Equivalent Finite Element Formulations for the Calculation of Eigenvalues Using Higher-Order Polynomials

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1 Appled Mahemacs: ; (: 3-3 DOI:. 593/.am.. Equvale Fe Eleme Formulaos for he Calculao of Egevalues Usg Hgher-Order Polyomals C. G. Provads Deparme of Mechacal Egeerg, Naoal echcal Uversy of Ahes, Ahes, GR-57 8, Greece Absrac hs paper vesgaes hgher-order approxmaos order o exrac Surm-ouvlle egevalues oe-dmesoal vbrao problems couum mechacs. Several alerave global approxmaos of polyomal form such as agrage, Berse, egedre as well as Chebyshev of frs ad secod d are dscussed. I a srucve way, closed form aalycal formulas are derved for he sffess ad mass marces up o he quarc degree. A rgorous proof for he rasformao of he marces, whe he bass chages, s gve. Also, a heorecal explaao s provded for he fac ha all he aforemeoed alerave pars of marces lead o decal egevalues. he heory s susaed by oe umercal example uder hree ypes of boudary codos. Keywords Fe Elemes, Galer/Rz, Global Approxmao, P-Mehods, Egevalue Aalyss. Iroduco I he framewor of he sadard Galer/Rz mehod for he soluo of oe-dmesoal boudary value problems govered by a dffereal equao wh he doma [,], he usual procedure cosss of subdvdg [,] o a cera umber of fe elemes for whch pecewse-lear (.e., local erpolao s assumed[]. I geeral, he shores he elemes are he more accurae he umercal soluo s (h-verso. Aleravely, hgher order p-mehods[] sugges he roduco of odeless bass fucos based o dffereces of egedre polyomals (up o he seveh degree ha cooperae wh he wo lear shape fucos,.e., N( x xn, ( x x, he laer assocaed o he eds x ad x. A leraure survey suggess ha for a cera dscrezao of he doma, he correspodg p-verso s geerally more accurae ha he h-verso[3-6]. he maer of usg hgher order approxmaos hrough compuer-aded-desg (CAD based Coos-Gordo macroelemes has bee recely dscussed for wo- ad hree-dmesoal problems[7-]. I hose wors some smlares ad dffereces of he so-called Coos macroelemes wh respec o he hgher order p-mehod have bee repored deal. Moreover, alerave CAD based NURBS or/ad Bézer echques have bee proposed * Correspodg auhor: cprova@ceral.ua.gr (C. G. Provads Publshed ole a hp://oural.sapub.org/am Copyrgh Scefc & Academc Publshg. All Rghs Reserved wh he las eghee years[-6]. I hs coex, hs paper coues he vesgao o he egevalue problem by movg from -D ad 3-D o -D egevalue problems, ad sees for ay smlares or esseal dffereces bewee fve alerave mehods. he sudy cludes classcal agrage polyomals ad exeds o he Berse polyomals ha are here he defo of Bézer CAD curves[7], as well as o Chebyshev polyomals ha have bee prevously used specral ad collocao mehods (e.g.[8]. I hs paper was foud ha all he aforemeoed polyomals are equvale he sese ha (afer he proper rasformao hey symbolcally cocde wh he classcal hgher order p-mehod (or p-verso[] as well as wh he class { }. Galer/Rz Formulao x (aylor seres... Geeral A geeral Surm-ouvlle problem ca be wre he followg dffereal equao d du px ( + ( rx ( λ q ( x U dx dx I ca be reduced o a sudy of he caocal ouvlle ormal form U + λ q x U ( ( ( Whou loss of geeraly, hs paper we deal wh he parcular case ha q( x, for whch Eq( degeeraes o he well-ow Helmholz equao : U x + λu x, x, ( ( ( [ ]

