Multiwavelets and multiwavelet packets of Legendre functions in the direct method for solving variational problems

ACME-219; No. of Pages 10 archives of civil and mechanical engineering xxx (2014) xxx–xxx

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Original Research Article

Multiwavelets and multiwavelet packets of Legendre functions in the direct method for solving variational problems K. Jarczewska *, W. Glabisz, W. Zielichowski-Haber Institute of Civil Engineering, Wrocław University of Technology, Wyb. Wyspiańskiego 27, Wrocław 53-370, Poland

article info

abstract

Article history:

A numerical technique for solving the linear problems of the calculus of variations is

Received 6 February 2014

presented in this paper. Multiwavelets and multiwavelet packets of Legendre functions

Accepted 19 April 2014

are used as basis functions in the Ritz method of formulation. An operational matrix of

Available online xxx

integration of multiwavelets and multiwavelet packets is introduced and is used to reduce

Keywords:

algorithm is applied to the analysis of mechanic problems which are formulated as func-

the calculus of variation problem to the solution of the system of algebraic equations. The Legendre multiwavelet packets

tionals. Two examples are considered in this paper. The first example concerns the stability

Wavelets

problem of a Euler–Bernoulli beam and the second one presents the calculation of the

Variation problems

extreme value of the functional which defines the potential energy of an elastic string. The

Operational matrix of integration

presented method yields the approximate solutions which are convergent to accurate results. # 2014 Published by Elsevier Urban & Partner Sp. z o.o. on behalf of Politechnika Wrocławska.

1.

Introduction

Variational principles are widely used in various fields of science: physics, mathematics and electrical engineering as well as many others [10,13,20]. The basic principles of variational calculus in the mechanics of systems and structures in which the volume and surface forces are in a state of equilibrium include, among others Lagrange's variational principle and the Castigliano's principle [13,20]. These principles are based on the theorem of minimum potential energy. A stationary condition of the functional which

describes the potential energy of the system expressed in the form of displacements (Lagrange's principle) or stresses (Castigliano's principle) which are represented by independent variables further leads to an admissible state of displacement and stress of the analyzed system [13,20]. Direct methods for solving the variational problems such as Ritz's, Kantorowicz's, Bubonow-Galerkin's, Treffz's and any other, generally reduce the issue to a system of algebraic equations for which the solution is given by the coefficients of assumed approximation [10,13]. By assuming the approximation functions which are kinematically admissible, the condition for the functional stationarity written in the form

* Corresponding author. Tel.: +48 71 320 48 56. E-mail address: [email protected] (K. Jarczewska). http://dx.doi.org/10.1016/j.acme.2014.04.008 1644-9665/# 2014 Published by Elsevier Urban & Partner Sp. z o.o. on behalf of Politechnika Wrocławska.

Please cite this article in press as: K. Jarczewska et al., Multiwavelets and multiwavelet packets of Legendre functions in the direct method for solving variational problems, Archives of Civil and Mechanical Engineering (2014), http://dx.doi.org/10.1016/j.acme.2014.04.008

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of potential energy or virtual work of the system, enables the approximate solution of a given problem to be obtained [12]. The advantage of variational formulation is that the integral variational principles define the problem globally. Generally, in the functional form all equations are set describing the phenomenon together with the boundary conditions that did not exist at the local (differential) formulations. Operational matrices of integration were first used in the direct methods for solving variational problems by Chen and Hsiao [4] by the taking of Walsh functions as the basis functions. Initial works that have given rise to the development of methods in which the integral operators are represented in the basis of orthogonal functions and have shown the application of the above method in different problems of science [8,5]. A solution to the heat flow problem in the variational formulation using operational matrices of Haar functions has been presented by Hsiao [15]. Glabisz [11,12] has developed a procedure that allows the solution of the variational problem to be obtained with operational matrix of integration of Walsh-wavelet packet functions and by introducing a dual approximation field, has also shown the solution of an extreme functional by assuming an additional internal condition. Razzaghi and Yousefi [23] presented the solution of the convection problem and as an example of optimal control solved the issue of brachistochrone [22]. In both cases, the authors of the work [24,22] used the operational matrices of integration of Legendre multiwavelet functions, for which the algorithm of creation has been presented [23]. Legendre multiwavelet functions have been used to find the extreme values of the functional by Khellat and Yousefi [19]. Sadek et al. [26] successfully used Legendre multiwavelets in the problem of optimal control of transverse vibration of a Euler Bernoulli beams. So far, multiwavelet bases have been rarely used in the analysis of mathematical models describing the problems of structural mechanics, and multiwavelet packets in such issues have not been used at all. In the article an algorithm for the solution of exemplary variational problems based on Ritz's direct method using the operational matrix of integration of Legendre multiwavelets and multiwavelet packets is presented. Two examples are analyzed. The first example concerns the stability of the equilibrium elastic simple beam and the second concerns the calculation of the deflection line of string which has a finite length. The influence of the form of the operational matrix of integration on the accuracy of the results is analyzed. The purpose of this article is to show that Legendre multiwavelets and Legendre multiwavelet packets taken as approximation functions can be an alternative to other basis functions such as Chebyshew functions [14], Laguerre [17], Jacobi series [21], and Haar functions [15,16] which have usually been used to obtain the solutions of such variational issues. In Section 2 basic information on classical and packet multiwavelet analysis is provided. The definition of the integral operational matrices of multiwavelet functions and multiwavelet packets based on Legendre functions is presented in Section 3. The formulation of the problem under consideration is given in Section 4. The application of an algorithm for determining the solutions of the discussed issues is presented in Section 5. Section 6 is a summary of the obtained results.

