35 YEARS OF ADVANCEMENTS WITH THE COMPLEX VARIABLE BOUNDARY ELEMENT METHOD

Transcription

1 N. J. DeMoes et al., It. J. Comp. Meth. ad Exp. Meas., Vol. 0, No. 0 (08) 3 35 YEARS OF ADVANCEMENTS WITH THE COMPLEX VARIABLE BOUNDARY ELEMENT METHOD Noah J. DeMoes, Gabriel T. Ba, Bryce D. Wilkis, Theodore V. Hromadka II, & Rady Boucher Departmet of Mathematical Scieces, Uited States Military Academy, Uited States ABSTRACT The Complex Variable Boudary Elemet Method, or CVBEM, was first published i Joural of Numerical Methods i Egieerig i year 984 by authors Hromadka ad Guymo []. Sice that time, several papers ad books have bee published that preset various aspects of the umerical techique as well as advaces i the computatioal method such as extesio to three or higher dimesios for arbitrary geometries, ohomogeeous domais, extesio to use of a Hilbert Space settig as well as collocatio methods, iclusio of the time derivative via couplig to geeralized Fourier series techiques, examiatio of various families of basis fuctios icludig complex moomials, the product of complex polyomials with complex logarithm fuctios (i.e., the usual CVBEM basis fuctios), Lauret series expasios, reciprocal of complex moomials, other complex variable aalytic fuctios icludig expoetial ad others, as well as liear combiatios of these families. Other topics studied ad developed iclude rotatio of complex logarithm brach-cuts for extesio of the problem computatioal domai to the exterior of the problem geometry, depictio of computatioal error i achievig problem boudary coditios by meas of the approximate boudary techique, mixed boudary value problems, flow et developmet ad visualizatio, display of flow field traectory vectors i two ad three dimesios for use i depictig streamlies ad flow paths, amog other topics. The CVBEM approach has also bee exteded to solvig partial differetial equatios such as Laplace s equatio, Poisso s equatio, usteady flow equatio, ad the wave equatio, amog other formulatios that iclude sources, siks ad combiatios of these equatios with mixed boudary coditios. I the curret paper, a detailed examiatio is made of the performace betwee four families of basis fuctios i order to assess computatioal efficiecy i problem solvig of two dimesioal potetial problems i a high aspect ratio geometric problem domai. Two selected problems are preseted as case studies to demostrate the differet levels of success by each of the four families of examied basis fuctios. All four families ivolve basis fuctios that solve the goverig partial differetial equatio, leavig oly the goodess of fit i matchig boudary coditios of the boudary value problem as the computatioal optimizatio goal. The modelig techique is implemeted i computer programs Mathematica ad MATLAB. Recommedatios are made for future research directios ad lessos leared from the curret study effort. Keywords: complex variables, complex variable boudary elemet method, CV BEM, basis fuctios, ideal fluid flow, Laplace s equatio. INTRODUCTION The Complex Variable Boudary Elemet Method (CVBEM) has bee the subect of umerous publicatios icludig papers ad books [ ]. The approximatio approach solves several partial differetial equatios (PDE) of high iterest to practitioers ad developers i various aspects of egieerig ad computatioal mathematics. The modelig approach fids good use i potetial flow problems, icludig heat, groudwater, diffusio ad may other problems ad applicatios of high iterest i sciece, techology, egieerig, ad mathematics related topics. Most recetly, the CVBEM has bee exteded ito three or higher spatial dimesios for arbitrary geometric spatial domais [4], ad has bee applied to usteady flow problems [8], ad also problems ivolvig the wave equatio []. The modelig approach to applyig the CVBEM to usteady flow problems ivolves determiatio of the steady state coditio ad the reformulatig the usteady flow PDE ito 08 WIT Press, ISSN: (paper format), ISSN: (olie), DOI: 0.495/CMEM-V0-N0--3

