A Review on Attempts towards CAD/CAE Integration Using Macroelements

Transcription

1 Computatoal Research (3): 6-84, 3 DOI:.389/cr A Revew o Attempts towards CAD/CAE Itegrato Usg Macroelemets Chrstopher Provatds Natoal Techcal Uversty of Athes, Greece *Correspodg Author: cprovat@gmal.com Copyrght 3 Horzo Research Publshg All rghts reserved. Abstract I ths paper we revew several aspects of older ad cotemporary attempts to tegrate computer-aded desg (CAD: geometrc model) ad computer-aded egeerg (CAE: fte elemets, boudary elemets, etc.). After a short revew o formulas for the descrpto of CAD surfaces, a systematc mechasm for creatg several types of correspodg soparametrc macroelemets s preseted. Gordo-Coos s tally appled coucto wth pecewse lear, Lagrage polyomals ad atural B-sples. The, t s exteded to more bass ad bledg fuctos. I addto to the well-kow Lagraga -type elemets, equvalet Bézera -type elemets are troduced. Tesor product B-sples ad aspects of NURBS sogeometrc formulato are gve. I addto to quadrlaterals, tragular macroelemets based o Barhll s terpolato are preseted for the frst tme. The revew covers applcatos of CAD-based macroelemets coucto wth the Galerk-Rtz formulato, the Boudary Elemet Method, as well as recet Global Collocato procedures. A umercal example o a vbratg membrae elucdates the performace of the CAD-based global terpolato ad depcts ts superorty over the covetoal fte elemet method. Keywords CAD, CAE, Global Iterpolato, Fte Elemet, Boudary Elemet, Collocato Techques. Itroducto Computer-Aded Desg (CAD) s the use of computer systems to assst the creato, modfcato, aalyss, or optmzato of a desg. Oe the earlest persoaltes who have cotrbuted ths area, ad had a vso of teractve computer graphcs as a powerful desg tool, were Steve Aso Coos (9 979), a professor the Mechacal Egeerg Departmet at Massachusetts Isttute of Techology (MIT) durg the 95s ad 96s. Durg World War II, he worked o the desg of arcraft surfaces, developg the mathematcs to descrbe geeralzed surface patches. At MIT s Electroc Systems Laboratory he vestgated the mathematcal formulato for these patches, ad 967 publshed oe of the most sgfcat cotrbutos to the area of geometrc desg, a treatse whch has become kow as The Lttle Red Book []. Hs Coos Patch was a formulato that preseted the otato, mathematcal foudato, ad tutve terpretato of a dea that would ultmately become the foudato for other surface descrptos that are commoly used today, such as B-sple surfaces, NURB surfaces, etc. Hs techque for descrbg a surface was to costruct t out of collectos of adacet patches, whch had cotuty costrats that would allow surfaces to have curvature whch was expected by the desger. Each patch was defed by four boudary curves, ad a set of bledg fuctos that defed how the teror was costructed out of terpolated values of the boudares. The terested reader may fd more detals []. Computer-Aded Egeerg (CAE) s the broad usage of computer software to ad egeerg tasks. It cludes computer-aded desg (CAD), computer-aded aalyss (CAA), computer-tegrated maufacturg (CIM), computer-aded maufacturg (CAM), materal requremets plag (MRP), ad computer-aded plag (CAP). I a more strct sese, CAE areas clude: Stress aalyss o compoets ad assembles usg FEA (Fte Elemet Aalyss); Thermal ad flud flow aalyss Computatoal flud dyamcs (CFD); Multbody dyamcs (MBD) & Kematcs; Aalyss tools for process (maufacturg) smulato for operatos such as castg, moldg, ad de press formg. Optmzato of the product or process. Safety aalyss of postulate loss-of-coolat accdet uclear reactor usg realstc thermal-hydraulcs code. I geeral, whe omttg the abovemetoed plag tasks, there are three phases to whch ay CAE procedure s dvded:

2 6 A Revew o Attempts towards CAD/CAE Itegrato Usg Macroelemets Pre-processg defg the model ad evrometal factors to be appled to t (typcally a fte elemet model, but facet, voxel ad th sheet methods are also used). Aalyss solver (usually performed o hgh powered computers). Post-processg of results (usg vsualzato tools). Ths cycle s terated, ofte may tmes, ether maually or usg bult- procedures, or fally by lkg the etre procedure to a commercal optmzato software such as [3, 4]. Heceforth, CAE wll restrct to the fte elemet (FEM/FEA) ad the boudary elemet (BEM) aalyss, whereas a very recet Global Collocato Method (GCM) wll be revewed ad commeted. It s remded that FEM decomposes the model of the structure small areas or volumes, called fte elemets [5, 6], whereas BEM deals mostly wth the boudary of the structure [7]. It s accepted that the poeers the FEA area are Joh HadArgyrs (93 4), Olgerd Cecl Zekewcz (9 9) ad Ray Clough (9 today). Some persoal vews are [8-]. The decade of 96s was very mportat for desgers, as sce the they have log used computers for ther calculatos. Ital developmets were carred out the 96s wth the arcraft ad automotve dustres the area of 3D surface costructo ad NC programmg, most of t depedet of oe aother ad ofte ot publcly publshed utl much later. Some of the mathematcal descrpto work o curves was developed the early 94s by Isaac Jacob Schoeberg, Apalatequ (Douglas Arcraft) ad Roy Lmg (North Amerca Arcraft), however probably the most mportat work o polyomal curves ad sculptured surface was doe by Perre Bézer (Reault), Paul de Castelau (Ctroe), Steve Aso Coos (MIT, Ford), James Ferguso (Boeg), Carl de Boor (GM), Brkhoff (GM) ad Garabeda (GM) the 96s ad W. Gordo (GM). It s argued that a turg pot was the developmet of SKETCHPAD system MIT 963 by Iva Sutherlad [3], who was a studet of S.A. Coos; however hs PhD thess was supervsed by C.E. Shao (96 ), the "father of formato theory". The dstctve feature of SKETCHPAD was that t allowed the desger to teract wth hs computer graphcally: the desg ca be fed to the computer by drawg o a CRT motor wth a lght pe. Effectvely, t was a prototype of graphcal user terface (GUI), a dspesable feature of moder CAD. Frst commercal applcatos of CAD were large compaes the automotve ad aerospace dustres, as well as electrocs. Oly large corporatos could afford the computers capable of performg the calculatos. Notable compay proects were at GM (Dr. Patrck J.Haratty) wth DAC- (Desg Augmeted by Computer) 964; Lockhead proects; Bell GRAPHIC ad at Reault (Bézer) UNISURF 97 car body desg ad toolg. Oe of the most fluetal evets the developmet of CAD was the foudg of MCS (Maufacturg ad Cosultg Servces Ic.) 97 by Dr. P. J. Haratty[4], who wrote the system ADAM (Automated Draftg Ad Machg) but more mportatly suppled code to compaes such as McDoell Douglas (Ugraphcs), Computer vso (CADDS), Calma, Gerber, Autotrol ad Cotrol Data. As computers became more affordable, the applcato areas have gradually expaded. The developmet of CAD software for persoal desk-top computers was the mpetus for almost uversal applcato all areas of costructo. Other key pots the 96s ad 97s would be the foudato of CAD systems Uted Computg, Itergraph, IBM, Itergraph IGDS 974 (whch led to Betley McroStato 984) CAD mplemetatos have evolved dramatcally sce the. Itally, wth D the 97s, t was typcally lmted to producg drawgs smlar to had-drafted drawgs. Advaces programmg ad computer hardware, otably sold modelg the 98s, have allowed more versatle applcatos of computers desg actvtes. Key products for 98 were the sold modellg packages -Romulus (ShapeData) ad U-Sold (Ugraphcs) based o PADL- ad the release of the surface modeler CATIA (Dassault Systemes). Autodesk was fouded 98 by Joh Walker, whch led to the D system AutoCAD. The ext mlestoe was the release of Pro/ENGINEER 988, whch heralded greater usage of feature-based modelg methods ad parametrc lkg of the parameters of features. Also of mportace to the developmet of CAD was the developmet of the B-rep sold modelg kerels (eges for mapulatg geometrcally ad topologcally cosstet 3D obects) Parasold (ShapeData) ad ACIS (Spatal Techology Ic.) at the ed of the 98s ad begg of the 99s, both spred by the work of Ia Brad. Ths led to the release of md-rage packages such as SoldWorks 995, SoldEdge (Itergraph) 996, ad IroCAD 998. Nowadays CAD s oe of the ma tools used desgg products. I order to aalyze mechacal compoets, whch are usually characterzed by complex shapes, the computer-aded-desg (CAD) model s usually exported the form of a eutral fle (IGES, DXF, STEP) that s further mported by a selectve fte elemet (FEM) or boudary elemet (BEM) code. Ths procedure duces some dffcultes such as the appearace of double pots (or double les) or loss of at least a few geometrcal data. Of course, specfc CAD coverters such as PATRAN ca export may types of FEM data that ca be mmedately feed commercal FEM codes. Specalzed CAD/CAM/CAE tegrated systems such as PRO/ENGINEER, IDEAS ad Dassault Systèmes (SOLIDWORKS, PLM: Product Lfecycle Solutos) exst for the last or 5years. However, all these tools the computatoal mesh geerato s a tme-cosumg task that has to follow the creato of the CAD model. A survey of thrty-four fte elemet systems utl 98 was reported by Brebba [5]. The term CAD/CAE tegrato may have multple meags, whch wll be better uderstood through a example. Suppose that a bcycle desger assgs a tetatve

