fourier transform in bounded domains

Meccanica 32: 197–204, 1997.c 1997 Kluwer Academic Publishers. Printed in the Netherlands.

Fourier Transform in Bounded Domains

F. M. E. DUDDECKTechnical University of Munich, Lehrstuhl fur Baumechanik, Arcisstr. 21; D-80333 Munich, Germany

(Received: 17 January 1997)

Abstract. The functional analysis, the concept of distributions u 2 D0 in the sense of Schwartz [7] and theirextension given by Gel’fand and Shilov [5] to ultradistributions u 2 Z 0, enables us to find by the means of theFourier transform a second ‘language’ to characterize physical behaviour. Almost any expression with physicalmeaning can be transformed, even if it is formulated in domains with complicated boundaries and even if it is notintegrable.

Numerical procedures in the transformed space can be developed in analogy to those well known in engineeringmechanics like the methods of Finite or Boundary Elements (FEM or BEM). Basis of the approaches presented hereis the analytical representation of characteristic distribution of a domain and the theorem of Parseval which statesthe invariance of energy in respect to the transformation. In addition, the concept of the characteristic distributionleads to a very simple derivation of the Green-Gauss formulas fundamental for the Boundary or Finite Elements(e.g. [6]).

Sommario. L’analisi funzionale, il concetto di distribuzione u 2 D0 nel senso di Schwartz [7] e la sua estensionealle ultradistribuzioni, u 2 Z 0, introdotte da Gel’fand e Shilov [5], permettono di caratterizzare, per mezzo dellatrasformata di Fourier, un secondo ‘linguaggio’ utile alla caratterizzazione del comportamento fisico.

Quasi tutte le espressioni con significato fisico possono essere ‘trasformate’, anche se sono formulate in dominidotati di frontiere complicate o anche se non sono integrabili. Procedure numeriche nello spazio ‘trasformato’possono essere sviluppate in analogia a quelle ben note nella meccanica applicata quali i metodi degli elementifiniti o degli elementi di contorno (FEM o BEM). Alla base degli approcci presentati nel contesto qui propostovi sono la rappresentazione analitica della distribuzione caratteristica di un dominio e il teorema di Parseval chestabilisce l’invarianza dell’energia rispetto alla trasformazione. Il concetto di distribuzione caratteristica consenteinoltre una derivazione molto semplice delle formule di Green-Gauss fondamenatali per gli elementi finiti o dicontorno (si veda per esempio [6]).

Key words: Functional analysis, Fourier transform, Finite elements, Transform methods, Solid mechanics

1. Introduction

On the one hand the applications of integral transform methods were in general restricted tovery regular geometries (e.g. full or half spaces). More complicated regions with irregularboundaries were only solved numerically in the untransformed domain. On the other handsome models in this original domain demand a high numeric effort which may be reduced byapplying integral transforms.

Therefore, one may wish to look for a possibility to combine the advantages of these twocomplementary modellings. The approach presented here may enable solutions of boundaryor initial value problems in solid mechanics in a similar way to Finite or Boundary Elements intransformed variables even for very irregular boundaries. Stiffness matrices can be evaluatedin the transformed domain.

As long as the shape functions consist of polynomials, trigonometric or exponential func-tions the corresponding integrations in the transformed domain can be interpreted as colloca-tions. Hence their evaluations demand very little numeric effort.

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198 F. M. E. Duddeck

Figure 1. Characteristic distribution of a circular domain �� (r0 = 1) (left) and sketch of its derivative �(1�x21�

x22)(�2x1) in respect to x1 (right).

2. Characteristic Distribution

Functional analysis enables the analytic treatment of singularities and jumps. Therefore,bounded domains can be easily described by the characteristic distribution � of the domain � R

n

� :=�

1 : : : x 2 0 : : : x 2 { = R

n � :(1)

If the boundaries are sufficiently smooth (Cm-regular domain in the sense of Adams [1]), �is equivalent to the n-dimensional Heaviside distribution H( ); (x) 2 C1(). A function ordistribution �(x) defined in the bounded domain can now be expressed by the scalar product1

h�; �i = hH( ); �i =ZRn

H( )� dx =Z >0

� dx; x 2 Rn: (2)

The hypersurface (x) = 0 describes the boundary of the domain � Rn with the gradient

of as inner normal � which may be defined uniquely everywhere on the boundary except atsingle countable points (r 6= 0).