2 4 C. G. Provads: Equvale Fe Eleme Formulaos for he Calculao of Egevalues Usg Hgher-Order Polyomals Equao ( covers may praccal problems physcs ad egeerg such as he axal elasc vbraos of a beam or he soud propagao alog a sragh acousc ppe. he soluo of Eq( s usually expressed as a seres expaso wo alerave ways: or ( ( U x N x U (3a ( ( U x f x a (3b where f (x are he bass fucos ad a he geeralzed co-ordaes ha refer o + odeless parameers. Also, N (x are he shape fucos ad U he odal dsplacemes a he posos xx,,,. I s remded ha N (x are cardal fucos,.e. N (X δ ( Kroecer s dela ad also paro he uy,.e. for all x [, ] N ( x. here are also some formulaos whch boh shape ad bass fucos parcpae such as hgher order p-mehods[]. I ay case, he Galer/Rz procedure[] cosss frs of he Galer mehod,.e. [ ( λ ( ] W U x + U x dx,,,, wh W ( x N ( x deog he weghg fuco, ad secod by he paral egrao of he secod order erm U ( x, whch fally leads o he alerave marx formulaos: ( K λ M U,,, (4a or ( K λ M a,,, (4b where he elemes of he mass ( MM, ad sffess ( KK, marces are gve by: or ( (, ( ( x (5a m N x N x dx N x N x d ( (, ( ( m f x f x dx f x f x d x (5b.. Bass Fucos ad Shape Fucos Below, occasoally eher he Caresa coordae x [, ] or s ormalzed value u x [,] wll be used. I s well ow ha ay erval [a, b] ca be reduced o[,] by a smple chage of varable x b a u+ a. ( able. agrage polyomals of low degree ( Frs Degree (ear Fe Eleme Accordg o he sadard leraure[], for a fe eleme of legh l, he wo lear shape fucos, ha s N( x xl, N( x xl, are assocaed o he eds x ad x l. I s well ow[] ha he bass fucos relaed o he aforemeoed shape fucos are: f ( x ad f ( x x, ha s U( x, b ( + b ( x.... agraga ype Macroeleme he doma [,] s subdvded o (preferably equdsa segmes, hus leadg o + odes. he, he varable U s globally erpolaed wh he ere doma [,] erms of + agrage polyomals assocaed o he aforemeoed + odes. For a o-decreasg sequece of + pos, x,, x, he agrage poly- omal ( x s defed as: ( x x ( x x ( x x ( x x ( x x + ( x ( x x ( x x ( x x ( x x ( x x + gvg uy a x ad passg hrough pos. ypcal ses of agrage polyomals are show able. herefore, Equao (3a holds wh N ( x,,, deo- g he h agrage polyomal ( N x x. As for each he deomaor Eq(6 s a cosa, whereas he omaor s a polyomal of degree, he laer ca be wre erms of s roos ( ρ x, ρ x,, ρ x, ρ x,, ρ x as follows: ( ( ( u ( u ( u ( u ( u u u ( u ( u 4u( u u( u - u 3u+ 4u + 4u u u 3 ( 3u ( 3u ( u 9u( 3u ( u 9u( 3u ( u u( 3u ( 3u u + 9u u u u + 9u u + 8u u u u + u 4 34 ( u ( u ( 4u 3( u 6 3u( u ( 4u 3 ( u 4u( 4u ( 4u 3( u 6 3u( 4u ( u ( u 3u( 4u ( u ( 4u u u + u u u 8u + 76u u u + 96u u + 6u u + u u + u u 6u + u u (6

3 Appled Mahemacs: ; (: ax + a x + + ax+ a ( ρ ( ρ ( ρ ( ρ ρ ( ρρ ρρ ρ ρ x a x x x a x + + x ( ρρ ρ..3. Berse Polyomals 3 Accordg o sadard compuer-aded-geomery owledge, for example[7], a Bézer curve of -h degree s defed erms of he ormalzed coordae u x as ( ( C u B u P u (7, I sadard mahemacal exs he varable C ( u Eq(7 s called Berse polyomal of degree ad s deoed by ( u [9, p. 36], whereas he bass or B ( bledg fucos, B u, are amed Berse bass, ad are defed as! B ( u u ( u u ( u, (8! (! he geomercal coeffces, { P }, are called corol pos. Equao (8 deoes ha for a curve ha s defed ( f ( x correspod o B ( u x by ( + corol pos, he hghes degree s u, whch meas ha he degree of a Bézer curve s defed by he umber of corol pos. I has bee show ha Berse polyomals have he