3 Computatoal Research (3): 6-84, 3 63 tube sze for the skeleto of the structure (CAD); the, a tegrated CAD/CAE system has to automatcally uderstad ot oly the dameter ad the thckess but also the secod momet of erta of ths tube, wthout the egeer havg to perform the relevat umercal calculatos. Closely related, the egeer ca geerate the mesh wthout extg the CAD system ad the eterg the CAE system; other words, both systems are tegrated oe, sharg the same database as s possble. A secod pot of vew s the usually reported fact that the CAD/CAE tegrato ad scetfc vsualzato s related to both the reducto of errors the formato trasfer from system to system as well as the reducto of memory resources ad overall computatoal tme [6, 7]. The vehcle for CAD/CAE tegrato s to use a commo platform to descrbe the geometry ad the ukow varable that satsfes a partal dfferetal equato (PDE) of a mult-physcs boudary-value-problem (e.g. heat trasfer, elastostatcs, thermo-elastcty, flud mechacs, electromagetc, etc). A thrd pot of CAD/CAE tegrato s to replace the usual fte elemets (of small-sze) wth others of larger sze applyg the same terpolato for both the geometry ad the varable. Relevat macro-elemets are usually of soparametrc type whereas sogeometrc oes have appeared [8]. The lack of a complete, sythetc ad crtcal report o the abovemetoed thrd pot of CAD/CAE tegrato, whch s drectly related to the developmet ad use of macroelemets, has motvated the author to wrte ths revew artcle. The paper s structured as follows. Secto summarzes the oe-dmesoal (uvarate) terpolato as well as the fve basc CAD formulatos. Secto 3 revews the usual applcatos whch Coos terpolato has bee used, as well as the dervato of global shape fuctos. Secto 4 provdes detals about the soparametrc or sogeometrc-lke approxmato that s used for the terpolato of both the geometry ad the varable or the ukow coeffcets. Secto 5 revews the three possble computatoal methods to be appled coucto wth the global terpolato wth subregos, ad provdes the umercal procedure for the estmato of mass ad stffess matrces of the structure. Secto 6 presets orgal umercal results cocerg the extracto of atural frequeces for a fxed rectagular membrae. Secto 7ad Secto 8are the Dscusso ad the Coclusos, respectvely.. CAD Surface Iterpolatos.. Geeral Hstorcally, there are rather fve ma terpolato formulatos CAD theory [9]: () Coos terpolato, () Gordo-Coos terpolato, () Bézer terpolato, (v) B-sples terpolato, ad (v) NURBS. The frst two of them drectly refer to the coordates of odal pots that belog to the boudary or the teror of a patch, whereas the last three refer to cotrol pots (at the edges of polygoal geerators that geerally do ot belog to the curvlear boudary). I all fve cases the terpolato of geometry refers to the coordates of ether odal pots or cotrol pots multpled by proper bass fuctos. For the sake of completeess, we start wth oe-dmesoal ad the exted to two- ad three-dmesoal terpolato... Oe-Dmesoal Iterpolato Let us cosder the doma a x b whch we wsh to terpolate the fucto f(x). Amog others, the most usual terpolatos of the fucto f(x) are: () Power seres, () Lagrage polyomals, () Legedre polyomals (p-method), (v) Chebyshev polyomals, (v) Berste polyomals (Bézer curve), (v) B-sples, ad (v) NURBS. All of them wll be brefly descrbed below.... Power Seres The fucto s approxmated by: f( x) = a ax ax ax =... Lagrage Polyomals The fucto s approxmated by: () = () f x L x f x L x f = = where L ( x ) deotes the well-kow Lagrage polyomal (see, for example, [])...3. P-Method The relevat seres expaso cludes the odal values at the eds, f(a) ad f(b) for whch the classcal lear hat fuctos are cosdered,.e.: N ( x) = xl ad N ( x) = xl where L = b a. I addto, we defe a certa polyomal degree p, whch leads to a umber of fuctos φ are defed terms of the Legedre polyomal P : φ x = P t dt = x,, 3, (3) It s remded that the Legedre polyomals are gve by m d m P ( x) = ( x m m m ),(4) m! dx ad they are solutos of the ordary dfferetal operato: x y xy m m y = (5)

