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THE APPLICATION OF THE COMPLEX VARIABLE BOUNDARY ELEMENT METHOD TO THE SOLUTION OF HEAT CONDUCTION PROBLEMS IN MULTIPLY CONNECTED DOMAINS By ALAIN JACQUES KASSAB A DISSERTATION PRESENTED TO THE GRADUATE SCHOOL OF THE UNIVERSITY OF FLORIDA IN PARTIAL FULFILLMENT nf TUV Dn ITTTPNPMNT f Infl TW nPCfi n0 nnl ff'TnOR John, Monique, Paul, Linda. ACKNOWLEDGEMENTS would like thank C.K. Hsieh suggesting this interesting topic, providing support recommendations throughout this research, neverending patience reading revising this manuscript guidance would throughout like especially my graduate studies thank spurring my interests various aspects computational heat transfer. have learned much benefitted many ways as Dr. Hsieh student. would teaching, like for thank providing Kurzweg with for first valuable engineering research opportunity, serving supervisory committee. been a pleasure have Roan as a teacher, during wish graduate thank studies, for for encouragement serving supervisory committee would also like thank Oliver Gater for graciously serving supervisory committee. C.Y. Choi kindly provided the RVBEM data reported in th h Th i i*n G r SrfnP rmcr catlrl w r'^Ca<^ r'rl Iiyin~y" + ^ TABLE OF CONTENTS ACKNOWLEDGEMENTS. . . . . . . . . LIST OF TABLES .. ... .. .. ... .. ................. LIST OF FIGURES .... . . .. ... . ABSTRACT .. ... .... ... .... ... ... CHAPTERS I INTRODUCTION. ..... ................ ... .... .... II LITERATURE REVIEW................ .. .......... MATHEMATICAL BASIS OF THE CVBEM CONDUCTION HEAT TRANSFER CVBEM IN DOUBLY CONNECTED DOMAINS Derivation the Nodal Equations for the CVBEM in D Modeling Heat Conditions oubly Connected Domains. Conduction Boundary for the CVBEM.... Assembly of the Equations. Boundary Element Solution the Boundary Element Equations...... CVBEM Approximation Interior for Points.... # and *. . I at . . .*. .* S NUMERICAL APPLICATIONS............................ Description Example es. Results and Discussion. EXTENSION OF THE CVBEM TO S. .. ... ... 54 ... .. .. ... 63 MULTIPLY CONNECTED DOMAINS.. . . .. . . . Page Basi for Formulation Page of CVBEM in a Triply Development Connected of Nodal Domain. Equations CVBEM Development Evaluati Development in a Triply Connected of Computer Terms of Equations Codes (61 for Domain for 6) . . Interior Points. . . Numerical Triply Application Connected the CVBEM Domain. Extension From Tripl to Multipl Connected CONCLUSIONS Domain.... RECOMMENDATIONS.. ..... ......... REFERENCES APPENDIX IMPLICIT PROGRAM DOMAINS AND FOR EXPLICIT DOUBLY CVBEM FORTRAN CONNECTED HYBRID CVBEM FORTRAN PROGRAM FOR DOUBLY CONNECTED DOMAINS. IMPLICIT PROGRAM DOMAINS AND EXPLICIT TRIPLY CVBEM FORTRAN CONNECTED BIOGRAPHICAL SKETCH ...... ..... .. ..... . LIST OF TABLES Table Page Expression Equation Equations coefficients C, (418) .... coding rules. Coefficients equations (423) and G. I (424).'' for the nodal Implicit, expli cit, hybrid solution methods........ Results of in a hollow the CVBEM solution heat flow square..... Equations for k ldk coding rules. . Results of CVBEM in a triply rification connected example for the domain . . . A^k ^k LIST OF FIGURES Figure The The CVBEM i linear Nj ,2(s) n a simply basis connected functions domain . Nj,1(s) Evaluation angle O(zk+1, Zkl; zk . Boundary CVBEM 42 discretizat ion in a doubly Pictorial used connected representation to derive domain. limiting process Zo~4Zj bzO New numbering system for boundary discretizat ion for the CVBEM in a doubly connected domain . . Exact potentials and streamlines for heat flow a hollow square region.. The CVBEM problems applied in the in a circular solution of three annulus Test solution convergence o of a problem the CVBEM in the in a circular annulus.... CVBEM domain applied imposed in the with solution three different irregular boundary condition combinations. Comparison of RVBEM applied results solution CVBEM an annulus imposed with a Robin condition. Comparison of three solution 57 Comparison exact and C' CVBEM generated 59 Page methods ........ Figure 63 A nodal point shared two boundary elements. New numbering system for boundary discretization for the CVBEM in a triple connected domain. Verification example for the CVBEM in a triply connected domain. A multiply holes. . connected domain containing four Page Abstract of Dissertation University Requ i rements of Florida for the Presented in Partia] e of Doct Degre the Graduate Sch L Fulfillment of or of Philosophy ool the THE APPLICATION METHOD T( OF THE COMPLEX VARIABLE BOUNDARY ELEMENT I THE SOLUTION OF HEAT CONDUCTION PROBLEMS IN MULTIPLY ALAIN JACQUES CONNECTED DOMAINS KASSAB August 1989 Chairman: Major Chung K. Department: Hsieh Mechanical Engineering complexvariable boundary element method (CVBEM) extended solution potential problems multiply connected domains. doubly connected domain taken for analysis, a finite width cut introduced domain. Linear basis functions are used derive CVBEM nodal equations through limiting procedure. was found that stream functions along cut not cancel out result additional term nodal equations. complex variable methods, Cauchy Riemann conditions must used gene rate additional equations relating stream functions heat fluxes resulting nodal equations are described. analysis shown reducible available simply connected formulation introducing a new node numbering system. The CVBEM equations are successfully tested solving example problems with available analytical solutions. Dirichlet, Neumann, Robin boundary conditions are tested using implicit method solution. The CVBEM shown converge boundary discretization scheme refined. variable example boundary comparing element method CVBEM also provided. real The three solution methods are compared, efficacy these methods is discussed. The multiply CVBEM connected