propery of paro of uy ad also he frs ( B ( u ad he las of hem ( B u gve uy a he frs ( B ( ad he las ed ( B (,,, (, of he erval [,], respecvely. I s remarable ha a ay eral corol po B ( u s less ha he uy, [7], ad also vashes a he edpos u,. I s rval o prove ha, a leas for he case of a sragh segme, whe he erval [,] s uformly dvded o a umber of so-called breapos, he corol pos cocde wh hem. Followg he deas of CAD/CAE egrao prevously proposed by he auhor he coex of closely relaed soparamerc Coos erpolao[7-] as well as he sogeomerc dea of Hughes e al.[4] referrg o NURBS represeao, hs wor we propose o subsue he Caresa coordae x C ( u Eq.(7 by he varable U u. herefore, wh respec o Eq.(3b, he bass fucos ad also he, corol pos P are replaced by he coeffces a...4. Hgher Order P-Mehod Followg Szabó ad Babuša[], he varable s expaded o a seres le ha Eq(3b. Accordg o hs echque [, p.38], he wo frs bass fucos, whch correspod o x ad x, are decal wh he lear shape fucos assocaed o he ed pos of he ere erval [,],.e., f ( x x ad f ( x x ad are called odal shape fucos. he res (- bass fucos are called eral shape fucos ad are ae as suable egrals of egedre polyomals properly mulpled by specfc coeffces depede o he ascedg order. I mus become clear ha he laer refer o odeless quaes (geeral coeffces ad, herefore, hey are also called bubble modes. By elaborag o [], cases of, 3 ad 4, he bubble bass fucos are gve by ( <u < : : f ( u 6u( u (9a 3: f ( u u( u ( u (9b 3 ( 5 5 ( : f ( u 5 4uu u 4 ( u (9c..5. Chebyshev Polyomals Chebyshev polyomals are caegorzed as frs ad secod d. I. Frs d U ( cos( cos x x x x x + x x 4 II. Secod d ( x ( ( 4 {( + cos x} s s cos ( x x x x + x x 3 5 ( ( 4 (a (b..6. Geeral Remars From he above aalyss, becomes obvous ha agrage, Berse, egedre, ad Chebyshev polyomals are dffere opos wh he followg characerscs: I. agrage polyomals are cardal shape fucos ha operae drecly o he odal values, U,,,. II. Berse (bass polyomals are geerally o-cardal bass fucos. he oly excepo are he wo bass fucos, B, ad B, whch are assocaed o he, eds (x, x ; hus hey operae drecly o he odal boudary values ad. he remag coeffces a,,, U U are assocaed o he eral corol pos P. III. egedre polyomals creae bubble bass fucos φ ( u ha operae o odeless geeralzed coeffces, a. IV. Chebyshev polyomals (boh of frs ad secod d dffer from he above cases he sese ha hey eher become u or vash a he eds of he doma.

4 6 C. G. Provads: Equvale Fe Eleme Formulaos for he Calculao of Egevalues Usg Hgher-Order Polyomals A careful speco of all above fve polyomal erpolaos [Eqs.(6,(8,(9 ad (], ad cosderg he well ow bomal expaso ( x + a x a, s easly cocluded ha all hese polyomals ca be expaded a aylor (power seres of he form: u x( u au [ a a a ] u ( u For each of he fve abovemeoed polyomals, dffere coeffces a [Eq(] are derved...7. Specal Cases I order o brg sgh he opc of subseco..6, some specal cases of small sze wll be dscussed full deal. I. Frs, he frs four (agrage, Berse, P-mehod, Chebyshev of s d alerave ypes of polyomals, he case of usg oly oe segme for he dscrezao ( s srcly relaed o he lear erpolao bewee he wo edpos. herefore, he same classcal fe eleme marces, whch are gve below by Eq.