2 N. J. DeMoes et al., It. J. Comp. Meth. ad Exp. Meas., Vol. 0, No. 0 (08) two portios ivolvig the steady state portio ad the usteady flow portio. Uder this scheme, the usteady portio problem defiitio ivolves homogeeous boudary coditios, whereas the steady state portio problem defiitio ivolves the origial problem boudary coditios. The usteady portio is the solved by use of geeralized Fourier series, ad the steady state portio that icludes the o-homogeeous boudary coditios, is solved by the CVBEM (or other Laplace s equatio solver). The total solutio is simply the sum of the solutios to the two problem portios idetified i the above. The geeralized Fourier series approach eables solutio of a wide class of PDE give the homogeeous boudary coditios afforded by the CVBEM solutio to the steady state portio of the problem. For example, a Poisso s equatio problem with a costat load value of - is ofte used i the modelig of torsio problems. Similarly, the wave equatio is solved by the Fourier series approach. Sice the itroductio of the CVBEM i the Joural of Numerical Methods i Egieerig [], CVBEM techology has advaced over the last 35 years. The CVBEM techological advacemets are briefly reviewed i this paper. The two-dimesioal (D) case is examied, although extesio to high dimesios follows [4]. The, focusig o the steady state portio of the PDE problem settig, the solutio of the steady state portio of the PDE problem settig is examied i detail regardig the computatioal efficiecy of differet basis fuctio families used i the CVBEM approach. The basis fuctio families examied i the curret paper are:. Complex Moomials, of the form (z z ).. Products of complex liear moomials with complex atural logarithm fuctios (with rotated brach cuts), of the form, (z z )l(z z ). 3. Liear complex poles (iverse factors of the form, z z, for several sigularity poits z, located exterior of the problem domai Lauret series, of the form, = ( z z ). Other complex aalytic fuctios may be used as alterative basis fuctio families, as ca liear combiatio or other combiatios of the basis fuctios that preserve the aalytic defiitio of the overall CVBEM approximatio fuctio. Demostratio of basis fuctio family computatioal efficiecy is provided by examiatio of potetial problems with respect to a high aspect ratio spatial domais located i the first quadrat of the complex plae. The motivatio for usig complex aalytic fuctios i the approximatio approach icludes the well-kow properties of aalytic fuctios. The aalytic fuctio ivolves two D real values fuctios that each solve Laplace s equatio. Furthermore, these two real fuctios are cougates which meas that plots of isocotours of both fuctios result i the well-kow flow et depictio of isopotetials ad stream lies. These two real fuctios are called the real ad imagiary compoets of the aalytic fuctio. Ad, these two compoets are related to each other by the Cauchy-Riema equatios, eablig both compoet fuctios to be examied give the other compoet fuctio. Boudary coditios of the Dirichlet or flux type or eve mixed types are all readily accommodated. Further details of the CVBEM are provided i the book, Foudatios of the Complex Variable Boudary Elemet Method [4].

3 N. J. DeMoes et al., It. J. Comp. Meth. ad Exp. Meas., Vol. 0, No. 0 (08) 3 The CVBEM ca be viewed as aother geeralized Fourier series that ca be coupled to the usual Fourier series. I the curret paper, the CVBEM approximatio fuctio is used to solve Laplace s equatio (for example, the steady state portio of the trasport equatio) over the arbitrary geometric domai ad with arbitrary but fixed boudary coditios. The usual Fourier series is used to solve the usteady compoet of the goverig PDE with homogeeous boudary coditios. Because of the use of the usual Fourier series, the geometric domais curretly are limited to -dimesioal bricks i order for the Fourier series elemet fuctios to satisfy the homogeeous boudary coditios. Load terms ivolvig sources ad siks are hadled as part of the steady state portio of the PDE. A derivatio ad detailed applicatio of the two-dimesioal usteady trasport modelig approach based upo the CVBEM (where, as described previously, the CVBEM is used to solve the steady state portio of the usteady flow problem, ad Fourier series is used to solve the homogeeous usteady portio of the trasport problem) is demostrated i the aalysis of a groudwater moud usteady flow problem [3, 4]. Although the refereced applicatio is focused upo a importat computatioal geoscieces problem, applicatio of the CVBEM approach to other computatioal topics is straight forward [4]. Review of the CVBEM The CVBEM is a aalytic method of solvig boudary value problems that differs from covetioal methods such as the fiite differece method (FDM) ad fiite elemet method (FEM) that require iterpolatio to determie a approximatio for the potetial fuctio. The CVBEM produces a cotiuous fuctio that approximates the solutio throughout the complete domai without the use of iterpolatio. The reaso the CVBEM does ot require iterpolatio is due to the method s reliace o aalytic basis fuctios to form the solutio approximatio. Because the basis fuctios are aalytic, the real ad imagiary parts satisfy the Laplace equatio. Because the basis fuctios are liear ad homogeeous, the priciple of superpositio states the summatio of the aalytic fuctios also satisfies the Laplace equatio that provides a stroger solutio to the boudary value problem. The real part of the CVBEM approximatio fuctio represets the potetial fuctio while the imagiary part represets the streamlies. The real ad imagiary parts of the approximatio fuctio are orthogoal to each other because the approximatio fuctio satisfies the Laplace equatio. The geeral CVBEM approximatio fuctio is wˆ( z) = cg () z () = where ŵ(z) represets the CVBEM approximatio for the potetial at locatio z, c represets the th complex coefficiet, ad g (z) represets the th fuctio i the set of complex basis fuctios chose to approximate the global trial fuctio, ad represets the umber of fuctios used to solve the global boudary value problem. I the followig, the four basis fuctio families cosidered i this paper will be examied. To solve the boudary value problem, the complex coefficiets must be foud. Each basis fuctio that is used i the summatio to approximate the potetial fuctio has a complex coefficiet, comprised of both a real ad imagiary part. Thus, there are degrees of freedom for each basis fuctio, ad degrees of freedom for the approximate solutio, where is the umber of basis fuctios. To determie all coefficiets there must be kow potetials, w(z), ad their

