†k1 h~12k1 hk2 ;

…19†

Consider a bounded function of the form f ˆ xV 0 f where xV 0 is the characteristic function of a bounded domain V 0 such that V , V 0 : We want to ®nd an approximation of f in the wavelet basis of Section 3. The natural way is to project f by performing the scalar product of f with the biorthogonal basis. However, the computation of inner products with dual wavelets is not an easy task since one needs suitable quadrature rules and in many cases, the dual wavelets are not known explicitly but only via certain functional equations from which the function values have to be computed or approximated. An alternative approach is to use the quasi-interpolation operator of Chui [10,14]. Now, let h . 0 be the distance between two interpolation points as shown in Fig. 2 (only a portion of the grid hZ2 has been represented for the sake of clarity). An approximation f0;h of f in the basis {f0;k uk [ Z2 } of the space of reference V0 is obtained by using the quasi-interpolation operator P h de®ned as follows [10] f0;h …x† ˆ Ph f …x† ˆ

X k

c0;k f0;k …h21 x†

…20†

where the coef®cients c0;k are derived from the values of f in a neighborhood of hk ˆ …hk1 ; hk2 †: In the case of the Bspline of order 2, the coef®cients c0,k are simply equal to the samples of f on the grid hZ2 and the scaling function is

E. Perrey-Debain, H.G. ter Morsche / Engineering Analysis with Boundary Elements 26 (2002) 1±13

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Fig. 2. Uniform grid points for surface interpolation.

an interpolating function. We have directly X f0;h …x† ˆ f …hk†f0;k …h21 x†:

…21†

k

This previous equation is called a one-point quasi-interpolation formula and the interpolation error is O…h2 † if f is C 2 [14]. After applying the forward Fast Wavelet Transform J times, f0;h can be rewritten as f0;h …x† ˆ

X k

21

c2J;k f2J;k …h x† 1

2J X 3 X X jˆ2 1 eˆ1

k

e e dj;k c j;k …h21 x†:

possible, is by simply picking the N coef®cients with largest absolute value. At this stage, it is worth to notice that since wavelets have vanishing integrals, the following equality holds for any J Z V0

f0;h …x†d2 x ˆ

Z V0

N f0;h …x†d2 x;

…25†

i.e. the coef®cients c2J;k keep the average value of the approximation of the original function f on the ®nest grid …J ˆ 0†:

…22† The basic reason why this new representation might be useful is that each wavelet contains information about f0;h essentially at location k and at the scale j. In part of the domain where f0;h has high frequency behaviors or discontinuities, a lot of wavelet coef®cients are needed, and where f0;h is smooth, we can use fewer coef®cients and still get a good quality of approximation. In other words, the FWT allows us to focus on the most relevant parts of the function. Now, let IM be the set of indices corresponding to the M non-zero wavelet coef®cients of f0;h : We want to ®nd a comN pressed version f0;h of f0;h by considering the subset of indices IN , IM where N ˆ #IN ; namely X X e e 21 N f0;h …x† ˆ c2J;k f2J;k …h21 x† 1 dj;k c j;k …h x†: …23† k

IN

Because wavelets are Riesz basis, we have the following inequality 0 11=2 X N e 2A uuL2 # C h@ …dj;k † : …24† uu f0;h 2 f0;h IM \IN

This reveals that the best way to pick N wavelet coef®cients making the upper bound of the L2 error as small as

4.2. Domain extension As the function f ˆ xV 0 f contains arti®cial singularities at the boundary of V 0, a lot of non negligible wavelet coef®cients are likely to increase in a certain neighbourhood of the boundary and the bene®t of the FWT may be drastically reduced or even completely deleted. This effect has certain similarities to the Gibbs phenomenon when using the Fourier technique [15]. To avoid this major drawback, we restrict ourselves to a rectangular domain V 0 ˆ ‰2A; AŠ £ ‰2B; BŠ and consider f ˆ xV 0ext f where V 0ext is the extended domain V 0ext ˆ ‰2…A 1 A 0 †; A 1 A 0 Š £ ‰2…B 1 A 0 †; B 1 A 0 Š: The original domain V must ®t in V 0 as shown in Fig. 3. The parameter A 0 depends on the coarsest scale 2J and the domain mesh size h and is found such that the wavelet representation (22) of f 0;huV 0 (i.e. the restriction of f0;h in V 0) is not changed by the arti®cial singularities at the boundary of V 0ext : We can show that A 0 is simply given by the following recursion formula A 0…J† ˆ 2A 0…J21† 1 2h

and

A 0…0† ˆ h:

…26†

Finally, an approximate particular solution up of Eq. (1)

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E. Perrey-Debain, H.G. ter Morsche / Engineering Analysis with Boundary Elements 26 (2002) 1±13

Fig. 3. Domain extension.

can be found from Dup …x† ˆ

nd f0;h uV 0 …x†

ˆ

X J0

1

21

c2J;k f2J;k …h x† X

I0nd ,I0

e e dj;k c j;k …h21 x†;

…27†

where J 0 and I 0 are the sets of indices J0 ˆ {kusupp f2J;k > V 0 ± B};

nc ˆ #J 0 ;

e I0 ˆ {…e; j; k†usupp c j;k > V 0 ± B}

We note that nc is not an arbitrary parameter and depends on J, h and the domain V 0 whereas nd is to be assigned by the user. The total degree of freedom to represent up is then n c 1 nd : In the previous discussion, the analysis was restricted to a single rectangular domain in order to obtain the simple recursion formula (26). When dealing with a complicated geometry, it suf®ces to consider a rectangular domains decomposition as illustrated in Fig. 4. Each rectangular domain is then extended according to Eq. (26). 4.3. The Newton potential

e and I0nd contains the nd largest udj;k u such that …e; j; k† [ I0 :

The classical method for obtaining a particular solution of

Fig. 4. Rectangular domains decomposition.

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Fig. 5. Integration scheme.

Eq. (27) is to construct the associated Newton potential (see Ref. [3]) given by the convolution integral up …x† ˆ …2p†21

Z R2

nd f0;h uV 0 …y† lnux 2 yud2 y:

…28†

Now let {fp ; c peˆ1¼3 } be associated potentials of {f; c eˆ1¼3 }: The particular solution up and its normal derivative are then given by up …x† ˆ

X J0

1

c2J;k 2J h2

X I0nd

ln…2J †h fp …22J h21 x 2 k† 1 2p

e 2j 2 e j 21 dj;k 2 h c p …2 h x

2 k†;

!

fp …x† ˆ …2p†21

…29†

X 2up 2fp 2J 21 …2 h x 2 k† …x† ˆ c2J;k h 2n 2n J0 1

X I0nd

e dj;k h

2c pe j 21 …2 h x 2 k†: 2n

4.4. Integration scheme The potential f p is given by the convolution integral Z R2

f…y†lnu x 2 yud2 y

…31†

The computation of Eq. (31) can be carried out by many techniques (see Ref. [6]). Among them, Atkinson's formula [3] is an ef®cient method provided that

Z suppf

f…y…r; u††r ln r d rdu;

where supp f ˆ ‰21; 1Š2 denotes the support of the scaling function f . This last integral is regular and is therefore computed via standard Gaussian quadrature formulas. Since the restriction of w to any interval ‰i; i 1 1† is in p 1 (i.e. the collection of algebraic polynomials of degree at most 1), it is judicious for better accuracy to divide the domain integral into 4 domain integrals

…30†

Let us note that the constant term ln…2J †h=2p is not useful here but becomes essential when dealing with domains of in®nite extent such as electrostatic ®elds associated with stationary distributions of charge in free-space or other similar potential problems.

fp …x† ˆ …2p†21

f is a smooth function. In our case, f is only continuous and its derivatives are piecewise continuous. If a high accuracy is required (say up to the machine precision) then an exact integration procedure is needed. This can be done by observing that f and its derivatives are piecewise polynomials. The singularity of the integrand can be removed by using polar coordinates,

fp …x† ˆ …2p†21

1 Z X i;i 0 ˆ0

where Ii ˆ ‰i 2 1; iŠ; obtained from

Ii £ Ii 0

f…y…r; u††r ln r dr du;

i ˆ 0; 1:

The

x1-derivative

…32† is

1 Z X 2f p 2f …x† ˆ …2p†21 …y…r; u††r ln r dr du: …33† 2x1 2y I i £ Ii 0 1 i;i 0 ˆ0

The numerical evaluation of Eqs. (32) and (33) involves integral like Z Ii £ Ii 0

g…y…r; u††r ln r dr du;

where g stands for either f or its derivative. Fig. 5 illustrates

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Fig. 6. Particular solutions (a) f p, (b) …2fp =2x1 †; (c) c p1 ; (d) c p3 :

the discretization of the unit square domain Ii £ Ii 0 according to the position of the evaluation point x with respect to the domain of integration. In each zone (I, II, III, IV) the integration is performed by using 30 points in both r and u directions giving a total of Gaussian integration points of 4 £ 30 £ 30 ˆ 3600: As explained before a high accuracy is expected since the restriction of g to the square domain is a polynomial of maximum order 1. The other potentials c peˆ1¼3 and their derivatives are directly evaluated by using Eq. (7)

c pe …x† ˆ …2p†21 ˆ …4p†21 ˆ

1 2

X k

X k

X k

bek bek

Z R2

Z R2

f…2y 2 k†lnux 2 yu d2 y f…y†{lnux 2 yu 2 ln 2} d2 y

bek fp …2x 2 k†;

where the constant term has been removed by invoking Eq. (10). By using symmetry properties, it can be shown that

fp …x1 ; x2 † ˆ fp …x2 ; x1 †; c p3 …x1 ; x2 † ˆ c p3 …x2 ; x1 †

c p2 …x1 ; x2 † ˆ c p1 …2x2 ; x1 †;

and consequently 2fp 2f p …x1 ; x2 † ˆ …x ; x †; 2x2 2x1 2 1 2c p2 2c p1 …x1 ; x2 † ˆ …2x2 ; x1 †; 2x1 2x2 2c p2 2c p1 …x1 ; x2 † ˆ 2 …2x2 ; x1 †; 2x2 2x1 2c p3 2c p3 …x1 ; x2 † ˆ …x ; x †: 2x2 2x1 2 1 Fig. 6 shows the graph of f p, …2fp =2x1 †; c p1 and c p3 . We note that all functions except f p all tend to zero as uxu goes to in®nity. 5. Storage The evaluation of the previous integrals is of course time consuming and our current method would lose its interest if a numerical integration was required to evaluate the particular solutions. We are now facing the following problem: ®nd a numerical scheme in order to quickly get a good approximation of the potentials and their derivatives. To achieve this, let us ®rst observe that, by using simple

E. Perrey-Debain, H.G. ter Morsche / Engineering Analysis with Boundary Elements 26 (2002) 1±13

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Fig. 7. Domain of storage considering the nine-noded quadratic Lagrangian element.

geometric transformations (translation, symmetry and rotation) and symmetry properties stated in Section 4, it is only required to have access to the set of functions ( ) 1 1 3 1 3 2fp 2c p 2c p 2c p S ˆ fp ; c p ; c p ; ; ; ; 2x1 2x1 2x2 2x1

Substituting this last expression into the convolution integrals (32) and (33) yields 0 21 @

fp …x† ˆ …2p†

lnuxu 2

1 X

1

n X

nˆ1

2nuxu4n

qˆ0

1 A; An;q x12…n2q† x2q 2

2

on the quarter-plane ‰0; 1† : Secondly, for points x far enough from the integration domain (i.e. uxu suf®ciently large), we may use simple asymptotic formulas by expanding the Green function into an in®nite series as shown in the next paragraph. Finally, for points lying in the integration domain or in its close vicinity, we may use a ®nite-element type approximation. In summary, our numerical sheme can be described as follows (see Fig. 7): 1. De®ne a square domain Da ˆ ‰0; aŠ2 outside which asymptotic formulas provide an accurate approximation of the potentials and their derivatives, 2. Use a ®nite-element type approximation to store the functions in Da. 5.1. Asymptotic behavior In our analysis below, we apply arithmetic to vectors x [ R2 by interpreting them as points in the complex plane. Thus, we have that ln x ˆ lnuxu 1 i arg…x†: So ln uxu is the real part of ln x; denoted as ln u xu ˆ R…ln x†: Now, for any uxu . uyu; we have the following expansion "   # 1 X 1 y n lnux 2 yu ˆ Rln…x 2 y† ˆ lnuxu 2 R n x nˆ1

An;q ˆ

n X

2n

q 0 ˆ0

2q

!

2n

!

