CHAPTER - 3 Orthogonal Collocation of Finite Elements using Hermite Basis

Transcription

1 CHAPTER - 3 Orthogonal Collocaton of Fnte Elements usng Hermte Bass Contents of ths chapter are publshed n: 1. Computers and Chemcal Engneerng, 58, 03-10, Appled Mathematcal Scences, 7(34), , Proceedngs of Internatonal Conference on Industral Compettveness,, Internatonal Congress of Mathematcans, Hyderabad, Inda, August 010.

2 3.1 Introducton As observed n Chapter, dfferent mathematcal models are encountered durng dsplacement of an ntally homogeneous solute from a medum of fnte length by the ntroducton of a solvent. Such partal dfferental equatons (PDEs) alongwth dfferent boundary and ntal condtons are solved analytcally (manly Laplace Transforms, Fourer Transforms etc.) and numercally (manly fnte dfference method, orthogonal collocaton method, ftted mesh collocaton method, orthogonal collocaton on fnte elements, Galerkn / Petrov Galerkn method) as well. The analytc soluton s always the best but due to ts complexty or non exstence (n many cases) one has to take the recourse of approxmaton technques. At the same tme, one must keep n mnd that dfferent numercal technques have certan lmtatons lke as n fnte dfference method, the soluton of the system s very unstable and requres strct selecton of step sze. Nevertheless, the accuracy of numercal soluton s not so hgh. The dscretzaton of even a few PDEs by the method of lnes can lead to an extremely large system of ODEs, the numercal soluton of whch may have severe cost and storage mplcatons. Ths stuaton can be tackled usng orthogonal collocaton method (OCM). For large values of the parameters, the nature of the equaton becomes stff; as a result the oscllatons ncrease. Therefore, n such a stuaton the OCM does not gve good results even for large number of collocaton ponts. To overcome ths stuaton, the method of orthogonal collocaton on fnte elements (OCFE) s used, n whch orthogonal collocaton method s assocated wth the fnte element method. In OCFE, the doman 0 x 1, s dvded nto small sub domans of fnte length Δx, called elements. Then the orthogonal collocaton s appled wthn each element. In ths process, t s mandatory that the tral functon and ts frst dervatve 44

3 should be contnuous at the nodal ponts or at the boundares of the elements. The tral functon s usually expressed n terms of Lagrangan nterpolaton polynomals. The collocaton ponts are the zeros of the orthogonal polynomals lke Jacob, Legendre and Chebyshev etc. In the present work the technque of OCFE wll be used wth a cubc Hermte bass nstead of Lagrangan bass. Wth the Hermte bass, the coeffcents of the bass functons are easly chosen so that the soluton and ts frst dervatve are automatcally contnuous at the boundary of the elements [Fnlayson, 1980]. Ths results n a sgnfcant savng n computatonal effort as compared to the Lagrange bass, where the contnuty equatons have to be solved. In ths technque the resdual functon s set orthogonal to the weght functon. The choce of the weght functon depends upon the type of orthogonal polynomal to be chosen. Carey and Fnlayson [1975] were the frst one to ntroduce orthogonal collocaton on fnte elements method. In ths method, the doman n dvded nto small elements and n each element collocaton ponts are chosen to be the zeros of orthogonal polynomals, taken n the expanson of the approxmatng functon, whch s expanded n terms of the seres of the orthogonal polynomals. The collocaton ponts are taken to be the zeros of these polynomals. Thus the resdual vanshes at the collocaton ponts gvng the desred accuracy. Hence, t s named as orthogonal collocaton method on fnte elements. The orthogonalty property of the polynomals ensures that the zeros are real and dstnct. The prncple of orthogonal collocaton s based on the orthogonalty of the polynomals that vanshes at the collocaton ponts and not of the orthogonalty of the resduals. 45

4 3. Descrpton of Orthogonal Collocaton Method General procedure for solvng the dfferental equaton usng orthogonal collocaton method s: Consder an orthogonal functon ( x, t) whch satsfes the lnear or non lnear dfferental equaton: N V ( ) = 0, n V, (3.1) and the lnear or non lnear boundary condton as: N B ( ) = 0, on B, (3.) where B s the boundary of the volume V, x s the poston vector and t represents the tme doman. The dependent varable s approxmated by the seres expanson of k ( x, t). These parameters can be determned by applyng equaton (3.1) and / or equaton (3.) at each of the k selected collocaton ponts. The key ponts of the soluton technque are: type of the collocaton ponts, type of the orthogonal polynomals, type of the bass functon Collocaton ponts Choce of the collocaton ponts s an mportant and senstve part of the orthogonal collocaton method. In general, collocaton ponts are taken to be the zeros of orthogonal polynomals lke Legendre polynomal or Chebyshev polynomal. These polynomals are partcular case of Jacob polynomals, gven as: 46

