《复变函数论》课程教学资源（书籍文献）The CVBEM for multiply connected domains using a linear trial function

The CVBEM for multiply connected domainsusing a linear trial functionR. R. Harryman III and T. V. Hromadka II*Williamson and Schmid,Irvine,CA,USAJ. L. VaughnTheOptical SciencesCompany,Placentia,CA,USAD. P. WatsonMartinMarietta,Baltimore.MD,USAThe objective of this paper is to present a modelling technique that approximates boundary valueproblems of the Laplace equation oveier fwo-dimensional multiply connected regions. By using thismethod,two-dimensional Laplace equation problems can be solved by use of analytic functions.Theflexibilityofthistechniqueis demonstratedonproblemswithmultiplyconnecteddomains,dissimilarmaterials,andmanytypesofboundaryconditionsthathavepreviouslybeen difficulttohandle.Keywords: boundary element methods, cauchy integral equation, boundary value problems, analyticfunctionsdissimilar materials, flux boundary conditions, andIntroductionmultiply connected domains, all with different typesThe complex variable boundary element methodof boundary conditions.(CVBEM)isamathematicalmodellingtechniquethatDetails regardingthemathematical underpinnings ofthe CVBEM, as well as a review of the literature, areapproximates boundary value problems of the Laplaceprovided inRef.I.Abriefdevelopment of theCVBEMor Poisson equation. The problems with which theCVBEM deals involve potential problems of the twois presented for the reader's convenience.dimensional Laplace equation.Specifically,theCVBEMhandles problems involving two-dimensional steadystatesoil waterflow,steady-stateheatflow,stress-Development of CVBEM approximationsstrain torsion effects,and other similar problems.Let Q be a multiply connecled domain enclosed byThe numerical technique follows from the Cauchyboundaries C1 and C2.Assume that C, and C2 areintegral formula. The produced approximation funcpolygonal lines composed of V, and Vz straight-linetions of the CVBEM are analytic in theregion enclosedsegments and vertices,respectively (Figure I). Ifbytheproblemboundary.Thereforethey exactly satw(z) = Φ(z) + i(z) is a complex variable function onisfythetwo-dimensionalLaplaceeguationintheentireR = C, U C, U , then d(z) can be defined tobe thedomain of the problem.The CVBEM integratesthestate variable and μ(z)to be the stream function. Conboundary integrals exactly along each boundary ele-sequently,@ and are related by the Cauchy-Reimannment:thusthemethod doesnotreguirenumerical inequationstegration.TheCVBEMcan solveproblems involvingdp二(1)axdydyax+Also at the Department of Applied Mathematics,California StateUniversity,Fullerton, CA, USAwhere @ and are real-valued functions that are har-monic functions for z E R:Address reprint requests to Mr. Harryman at Williamson and Schmid.17782 Sky Park Boulevard, Irvine, CA 92714, USA.2+da中00(2)ax2ax2ay2ay2Received 19 June 1989:accepted 27 September 1989104Appl.Math.Modelling,1990,Vol.14,February@1990ButterworthPublishers

The CVBEM for multiply connected domains using a linear trial function R. R. Harryman III and T. V. Hromadka II* Williamson and Schmid, Irvine, CA, USA J. L. Vaughn The Optical Sciences Company, Placentia, CA, USA D. P. Watson Martin Marietta, Baltimore, MD, USA The objective of this paper is to present u modelling technique that approximates boundary value problems of the Laplace equation over two-dimensional multiply connected regions. By using this method, two-dimensional Laplace equation problems can be solved by use of analytic functions. The flexibility of this technique is demonstrated on problems with multiply connected domains, dissimilar materials, and many types of boundary conditions that have previously been dtfficult to handle. Keywords: boundary element methods, cauchy integral equation, boundary value problems, analytic functions Introduction The complex variable boundary element method (CVBEM) is a mathematical modelling technique that approximates boundary value problems of the Laplace or Poisson equation. The problems with which the CVBEM deals involve potential problems of the two￾dimensional Laplace equation. Specifically, the CVBEM handles problems involving two-dimensional steady￾state soil water flow, steady-state heat flow, stress￾strain torsion effects, and other similar problems. The numerical technique follows from the Cauchy integral formula. The produced approximation func￾tions of the CVBEM are analytic in the region enclosed by the problem boundary. Therefore they exactly sat￾isfy the two-dimensional Laplace equation in the entire domain of the problem. The CVBEM integrates the boundary integrals exactly along each boundary ele￾ment; thus the method does not require numerical in￾tegration. The CVBEM can solve problems involving * Also at the Department of Applied Mathematics, California State University, Fullerton, CA, USA. Address reprint requests to Mr. Harryman at Williamson and Schmid, 17782 Sky Park Boulevard, Irvine, CA 92714, USA. Received 19 June 1989; accepted 27 September 1989 dissimilar materials, flux boundary conditions, and multiply connected domains, all with different types of boundary conditions. Details regarding the mathematical underpinnings of the CVBEM, as well as a review of the literature, are provided in Ref. I. A brief development of the CVBEM is presented for the reader’s convenience. Development of CVBEM approximations Let R be a multiply connected domain enclosed by boundaries Cl and C2. Assume that C, and Cz are polygonal lines composed of V, and V2 straight-line segments and vertices, respectively (Figure I). If w(z) = 4(z) + i+(z) is a complex variable function on R = Cl U Cz U ll, then $(z) can be defined to be the state variable and +!&I) to be the stream function. Con￾sequently, 4 and Ic, are related by the Cauchy-Reimann equations where (b and + are real-valued functions that are har￾monic functions for z E R: a*+ a*+ 2+7=0 a2* a2* ay y-$+1=0 ay (2) 104 Appt. Math. Modelling, 1990, Vol. 14, February 0 1990 Butterworth Publishers