4 4 N. J. DeMoes et al., It. J. Comp. Meth. ad Exp. Meas., Vol. 0, No. 0 (08) correspodig locatios. To solve for the coefficiets, create the followig matrix equatio of the form Gc = w: g g g L g g g g L g g g g L g c c c M c 0 w w = w M w 0 () where G represets the value of each basis fuctio at z, c represets the vector of complex coefficiets, ad w represets the kow potetial values. Solve for the coefficiets by performig liear algebra operatios to row reduce the matrix to fid c. Oce the coefficiets are foud ad back substituted ito the CVBEM approximatio fuctio, the real part is used to approximate the potetials i the problem domai. I this paper, basis fuctio families are used without combiatio. However, aother optio for use i the approximatio approach is to use a liear combiatio of the various basis fuctio families. Additioally, other aalytic fuctios are available to use as basis fuctios such as the complex expoetial fuctio ad complex trigoometric fuctios (that are, i tur, related to the complex expoetial fuctios). Research ito the CVBEM appears to idicate that use of the basis fuctios developed by the umerical itegratio of the Cauchy itegral formula results i the most stable ad usable basis fuctios. 3 CVBEM Basis Fuctio Families Cosidered The geeral CVBEM fuctio is 3. Complex Moomials wz ˆ () cg () z (3) For this basis fuctio family, complex fuctios of the form (z z ) are used where coordiates z are expasio poits as utilized i the Taylor series for complex variable fuctios [5]. Expasio of these basis fuctios reduces to the simpler complex moomials of the form z, where as i the prior form, are positive iteger expoets. This formulatio is aother represetatio of the complex variable polyomial method or CPM, as examied i a umber of publicatios [6]. The CPM approximatio fuctio is wz ˆ () = c( z z ) (4) = 3. Numerical Itegratio of Cauchy Itegral Usig Polyomial Iterpolatio Fuctios (Stadard CVBEM Basis Fuctios) The more traditioal basis fuctio family is of the form (z z ) l (z z ) where poits z are CVBEM odes ad the l (z z ) is the complex atural logarithm fuctio defied