2. Multiwavelets and multiwavelet packets of Legendre functions In the multiwavelet analysis the set k2j of scaling functions ’njl ðxÞ define the space Vkj and the set of k2j wavelet functions cnjl ðxÞ define space Wkj [1–3,6,18,27]. Basis functions ’njl ðxÞ and cnjl ðxÞ are obtained by dilation (2j/2) and translation (l) of each from k functions which generate the set of fundamental multiscaling functions ’n ðxÞ ¼ f’0 ðxÞ; ’1 ðxÞ; . . . ; ’k1 ðxÞg from the space V0k and fundamental multiwavelets functions cn ðxÞ ¼ fc0 ðxÞ; c1 ðxÞ; . . . ; ck1 ðxÞg from the space W0k satisfying the following relations ’njl ðxÞ ¼ 2 j=2 ’n ð2 j x  lÞ;

x 2 ½2 j l; 2 j ðl þ 1Þns 

(1)

cnjl ðxÞ ¼ 2 j=2 cn ð2 j x  lÞ;

x 2 ½2 j l; 2 j ðl þ 1Þn f 

(2)

where ns is the compact support length of fundamental multiscaling functions, nf is the compact support length of fundamental multiwavelet functions, n ¼ 0; . . . ; k  1; l ¼ 0; . . . ; ð2 j  1Þns; f ; j denotes the level of approximation and parameter k determines the number of fundamental multiwavelet function sets as well as the number of fundamental multiscaling function sets. Number k does not depend on the assumed level j. set of multiwavelet functions cn ðxÞ ¼ The 0 1 k1 fc ðxÞ; c ðxÞ; . . . ; c ðxÞg is generated on the basis of multiwavelet scaling functions ’n ðxÞ ¼ f’0 ðxÞ; ’1 ðxÞ; . . . ; ’k1 ðxÞg [1,2,9,27]. The equations which define the functions cn ðxÞ and ’n ðxÞ take the following forms cn ðxÞ ¼

k1 pffiffiffiX ð0Þ ð1Þ 2 ðgn; j ’ j ð2xÞ þ gn; j ’ j ð2x  1ÞÞ;

n ¼ 0; . . . ; k  1

(3)

n ¼ 0; . . . ; k  1

(4)

j¼0

’n ðxÞ ¼

k1 pffiffiffiX ð0Þ ð1Þ 2 ðhn; j ’ j ð2xÞ þ hn; j ’ j ð2x  1ÞÞ; j¼0

Multiwavelet functions cn ðxÞ and scaling functions ’n ðxÞ defined by the relations (3) and (4) are presented as linear combinations of the same functions, but their differences determine ð0Þ ð1Þ respectively low and high-pass filter coefficients fhn; j ; hn; j ; ð0Þ ð1Þ gn; j ; gn; j g. The function space Wkj created by multiwavelets cnjl ðxÞ is orthogonal complementation of the function space Vkj to the space of the upper level of approximation V kjþ1 and can be defined as V kjþ1 ¼ Vkj Wkj ¼ Vkj1 Wkj1 Wkj ¼ Vkjm Wkjm  . . . Wkj1 Wkj

(5)

According to expression (5), any square integrable function from the j level of approximation, when the multiscaling functions are taken from the m level of approximation, can be expressed in multiwavelet bases [9] in the following formula

j

fm ðxÞ ¼

m 2X 1X k1

v j1 X 2X 1X k1

l¼0 n¼0

l¼0 v¼0 n¼0

snm;l ’nm;l ðxÞ þ

dnv;l cnv;l ðxÞ

(6)

Please cite this article in press as: K. Jarczewska et al., Multiwavelets and multiwavelet packets of Legendre functions in the direct method for solving variational problems, Archives of Civil and Mechanical Engineering (2014), http://dx.doi.org/10.1016/j.acme.2014.04.008

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n o n o ð0Þ ð1Þ ð0Þ ð1Þ Table 1 – Low-pass filter coefficients hn; j ; hn; j and high-pass filter coefficients gn; j ; gn; j of Legendre multiwavelet functions for k = 5. ð0Þ

2

0:707106 6 0:612372 6 6 0 6 4 0:2338535 0

0 0:353553 0:684653 0:405046 0:153093

2

0:14664 6 0 6 6 0:12201 6 4 0 0:18417

0:254000 0:070243 0:211331 0:053649 0:318998

hn; j 0 0 0:176776 0:522912 0:592927 ð0Þ gn; j 0:163956 0:272055 0:208908 0:207784 0:395016

ð1Þ

0 0 0 0:088388 0:350780 0:581987 0:429197 0:055339 0:439024 0:387834

hn; j 0:707106 0 0 0 6 0:6123724 0:353553 0 0 6 6 0 0:684653 0:176776 0 6 4 0:2338535 0:405046 0:522912 0:088388 0 0:153093 0:592927 0:3507803 ð1Þ gn; j 2 0:14664 0:254000 0:163956 0:581987 6 0 0:070243 0:272055 0:429197 6 6 0:12201 0:211331 0:208908 0:055339 6 4 0:053649 0:207784 0:439024 0 0:18417 0:318998 0:395016 0:387834