4 64 A Revew o Attempts towards CAD/CAE Itegrato Usg Macroelemets It s also remded that the well-kow Gauss pots are roots of the above Legedre polyomals. The bass fuctos N, N are called odal shape fuctos or exteral shape fuctos or exteral modes. The bass fuctos N = φ, 3, 4,, p = are called teral shape fuctos or teral modes, or, sometmes, bubble modes. Therefore: p f x N x f N x f N x a = (6) = 3 The advatage of these fuctos s that they are orthogoal thus leadg to baded mass ad stffess matrces [, pp ]...4.ChebyshevPolyomals Although these polyomals do ot drectly appear CAD formulatos, t s structve to meto that they ca approxmate a fucto wth the same accuracy wth the power seres. They are categorzed polyomals of frst ad secod kd. For example, the Chebyshev polyomals of the secod kd are defed as: U m ( x) {( m ) cos x} s = s cos ( x) m m m m = x x ( x ) (7) 3 m m 4 x ( x ) 5 ad ther roots are gve by ˆ π ξ = cos, =,, m. (8) m As we wll see later, the above roots ca be used as collocato pots a global collocato method, whch s a alteratve to the Galerk-Rtz procedure [, 3]...5. Berste Polyomals (Bézer curve)..5.. Noratoal Bézer Curve A curve of -th degree s defed as C u = B u P u (9) =, where ( u) = [ x( u) y( u) ] T C are the Cartesa coordates o the xy-plae. The bass fuctos, B ( u ), are the classcal Berste, polyomals that are defed as! B u = u u = u u,!! ( ) () The geometrcal coeffcets, { P }, are called cotrol pots. Equato () deotes that for a curve that s determed by ( ) cotrol pots, the term of hghest degree s u. Ths mples that the polyomal degree of Bézer curve s determed by the umber of cotrol pots. The basc propertes of a Bézer curve are may ad ca be foud elsewhere [4-7]. I bref: The curve passes through the frst (Ρ ) ad the last (Ρ ) cotrol pots ad ts taget at these eds has the drecto of the frst straght segmet (Ρ Ρ ) ad the last oe (Ρ - Ρ ), respectvely, of the cotrol polygo (Ρ Ρ Ρ - Ρ ). The Berste polyomals are oegatve: B u, They have the rgd-body property (partto of uty: B ( u) =, u [,] )., They = are smaller tha ut except of those assocated to the eds where they equal to uty, ad appear a maxmum the terval [,] at u =. ta= They are symmetrc wth respect to the posto u =,. They are rapdly calculated by the followg recursve formula: = ( ) ( <, > ) B u u B u ub u,,,, B u,.() Ther dervatve s also calculated by the followg recursve formula: ( u) db, B ( u) = =., ( B, ( u) B, ( u) ), du B u B u,, () It s also remarkable that: The tme-cosumg computato of the -combatos s smplfed by usg Castelau s algorthm. Also, the terpolato of Bézer curve s mathematcally equvalet wth the power seres (), for the same value of. Smlarly, both the p-method ad the Berste polyomals are equvalet wth the Lagrage polyomals [see, ()], of course for the same value of Ratoal Bézer Curve Sce the cocal surfaces,.e. crcles, ellpses, hyperbolas, cylders, coes, spheres, et cetera, are ot descrbed precsely by meas of polyomals, but requre the use of ratoal fuctos, the form: X u Y u x( u) =, y( u) =, (3) W u W u

5 Computatoal Research (3): 6-84, 3 65 t became ecessary to troduce the -th ratoal Bézer curve, whch s defed as: or = {, } C u x u y u where {, } B u w P x y, = = u..6. B-Sples =, B u w =,, (4) C u = R u P, u ( (5) B u w =,, R u u, =..6.. Older Deftos, B u w (6) The meag of sples was publshed 946 ad later by Schoeberg [8]. It refers to the pots ( x, f ), ( x, f ),, ( x, f ) wth x < x < < x, whch we wsh to terpolate through a multply-defed fucto f ( x ). The pots x, x,, x are called breakpots. For the sake of brefess ad wthout loss of geeralty, we reduce to (cubc) polyomals of thrd degree. The desred propertes are as follows: I each terval x x x, wth =,,,, f x s a cubc polyomal. the fucto The fucto f x, as well as the frst ad secod dervatves, are cotuous at the above pots. Itroducg the trucated power as:, m x x x x =, (7) m ( x x ), x > x whch has C (m-) -cotuty, the orgal expresso cossts of a power seres the form [8]: 3 f ( x) = a a x a x a x b x x 3 (8) = It s apparet that (8) cludes (3) terms ad esures C -cotuty, because for smplcty we cosdered m = 3. Alteratvely, (8) ca be modfed so as to clude addtoal trucated polyomals of secod degree,.e.: f x = a ax ax ax b x x c x x = = 3, (9) Obvously, (9) cludes () terms ad esures C -cotuty Cotemporary Procedures The breakthrough was made 97, depedetly by Cox [9] ad DeBoor [3], who both acheved the B-sples ad ther dervatves to be rapdly computed through recursve formulas [cludg () ad ()]. Charles DeBoor [6] cotued hs research ad hs tal FORTRAN codes exst today MATLAB (sple toolbox) uder the ame spcol, amog others [3]. I bref, for a odecreasg set of () breakpots ( x, x,, x, x), ad for a cocrete polyomal degree p, we defe the kot vector U, whch strogly depeds o the multplcty of the teral odes, as follows: x, x, x, x, x,, x, multplcty = ( p ) terms ( p ) terms U = x, x, x, x,, x, x, x,, x, multplcty = ( p ) terms terms terms ( p ) terms () Based o ether of the kot vectors of (), (m) cotrol pots (Ρ Ρ Ρ m- Ρ m ) are costructed, so as the followg relatoshp always holds: m = p () Let us cosder a odecreasg sequece U = { u,, u } of real umbers, for example, m u u, =,, m. The elemets u are called kots, ad U s the kot-vector. The -th B-sples bass fuctos N u, s defed as: of degree p (order p), deoted by, p f u u < u N, ( u) = otherwse u u u u N u N u N u p =, p, p, p u u u u p p () Cotemporary Versus Older Deftos Some youg readers may beleve that () s dfferet tha (8) or (9). However, for example case of p = 3 (cubc sples), t s trval to prove that: Whe the multplcty equals to oe, the (3) bass fuctos of () are equvalet (ot detcal) to those bass fuctos gve by (8). Whe the multplcty equals to two, the () bass fuctos of () are equvalet (ot detcal) to those bass fuctos gve by (9). Whe there are o teral pots ( = ), B-sples cocdes wth p-thbézer.