extended domains triply using generalized development doubly connected domains. The mechanism leading formation analyzed doublevalued applied stream the functions critically formulation stream functions along multiple cuts . General nodal equat ions are also derived an extension the CVBEM formulation. CHAPTER INTRODUCTION mathematical analysis potential problems, one encounters differential conditions. conduction parabolic steadystat dimensions. equations solution subject example, analysis, partial c e heat ordinary appropriate partial boundary unsteadystate temperature differential conduction temperature equation, analysis governed heat governed whereas multiple elliptic partial difficult nonlinear differential solve; equation. however, system These equations boundary domains are are conditions irregular shape, problems cannot solved exactly. Numerical methods must then used to solve them. numerical solution heatconduction equations can domain approach methods classified approach consists (FDM)1',2a into two broad boundary ever finite categories, approach. element popular methods namely whole finite whole domain difference (FEM) These . 1 1 usually 1 . 4m I 1 finite difference finite element equations are derived solution. FDM, local energy balance invoked nodal points order derive set algebraic equat ions equations; are whereas derived FEM, basis algebraic sat i sfact ion govern 1 ng partial equation global sense. more difficult derive finite element equations, are more convenient use solution problems irregular recently domains. however, Such advantage because diminishing advance grid generation techniques. The primarily boundary approach represented numerical boundary solution integral equation methods (BIEM), which are commonly known boundary element partial methods (B] differential ) In t equations hese are methods, used the governing conjunction with boundary conditions derive integral equation, which consists contour domain integrals Great simplicity arises situations where there are no heat sources sinks vanishes that Dealing only only domain; contour system solel with then integrals boundaries boundary, domain remain. need boundary integral follows discretized. approach EM less than whole domain approach. However, coefficient matrices generated the boundary approach are unsymmetric matrix elements are nearly fully populated. There have been number studies devoted comparison boundary approach with whole domain approach. They are reviewed together with others next chapter. There been much attention given recently stud ies BEM. Indeed, have now been developed level that competitive with FEM. However, most studies are confined real domain analyzed with real variables. This thesis is concerned with particular form called complex variable boundary boundary Cauchy element integral integral methods equation formula. (CVBEM) in a cor Using which I ex plane complex o rmu lat e based variables, the the the methods conduction domains will used field problems will shown solve simply later, twodimensional and multiply the complex heat connected methods possess element advantages methods (] approximating not found RVBEM). functions that real example, are variable the analytic CVBEM thus boundary generate satisfy governing equation throughout system domain More mpl< Errors CVBEM can assessed, which further adds confidence these methods. thrust this work extend the CVBEM simply connected domains doubly connected domains. methods will used solve several verification examples heatconduction problems doubly connected domains. The CVBEM will then extended solution potential problems in multiply connected domains. CHAPTER LITERATURE REVIEW Boundary element methods (BEM) can categorized real variable variable boundary boundary elemel element me nt methods :thods (RVBEM) (CVBEM). There complex are three types RVBEM formulations. Direct RVBEM are formulated using Green's expressed in identity real numbers with that dependent appear variables integral equations. equations Semidirect terms RVBEM unknown formulate functions, the such integral stream functions potential flows; these functions are then related dependent variables. Indirect RVBEM express dependent distribution solution variables over governing terms boundary partial integrals unit differential a density singular equation. contrast, are the CVBEM based are Cauchy formulated integral complex formula. plane unknowns appearing in the CVBEM integral equations are potential stream function components complex variable. such, CVBEM can considered semid i rect boundary 1 n+nfl'rn 1 m + 1,h rn r' ,, 1F 4 n r I ,, a . . 1 numerical modeling. contributed 911 maturity great of the many technique, authors and have RVBEM have now been firml rooted as a useful numerical method solution mechanics, field problems geomechan ics, among solid n others. mechanics 6,1217 , fluid Cruse18 reviewed current advances RVBEM solid mechanics, comparing the state art the RVBEM with FEM. There been relatively little attention given use RVBEM solution heat transfer problems Rizzo Shippy used RVBEM solve transient heat conduction problems while Chang developed constant element analysis to solve heat conduction problems anisotropic media. The RVBEM have been applied solution conditions conduction (e.g ., problems Dirichlet with linear Neumann, boundary Robin conditions)6'10,21 nonlinear boundary conditions (e.g. , radiative conditions) They have also been used solve steady instead problems two three dimensions. 2429 More recently, RVBEM have been used conjunction with optimization methods design thermal systems. 