5 n (, ) ( n 1) ( n 1) n ( n m 1) z 1 Pn ( z) n! ( n 1) m0 m ( m 1) (3.3) m The Legendre and Chebyshev polynomals are specal cases of Jacob polynomals for 0 and 1, respectvely. In axal doman, the zeros of shfted Legendre polynomal have been taken as collocaton ponts. The zeros of Legendre polynomal are calculated from the followng recurrence relaton: ( j1 j j3 x j 1) P ( x) ( j 3) xp ( x) ( j ) P ( ) ; j =,, m+1, (3.4) where P 0 (x) = 1 and P -1 (x) = 0. In case of Legendre polynomal, 0 and 1 are taken to be the boundary ponts. The collocaton ponts are obtaned by mappng the computatonal doman of the nterval [-1, 1] to [0, 1] wth the help of followng relatonshp: x j 1 um 3 j, (3.5) where x j s the j th collocaton pont n the nterval [-1, 1]. Chebyshev polynomal has the tendency to keep the error down to a mnmum at the corners (Chen, Lee and Chang, 003, Nelsen and Hesthaven, 00). Therefore, the zeros of shfted Chebyshev polynomal have been used as collocaton ponts n the radal doman, because the results are requred at the corner n radal doman. The m+ nterpolaton ponts are chosen to be the extreme values of an (m+1) th order shfted Chebyshev polynomal: x j ( j 1) cos m 1 ; j = 1,,, m+. (3.6) The dscretzaton end ponts are fxed as 0 and 1. 47

6 3.. Orthogonal polynomals The oscllatory behavor of Legendre and Chebyshev polynomals has been dscussed by Arora et al. [005]. In Fgure 3.1 and Fgure 3. the behavor of these orthogonal polynomals for order 5 and 7 s shown. Due to the fluctuatons n the doman of x [1,1], Chebyshev polynomal does not gve satsfactory results on the average as compared to Legendre polynomal. The weght functons are zero at x = 0 and x = 1 n case of Legendre polynomal whereas these are non zero n case of Chebyshev polynomal. Ths factor also corresponds to the fact that Legendre polynomal gves good results on the average. The collocaton ponts n case of Chebyshev polynomal are more or less equdstant near to centre whereas ths s not so n case of Legendre polynomal. However, the Chebyshev zeros are closer to the boundary as compared to Legendre polynomal, due to ths reason former mnmzes the error at corners than the latter and gves good results at the corners. So, Chebyshev polynomal gves good results at the corners as compared to the Legendre polynomal. Fgure 3.1: Behavor of Legendre and Chebyshev polynomals of order 5 48

7 Fgure 3.: Behavor of Legendre and Chebyshev polynomals of order Bass functon Two type of bass functon can be chosen Lagrange or Hermte. Hermte bass s qute useful than Lagrange bass because of the C 1 contnuty of Hermte polynomals, whch n result reduces the number of equatons by one thrd than usng Lagranges bass functon. 3.3 Orthogonal Collocaton on Fnte Elements The technque of orthogonal collocaton on fnte elements (OCFE) s a combnaton of two technques vz: fnte element method (FEM) and orthogonal collocaton method (OCM). The doman of nterest 0 x 1 s dvded nto subdomans : 0 1 x1 x x ne 1 called elements wth hk xk 1 xk as shown n Fgure 3.3. The global varable x vares n the k th element, where k = 1,,...,ne. A new varable u ( x xk ) / hk s ntroduced n such a way that as x vares from x k to xk 1, u vares 49

8 from 0 to 1 as shown n Fgure 3.4. Then orthogonal collocaton method s appled on the new varable u. x1 0 x x 3 x x 1 1 ne ne Fgure 3.3: Fnte elements on global doman u1 0 u u m um 1 1 Fgure 3.4: Orthogonal collocaton on local doman In the present study, among the Lagrange and Hermte bass, the technque of OCFE s appled wth Hermte bass because of the advantage that t reduces the number of equatons due to C 1 contnuty of Hermte polynomals. 3.4 Cubc Hermte Polynomals We start wth the defnton of cubc Hermte polynomals and dscuss ts fundamental propertes Defnton 1 The cubc Hermte polynomal for a functon f : a, b R s a unque 3 rd degree polynomal s whch satsfes the followng constrant at x j j a and x b : s x f x, 1, ; j 0,1. (3.7) 50

