《复变函数论》课程教学资源（书籍文献）Complex variable element solution of potential flow problems using Taylor series for error analysis

Complex variableelement solution ofpotential flow problems using Taylor seriesfor error analysisT. V. Hromadka IIDepartment of Mathematics,California StateUniversity,Fullerton,CA, USAR. J. WhitleyDepartmentofMathematics,UniversityofCalifornia,Irvine,CA,USAThe complex variable boundary element method(CVBEM) is a numerical approach to solving boundaryvalue problems of wo-dimensional Laplace and Poisson equations.The CVBEM estimator exactlysolves the governing partial differential equations in the problem domain but only approximatelysatisfies theproblemboundary conditions.In this papera new CVBEM errormeasure is used in aidingin the development ofimprovedCVBEM approximators.The newapproach utilizes Taylor series theorand can be readily programmed into computer software form.On the basis of numerous test appli-cations it appears that use of this new CVBEM error measure leads to the developmentof significantlyimprovedCVBEMapproximationfunctions.Keywords: complex variable boundary element method,Taylor series, error analysis,potential flowproblems1.Introductionand real variable boundary element methods.2 How-ever,issues regarding conditioning of the stiffness ma-The objective in using the complex variable boundarytrix for cases of small discretization remain open. Theelcmcntmethod (CVBEM)istoapproximatcanalyticCVBEM resultsina well-conditionedmatrix systemcomplexfunctions.Given that w is a complex functionthat may provide an alternative to highly discretizedwhich is analytic over a simply connected domain 2conditioningproblemswith boundaryvaluesw()for rEF(TisasimpleIn this paper the CVBEM is expanded as a gener-closed contour), then both the real () and the imag-alized Fourier series but introduces the use of Taylorinary()partsof w=+satisfytheLaplaceseriesdefinedoneachboundaryelement,expandedequation over2.Thus two-dimensional potentialflowwith respect to each nodal point.Boundary conditionsproblems can be approximated by the CVBEM, in-areapproximatedina"mean-squareerrorsenseincluding steady-state heat transport, soil water flowthat a vector space norm is defined which is analogousplanestress,and elasticityto the lznorm and then minimized by the selection ofThe development of the CVBEM for engineeringcomplex coefficients to be associated to each nodalapplications is detailed by Hromadka and Lai.Thepoint located on the problem boundary, F.For prob-CVBEM is a boundary integral technique,and con-lems in whichtheboundary conditionvaluesarevaluessequently,a literature review of this class ofnumericalof a function analytic on n UT the CVBEM approx-methods can be found in works such as the one byimation function converges almost everywhere (ae)Lapidus and Pinder.2The Laplaceand Poisson equaonTtions have been solved numerically with a high rate ofThe CVBEM generalized Fourier series approachconvergence by the finite element, finite difference,will be developed before the development of the nu-merical techniqueis presented, Tokeep the paper con-cise, the development of the CVBEM approach, theAddress reprint requests to Prof.Hromadka at the Dept. of Math-definition of the working vector spaces,proofs of conmatics,CaliforniaStateUniversity,Fullcrton,CA92634,USAvergence of the generalized Fourier series expansion,and theproof of boundarycondition convergence areReceived 16 April 1991; revised 15 October 1991; accepted 12 Noallbrieflypresented.vember19911992Butterworth-Heinemann114Appl.Math.Modelling,1992,Vol.16,March

Complex variable element solution of potential flow problems using Taylor series for error analysis T. V. Hromadka II Department of Mathematics, California State University, Fullerton, CA, USA R. J. Whitley Department of Mathematics, University of California, Irvine, CA, USA The complex variable boundary element method (CVBEh4) is a numerical approach to solving boundary value problems of two-dimensional Laplace and Poisson equations. The CVBEM estimator exactly solves the governing partial differential equations in the problem domain but only approximately satisfies the problem boundary conditions. In this paper a new CVBEM error measure is used in aiding in the development of improved CVBEM approximators. The new approach utilizes Taylor series theory and can be readily programmed into computer software form. On the basis of numerous test appli￾cations it appears that use of this new CVBEM error measure leads to the development of significantly improved CVBEM approximation functions. Keywords: complex variable boundary element method, Taylor series, error analysis, potential flow problems 1. Introduction The objective in using the complex variable boundary element method (CVBEM) is to approximate analytic complex functions. Given that u is a complex function which is analytic over a simply connected domain 52 with boundary values w(l) for C E I (I is a simple closed contour), then both the real (4) and the imag￾inary (4) parts of w = 4 + it,h satisfy the Laplace equation over 1R. Thus two-dimensional potential flow problems can be approximated by the CVBEM, in￾cluding steady-state heat transport, soil water flow, plane stress, and elasticity. The development of the CVBEM for engineering applications is detailed by Hromadka and Lai.’ The CVBEM is a boundary integral technique, and con￾sequently, a literature review of this class of numerical methods can be found in works such as the one by Lapidus and Pinder.* The Laplace and Poisson equa￾tions have been solved numerically with a high rate of convergence by the finite element, finite difference, Address reprint requests to Prof. Hromadka at the Dept. of Math￾ematics, California State University, Fullerton, CA 92634, USA. Received 16 April 1991; revised 15 October 1991; accepted 12 No￾vember 1991 and real variable boundary element methods.2 How￾ever, issues regarding conditioning of the stiffness ma￾trix for cases of small discretization remain open. The CVBEM results in a well-conditioned matrix system that may provide an alternative to highly discretized conditioning problems. In this paper the CVBEM is expanded as a gener￾alized Fourier series but introduces the use of Taylor series defined on each boundary element, expanded with respect to each nodal point. Boundary conditions are approximated in a “mean-square” error sense in that a vector space norm is defined which is analogous to the l2 norm and then minimized by the selection of complex coefficients to be associated to each nodal point located on the problem boundary, I. For prob￾lems in which the boundary condition values are values of a function analytic on R U r the CVBEM approx￾imation function converges almost everywhere (ae) on r. The CVBEM generalized Fourier series approach will be developed before the development of the nu￾merical technique is presented. To keep the paper con￾cise, the development of the CVBEM approach, the definition of the working vector spaces, proofs of con￾vergence of the generalized Fourier series expansion, and the proof of boundary condition convergence are all briefly presented. 114 Appl. Math. Modelling, 1992, Vol. 16, March 0 1992 Butterworth-Heinemann