TheCVBEMformultiplyconnected domains:R.R.Harryman Illetal.the line segment joining nodes z, and zj+i,j = I,..m.m+2....., m + n + 2.Noticethat Imis theline segment joining nodes Zm and zi and In+n+t joinsnodes Zm+n+, and zm+2 (Figure 1). Therefore C, =Um-, I, and C, - Umm+? We can nowdefine a continuous global trial functionGi(z) on C, U C2 bynm(+)(3)G;(2) = N(2)3mAn analytic approximation function'is determined by1Gi(a)(4)o(z) =daZEO2mia-zn+!11mtn+1Since o(z) is analytic in , its real and imaginary parts+n+2individuallysolvethetwo-dimensional Laplaceequation in 2. Simplifying (z), we obtain1Gi(a)ZEQo(z)daBoundary of multiply connected domainFigure 1.2ia-zCiUC21G,(a)Define nodal points[z,j= 1,.m+1]onthedaZES2ioutercurveC,suchthatm≥V,andanodal pointislocated at each boundary vertex, z... =- zi, and theseur,f=1points are located on C,in the counterclockwise di-Jtm+1rection.Similarly,define nodal points[zi,j = m+2n+1m +n +2l on the inner curve C,suchthatG,(a)ZdaZEQ(5)n ≥Vz,a nodal point is located at each boundary ver-2ia7j11lex, zm+2 = Zm+n+2, and these points are located onj+m+1C2 in a clockwise direction (Figure 1).Ateach nodal point zpj=.,m+n+2,letWedefinethebasis function N,(z)by lineartrial funcΦ, and ,be the specified real nodal values.Let ,betions,Zj-1ZET-Izj zj-1ZETUT+I?N,(z) =0j=1,...,m+n+ 1j手m+1(6)Zj+1 - zZET,(j+1 - zjwhereI,=Im+1,Im+2=Fm+n+2.ThenonTG(z) = N(z)@, + Nj+i(z))+i= (N(z), + N+1(z)+) + i(N(z)山 + N+i+1)wherew,= Φ,+ i.Therefore( G(a) da = [(+a) + (a-) daa-zo(zj+1 - z,)(a - zo)Tda-ada3+10-20+!1a-zozj+1 - zja-zoZj+1 -rAppl. Math, Modelling, 1990, Vol. 14, February105