The left part of Figure 1 shows as an example a circular domain �� R2 with radius

r0 = 1. The characteristic distribution is �� = H( ) = H(r20 � x2

1 � x22). The support of

the derivative Dj in the distributional sense of � is limited to the boundary @ and can bedescribed by a Dirac distribution �( ). The chain-rule leads to

Dj� = DjH( ) = �( )Dj = �j d�: (3)

�j is the component of the inner normal in xj-direction, d� is the Euklidian measure. Thefunctional represents an integration along the boundary

hDj�; �i = hDjH( ); �i = h�( )Dj ; �i =

Z =0

�j� d�: (4)

The gradient r� is analogously hr�; �i = h�( )r ; �i with � = �1; : : : ; �n; �j 2C1(). The right part in Figure 1 gives a rough sketch of the derivative of the circular domainalthough the singular character of the Dirac distributions cannot be visualized properly.

1 See e.g. Constantinescu [3], Hormander [6] and Gel’fand and Schilov [5].

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Fourier Transform in Bounded Domains 199

3. Formulas of Green–Gauss

The Green or Gauss formulas can be formulated in the Fourier transformed space. A physicalproblem, e.g. the evaluation of a displacement vector field u, in a bounded domain demandsa restriction of u and its derivatives to . That can be achieved by a multiplication with thecharacteristic distribution or, in the Fourier transformed space, by an-dimensional convolution(� ; x 2 R

n ), see the convolution theorem of the fourier transform, e.g. [2],

u! �uF

$ u!1

(2�)n� � u =

1(2�)n

ZRn

�(�)u(x� �) d�: (5)

The complementarity of the two descriptions, one in the original, the other in then-dimensional

spatial Fourier-transformed space, is expressed by the link F

$. The application of a lineardifferential operator P(D) to a distribution h�u; ��i with �� as the complex conjugate of a testfunction � leads to

P(D)f�ug; ��=�u;P�(D)��

�(6)

F

$1

(2�)2n

DP(x)f� � ug;

��

E=

1(2�)2n

D� � u; P�(x)

��

E;

where P�(D) is the adjoint operator.The four scalar products have the same values which can be found by partial integration and

the theorem of Parseval (cf. [4]). The application of the generalized Leibniz formula for thederivation of products leads, according to Gel’fand & Shilov [5], Volume 1, to Green–Gaussformulas or respectively to the principle of reciprocity known in the method of BoundaryElements. For example, Green’s second identity with the Laplace-operator� in n-dimensionsis

�fH( )ug; ��=H( )u;��

�=H( )�u; ��

�

+nXk=1

1Xj=0

DD

1�jk u

�� =0

�( )Dk ; (�Dk)j �� =0

+: (7)

This can be generalized to arbitrary operators2. The conventional form of this identity isZ

(u1�u2 � u2�u1) dx = �

Z

(�( )r )T (u1ru2 � u2ru1) dx

= �

Z@

�u1@u2

@�� u2

@u1

@�

�d�: (8)

4. Approximation of the Domain

The process of approximation has three steps, the approximation of the characteristic distrib-ution by a triangulation, the choice of shape functions for the unknown displacement vectorfield u and finally the conception of a sufficient number of test functions �k. This paper doesnot discuss solution routines as the inversion of a stiffness matrix.

2 The formal treatment of the domain by the characteristic function and its derivatives leads to a simpleevaluation of the C-matrix for the BEM, [4]. The points where the boundary is not smooth lead analogously tospecial corner integrals and not only boundary integrals.

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Figure 2. Characteristic distribution of the basic trianglein the original space (left) and in the transformed space(right).

Figure 3. Example for an assembled characteristic function.