2q 0 0



…21†q1q : …n 2 q 0 1 1†…2…n 2 q 0 † 1 1†…q 0 1 1†…2q 0 1 1†

and 1 X 2fp 1 …x† ˆ …2p†21 lnux 1 …s; 0†u 2 2x1 2nux 1 …s; 0†u4n nˆ1 ! n X 2…n2q† 2q  …x1 1 s† x2 Bn;q qˆ0

2 …2p†21 lnux 2 …s; 0†u 2

1 X nˆ1



n X qˆ0

! …x1 2

s†2…n2q† x2q 2 Bn;q

;

1 4n

2nux 2 …s; 0†u

10

Bn;q ˆ

E. Perrey-Debain, H.G. ter Morsche / Engineering Analysis with Boundary Elements 26 (2002) 1±13 n X

2n

q 0 ˆ0

2q



2n

!

!

2q 0

0

…21†q1q 0 ; 0 …2…n 2 q † 1 1†…q 0 1 1†…2q 0 1 1†4n2q

where s stands for the shift s ˆ 1=2: In practice, the previous in®nite series are truncated at n ˆ nmax : It is clear that the accuracy improves as nmax increases but this give rise to more computation since the number of terms in the series grows as n2max : Numerical experiments showed that, by using nmax ˆ 5; the truncated series are accurate enough outside of the domain D3=2 ˆ ‰0; 3=2Š2 for our applications (maximum 10 25 of absolute error). The asymptotic series for the potentials c p1 and c p3 have a `slow' convergence and the leading order terms …nmax ˆ 1† …c p1 †nmax ˆ1 …x 1 …0; s†† ˆ 2 …c p3 †nmax ˆ1 …x 1 …s; s†† ˆ

3 32puxu4 27 8

256puxu

…x22 2 x21 †;

…x41 2 6x21 x22 1 x42 †

are suf®ciently accurate outside of the domain D3 ˆ ‰0; 3Š2 : The same holds for the corresponding derivatives obtained by differentiating the previous formulas. 5.2. Storage with ®nite-element approximation The quality of the approximation scheme is of course strongly related to the smoothness of the functions to be stored. Fortunately, functions of S are relatively smooth in their domain of storage (see Ref. [16] for regularity properties). Finding the optimal scheme is not the primary issue of this work. Here, we shall use a simple ®nite-element type approximation on a regular grid. Let gp be a function of S. We consider the partition of Da into a regular distribution of square subdomains. On each subdomain, gp is approximated as g^p …x† ˆ

d X d X iˆ0 jˆ0

gp …xij †lij …x†

where xij is a collection of points evenly distributed over the subdomain, with stepsize hs (see Fig. 7 in the case d ˆ 2). Functions lij are the typical bivariate Lagrangian polynomials of degree d satisfying the interpolation property lij …xi 0 j 0 † ˆ dii 0 djj 0 : Values for gp …xij † are computed from the integration procedure of Section 4.4 and stored in single precision. Among the possibilities for d and hs, the use of the nide-noded quadratic Lagrangian element have been shown to perform well. For all our functions, we used a mesh spacing hs ˆ 0:01 and numerical experiments showed that the approximation error does not exceed 10 25 which is suf®cient for our numerical applications as shown in Section 6. The cost of storage for f p and …2fp =2x1 † in D3=2 is 2…1:5=hs †2 £ 4 bytes. The cost for the 5 other functions in