6 66 A Revew o Attempts towards CAD/CAE Itegrato Usg Macroelemets Therefore, a lot of research coducted 96s usg the older framework ((8) ad (9)) may be repeated o the ew framework of ()...7.NoUform Ratoal B-Sple (NURBS) Curve Smlarly to the ratoal Bézer curve [see, (6)], o uform ratoal B-sples (NURBS) s a exteso of the B-sples the followg form: =, p N uwp, p = C ( u) = (3) N uw where {w } are the weghts. Usually, a =, b = ad w > for all. Takg the ratoal bass fuctos as, R, p ( u) the curve s expressed as: = N =, p N =, p u w uw, (4) C( u) = R ( uw ) P (5), p ad they are pecewse ratoal fuctos the terval u,. [ ]..8. Comparso betwee Dfferet Sets Of Bass Fuctos It s well kow that power seres s detcal wth Lagrage polyomals [4, pp ]; the oly dfferece s that the frst cludes arbtrary coeffcets, whereas the secod cardal shape fuctos (of [,]-type) that are multpled by the assocated odal values. Chebyshev polyomals are also multpled by arbtrary coeffcets ad they may have some advatages whe collocatg at ther roots [, 3]. The use of Legedre polyomals s somehow dfferet. The relevat seres expaso cludes the odal values at the eds, f(a) ad f(b) for whch the classcal lear hat N x = xl ad fuctos are cosdered,.e.: N( x) = xl where L b a =. I addto, a certa desred umber of bubble fuctos, the form of dffereces betwee Legedre polyomals, are usually used. The advatage of these fuctos s that they are orthogoal thus leadg to baded matrces [, pp ]. Berste polyomals have bee used the costructo of the well-kowbézer curves, ad ca be foud may textbooks such as [5-7]. B-sples s othg ew but the usual sple terpolato, whch ca be foud every textbook of umercal aalyss such as [], whereas detaled formato s gve elsewhere [7-3]. The superorty of B-sples s that t ca hadle every hgh umber of data, cotrast to Lagrage polyomals that caot hadle more tha te to twelve pots at maxmum. Fally, NURBS s a exteso of B-sples as t uses weghts for every cotrol pot. As oe ca otce Fg., whle Lagrage polyomals may exceed the uty ad be characterzed by hgh values partcularly the eghbourhood of the eds, Berste polyomals (or eve geeral B-sples, ot show) are always less tha or equal to uty (at the ed odes). It s worth-metog that, both cases, the sum of the shape fuctos s equal to uty (rgd-body property, partto of uty). Smlarly, the p-method the sum of the frst ad last bass fuctos (N ad N, assocated to the eds of the doma) s equal to uty, whereas the termedate bubble fuctos, φ, vash at the eds but ther sum s dfferet tha zero at every teral pot (see, []). A sde ascertamet o the equvalecy betwee the three fuctoal sets power seres,.e. Lagrage polyomals, Berste polyomals ad the p-method s the umercal fdg that the calculated egevalues are detcal [3, 33]..3. Two-Dmesoal (D) Iterpolato I bref, surfaces are categorzed to quadrlateral ad tragular patches,.e. wth four ad three sdes, respectvely. The terpolatos ca treat ether straght or curvlear sdes of the patch. The oldest way to create a D patch s Coos terpolato []. Later, tesor products of Bézer, B-sples ad NURBS were appled [9, 7]..3.. Quadrlateral Patches We dstgush two categores. The frst deals wth data alog the four boudares of the quadrlateral; other words, t cossts the boudary-oly formulato wth odal pots alog the boudary oly. The secod formulato uses the aforemetoed boudary data as well as addtoal data the teror of the quadrlateral; therefore t cludes teral odes as well, ad s called trasfte terpolato..5 Bezer, p = 7, C 6 -cotuty Lagrage-polyomals, 7 uform subdvsos, p = P-method, p = Fgure. Comparso betwee three equvalet sets of bass fuctos for seve uform subdvsos usg: () Berste (Bézer) polyomals, ()

7 Computatoal Research (3): 6-84, 3 67 Lagrage polyomals, ad () p-method.3... Coos Iterpolato A four-sded rego ABCD, as show o the left of Fg., ca easly be mapped to a ut square the ξη parametrc doma show o the rght of the same fgure by the method of Coos patch []. For purposes of geeralzato, the relevat theory s gve below usg sutable proectos. or, usg covetoal otato, as: x ξη, = ξ x, η ξx, η ( η) x( ξ, ) ηx( ξ,) ( ξ)( η) x(,) ξ( η) x(,) ξηx(, ) ( ξ ) ηx(,) (3).3... Gordo Iterpolato Accordg to Gordo s terpolato formula [34], the T Cartesa coordates x ( ξη, ) = [ x( ξη, ), y( ξη, )] of each pot P of the patch ca be approxmated by ts boudares ( x( ξ, ), x( ξ, ), x(, η), x (, η) ) as well as by ts ter-boudares (f ay) as follows: ( ξη, ) = P P P x x x x (3) ξ η ξη ad ( ξη, ) = P P P u u u u (3) ξ η ξη Fgure. Defto of the Coos-patch macroelemet ABCD Frst, the cocept of the loftg proector P s troduced. Ths proector s ay dempotet lear operator, whch maps a true surface to a approxmate surface, subect to certa terpolatory costrats. Let us assume that the Cartesa co-ordates T x( ξη, ) = { xyz,, } A, wth ξ ad η deotg ormalzed co-ordates, are kow at the boudares ( ξ =, ; η =, ) of a curvlear patch of area A. Let us also defe the well-kow cardal bledg fuctos: E ( ξ) = ξ, E ( ξ) = ξ, E ( η) = η, E ( η) = η (6) Now, the followg udrectoal, or loftg, operators P ξ { x } ad P η { x } may be costructed (summato over repeated dces s uderstood): P ξ P η { x} = x( ξ η) E ( ξ) { x} = x( ξη) E ( η),,,. (7) The above loftg operators form the bass for the defto of more complex operators wth bledg terpolato propertes more tha oe drecto. So, the two-dmesoal loftg operator: { } = { } = (, ) P x PP x x E E (8) ξ η ξ η ξη ξ η ca be costructed wth the ad of the udrectoal operators P ξ { x } ad P η { x }. Fally, the co-ordates of ay pot the teror of the curvlear patch s approxmated as: ( ξη, ) = ( P P P ){ } x x (9) ξ η ξη For example, based o Gordo s terpolato, t s easy to derve the shape fuctos for a fve-ode elemet havg oe teral oded at ts cetre ( addto to the four oes at the corers) [35, 36]. Remark: It has bee foud that whe the odal pots o the opposte sdes of a patch possess detcal ormalzed postos wth those odes the teror, the Gordo-Coos terpolato degeerates to the cross product P ξη [37, p.38]. Uder the latter codtos, whe Lagrage polyomals have bee used, Coos terpolato cocdes wth the well-kow terpolato fte elemets of Lagrage type (or Lagrage famly) [5, p.53]. I cocluso, patches where structured meshes of odal pots are used, all three basc CAD terpolatos (Coos, Bézer, ad B-sples) are expressed by a tesor product of -D ξ- ad η-terpolatos. Detals are gve below. Fgure 3. Defto of the Gordo-Coos macroelemet ABCD Tesor Products By defg the breakpots alog each sde of a