The CVBEM for multiply connected domains: R. R. Harryman III et al. Figure 1. Boundary of multiply connected domain Define nodal points [zi, j = 1, . . . , m + I] on the outer curve C, such that m 2 V, and a nodal point is located at each boundary vertex, z, + , = zI, and these points are located on C, in the counterclockwise di￾rection. Similarly, define nodal points [zj, j = m + 2, . . . , m + n + 21 on the inner curve Cz such that n L Vz, a nodal point is located at each boundary ver￾tex, zm+2 = z,+~+~, and these points are located on C2 in a clockwise direction (Figure I). At each nodal point Zj,j = 1, . . . , m + n + 2, let $j and ICI, be th e specified real nodal values. Let f’j be I Z - Zj-I Zj - Zj- I ZET,-, 1 the line segment joining nodes Zj and zj+ 1, j = 1, . . . , m, m + 2, . . . , m + n + 2. Notice that f, is the line segment joining nodes z_~ and zl and r, +, + I joins nodes z,+~+~ and zm+2 (Figure I). Therefore Cr = U,$, rj and Cz = Ujmz,?J rj. We can now define a contmuous global trial function G,(z) on C, U C2 by m+n+1 G,(Z) = N,(Z) z C&j + iii&) (3) j=l j#m+l An analytic approximation function’ is determined by (4) Since G(z) is analytic in LR, its real and imaginary parts individually solve the two-dimensional Laplace equa￾tion in 0. Simplifying G(z), we obtain 1 h(z) = g I G(a) da - ZER c.“Ca-z I * I =- 27ri I- G(a) da ZEfi m+n+la -z u I; j= I .jfm + I We define the basis function N,(z) by linear trial func￾tions, N,(Z) = 0 zErjUrj+, j= l,.,m+n+ 1 j#m+l Zj+l - Z Zj+ I - Zj Z E rj where r, = rm+, rm+Z = I’,+,+*. Then on fj, G,(Z) = Nj(Z)Bj + Nj+ I(Z)aj+ 1 = (Nj(z)$j + Nj+ l(Z)$j+ 1) + i(Nj(Z)$j + Nj+ I%+ 1) where aj = qj + i$j. Therefore Gl(a) da = (Zj+ I - a)L3j + (a - Zj)aj+r da r, a - zo _I￾I (Zj+ 1 - ZjHa - Zo) J - - = Zj+ 1 wj - Zjwj+ 1 da +Bj+l -Tsj ada Zj + t - Zj _I￾b a - zo Zj+l -Zj I ~ a-Z0 I (6) Appl. Math. Modelling, 1990, Vol. 14, February 105

The CVBEM for multiply connected domains: R. R. Harryman Ill et al.Simplifying the last integral, we getada(u - Zo + zo)doa-Zoa-zoQZoda-Zoda+a-zZ0j.j+1da10-Z2j-1x2=zj+1-zj+ zoln(azo)j-1j+12jZ5+15=Zj+1-Zi+Zo[In+ioiitFigure2.Central angle ii1(7)where ejy+1,j - l,..,m,m+2.We define the term H, by,m+n+Iisthecentral anglebetweenthe straight-linesegmentsjoining Z, and zi+i to interior point zo (Figure 2).(8)+ i0j,j +H,=InZiZoThen,[Gada-3-H+0+1-(Zj+1 Zj + ZoH)u-zoZj+1- ZoZj+I Zj-3+0 2 H,+ 可+ -0,+ @+0-0]Zj+1 - ZjZj+1 -Zj(zo -zj+m)ZoZj(9)= +1 @, +@+0THZi+1Since o(zo) is the sum of the contributions of each T, divided by 2i,1m+n+Gi(a)>dao(Zo):2mi.Zoi=fx1-Z0-Z(30-3+))H20U2元jtm+In1Zn:ZiZo-22i2Zi+Zi+ 1H2(10)[j+1(zo ~z) (zo Zj+1)]2元Zj+1 ZiitThis canberepresented asthecomplexfunctiono(zo) = (zo) + i(zo)=Φ(zo,....m，m+2,..,m+n+1,1，..，m，m+2...,m++1)+i(zo,1...mm+2....m+#+1,..*，m，m+2..*,m++1)106Appl.Math.Modelling,1990,Vol.14,February

The CVBEM for multiply connected domains: R. R. Harryman Ill et al. Simplifying the last integral, we get I ada -ZZ a - z. I (a - z. + ZO) da L r; a - z. = zj+, - zj + zoh (a - zo) ZJ = zj+ 1 - Zj + Zo [ In /'~':"/ + i@j,+i] (7) where Oj,‘j.i+ , j = 1, . . . , m, m + 2, . . . , m + n + 1 is the central angle between the straight-line segments joining zj and z,+ 1 to interior point z. (Figure 2). Then, 2. J r 2 J 'j+l Figure 2. Central angle O, We define the term Hj by Hj = In Zjtl - ZO i i Zj - ZO + iej,+, I G,(a) da = Zj+ ISS, - Z@j+ I H_ + Fj+ I - a, a - zo zj+, - zo J r, zj+, _ zj (Zj+l - zj + zOH,) Zj+ IzSj - Zjwj+ I zjt, - <i ffi + aj+, _ aj + %+ Izo - %” Hj zj+ I - 22 Since &(zo) is the sum of the contributions of each rj divided by 2ri, j#m+ I = &.m~'[ZJj+~(ZO - Zj) - Tjj(ZO - <j+l)l HJ J-1 Zj+l - ZJ j#m+l This can be represented as the complex function G(zo) = $4~0) + &z0) = &ZO,&. . . ,TLr~m+2,~~. .&,+n+l, 3’,. . ,~m71cl,+zl.~~ ,&in+l) +i~(z0,~l,.~.r~~,~m+~,.,~~+n+~,~~,.,;i;m,~m+2,.,~~+.+~) (8) (9) (10) 106 Appl. Math. Modelling, 1990, Vol. 14, February