The only part which requires a treatment in the original space is the definition of a basicnormed triangle element (left part in Figure 2)3

�0 := H(x1)H(x2)H(1 � x1 � x2) (9)

F

$ �0 =1

x2 � x1

�1x2(1� e�ix2)�

1x1(1� e�ix1)

�:

Arbitrary triangles are constructed by affin transformations Da and translations Tb of thebasic element in both spaces (b 2 R

n ; a 2 Rn � R

n )

Da : �0(x)! �j1(ax)F

$ Da : �0(x)! jdet aj�1�j1(a

�Tx)

Tb : �0(x)! �j1(x� b)F

$ Tb : �0(x)! �j1(x) e�ib�x: (10)

The assemblage of several triangles can be done in the transformed space. The n-dimensional domain with piecewise linear boundaries � R

n , fulfilling the Lipschitz condi-tion, can be represented by the unification of m particular domains j . In analogy to this the

3 All mathematical expressions are evaluated with the aid of the formel manipulator MAPLE V/3 [8].

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Figure 4. Basic threedimensional normed element.

global characteristic function � can be represented by the sum of elementary characteristicfunctions �j ; j = 1 : : : m (see the example in Figure 3)

=m[j=1

j ! � =mXj=1

�jF

$ � =mXj=1

�j : (11)

The same procedure can be applied for three dimensional problems or for boundaryelements (cf. [4]). The basic three dimensional element, a normed tetrahedron (Figure 4), canalso be expressed by Heaviside distributions

�0 := H(x1)H(x2)H(x3)H(1 � x1 � x2 � x3) (12)

F

$ �0 =i(x1x2 + x1x3 � x2

1)(e�ix2 + e�ix3)

x1x2x3(x1 � x2)(x1 � x3)

�i(e�ix1 + ei(x1�x2�x3))

x1(x1 � x2)(x1 � x3)+i(1 + e�i(x2+x3))

x1x2x3:

The explicit transformation with the aid of MAPLE V/3 [8] may be done by the alternativeform of �0:

�0 = (H(x1)� H(x1 + x2 + x3 � 1))

�(H(x2)� H(x2 � 1))(H(x3)� H(x3 � 1)):

5. Approximation of the Unknown Displacement Vector

The solution u belonging to the infinite dimensional space U � D0() is approximated by aprojection of U on a finite dimensional subspace Uh � U . The latter space includes the Nlinear independent shape functions uj(x) 2 Uh; j = 1 : : : N , normally polynoms or (complexor real) exponential functions4. Both can be transformed easily into Dirac distributions or theirderivatives �(�), e.g. for the R1

xk F

$ 2�ik�(k)(x); k = 0; 1; 2; : : :

ebx F

$ 2��(x+ ib); b 2 C : (13)

4 The shape functions may be chosen globally for the whole domain or locally for one or a group of elements.

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Figure 5. Linear shape functions �i for the triangular membrane element.

These Dirac distributions simplify all integrations or convolutions. Avoiding all numericaltreatment the integration can be interpreted as a collocation procedure in the transformedspace and the convolution as a derivation or translation, e.g. in the R1

Z +1

�1

f(x)�(�)(x� b) dx = (�1)�D�f(b): (14)

f(x) � �(�)(x) =

Z +1

�1

f(�)�(�)(x� � � b) d� = (�1)�D�f(x� b):

Approximating u by u �Paj uj(x) the convolution in (6) � � u results in a derivation or

translation of the characteristic distribution �. It remains the evaluation of the scalar productwith the chosen test functions �k. As long as polynomial or exponential functions are chosenthese test functions lead to a second collocation with once more negligible numeric effort.

6. Choice of the Test Functions

The solution of the problem, i.e. the determination of theN unknown coefficients aj , demandsN test functions. They may be chosen similar to the uj(x); the isoparametric choice is inmost cases the most convenient one. The polynomials as well as the exponential functionslead again in the transformed space to Dirac distributions. Therefore, the integration of thescalar product is converted into a collocation. Both procedures, the convolution of the shapefunctions uj and the integration with the test functions �k, demand almost no computationalefforts even for elements with a complicated shape.

The accuracy of the chosen approximation depends strongly on the number of shape and testfunctions as well as on the quality of these functions, e.g. the degree of the polynomials. Thisis already discussed in the literature. The goal of this paper is to show that the correspondingmethods, like Galerkin, Ritz, Trefftz etc., in the literature for the original space, can betransferred to the Fourier transformed space.