3

0 7 0 7 7 0 7 5 0 0:044194 3 0:219970 0:486664 7 7 0:627492 7 7 0:511081 5 0:240553

where coefficients of approximation are given below Z snm;l ¼ Z ¼

2m ðlþ1Þ 2m l 2m ðlþ1Þ 2m l

wn2iþ1; j ðxÞ ¼ 21=2 j

j

dnv;l

j fm ðxÞcnm;l ðxÞdx

1=2 j

(7)

X k

j

Gk wi ð2 x  kÞ

ð0Þ

hn; j ¼

0

ð1Þ

’n ðxÞ ¼ f’0 ðxÞ; ’1 ðxÞ; . . . ; ’k1 ðxÞg

(10)

wn1;0 ¼ cn ðxÞ;

cn ðxÞ ¼ fc0 ðxÞ; c1 ðxÞ; . . . ; ck1 ðxÞg

(11)

ð0Þ

gn; j ¼

pffiffiffi Z 12 n 2 c ðxÞ’ j ð2xÞdx

(15)

0

ð0Þ

ðh0n; j wnj ð2 j xÞ þ h1n; j wni ð2 j x  1ÞÞ

ð1Þ

ð0Þ

gn; j ¼ ð1Þnþ jþk gn; j

(16)

The exact description for obtained Legendre multiwavelet functions is given in [1–3]. For example in Table 1, values of low and high pass filter coefficients for Legendre multiwavelet functions for k = 5 are shown. In Fig. 2 fourth components of Legendre multiwavelet packet functions for k = 5 are shown which form the basis of the functions from subspace V k¼5 j¼4 according to the diagram in Fig. 1.

3. Operational matrix of integration for Legendre multiwavelet functions

When using formulas (3) and (4), expressions (8) and (9) take the form

j

(13)

(9)

wn0;0 ¼ ’n ðxÞ;

n¼0

pffiffiffi Z 12 n 2 ’ ðxÞ’ j ð2xÞdx;

hn; j ¼ ð1Þnþ j hn; j ;

k X X

ðg0n; j wni ð2 j xÞ þ g1n; j wni ð2 j x  1ÞÞ

Filter coefficients which are necessary to determine the fundamental multiwavelets and fundamental multiscaling functions are calculated from the formula

Functions wn0;0 and wn1;0 are the initial points of recursive expressions (8), (9) and enable successive functions wni; j ðxÞ to ð0Þ ð1Þ be determined. Coefficients of matrices Hk ¼ fhn; j ; hn; j g and ð0Þ ð1Þ Gk ¼ fgn; j ; gn; j g are treated as low and high pass filters respectively and are used in classical multiwavelet analysis. The set of n ¼ k  1 multiscaling functions from space V0k creates the fundamental multiscaling packet functions wn0;0 , and the set of n multiwavelet functions from space W0k creates the fundamental multiwavelet packet functions wn1;0 :

wn2i; j ðxÞ ¼ 21=2 j

j

3 0:219970 0:486664 7 7 0:627492 7 7 0:511081 5 0:240553

In the presented paper the set of multiscaling functions ’n ðxÞ is using Legendre polynomials Pn ðxÞ ¼ defined n ð1=2n n!Þðð@n ðx2  1Þ Þ=@xn Þ which are scaled up to the range of x 2 ½0; 1 according to the expression  pffiffiffiffiffiffiffiffiffiffiffiffiffiffi  2n þ 1Pn ð2x  lÞ; x 2 ½0; 1 ; n ¼ 0; . . . ; k  1 (14) ’n ðxÞ ¼ 0; x2 = ½0; 1

In the frame of multiwavelet packet analysis and in contrast to multiwavelet analysis, subspaces of multiwavelet functions Wkj are decomposed on subsequent stages of decomposition [12,18,25]. The functions which create the subspaces of decomposition Wkj are packet functions defined as X (8) wn2i; j ðxÞ ¼ 21=2 j k Hk wi ð2 j x  kÞ wn2iþ1; j ðxÞ ¼ 2

k X X n¼0

fm ðxÞ’nm;l ðxÞdx;

3 0 7 0 7 7 0 7 5 0 0:044194

2

Presented in this article the algorithm for determining the extreme value of the functional is based on the operational matrix of integration of multiwavelet functions and multiwavelet packet functions. The operational matrix of integration Pm is a matrix that enables the approximation of the

(12)

V4 w0,3

w1,3

w0,2

w1,2

w0,1 w0,0

w1,1 w1,0

w2,0

w2,1 w3,0

w4,0

w5,0

w2,2 w3,1

w6,0

w7,0

w4,1 w8,0

w9,0

w3,2 w5,1

w10,0

w11,0

w6,1 w12,0

w13,0

w7,1 w14,0

w15,0

l=3 l=2 l=1 l=0

Fig. 1 – Diagram of the wavelet packets after decomposition of subspace Vj = 4. Please cite this article in press as: K. Jarczewska et al., Multiwavelets and multiwavelet packets of Legendre functions in the direct method for solving variational problems, Archives of Civil and Mechanical Engineering (2014), http://dx.doi.org/10.1016/j.acme.2014.04.008