D3 is 5…3=hs †2 £ 4 bytes. The total cost of storage is approximately 2 Megabytes. Nowadays, such a computational price to pay is affordable considering resources of a simple PC. The storage procedure can be generalized to R3 : The memory space required is then proportional to h23 s and the number of functions to be stored increases because of the derivative in the x3 direction and the number of `mother' wavelets is 23 2 1 ˆ 7 instead of 3 in the 2D case (see Appendix A). This will certainly limitate the potential use on a normal personal computer. The alternative would be to store f p and …2fp =2x1 † only; the other functions can be deduced by symmetry and by using the re®nement Eq. (7). If the same mesh spacing hs is used, the cost of storage in the cube ‰0; 3=2Š3 is approximately 27 Megabytes. This is still affordable on a PC. The price to pay is an increase of the computational time when using Eq. (7). 6. Numerical examples Three problems of increasing complexity are considered. Computations were performed on a SUN WorkStation 300 MHz. The algorithm for the computation of the particular solution was implemented in single precision, the algorithm for the BEM Laplace's solver was implemented in double precision. A simple adaptative quadrature has also been implemented to insure a good numerical evaluation for internal values close to the boundary. Finally, the decreasing rearangement of the wavelet coef®cients is performed by the Heapsort algorithm from Numerical Recipes [17]. Example 1 To test the effectiveness of the method we solve the Poisson's equation on the ellipse V ˆ {…x; y†ux 2 =4 1 y2 # 1} with the boundary conditions uuG ˆ 0: For f, we chose the three values f ˆ 22; 2x and 2x 2. These problems have been solved by Golberg [18,19], Partdridge et al. [20] and Alessandri and Tralli [21]. Numerical results comparing these four different approaches are given in Tables 8.4±8.8 in Ref. [22]. In our calculations, we use 20 quadratic elements for the boundary. Solutions for the two ®rst cases are given in Tables 1±2. These results have been obtained with h ˆ 1; J ˆ 1; …nc ˆ 15† and nd ˆ 0: For convenience, all numbers have been rounded to four decimal places. The excellent accuracy obtained is not surprising since the 2nd order Bspline approximates exactly any constant or linear pro®le. Table 1 Solutions of Du ˆ 22 on V x1

x2

Computed

Exact

1.5 1.2 0.6 0.0 0.9 0.3 0.0

0.00 0.35 0.45 0.45 0.00 0.00 0.00

0.3500 0.4140 0.5660 0.6380 0.6380 0.7820 0.8000

0.3500 0.4140 0.5660 0.6380 0.6380 0.7820 0.8000

E. Perrey-Debain, H.G. ter Morsche / Engineering Analysis with Boundary Elements 26 (2002) 1±13 Table 2 Solutions of Du ˆ 2x1 on V

11

Table 3 Solutions of Du ˆ 2x21 on V

x1

x2

Computed

Exact

x1

x2

Comp. (a)

Comp. (b)

Comp. (c)

Exact

1.5 1.2 0.6 0.0 0.9 0.3 0.0

0.00 0.35 0.45 0.45 0.00 0.00 0.00

0.1875 0.1774 0.1213 0.0000 0.2051 0.0838 0.0000

0.1875 0.1774 0.1213 0.0000 0.2051 0.0838 0.0000

1.5 1.2 0.6 0.0 0.9 0.3 0.0

0.00 0.35 0.45 0.45 0.00 0.00 0.00

0.2547 0.2137 0.1494 0.0963 0.2422 0.1509 0.1289

0.2602 0.2198 0.1432 0.1031 0.2406 0.1517 0.1360

0.2599 0.2201 0.1437 0.1036 0.2402 0.1513 0.1365

0.2598 0.2201 0.1437 0.1037 0.2402 0.1514 0.1366

For the quadratic pro®le, three tests have been carried out: in all cases, h ˆ 0:01 and nd ˆ 0 and we chose (a) J ˆ 7 …nc ˆ 15†; (b) J ˆ 6 …nc ˆ 45†; (c) J ˆ 5 …nc ˆ 135†: Solutions are given in Table 3. Note that the test (a) gives reasonable results with very few coef®cients. This is because these coef®cients, obtained after 7 forward FWT, keep the average value of the function on the ®nest grid f0;h uV 0 : For example, by choosing the parameters as in the two ®rst cases, the solution would be disatrous even if the computational cost is roughly the same. Example 2 We consider the problem Du…x† ˆ 2ex1 2x2 ; u…x† ˆ e

x 1 2x2

x [ V; x1

1 e cosx2 ;

neglected …nd ˆ 0†: In Fig. 8 we show graphs of the absolute value of the error (in decimal logarithmic scale) on the x1-axis. Here again, reasonable results (maximum 5% of relative error) are obtained with very few coef®cients (case (a)). The accuracy obtained with test(d) is almost comparable with Golberg's results (see Table 6 in Ref. [19]) using the Multiquadrics interpolation.) Example 3 In the previous examples, the source terms f are in®nitely smooth and best results have been obtained by only considering the coef®cients for the scaling function at different levels of resolution. To see the bene®t of the wavelet coef®cients we consider the following problem Du…x† ˆ f …r† ˆ 2…1 1 50ur 2 1=2u††21 ;

x [ G;