CVBEM using Taylor series:T.V.Hromadka ll and R. J. WhitleyIn this paper a new CVBEM error measure is usedProofFor wEWa, then w EE2(),and the result followsinaidinginthedevelopmentofimprovedCVBEMap-proximators.The new approach utilizes Taylor seriesimmediately.theory and can be readily programmed into computersoftware form.This new approximation error evaluation technique provides a convenient-to-use measure2.2.Almost-everywhere (ae) equivalenceinimprovingCVBEMmodelsbyfurtherdiscretization.For w EWo,functions x E Wn cqual to w ae on Irepresent an equivalence class of functions which may1.1. Definition of working space, Wobe noted as [w]. Therefore functions x and y in WnareLet be a simply connected convex domain within the same equivalence class whena simple closed piecewise linearboundarywith centroid located at O + oi. Then in this paper, w E WoIx-ydμ=0has the property that w(z) is analytic over U T.For simplicity, w E Wo is understood to indicate [a]I.2.Definition of thefunctionoThis follows directly from the fact that integrals overFor o E W the symhol all is notation forsets of measure zero have no effect on the integralvalue.(Rea)2 dμ +/ (Im a)2 dμr.where both Ta and μ are a finite number of subsets2.3. Theorem (uniqueness of zero element in Wo)of r that intersect only at a finite number of pointsLetwEWnandΦ=Oaeon Tand=oaeonin r.T.Then (w,w)= 0→ w =[0].The symbol oll, for w E Wa is notation forGreen's theorem states, let F and G be continuousand have continuous first and second partial deriva-[ lo(s)e du0=tives in a simply connected region R bounded by ap≥1simple closed curve C. ThenLaG-0)--Of importance is the case of p = 2: [( +)PF0C(ay([Rew]2 + [1mw]) dμ[02=(aFaGFa)dxddxaxaydy1.3.Almost-everywhere(ae)equalityLet F,G-.ThenA property that applies everywhere on a set E ex-adcept for a subset E' in E such that the Lebesgue mea-drsure m(E') =o is said to apply almost everywhereanS(ae).Because sets of measure zero have no effect onintegration,almost-everywhereequality onFindicatesdothe same class of element.Thus for w E Wo,[] =(w E Wa:w(g) are equal ae for g E j. For example,[0] =(wEW,:w()=0ae,Ei).When understood,ButV2 =0in 2.Thusthe notation[I will be dropped.ad-dP=+b-an2.Mathematical development2The Hp spaces (or Hardy spaces)are welldocumentedBut (w,w)=0 impliesΦ=0 on Fand=0 on Tinthe literature.3Of spccial intcrest arc thc Ep(Q)spaces(henceaulas=0aplan=0),andof complex valued functions. If w E E2(2), then )satisfies the conditions of the definition of working$pndr +Ippndr-(+ )space on Wn,where lo(8)llz is bounded as1.Finally,if E E(),then the Cauchy integral rep00resentation of w(z)for z E 2 applies.It is seen thatWn C E2(2).Thus (w,w) = 0 Φx - 0 - dy on 0.2.1.Theorem (boundary integral representation)Thus Φ(x,y) is a constant in 2. ButLet aE Waand zE Q. ThenlimΦ=0Φ=0ECo(r)dy1(z) =2miJ5-ZSimilarly,=0. Thus w= [0].Appl.Math.Modelling,1992,Vol.16,March115

CVBEM using Taylor series: T. V. Hromadka II and R. J. Whitley In this paper a new CVBEM error measure is used in aiding in the development of improved CVBEM ap￾proximators. The new approach utilizes Taylor series theory and can be readily programmed into computer software form. This new approximation error evalua￾tion technique provides a convenient-to-use measure in improving CVBEM models by further discretization. Proof For o E Wn, then w E E*(a), and the result follows immediately. 2.2. Almost-everywhere (ae) equivalence For w E Wn, functions x E Wn equal to w ae on r represent an equivalence class of functions which may be noted as [w]. Therefore functions x and y in Wn are in the same equivalence class when 1 .I. Definition of working space, Wn Let R be a simply connected convex domain with a simple closed piecewise linear boundary r with cen￾troid located at 0 + Oi. Then in this paper, w E Wn has the property that o(z) is analytic over fl U r. I .2. Definition of the function /wll For w E Wn the symbol ((w(( is notation for l/2 II 0 II = [/ (Re o)? dp + (Im w)* dp I‘+ I-+ 1 where both r+ and re are a finite number of subsets of r that intersect only at a finite number of points in r. The symbol ]/ml], for w E Wn is notation for ll4lp = [ j- I4CP b] I” p21 1‘ Of importance is the case of p = 2: I I/2 Ibll~ = _( We WI* + Pm 4’) dcL I I .3. Almost-everywhere (ae) equality A property that applies everywhere on a set E ex￾cept for a subset E’ in E such that the Lebesgue mea￾sure m(E’) = 0 is said to apply almost everywhere (ae). Because sets of measure zero have no effect on integration, almost-everywhere equality on r indicates the same class of element. Thus for w E Wn, [o] = {o E W,:&) are equal ae for 5 E r}. For example, [O] = {w E W,:w([) = 0 ae, b E r}. When understood, the notation [ ] will be dropped. 2. Mathematical development The HP spaces (or Hardy spaces) are well documented in the literature.3 Of special interest are the Ep(S1) spaces of complex valued functions. If w E E*(R), then w satisfies the conditions of the definition of working space on Wn, where ]]o(S[)]]~ is bounded as 6 -+ 1. Finally, if w E E2(0), then the Cauchy integral rep￾resentation of w(z) for z E fl applies. It is seen that Wn C E2(sL). 2.1. Theorem (boundary integral representation) Let w E Wn and z E R. Then Ix-y(dp=O r For simplicity, w E Wn is understood to indicate [w]. This follows directly from the fact that integrals over sets of measure zero have no effect on the integral value. 2.3. Theorem (uniqueness of zero element in W,) Let o E Wn and 4 = 0 ae on r+ and CF, = 0 ae on rB. Then (w,w) = 0 + o = [O]. Green’s theorem states, let F and G be continuous and have continuous first and second partial deriva￾tives in a simply connected region R bounded by a simple closed curve C. Then Edx-$dy) = -[,[F($+$) aFaG aFaG + axdx+- a~ ay )I dx dy Let F = 4, G = 4. Then But V24 = 0 in a. Thus But (w,w) = 0 implies 4 = 0 on I’, and Cc, = 0 on r+ (hence a$/& = 0 3 at#dan = O), and Thus (w,w) = 0 3 & = 0 = & on a. Thus #(x,y) is a constant in a. But Similarly, Ic, = 0. Thus w = [O]. Appl. Math. Modelling, 1992, Vol. 16, March 115