3032 only recently that complex variables have been linear trial function between boundary nodes investigate multizone, anisotropic, twodimensional flows porous media. zones Anisotropic through principal simply zones rotation axes connected order zones were transformed stretching obtain isotropic transformations domain governed about consisting Laplace equation. These zones were then coupled imposing compatibility conditions interfaces. means examples, they were able to show results close agreement with available analytical extended analysis FEM this was analysis later solutions. moving generalized Hromadka boundary a formalism and problem! now Guymon34 s: their referred to as the complex variable boundary element method Brebbia RVBEM, could then credited Hromadka principal could promoter considered major contributor development CVBEM. idea expanding CVBEM approximation finite series analytic approximation functions CVBEM that linked analytic function linear method (AFM) Veer. 3637 Having established theoretical basis CVBEM, Hromadka38 developed approximation technique a heuristic error minimization. Here, through considerable computer effort, an analytic (within given tolerance) were identified approximate reduced where boundary. concentrating approximate error additional boundary CVBEM nodes deviated was along from then regions known system boundary. Hromadka was able establish relative error bounds CVBEM.39 also showed that, limit infinitesmally small 1 discretization boundary, CVBEM solution approached exact solution limit. 4042 Hromadka successfully tested CVBEM variety steady twodimensional fluid flow heat transfer problems for which analytical solutions were available. The CVBEM were found accurate results converged number collocation points boundary was increased. addition, integrated measure relative error was shown an effective adapt ive scheme useful judiciously locating add itional nodal points reduce overall 1 error. The CVBEM have also been applied solution steady groundwater flow soil freezing problems. 4346 such applications CVBEM can accommodate largescale inhomogeneous subdomains Lai47 applied CVBEM solution densitystratified inviscid fluidflow conditions. was found that iterative divided matrix approach However, multilayer solution flow the gave s. global at isfactory set results. equations generated from assembly subdomai n equations boundary conditions was unsatisfactory. This can ascribed fact that, limit large number subdomains, CVBEM approach formulation with each The CVBEM CVBEM subdomain literature becoming akin growing recently. superelement The interested reader referred to Reference a detailed review literature References full exposition of the methods. clear that result both RVBEM CVBEM a set algebraic equations. been pointed out Chapter set equations generated boundary approach inherently smaller than those generated structures the these whole domain equations approach. are quite However, different. coefficient matrix algebraic equations generated domain approach, although large, banded. This allows efficient storage coefficient matrix boundary solution of the approach algorithms. algebraic smaller, equations yet contrast, generated fully square operation cubic counts domain between reported actually that f avor unless large problems are tackled major contributions operation counts considered study were solution originated phase from arithmetic number performed. storage However, Moukerjee Moraj aria pointed out that, for same level discretization, provide higher accuracy than FEM. This them believe that are computationally more efficient than FEM, a finding also supported Bane rj Butterfield (Reference sections addition, 8.11), level Ligett presolution Liu, Cruze. phase preparation discretization element generation also strongly favors boundary element methods. date, solution imposed CVBEM potential with have been problems Dirichlet primarily simply Neumann used connected boundary domains conditions. There lack studies dealing with solution potential problems multiply connected domains whose boundaries developed are imposed this work with thus Robin fills conditions. practical CVBEM need solution such problems. CHAPTER MATHEMATICAL BASIS OF THE CVBEM IN CONDUCTION HEAT TRANSFER theoretical basis CVBEM Cauchy integral formula, w(zo) 21 S2ri I r w(z Ezo ZoEQ (31) which relates the value of a complex analytic function w at point inside kconnected Jordan domain complex plane integral that function along boundary that domain; see Figure 31. The contour integral performed that domain lies the left of the contour integration. Steady state heat conduction problems generation thus with are construct constant thermal governed complex conductivity Laplace potential, no equation. w(z)=t(z)+it(z), heat can when solving these problems a twodimensional plane. real part potential represent i ng state variable, imaginary identified part temperature; potential whereas representing zo r (z), (z), ~~>1 Figure . The CVBEM in a simply connected domain. (32) which field relation (33) 33), n represents outward drawn normal to r, tangential coord i nate along again positive direction defined earlier. Using Fourier heat conduction lefthand side (33) enables expressed an integral heat flux. This provides link between stream function heat flow. CauchyReimann relations also imply that temperature stream function form an orthogonal net CVBEM approach solution boundary integral equation, two basic approximations are made Cauchy integral formula as follows: The boundary is discretized into finitelength segments denoted The entire boundary union these segments, i.e. , r. r^U rJ see Figure 31. In analogy to finite element terminology endpoints of these segments are called nodal points. Since domain P is simply =_ 0 as (ii)The potential w along the discretized boundary expanded used analytic for this expansion. series. The Polynomials order can the CVBEM approximation thus hinges on the degree polynomial used. this develop chapter, CVBEM. linear this elements effort, will used piecewise linear interpolating