5 N. J. DeMoes et al., It. J. Comp. Meth. ad Exp. Meas., Vol. 0, No. 0 (08) 5 with its brach cut selected to lie exterior of the problem domai uio boudary. Although odal coordiates z typically are defied o the problem boudary [4], placig CVBEM odes to be located exterior of the problem domai uio boudary have also bee used with good computatioal results [7]. These basis fuctios result from the umerical itegratio of the Cauchy itegral formula where liear trial fuctios are used to iterpolate the complex fuctio betwee odes located o the problem boudary. Depedig o mathematical formulatio, differet umerical statemets ca result that all have i commo the basis fuctios as described above. The stadard CVBEM approximatio fuctio is wˆ ( z) = c ( z z )l ( z z ) (5) = 3.3 Complex Poles Aother basis family family examied are the traditioal complex variable poles of the form z z. The poits z are located o or the problem boudary or exterior of the problem domai uio boudary. Theoretical cosideratios ivolvig complex fuctio poles of higher orders are examied i stadard texts o complex variables [5]. The complex poles CVBEM approximatio fuctio is 3.4 Lauret Series wz ˆ () = c (6) z z = The fourth family of basis fuctios examied is the terms of the Lauret series ad are of the form, where are positive itegers ad z ( z z0 ) 0 is a fixed expasio locatio exterior to the problem domai ad usually also exterior to the problem boudary. The locatio of z 0 impacts the approximatio error. I the followig, z 0 = i because this value reduced the approximatio error sigificatly compared to other locatios such as 40 40i or 0 + 0i. Several Lauret series ca be examied together, meldig together some of the advatages of the complex fuctio poles basis fuctios. However, z 0 must be located outside the problem domai to eable the basis fuctio to be completely aalytic o the problem domai. The theoretical exploratio of the Lauret series ca be foud i other books [5]. The Lauret series CVBEM approximatio fuctio is wz ˆ () = c ( ) = z z0 (7) 3.5 Mixed Basis Fuctio Approximatios ad other Families of Basis Fuctios Aother optio for use i the approximatio approach is to use a liear combiatio of the various basis fuctios examied i the above. Additioally, other aalytic fuctios are

6 6 N. J. DeMoes et al., It. J. Comp. Meth. ad Exp. Meas., Vol. 0, No. 0 (08) available to use as basis fuctios such as the complex expoetial fuctio ad complex trigoometric fuctios (that are, i tur, related to the complex expoetial fuctios). Research ito the CVBEM appears to idicate that use of the basis fuctios developed by the umerical itegratio of the Cauchy itegral formula results i the most stable ad usable basis fuctios. 3.6 Example Problems I order to demostrate the utility ad efficiecy of the CVBEM ad the various basis fuctios cosidered i the above, two test problems are examied:. Heat trasfer i a two dimesioal rod.. Ideal fluid flow i a 90-degree bed. Both demostratios are cosidered i detail i Sectio 4. To determie the accuracy of the CVBEM usig differet basis fuctios, the exact solutio is compared to the CVBEM approximatio fuctio alog the ceter lie of the domai. The reaso the ceter lie is chose is twofold. The first reaso is because the ceter lie ecompasses the largest deviatio to the exact solutio due to the decrease i proximity to the boudary. The secod reaso is because the ceter lie is the most accurate depictio of the iterior which is the part of the domai that the CVBEM is desiged to model. The first problem tests the computatioal accuracy of the CVBEM agaist the FEM i high aspect ratio domais with simple boudary coditios. The secod example problem tests the effectiveess of differet basis fuctios as the boudary coditio complexity icreases i the high aspect ratio domai to determie if certai basis fuctios are more accurate for high aspect ratio domais. The last two example problems examie other features of the CVBEM, such as the iclusio of the time derivative as used i trasiet potetial flow problems or the wave equatio, amog other formulatios, as well as sources ad siks, ad cosideratios of three or higher spatial dimesios, are preseted i the refereced publicatios above. 4 Example Problem Domai Defiitio The problem domai is a two-dimesioal spatial rectagle of high aspect ratio. The stadard domai has top ad bottom boudaries of legth 0 ad the left ad right boudaries of legth. All example problem places the problem domai etirely i the first quadrat. The stadard problem domai is depicted i Fig. where the boudary coditios are represeted as u, for the th test problem. Figure : Stadard problem domai for both test problem.

