《复变函数论 Functions of a Complex Variable》课程教学资源（参考文献）复数边界元经典 The CVBEM for multiply connected domains using a linear trial function

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 a 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 difficult to handle. Keywords:boundary element methods.cauchy integral equation.boundary value problems,analytic functions Introduction dissimilar materials,flux boundary conditions,and multiply connected domains,all with different types The complex variable boundary element method of boundary conditions. (CVBEM)is a mathematical modelling technique that Details regarding the mathematical underpinnings of approximates boundary value problems of the Laplace the CVBEM,as well as a review of the literature,are or Poisson equation.The problems with which the provided in Ref.1.A brief development of the CVBEM CVBEM deals involve potential problems of the two- is presented for the reader's convenience. dimensional Laplace equation.Specifically,the CVBEM handles problems involving two-dimensional steady- state soil water flow,steady-state heat flow,stress- Development of CVBEM approximations strain torsion effects,and other similar problems. The numerical technique follows from the Cauchy Let n be a multiply connected domain enclosed by integral formula.The produced approximation func- boundaries C1 and C2.Assume that C and C2 are tions of the CVBEM are analytic in the region enclosed polygonal lines composed of V,and Vz straight-line by the problem boundary.Therefore they exactly sat- segments and vertices,respectively (Figure 1).If isfy the two-dimensional Laplace equation in the entire (z)=(z)+i(z)is a complex variable function on domain of the problem.The CVBEM integrates the R=C:U C2 Un,then (z)can be defined to be the boundary integrals exactly along each boundary ele- state variable and (z)to be the stream function.Con- ment;thus the method does not require numerical in- sequently,and are related by the Cauchy-Reimann tegration.The CVBEM can solve problems involving equations dφa地 d地。_d业 (1) *Also at the Department of Applied Mathematics,California State ax dy ay University,Fullerton,CA,USA. where and are real-valued functions that are har- Address reprint requests to Mr.Harryman at Williamson and Schmid. monic functions for z e R: 17782 Sky Park Boulevard,Irvine,CA 92714,USA. 8+3=0 2业，2业 =0 (2) Received 19 June 1989:accepted 27 September 1989 104 Appl.Math.Modelling,1990,Vol.14,February 1990 Butterworth Publishers

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

The CVBEM for multiply connected domains:R.R.Harryman Ill et al. the line segment joining nodes z;and+j=1,..., m,m 2,...m n 2.Notice that Im is the line segment joining nodes zm and zi and Tm+joins nodes Zmn+and zm+2 (Figure 1).Therefore C= U座，T,and C2-Umm2r We can now define a continuous global trial function Gi(z)on CIUC2 by +用+1 G()=N()∑ (Φ+) (3) j=l j≠m+I An analytic approximation function'is determined by 1 Gi(a m+3 (z)= da z∈n (4) 2mi a-z CIUC? im+n 2m+3 Zm+n+1 m+n 2m+2m+n+2 Since (z)is analytic in n,its real and imaginary parts individually solve the two-dimensional Laplace equa- tion in Simplifying (z),we obtain 1 Gi(a) Figure 1.Boundary of multiply connected domain z）= a z∈2 a-z CIUC? Define nodal points [,j=1,...m+1]on the 1 G(a) da z∈2 outer curve C,such that m =V and a nodal point is 2mi a-z m+n+l located at each boundary vertex,z+.=z,and these I points are located on C,in the counterclockwise di- f=1 rection.Similarly,define nodal points [z j=m +2, m+1 ...m +n+2]on the inner curve C2 such that 1 mG(a) n=V2,a nodal point is located at each boundary ver- 2mi a z∈2 (5) Ja-z tex,m+2=m+2 and these points are located on m+1 C2 in a clockwise direction (Figure 1). At each nodal point zj=1,....m +n 2,let We define the basis function N,(z)by linear trial func- and be the specified real nodal values.Let I,be tions, (2-i-1 3-3-1 z∈-1 N(z)= 0 z∈Ur+1 j=1,.,m+n+1j≠m+1 (6) Z+1一Z 对+1一对 z∈「 where T-「m+l,「m+2=Tm+ntz.Then on[, G()-Nz)@+N+(z)ō+1 =(N(z)Φ+N+1(z)吨+)+iN(z)西+N+1西+) where ;=;iv.Therefore Gi(a)da= f+1-a)画+(a-》@出da a-zo (z+1-(a-0) =西-山「da++1-@∫ad血 3+1-3 J-0+1-Z a-zo Appl.Math.Modelling,1990,Vol.14,February 105