boundary polynomial complex will used potential. represent Referring Figure 32, linear basis functions Nj,l(s)) are defined on each element (34) (35) where O Using these basis functions, boundary coordinate is expressed i(s) parametrically on Nj ,2(s) (36) Notice that parameter related nodal points Nj,()= s)= Nj,2( s= s=l +(s) Iz j ++2 fzj j +l1 1,2 NJ2 Sj+1,1 II z z. z. 1 j1 z j+1 z j+2 ZN gu re .The near funct ions NJ Nj,i Szi Zj+l1 (37) The differential coordinate along element follows from (36) (38) j+z.j) similar fashion, complex potential expressed parametrically w[E(s)] Nj ,(s) (39) Here, using specified Hromadka s complex notation, potential .j=$(z.j)+i (zj) nodal point Introducing integral (36), formula, (38), equation (39) (31), into leads Cauchy firstorder approximation to w(zo) denoted A o W(Zo) N fl N 2rit(zo) = [ i ( O1 S~f~  zo j+lZ J (310) Then, expanding terms integrand rearranging, can expressed sum first j+i j s> (")"j+l Zj+l] where (1s) Ij(zo)= s7 (312) (zoz . (313) integral (312) can performed to yield Ij(zo)=(j +i 3 +1 J+1  *) + [7 j.0 +w .(l7 j)]1n( \ 3 /+ 7 upon introducing from (313), j+l (Zozj) Wj( zj ) zoZ Sz.i+ I ^J zo zo (314) Now fully specified each node then equat ion; b(zo)a boundary explicitly. (311) However r conditions boundary (314) regular specify are sufficient heat conduction any integral estimate problems, none problem thus them formed using equation (311) generate nodal equations each point to solve for unknown and/or Cauchy integral formula relation evaluating tl analytic function any interior point; Ij(zo) = (.j+ j)+ "j+l (z.jlZj) (Zj+1 Hromadka Guymon37 replace boundary around node j with limit a small e 0. circle Another radius technique E and I also evaluate used A(zo) w(zo) Hromadka Guymon, approach boundary node from interior; t(zk)=l im ^c2 zo*11 A (o) w(Zo) that The iS, latter one approach takes will limit now used develop the CVBEM nodal equations. Notice that terms which lead singularities equation (311), nodal point approached from interior are contributed two adjacent elements arguments approaches logarithmic term (314) vanishes equals unity, thereby leading singularities integrals Fortunately, Ik1 these singular terms can manipulated cancel each other out. With this mind, contour integral equation (311) first recast Ij(zo) +Ik_1(zo) +Ik(zo)) (315) S(zoI=kk ( J=1 j k,kl Then limit applied A \ w(zk)= Sim '7'/& 97. I (zk) or r 1 (zo)= Finally, (314) introduced into (316), after much manipulation 2ri 1 this (zk)= yields Zk+1 Zk Wkn lzk1Zk N + J k,k1 j+1 (ZkZj) j (ZkZj +1) (z +1z j z +1 kz< z.zk (317) Zj k Equation (317) useful deriving doubly connected domains as will shown the next chapter. time being, well that Hromadka studied existence, continuity, convergence this approximation presented examples Pe reforming equations complex each algebra nodal (317) point ields pair These k on nodal equations, one for one can used along with boundary conditions gene rate set simultaneous equations solve unknowns inherent coupling nodal variables (317) that leads nearly full populated unsymmetric coefficient matrix, discussed literature review. Once have been fully determined boundary nodes, temperature can evaluated closing this chapter, some comments are order examine terms (317) clear that (317) complex logarithr expansion ns and potential complex polynomials. terms This natural expansion analytic logarithm satisfies is under the the Laplace summation equation can throughout expressed fur. using complex variables ,zk) In Z+1 k z zk k d(zj,zk) +iO(zj+1 ,zj;zk) (318) where refers distance refers angle. They are in turn related nodalpoint locations d(zj+1 ,zk) xk)2+ (yj+lk)21/2 (319) d(zj,zk) =[(xxk)2+ (yjyk)21/2 (320) arg(z zk) (321) Here principal value ( r taken; see Figure 31. However, confusion may arise evaluating logarithm outside summat ion (317) this outside logarithm, 0(zj +1,Zj ; zk)=arg(Zj +1zk) d(zj+l =[(Xj+1 where d(zk+l ,zk) =[(Xk+lk)2+ (Yk+ lYk)2]1/2 (323) d(zkl,Zk) =[(xk1 Xk)2+ (kl (324) and, angle re lated interior angle 0int O(zk+1 ,Zkl k) = 20nt lnt (325) illustrated Figure 33. This notation will used throughout this work. development CVBEM simply CVBEM connected doubly domains connected now complete domains Extension follows next chapter. as yk)2l1/2 int Zk+l Zk / Zk k+1 k1 3 x figure 33. Evaluat ion angle 8(zk+l,Zk1Zk). CHAPTER CVBEM IN DOUBLY CONNECTED DOMAINS Cauchy integral formula can applied solution potential problems a doubly connected domain introducing boundaries cut domain. connecting However, inner stream outer function component of the complex potential is double valued along line analysis. connected cut, 5456 domain this Further, must must CVBEM reducib] accounted analysis that for doubly a simple connected domain the inner boundary vanishes, (ii) extendable that multiply connected domain more holes are added domain. With these serving guidelines CVBEM doubly connected domains are developed this chapter. Derivation Nodal Equations for the CVBEM in Doubly Connected Domains discretization scheme illustrated Figure useful A cut developing made the the CVBEM doubly domain connected discrete domains. closed gure Bound CVBEM ary secret a doubly izat ion used connected derive the domain. numbered counterclockwise direction. inner boundary numbered discretized in a clockwise into direction. (NM) . The elements, cut which is shared are by two elements they are equal length opposite direction. convenience later analysis, these two elements are separated small gap 2z=6x+i6y, which will eventually order taken close zero gap. This limiting approach process permits consideration double values complex potentials along the line cut. Using 32) Fourier conduction recognizing that stream points are functions double points valued, M and complex potentials at these points are related to each other &(zM) U(zN) (41) (42) Here, related total heat flow rate per unit thermal conductivity (Q/k) across either system boundaries or _= (Zl) _= (ZM+I) rl end startingpoint nodal locations; hence, QM,1 denotes boundary total that rate lying from heat flow point across point outer Taking normal positive pointing outward considering basic problem which heat generated within QM,1 are QN,M+1 related must opposite energy sign, conservation their principle magnitudes as general value this not QM,nw known unless flux fully specified over either boundaries. with uniform boundaries, Then, unknown temperature then an unknown 4 's. domain s vanish determined trivial enclosed a priori along case insulated definition, does double connected inner connected boundary domain vanishes, becomes this simply becomes a special case analysis that follows. linear basis boundary doubly element connected method domain now shown developed Figure 4 Following the boundary limiting node process described approached in Chapter from III, interior. a doubly connected gap cut domain, also zero limit. necessary to Thus, take following (310), linear CVBEM approximation Complex so QNM+1 I 2riw(zk)= Sim 6x, 6yO Z,"Zk 1o NN (s)w +N. j, 1 (j j,2 I Lj,1(s)z+Nj,2( 2(s)j+1+l s)zj+l] z.)ds) (44) J/ Equation (44) can recast 2ri(zk) = lim 6x, 6yO\ lim Zo Zk 2ri(zo) (45) Here limit inside parentheses recognized that time 2iriw(zk) terms previously for derived j=k1 (317) fact summation this (317) can recovered, inner limit expressed a summation of Hj(zk) N N = Hj (zk)= =1 i =1 Ji a= j+l(zkzj) z (Zkj+) Zj +lZk z j+1 J ]1 z j zk z J+l J J Notice that j=k1 terms have now been included summat ion because kl+ Zk+l  Hk=&kln ZkI k k zk1 Zk zk (47) outer limit (45) is applied next, notice (46) (zj+1 notice line reasoning, general expression for w(zk) can derived, this will based general izat ion the derivation which follows. For nodal point one may use (45) (46) write 2xri(z1) = 1rim 6x, 6y&O N Hj(zl) (47) Since zM=Zl+6xi by zM+ 1=ZN+6x i 6y Figure 42, terms j=1,M1, must sorted out from summation, 2rii (z1) im y 6x, 6ya0 H1(Z1)+ HM1(zl)+ HM(z) )+ HN(Z1)+ H(zl)) (48) j#1,Ml,M,N Then applying limit as indicated using (41) relations zN+l=zl WN+1 W, there is derived / z2Z 2riu(z1)= 1ln ZM_z1 + ji1 Hj(z1) ,M,N Z 1Nl1 lM+1Zl zM+1 ZN1 ZNl Z1j (49) l zM+1Mlz ilnzMl1Zl ] P Hi P1 zi ZM+1 ^Sz M1 " L.\ Figure 42. Pictorial representation of the limiting process and bz+O. I _ Zo+Z 6y 8x M+l where natural logarithm term inside bracket comes from HN1 (Zl) term summation. This related term . WM+1 is noted (42), that because thus producing way this (49) extra is derived, &(zM) z(zN) set in the equal summation W(ZM+1) set equal this equation. identical procedure can used derive 2riA(zM+1) , 2iri and (zk then a general w(zk) Zk in Zk equation fI Zk 1 Zk j#t can N + S =l1 kl,k be deduced Hj(zk) ,M,N +. i l M+nl(Zk 1 zk (410) Equation (410) can used for any point boundaries domain Figure long following rules are adopted: When k=l, kl=M1 when k=M+1, kl=N1. (ii) Singularities logarithmic terms can cancelled out limiting procedure. For example, when k=l, term brackets (410) becomes zl1ZN1 i M+lZ)+ zl:Nl zlzl 1 ilnzM+1z~ 191 242 w(zl) ZkZN1 ZM+1Zkh_ ZkZM1nl ( Z1Zk M+1 ZN1 N1 k 1 Ml Ml kj (iii)Rule above also applies summation. example, when k=1, kl=M1; then, HMl should excluded from summation (49) (iv) summation terms Hj(zk) (410) , (zM) is set equal to ((zl) W(zM+1) set equal O(zN) for the reasons previously explained; see paragraph following (49). Since Z(zM) related to 3(zl) &(ZM+1) related to c(zN) equations (41) (42), AM w(ZM) i(zN) equations are redundant. CVBEM developed above are strictly valid doubly connected domain shown Figure 41. order develop connected general domai n method analyzed that earlier, reducible new numbering simply system introduced cut as shown effectively Figure eliminated 43. the new doubly system, connected domain The point now takes place point Ml, point takes place point N1 (Figure 41). With these changes, the new nodal equation can then be derived 2ri0 (zk) an^ {Jzk) z k+ Zk  Zk/ FN r +1 rM+1 ['I . Figure 43. New numbering system for boundary discretizat ion for the CVBEM in a doubly connected domain. iy X Z)1 where, depending location the nodal point term is given one following expressions k#l,M M+1 zk z+1 Zk zk] 1 ZlZk ZkZN i ZM+1lZk ZM+lZ ZNZk M1C+1 N7 ZN k for (412) special  (ln(  [I( ZM+1z1 M 1 ZZNN InzM+1z1 M+1 N ZNZ1 z Z (413) z AM+1= _( ZMZN ln(ZMlN _M+lN I N z ZM+1 ZM Z Z M1 i ZNZM+ Z1ZM ZMZM+1 I ZM+l ,/ZN ZMM I1 [ZzM nzMI )+ln( (414) (415) (416) Notice that equations (411) (416) are val id doubly connected domain new numbering system Figure 43. new numbering system when k=1, kl=M when k=M+1 kl=N. informative compare equations (411) 317). Clearly righthand side exactly ZkZM nfzl ZlzM In zM M+1ZN > zlzN f where for term subscripts doubly ilAk and s plays simply the refer connected role nodal domains, res pertu rbat i on equations pectively. The accounts double value along cut. simply connected domains, such a cut does not exist; summation braces terminates j=M, i'kAk term drops out. Equation (411) thus reducible simply connected domain equation (317) is noted that, development follow remainder this chapter, node numbering will refer new system Figure 43, unless noted otherwise now estimated possible temperature to derive stream equations function, relating (zkk) specified temperature stream function, first term righthand side (417), associated with single valued nodal equations, expressed 27ri [(zk)]S= w(Zk)1ln d(Zk+l,Zk) d(zkl ,Zk) +iG(zk+lZk1 ;zk) N +k j j+lk L[j+1(zkzj) W(zk Zj+l)](C+iD) (418) Here, have been given 1 , (323) 325). A #(k) ' 35 Table 41. Expressions for coefficients C, D, A, B, and F in Equation [A(xj+1X) +B(yj+1 [B(xj+1 xj) A(yj+1 (418) Yj)]/F y )]/F J j~ ]/ where d(zj+1 ,zk) d(zj,zk) o (zj+1,Zj;zk) (xj+1Xj)2+ (Yj+1Yj)2 Next term Ak is decomposed Ak Ak +iAk (419) components can derived expanding (412) (416) into their real imaginary components with help explicit expressions for (419), giving = [EkCN, k (xkXN) +DN k (YkN) +CM, k (xkXM) DM,k (YkM) (420) = [FkCN,k (YkYN)DN,k(xkXN)+CM,k (YkYM)+DM,k(xkXM)] (421) where coefficients D's, are listed Table 42. Coding rules for subscripts encountered general expressions C's are given table. Provisions are also made table for modifications required to evaluate nodal equation (417) can now be written Ai[(z ( 2xi[4 (zk) + i (zk) D +iAk) (422) as Atk A k as C's, , AM, M+1' 2ri[ (zk)]S+i (A k Tab 1 Equat ions 42. coding rules. Equat ion =[EkCN, k (kN) +DN,k YkN) +CM, k (XkM) DM,k Yk (420) =[FkCN, k Yk YN)DN, k (xk xN ) +CM k (YkYM) +DM, k (XkXM) (421) where (ZM+1 ,zk) ,zk) (ZM+1 ,zl;zk) k (Xf e) k(xfxe) e)l / A,k (Yf d(zf,zk) d(ze,zu) ,Ze; = arg(zfzk) arg(zezk) Xf Xe (YfYe) Coding rules subscripts Ce,k through equations given above: e=M f=1l k= [ A^k %k YM) ^k k(YfY Table 42 (Continued). Modifications in Ek through Fe equations for Al' 1M, AM+1' k=l :Ek =ln d(z d(zM ,z1) , Fk=O(ZM+lzM zl), (ii) k=M (iii)For k=M+1 : Ek=ln d(z d(z NZM+l) 1,ZM+l) , Fk=O(zN,Z1 zM+1) = N,M+1=0 (iv) k=N CN,N : CM as the following pair of algebraic equations: A_ N ' (zk) =A*'k+B*k .+ j ,j+lk +G3, jj +4 j j +1+ 5 k (423) 4(zk)=B *k N A*ak+E j=1 [G3, j+G4,j j+l G1 (424) ,* jjG 2, jj +1 &5+ 2k where, expressions A's ,Gi, j are given explic (423) itly Tables (424) 43. relate The temperature nodal t(zk) equations stream function 1l(zk) nodal point the specified nodal temperatures stream functions, along boundary. The solve nodal most equations potential derived problems above are involving sufficient only However, heat conduction problems involving Neumann Robin boundary derived conditions, relate additional the heat equations flux must shown next section. [G1 .+Gj+i Tabi 43. Coefficients equations (423) G for the and (424). nodal d(zk+l ,zk) d(zkl,k) Zk+l Zkl; zk) G2 j 3,j Y j+l) C(kj+l) 2 [C(xk x ) D(Yk  [C(yk Xj+1) Yj+1) +D(xk 2[D(xk xj) +C(yk yj)] where C and are given Table 41. yj) [D(Yk values boundary. However, when a Neumann condition imposed boundary, CauchyReimann conditions can used to derive j+1= (425) Consistent function with the the heat linear flux approximation approximated the complex linear over element equat ion (425) becomes = J1 *sb. =0. , J+1 'J O0 [N.j, ( ) .j+N.j 3 3 J zj)ds 3_ (426) Evaluating the integral equation (426) summing over elements gives a relation between j +1 and .j+3 =ij+E il (427) where (428) (As)j+i=d(zj+iz j+i 1)= Z+Z.+i (429) (kqa,j+i1 2Lkk/J+i k +1 k(0) () +l(zj +l k a, j+11(+11] problem with heat flux spec ified along a boundary, equation (427) can used prepare a data set for direct input into nodal equation (423) (424), depending method assembly equations chosen solving problem. Robin convective) condition imposed boundary is modeled by the relation  4o0 = 0 (430) Here, convective coefficient (which may also account ambient for source similar radiation or that sink linearized), temperature. leading equation Following procedure (427), stream functions are related Robin condition following expression: 1 1 _ h 1 (431) Here (h , (432) on j1 h 1Lh4 i] j ti1 2 +1 ,k j+i f( h again inner relation M>j+l>1 boundary. between boundary; outer boundary, Equation (431) stream however, N>j +>M+ 1 provides function because on additional temperature complexity, not possible use as direct input nodal equations (423) (424). use solution will discussed next section. Assembly Boundary Element Equations close examination (423) (424) reveals that potential estimated function stream weighted every point on function sum the point potential boundary. Notice are stream that, these equations, estimated quantiti es are marked with caps (A) , whereas prescribed quantities are marked with bars () C lefthand Then, estimated side equations that also appear depend prescribed same point righthand side these equations. This a distinct feature CVBEM that leads diffe rent solution methods will discussed later. also noted that coefficients G's (423) (424) are solely dependent nodal on whose boundaries are fully specified, coefficients can be evaluated. final point related those source terms, 2rA 2Ak, 2rAk (423) (424). They are course, result perturbat ion term (417). shown (420) (421), these source terms are related positions ZM+1, Specifically, they include location point where are eva luated, also those four points which are along original line cut. Calling these source terms STk ^k 2Jk 2A , (433) STk 2rtk (434) possible combine (423) (424) matrix equation CR, CR, C C I, CI ST, A A (435) Here subscripts R and for partitioned matrices as equation. matrix desirable These formulation that part it ioned given matrices are equation elements each order (435) row highly global coefficient matrix sum one, outcome the uniform potential domain. 50 Furthermore, partitioned matrices, whereas elements elements each each row row must must sum sum zero, one. These results are handy checking computer coding. Prior examination planning equat ion solution (435) strategies, order. re doubly connected domain with prescribed inner outer boundary es, there are unknowns in the global coefficient matrix. However, each component of the vector contains which, according equations (43) (427), related heat flux boundaries. the solution heat conduction problems with prescribed Dirichlet Neumann are conditions, given either temperature boundaries. heatflux heat fluxes values are, nevertheless, related (427). However, Robin condition imposed on boundary, none temperatures or heat fluxes are given, yet they are still related (431). With these serve as guidelines, three solution methods are developed. They are, namely, Solution of the Boundary Element Equations The type