9 The four constrants mposed on the structure of a cubc Hermte polynomal gve rse to four fundamental polynomals Defnton A cubc Hermte polynomal of a gven functon n the nterval a x0 x1... xn b can be expressed n terms of four fundamental polynomals as: s ( x) H ( x) f ( x ) H ( x) f ( x ) Hˆ ( x) f ( x ) Hˆ ( x) f ( x ), (3.8) j 1 j j1 1 j j1 where the fundamental polynomal H, ˆ H 3, 1,, satsfy the condtons: ˆ H x j j, H x j 0, ' ' 0, ˆ H x j H x j j. (3.9) The polynomal H1, H ˆ ˆ, H1, H and ther frst dervatves takes the values 1 and 0 at x x j and x x j 1. The graphcal representaton of H1, H ˆ ˆ, H1, H s gven n Fgure 3.5 and for convenence, these terms are wrtten as H ˆ ˆ L,, HR,, HL,, HR, respectvely. The structure of the fundamental polynomals s descrbed as follows: (1) ( H -Left) H1 H L, satsfes (0,0,1,0) condtons () ( H -Rght) H H R, satsfes (1,0,0,0) condtons (3) ( Ĥ -Left) H ˆ ˆ 1 H L, satsfes (0,1,0,0) condtons (4) ( Ĥ -Rght) H ˆ ˆ H R, satsfes (0,0,0,1) condtons 51

10 H (b) H, (a) L, R (c) H ˆ L, (d) H, ˆ R Fgure 3.5: Graphcal representaton of cubc Hermte polynomal 5

11 Usng the above condtons, an explct representaton of each fundamental cubc Hermte polynomal can be expressed as: a) H -Left: H L, x x 3 3 x x x x 1, xx, x 1 HL, x h h (3.10) 0, otherwse b) H -Rght: H R, 1 3 HR, x h h 0, otherwse 3 x x x x x x, xx, x (3.11) c) Ĥ -Left: H ˆ L, 1 Hˆ x h L, x x x x, xx, x 1 1 (3.1) 0, otherwse d) Ĥ -Rght: H ˆ R, 1 Hˆ x h R, x x x x, xx, x 1 1 (3.13) 0, otherwse 3.5 Dscretzaton Usng Approxmatng Functon The dmensonless form of the model equatons along wth boundary and ntal condtons are dscretzed usng the approxmatng functon. The approxmate soluton cu, t at the by: th j collocaton pont n the th element s gven 4 c u, t a ( t) H u ; 1,,..., k; j,3. (3.14) j j l( 1) l l1 53

12 j where the four coeffcents al ( 1) ( t ) n each element are unknown parameters. The labelng of the parameters are such that the solutons c u, t and c u t k k are contnuous at ther common node ponts. The dervatves of tral functon (3) can be wrtten as follows: 1, c 4 j 1 j dhl al ( 1) l1 (3.15) u h du c 4 j 1 j d Hl a l ( 1) u h l 1 du (3.16) c 4 j j dal ( 1) Hl t l 1 dt (3.17) The tral functon s composed of value of the functon and frst dervatve at each end n such a manner that three of these quanttes are zero and fourth s one. In partcular, for k th element [ xk, xk 1] cubc Hermte bass functons are gven as follows: H H H H k 1 k k 3 k (3.18) Detaled descrpton about cubc Hermte functons s avalable n Fnlayson [1980]. 3.6 Convergence Crteron The convergence of the method, for steady state, s checked by the followng formula: 1 L x M C, wth L 1 (3.19) k 54

13 The value of should be kept small enough to make L 1 or more elements xk should be nserted. Wth the ncrease n number of elements the method converges asymptotcally (Arora, Dhalwal & Kukreja, 005). 3.7 Algorthm of the Method Step 1. Dvde the doman 0 x 1 nto a mesh 0 x1 x xne 1 1. Step. Transform the global varable x nto local varable u ( x x ) / h. k k Step 3. Express approxmate soluton, c u t at j th j collocaton pont n the th element usng cubc Hermte bass. c j c Step 4. Obtan the dervatves j, u u and c j t of the tral functon. Step 5. Carry out dscretzaton of the model usng step 4. Step 6. Evaluate step 5 at collocaton ponts u (1 ) and 1 1 u (1 ) Step 7. The dfferental algebrac system obtaned n step 6 s solved usng MATLAB. 3.8 Applcatons of Hermte Polynomals The use of Hermte polynomals n the soluton of mathematcal models numercally s tself very vast as a large number of research papers are avalable n whch varous type of mathematcal models are solved usng Hermte Polynomals. Housts [1978] appled collocaton methods based on pecewse Hermte cubc polynomals to lnear ellptc problems subject to Drchlet and Neumann boundary condtons on rectangular domans. A pror estmates are obtaned for the error of approxmaton. The lnear ellptc problem consdered was: 55