CVBEM using Taylor series: T.V.Hromadka Il and R.J.Whitley2.4.Theorem(W,isvectorspace)3.4.DefinitionofN()W is a linear vector space over the field of realA linearbasisfunctionN()is definedforEbynumbers.SErj-1(( - zj-1)/(zj zj-1)Proofger,N(g) -(2g+1 -g)/(z)+1 - 2)This follows directly from the character of analytic0functions. The sum of analytic functions is analytic,gE,-TUT,and scalar multiplication of analytic functions is ana-The value of N(g) is found to be real and bounded aslytic.The zero element has alreadybeen noted by [0]indicated by the next theorem.intheorem2.33.5.Theorem2.5.Theorem (definition of the inner product)Let N,(g) be defined for node P, E r. Then O Let x, y, z, E Wo.Define a real-valued functionN(9) ≤ 1.(x.y) by3.6.Definition of Gm(g)[RexReydu+ImxImyduLetanodalpartitionof mnodes(P)bedefinedon(x,y) =Iwith m≥Aand with scalel.At each nodeP,define0nodal values w, = Φ, + ii,where Φ, and , are realThen (, ) is an inner product over Wo.numbers. A global trial function Gm(c) is delined on rProofforErbyItis obvious that (x,y)=(y,x); (kx,y)=k(x,y)formk real: (x + y,z) = (x,z) + (y,z): and (x,x) = Ixll ≥ 0Gm() = ZN(9)a)By theorem 2.3, (x,x) = 0 implies Re x = 0 ae on Taj=1and Im x = 0 ae on Fand x =[0] E Wn3.7.TheoremThree theorems follow immediately from the above,and hence no proof is given.From definition 3.6, Gm() is the sum of integrablecontinuous functions, and hence (a)Gm()is contin-uouson F and (b) for w(g) EWn,w(g) EL2(r)2.6.Theorem (Woisan innerproduct space)For the defined inner product, Wo is an inner prod-3.8.Discussionuct space overthefield ofreal numbers.As a result of w() E L2(r), then w() is measurableon I, and for every e > o there exists a continuouscomplex-valued function g() such thatIo() - g(g)l O there exists a G(g) such thatIlo() -- G(gl O thereexistsa CVBEMapproximationm(z)suchthata(z)wheremisthenumberof complexvariableboundaryelements (CVBEs)@m(z)l <e.116Appl.Math.Modelling,1992,Vol.16,March

CVBEM using Taylor series: T. V. Hromadka Ii and R. J. Whitley 2.4. Theorem (W, is vector space) Wn is a linear vector space over the field of real numbers. Proof This follows directly from the character of analytic functions. The sum of analytic functions is analytic, and scalar multiplication of analytic functions is ana￾lytic. The zero element has already been noted by [0] in theorem 2.3. 2.5. Theorem (de$nition of the inner product) Let X, y, z, E WQ. Define a real-valued function (x,Y) by (x,y) = JRexReydp + IImxImydp r, r* Then ( , ) is an inner product over Wn. Pro@ It is obvious that (x,y) = (y,x); (kx,y) = k(x,y) for k real; (x + y,z) = (x,z) + (y,z); and (x,x) = ((xl] 2 0. By theorem 2.3, (x,x) = 0 implies Re x = 0 ae on I+ and Im x = 0 ae on I+ and x = 101 E Wn. Three theorems follow immediately from the above, and hence no proof is given, 2.6. Theorem (W, is an inner product space) For the defined inner product, Wn is an inner prod￾uct space over the field of real numbers. 3. The CVBEM and Wn 3.1. Definition of A Let the number of angle points of I be noted as A. By a nodal partition of I, nodes {Pj} with coordinates {zj} are defined on I such that a node is located at each vertex of I and the remaining nodes are distributed on I?. Nodes are numbered sequentially in a counterclock￾wise direction along I. The scale of the partition is indicated by 1, where I = max lZj+i - Zjl. Note that no two nodal points have the same coordinates in I?. 3.2. Definition of I’j A boundary element Ii is the line segment joining nodes zj and Zj+ 1; r. ,= {Z: Z = Z(t) = Zj(1 - t) + Zj+lf, 0 5 t 5 1). (Note for m nodes on I that z,+i = z,.) 3.3. Discretization of r into CVBEs Let a nodal partition be defined on I. Then I= ;jIj i=l where m is the number of complex variable boundary elements (CVBEs). 116 Appl. Math. Modelling, 1992, Vol. 16, March 3.4. Definition of N,(l) A linear basis function N&J is defined for 5 E I by Nj(0 = { (5 - Zj- l)/(Zj - Zj- 1) 5E rj-1 (Zj+l - 0/(Zj+1 - Zjl 5E rj 0 54rj_lUrj The value of Nj([) is found to be real and bounded as indicated by the next theorem. 3.5. Theorem Let Nj(5) be defined for node Pj E r. Then 0 5 NJ[) 5 1. 3.6. Definition of G,(l) Let a nodal partition of m nodes {Pi) be defined on I? with m zz A and with scale 1. At each node_Pj, define nodal values Oj = $j + i(clj, where j and $j are real numbers. A global trial function G,(l) is defined on I for I E I by j=l 3.7. Theorem From definition 3.6, G,(J) is the sum of integrable continuous functions, and hence (a) G,(l) is contin￾uous on I and (b) for ~(5) E Wn, ~(5) E L’(r). 3.8. Discussion As a result of w(l) E L*(I), then w(C) is measurable on I, and for every E > 0 there exists a continuous complex-valued function g(l) such that II40 - sK)IL ll, 0 there exists a G(c) such that II40 - ‘XIII 0 there ex￾ists a CVBEM approximation &J,(Z) such that I&) - &Jz)( < E