solution method developed hinges specified evaluated are related. example, given nodal point Dirichlet boundary condition), then unknown this point. The implict solution method formulated setting equal N+kth equation (435). meantime, since kth given, equation longer deleted from unknown; (435). this permits implicit equation for node then obtained rearranging nodal equation (424) N j=1 j ,j+l1k [G3, j 4j+G4, j+1 G1 Sj G2 j]+ ,) 3 02,1 3+11+ (436) This equation can then used together with other equations unknown *1 solve appears unknown both sides Inasmuch (435), this as method called implicit. The method explicit that, solution since method differs given, from can implicit set equal kth equation (435). other hand, A* k = (B* 2i ^, equation node can obtained from the nodal equation (423) (1B*) k N =A*gk+E j l=1 j ,j+l1k [G1, j+G2, j+1 +G3,j j+G4, j +1] 2A k 2r.k (437) This equation can then used together with other equations solve unknown Notice that this time unknown appears only lefthand side equation (435), method thus named explicit. implicit equations are generated explicit methods solve for described unknowns above, . This occurs when Dirichlet Neumann conditions are imposed boundaries. However, when Robin condition imposed boundary, both become unknowns. Although one may still use either implicit or explicit method solve such a problem, there are now unknowns, one must rely on (431) supply additional equations. discussed equation (431) complicated because contains both unknowns direct input into equations (423) thus impossible. must then append (431) (435) raise ; number equations (435). same example sake discussion, discussed use earlier. still hybrid made method, unknown set equal this yields equations unknowns This occurs even Dirichlet Neumann condition imposed boundaries. equations Hence, solved hybrid method, simultaneously number always doubled compared with implicit explicit schemes. This adds considerable effort computation solution. Yet, evaluated even though given, this provides means for checking accuracy computation. Hromadka developed a computationally efficient iterative method for solution hybrid scheme. methods equation 49,50 should described (435) noted above, very that global nearly solution coefficient fully matrix populated unsymmetric Direct methods solving simultaneous equations are thus useful. faci 1 itate readers structure matrix equations solution problems with three CVBEM methods described above, coding suggestions are provided Table 44. addition, implicit explicit formulations are implemented FORTRAN program provided Appendix Table 44. Implicit, explicit, hybrid solution methods. Implicit Method: 1S sol\ (424) specified le for Ck zk, then loading the set tk= following . Also equation the matrix equations N j kk1 [G3, jj+G4, j+1 G1, jG2, j+1 A 1Ck A(Z w(zk) k + Ck Upon solution (423) to the matrix the matrix equations, is specified solve for ~1 zk, then loading th set set k= following so g equation equations N 1)ak+ j kkl [G1 jj+G2, jj+G3,+G4, j +G, jj+1 k Upon solution the matrix equations, set w(zk) =k+ k' Explicit Method: (423) to sol in the matrix spec ified ve for k equations: at by z then loading set the following Also use equation N (1B*) k=A*k + j kkl [Gl jj+G2, jj +1+G3, j j+G4, jj +1] + 4 Upon solution the matrix equations, set A A w(zk) =k +kk 2. If tk i (424) to the matrix LS specified solve for zk, then sei loading the t 3 fO lowing Also use equation equations A* k =(B* A*#k J 1 Table 44 (Continued) Hybrid Method: S. Also loading equations is use the specified (423) and following z then (424) t4 two equat ions Ck tand r k a n the A set_ = d k by matrix k= A*#+B*k+ j kk1 Gl ,I j +G2, j3+l .iS + G4 , J4+ O= (B* 1) kA* + + Jkk1 [G3, j +G4, j J+1 1 J J2, jS +l] + ah, Upon solution matrix equations, set w(zk) p. If ok=ad" 1 oad 1 ng specified use (423) following z , (4 4) two then set to solve equations set for in matrix equations: ,jj++G2 + kE jk, k1 N k =B kA k+ t j Sk ,kl +G3 j S+G4 , J++11 2 '~k , j +G4, j j+l G1 ,j+G2, j+l] + ,] 02,]]1 2Ak 2 k Upon solution of the matrix equations, set w(zk) 3, j++1 +2A k ,J J+1 =k^ + k+k +G3 O=A* k+(B* _+ ^ Ck+k CVBEM Approximation for and at Interior Points Once where the can loublevalued are known derived potentials boundary focusing nodes nodes attention , zo) w{Zo) only Figure 41. Cauchy integral evaluated limit small gap goes zero A(zo) lim Ex by0 PZo (438) Using linear elements, (438) can integrated to yield A(zo) lim 6x, byO0 N 1 V' 27rni I j=1 Hj(zo) J (439) account for doublevalued complex potential points Figure 41, HM, are sorted out from summation, they are evaluated complex indicated potential limit. (zo) The CVBEM the approximation numbering system of Figure 41 then A / W;(zo) N j=1 J M,N H (zo)+2(zM_ 1 zM_ zo) _ ZozN1  z z i ZM+1lzo z_zo (440) as HM+1' HN1' + In(ZM+1Zo 2 zzo I 2iriw(zo) = Hj(zo) i A (441) where, A =(zozM 1n(Z)zo+ 1 /zM+lzo zIZM/ n zMzo/ +n zlzo ) / ZoZN ZM+1zo ZM+1ZN)\ ZNo ) (442) Notice here that zo r'. With derivation of the boundary interiornode nodalequation assembly equations procedure, presentation development linear CVBEM doubly connected domains now complete. CVBEM will now applied solution heat conduction problems. zoet, CHAPTER NUMERICAL APPLICATIONS any numerical method, concern arises over accuracy convergence CVBEM solving problems doubly connected domains. this end, heat conduction problems with available exact solutions will solved using the CVBEM, numerical results will compared with exact solutions error. These examples will consider three type boundary conditions, order compare results, one examples will solved using both the RVBEM CVBEM. addition, another example solved using three solutions methods described previous chapter. Description Examples Three geometry ies are used test. They include square region with concentric square hole