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 get [(a-zo+2o da a-0 4-Z0 2 ∫a-oda+ zoda a-a09 a-Zo j,j*1 =da+io)a-zo r da + 2j+2 =3+1-3+oln(a-o) j+1 2j+1 =+1-3+20 3+120 3-0 +i0+1 (7) Figure 2.Central angle where9w+1,j=l,·..,m,m+2,.·.,m+n+ We define the term H,by I is the central angle between the straight-line segments joining z and to interior point zo(Figure 2). H;=In 3i+1-20 +i0+1 (8) Z-20 Then, Gi@d血=3画-30H+1二0g*1-+ 对+1-0 3+1-3 =+回西H,+可+1-可+可：-可@H +1- 3+1-3 =而+1一 ,+ [()(]4 (9) Since @(zo)is the sum of the contributions of each Ti divided by 2mi, 】m+n+1 (z0）= Ga)da 2mi ≠m+1 1m+”+1 2m -西+[s()-台 ≠m+1 1 i=1 (）(]4 ≠m+1 [a+1(0-3)-可(z0-3+川 H =2 (10) 3+1-乙 *n+ This can be represented as the complex function (zo）=(zo)+i(zo) =(20,可1，，币m,本m+2,币m+n+1,,,亚m,中m+2,,中m+n+) +i0(20,币1，，中m,中m+2,,中m+n+1,1,,中m,中m+2,m+n+i) 106 Appl.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

The CVBEM for multiply connected domains:R.R.Harryman Ill et al. where zo is the interior of n,andand are real- BOUNDED STEP valued functions representing the real and imaginary components of (z). -T- lf可=西+i证is known at each3,j=l,,. O FLUX m,m 2,...,m n 1,then equation (10)is 巾= analytic inside n,so d(x,y)and (x,y)both satisfy the Laplace equation in n.If (z)=@(z)everywhere =0 on CIU C2,then (z)=w(z)in n,and (z)is the φ.0 exact solution of the boundary value problem. Actually,usually only one,and occasionally nei- =0 ther,of the two specified nodal values (is known at each z,j=l,..·,m,m+2,·.·,fm+n+1 -0 and we must estimate values for the unknown nodal values.Using an implicit method,we can evaluate (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. 2 The above values as estimates of the unknown nodal values can be used along with the known nodal values Figure 3.Bounded step boundary conditions to define @(z)by equation (10). BOUNDED STEP Examples We now apply the CVBEM to two example problems. For each problem a diagram of the boundary conditions and the CVBEM generated flow net will be presented. The problems considered are 1.flow over a bounded step and =8中6 2.flow around objects in two regions. Flow over a bounded step 0 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 Figure 4.Bounded step flow net 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) The output results of the CVBEM are evaluated on are the lines of equal potential (Figure 4). 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 Flow around objects in two regions state function is negative for x4.At x 4 the stream variable "jumps.'The the area enclosed by the lines x =-10,x =10,y state function is orthogonal to the stream variable and -5,and y =5.This area is split by a line at x =4, represents lines of equal potential,while the stream the area wherex 4 having twice the conductivity of variable represents the flow lines (Figure 6). the arca where x>4.There are two holes in the region x 4 that is a square.It has vertices at (6,-1),(8,-1), The CVBEM develops approximate solutions to two- (8,1),and (6,1)(see Figure 5). dimensional Laplace problems.For problems dealing Appl.Math.Modelling,1990,Vol.14,February 107

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

The CVBEM for multiply connected domains:R.R.Harryman Ill et al. 3-HOLE 2-REGION PROBLEM 3-HOLE 2-REGION PROBLEM loPTmmmpompprmpmmmy alo -5 Ψe4 0 业=0 Ψ5 Φ=0 0 =O around around 山=-5s-5 TIUX 中-5/ 小-5 中5 5 -0 -5 0 5 10 Figure 5.Three-hole,two-region problem boundary conditions Figure 6.Three-hole,two-region problem flow net with linear boundary conditions.the results of the ilar materials;two example problems demonstrate the CVBEM are exact.For more difficult spaces (that is, utility of the numerical techniques. nonlinear boundary conditions)the CVBEM exactly solves the Laplace equation over the problem domain References but approximates the problem boundary conditions.In I Hromadka,T.V.,Il and Lai,C.The Complex Variable Bound- this paper the CVBEM is cxtended to multiply con- ary Element Method in Engineering Anulysis.Springer-Verlag, nected regions with applications to domains of dissim- New York,1987 108 Appl.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