CVBEM using Taylor series:T.V.Hromadka lIl and R.J.WhitleyButProofLet d = min / - zl. E T. Then for a global trialN+L/m)5- Zfunction Gm()defined onF,FZ-Z1 r[w(g) - Gm(g)]dg[o(z) - 0m(z)] =and thus2i.-zM2㎡REN()≤2N1R/22个(mRmRGmll-2元(7)Choosing Gm (see section 3.8) such that o - Gmll<whichis aresult independent of j.Notethatasthe2 de guarantees the desired result.partition of into CVBEs becomes finer,i.e., maxiT/l→0, then m → and [EN() -→ 0. Also, as theorderof theTaylor seriespolynomial increases,N→4. Taylor series expansions on CVBEs, and recalling that (L/m) < R/2, then [EN(g)| -→ 0.4.1.Construction4.3.CVBEM erroranalysisIet a EWo.Then wis analytic on an open domainFrom Cauchy'stheorem,for zE.Q4suchthatQUFisentirelycontainedintheinterior1Cw(r)dyof Q4,LetF*be in Q4 such thatF*is a finite length(8)(z) =2mi.1-zsimple closed contour that is exteriorto U F.Thenw is analytic on r*,and by the maximummodulusOnr,lettheorem,mZEr*(2)0(z)/≤Mo(t) = Ex,T(r)ter(9)j-1for somepositive constantM.where X, is the j-element characteristic function (i.e.,Also,X, = 1 for E;o, otherwise).Then for z E ,(3)ZEQUr[o(z)]≤M2xT()dDefine a nodal partition of m nodes on T. Complex1w(z) =variable boundary elements are defined to be the straight2元iJS-zline segments , = [zi, zi+i] where, for m nodes,Zm+1 = ZrAt the midpoint z, =(z, + Zj+1) of each)=(10)T.expand w(z)intoaTaylor series T(z-z).Each=/2元JL-zT,(z-z)has anonzero radius of convergenceRi,andz)=w(z)intheinteriorof circleC,=(z:zTz-For T() = PN() + EN(),z=R,l.The C,all minimally have radii R,where R 42l such that , E T and (2 E T*. Descretize1minS-(z)=T into m CVBEs, I,j = 1, 2,,m, such that the2m212m]1-zS-zlengthofT,F≤2L/mwhereL=Jrldzland2L/m<R, and the other conditions regardingplacement of= 0(z) + E(z)(11)nodes at angle points of T are satisfied.The v(z) is the CVBEM approximation based onorderNpolynomials,where itis understood mnodes4.2.Taylor series expansionarc uscd. The crror, Ev(z), is cvaluated in magnitudcForEf,for z E and using (12) to be(EN()dyT( - Z) =PN()+EN()(4)[E~(z)] :2TI-zwhereNisthepolynomial degree,andfrom Cauchy'stheorem,1 (m)(max ITJD(maxE)(g)N+w(z)dz2元min[2EN() =(5)2Tiz-r2(2）()(2M/mRThemagnitude of EN()l is,forevery j,DN+:-Z|N+1 max |0(2)2mR2LMEN()] ≤(12)2元z-元minlz -lTDmRZEC,gEr (6)where D = min - z for ErAppl.Math.Modelling.1992,Vol.16,March117

CVBEM using Taylor series: T. V. Hromadka II and R. J. Whitley Proof Let d = mitt (5 - z(, J E r. Then for a global trial function G,(J) defined on r, Choosing G, (see section 3.8) such that J/w - G,& / -+ 0. Also, as the order of the Taylor series polynomial increases, N + cc), and recalling that (L/m) l> 2rr mm JJ - 21 (12) Appl. Math. Modelling, 1992, Vol. 16, March 117

CVBEM using Taylor series:T.V.Hromadka Il and R.J.WhitleyRecalling that (L/m) 1 a higher-order poly-For order N Taylor series expansions the CVBEMnomial expansion is used, and consequently,ad.ditional interpolation nodes are defined in each FisetsintheCauchy limitFor example,for N = 2 a midpoint node is addedNPN(g)dgto each I; for N = 3, two additional nodes arePN(z) =(14)2mij=11s-zdefined in the interior of each ,3. Given N, a matrix solution provides the coefficientsas z-→F while zEneeded to define interpolating polynomials foreachIf collocation is used, the numerical approach is toCVBE,using splines.set!4.The unknown nodal values are estimated by meansof collocation or least-squares errorminimization.PN(z) = (z)(15)5.Usingthe estimatesfortheunknownnodal valuesfor each nodal coordinate z, E ry.a CVBEM approximation (z)is well defined forIf a least-squares approach is used, the numericaestimating w(z) values in the interior of 2.approachisto minimize46.CVBEM error is evaluated by comparing o(z)andw(z)withrespecttotheknownboundaryvaluesofTErJIPN() - (j= 1,2,...,m (16)w(z) on F; that is, compareto on Fa,and com-Lettingpareto on F.(From the previousmathematicaldevelopment, if = 中 on and = on IgGm() =≥N(C)a)then w(z) =w(z) for all z E Q, if w E Wn.)7. After and are compared as to boundary con-j1dition values, then the CVBEM program user canwhere it is recalled , = w(z,),decrease thepartition scale (i.e.,increase the num)limG() = ()berof nodesuniformly)and/orincreasetheCVBE0interpolating polynomial order,N.The modellinggoal is to increase (m,N) until the boundary con-andditionsarewell approximatedbythe CVBEM (z)1G)dyIt is recalled that regardless of goodness of fit ofw() = lim,ZEn(17)02miJg-za(z) to thc problcm boundary conditions, the com-ponents of o(z), i.e., the functions Φ(z) and μ(z)where l is the scale of the nodal partition of T.(where(z)=(z)+i(z))exactlysatisfytheLaplacian V25 - 0 and v2 = 0 for ali z E 0. Thusthereis no error in satisfying the Laplacian equation5.Implementationin 2.This feature afforded bythe CVBEM is notIn general,onedoes not haveboth andvaluesachieved by use of the usual finite element or finitedifference numerical techniques, which have errorsdefined on F but instead has valucs defined only ona portion of I, specified as F, and values definedin satisfying theproblem's boundary conditions asonly on the remaining portion of T, Tu, where , Uwell as errors in satisfying the flow field LaplacianT = . That is, we have a mixed boundary valuein .8. A new approach to evaluating CVBEM approxi-problem.The numerical formulation given in the above equa-mation error is to examine the closeness betweenvaluesofthe interpolatingpolynomial ineachCVBEtions solves for the unknown values on Fand theand the CVBEM o(z) function, for z in I.That is,unknown valucs on F.Once the unknown and examine in a Cauchy limit PN(g)-()ll2,values are estimated, denoted as and , then theglobal trial functions are well defined and can be usedforallCVBETi.AsPN()-o(llz-→0(i.e.,byin (z) estimates for the interior of . The possibleincreasing m and N) for all j and all E F,thennecessarily, a(z) -→ w(z) for all z E , if o(z) E Wn.variations in such boundary condition issues are ad9.ThechoicetoincreasemorNismadebyincreasingdressed by Hromadka and Lai.118Appl.Math.Modelling,1992,Vol.16,March