where V is the same ellipse as introduced in Section 5. The solution in V is simply given by u…x† ˆ e x 1 2x2 1 ex1 cosx2 : This problem has been chosen so we could compare our results with previously published BEM results in Ref. [19]. The boundary is discretized with 40 quadratic elements. To compute the particular solution, we chose h ˆ 0:01 and considered the four following tests: (a) J ˆ 7 …nc ˆ 15†; (b) J ˆ 6 …nc ˆ 45†; (c) J ˆ 5 …nc ˆ 135†; (d) J ˆ 4 …nc ˆ 405†: In all cases, the wavelet coef®cients are

x[V

x [ G;

u…x† ˆ 0; …x 12

where r ˆ 1 x22 †1=2 and V is the unit circle. Graph of the source term f …r† in Fig. 9 shows the kind of singularity we are facing in this example. For this particular problem, the solution u is radial and is given by the following integral u…r† ˆ ln r

Zr 0

f …t†t dt 1

Z1 r

f …t†t ln t dt;

r , 1:

…34†

The calculations were done using 40 quadratic elements

Fig. 8. Absolute errors along the x1-axis for Example 2.

12

E. Perrey-Debain, H.G. ter Morsche / Engineering Analysis with Boundary Elements 26 (2002) 1±13 Table 4 Solutions ( £ 10 22) for Example 3(c)

Fig. 9. Source term f …r† for Example 3.

for the boundary. Internal values are computed at approximately 2000 nodes. Three tests have been carried out by ®xing the parameters h ˆ 0:01 and J ˆ 6 …n c ˆ 25† and by choosing respectively (a) nd ˆ 10; (b) nd ˆ 100; (c) nd ˆ 1000: These different values for nd are estimated after examination of the decreasing rearangement of the wavelet coef®cients. A plot of u from case(c) is provided in Fig. 10. The solution is radially symmetric, as expected. In Table 4, we show computed values for u at selected points. Results in the last column are obtained from Eq. (34) by using MAPLE

x1

x2

Comp. (a)

Comp. (b)

Comp. (c)

Maple V

0.00 0.76 0.28 0.32 0.52 0.96 ±

0.04 20.44 0.00 20.68 0.48 0.24 CPU

3.670 0.762 3.403 1.577 1.775 0.066 10.4 s

4.007 0.825 3.901 1.714 2.045 0.067 11.4 s

3.990 0.814 3.889 1.703 2.020 0.068 19.9 s

3.989 0.814 3.889 1.703 2.020 0.068 ±

V. These results clearly shows the good level of accuracy obtained (up to 4 signi®cant ®gures for the last case) when dealing with singular functions. To our knowledge no such attempts have been tried when using the DRM with RBFs. The ef®ciency of the method in terms of total CPU time is illustrated in the last row. It should be mentioned that this time is mostly consumed for the evaluation of the internal values and the CPU time required for performing 6 forward FWT with about 50,000 original samples of f is less than 1 s. 7. Conclusion In this paper, we have shown how to replace the traditional Radial Basis Functions by B-spline wavelets [13] to solve 2D Poisson's equation. As there is no analytical form for the associated particular solutions, these last are stored in a square domain by using ®nite-element approximation.

Fig. 10. Solution for Example 3(c).

E. Perrey-Debain, H.G. ter Morsche / Engineering Analysis with Boundary Elements 26 (2002) 1±13

Asymptotic expressions are then available outside of the domain of storage. The main advantages of such a construction are (i) the computational cost to interpolate the source term is negligible even for very high degree of freedom, this is not the case when using the RBFs since the interpolation matrix has to be inverted, (ii) the Fast Wavelet Transform permits to obtain high level of compression, especially for smooth functions, and a good accuracy (4 digits) can be obtained even for singular functions, (iii) the global cost, in terms of CPU time, is relatively cheap and does not penalize our current approach in comparison with standard DRM. Moreover, as evoked in Section 4.3, our technique could be easily extended for free-space problem. Although we have only considered linear approximation in the previous applications, the use of higher order might be fruitful especially when dealing with large scale problems and smooth functions. Finally, the storage scheme adopted in this work has been chosen for its simplicity. In order to reduce the approximation errors for the potentials and their derivatives, an optimization of this scheme should be carried out and this could be the subject of further investigation. Appendix A In this appendix, we present some useful de®nitions concerning wavelets basis. For the sake of generality, the concepts are presented in Rn : As wavelets are usually obtained from the univariate case via tensor product as in Section 3.4. It is straightforward to see that the number N of `mother' wavelets {c e ue ˆ 1¼N} is simply given by N ˆ 2n 2 1: Riesz basis Let H be the Hilbert space L2 …Rn † with the usual inner product k´; ´l and associated norm uu´uu ˆ k´; ´l1=2 : Let f be a function of H. The collection {fk ˆ f…´ 2 k†uk [ Zn } is a Riesz basis of the subspace V , H if the linear span of f k is dense in V and strictly positive constants A and B exist such that