8 68 A Revew o Attempts towards CAD/CAE Itegrato Usg Macroelemets curvlear quadrlateral ABCD or a box-lke volume ABCDEFGH, for a certa polyomal degree (p) ad for a certa multplcty (usually oe or two), the bass fuctos N, p( u ) are determed. It s oted that here u s ether of the ormalzed coordates ξ, η, or ζ, whereas the umber of bass fuctos equal to the (m) cotrol pots the partcular drecto (m ξ, m η, ad m ζ, the ξ-, η-, or ζ-drecto, respectvely). The, the aforemetoed oe-dmesoal cotrol pots produce = ( m ξ ) ( m η ) or = ( m ξ ) ( m η ) ( m ζ ) cotrol pots for D ad 3D problems, respectvely. Each of the aforemetoed cotrol pot s assocated wth oe global shape fucto gve by: N I ( ξ) N ( η), p ( ξ) ( η) ( ζ ) Np = N N N p p kp, for D problems for 3D problems (33) Therefore, the geometry s terpolated by the expressos: ( ξη) = I ( ξη) x, N, x, I = ( ξη) = I ( ξη) y, N, y, I = ( ξη) = I ( ξη) z, N, z. I = I I I (34) The parametrc patch of (36) s expressed as: u v w= u, v, w (37) I ths case, the parametrc patch ca be tersected wth creasg values of the varables u,v that vary betwee ad, ad the calculatg the correspodg values of w from each set of u ad v. Apart from CAD applcatos, o applcato of Barhll s formula to egeerg aalyss has bee reported so far. For the frst tme, t s show that, (36) heretly cludes the covetoal three- ad sx-odedtragular fte elemets. Detals are gve Appedx A ad Appedx B, respectvely. Of course, (36) allows for the dervato of ovel arbtrary-oded tragular macroelemets, that have bee successfully appled to stress aalyss problems [9]..3.. Tragular Patches We wll preset two alteratves as follows Sde Degeerato Oe possblty s to use the formulas prevously used quadrlateral patches, by degeeratg oe of the four sdes to oe pot [, p.3]. For example, whe degeeratg the etre sde AD to a uque pot A, the quadrlateral degeerates to a tragle ABC, (3) becomes [38, p.35]: ( ξη, ) = ( η) ξ ( ξ) ( η BC AB ) u ( ξ) η ξ[ ( η) u ηu ] u u u AC B C (35).3... Barhll s Formula The mathematcal descrpto of tragular patches has bee publshed by Barhll [39-4]. I those works a trlearly bleded terpolat formulato s troduced, whch s qute dfferet tha Gordo-Coos; today ths formula ca be foud eve CAD textbooks such as [4, p.44]. The tragular patch show Fg. 4a ca be mapped to the parametrc patch of Fg. 4b through the relatoshp: (,, ) P uvw ( ) ug v wh v vh w = v v w u f w wf u vg u w u u wf ug vh ( ) ( ) ( ) ( ) ( )] (36) Fgure 4. Defto of the tragular parametrc Barhll s patch.4. Three-Dmesoal (3D) Iterpolato Volume blocks are categorzed to hexahedral (eght vertces, ad twelve edges) ad tetrahedral (four vertces, ad sx edges) blocks. The terpolatos ca treat ether straght or curvlear edges of the volume..4.. Geeral A boxlke rego ABCDEFGH, show Fg. 5a, ca easly be mapped to a ut cube the rts parametrc doma ( r,s,t ) show Fg. 5b. The relevat formula may be foud [, p.4] ad, as prevously metoed, s has bee appled by Cook [43] for fte elemet mesh geerato purposes. For more detals the terested reader may also cosult a textbook [44]. The oly dfferece wth a quadrlateral patch s that, here, sx fuctos (assocated to the sx faces) are bleded, stead of four fuctos (assocated to the four boudary edges of the D patch). Below, the same formula s wrtte terms of proectos. Besdes the above-metoed P { x }, P { x } ad r s Prs { x } used the D problem, ow, oe-, two- ad three-dmesoal operators are further troduced as

9 Computatoal Research (3): 6-84, 3 69 follows: P x = x rst,, E t { } { x} = { x} = x(,, ) { x} = { x} = x(,, ) { x} = { x} = x(,, ) t k k P PP r s t E s E t st s t k k P PP r s t E r E t rt r t k k P PPP r s t E r E s E t rst r s t k k (38) Aga, summato over repeated dces s uderstood. Havg troduced the oe-, two- ad three-dmesoal operators, the the followg formula descrbes the terpolato of the co-ordate vector [,p.4]: The, substtutg (4) ad (4) to (39), oe ca fally derve three equvalet expressos of the three-dmesoal Coos terpolato formula, as follows: x x( r s, t) = S{ x} E{ x} C{ x} ( r, s, t) S{ x} C{ x} x( r, s, t) E{ x} C{ x}, (4) = (43) = (44).4... Commets o 3D Equvalet Expressos Obvously, (4) s the most geeral expresso because t cludes ay type of the surroudg surfaces S (edge-oly oded ad/or terally oded). Moreover, (43) s obtaed by elmatg the edges (E) ad t s based o the surface S, the last beg corrected by the co-ordates of the corers C. Fally, (44)cludes oly the twelve edges E ad eght corers C, or other words, the absolutely ecessary data for the costructo of a Coos block made of Coos surfaces; t s evdet that (44) uses odal pots arraged alog the twelve edges oly. Fgure 5. Three-dmesoal superbrck macroelemet aalyzed usg boudary-oly Coos formulato x ( rst,, ) = ( ) ( P P P){ x} r s t ( ) ( P P P ){ x} rs st st 3 ( ) P { x}. rst (39).4.. Equvalet Expressos of Trvarate Coos Formula.4... Geeral Remarks Equato (39) s geerally applcable but t ca be further smplfed ad be wrtte more readable expressos. So, the case of a geeralzed curvlear hexahedral /parallelod (boxlke rego), the geometry cludes: Sx surfaces, S Twelve edges, E, ad Eght corers, C Obvously, (39)cludes all three quattes: Surfaces (S), edges (E) ad corers (C). For coveece, the proectos related to the S, E ad C are deoted as follows: S E C { x} = ( Pr Ps Pt ){ x} { x} = ( Prs Pst Ptr ){ x} { x} = P { x}. rst (4) However, the case of adequately smooth ad regular surfaces, the edges E ca suffcetly descrbe S. I fact, applyg (3) o the sx surfaces of the superbrck, oe ca easly derve the followg relatoshp: { x} 3 C{ x} = E{ x} S (4) Fgure 6. Overall applcablty of Coos-terpolato I cocluso, cases where the superbrck s suffcetly regular, (44) s the most advatageous. Usg covetoal otato, (44) becomes: x( r, s, t) = ( s)( t) x( r,,) r ( s) x(,, t) ( s) t x( r,,) ( r)( s) x(,, t) s ( t) x( r,,) r s x(,, t) s t x( r,, ) ( r) s x(,, t) (45) ( r)( t) x(, s,) ( r) t x(, s,) r ( t) x(, s,) r t x(, s,) [( r)( s)( t) x(,,) ( r)( s) t x(,,) ( r) s ( t) x(,,) ( r) s t x(,, ) r( s)( t) x(,,) r( s) t x(,,) r ( s)( t) x(,,) r s t x(,, ) ].4.3. The Three Alteratve Uses of Coos Iterpolato I more detal, bvarate (D) Coos terpolato o curvlear surfaces s useful for () mesh geerato for two dfferet purposes (D FEM or 3D BEM applcatos), ad () to derve D fte macroelemets. Also, trvarate (3D) Coos terpolato o curvlear volumes s useful for () mesh geerato (3D FEM applcatos), ad () to derve 3D fte macroelemets, as show Fg.6.