7 N. J. DeMoes et al., It. J. Comp. Meth. ad Exp. Meas., Vol. 0, No. 0 (08) 7 5 Test Problem : Heat Trasfer i a Oe Dimesioal Rod 5. Descriptio of Test Problem - Numerical Solutios ad Assessmets Heat trasfer is oe of the may applicatios that Laplace s equatio ca model. Laplace s equatio is of the form u x + u = 0 (8) y ad is used i combiatio with the boudary coditios to solve for u (x, t), the approximate solutio to Test Problem. I this oe dimesioal rod, heat travels liearly through the rod o the top ad bottom edges of the rod for this problem while the temperatures o the left ad right boudary coditios are held at costat temperatures, 0 ad, respectively. The boudary coditios o the top ad bottom must equal the boudary coditios o the left ad right at each corer of the rod. The global boudary value problem boudary coditios satisfy these coditios are: x u(, y) = 0, u( x, ) = 0 0 x u(, y) =, u( x, ) = 0 0 (9) Because the legth is kow ad the boudary coditios are a fuctio of locatio ad legth all temperatures are kow alog the boudary. To solve for the temperatures o the iside of the rod, collocatio poits are chose at evely spaced locatios alog the boudaries ad the kow boudary temperatures are used to solve for the complex coefficiets i the approximatio fuctios. The followig sectios provide the results of the approximatio fuctios compared to the exact solutio alog the ceter lie of the domai. The ceter lie is used for accuracy aalysis withi the domai because that lie is furthest from either boudary, makig the ceter lie the lie i which approximatio error will be the greatest withi the boudary domai. Examiatio of the domai where the maximum error occurs provides clarity o the accuracy of the approximatio withi the etire domai because all other approximatio error withi the domai is less. The followig sectios display the results. 5. Compariso of CVBEM Basis Fuctios with Liear Boudary Coditios The CPM, CVBEM ad complex poles basis fuctio approximatio fuctio use = 3 collocatio poits to solve for the approximatio fuctios. The Lauret series uses = 7 istead of = 3 because = 7 provides a more approximate solutio. Figure ad 3 break up the four approximatio fuctios ito two groups of two plotted agaist the exact solutio. I Fig., the exact solutio of the problem which is kow because test problem s boudary coditios are cotiuous throughout the problem domai is plotted agaist the CPM ad CVBEM approximatio method. There appears to be o sigificat error betwee the CPM ad CVBEM approximatios ad the exact solutio. The same coclusio is made about the Lauret series approximatio ad the complex poles basis fuctio approximatio i Fig. 3.

8 8 N. J. DeMoes et al., It. J. Comp. Meth. ad Exp. Meas., Vol. 0, No. 0 (08) Figure : CPM ad CVBEM vs exact. Figure 3: Lauret ad complex poles vs exact. Computatioal error aalysis is depicted i Figs For this paper computatio error is defied as f( x, y) u ( xy, ) CE = f( x, y) where CE is computatioal error, f(x, y) is the exact solutio, ad u (x, y) is the type of basis fuctio approximatio. Figure 4 shows the amout of computatioal error for the etire legth of the domai o the ceter lie for the CPM ad CVBEM approximatio. Figure 5 provides the same iformatio for the Lauret ad complex poles basis fuctio approximatios. The CPM, CVBEM, ad complex poles basis fuctio approximatio all produce better approximatios for the boudary value problem tha the Lauret series approximatio. Each (0) Figure 4: CPM ad CVBEM error

9 N. J. DeMoes et al., It. J. Comp. Meth. ad Exp. Meas., Vol. 0, No. 0 (08) 9 Figure 5: Lauret ad complex poles Error. type of basis fuctio acts differetly. Thus, some basis fuctios are more suitable for specific types of modelig problems tha other basis fuctios. Additioally, the figures above depict the error alog the ceter lie. This error distributio is i lie with the Mi/Max Modulus Theorem. Because the max error occurs alog the boudary, examiatio of the error alog the boudary is oly ecessary, reducig computatioal cost. 6 Test Problem : Fluid Flow i a 90-Degree Bed Ideal fluid flow i a 90-Degree Bed ca be modeled by the fuctio f(z) = z. The real part of f(z), x y represets the isopotetial values. Because fluid flow potetials are depedet o locatio, the boudary coditios o all four sides of the domai are of the same form, x y. The domai i this problem is iitially set to a :0 ratio. Sectio 6., depicts the accuracy of the four basis fuctio approximatios to the exact solutio alog the ceter lie. The same boudary coditios are applied to two other high aspect ratio domais or ratios :5 ad :50 to determie the maximum absolute error for each approximatio compared to the error of the FEM discrete approximatio. 6. Descriptio of Test Problem - Numerical Solutios ad Assessmets Similar to Test Problem, Laplace s equatio u x + u = 0 () y is the goverig PDE that must be solved to create a approximatio for test problem. The global boudary value problem boudary coditios are give by: u (, y) = x y, u (, y) = x y, u ( x, ) = x y, u ( x, ) = x y. () 6. Compariso of CVBEM Basis Fuctio Performace Figure 6 shows the compariso of the two most accurate basis fuctio approximatios plotted agaist the exact solutio, x y. Ocular aalysis determies o sigificat differece betwee the exact solutio ad the approximatio solutios for the CPM ad CVBEM basis fuctios. Similarly Fig. 7 depicts the Lauret series ad Complex Poles approximatio

