Please cite this article as: Staisław Kula, Method of fudametal solutios for Helmholtz eigevalue problems i elliptical domais, Scietific Research of the Istitute of Mathematics ad Computer Sciece, 009, Volume 8, Issue, pages 85-90. The website: http://www.amcm.pcz.pl/ Scietific Research of the Istitute of Mathematics ad Computer Sciece METHOD OF FUNDAMENTAL SOLUTIONS FOR HELMHOLTZ EIGENVALUE PROBLEMS IN ELLIPTICAL DOMAINS Staisław Kula Istitute of Mathematics ad Computer Sciece, Czestochowa Uiversity of Techology, Polad ula@imi.pcz.pl Abstract. I this paper, the method of fudametal solutios for Helmholtz eigeproblems i a elliptical domai is preseted. To fid the approximate solutio of the problem, the Hael fuctio of the first id ad zero order as the fudametal solutio of the Helmholtz equatio i ubouded domai o the plae was used. Numerical examples illustratig the accuracy of the preset method are give. Itroductio The trasverse vibratio of a membrae which occupy the domai S, is govered by the followig equatio [] where is the Laplace operator, w w = () c t c = T ρ is the speed of soud, T is the uiform tesio per uit legth of the edge S, w x, y, t is the displacemet of the plate poit (x,y) at time t. Whe the free vibratio of the i t membrae is cosidered, a harmoic time depedece of the form e ω is assumed, t i.e. (,, ) (, ) e i ω w x y t = W x y. Substitutig the form of the displacemet ito (), the Helmholtz equatio is obtaied where Ω = ω ρ T assume here the Dirichlet coditio + W = ρ is the desity per uit area, ( ) W Ω, ( x y) S 0, (). The equatio () is completed by boudary coditios. We W ( x y) = 0, ( x, y) S, (3) The differetial equatio () ad boudary coditio (3) form the Helmholtz eigevalue problem. The problem for elliptic domai S is the subject of this paper.
86 S. Kula. The method of fudametal solutios We solve the eigevalue problem ()-(3) by usig the method of fudametal solutios (MFS). The MFS is a boudary method which does ot ivolve discretizatio ad itegratio. The method is applicable to problems for which a fudametal solutio of the goverig equatio is ow. The basic idea of the method is the usage of a liear combiatio of fudametal solutios with sources located at fictitious poits outside the domai of the cosidered problem. The fudametal solutio of a differetial equatio is ot uique. For the Helmholtz equatio () we tae the Hael fuctio [, 3] where i ( ) ( ) G P, Q = H 0 ( Ω d( P, Q) ) (4) 4 ( H ) 0 ( ) is the Hael fuctio of the first id ad zero order, ( P Q) i =. The fuctios G ( x y; ) distace betwee poits P ad Q, Q d, is the, satisfy the Helmholtz equatio i the domai S for each source poits Q located outside S. I the MFS, we approximate the solutio of the problem by a fuctio of the form w ( P) = c G( P Q ) =, (5) The approximate solutio w satisfies the differetial equatio (), ad it does ot satisfy the boudary coditio (3). The coditio ca be satisfied approximately by a suitable determiatio of the coefficiets c, =,,,. For this purpose we use the last square method. First we choose the poits P i, i =,,,, located o boudary S of the domai S. We determie the coefficiets c so that the fuctio f (( c c,..., c )) = [ w ( Pi ; c, c,..., c )] i=, (6) has a miimum. This leads to a homogeeous liear system of equatios f c j = c = i= (, Q ) G( P, Q ) = 0, G P i j i j =,,, (7) which ca be writte i the matrix form as A c = 0 (8) where A = [ a ], j a ( ) ( ) j, j G Pi, Q j G Pi, Q =, = [ c c ] T i= c... c
Method of fudametal solutios for Helmholtz eigevalue problems i elliptical domais 87 For a o-trivial solutio of equatio (8), the determiat of the coefficiet matrix A is set equal to zero, yieldig the eigevalue equatio det A(Ω ) = 0 (9) Equatio (9) with the uow Ω, is solved umerically. The eigefuctios for the determied eigevalues Ω m, m =,,, are give by (5) where the coefficiets c, =,,, are derived i depedece of c from - equatios of the system (7). For the cosidered Helmholtz eigevalue problem the fuctio w defied by (5), after usig (4) has the form w ( ) ( ) x, y = c H Ω ( x ξ ) + ( y η ), ( x y) S = 0 where the source poits Q ( ξ, η ) coefficiets of the matrix A are a j = i= H, (0), =,,,, are over the domai S. But the Ω x i ξ j + yi η j H0 Ω xi ξ + yi η () ( ) ( ) ( ) ( ) ( ) ( ) 0 where the collocatio poits ( ) S of the domai S. P x i, y i, i =,,,, are located o the boudary. Numerical examples We assume that the boudary of the cosidered membrae domai is a ellipse with major ad mior semi-diameters equal a ad b, respectively. The collocatio poits are uiformly distributed o this ellipse ad the source poits are uiformly distributed o a circle with the radius r = α max{ a, b}, α >. Geometry cofiguratio of a elliptical domai with the collocatio poits o the boudary ad a circle with the source poits is preseted i Figure. The Hael fuctio ( H ) 0 occurrig i equatios (0, ) assumes complex values ad that way the elemets of the matrix A are complex. Our goal is to evaluate real roots of the complex equatio (9). For this purpose the ew fuctio F is defied as where the symbol ( Ω ) = det A( Ω ) F () deotes a modulus of complex umber. It is clear that the roots of equatio (9) are determied by miima of the fuctio F. I Figure a