TheCVBEMformultiply connected domains:R.R.Harryman Illet al.BOUNDEDSTEPwhere zo is the interior of , and and are real-valued functions representing the real and imaginary...componentsofa(z).1.-1...-...1f,=+isknownateachzj=1,OFLUX.,m +n + 1,thenequation(10)ism. m+2.@=analytic inside , so (x,y) and (x,y) both satisfy*=0the Laplace equation in Q. if (z) w(z) everywhere中=oonC,UCz,then(z)=w(z)in,and (z)isthe--..exact solution of the boundaryvalueproblem.-...-1..Actually,usually only one,and occasionally nei-4=0ther,ofthetwospecifiednodalvalues(@,)isknown4-0.,m+n+/..m,m+2.at each zi,j=and we must estimate values for the unknown nodaloFvalues. Using an implicit method, we can evaluate o(z)arbitrarilycloseto each nodal point and thengeneratethe unknownnodal variableas functions of all theknownnodal variables. This results in m + n equations form+nunknownnodalvariables,whichcanbe solvedwithmatrices.O2The abovevalues as estimates of theunknown nodalFigure3.Bounded step boundary conditionsvalues can be used along with the known nodal valuesto define o(z) by equation (10).BOUNDEDSTEPExamplesWe nowapplytheCVBEM totwoexampleproblems.A--,3-For each problem a diagram of the boundary conditionsA.andtheCVBEMgeneratedflow netwill bepresented.Theproblemsconsidered are1. flow over a bounded step and0-8/0=6-2. flow around objects in two regions.tb=.4-1.Flow overa boundedstepaIn cartesian coordinates the problemboundaryisa2-containedby the lines x=0,x=3.y=0,and y2, with the vertex of the step at the point (l, 1). (Seethe appendixes and Figure 3.)The results of theCVBEMapplied tothis problem-yield a solution for both the boundary and interioro2points (see the appendixes). Upon evaluating bothboundaryand interiorpoints,theresults canbeplottedFigure aBounded step flow netas in the output diagram. In this example the stream-lines are the flow lines over the step, and the statefunction lines (which are orthogonal to the strcamlincs)The output results of the CVBEM are evaluated onare the lines of equal potential (Figure 4).both the boundary and the interior in order to plot thevariables.The streamlines run horizontally through theregion,and thestatefunctionlinesrun vertically.TheFlow around objecis in two regionsstate function is negative for x 4. At x =4 the stream variable ""jumps."*Thestate function is orthogonal to the stream variable andthe area enclosed by the lines x-10.x=10.y-5, and y = 5. This area is split by a line at x = 4represents lines of equal potential, while the streamthe area where x 4. Thcrc are two holes in the regionxThe CVBEM develops approximate solutions to two-4 that is a square. It has vertices at (6, -1), (8, -1),dimensional Laplace problems.For problems dealing(8, 1), and (6, 1) (see Figure 5).Appl.Math.Modelling,1990,Vol.14,February107