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7. Example

The vertical displacementw of a membrane with the constant tensionN due to vertical loadsis described by the differential operator �N�

�N�fw�g; ��

�F

$N

(2�)4

Djxj2(w � �);

��

E: (15)

The membrane is triangulated and the unknown local displacement w of the element isapproximated by linear shape functions

Pwj �j(x1; x2) (Figure 5)

�1 = x1F

$ �1 = 4�2i�0(x1)�(x2)

�2 = x2F

$ �2 = 4�2i�0(x2)�(x1)

�3 = 1� x1 � x2F

$ �3 = 4�2(�(x1)�(x2)� i�0(x1)�(x2)

� i�0(x2)�(x1)):

The isoparametric shape functions �k = �k lead to a scalar product which is zero for eachj; k combination

�N h�fwj�j�g; �ki = 0 (16)

F

$N

4�2

D(x2

1 + x22)(wj �j � �); �k

E= 0;

because the �j are homogeneous solutions of the differential operator �N�. Separated intosingular terms at the boundaries and regular inner terms one obtains the stiffness matrix of theelement well known in the theory of Finite Elements. For the inner part these values are

K(i) =N

2

2664

1 0 �1

0 1 �1

�1 �1 2

3775 : (17)

The two parts correspond to those obtained by the application of the Leibniz rule for thederivation of products, e.g. (7). The singular external part leads to the complementary matrixK(i) = �K(e). The shape functions are homogeneous solutions. The total scalar product< �f�wg; �� > is zero. The approach is a particular form of the method given by Trefftz.It remains the evaluation of the other scalar products in (7), the boundary integrals and thedomain integral < ��w; �� >=< f; �� > with the volume forces f . They can be handled in asimilar manner, cf. [4].

8. Conclusions

Introducing the mathematical tools of the functional analysis the field of applications for theFourier transform is enormously expanded. Every – in a physical sense – meaningful expres-sion can be transformed. This is shown here for some of the standard numerical procedures(Finite Elements) of engineering mechanics. It may easily be transferred to boundary typeelements. The application of the Fourier transform is extended to bounded domains. This maybe advantageous. For further investigations the following hypotheses may give some ideasabout the application and relevance of the presented approach:

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� The shape functions chosen in the normal approximation methods, like polynomialsor exponential functions, are transformed into Dirac distributions and their derivatives,facilitating the evaluation of all integrations. The common integrals can be interpretedas collocations. More complex problems as well as curvilinear or non linear ones can betreated similarily, although this demands numerical integrations like in the untransformedspace. The question of eventual better convergence may be interesting to discuss.

� The boundary element method can be applied even if the fundamental solution is onlyknown or treatable in the transformed space. Some or all integrals may be easier solvedin the transformed space. The fundamental solution is in general very easy to obtainin the transformed domain. The common difficulties occur in the process of inversetransformations which is – as this paper has tried to show – not necessary any more.

� Some optimization problems may refer to hybrid methods, global optimization in theoriginal and local optimization in the transformed space.

References

1. Adams, R.A., Sobolev Spaces, Academic Press, New York, San Francisco, London, 1975.2. Bracewell, R.N., The Fourier Transform and its Applications, McGraw-Hill, New York, 1986.3. Constantinescu, F., Distributionen und ihre Anwendung in der Physik, Teubner, Stuttgart, 1974.4. Duddeck, F.M.E., Funktional-Analysis der Kontinuumsmechanik - Fouriertransformation bezuglich Raum

und Zeit der Energiemethoden, PhD-thesis, Technical University Munich (to be published 1997).5. Gel’fand, I. M. and Shilov, G. E., Verallgemeinerte Funktionen (Distributionen), Volume I-V, Deutscher

Verlag der Wissenschaften, Berlin, 1960–1964.6. Hormander, L., The Analysis of Linear Partial Differential Operators, Volume I-IV, Springer, Berlin, Heidel-

berg, 1983.7. Schwartz, L., Theorie des Distributions, Volume I-II, Hermann & Cie, Paris, 1950/51.8. Waterloo Maple Software, University of Waterloo, Maple V Release 3, 1994.

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fourier transform in bounded domains

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