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Table 2 – Standard operational matrix of integration of Legendre multiwavelets for k = 2 and j = 2. 2

0:5 6 0:29093 6 6 0:00390624 6 6 0 P8 ¼ 6 6 0:00552427 6 6 0:00552427 6 4 0 0

0:288675 0 0 0:128906 0:0047842 0:0047842 0:02485922 0:02485922

0 0 0 0:0744241 0:00276214 0:00276214 0:0430574 0:0430574

0 0:125 0:0721688 0 0 0 0:0497184 0:0497184

0 0 0 0 0 0 0:0405949 0

0 0 0 0 0 0 0 0:0405949

0 0:022097 0:038273 0:0441942 0:0360844 0 0 0

3 0 0:022097 7 7 0:038273 7 7 0:0441942 7 7 7 0 7 0:0360844 7 7 5 0 0

Table 3 – Standard operational matrix of integration of Legendre multiwavelet packets for k = 2 j = 2 and l = 0. 2

0:5 6 0:29093 6 6 0:00390624 6 6 0 P8 ¼ 6 6 0:01104854 6 6 0 6 4 0 0

0:288675 0 0 0:128906 0 0:0117187 0:0263128 0

0 0 0 0:0744240 0 0:0338291 0:0345267 0

0 0:125 0:0721688 0 0 0 0 0:0351562

0 0 0 0 0 0:0287049 0:0351562 0

integral from the base of wavelet functions in the base of these R xk where wm ðxÞ ¼ functions: xo wm ðxÞdx ffi Pm ðxÞwm ðxÞ, ½w0 ðxÞ; w1 ðxÞ; . . . ; wm1 ðxÞT is a vector of taken basis functions, and each element of the vector wm ðxÞ is a discrete representation of a particular function [4,5,7,8,12,16]. The location of the discretization points depends on the number of m section divisions of independent variable x 2 ½x0 ; xk . Number m = k2j is associated with the level of approximation j and is equal to the number of assumed basis functions. When the endpoint of integration limit is in the middle of each of the m sections of the independent variable, the matrix Pm is called the standard operational matrix of integration, as opposed to the modified operational matrix of integration which is defined by parameter g 6¼ 0:5 ðg 2 ½0; 1Þ. The base for defining the operational matrix of the integration of multiwavelet functions is expression (17) which results from Eq. (4) after applying an operation of integration on both its sides ! Z 1=2 Z 1 Z 1 k pffiffiffiX ð0Þ ð1Þ n l l c ðxÞdx ¼ 2 gn;l ’ ð2xÞdx þ gn;l ’ ð2x  1Þdx 0

n¼0

0

1=2

(17) Assuming that the bases functions from resolution level j = J are used, the fundamental multiscaling functions which are Legendre polynomials of the kth order have to be integrated and then rescaled so many times until the level j = J is reached. Each scaled integral has to be shifted to the value ns =2 j as many times as the number of low-pass filter matrices are used. After the discretization process of these functions, matrices are obtained which represent the integrals of a scaling function of dimensions m  k. By multiplying the matrices by highpass coefficients according to (17) the set of integrals from multiscaling functions are obtained, from every function space which is taken into account in the analysis. When the obtained sub-matrices have been shifted and supplemented by the matrices, which represent the integrals from the multiscaling functions, the matrix form of integrals from bases functions is obtained. Finally, the operational matrix of integration for

0 0:015625 0:0270633 0 0:0255155 0 0 0:0175781

0 0:0382733 0:0662913 0 0:0625 0 0 0:0143525

3 0 7 0 7 7 0 7 0:0625 7 7 7 0 7 0:03125 7 7 0:0255155 5 0

multiwavelet functions can be calculated numerically as R xk 1 x0 wm ðxÞdxwm ðxÞ ffi Pm ðxÞ Operational matrices of integration for packet multiwavelet bases are determined in the same way as in the case of multiwavelet bases. The basis of formulating the method, are the expressions (12) and (13) which after integration of both their sides take the following forms Z 1 wn2i; j dx 0 ! Z 1=2 Z 1 k j XX ð0Þ ð1Þ hn; j wni ð2 j xÞdx þ hn; j wni ð2 j x  1Þdx ¼ 2 j=2 n¼0

0

j

1=2

(18) Z 0

1

wn2iþ1; j dx

¼2

j=2 j

k X X n¼0

j

ð0Þ gn; j

Z 0

1=2

wni ð2 j xÞdx

þ

ð1Þ gn; j

Z

1

1=2

! wni ð2 j x

 1Þdx (19)

For example, Table 2 shows the operational matrices of integration of Legendre multiwavelet functions of order k = 2 at the level of approximation equal to j = 2. Table 3 contains the operational matrix of integration of Legendre multiwavelet packet functions at k = 2 and j = 2.

4.