10 0 N. J. DeMoes et al., It. J. Comp. Meth. ad Exp. Meas., Vol. 0, No. 0 (08) Figure 6: CPM ad CVBEM vs exact. Figure 7: Lauret ad complex poles vs exact. fuctio with = 7 collocatio poits ad = 3 collocatio poits, respectively. The reaso that the Lauret series used 7 collocatio poits istead of = 3 like the other three basis fuctios is do to the icreased error whe icreases for the Lauret Series. This is due to the computers ability to hadle the ill-coditioed matrix caused by the ature of the basis fuctio. Figures 8 ad 9 depict the error, defied to be the differece betwee the exact solutio ad the approximate solutio o the ceter lie, for each approximatio fuctio. The ceter lie is defied to be the the lie bisectig the problem domai at y =.5. The error is sigificatly less for the CPM tha for ay other basis fuctio approximatio. The reaso for this is that the problem that is beig modeled u (x, y) = x y is a part of the basis fuctio family that makes up the CPM approximatio. Thus, the exact solutio is cotaied i the CPM approximatio, resultig i zero error. This would ot occur i a applicatio problem, but is importat to uderstad. Figure 8: CPM ad CVBEM error.

11 N. J. DeMoes et al., It. J. Comp. Meth. ad Exp. Meas., Vol. 0, No. 0 (08) Figure 9: Lauret ad complex poles error. 6.3 Error Compariso for Differet Domai Ratio Additioal aalysis o the impact domai ratio has o the accuracy of the CVBEM resulted i testig three differet domai ratios across the four basis fuctio families used i the CVBEM, complex moomials, traditioal CVBEM, complex poles, ad Lauret series usig = 8 collocatio poits. The maximum computatioal error of each approximatio is compared to the computatioal error of the FEM that uses a step size of 0.. The error aalysis is coducted o Test Problem. The results are listed i Table. The error data reveals that the higher the domai ratio, the greater the error for each. While the CPM ad complex poles basis fuctio family error decrease sigificatly, the CVBEM appears to maitai its validity with limited chage i error. Additioally, the Lauret series basis family produces the most iaccurate model. However, the FEM is the most accurate method of modelig BVPs i high aspect ratio domais out of the basis fuctio families tested. There are two reasos for why the FEM produced a better approximatio for this problem. The computer ad software used to create our approximatio fuctio fails to accurately solve for the ill-coditioed matrix that is created whe solvig for the coefficiets. As a result, the complex approximatio fuctios are limited to = 8 basis fuctios. If the program ad computatioal ability icreased ad the umber of basis fuctios icreased there would be higher accuracy i the complex approximatio fuctios that may be greater tha the FEM. The secod effect of havig limited to 8 is that there is a greater gap betwee each of the collocatio poits alog the problem domai. This results i greater ambiguity i the approximatio fuctio as the space betwee kow potetial values icreases with a greater aspect ratio. Although these coclusios are true, ode cout is ot totally idicative of modelig performace because the positioig of the model odes ad collocatio poits ca have Table : Display of error for differet basis families. Maximum Relative Error o the Boudary =8 Domai Ratio CPM CVBEM Complex poles Lauret FEM :0.03x0-6.8x x0-3.4x x0-3 :5.743x x x0-0.94x x0 - : x x x0-0.5x x0-0