CVBEM using Taylor series: T. V. Hromadka II and R. J. Whitley Recalling that (L/m) 1 a higher-order poly￾nomial expansion is used, and consequently, ad￾ditional interpolation nodes are defined in each Ij. For example, for N = 2 a midpoint node is added to each Ij; for N = 3, two additional nodes are defined in the interior of each Ij. Given N, a matrix solution provides the coefficients needed to define interpolating polynomials for each CVBE, using splines. The unknown nodal values are estimated by means of collocation or least-squares error minimization. Using the estimates for the unknown nodal values, a CVBEM approximation h(z) is well defined for estimating w(z) values in the interior of 0. CVBEM error is evaluated by comparing h(z) and o(z) with respect to the known boundary values of o(z) on I; that is, compare 4 to $J on I+, and com￾pare $ to $ on I,.*(From the previousAmathematical development, if 4 = b, on I+ and Cc, = I,!J on Ia, then h(z) = w(z) for all z E a, if w E Wo.) After 3 and w are compared as to boundary con￾dition values, then the CVBEM program user can decrease the partition scale (i.e., increase the num￾ber of nodes uniformly) and/or increase the CVBE interpolating polynomial order, N. The modelling goal is to increase (m,N) until the boundary con￾ditions are well approximated by the CVBEM &j(z). It is recalled that regardless of goodness of fit of i;(z) to the problem boundary conditions, the com￾ponents of h(z), i.e., the functions 4(z) and 4(z) (where h(z) r 4(z) + i$(z)) exactly satisfy the Laplacian V*4 = 0 and V2$ = 0 for all z E R. Thus there is no error in satisfying the Laplacian equation in a. This feature afforded by the CVBEM is not achieved by use of the usual finite element or finite difference numerical techniques, which have errors in satisfying the problem’s boundary conditions as well as errors in satisfying the flow field Laplacian in R. A new approach to evaluating CVBEM approxi￾mation error is to examine the closeness between values of the interpolating polynomial in each CVBE, and the CVBEM h(z) function, for z in Ij. That is, examine in a Cauchy limit IPj”(Q - &(6)112, 5 E Ij, for all CVBE Ij. As llpy({) - &({)I12 -, 0 (i.e., by increasing m and N) for all j and all t E Ij, then necessarily, ;(z> -+ o(z) for all z E IR, if w(z) E Wn. The choice to increase m or N is made by increasing 118 Appl. Math. Modelling, 1992, Vol. 16, March

CVBEM using Taylor series:T.V.Hromadka Il and R.J.Whitleyafter the software generates successively finer CVBEMboth m andNinthoseboundaryelementsthathaveestimates, by discretization, by use of the error mea-the most approximation error of P(g)- ()ll forsure between PN()and w() on T.The maximal errorgEF.In this way,o(z)approximations improveE=o(z)-a(z)isthencomputedfordemonstrationin accuracy without excessive additional compupurposes, as w(z) is known.Plots of error for 25-andtation. Generally, three or four attempts in devel-40-node discretization are provided for each applica-oping o(z) functions may be needed for difficultpotential flow problems, each successive CVBEMtionTo demonstrate the error analysis procedures dis-approximator being based upon the prior attemptcussedabove,twomixed boundary valueproblems arebut with localized increases in mandN where ap-considered in which analytic solutions are known. Aproximation error was largest.FORTRAN computer program, based on the complexvariable boundary element method, which allows an6.Applicationincrease in Taylor series polynomial order or an in-crease in nodal density is used.A CVBEM computer program was prepared that in-Forbothproblems considered,an initial nodal pointcluded the ability to increase the number of boundaryschemeof12nodes is defined on eachof theproblemelements by discretizing specific elements into moreboundaries.Boundary conditions of specified streamelements, and also to increase the interpolating func-functionor specifiedpotentialfunctionvaluesareusedtion polynomialorder,N,within specificboundaryele-(even though flux typeboundaryconditions arements.Theprogram included in its output the comstraightforward to include).ACVBEM approximationputation of Py() - w(g)lz, EF, for each boundaryfunction is developed based on the initial nodal pointclcmcnt.For each CVBEM attempt, Nis increased byI andplacement, and streamlines are automatically gener-ated and plotted as solid lines within theproblem do-the boundary element halved in length (to produce twomain (recall the CVBEM develops a function (x,y)elements)thathadthelargestvaluesofPN(g)-()l2insidetheproblemdomain,2).Forcomparisonpur-The modelling process continues until a reasonableposes,associated streamlines for the analytic solutiono(z)fit to the problem boundary conditions is achieved.w(z) also are plotted, as dashed lines, within 2.Applications demonstratingtheCVBEMtonumer-Because application of the CVBEM necessarily in-ically solve boundary value problems involving the twovolves problems in which the exact solution is un-dimensional LaplaceorPoissonequationcanbefoundin several publications.I-5 The focus of this paper isknown.one of thetwo boundary condition functionvalues is left"unknown"'along theproblemboundary.thepresentation of another CVBEM error evaluationBy using the error between the approximated boundarytechnique that appears to provide a more robust guidevalues and the known boundary values,additionalin developing subsequent improved CVBEM approx-CVBEM model complexity can be introduced.Theimators than the other techniques in use, such as theerrors computed are the usual integrated root-mean-approximate boundary technique,, and the usual eye-fitcomparisonsbetweenversusonFandversussquareerror andmagnitudeerror.Plots of these errors,tonas computed along theproblemboundary,areincludedintheapplication figures.In numerous test problems it was found that use ofSubsequent CVBEM approximations,ultimatelythePN(g)-(llzerrortopinpointlocalizedCVBEMleadingtouseof25and40nodesonT,areshowninapproximation error provided,in general, a better ap-the attached figures, along with a comparison ofproachtoimprovingCVBEMfunctions than the ap-strcamlines between approximation and solution re-proximate boundary approach. The following prob-sults, and the error plots. A quartic trail function islems demonstrate application of the P(g) - w(g)l2used in the 40-point discretization.error measure technique to locate where additional no-dal points need to be added to I in order to developApplication Amore refined CVBEM approximations. In each appli-Solve ap/ax? + a'olay? = 0 in , where is thecation a mixed boundary value problem is defined byprescription of either Φ,,or a/analong portions ofdomain shown in Figure I.Stream function values areI.The CVBEM is applied to an initial nodal pointspecified along the horizontal lines of r,and potentialdistribution along I, and then the error measure isfunction values are specified along the horizontal linesevaluated for each boundary element. The boundaryof F, forming a mixed boundary value problem, Theelement that manifests the largest value of error is thenanalytic solution used is w(z) = In [(z + 1)/(z - 1)].further discretized,or the Taylorpolynomial order in-Figures 1,2,and 3 show approximation results versuscreased by1(uptoamaximumorder of8 inthepre-exact values for 12-,25-, and 40-boundary nodes,re-pared computer software). The program user selects,spectively.Theaccompanyingfigures show magnitudeup front, the order of the Taylor polynomial to be used;and integrated root-mean-square error plots alongTthe program conducts the discretization.for theboundary values.Foreach problem shown theexact solution used toApplicationBgenerate the test problem is given. Initially, nodes areonly defined to be located at the vertices of I. Also,Figure 4 solves the Laplace equation for ideal fluidflow over a cylinder on the shown domain, 2. Thea quadraticpolynomialis usedforeachelement.There-Appl.Math.Modelling,1992,Vol.16,March119