Problem formulations

This section presents a general method for determining the extreme of the functional using the operational matrix of integration of multiwavelet functions and multiwavelet packet functions. The necessary condition for the existence of stationarity point of the functional is zero value of the variation of the functional obtained at each point of integration interval [13,10,20]. Assuming the functional form Z J½uðxÞ ¼

xk

Fðx; uðxÞ; u0 ðxÞÞdx

(20)

xo

Please cite this article in press as: K. Jarczewska et al., Multiwavelets and multiwavelet packets of Legendre functions in the direct method for solving variational problems, Archives of Civil and Mechanical Engineering (2014), http://dx.doi.org/10.1016/j.acme.2014.04.008

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w40,0

w41,0

2

4 3 2 1

1 −1

0.2

0.4

0.6

0.8

1.0

x

−2

−1

w42,1,1

w42,1,2

w43,0

w42,0 1.5 1.0 0.5

0.2

0.4

0.6

0.8

1.0

x

− 0.5

4 2 0.2

0.4

0.6

0.8

1.0

x

−2

− 1.0 − 1.5

w47 ,0 1

2

2

1

1

1

−2

−2

−2

−1 −2 −3 −4

w42,2,1

w42,2,2

w42,2,3

w42,2,4

−1

0.2

0.4

0.6

0.8

1.0

−1

0.2

0.4

0.6

0.8

1.0

x

w46 ,1,1

0.4

0.6

0.8

1.0

−1 −2 −3

0.2

0.4

0.6

0.8

1.0

x −1

x

w46 ,1,2

3 2 1 −1 −2 −3

0.2

3 2 1

3 2 1 −1 −2 −3

x

0.2

0.4

0.6

0.8

1.0

x

0.4

0.6

0.8

1.0

0.2

0.4

0.6

0.8

1.0

x

x

0.2

0.4

0.6

0.8

1.0

x

− 0.5 − 1.0 − 1.5 − 2.0

0.6

0.8

1.0

0.2

0.4

0.6

0.8

1.0

0.2

0.4

0.6

0.8

1.0

0.2

0.4

0.6

0.8

1.0

3 2 1 −1 −2 −3

w414,0

x

x

x

w415,0 4

1.5 1.0 0.5

3 2 1 −1 −2 −3

0.2

3 2 1 −1 −2 −3

0.4

−4

w46 ,0

2

0.2

2

0.2

0.4

0.6

0.8

1.0

x −2

x

−4

Fig. 2 – The fourth component of Legendre multiwavelet packet functions for k = 5, which form the function basis according to the diagram presented in Fig. 1.

and determining the variation of the functional as a major linear part of the increment of the functional one can obtain the Euler–Lagrange equation (21), the solution of which is the form of function accomplishing the functional extreme (20): 0

@F=@uðxÞ ¼ d@F=dx@u ðxÞ ¼ 0

(21)

In the approach based on direct methods for solving variational problem, functional J[(u(x))] for every admissible function u (x) becomes a function J[(u(x))] = [J(c1, c2, . . ., cm)]. The variables of function are coefficients of the expansion of the admissible function in the base of approximation functions ðuðxÞ ¼ P i ci ’i ðxÞÞ [10,13,20]. The vector of sought coefficients can be obtained by solving the algebraic system of equations resulting from the necessary condition of existence of a functional extreme for which the first variation of the functional is equal to zero dJ½uðxÞ ¼ d½Jðc1 ; c2 ; . . . cm Þ ¼ 0. In the Ritz approach if the variation is equal to zero it means that the first derivative of the system of the functions J½uðxÞ ¼ Jðc1 ; c2 ; . . . ; cm Þ with respect to the variables ci is equal to zero. Therefore, to determine the values of the coefficients cm, one has to solve the system of m equations in the form @Jðc1 ; c2 ; . . . cm Þ=@c1 ¼ 0; @Jðc1 ; c2 ; . . . cm Þ=@c2 ¼ 0; . . . ; @Jðc1 ; c2 ; . . . cm Þ=@cm ¼ 0

(22)

In the paper as the basis functions of the direct Ritz method, multiwavelets and multiwavelet packets of Legendre

functions are used. It is assumed that as the basis of these functions a derivative of function u(x) is presented as u0 ðxÞ ¼

m X

ci ’i ðxÞ ¼ cTm wm ðxÞ

(23)

i¼1

The coefficients ci are the expansion coefficients of the derivative of search function on the basis of approximation functions. After integration of expression (23) and using the operational matrix of integration Pm the relationship which is the solution to the issue under consideration is obtained as Z xk u0 ðxÞdx þ uðx0 Þ ¼ cTm Pm wm ðxÞ þ uðx0 Þf T wm ðxÞ (24) uðxÞ ¼ xo

where the vector components f T ¼ ½e1 ; e2 ; . . . ; el  ; 0; 0; . . . ; 0 |fflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflffl}l |fflfflfflfflfflffl{zfflfflfflfflfflffl}ml are the values of integrals of bases functions, l is the number of scaling functions from space Vn of the lowest order of multiwavelet decomposition. If any independent function exists in the functional, this function is also approximated in the base of multiwavelets or multiwavelet packet functions as f ðxÞ ¼ dTm wm ðxÞ;

dTm ¼ ½d1 ; d2 ; . . . dm ;

di ¼ h f ðxÞ; ’i ðxÞi

(25)

where hf(x), wi(x)i denotes the scalar product of function f(x) and the ith basis function wi(x). In the case of determining the functional extreme by imposing additional s constraints, the Lagrange multipliers method is used. The variation problem at any boundary

Please cite this article in press as: K. Jarczewska et al., Multiwavelets and multiwavelet packets of Legendre functions in the direct method for solving variational problems, Archives of Civil and Mechanical Engineering (2014), http://dx.doi.org/10.1016/j.acme.2014.04.008

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The function u(x) defines the state of the displacement of system shown in Fig. 3. For example, assuming that N = 18, q(x) = 1 and EI = 1 the function u(x) takes the following form uðxÞ ¼ 0:0395 þ 0:071x þ 0:0278x2 þ 0:0395cosð4:2426xÞ 0:016584sinð4:2426xÞ. This result was obtained by solving the Euler–Lagrange equation of analysis problem. Error Dg occurring in the approximate solutions of function u(x) is defined by the following relationship (31)

Fig. 3 – Diagram of the beam under consideration.