The CVBEM for multiply connected domains: R. R. Harryman Ill et al. where za is the interior of 0, and & and 4 are real￾valued functions representing the real and imaginary componentsof h(z). IfZGj=+j+i$jisknownateach?i,j= l,., m, m + 2, . . . , m + n + 1, then equation (IO) is analytic inside Q, so 4(x, y) and +(x, y) both satisfy the Laplace equation in a. If h(z) = o(z) everywhere on C, U Cz, then G(z) = w(z) in 0, and h(z) is the exact solution of the boundary value problem. Actually, usually only one, and occasionally nei￾ther, of the two specified nodal values (&, $,I, is known ateachzi,j = 1,. _. ,m,m + 2,. . . ,m + n + 1, and we must estimate values for the unknown nodal values. Using an implicit method, we can evaluate G(z) arbitrarily close to each nodal point and then generate the unknown nodal variable as functions of all the known nodal variables. This results in m + n equations for m + n unknown nodal variables, which can be solved with matrices. The above values as estimates of the unknown nodal values can be used along with the known nodal values to define h,(z) by equation (10). Examples We now apply the CVBEM to two example problems. For each problem adiagram of the boundary conditions and the CVBEM generated flow net will be presented. The problems considered are 1. flow over a bounded step and 2. flow around objects in two regions. Flow over a bounded step In Cartesian coordinates the problem boundary is contained by the lines x = 0, x = 3, y = 0, and y = 2, with the vertex of the step at the point (1, 1). (See the appendixes and Figure 3.) The results of the CVBEM applied to this problem yield a solution for both the boundary and interior points (see the appendixes). Upon evaluating both boundary and interior points, the results can be plotted as in the output diagram. In this example the stream￾lines are the flow lines over the step, and the state function lines (which are orthogonal to the streamlines) are the lines of equal potential (Figure 4). Flow around objects in two regions The problem boundary in Cartesian coordinates is the area enclosed by the lines x = - 10, x = 10, y = -5, and y = 5. This area is split by a line at x = 4, the area where x 4. There are two holes in the region x 4 that is a square. It has vertices at (6, - I), (8, - l), (8, l), and (6, 1) (see Figure 5). BOUNDED STEP ’ ’ ’ ’ 1 ’ ’ ’ _a_O_O_.a_ _*_a-0- 0 FLUX I 0- I : qL0 i + A_ :_ ; >$-0 d- $=o / / 0 a- * , / o- _ ~_ ~_D_~-o-o-~-o-~- -I 7 I I I I1 i ‘4 I, I I L 0 I 2 3 Figure 3. Bounded step boundary conditions BOUNDED STEP 0 I 2 3 Figure 4. Bounded step flow net The output results of the CVBEM are evaluated on both the boundary and the interior in order to plot the variables. The streamlines run horizontally through the region, and the state function lines run vertically. The state function is negative for x 4. At x = 4 the stream variable “jumps.” The state function is orthogonal to the stream variable and represents lines of equal potential, while the stream variable represents the flow lines (Figure 6). Conclusions The CVBEM develops approximate solutions to two￾dimensional Laplace problems. For problems dealing Appl. Math. Modelling, 1990, Vol. 14, February 107

TheCVBEMformultiply connecteddomains:R.R.Harrymanlll etal.3-HOLE2-REGIONPROBLEM3-HOLE2-REGIONPROBLEMTTTTTTTTTTTTTTTTTTTYTTTTTYT1oEOFTTTIyaoW-534-0-Wso0=0oF0loy=oaroundyonearoundW-5y--52Wa-=-10flux5F64:5a-15T.TTTTTT-10E10bREo5-10-51010.1O-5Figure5.Three-hole,two-regionproblemboundaryconditionsFigure6.Three-hole,two-region problem flow netilarmaterials;two example problems demonstrate thewith linear boundary conditions. the results of theutility ofthe numerical techniques.CVBEM areexact.For more difficult spaces (that is)nonlinearboundary conditions)theCVBEMexactlyReferencessolvestheLaplaceequationovertheproblemdomainbutapproximatestheproblemboundaryconditions.InHromadka,T.V.,II and Lai, C.The Complex Variable Bound.1this paper the CVBEM is cxtcnded to multiply con-ary Element Method in Engineering Anulysis.Springer-Verlag,NewYork,1987nectedregionswithapplicationstodomainsofdissim-108Appl.Math.Modelling,1990,Vol.14,February

The CVBEM for multiply connected domains: R. R. Harryman III et al. 3-HOLE 2-REGION PROBLEM ‘“II c=-25’ .+-IS’ +=5’ - IOL, ,., , , , , j -10 -5 0 5 IO Figure 5. Three-hole, two-region problem boundary conditions with linear boundary conditions, the results of the CVBEM are exact. For more difficult spaces (that is, nonlinear boundary conditions) the CVBEM exactly solves the Laplace equation over the problem domain but approximates the problem boundary conditions. In this paper the CVBEM is extended to multiply con￾nected regions with applications to domains of dissim￾J-HOLE 2-REGION PROBLEM -10 -5 0 5 IO Figure 6. Three-hole, two-region problem flow net ilar materials; two example problems demonstrate the utility of the numerical techniques. References 1 Hromadka, T. V., II and Lai, C. The Complex Variable Bound￾ary Element Method in Engineering Analysis. Springer-Verlag, New York, 1987 108 Appl. Math. Modelling, 1990, Vol. 14, February