(a ad (b, are obaed. For Chebyshev of d d ( U ( x, U ( x x, he facor U ( x duces a slgh modfcao. II. Secod, he case of wo subdvsos, equvale polyomals whch clude erms up o he secod degree, s represeed able. From able, becomes obvous ha all fucoal ses are complee, as hey clude all he erms of he se {, u, u }. Moreover, whle agrage ad Berse (bass polyomals esure he paro of uy (rgd body propery, he p-mehod ad Chebyshev polyomals do o. 3. Aalycally Calculaed Marces Based o Eqs(5, he sffess ad mass marces are aalycally calculaed hrough mapulao ad are show below. 3.. ear Ierpolao ( I hs case, all hree formulaos of hs paper degeerae o he smple case of he coveoal lear fe eleme of he leraure[]. he marces are gve by: l K, M cosse 6 ( l where deoes he legh of he fe eleme. I s l well ow ha accuracy creases whe he oal legh of he doma s dvded o a creasg umber of such elemes (h-verso. I hs case, he oal mass ad sffess marces are obaed from he assemblage of all fe elemes volved []. 3.. Quadrac Approxmao ( he releva marces are:. agrage polyomals: K 8 6 8, M 6 (3a Berse polyomals: K 4, M (3b Hgher order P-mehod: (3c K, M Cubc Approxmao ( 3 Based o Eqs(5, he sffess ad mass marces are gve by:. agrage polyomals: K , M (4a Berse polyomals: K, (4b M Hgher order P-mehod: / /, K / / /3 6 3 /3 6 /5 6 M 3 / 3 /3 6 3 /3 (4c Oe ca oce ha he case of he p-mehod, o symmery exss alog he dagoal erms of he mass marx whch correspod o eral modes ( 5. hs fac ca be explaed o he bass of able 3, where oe ca oce ha he shape fucos N ad N of agrage ad Berse polyomals are symmerc wh respec o he ceral po u/, whle he same does o hold for he

5 Appled Mahemacs: ; (: p-mehod Quarc Approxmao ( 4 M (5b Based o Eqs(5, he sffess ad mass marces are gve by: agrage polyomals: Hgher order P-mehod: / /, K K, / / (5a /3 6 3 /3 M / M 3 / 3 (5c. Berse polyomals: 5 / /3 6 3 / , Oe ca oce ha he case of he p-mehod, aga o K symmery exss alog he dagoal erms of he mass marx whch correspod o eral modes ( Formulao able. Alerave polyomals volved case of (seres expaso s cluded he lower row. Formulao Shape of Bass fucos Sum N or f N or f N or f agrage ( u ( u 4u( u u( u u 3u+ 4u + 4u u u Berse ( u u( u u u u+ u u + u P-mehod -u 6 ( u u u 6 ( u u Chebyshev of s d u u Chebyshev of d d u 4u able 3. Alerave polyomals volved case of 3. u + u 4u + u Shape of Bass fucos N or f N or f N or f N 3 or f 3 3u 3u u 9u 3u u 9u 3u u u 3u 3u agrage ( ( ( ( ( ( ( ( ( Berse ( u 3 3u( u 3u ( u 3 u P-mehod -u 6u( u u( u ( u u 3.5. Chebyshev Polyomal Based Marces Below we prese he marces for he mos geeral case deal hs sudy, whch s for he quarc approxmao ( 4. Obvously, for ay oher m < 4, he upper lef submarx should be ae (,, m., Chebyshev Polyomal of Frs Kd 4 3 8( K 6 3 6( 64 ( ( 9+ 4 ( ( ( 64 ( ( (

6 8 C. G. Provads: Equvale Fe Eleme Formulaos for he Calculao of Egevalues Usg Hgher-Order Polyomals ( 4 ( ( 4 ( 3 5 ( M ( ( ( (5d Chebyshev Polyomal of Secod Kd (4 3, K 8 (64 / 3 6 (3 (64 (8 5 / (3 6 + (64 (9 5 / 5 6 ( (4 3 (64 (8 5 / 5 6 ( (4 ( /35+ 9 (4 / 3 ( + (4 (4 5 / (4 / 3 (8 (6 5 / 5 ( (8 9 / M (4 / 3 + (8 (6 5 / 5 ( (8 9 / 3 + ( ((64 / 7 64 / / ( (8 (6 5 / 5 ( (8 9 / 3 + (64 (5 7 / 35 + (6 / 3 ( (6 8 / (4 (4 5 / 5 ( (8 9 / 3 + ( ((64 / 7 64 / / 3 ( (6 8 / ( ( ((56 / / / 5 8 (5e 3.6. Implemeao of he Boudary Codos I hs wo-po boudary value problem, hree alerave ypes of boudary codos may exs, as follows:. Free-free (F-F boudary codos: U (, U ( (6a. Boh eds uder Drchle (D-D boudary codos: U(, U( (6b. Oe ed uder