10 7 A Revew o Attempts towards CAD/CAE Itegrato Usg Macroelemets It has bee prevously reported that, the mesh produced by D-Coos terpolato ca be easly smoothed by takg the average of the eght eghbourg odes. Also, the mesh produced by 3D-Coos terpolato ca be easly smootheed by takg the average of the twety-sx eghbourg odes [45-47]. 3. Use of Coos Iterpolato 3.. Older Fdgs From the aalyss sted Secto, t becomes clear that 967 Coos terpolato was the frst achevemet CAD surface theory, prmarly amg at descrbg surfaces or volumes []. I early 97s, FEM aalysts uderstood that the same terpolato mght have a secod usage, as a geerator of structured fte elemet meshes [43, 48-54]. I more detals, frst a mesh geerato s costructed o the paret ut square or cube for D ad D problems, respectvely. Secod, odal pots are costructed alog the boudares of the vrtual structure. Thrd, (3), or (39) ad ts varatos, are appled ad thus establsh a mappg betwee the aforemetoed odes the paret elemet ad the vrtual structure. As a result, the desred fte elemet mesh s geerated. Moreover, t was early uderstood that Coos terpolato could be used for egeerg aalyss purposes as well [55, 56]. 3.. Cotemporary Fdgs 3... Cotrbuto o Covetoal Fte Elemet Meshes I subsecto.4.3 we metoed that, due to the fact that the mesh geerated usg (3) or (39) for D ad 3D problems, respectvely, s usually of low qualty, smoothg schemes have bee successfully appled [57, 45-47]. The covergece leads to almost rectagular elemets Costructo of Gordo-Coos Macroelemets for Egeerg Aalyss 3... Lterature Survey Early thoughts o the costructo of large elemets were made by the poeer Bruce Iros [58]. Expressos of lower order large elemets based o Coos-Gordo deas are [55, 59-6]. Numercal examples o D potetal problems were preseted 989 by Kaarachos ad Derzots [6] at the Natoal Techcal Uversty of Athes (NTUA); as a member of NTUA s FEM/BEM group, the author ca cofrm that relevat attempts started there 98. Ths work was exteded to D elastcty [63]; the relevat coferece the author aouced that the same methodology s ot applcable oly to Coos but also coucto wth all other CAD formulatos such as Gordo-Coos, Bézer, B-sples ad NURBS. Boudary-oly Coos terpolato was appled to D potetal problems [37, 64-68], axsymmetrc elastostatcs [69], ad D elastodyamcs [7-73]. I the process of the research t was foud that boudary-oly Coos terpolato correspods to a seres expaso of the soluto wth those terms appearg Pascal s tragle wth a surplus of two (Seredpty famly), a fact that was later foud as a ote [74]. I some examples, the boudary-oly Coos formulato was more accurate ot oly tha the covetoal FEM, wth the same mesh desty, but also tha BEM [75]. It was later uderstood that, t s ot always possble to accurately solve all egeerg problems uless a suffcet umber of teral odes are used. The latter s easly acheved usg Gordo-Coos (trasfte) terpolato [35-37, 7, 76]. Boudary-oly Coos terpolato was successfully appled to 3D potetal ad elastcty problems [77-79]. The theme deftely closes wth [8], whch compares twelve alteratve models oe aother; although t focuses o D elastostatcs, the coclusos are of geeral mportace. Cocerg BEM ad sold modelg, tal attempts for CAD/CAE tegrato were made by Casale [8-83]. Applcato of Coos terpolato 3D BEM problems was preseted durg the years -3 [84-87]; these cofereces the geeral applcablty coucto wth Bézer, B-sples ad NURBS was repeated. The overall cocluso s that boudary-oly formulato sometmes caot coverge to the accurate soluto, whereas the Gordo-Coos [88] always acheves t [89]. Cocerg structures of tragular shape, the work of Kaarachos ad Dmtrou [38, 89] has show relable results whe a sde of the quadrlateral s degeerated. I addto to the latter works, Provatds ad Atoou [9] have foud ecouragg results whe Barhll s terpolato [39-4] s used for the egeerg aalyss. As oe ca see spectg [9-9], Perre Bézer was a employee of Reault ( Frace) ad started publshg sce 966, whch s almost the same perod whch Steve Coos was workg at MIT ( USA). Bézer s theory s related to the dscovery of the cotrol pots, whch are vertces of a polygo (geerator) that cotrols a curve. The same theory s also exteded to curved surfaces, usg a set of geerators [9, 7]. Exteso to volumes s possble []. I the cotext of egeerg aalyss, t has bee foud that Bézer formulato s equvalet to the use of moomals or Lagrage polyomals [], a matter that has bee also treated D problems [3, 33, ]. Later, ratoal Bézer curves [3-5], whch have several advatages over polyomal Bézer curves, appeared. Clearly, ratoal Bézer curves provde more cotrol over the shape of a curve tha does a polyomal Bézer curve. I addto, a perspectve drawg of a 3D Bézer curve (polyomal or ratoal) s a ratoal Bézer curve, ot a polyomal Bézer curve. Also, ratoal Bézer curves are eeded to exactly express all coc sectos. A degree two polyomal Bézer curve ca oly represet a parabola, a smlar way wth the Lagraga polyomals. Exact

11 Computatoal Research (3): 6-84, 3 7 represetato of crcles requres ratoal degree two Bézer curves. Ratoal Bézer curves permt some addtoal shape cotrol: by chagg the weghts, you chage the shape of the curve. It s possble to reparametrze the curve by smply chagg the weghts a specfc maer. Fally, the process of CAD evoluto, the use of B-Sples was proposed [6,7]. It s remded that B-sples s othg dfferet tha usual sples but they are replaced by o-cardal bass fuctos, Np. ( ξ ) of polyomal degree p, whch are calculated for every ormalzed coordate, ξ, through fast teratve formulas developed depedetly by Cox [9] ad DeBoor [3] 97, as metoed by the latter researcher hs moograph [6]. B-sples appear a local support ad clude, as a specal case, the abovemetoed Bézer curves. Despte the local support of B-sples, a better cotrol o regoal chages s acheved whe usg the o-uform ratoal B-sples (NURBS) that were developed md-sevetes tll early eghtes [8,9]. The terested reader may also cosult a later survey [] ad a useful moograph of 646 pages [7]. Accordg to the abovemetoed sequece CAD evoluto,.e. Gordo-Coos Bézer B-sples NURBS, a relevat cotrbuto structural (geerally egeerg) aalyss s atcpated, as follows. The commo characterstc of all relevat works egeerg aalyss s that, they all am at decomposg the doma to CAD-cotrolled subregos [-4]. Cocerg Gordo-Coos (trasfte) terpolato, relevat research o graphcs cotued at least utl 4 [5-8], whereas egeerg aalyss was made electrcal egeerg [9] ad soft-tssue bomechacal [3] applcatos. Ital works are [55, 59-6]. As regards the FEM/BEM group at NTUA, soparametrc Gordo-Coos macroelemets were publshed the perod 989- [38, 57, 6-73, 75-8, 84-9]. I ths paper we wll use the term sogeometrc whe referrg to terpolatos related to cotrol pots,.e. Bézer, B-sples ad NURBS. Cocerg Bézer represetato, t s remded that t s equvalet wth the use of Lagrage polyomals []. I other words, for ay selected degree of the polyomals volved, despte the fact that the cotrol pots sweep the etre doma, fact t s as f odal pots alog oly the sdes were used (D: 4 sdes of the quadrlateral, 3D: edges of the hexahedro). Cocerg B-sples egeerg aalyss, a forgotte but ( author s opo) poeerg work s due to Arstodemo [3] ad later a lot of papers appeared [3-4]. Amog these works, [37-4] refer to the boudary elemet method (BEM). Cocerg the evoluto NURBS, the terested reader may cosult [4], amog others. Oe of the frst attempts to corporate NURBS structural aalyss, ad partcularly shape optmzato, s due to Schramm ad Plkey [43], 993, followed by other vestgators such as [44-46]. I 5, the group of Prof. T. J. R. Hughes troduced the term sogeometrc elemets [47-49], ad later they appled a effcet quadrature to reduce the CPU-tme [5]. The author [68] had tmely dscovered that, despte ts elegace, the global terpolato s geerally slower tha the commo fte elemet method (Galerk-Rtz), of course for the same mesh desty. I 5, he proposed to preserve the CAD-based global terpolato ad replace the doma tegrato through a global collocato scheme [73, p. 674]. Cocerg the Global Collocato Method (GCM), a tal cotrbuto s due to Blerk ad Botha [5]. The problem was systematcally tackled startg from oe-dmesoal problems, frst startg wth Lagrage polyomals [, 3, 5] ad the cotug wth B-sples [53, 54]. The, the same dea was exteded to D potetal problems [55, 56], elastostatcs [57], as well as egevalue problems acoustcs ad elastodyamcs [58, 59]. It s worth-metog that the applcato of sogeometrc collocato methods was later proposed by others [6] Global Shape Fuctos Followg [8], let us assume that the sdes AB, BC, CD ad DA of the patch ABCD show Fg. 3 clude q, q, q 3 ad q 4 odes, respectvely. I ths case, the correspodg umber of subdvsos per sde are =q -, =q -, 3 =q 3 - ad 4 =q 4 -, respectvely. The, the umber of odes alog the boudary of the patch becomes: q = = q q q q (46) b whle cosderg the addtoal umber of odes becomes: e b I teral odes, the total q = q (47) I the sequece, the boudary values of the dsplacemet vector,.e. u(ξ,), u(,η), u(ξ,) ad u(,η) (3) or fally (3), are terpolated by ay set of bass fuctos B ( ˆ ξ ) lke those preseted Secto ( ˆ ξ s ether ξ or η ; the upper dex B below correspods to the relevat sde of the Coos patch): Sde I AB ( ξ ) = B ( ξ) u( ξ ) AB: u,, = BC ( η) = B ( η) u( η) Sde BC: u,, = 3 CD ( ξ ) = B ( ξ) u( ξ ) Sde CD: u,, = 4 DA ( η) = B ( η) u( η) Sde DA: u,, q q q q = (48) Typcal boudary approxmato the above bass fuctos B ( ˆ ξ ) mght cosst of () pecewse lear terpolato (kode = ), () B-sple terpolato (kode = ), () Lagrage polyomal (kode = 3), (v) Berste