CVBEM using Taylor series: T. V. Hromadka II and R. J. Whitley both m and N in those boundary elements that have the most approximation error of jIPj”({) - &.J([)II for 5 E rj. In this way, &J(Z) approximations improve in accuracy without excessive additional compu￾tation. Generally, three or four attempts in devel￾oping h(z) functions may be needed for difficult potential flow problems, each successive CVBEM approximator being based upon the prior attempt but with localized increases in m and N where ap￾proximation error was largest. 6. Application A CVBEM computer program was prepared that in￾cluded the ability to increase the number of boundary elements by discretizing specific elements into more elements, and also to increase the interpolating func￾tion polynomial order, N, within specific boundary ele￾ments. The program included in its output the com￾putation of I/P?(b) - w(J)lh, i E rj, for each boundary element. For each CVBEM attempt, N is increased by 1 and the boundary element halved in length (to produce two elements) that had the largest values of I/P,“([) - w(&‘)112. The modelling process continues until a reasonable &J(Z) lit to the problem boundary conditions is achieved. Applications demonstrating the CVBEM to numer￾ically solve boundary value problems involving the two￾dimensional Laplace or Poisson equation can be found in several publications. 1-5 The focus of this paper is the presentation of another CVBEM error evaluation technique that appears to provide a more robust guide in developing subsequent improved CVBEM approx￾imators than the other techniques in use, such as the approximate boundary technique,’ and the usual eye￾fit comparisons between 4 versus 4 on l+, and Cc, versus * on r+. In numerous test problems it was found that use of the llPN(l) - &([)[I2 error to pinpoint localized CVBEM approximation error provided, in general, a better ap￾proach to improving CVBEM functions than the ap￾proximate boundary approach. The following prob￾lems demonstrate application of the IIPy(J) - w([)/h error measure technique to locate where additional no￾dal points need to be added to r in order to develop more relined CVBEM approximations. In each appli￾cation a mixed boundary value problem is defined by prescription of either 4, I+/J, or a$/& along portions of r. The CVBEM is applied to an initial nodal point distribution along I’, and then the error measure is evaluated for each boundary element. The boundary element that manifests the largest value of error is then further discretized, or the Taylor polynomial order in￾creased by 1 (up to a maximum order of 8 in the pre￾pared computer software). The program user selects, up front, the order of the Taylor polynomial to be used; the program conducts the discretization. For each problem shown the exact solution used to generate the test problem is given. Initially, nodes are only defined to be located at the vertices of r. Also, a quadratic polynomial is used for each element. There￾after the software generates successively finer CVBEM estimates, by discretization, by use of the error mea￾sure between PIN(l) and w(l) on r. The maximal error E = /C%(Z) - w(z)11 is then computed for demonstration purposes, as w(z) is known, Plots of error for 25 and 40-node discretization are provided for each applica￾tion. To demonstrate the error analysis procedures dis￾cussed above, two mixed boundary value problems are considered in which analytic solutions are known. A FORTRAN computer program, based on the complex variable boundary element method, which allows an increase in Taylor series polynomial order or an in￾crease in nodal density is used. For both problems considered, an initial nodal point scheme of 12 nodes is defined on each of the problem boundaries. Boundary conditions of specified stream function or specified potential function values are used (even though flux type boundary conditions are straightforward to include). A CVBEM approximation function is developed based on the initial nodal point placement, and streamlines are automatically gener￾ated and plotted as solid lines within the problem do￾main (recall the CVBEM develops a function &(x,y) inside the problem domain, a). For comparison pur￾poses, associated streamlines for the analytic solution w(z) also are plotted, as dashed lines, within R. Because application of the CVBEM necessarily in￾volves problems in which the exact solution is un￾known, one of the two boundary condition function values is left “unknown” along the problem boundary. By using the error between the approximated boundary values and the known boundary values, additional CVBEM model complexity can be introduced. The errors computed are the usual integrated root-mean￾square error and magnitude error. Plots of these errors, as computed along the problem boundary, are included in the application figures. Subsequent CVBEM approximations, ultimately leading to use of 25 and 40 nodes on r, are shown in the attached figures, along with a comparison of streamlines between approximation and solution re￾sults, and the error plots. A quartic trail function is used in the 40-point discretization. Application A Solve ~*~/&K~ + a2$/ay2 = 0 in R, where fl is the domain shown in Figure 1. Stream function values are specified along the horizontal lines of r, and potential function values are specified along the horizontal lines of r, forming a mixed boundary value problem. The analytic solution used is o(z) = In [(z + l)/(z - l)]. Figures 1, 2, and 3 show approximation results versus exact values for 12-, 25, and 40-boundary nodes, re￾spectively. The accompanying figures show magnitude and integrated root-mean-square error plots along r for the boundary values. Application B Figure 4 solves the Laplace equation for ideal fluid flow over a cylinder on the shown domain, s1. The Appl. Math. Modelling, 1992, Vol. 16, March 119