14 Lu ad u bd DeD u cd u dd u ed u fu g n (0,1) (0,1), (3.0) x x x y y x y alongwth wth boundary condton: u Bu u 0, on = boundary of (3.1) x Wth ether 0 or 0 but not both and f 0. Gher and Marzull [1986] solved an ntal value problem wth the use of orthogonal collocaton wth Hermte bass functon. The method s shown equvalent wth a classcal collocaton method nvolvng a larger collocaton system. The order of the method s nvestgated brngng out some connecton wth a class of hybrd multstep formulas. Suffcent condtons are gven guaranteeng convergence of teratve method for the soluton of the nonlnear collocaton system. The resultng nterpolatng pecewse cubc curve s twce contnuously dfferentable wth respect to arc length and can be constructed locally. The Intal value problems solved are: y f x, y, y x y, x x X and (3.) y f x y y y x y y x y (3.3) Second order equaton,,,, on the regon x0 x X, y, x (3.4) Baleck and Dllery [1993] analyzed the convergence rates of two Schwarz alternatng methods for the teratve soluton of a dscrete problem whch arses when orthogonal splne collocaton wth pecewse Hermte bcubcs s appled to the Drchlet problem for Posson's equaton on a rectangle. Fourer analyss s used to obtan explct formulas for the convergence factors by whch the H1-norm of the errors s reduced n one teraton of the Schwarz methods. It s shown numercally that whle these factors depend on the sze of overlap, they are ndependent of the partton step sze. 56

15 u f n, u 0 on (3.5) where (0,1) (0,1) and s ts boundary. Rogers and McCulloch [1994] modeled the effects of geometrc complexty, non unform muscle fber orentaton, and materal n homogenety of the ventrcular wall on cardac mpulse propagaton. Spatal varatons of tme-dependent exctaton and recovery varables were approxmated usng cubc Hermte fnte element nterpolaton and the governng fnte element equatons were assembled usng the collocaton method. u. D u c1u u a1 u cu, t v bu dv, t (3.6) (3.7) wth boundary condton: u n 0. (3.8) Duarte and Portugal [1995] presented a movng fnte element method based on cubc Hermte polynomals. Ths methodology s appled to the soluton of a front reacton model - the caustc zng reacton wth boundary condtons beng tme derved, arsng from the basc formulaton. The soluton of ths problem s obtaned va the Lagrange Multplers Method. The results obtaned are n close agreement wth those publshed n the lterature where orthogonal collocaton n Fnte Elements was used. 1 C OH D C e OH I I C De, 0 r z t & z t r 1 OH r 1 C CO D C 3 e CO3 I I C De, 0 r z t & z t r 1 CO3 r (3.9) (3.30) 57

16 wth boundary condtons: OH CO3 OH 0, t CO3 0, t 1, t 1, t 0, 0, kl D e kl D e C C 0, t, 0 OH OH C C 0, t, 0 CO3 CO3 (3.31) (3.3) (3.33) (3.34) C C C C OH OH CO3 CO3, I I Z Z I I Z Z (3.35) And ntal condtons: C r,0 C, OH OH n C r,0 C, CO3 CO3 n I C r,0 C, 0 r Z ca OH ca OH t n I C r,0 0, Z ca OH t r R I I 0 Z 0 Z 0.8R (3.36) Lopez and Temme [1999a, 1999b] has gven asymptotc representatons of Bernoull, Euler, Bessel, Jacob, Laguerre and Buchholz polynomals n terms of Hermte polynomals. These asymptotc approxmatons are vald for large values of a certan parameter. Edoh, Russell and Sun [000] presented a collocaton method of O(h 4 ) for solvng nonlnear frst order PDEs wth perodc boundary condtons. A p-dmensonal Hermte cubc collocaton method was consdered for dscretzaton. An adaptve grd 58