Drchle, he oher free (D-F: U(, U ( (6c I geeral, he mahemacal problem becomes: de K λm (7 I s well ow ha for small problems Eq(7 ca be solved eve hrough he characersc polyomal, whereas for large-scale problems s usually elaboraed usg ay ow algorhm such as subspace erao, aczos, QR, ec. Deals abou he reame of boudary codos wll be dscussed Seco Numercal Resuls he doma [, ] s subeced o eher of boudary codos (b.c. gve by Equaos (6 a x ad a x. For smplcy, he compuaos were performed for. Accordg o he boudary codos, he exac egevalues are: ( Free-free b.c. (F-F: π λ,,, (8a, exac ( Boh eds uder Drchle b.c. (D-D: π λ,,, (8b, exac ( Oe ed fxed (Drchle b.c., he oher free (D-F: ( π λ,,, (8c, exac 4 All umercal ad symbolc (Symbolc oolbox calculaos were performed o a PC (DE aude E65 usg MAAB 7.. (Ra. Gve he sffess K ad mass marces M, he symbolc egevalues were foud usg he commad eg(v(m*k, whereas he alerave commad eg(k,m wors for he umercal operaos oly. For a gve polyomal of degree, he error orm of he -h calculaed egevalue s deermed as: ( e r λ calculaed, ( λ λ, calculaed, exac,,, (9 λ, exac 4.. Free-Free (F-F Boudary Codos hs case requres o specal care as oe row or colum has o be elmaed. I all sx cases,.e. ( agrage, ( Berse, ( p-mehod, (v Chebyshev of s d, (v Chebyshev of d d, ad fally (v aylor seres, he egevalues were foud o be decal. Moreover, he covergece qualy s excelle (he same qualy wh he resuls pre-

7 Appled Mahemacs: ; (: seed able 4 bu s o preseed for he sae of brefess. 4.. Oe Drchle (D-F or wo Drchle (D-D Boudary Codos For he las wo ypes of boudary codos [D-D ad D-F: Eq(6b,c], ad for he frs hree ypes of polyomals [.e. ( agrage, ( Berse, ad ( p-mehod], he row(s ad colum(s ha correspod o he resrced ed(s s (are smply elmaed. As prevously happeed, all hese hree ypes of polyomals he egevalues were foud o be decal. Moreover, covergece qualy s show able 4, labeled as D-D ad D-F. I coras, case of eher Chebyshev polyomals or aylor seres he way of elmao of row(s ad colum(s s o appare ye. 5. A heorecal Explaao Sce all ypes of polyomals deal hs sudy [.e. ( agrage, ( Berse, ( p-mehod ad releva egedre polyomals, as well as (v Chebyshev of frs ad (v Chebyshev of secod d] spa he bass fucos cluded he class { x } (aylor power seres: (,,, f x x ( becomes ecessary o oba he correspodg sffess K ad mass M marces. I s clarfed ha he- ceforh he upper lef superscrp Eq( wll sad for he word aylor. able 4. Calculaed egevalues usg polyomals of h degree ( uform subdvsos. Resuls are show as errors ( % for wo suppor codos: (a boh eds are fxed (D-D, ad (b oe ed s fxed ad he oher s free (D-F. MODE Exac ω Eq(7a (a : D-D ERRORS (% Degree of polyomal MODE Exac ω Eq(7b (b : D-F ERRORS ( % Degree of polyomal

8 C. G. Provads: Equvale Fe Eleme Formulaos for he Calculao of Egevalues Usg Hgher-Order Polyomals 5.. he Class { x } I hs case he varable U s expaded a aylor seres: U( x a x a,, a ( x Accordg o Eq.