12 N. J. DeMoes et al., It. J. Comp. Meth. ad Exp. Meas., Vol. 0, No. 0 (08) cosiderable ifluece o computatioal results. Curret research shows that similar computatioal error may be achieved with fewer odes ad collocatio poits by careful positioig of the odes ad collocatio poits. 7 Coclusios ad Recommedatios for Future Research The Complex Variable Boudary Elemet Method has the capability to use differet basis fuctios as the aalytic fuctios to approximate the potetial fuctio. I this paper, error for each basis fuctio was examied as the domai ratio decreases. The reaso for the icrease i error occurs because of the limitatios i computig power. The computers used i these tests were oly able to use up to = 8 collocatio poits. Thus, as the domai stretches with higher aspect ratios, the distace betwee each collocatio poit icreases, icreasig the probability that the approximate solutio estimates a icorrect potetial value. Although error icreased for each basis fuctio family that was tested, the traditioal CVBEM model with basis fuctios of the form (z z ) l (z z ) decreased i accuracy the slowest as the domai ratio icreased; therefore, the traditioal CVBEM holds its accuracy the best i problems with high aspect ratios. I Table, the CVBEM demostrated that the error decreased betwee the ratios of : ad :0 before icreasig for higher domai ratios. Future research icludes determiig the domai ratio that optimizes accuracy for each basis fuctio ad the specific boudary coditios. Oce the basis fuctio is paired with the domai ratio that optimizes CVBEM performace, the research ca be exteded to applyig the basis fuctio family to 3 dimesioal domais with the same high aspect ratio. Refereces [] Hromadka, T.V. & Guymo, G.L., Numerical aalysis of ormal stress i o- Newtoia boudary layer flow. Joural of Numerical Methods i Egieerig, 0(), pp. 5 37, 984. [] Hromadka, T.V. & Lai, C., The Complex Variable Boudary Elemet Method. Spriger- Verlag: New York, 984. [3] Hromadka, T.V. & Whitley, R.J., Approximatig three-dimesioal steady state potetial flow problems usig two-dimesioal complex polyomials. Egieerig Aalysis with Boudary Elemets, 9, pp , [4] Hromadka, T.V. & Whitley, R.J., Foudatios of the Complex Variable Boudary Elemet Method. Spriger: New York, 04. [5] Hromadka, T.V. & Whitley, R.J., A ew formulatio for developig CVBEM approximatio fuctios. Egieerig Aalysis with Boudary Elemets, 9(), 39 4, [6] Dumar, P. & Kumar, R., Complex variable boudary elemet method for torsio i aisotropic bars. Applied Mathematical Modelig, 7(), pp , [7] Hromadka, T.V. & Whitley, R., Advaces i the Complex Variable Boudary Elemet Method. Spriger-Verlag: New York, 998.

13 N. J. DeMoes et al., It. J. Comp. Meth. ad Exp. Meas., Vol. 0, No. 0 (08) 3 [8] Wilkis, B.D., Greeberg, J., Redmod, B., Baily, A., Flowerday, N., Kratch, A., Hromadka, T.V., Boucher, R., McIvale, H.D. & Horto, S., A usteady two dimesioal complex variable boudary elemet method. Applied Mathematics, 8(6), pp , [9] Hromadka, T.V., A Multi-Dimesioal Complex Variable Boudary Elemet Method. WIT Press: Southampto, Eglad, 00. [0] Hromadka, T.V. & Lai, C., The Complex Variable Boudary Elemet Method i Egieerig Aalysis. Spriger-Verlag: New York, First prit 987, Republished Digitally, 0. [] Hromadka, T.V., Isehour, M., Rao, P., Ye, C.C. & Crow, M., Computatioal biopsy to assess accuracy of large scale computatioal groudwater flow models. Iteratioal Joural of Egieerig Scieces ad Maagemet, 7(), pp. 6, 07. [] Wilkis, B.D., Hromadka, T.V. & Boucher, R., A coceptual umerical model of the wave equatio usig the complex variable boudary elemet method. Scietific Research Publishig, 8(5), pp , [3] Wilkis, B.D., Flowerday, N., Kratch, A., Greeberg, J., Redmod, B., Hromadka, T.V., Hood, K., Boucher, R. & McIvale, H.D., A computatioal model of groudwater moud evolutio usig the complex variable boudary elemet method ad geeralized fourier series. The Professioal Geologist, 54(), pp. 5, 07. [4] Hood, K., Wilkis, B.D., Hromadka, T.V., Boucher, R. & McIvale, H.D., Developmet of a earthe dam break date base. The Professioal Geologist, 54(), 0, 07. [5] Matthews, J.H. & Howell, R.W., Complex Aalysis for Mathematics ad Egieerig. Times Mirror Compay: Iowa, 996. [6] Bohao, A.W. & Hromadka, T.V., The complex polyomial method with a leastsquares fit to boudary coditios. Egieerig Aalysis with Boudary Elemets, 33(8 9), pp. 00 0, [7] Bloor, C., Hromadka, T.V., Wilkis, B.D. & McIvale, H.D., Cvbem ad fvm computatioal model compariso for solvig ideal fluid flow i a 90-degree bed. Ope Joural of Fluid Dyamics, 6, pp , 06.