2

X X X

2 A uck u # ck fk # B uck u2 :

n n n k[Z

k[Z

k[Z

Biorthogonality and dual pairs Let a dual scaling function f~ and dual wavelets c~ e generate a dual multiresolution of H. The scaling functions f , f~ and wavelets e ~ e0 c , c are said to form a dual pair if kf; f~ …´ 2 k†l ˆ d0;k ;

0

kc e ; c~ e …´ 2 k†l ˆ d0;k de;e 0

in which d denotes the Kronecker symbol.

13

Moreover, they satisfy the biorthogonality conditions if kf; c~ e …´ 2 k†l ˆ kf~ ; c e …´ 2 k†l ˆ 0: References [1] Cummins PF, Vallis GK. Solvers for self-adjoint elliptic problems in irregular two-dimensional domains. ACM Trans Math Soft 1994; 20(3):247±61. [2] Chen CS. The method of fundamental solutions for non-linear thermal explosions. Commun Num Meth Engng 1995;11:675±81. [3] Atkinson KE. The numerical evaluation of particular solutions for Poisson's equation. IMA J Num Anal 1985;5:319±38. [4] Nardini D, Brebbia CA. A new approach to free vibration analysis using boundary elements. Southampton: Computational Mechanics Publications, 1982. [5] Golberg MA, Chen CS. The theory of radial basis functions applied to the BEM for inhomogeneous partial differential equations. Bound Elements Commun 1994;5:57±61. [6] Golberg MA. The numerical evaluation of particular solutions in the BEM Ð a review. Bound Elements Commun 1995;6:99±106. [7] Golberg MA, Chen CS, Bowman H. Some recent results and proposals for the use of radial basis functions in the BEM. Engng Anal Bound Elements 1999;23:285±96. [8] Chen CS, Brebbia CA, Power H. Dual reciprocity method using compactly supported radial basis functions. Commun Num Meth Engng 1999;15:137±50. [9] Brebbia CA, Dominguez J. Boundary elements Ð an introductory course. Southampton: Computational Mechanics Publications, 1989. [10] Chui CK. An introduction to wavelets. New York: Academic Press, 1992. [11] Dahmen W. Wavelet and multiscale methods for operator equation. Acta Num 1997;6:55±228. [12] Jawerth B, Sweldens W. An overview of wavelet based multiresolution analyses. SIAM Rev 1994;36(3):377±412. [13] Cohen A, Daubechies I, Feauveau J. Bi-orthogonal bases of compactly supported wavelets. Comm Pure Appl Math 1992;45:485±560. [14] Chui CK. A characterization of multivariate quasi-interpolation formulas and its applications. Numerische Mathematik 1990; 57:105±21. [15] Kassab AJ, Nordlund RS. Ef®cient implementation of the Fourier dual reciprocity boundary element method using two-dimensional fast Fourier transforms. Engng Anal Bound Elements 1993;12:93±102. [16] Doob JL. Classical potential theory and its probalistic counterpart. New York: Springer, 1984. [17] Press WH, Teukolsky AA, Vetterling WT, Flannery BP. Numerical recipes in fortran. Cambridge: Cambridge University Press, 1992. [18] Golberg MA. The method of fundamental solutions for Poisson's equation. Engng Anal Bound Elements 1995;16:205±13. [19] Golberg MA, Chen CS, Karur SR. Improved multiquadric for approximations for partial differential equations. Engng Anal Bound Elements 1996;18:9±17. [20] Partridge PW, Brebbia CA, Wrobel LC. The dual boundary element method. Southampton: Computational Mechanics Publications, 1992. [21] Alessandri CA, Tralli A. A spline based-approach for avoiding domain integrations in BEM. Comp Struct 1991;41:859±68. [22] Golberg MA, Chen CS. Discrete projection methods for integral equations. Southampton: Computational Mechanics Publications, 1997.