Dg ¼ condition is extended to the equation constraints Gi(x, u(x), u0 (x)) = ai which is multiplied by the appropriate multiplier li and summed, which finally leads to the functional form J½uðxÞ;l ¼ J½uðxÞ þ V½uðxÞ;l Zxk s X ¼ ½Fðx; uðxÞ;u0 ðxÞÞ þ li ðGi ðx;uðxÞ;u0 ðxÞÞ  ai Þdx

(26)

i¼1

x0

Unknown coefficients of expansion of the function that minimize the functional (26) are obtained by using the Ritz variation formulation in the assumed approximation basis as the result of the solution of the m + s system of algebraic equations in the form: ( @~Jðc1 ; c2 ; . . . ; cm ; l1 ; l2 ; . . . ls Þ=@ci ¼ 0; i ¼ 1; . . . m (27) @~Jðc1 ; c2 ; . . . ; cm ; l1 ; l2 ; . . . ls Þ=@li ¼ 0; i ¼ 1; . . . s

5.

Let us consider the linear problem of determining the critical load Ncr of a beam which is loaded as shown in Fig. 3. Using the theorem of minimum potential energy, the functional which describes the analyzed problem is formulated as follows Z 1 Z 1 J½uðxÞ ¼ ð1=2Þ EIu00 ðxÞ2 dx  ððN=2ÞuðxÞ2 þ qðxÞuðxÞÞdx (28) 0

On the assumption that in the base of multiwavelet or multiwavelet packet functions the derivative of function u(x) is sought in accordance with expression (23), then the integrand functions which appear in expression (28) take the following form uðxÞ ¼ cTm Pm wm ðxÞ þ uðx0 Þ;

u00 ðxÞ ¼ cTm Pm wm ðxÞ þ u0 ðx0 ÞP1 m

i¼1

(31)

i¼1

~ ðxi Þ denote the exact and approximate Symbols u(xi) and u values of the solution at the ith point of variable x respectively. The values of global approximation errors of the function u (x) at the sample axial force N = 18, q(x) = 1 and EI = 1 are presented in Tables 4–6. The results in Tables 4–6 are obtained by using operational matrix of integration of Legendre multiwavelets, Legendre multiwavelet packet functions and Walsh wavelet packets respectively. The approximation solutions depend on the level of approximation j and also on the kth order of approximation functions. The results confirm the convergence of approximate solutions to the exact ones and

Table 4 – Global errors in variational solutions of problem (28) depending on the parameter j (m = k2j) and kth order of Legendre multiwavelets assuming the axial force N = 18.

Numerical examples

0

m m X X ~ ðxi Þj2 = juðxi Þj2 juðxi Þ  u

(29)

The system of equations resulting from the condition of stationary potential energy and the boundary conditions (u(L = 0) = 0, u0 (L = 0) = 0, u(L = 1) = 0, u00 (L = 1) = 0) is presented as follows 8 @~Jðcm ; ls Þ=@ci ¼ 0 i ¼ 1; . . . m > > > > > T T 1 1 T > > > ) EIcm Pm Pm I  Ncm I  dm Pm I þ l1 Pm wm ðxL¼0 Þ þ l2 wm ðxL¼0 Þ > > 1 > > > þ l3 Pm wm ðxL¼1 Þ þ l4 Pm wm ðxL¼1 Þ ¼ 0 < (30) @~Jðcm ; ls Þ=@l1 ¼ 0; ) cTm Pm wm ðxL¼0 Þ  uðxL¼0 Þ ¼ 0 > > > T 0 ~ > @Jðcm ; ls Þ=@l2 ¼ 0; ) cm wm ðxL¼0 Þ  u ðxL¼0 Þ ¼ 0 > > > > > > @~Jðcm ; ls Þ=@l3 ¼ 0; ) cTm Pm wm ðxL¼1 Þ  uðxL¼1 Þ ¼ 0 > > > : ~ @Jðcm ; ls Þ=@l4 ¼ 0; ) cTm Pm wm ðxL¼1 Þ  u00 ðxL¼0 Þ ¼ 0 ~Jðcm ; lk Þ ¼ J½uðxÞ þ l1 uðxL¼0 Þ þ l2 u0 ðxL¼0 Þ þ l3 uðxL¼1 Þþ where 00 l1 u ðxL¼1 Þ.

j

D [%]