12 7 A Revew o Attempts towards CAD/CAE Itegrato Usg Macroelemets (Bézer) polyomals (kode = 4), ad (v) NURBS (kode = 5). The meag of assocate code wll be explaed subsecto Smlar terpolatos are also cosdered alog the ter-boudares (f ay). Fally, f all these terpolatos are substtuted to (48), ad subsequetly to (3) or (3), the soluto u(ξ,η) s approxmated by global cardal shape N ξη, wth the Coos-patch, as follows: fuctos q e u( ξη, ) = N ( ξη, ) u, (49) = wth u deotg the dsplacemet vector at odal pot, appearg at the boudares or/ad the teror of the Coos macro-elemet. I case of Bézer B-sples or NURBS, u (49) should be replaced by a. For example, the global shape fucto assocated wth the corer ode A s gve by: A DA ( ξη, ) = ( ξ) ( η) q4 AB E ( η) B ( ξ) E ( ξ) E ( η) N E B. (5) Also, the global shape fucto assocated wth a termedate ode alog the sde AB s gve by: AB ( ξη) ( η) ( ξ) N, = E B, =,, q. (5) Fally, the global shape fucto assocated wth a teral ode s gve by: N ( ξη, ) = E ( ξ) E ( η), k l (5) k =,, ; l =,, ξ wth the subscrpt deotg the th ode havg ξ, η. ormalzed coordates k The Role of Bledg Fuctos I addto to the dex that determes the bass fuctos alog each of the four sdes of a patch ABCD, ow we deal wth the type of the bledg fuctos. Whe the bledg fuctos are lear fuctos fluecg the etre sde: E ( ξ) = ξ ad E ( ξ) = ξ, we assg them the kode =. Ths case correspods to the boudary-oly Coos macroelemet, uder the otato C. Aga, C meas that all four sdes are terpolated wth the same set of bass fuctos,.e. = for pecewse lear, = for B-sples, = 3 for Lagrage polyomals, = 4 for Berste (Bézer) polyomals, ad = 5 for NURBS. I addto to the above lear E ( ξ ) ad E ( ξ ), the bledg fuctos E ( ξ ) ad E ( η ), k =,, ; k l ξ l =,, η, ca be ay reasoable cardal fuctos that obey the delta-kroecker codto: E ( ξ ) = δ. I more detals, they have to vash alog the opposte boudary as well as alog those terboudares to whch they are ot assocated. Typcal bledg fuctos are () lear fuctos for the etre sde: E ( ξ) = ξ ad E ξ = ξ (kode = ), () the hat (pecewse lear) l η fuctos wth compact support betwee two adacets odes (kode = ), () cardal cubc B-sples (kode = ), ad (v) Lagrage polyomals (kode = 3). Based o the above deftos ad code pars (, ), we ca costruct 5 4 = macroelemet models, deoted by C, =,, 3, 4, ad 5, whereas =,,, 3, ad 4 (C stads for Coos ). Aga, the frst dex,, refers to the type of the bass fuctos, whereas the secod dex,, refers to the type of bledg fuctos. It s easy to uderstad that the umber of possble models C ca be hghly creased whe some of the sdes have a dfferet set of bass fuctos that what aother sde has. Moreover, f opposte sdes have the same umber of odal pots ( = = 3 ξ, = = 4 η ) ad all odes (boudary ad teral oes) are dstrbuted alog les of costat ξ - ad η -values. The: Whe usg pecewse-lear terpolato for both the bass ( B ( ˆ ξ ) ) ad the bledg fuctos ( E ( ˆ ξ ) ), (5) up to (5) produce a mesh of covetoal four-ode blear fte ξ η elemets. I fact, cosderg the compact support of the pecewse-lear fuctos, the proof s qute trval. Whe usg Lagrage polyomals for both the bass ad the bledg fuctos, (3) ad (3) degeerate P ξη u, respectvely. I other words, the shape fuctos become the well-kow tesor products of Lagrage polyomals. For a proof we refer to Referece [37]. to P ξη ( x) ad 4. Isoparametrc ad Isogeometrc-lke Approxmatos Gordo-Coos ad Lagraga type elemets are based o the followg seres expaso: x ( ξη, ) N ( ξη, ) = x = ( ξη, ) ( ξη, ) =, (53) U = N U whch characterzes all soparametrc elemets. I cotrast, sogeometrc-lke approxmatos (.e., tesor products of Bézer, B-sples ad NURBS) are based o the followg seres expaso: x ( ξη, ) N ( ξη, ) = ( ξη, ) ( ξη, ) U = N a = = x, (54) where a are ot geerally the odal values at the cotrol pots but geeralzed coeffcets. Oly at the very eds of the boudary the extreme cotrol pots (P ad P ) cocde

