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 applications 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 imaginary (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, including 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 consequently, 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 equations 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 Mathematics, California State University, Fullerton, CA 92634, USA. Received 16 April 1991; revised 15 October 1991; accepted 12 November 1991 and real variable boundary element methods.2 However, issues regarding conditioning of the stiffness matrix 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 generalized 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 problems in which the boundary condition values are values of a function analytic on R U r the CVBEM approximation function converges almost everywhere (ae) on r. The CVBEM generalized Fourier series approach will be developed before the development of the numerical technique is presented. To keep the paper concise, the development of the CVBEM approach, the definition of the working vector spaces, proofs of convergence 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 approximators. The new approach utilizes Taylor series theory and can be readily programmed into computer software form. This new approximation error evaluation 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 centroid 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 except for a subset E’ in E such that the Lebesgue measure 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 representation 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 derivatives 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 <e/23. The CVBEM and WoChoosingGm()toapproximateg()by3.1. Definition of AIIGm(z) - g(z)l, <e/2Let the number of angle points of F be noted as A.By a nodal partition of,nodes (P) with coordinatesthen(zj are defined on F such that a node is located at eachNo() - Gm(llvertex of and the remaining nodes are distributed onF. Nodes are numbered sequentially in a counterclock-<o()- g(Ol + g( - G(Ol<wise direction along T.The scale of the partition isThe CVBEM approximation function, m(z),is de-indicated by I, where I = max Ikj+!ziNotethatveloped from Gm(z)for mnodes onT byno two nodal points have the same coordinates in T.1rGm()drZEn(1)0m(z):2miJS-z3.2.Definition ofrA boundary element T, is the line segment joiningwhere the , values used in Gm() are given by , nodes zjand zj+r; I, = (z: z = z(t) = zj(I - t) + zj+1l,w(z), w E Wn.0 ≤ t 1).(Note for m nodes on r that zmi= z.)3.9.Theorem3.3.Discretizution ofr into CVBEsLet w E Wn. For e > O there exists a G(g) such thatIlo() -- G(gl < e.Let a nodal partition be defined on T. ThenProof follows from the discussion in section 3.8FFEr=Ur3.10.TheoremLet w E Wn and z E. For every e> 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 analytic. 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 product 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 counterclockwise 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 continuous 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 < 42 Choosing G,(l) to approximate g(J) by IlG,(z) - dz>ll, < 42 then Il4) - G&)llr < II&) - g(5IL + k(5) - GUIL < E The CVBEM approximation function, G,(z), is developed from G,(z) for m nodes on I by A,(z) = J- GA3 dl 277i I ~ 5-z ZER (1) where the T3j values used in G,(l) are given by Tjj = W(Zj), W E Wfl. 3.9. Theorem Let w E Wo. For E > 0 there exists a G(c) such that II40 - ‘XIII < E. Proof follows from the discussion in section 3.8. 3.10. Theorem Let w E Wo and z E 42. For every E > 0 there exists 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,& < 2a de guarantees the desired result. 4. Taylor series expansions on CVBEs 4.1. Construction Let w E Wn. Then w is analytic on an open domain flA such that fi U r is entirely contained in the interior of RA. Let E* be in RA such that r* is a finite length simple closed contour that is exterior to R U IT. Then w is analytic on r*, and by the maximum modulus theorem, (~zI( 5~ z E r* for some positive constant M. (2) Also, jo(z)j5~ ZERU~ (3) Define a nodal partition of m nodes on r. Complex variable boundary elements are defined to be the straight line segments r, = [zj, zj+,] where, for m nodes, Zm+1 = zr. At the midpoint zj = d(Zj + Zj+l) of each rj, expand w(z) into a Taylor series T,(z - Zj). Each Tj(z - IQ has a nonzero radius of convergence Rj, and Tj(z - @ = w(z) in the interior of circle Cj = {z:)z - zjl = Z$}. The Cj all minimally have radii R, where R = min ICI - &) such that l1 E r and 5; E I’*. Descretize r into m CVBEs, rj,j = 1, 2, . . . , m, such that the length of rj, I/r,/ I 2Llm where L = Jr ld[l and 2Llm < R, and the other conditions regarding placement of nodes at angle points of r are satisfied. 