17 refnement scheme was ntroduced to study the torus breakdown. The numercal results for the method are compared wth the analytc results. j1 p r f j, r g, r, 1,... q, (3.37) Wth perodc boundary condton: j r 1, 1 1 r 1, ,,0,......,,,..., j j p j j p for 1,..., q, j 1,..., p. (3.38) Djkstra [00] presented a conventonal pseudo-spectral collocaton method to solve an ordnary frst order dfferental equaton. The dervatve s obtaned from Lagrange nterpolaton and has degree of precson N for a grd of (N+1) ponts. Hermte nterpolaton s used as pont of departure. The second order dervatve s obtaned wth degree of precson (N+1) for the same grd. The double collocaton leads to a soluton accuracy whch s superor to the precson obtaned wth the conventonal method for the same grd. The accuracy of the present method s found comparable to the soluton accuracy of the standard method wth twce the number of grd ponts. Ghavam et al. [00] proposed novel modfed Hermte polynomal functons for use n mpulse rado communcatons. Wth these functons pulse shapes whch are orthogonal and have nearly constant pulse wdth regardless of the pulse order are generated. A MATLAB model for generatng the pulses s desgned. Leao and Rodrgues [004] have compared dfferent strateges for the numercal soluton of smulated movng bed processes to predct the transent and steady-state behavor. The model assumes axal dsperson flow for the lqud phase and plug flow for the sold phase. The models for transent stuatons were wrtten n terms of system of partal DAEs or PDEs and for steady-state, the models were represented by a system of DAEs or ODEs. Dfferent publc doman solvers lke DASSL, PDECOL, COLDAE 59

18 and COLNEW were used to numercally solve the models by consderng both lnear and non-lnear sotherms. C C C Pe x x j 1 j j 1 * j j qj qj, (3.39) q j qj q q x * j j j, (3.40) Wth Boundary and Intal Condtons: C j,0 C j Pe 1 j j C x, at x = 0 (3.41) C j x q j 0, at x = 1 (3.4) qj 1,0, (3.43) C x, q x, 0. at 0 (3.44) j j Ebed and Mory [005] and Ma and Khorasan [005] employed Hermte polynomals for mage analyss. Zhan and L [006] proposed a generalzed fnte spectral method for 1D Burgers and Korteweg-de Vres equatons. The Legendre, Chebyshev and Hermte polynomals were used as the bass functons. To attan hgh accuracy n tme dscretzaton, the fourth-order Adams-Bashforth-Moulton predctor and corrector scheme was used. Cermakova et al. [006] concluded that twn system conssts of two vessels and external ppng n total recycles. Expermental results from ths system can be evaluated usng Z- transforms to derve partcle RTD for subsequent testng of alternatve flow models. The axal dsperson model was appled usng the advecton dffuson equaton (sometmes called the dffuson wth bulk flow equaton ) derved thereof, whch was solved numercally. Ths contrbuton presents an analytcal soluton of analogous equatons, whch enables drect and precse evaluatons of ths problem. 60

19 The model equatons are: c v c D c a 0, t x x (3.45) Wth Dankwerts boundary condtons: c vc Da vc t x x0 1 0 x 0, at x = 0 (3.46) c 0, x at x = L (3.47) and ntal Condton: c x,0 c f x. (3.48) 1 0 Jang, Zhu and Cao [007] utlzed two famles of Hermte polynomals and ther Fourer transform to express tme-doman sgnal and correspondng frequency response as a weghted sum of these quanttes. The general propertes, whch affect the performance of the proposed method, of the two famles of Hermte functons are studed. The performance of algorthm s found senstve to the choce of orgn and scalng factor. Rocca and Power [008] presented a double boundary collocaton approach based on the meshless radal bass functon Hermtan method (symmetrc method). In the double collocaton approach, the boundary condton and the governng PDEs are requred to be satsfed smultaneously nstead of only the boundary condton as requred n the sngle collocaton. The results obtaned wth ths method are characterzed by a hgher precson especally for the predcton of the fluxes at the boundares. Ths s due to the hgher order of contnuty of the approxmaton at the boundary ponts mposed by the double collocaton. 61

20 Stevens, Power and Morval [009] presented a meshless algorthm for transport-type PDEs, based on local Hermtan nterpolaton of functon values and boundary operators, and usng an explct tme formulaton. Computatonal cost to advance the soluton n tme s mnmal, and s largely dependent on local system support sze. The performance of the method s examned for a varety of lnear convecton dffuson reacton problems, featurng both steady and unsteady solutons. The method s also demonstrated wth a nonlnear Rchards model, solvng an unsaturated flow n porous meda problem. Besde the above nvestgators, other researchers lke Wtschel [1973], Noschese [1997], Black and Geddes [001], Vskov [008] etc. presented dfferent propertes related to Hermte polynomal. 6