13 Computatoal Research (3): 6-84, 3 73 wth the correspodg eds of geerator curves ad, therefore, the coeffcets a ad a cocde wth the values U ad U, respectvely. It s remarkable that all sogeometrc-lke approxmatos the key problem s to mpose the boudary codtos. Ths dffculty s ot apparet D problem, but t appears D ad 3D oes. For example, f the etre sde AB of the quadrlateral ABCD s gve homogeeous Drchlet codtos (U = ), ths mples that all coeffcets a (54) must vash. I cotrast, whe half of the sde AB s gve the zero value whereas the rest s gve for example by U = ξ- /, we caot a-pror calculated these coeffcets, uless we fulfll the codto at so may pots as the umber of cotrol pots alog AB, ad the vert the produced matrx. I ths cotext, accordg to Natekar et al. [4], bvarate NURBS represetato s appled to derve shape fuctos N I ( ξ,η) that are based o the set of I-th cotrol pots defg the system geometry. The same shape fuctos are also used to approxmate the dsplacemet feld wth the doma. It s remarkable that at ay pot wth the doma the sum of these shape fuctos equals to the uty but the value of N I ( ξ,η) at the I-th cotrol pot s ot uty to that ode. Therefore, eve f the cotrol pot were to be cocdet wth the locato of the boudary codto, drect applcato of the boudary codto s ot possble sce the specfed feld value wll be dstrbuted to cotrol pots fluecg the pot uder cosderato [4]. The abovemetoed dffcultes were the reaso that the author preferred to work maly wth Coos terpolato (although very early he suggested the applcato of NURBS [45, p. ad pp. 3-33], whle other research teams drectly followed the tedecy of the moder CAD systems ad worked wth NURBS [8], amog others. 5. Computatoal Procedure 5.. Geeral The vbrato of a elastc sold structure s descrbed through the stress equlbrum equato [5]: T L σ f = ρu (55) whle for acoustcal cavtes (or membraes) t holds: c 5.. Numercal Procedure u t u = (56) I ths work we cosder three umercal procedures: () Galerk/Rtz, () Collocato, ad () Bouday Elemet Method, as follows Galerk/Rtz Applyg the Galerk method to (55) or (56), oe obtas the well-kow matrx formulato [5]: [ M] { a( t) } [ K] { a( t) } = { f( t) } M ad [ ] (57) where[ ] K are the mass ad stffess matrces, respectvely, whch are gve by: T N ρndω, elastcty Ω m = NN dω, acoustcs Ω c (58) T ( LN) E( LN) dω, elastcty Ω k = N N dω, acoustcs Ω where N deote the shape fuctos by whch the soluto (dsplacemet compoet or acoustc pressure) s terpolated as follows: e (, ) = u x t N x a t, (59) = where e s the umber of DOF per elemet. Also, { f ( t) } s the force vector ad { a ( t) } deotes the vector of degrees of freedom,.e. t cossts of ether the dsplacemet compoets, u, or the geeralzed coeffcets whch the soluto s expaded Collocato Usg the same global shape fucto as above (secto 4.), ad fulfllg the PDE (55) or (56), oe obtas a smlar matrx formulato wth (57) wth the dfferece that ow o doma tegral appears, that s: ρ N ( x ), elastcty m =, N ( x ), acoustcs c k = E L ( x ) T L N x N x N, elastcty, acoustcs (6) where correspods to the collocato pots ad to the cotrol pots Boudary Elemet Method (BEM) I the case of elastcty problems, the co-ordate vector wth the l-th patch, possessg q l odes, s terpolated o the bass of the boudares of the patch as follows: q l ( r s) N ( r, s) x N x x, = = (6) = Sce each patch s cosdered as a soparametrc elemet t

14 74 A Revew o Attempts towards CAD/CAE Itegrato Usg Macroelemets holds that: u p q l ( r, s) = N ( r, s) = q = Nu l ( r, s) = N ( r, s) p = Np = u (6) By substtutg (6) the usual tegral equato (see for example [7]) ad summarzg over the N p patches whch the boudary s dvded, oe obtas: Np * c u plk Nk dγ up p= Γ p (63) Np * = ulk Nk dγ pp p= Γ p I (63) the ftesmal area dγ s gve by: where the Jacoba s calculated as: (, s) dr ds dγ = G r (64) 3 (, ) = ( ) / G r s g g g where 3 x x x x 3 3 g =, r s s r g g x x x x = r s s r 3 3 x x x x = r s s r, (65) Now, for the purposes of the umercal tegrato oly, the patch s dvded to N r N cells where a secod set of s ormalzed co-ordates ( r, s ) s troduced [84]. Therefore, the term G ( r, s) dr ds (64) s replaced by G ( r, s) G ( r, s ) dr ds, whch requres a trval (e.g.,, 3 3, 4 4 ) Gaussa quadrature. A selectve tegrato scheme has bee recetly developed. Therefore, the fal algebrac system obtas the form: Np Np p p p p C U Hˆ U = G P (66) p= p= where U s the dsplacemet vector of all odes o the p boudary of the structure (alog the patch edges), U ad p P are dsplacemet ad tracto vectors referrg to the p p p-th patch as well as H ad G are the osymmetrc fluece matrces. By properly assemblg the submatrces (66) we obta: H U = G P (67) If q s the total umber of odes alog all edges of the patches o the boudary of the structure, the dmesos of the vectors ad matrces (67) are as follows: U : dsplacemetvector ( 3 q ) P : tracto vector ( 3 q ) H : total dsplacemet fluece matrx ( 3 q 3q ) G : total tracto-fluece matrx ( 3 q 3q ) The above symbol q s larger tha q ( q = q q ) wth Δq depedg o the umber of sharp corers ad ther multplcty tracto dscotuty [7]. The above statc aalyss descrbed by (67) ca be also exteded to the soluto of dyamc problems too. Brefly, usg a set of radal bass fuctos a mass matrx s costructed for the etre structure ad t s combed wth the statc matrces H ad G show (67) [86, 87]. 6. Numercal Results I addto to prevous results, ths secto we preset addtoal results that elucdate the computatoal performace of the proposed techques. Example: Egevalues of a fxed rectagular membrae The dmesos of the membrae the x- ad y-drectos are a b =.5. m, the wave velocty s take as c = m/s, whereas the etre boudary s cosdered to be fxed. The free vbratos are govered by (56) ad the exact egevalues are gve by: ω = π c,,,,, = (68) a b The rectagular doma,.5., s uformly dvded to x y subdvsos, preservg the same rato,.e. x y = 4, 6 3, 8 4, 5, 6, 4 7, ad 6 8, thus leadg to ( x )( y ) = 5, 8, 45, 66, 9,, ad 53 odal pots, respectvely. I all methods the umber of ukows s the same, whereas ( x y ) =, 8, 4, 3, 36, 4, ad 48 odal pots belog to the boudary ad have to be elmated. Thus, the partcular case of Drchlet problem, the umber of ukows s commo for all the three methods as follows: Covetoal fte elemet (FEM) usg four-ode blear elemets. (Gordo)-Coos Patch Macroelemet (CPM), coucto wth the Galerk-Rtz formulato (doma tegrato), accordg to (58). The abovemetoed Gordo-Coos approxmato, coucto wth the Global Collocato Method (GCM), that s wthout doma tegrato, accordg to (6). The computatoal error was determed by the formula ω ω ω, percet (%), ad the ( calculated exact exact ) qualty of the umercal soluto s llustrated Fg.7, where oe ca otce that the proposed CPM ad GCM methods are far superor to the covetoal FEM. Although CPM s the overall best t eeds hgh computer tme, whereas the GCM s much more effcet because t eeds o doma tegrato at all.