(5b, he elemes of he mass ad sffess marces wll be gve by: ( m ( + + ( + + ( +,,, +,,,, 5.. A Proof for he Idecal Egevalues ( For a sgle square marx A of order (+, Ralso ad Rabowz[9, p. 484] (heorem.3 have demosraed ha ay smlary rasformao PAP appled o A leaves he egevalues of he marx uchaged. he same s foud he exboo of Bahe [, p. 45] where, provded P s orhogoal, a poof s gve hrough he characersc polyomal. I hs paper we geeralze he proof o ay par of sffess K ad mass M marces (heorem 3; he upper lef superscrp aes he symbolc values, l, b, p as well as c ad c, whch correspod o he als of he sx alerave polyomals (aylor seres, agrage polyomals, Berse bass, P-mehod/egedre, as well as Chebyshev of s ad d d. heorem ad heorem am o suppor heorem 3. HEOREM-. he chage of bass bewee a bass polyomal f ( x,,, [such as agrage, Berse, P-mehod, Chebyshev e ceera], ad he class { x } (aylor seres, duces a relaoshp of quadrac form bewee he mass marces M ad M. PROOF. Frs, we cosder he bass vecors of order (+, for boh abovemeoed fucoal ses: f ( x N, N (3 x f ( x Based o Eq(3, he correspodg mass marces [cf. Eq(5b] are gve compac form as follows: ( x ( M N N d, M N N dx (4 Secod, we cosder he equvale expressos of U(x usg eher aylor seres or he -ype polyomal bass, for whch holds: U ( x ( ( ( ( a N a N (5 Sce U(x s a scalar, equals o s raspose, herefore: U ( x ( N ( a ( N ( a (6 ef-mulplcao of Eq(5 by Eq(4 pars, leads o: [ U( x ] ( ( ( ( a N N a (7 ( a ( N( N ( a Iegrag he las wo pars of Eq(7 over he erval [,], ad cosderg ha he vecors ( a ad ( a are cosa (so hey ca ex he egral, we oba: ( a ( N( N dx ( a (8 ( a ( N( N dx ( a Subsug Eq(4 o Eq(8, oe obas he desred marx dey: ( ( ( ( a M a a M a (9 Equao (9 cosues he proof of he heorem. Remar: I s remded ha Eq(9 s he well ow marx rasformao (from local o global sysem he fe eleme praxs (e.g., [,]. Neverheless, hs sudy oly mahemacal cosderaos have bee ae o accou, whereas egeerg boos s usually derved o he bass of vrual wor (he laer s cosdered o be varable boh local ad global orhogoal co-ordae sysems. I coras, hs wor o orhogoaly relaoshp bewee he dffere bases was cosdered. HEOREM-. he coes of heorem- s exeded o he sffess marces K ad K as well. PROOF. ag he dervaves x of he vecors Eq(3, we oba he wo ew vecors: ( f x ( N ( N, N N (3a x x x x x f x ( herefore, he correspodg sffess marx [Eq(5b] s wre as: ( K N N ( K N N d, x dx (4a Heceforh he proof s he same as heorem-, leadg o he dey: ( ( ( ( a K a a K a (3 HEOREM-3. For ay par of marces ( K, M, whch ( correspods o a bass polyomal f x,,,, he correspodg egevalues ( λ are decal wh he egevalues ( s produced by he class λ,,, of he par ( K, M ha { x } (aylor seres.

9 Appled Mahemacs: ; (: 3-3 PROOF. Cosderg he sffess K ad mass M marces [Eq(], he referece egevalue problem becomes ( a sads for he coeffces aylor seres: K λ M a (3 ( (,,, Furhermore, cosderg he sffess K ad mass M marces, he ew egevalue problem becomes ( a sads for he correspodg coeffces -ype polyomal bass: ( ( λ,,, K M a By lef-mulplyg Eq(3 by ( a we oba: ( ( ( λ,,, a K λ M a (35 Fally, subracg Eq(34 ad eq(35 pars, oe obas: ( λ ( λ ( ( (3 a K M a (33 Equaos (9 ad (3 of heorems ad, respecvely, are geerally applcable ad herefore hey hold eve for he -h egevecor. Uder hese crcumsaces, subsug hem o Eq(33 we oba: ( ( ( a K λ M a,,, By lef-mulplyg Eq(4 by ( a, oe obas: ( ( (,,, (34 a M a (36 Sce he quadrac form cao vash, s cocluded ha: ( λ ( λ ( ( a M a (37 whch cosues he proof of he heorem. λ λ λ Corollary. he decal egevalues ( ( ca be expressed by commo Raylegh quoes: ( a ( M K ( a λ a a ( ( ( a ( M K ( a ( a ( a,,, 5.3. Applcao of Arbrary Boudary Codos (38 Whaever follows s geerally applcable bu for he purposes of hs paper has parcular value for aylor seres