88 S. Kula exemplary graph of the fuctio i logarithmic scale for a elliptical domai with semi-diameters ratio a/b =.0 ad = 4 is preseted. Locatios of the distict drops idicate the roots of equatio (9). y Q j = (ξ j,η j ) b P i = (x i, y i ) a x Fig.. Geometry cofiguratio of the cosidered domai with collocatio poits P i o the ellipse ad source poits o Q j the circle 0 0 30 40 50 60 4 6 8 Ω 70 log det A( Ω ) Fig.. Logarithmic values of modulus of the determiat deta(ω ) as a fuctio of Ω for the Helmholtz operator i the elliptical domai with semi-diameters ratio a/b =.0
Method of fudametal solutios for Helmholtz eigevalue problems i elliptical domais 89 To estimate the MFS accuracy, a example of the Helmholtz eigevalue problem i a circular domai will be solved. I this case the eigevalues are the roots of the equatio [3] m ( Ω ) = 0 J, m =,, (3) A sequece of the eigevalues for each m ca be obtaied. First members of the sequeces are umerically determied by fidig the local miima of the fuctio F Ω F Ω is give by equatio (). The calculatios were performed log ( ), where ( ) for various umbers of source poits ( =6; 0; 4). The compariso of the results obtaied by the MFS with exact values obtaied as roots of equatio (3) is preseted i Table. Note that the umerical calculatio by MFS with small umber of the source poits gives the results with a larger error ad ca lead to omissio of eigevalues. The compariso of the results obtaied by MFS ad the exact eigevalues show that the absolute error for = 4 is ot larger as 0 5. The eigevalues of the Helmholtz operator are the frequecy parameters of free vibratio of a membrae. The first seve frequecy parameter values of the elliptic membrae with clamped edge are preseted i Table. The calculatios are performed for various values of semi-diameters ratio a/b of the ellipse. The frequecy parameters obtaied by usig MFS were compared with the eigevalues determied by the fiite elemet method i the paper [] by Buchaa ad Peddieso. For assumed umber of sources = 4 i MFS a small differeces of the results calculated by usig both methods are observed. Table. Eigevalues Ω ( =,,0) of the Helmholtz operator i a circular domai obtaied by the MFS for = 6; 0; 4 ad exact eigevalues obtaied as roots of equatio () MFS = 6 MFS = 0 MFS = 4 Exact eigevalues Ω.40483.40483.40483.40483 Ω 3.837 3.837 3.837 3.837 Ω 3 5.3560 5.356 5.356 5.356 Ω 4 5.5005 5.5008 5.5008 5.5008 Ω 5 6.37893 6.3804 6.3806 6.3806 Ω 6 7.05 7.0558 7.0559 7.0559 Ω 7 7.5899 7.58809 7.58834 7.58834 Ω 8 8.40860 8.476 8.474 8.474 Ω 9 8.6537 8.65374 8.65373 8.65373 Ω 0 8.76646 8.7748 8.7748
90 S. Kula Table. Frequecy parameters Ω ( =,,7) of the elliptic membrae for various values of semi-diameters ratio b/a of the ellipse a/b =.5 a/b =.0 a/b = 3.0 MFS FEM * MFS FEM * MFS FEM * Ω.039.039.88858.889.763.763 Ω.956.97.50508.508.4359.48 Ω 3 3.84097 3.850 3.4590 3.47.98058 3.07 Ω 4 4.7896 4.88 3.99047 4.000 3.4586 3.477 Ω 5 5.079 5.04 4.5856 4.584 3.6849 3.697 Ω 6 5.74644 5.769 4.98843 4.990 4.0443 4.30 Ω 7 6.0787 6.7 5.545 5.56 4.8454 4.8 the results by FEM are give i the paper [4] Coclusios I the paper, the applicatio of the method of fudametal solutios to the Helmholtz eigevalue problem i the elliptical domai has bee preseted. As the fudametal solutio of the Helmholtz equatio i the plae, the Hael fuctio of the first id ad zero order was used. I order to determie the eigevalues, the miimum of a real fuctio which is the logarithm of modulus of a complex fuctio was foud. The source poits occurrig i the approximate formula of the solutio ca be selected o a circle outside of the cosidered elliptical domai. The compariso of umerical results shows that high accuracy of the calculatio is achieved for 4 sources. Refereces [] Pietrowsi L., Rówaia róŝiczowe cząstowe, PWN, Warszawa 955. [] Che J.T., Che I.L., Lee Y.T., Eigesolutios of multiply coected membraes usig the method of fudametal solutios, Boudary Aalysis with Boudary Elemets 005, 9, 66-74. [3] Reutsiy S.Yu., The method of fudametal solutios for Helmholtz eigevalue problems i simply ad multiply coected domais, Boudary Aalysis with Boudary Elemets 006, 30, 50-59. [4] Buchaa G.R., Peddieso J., A fiite elemet i eliptic coordiates with applicatio to membrae vibratio, Thi-Walled Structures 005, 43, 444-454. [5] Duffy D.G., Gree s fuctios with applicatios, Chapma & Hall/CRC, Boca Rato, Washigto DC 00.