4.2. Taylor series expansion For 5 E lYj, Tj([ - q) = P,“(l) + E?(5) (4) where N is the polynomial degree, and from Cauchy’s theorem, The magnitude of IEy({)l is, for every j, lE?y(5), I _I_ C - zj N+’ max ldz)l2~R J I I 2r z-q min )z - 51 (5) z E cj 5Er (6) where D = min (5 - z( for 5 E r. But and thus which is a result independent of j. Note that as the partition of r into CVBEs becomes finer, i.e., max llf’jll + 0, then m -+ 0~1 and jE,“(S>/ -+ 0. Also, as the order of the Taylor series polynomial increases, N + cc), and recalling that (L/m) < R/2, then IEr(<)l+ 0. 4.3. CVBEM error analysis From Cauchy’s theorem, for z E a, w(z) = JI 45) 4 _ 2rrii 5-z On r, let 4C) = 2 XjTj(l) 5= (9) j=l where Xj is the j-element characteristic function (i.e., Xj = 1 for 5 E rj; 0, otherwise). Then for z E J2, m 4.4 = y$ I x XjTj(l) d5 j=l I- 5-z (10) = h&) + EN(z) (11) The h)N(z) is the CVBEM approximation based on order N polynomials, where it is understood m nodes are used. The error, EN(z), is evaluated in magnitude for z E fi and using (12) to be s L (m)(max Ilrjll)(max IE~((>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)< R/2, [E~(z)/-→0 as eitherIn this paper we focus upon the Taylor series ex-m -→ oo or N -→ oo. Thus as the number m of CVBEspansions in each Fi,as the interpolation polynomialincreases,or the order of the interpolating polynomialorder,N,increasesand alsoas thenumber ofCVBEs,N increases, error |Ex(z)] → 0.m, increases.Thus the numerical approach used in the CVBEM4.4.CVBEMnumericalanalogcomputer program formulation is outlined by the fol-lowing steps:As z- ge F, for zE Q, then w(z)-→ w(s) - T(),The CVBEM procedure is to set in a Cauchy limit1.Discretize the problem boundaryT (whichis a finitesense,unionof straight line segments)into mCVBEsby"()duse of nodal points distributedonFwhere minimally(13)T(z) =2m台-za node is placed at each corner of r; i.e., rn ≥ A.2. For N =1 a linear interpolating polynomial is deas z-rwhilezEnfined on each I, For N > 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) < R/2, (E,&)( + 0 as either m -+ ~0 or N + ~0. Thus as the number m of CVBEs increases, or the order of the interpolating polynomial N increases, error IEN -, 0. 4.4. CVBEM numerical analog As z + 5 E Ij, for z E a, then W(Z) -+ ~(5) = q(c). The CVBEM procedure is to set in a Cauchy limit sense, (13) as z -+ I while z E 0. For order N Taylor series expansions the CVBEM sets in the Cauchy limit (14) as z + I while z E 0. If collocation is used, the numerical approach is to set’ Py(Zi) = W(Zi) (15) for each nodal coordinate zi E l?j. If a least-squares approach is used, the numerical approach is to-minimize4 llPY(5) - &)I1 5= Letting G,(3) = 5 Nj(l) zj j=l where it is recalled Wi = w(zJ, limG([) = o(f) l-+0 and j= 1,2 3. ., m (16) ZER (17) where 1 is the scale of the nodal partition of I. 5. Implementation In general, one does not have both 4 and I,!J values defined on I but instead has 4 values defined only on a portion of I, specified as I+ and $ values defined only on the remaining portion of I, I+, where I+ U Ts = I’. That is, we have a mixed boundary value problem. The numerical formulation given in the above equations solves for the unknown I/J values on I+ and the unknown r$ values on I+. Once theAunknown + and i,f~ values are estimated, denoted as 4 and I/J, then the global trial functions are well defined and can be used in G(z) estimates for the interior of Cn. The possible variations in such boundary condition issues are addressed by Hromadka and Lai.’ In this paper we focus upon the Taylor series expansions in each Ii, as the interpolation polynomial order, N, increases and also as the number of CVBEs, m, increases. Thus the numerical approach used in the CVBEM computer program formulation is outlined by the following steps: 1. 2. 3. 4. 5. 6. 7. 8. 9. Discretize the problem boundary I (which is a finite union of straight line segments) into m CVBEs by use of nodal points distributed on I where minimally a node is placed at each corner of I; i.e., m 2 A. For N = 1 a linear interpolating polynomial is defined on each Ij. For N > 1 a higher-order polynomial expansion is used, and consequently, additional 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 compare $ 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 condition values, then the CVBEM program user can decrease the partition scale (i.e., increase the number of nodes uniformly) and/or increase the CVBE interpolating polynomial order, N. The modelling goal is to increase (m,N) until the boundary conditions 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 components 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 approximation 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