ad Chebyshev polyomals, whch do o drecly apply for he boudary codos. Equao (5 s collocaed a ( + pos wh he erval [,]. Oe possble, bu o resrcve possbly s o uformly dvde he doma [,] o segmes, hus roducg he breapos: ( x, x,, x, x (39 herefore, we derve a relaoshp he form: ( ( U f x f x a U (4 U f ( x f( x a [ A] whch vecor form s wre as: or equvalely: { } [ ] { } a U A a (4a { } [ ] { } a A U (4 Subsug Eq(4 o Eq(35 we derve: where ( * λ * ( ( K M U (4 M ( A M ( A * * K A K A, ad (43 I hs framewor, s ow rval Eq(4 o elmae he row ad colum ha correspods o he Drchle boudary codo. 6. Dscusso he movao of hs wor was he eresg paper by Çel[8] ha deals wh he collocao mehod usg Chebyshev polyomals, whch cosue a complee fucoal se. Prevously, he sae-of-he-ar was he use of bass fucos he form of algebrac polyomals he form f x a x ρ x ρ x ρ, wh ρ a ad ( ( ( ( ρ b, so as o esure sasfaco of he homogeeous Drchle boudary codos he erval [a,b] (for example, []. A ha perod he auhor had accumulaed umercal experece o he excelle behavor of global approxmao, eher usg B-sples or agrage polyomals ([7-], amog ohers. aer, whe he red o compare he egevalues obaed usg ( agrage polyomals, ( Berse (bass oes, ad ( aylor seres, a umercal cocdece was remared whe he same collocao pos were used []. he abovemeoed umercal cocdece pushed he auhor o exed hs research from collocao o he mos popular fe eleme mehod whch closed form aalycal formulas of he marces ca be derved. I hs formulao, o oly he prevously foud cocdece[] was repeaed, bu also was furher foud ha he famous P-mehod[] s also decal wh all ohers. Alhough oe could smply sae ha all hese polyomals (agrage, Berse, egedre, ad Chebyshev have he same basc bass, whch s he class { } x (aylor seres, he possble saeme ha he Raylegh quoe s he same has o be mahemacally prove. Aoher eresg po s he way ha he boudary codos are mposed. As prevously meoed, agrage polyomals are assocaed drecly o he odal values of he

10 C. G. Provads: Equvale Fe Eleme Formulaos for he Calculao of Egevalues Usg Hgher-Order Polyomals prmary varable U all-over he doma, whereas he Berse (bass polyomals refer o corol pos. However, sce he exreme corol pos, P ad P, cocde wh he wo eds, here s o problem o mpose free-free or Drchle boudary codos. Smlarly, he p-mehod s based o he well ow lear shape fucos assocaed o he eds of he erval [a,b], ad smply s fucoal se s erched by he odeless bubble fucos. I coras o he wo aforemeoed polyomals, smlar o he aylor seres, Chebyshev polyomals have a more specral characer, as hey refer o arbrary coeffces sead of he pure values of he varable U,.e. U ad U. I s worhy o meo a commo characersc smlary bewee Chebyshev polyomals ad P-mehod. I more deals, as he degree of he polyomal creases, he sffess marx K ad mass max M (boh of order (+ ca be mmedaely derved from he prevously calculaed submarces K - ad M - (boh of order, by smply compleg he (+-h row ad he (+-h colum ( s remded ha Seco he dces vary bewee,,. Despe he cocdece foud oe-dmesoal problems, prelmary comparsos wo-ad hree-dmesoal problems sugges ha here s a sgfca dfferece bewee agraga ype fe elemes ad he p-mehod, as he frs occupy a broader space ha he secod oes [9,]. 