CVBEM using Taylor series:T.V.Hromadka Il and R.J.WhitleyFigure3.ApplicationAwith40nodesFigure1.ApplicationAwith12nodesFigure2.ApplicationAwith25nodesFigure4.ApplicationBwith12nodes120Appl.Math.Modelling,1992,Vol.16,March

CVBEM using Taylor series: T. V. Hromadka II and R. J. Whitley 5 , h, : ; i#.l, ._. . . . . . . .,.,. I\ :. .,. /. l.I* g ., ._._. ., ._ 4 i i %I _. -. i 5 ; f Figure 1. Application A with 12 nodes Figure 2. Application A with 25 nodes 120 Appl. Math. Modelling, 1992, Vol. 16, March * 3. *. 3 . * * I. II 0. I, ‘* .,. . .I . .I , * ,. II. ‘L a, . . Figure 3. Application A with 40 nodes Figure 4. Application E? with 12 nodes

CVBEMusing Taylor series:T.V.Hromadka Il and R.J.WhitleyCVBEM versus analytic results are compared in Figures4, 5,and 6for 12-,25-,and 40-node placements,respectively.Alsoshownarecorrespondingerrorplotsin meeting boundary condition values along . Theexact solution is w(z)= z + 1/z.Stream function val-ues are specified along the arc and also on xO: otherwise, potential function values are specifiedalong .Discussionof resultsThe two application problems demonstrate using twocommonlyemployed errorevaluationtechniques inhandlingapproximation errorin meetingtheproblemboundary conditions.Because the CVBEM leads toexact solution of the partial differential equation,onlyboundary value approximation error exists. The ap-proach to add CVBEM model complexification by eithermore nodes or higher Taylor series order expansionsencompassestwo viabletechniques used inthis paper.The complexification is added,however, where theboundary condition approximation error is relativelylargeBecause the CVBEM developsa two-dimensionalapproximation function, precise flow nets can be de-veloped insidetheproblemdomain,whichcanbecom-paredtoknownproblemsolutionswhenavailable.Forexample,Figure 7 shows a plot of stream functionFigure6.ApplicationBwith40nodesvalues for Application A, while Figures 8 and 9 show25-and40-node CVBEMapproximations,respec-tively.Figure l0 shows Application A,a statevariable0.8.Figure7.Stream function surfaceplot, In[(z+1)/(z-1]plot,whereasFigures 11 and 12 show25-and 40-nodeCVBEMapproximations.The comparability of the approximated flow nettothe solution's flow net is of importance due to thenced for computing higher-order derivative functionsfrom the CVBEM approximation function, o(z).Forexample, we know that (z) = (x,y) + ii(x,y) where2=0andv2-0 inside2.Then otherdifferentialquantitiesmaybe evaluatedby directlydifferentiatingFigure5.ApplicationAwith25nodeso(z)(thisdiffersfromdomaindiscretizationtechniquesAppl.Math.Modelling,1992,Vol.16,March121

CVBEM using Taylor series: T. V. Hromadka II and R. CVBEM versus analytic results are compared in Fig￾ures 4, 5, and 6 for 12-, 25, and 40-node placements, respectively. Also shown are corresponding error plots in meeting boundary condition values along r. The exact solution is o(z) = z + l/z. Stream function val￾ues are specified along the arc and also on x = 0; otherwise, potential function values are specified along E. Discussion of results The two application problems demonstrate using two commonly employed error evaluation techniques in handling approximation error in meeting the problem boundary conditions. Because the CVBEM leads to exact solution of the partial differential equation, only boundary value approximation error exists. The ap￾proach to add CVBEM model complexitication by either more nodes or higher Taylor series order expansions encompasses two viable techniques used in this paper. The complexification is added, however, where the boundary condition approximation error is relatively large. Because the CVBEM develops a two-dimensional approximation function, precise flow nets can be de￾veloped inside the problem domain, which can be com￾pared to known problem solutions when available. For example, Figure 7 shows a plot of stream function values for Application A, while Figures 8 and 9 show 25 and 40-node CVBEM approximations, respec￾tively. Figure IO shows Application A, a state variable Figure 5. Application A with 25 nodes J. Whitley ,~_._._._ ~ . . ‘. t, -‘L ,* . . Figure 6. Application B with 40 nodes Figure 7. Stream function surface plot, In [(z + IMz - 111 plot, whereas Figures II and 12 show 25- and 40-node CVBEM approximations. The comparability of the approximated flow net to the solution’s flow net is of importance due to the need for computing higher-order derivative functions from the CVBEM approximation function, h(z). For example, we know that &i(z) = +(x,y) + i&x,y) where V2@ = 0 and V2$ = 0 inside R. Then other differential quantities may be evaluated by directly differentiating h(z) (this differs from domain discretization techniques Appl. Math. Modelling, 1992, Vol. 16, March 121

CVBEMusing Taylor series:T.V.Hromadka lIl and R. J.Whitley0.5D.8aa-182254uFigure8.Streamfunctionsurfaceplot,25-neapproximationFigure11.Statevariablesurfaceplot,25-nodeapproximation,In [(z + 1/z -1)]In [(z + 1)/(z - 1)]S3-Figure12.Statevariable surfaceplot,40-nodeapproximation,Figure9.Streamfunction surfaceplot,40-nodeapproximation,In [(z + 1)/(z-1)]In[(z+ 1)/(z-1)]that useinterpolation functions in the problem interior),resulting in a two-dimensional function definedinside.Forexample,given(z)foramixedboundary25valueproblem.[dmo(z)l/dzmis readilycomputed and-evaluated for z E .167.ConclusionsaOTheCVBEMisanumcricalapproachtosolvingboundary valueproblems of two-dimensionalLaplace and4Poisson equations. The CVBEM estimator exactly1solves the governing partial differential equations intheproblemdomainbutonlyapproximatelysatisfiesthe problem boundary conditions. The CVBEM ap-proximator can be improved by developing abetterfitto the problem boundary conditions. In this paper anew CVBEM error measure is used in aiding in thedevelopmentofimprovedCVBEMapproximators.TheFigure 10. State variable surface plot, In (z + 1)/(z-1)]new approach utilizes Taylor series theory and can be122Appl.Math.Modelling,1992,Vol.16,March