3 4 5 6 7

k=2

k=3

k=4

k=5

17.081 3.8876 0.899596 0.214597 0.0840358

6.82042 2.34003 0.95312 0.485847 0.310763

3.73991 0.973563 0.247268 0.0630514 0.0083241

2.08541 0.525276 0.131531 0.0329012 0.0174035

Table 5 – Global errors in variational solutions of problem (28) depending on the parameter j (m = k2j) and kth order of Legendre multiwavelet packets; assuming the axial force N = 18. D [%]

j

2 3 4 5 6

k=2

k=3

k=4

k=5

3.081 0.8876 0.00899596 0.000114523 0.0000840358

2.82042 0.64245 0.0075312 0.000325841 0.0000310763

1.73991 0.373563 0.00247268 0.000430514 0.0000083241

1.08541 0.125276 0.00131531 0.000329012 0.0000023241

Table 6 – Global errors in variational solutions of problem (28) depending on the parameter j for Walsh wavelet packets; assuming the axial force N = 18. D [%] j

m=2

m = 32

m = 64

m = 128

m = 256

2.2314

0.683923

0.187327

0.0487646

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(b)

(a)

Fig. 4 – The dependence of the axial force on (a) the determinant of the matrix ½d~J, (b) the condition number of the matrix ½d~J for Legendre multiwavelet functions for k = 5, j = 3 and g = 0.5.

(a)

(b)

Fig. 5 – The dependence of the axial force on (a) the determinant of the matrix ½d~J, (b) the condition number of the matrix ½d~J, for Legendre multiwavelet packets for, k = 5, j = 2 and l = 0.

smaller errors in approximation are obtained in the considered case when using Legendre multiwavelet packets bases than Legendre multiwavelets and Walsh wavelet packets. Theoretical values of the critical load according to [23] are: NIcr ¼ p2 EI=ð0:69916lÞ2 ;

NIIcr ¼ p2 EI=ð040667lÞ2 ;

NIII cr

¼ p2 EI=ð0:28811lÞ2 The approximate values of the critical load have been obtained by using the operational matrix of integration of Legendre multiwavelet functions and Legendre multiwavelet packet functions on the assumed level of approximation j, as a result of the analysis of the determinant of the matrix ½d~J and of the analysis of the condition number of matrix ½d~J. The coefficients

of matrix ½d~J are the coefficients of the system of equations (30). In Figs. 4 and 5 examples of the functions of a determinant of matrix ½d~J and functions of condition number of matrix ½d~J are presented which depend on the value of the axial force N. As example approximation bases, Legendre multiwavelets for k = 5 on the level of approximation j = 3 and the Legendre multiwavelet packets for k = 5 at j = 3 are adopted respectively. Multiwavelet packets were taken from the last stage of decomposition (l = 0) of the analyzed subspace Vkj . Using adopted multiwavelet bases, the critical loads and the values of the coefficient of condition number of matrix ½d~J have been presented in Tables 7 and 8. In both cases, the results are consistent with the theoretical values of the critical force.

Table 7 – The approximate values of critical force and the condition number of the matrix ½d~J for Legendre multiwavelet functions for k = 5, j = 3 and g = 0.5. NIcr 19.9019EI/l 2

CondNumb (NIcr )

NIIcr

CondNumb (NIIcr )

NIII cr

CondNumb (NIII cr )

1.68129  10 12

59.9749EI/l 2

6.75049  10 13

117.132 EI/l 2

1.39758  10 13

Table 8 – The approximate values of critical force and coefficients of conditioning of matrix ½d~J for Legendre multiwavelet packets for k = 5, j = 2, g = 0.5 and l = 0. NIcr 19.6234EI/l

CondNumb (NIcr ) 2

2.42322  10

12

NIIcr 60.5261EI/l

CondNumb (NIIcr ) 2

5.50268  10

12

NIII cr 118.493EI/l

CondNumb (NIII cr ) 2

5.54593  10 13

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Table 9 – Relative power deviation D of the theoretical and numerical values for the first critical force of the system (28) for Legendre multiwavelets for k = 3 at various parameters j and g. j

2 3 4

~J½uðxÞ; lÞ ¼ ðN=2Þ

D [%] g = 0.2

g = 0.4

g = 0.5

g = 0.60

g = 0.80

0.6613283 8.8136823 1.8586101

0.2658496 0.0606024 0.0111536

0.179774 0.034524 0.005607

0.135149 0.021756 0.003129

0.208348 0.021514 0.003134

j

D [%] g = 0.2

g = 0.4

g = 0.5

g = 0.6

g = 0.80

0.0418658 1.3923132 1.0872599

0.006105 0.000779 0.000313

0.0004527 0.0003149 0.0002110

0.00652228 0.00214743 0.00353492

0.20980605 0.01428111 0.01072542

In Tables 9 and 10 the approximate error of the critical force is summarized. The first critical force obtained by using Legendre multiwavelet functions for k = 3 according to g and j is presented in Table 9. The second critical force obtained by using Legendre multiwavelet packets for k = 3 according to g and j is presented in Table 10. The values of errors were ~ i j= from the relationship DNicr ¼ ðjNicr  N calculated cr jNicr jÞ  100%. In the problem of the stability, the form of operational matrix of integration which depends on the parameter g has a significant influence on the obtained solution. With a standard operational matrix of integration, the errors of approximate solutions are not always smaller than for the suitably modified operational matrix of integration. The next task considers the problem of finding the extreme of the functional in the following form Z x1 Fðx; uðxÞ; u0 ðxÞÞdx J½uðxÞ ¼ x0 x1

Z ¼

ððN=2Þu0 ðxÞ2 þ qðxÞuðxÞÞdx

(32)

x0

uðx1 ¼ 1Þ ¼ 0

(33)

Fig. 6 shows a diagram of a static string of which deflection can be described by the extreme value of the functional (32). Taking into account the overall relationships (23)–(25) and using Legendre multiwavelet functions or Legendre

Fig. 6 – String problem under consideration.