7. Coclusos I hs wor, he ere oe-dmesoal doma [,] was cosdered as a sgle macroeleme (global approxmao, a he eds of whch hree (all possble dffere ypes of boudary codos were mposed. Also, for he spaal approxmao of he varable U fve dffere ypes of global hgher order polyomals were cosdered. I was foud ha all hese fve bass fucos,.e. ( classcal agrage polyomals, ( Berse polyomals (useful CAD curve represeao, ( bubble fucos based o egedre polyomals (p-mehod, as well as Chebyshev polyomals of (v frs d ad (v secod d, share he same fucoal space,.e. {, x,, x }. Alhough he coeffces of seres expasos ad releva marces hghly deped o he parcular polyomal chose, he calculaed egevalues were foud always he same, a fac ha was also rgorously prove ally case of free-free ad he for arbrary boudary codos. REFERENCES [] Zeewcz, O. C., 977, he Fe Eleme Mehod, hrd ed., McGraw-Hll, odo [] Szabó, B., ad Babuša, I., 99, Fe Eleme Aalyss, Joh Wley & Sos, Ic., New Yor [3] Szabó, B. A., ad Meha, A. K., 978, p-coverge fe eleme approxmaos fracure mechacs, Ieraoal Joural for Numercal Mehods Egeerg, (3, [4] Babuša, I., Szabó, B. A. ad Kaz, I. N., 98, he p-verso of fe eleme mehod, SIAM Joural o Numercal Aalyss, 8(3, [5] Zeewcz, O. C., Gago, JPdeSR, ad Kelly, D. W., 983, he herarchcal cocep fe eleme aalyss, Compuers & Srucures, 6(-4, [6] Frberg, O., Möller, P., Maovča, D., ad Wberg, N. E., 987, A adapve procedure for egevalue problems usg he herarchcal fe eleme mehod, Ieraoal Joural for Numercal Mehods Egeerg, 4(, [7] Provads, C. G., 6, rase elasodyamc aalyss of wo-dmesoal srucures usg Coos-pach macroelemes, Ieraoal Joural of Solds ad Srucures, 43(-3, [8] Provads, C. G., 6, Free vbrao aalyss of wo-dmesoal srucures usg Coos-pach macroelemes, Fe Elemes Aalyss ad Desg, 4(6, [9] Provads, C. G., 6, hree-dmesoal Coos macroelemes: applcao o egevalue ad scalar wave propagao problems, Ieraoal Joural for Numercal Mehods Egeerg, 65(, -34 [] Provads, C. G., 6, Coos-pach macroelemes wo-dmesoal parabolc problems, Appled Mahemacal Modellg, 3(4, [] Provads, C. G., 9, Egeaalyss of wo-dmesoal acousc caves usg rasfe erpolao, Joural of Algorhms & Compuaoal echology, 3(4, [] Clar, B. W., ad Aderso, D. C., 3, he pealy boudary eleme mehod for combg meshes ad sold models fe eleme aalyss, Egeerg Compuaos, (4, [3] Corell, J. A., Real, A., Bazlevs, Y., ad Hughes,. J. R., 6, Isogeomerc aalyss of srucural vbraos, Compuer Mehods Appled Mechacs ad Egeerg, 95(4-43, [4] Hughes,. J. R., Corell, J. A. ad Bazlevs, Y., 5, Isogeomerc aalyss: CAD, fe elemes, NURBS, exac geomery ad mesh refeme, Compuer Mehods Appled Mechacs ad Egeerg, 94(39-4, [5] Ioue, K., Kuch, Y., ad Masuyama,., 5, A NURBS fe eleme mehod for produc shape desg, Joural of Egeerg Desg, 6(, [6] Schramm, U., ad Pley, W. D., 993, he couplg of geomerc descrpos ad fe elemes usg NURBS A sudy shape opmzao, Fe Elemes Aalyss ad Desg, 5(, -34 [7] Pegl,., ad ller, W., 997, he NURBS Boo, d edo, Sprger, Berl [8] Çel, I., 5, Approxmae calculao of egevalues wh he mehod of weghed resduals-collocao mehod, Appled Mahemacs ad Compuao, 6(, 4-4 [9] Ralso, A., ad Rabowz, P.,, A Frs Course Numercal Aalyss, Dover Publcaos, Meola, New Yor

11 Appled Mahemacs: ; (: [] Bahe, K. J., 98, Fe eleme procedures egeerg aalyss, Prece-Hall, Eglewood Clffs, New Jersey [] Flayso, B. A., 97, he Mehod of Weghed Resduals ad Varaoal Prcples, Academc Press, New Yor [] Provads, C. G., 8, Free vbrao aalyss of elasc rods usg global collocao, Archve of Appled Mechacs, 78(4, 4-5