CVBEM using Taylor series: T. V. Hromadka II and R. J. Whitley Figure8. Stream function surface plot, 25-node approximation, In [(z + 1 )/(I - 111 Figure9. Stream function surface plot, 40-node approximation, In [(z + l)/(z - l)] 0-w Figure 10. State variable surface plot, In i(z + l)/(z - l)] 122 Appl. Math. Modelling, 1992, Vol. 16, March Figure 11. State variable surface plot, 25-node approximation, In [(z + l)/(z - 111 Figure 12. State variable surface plot, 40-node approximation, In [(z + l)/(z - 111 that use interpolation functions in the problem inte￾rior), resulting in a two-dimensional function defined inside 0. For example, given i3(z.) for a mixed boundary value problem, [d”‘&(z)]/& is readily computed and evaluated for z E 0. 7. Conclusions The CVBEM is a numerical approach to solving bound￾ary value problems of two-dimensional Laplace and Poisson equations. The CVBEM estimator exactly solves the governing partial differential equations in the problem domain but only approximately satisfies the problem boundary conditions. The CVBEM ap￾proximator can be improved by developing a better fit to the problem boundary conditions. In this paper a new CVBEM error measure is used in aiding in the development of improved CVBEM approximators. The new approach utilizes Taylor series theory and can be

CVBEM using Taylor series:T.V.Hromadka Il and R.J.WhitleyAreadily programmed into computer software form.Onnumber of angle points on Idthe basis of numerous test applications it appears thatRe w, w = Φ + id中use of this new CVBEM error measure leads to theImw2development of significantly improved CVBEM ap-convex, simply connected domain with cen-troid 0 + Oiproximation functions.2aurDa(zEQ:z enclosed byTa)NotationsQsUTsdμ(+±2)1/2Idg,gerJolLengthofr(r.(Re )2 dμ + Jr,(Im w)2 dμ)1/2ilall1max zj+1 - zw;nodal value w(z), w, = Φ, + idjN(S)olinear basis function defined on EFCVBEM approximation evaluaied at z, ETP;nodal point j,P,E(Jrlo(g)p dp)/plollpcentroid of Q(z. = 0 + 0i)Z(w,w)Jr(Rew)2dμ+ Jr(Im )2dμ3nodal point coordinates defined on Fsimple closed contour forming the boundaryofReferencesIboundary element (line segment) connectingHromadka.T.V..Il and Lai. C. The Complex Variable Bound-Inodal points with coordinates zj,zj+1ary Element Method in Engineering Analysis.Springer-Verlag,TszEQ:z=o,TEr)NewYork,19872Lapidus, L.and Pinder, G.F, Numerical Solution of PartialTo,FsFUF-F and Fn Iat finite numberDifferential Equation in Science andEngineering.JohnWileyof points.Here, is known on Ts,and New York,1982isknown onFa.wherew=@+ib.Both3Duren,P.L.Theory of H Spaces.Acadcmic, San Dicgo, CA,Fand Fare simply connected contours197084a coordinatereduction factor,0<<1Hromadka, T.V.,Il and Whitley.R.J.Numerical approxi-mation of linear operator equations using a generalized Fourier3, zEI,zEn;=Reforo≤<2m(noseries: Ordinary and partial differential equations with boundarytwo pointsSiand 2onFhavethe sameconditions.Appl.Math.Modelling1989,13,601-614angle )Mathews,J.H.BasicComplexVariablesforMathematicsand0,branch-cut angle of In, (z - z)Engineering.Allyn and Bacon,Boston,MA,I982Appl.Math.Modelling,1992,Vol.16,March123

CVBEM using Taylor series: T. V. Hromadka Ii and R. J. Whitley readily programmed into computer software form. On the basis of numerous test applications it appears that use of this new CVBEM error measure leads to the development of significantly improved CVBEM ap￾proximation functions. Notation 6 l4L 5 E r L length of I fhi([) 1' max (Zj+ 1 - Zj( P,’ mear basis function defined on J E I nodal point j, Pj E r $jl centroid of fi(z, = 0 + Oi) nodal point coordinates defined on I simple closed contour forming the boundary ofn rj boundary element (line segment) connecting nodal points with coordinates Zj, zj+r rs (2 E fi2: z = SC, 5 E rj I?+, r+ r+ u r+ = r and I+, n I+ at finite number of points. Here, 4 is known on I+, and I,!I is known on I, where w = 4 + i$. Both I4 and I, are simply connected contours f,Z a coordinate reduction factor, 0 < S < 1 [ E r, Z E a; 6 = Reie for 0 5 0 < 2~ (no two points l1 and & on I have the same angle 6) 4 branch-cut angle of In, (z - zj) A 4 cc, n number of angle points on I Re o, w = 4 + ilc, Im w convex, simply connected domain with cen￾troid 0 + Oi wr {Z E R: z enclosed by I,} fls u Is ($2 + $2)“2 (Sr,(Re a)* d/L + Sr,(Im 4’ 4-P* nodal value w(z~), wj = $j + i$j CVBEM approximation evaluated at zj E I (S&&)J” 4P Sr,(Re 4’ & + Sr,(Im 4’ & References 1 Hromadka, T. V., II and Lai, C. The Complex Variable Bound￾ary Element Method in Engineering Analysis. Springer-Verlag, New York, 1987 2 Lapidus, L. and Pinder, G. F. Numerical Solution of Partial Differential Equation in Science and Engineering. John Wiley, New York, 1982 3 Duren, P. L. Theory of HP Spaces. Academic, San Diego, CA, 1970 4 Hromadka, T. V., II and Whitley, R. J. Numerical approxi￾mation of linear operator equations using a generalized Fourier series: Ordinary and partial differential equations with boundary conditions. Appl. Math. Modelling 1989, 13, 601-614 5 Mathews, J. H. Basic Complex Variables for Mathematics and Engineering. Allyn and Bacon, Boston, MA, 1982 Appl. Math. Modelling, 1992, Vol. 16, March 123