1

þ 0

1

Z cTm wm wTm cm dx þ

1

0

cTm Pm wm wTm dm dx

uðx0 Þf T wm wTm dm dx þ l1 ðcTm f

þ uðx0 Þf T wm ðx0 Þ  uðx0 ÞÞ þ l2 ðcTm f þ uðx1 ÞÞf T wm ðx1 Þ  uðx1 ÞÞ

(34)

The stationary condition of the functional (34) according to (27) leads to the algebraic equations whose solution are approximation coefficients determined by expression (23). By substituting the obtained values cm of the coefficients into relationship (24) one can obtain approximate solutions of the deflection line of the analyzed string. For example, assuming that the load is N = 2 and q(x) = sin(5x), the function that implements the functional extreme (32) is u(x) = (1/50) (0.958924x + sin(5x)). This result was obtained by solving the Euler–Lagrange equation of analysis problem. The values of errors of the approximation of the extreme value of the functional (34) are summarized in Table 11. In this case, the bases of Legendre multiwavelet packet functions for k ðk ¼ 1; 2; . . . ; 5Þ were taken in the analysis. Errors in approximate solutions of (31) decrease with the increase of the scaling parameter j. These errors are small enough that each of the adopted bases allows an accurate solution for the exact solution to be obtained. The results shown in Fig. 7 and Table 12 confirm greater accuracy of the method which is measured by an approximation error and achieved using multiwavelets bases and packet multiwavelet Legendre bases, when compared with Haarwavelet and Walsh-wavelet functions. For example, in order to obtain a comparative error in the approximation of the variational solution of problem (34) and its derivative, the operational matrices of integration Walsh wavelet functions of more than double in size than when using the Legendre multiwavelet packets order k = 3 (Fig. 7 and Table 12), must be

Table 11 – Global errors in variational solutions of problem (34) depending on the parameter j for Legendre multiwavelet packets of type k at l = 0. D [%]

j

with given boundary conditions uðx0 ¼ 0Þ ¼ 0;

Z 0

Z

Table 10 – Relative power deviation D of the theoretical and numerical values for the second critical force of the system (28) for Legendre multiwavelet packets for k = 3 at various parameters j, g, and at l = 0.

2 3 4

multiwavelet packets, the general formula of the functional can be expressed in the form

1 2 3 4

k=2

k=3

k=4

k=5

0.31828 0.00712717 0.0037472 0.00050542

0.490318 0.0011584 0.0009908 0.00051556

0.089726 0.0050303 0.00051248 0.00050041

0.027385 0.0050070 0.00052114 0.00043213

Table 12 – Global errors of the approximate solution of variational problems (34) depending on the type and the number m of wavelet functions.

Legendre multiwavelet packets k = 3 Walsh wavelet packets Legendre multiwavelets k = 3 Haar wavelets

m

D [%]

12 32 48 32

0.00115840 0.00198336 0.00051556 0.00199401

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Fig. 7 – The solution of the variational problem (34) (a and c), the derivative of the solution of the variational problem (34) (b and d). (a and b) A solution obtained by using 32 Walsh-wavelet packet functions; (c and d) solution obtained with the base of Legendre multiwavelet packet functions for k = 3 from the last stage of decomposition of the space V j¼2 (exact solutiondashed line, approximate solution-solid line).

used. The use of a comparable number of Haar wavelet basis functions, and for example Legendre multiwavelet functions for k = 3, results in an error of approximation of the deflection function of the analyzed string of one order higher than in the case of the Haar wavelet functions.

6.

Conclusions

In the paper the method for obtaining the operational matrix of integration for Legendre multiwavelet packet functions, and the application of the operational matrix of integration for Legendre multiwavelet and packet multiwavelet functions to solve various variational problems is presented. In order to determine the extreme value of the functional the Ritz direct method has been used. Applied boundary conditions have been taken into account using the Lagrange multipliers method. The effectiveness of the algorithm measured as the convergence of approximate solutions to the exact solutions has been tested using multiwavelet functions and Legendre packet multiwavelet functions of different order and also using different levels of approximation and standard and modified operational matrices of integration of bases functions. The results obtained using the operational matrix of integration and from the classical wavelet analysis and also multiwavelet analysis have been compared. Performed numerical analysis became the basis for the following conclusions: - basis Legendre multiwavelet functions and Legendre multiwavelet packet functions are an effective approximation basis for variational problems of the issues discussed,

- the form of the operational matrix of the integral Legendre multiwavelet functions and multiwavelet packet functions has a significant impact on the achieved accuracy of the calculations, - application of the operational matrix of integration for multiwavelet functions and Legendre multiwavelet packet functions results in smaller approximation errors than when using the operational matrix of integration for Haar-wavelet functions and Walsh-wavelet packets at the same number of approximation functions.

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