punched out, (ii) circular annulus, (iii) Irregular geometry enclosing an irregular hole. conditions imposed on square region are 1i 1l1 uuCt rnt P ~ tnt rm pE.1 Pn r' cran ], 11n Ul ^l rf h n ^d j v fI I [ S  r1 I j I Exact solution: (z)=z 2 1 2 1 1 2 Equipotential lines Figure 51. Exact enti store amlines for heat flow a hollow square region ( r equipotential lines. Physically, horizontal boundaries are insulated, vert ical boundaries are imposed with uniform temperatures. numerical solut ion boundaries are d iscret ized using ten nodes, Dirichlet boundary conditions are spec ified nodal points. With chosen temperature field, total heat zero; flow i.e., rate across the either the (43) are inner zero. or outer Yet, boundary sake of generality, taken as unknown, its value computed for comparison error. addition, value point taken zero. This serves purpose picking constant integration relative which remaining stream functions are evaluated noted that since much known about the heat transfer this square, this problem serves as excellent first example testing the CVBEM. second geometry sketched inset Figure 52. Here three cases are tested, their exact solutions are given three w(z) equations legend. circular each boundaries boundary boundaries they are, are are discretized consisting computed turn, into points. with used elements, values equations input with the given, solving 0.15  0.14 . 0.13  0.12  0.11  0.10 0.09 0.08 0.06  0.065 M12 N24 Ro1.0 R10.6 Exaot solution o0 (z) z + J (z) z2 o (s) ez 0speaified on lnner and outer boundarte of annulus 0 UW El T~ L  Node number Figure .The CVBEM oblems applied in t n a circular solution annul ree for heat the annulus flow rates share over the their same feature boundaries are that zero; the 1 i.e., total Q's are zero (43) Yet, like previous example, taken to be unknown computed the solution. different set Dirichlet conditions imposed boundaries annulus shown Figure 53. Here exact solution taken ln(z/Ri) w(z) In(Ro/Ri)' this represents situation where heat supplied outer boundary maintain uniform temperature where heat removed from Inner boundary maintain uniform temperature <=0. Here total heat flow rates across boundaries are not zero. With total number this nodes example sequentially serves purpose picked testing convergence of the numerical solution. third geometry irregular domain sketched Figure 54. tested three different boundary condition combinations given legend. exact temperature distribution w(z)=ln(z used given relation generate conditions imposed boundaries. this example, a total nodal points used to discretize boundaries. examples tested above, conditions =1 , 1 3 5 7 9 11 Node number Figure Test t solution he convergence of a problem i n a circ CVBEM ular annul us. 0.07 1 3 7 0 11 Node number Figure CVBEM domain applied impose solution with three diff erent irregular boundary condition combinations. annulus shown Figure 55 placed environment so that a Robin condition imposed on the outer boundary. The inner boundary maintained uniform temperature analytical solution for this problem taken t(r) #Poo iumbe number ln(r/Ro) ln(Ri/Ro) (1/Bi) (1/Bi)' where temperature Bi=hRo/k Biot surroundings. Notice that this problem actually serves two purposes: not only tests Robin condition it also compares results same CVBEM discretization with RVBEM. schemes are used this on comparison, boundaries both methods, two cases are tested follows. first case, elements are used boundaries, with second each boundary case, consisting number nodal elements points. boundaries are doubled. noted that the examples described above are solved using implicit method described Chapter order test other solution methods given that chapter final example provided shown Figure 56. Here w(z)=z2 test case Figure 52 restudied effects solution methods used. The cross plotted for implicit method Figure is plotted as square Figure 56. a a ag act Solution 0(r#) 0l#* U12 Ina(r/Ro)(1/Bi) ln(B/Na)(1/B1) e,100 'C a110 o eve SN=4 > TBKU & 5VBKU 0.01 0 0 0.00 Node number Figure mpar applied ison res in the solution of the CVBEM an annulus impo RVBEM sed with a Robin condition 0.06 0.0o 0.04 1 3 5 7 9 11 Node number Figure Compare son of three solution methods. er= I w(z)w max[ I w(z) (51) where w(z) exact solution, A \ w(z) represents CVBEM solution. computations were performed double precision Microsoft elimination 8Mhz Fortran algorithm 80286 optimizing with microcomputer compiler. partial using Gaussian pivoting equilibration was used to solve simultaneous equations. singular value decomposition algorithm was also invoked dealing with solution matrix equations whose coefficient matrices are poorly conditioned. Results Discussion The values square region are accurately computed CVBEM; see Table 51. addition, total heat flow rates across boundaries are computed '=7.099x1016 which excellent agreement with exact value, Fact Exactt0 that, This this good ac example, curacy linear can ascribed elements have been used w(z)= z, model (ii) (i) the firstdegree piecewise linear analytic boundary function domain. For the al nnulus shown in Figure 52, the first case , Table 51. Resu 1 ts of the CVBEM so a hollow lution f square. heat flow Boundary Conditions Imposed 4 on To and Fi Node eg e Number 1 0.OOOOE+O0 0.3706E14 2 0.1776E14 0. 1332E14 3 0.6661E15 0.1110E14 4 0.4441E15 0.1332E14 5 0.1776E14 0.8882E15 6 0.1110E15 0.1457E13 7 0.6328E14 0.2220E15 8 0.8882E15 0.1776E14 9 0.7772E15 0.1887E14 10 0.6217E14 0.1110E15 Definition of the errors eC and e, above: e I=CVBEM tEXACTI e' = CVBEM DEXACT1 w(z)=z2 problem are plotted Figure 57. The results show good agreement between CVBEM exact solution regions small 1 temperature gradients. The values for are also computed cases SI2z=0.000035, z2=0.0176, ez=0.0197 e Given small number nodal points (N=24) discretized over two boundaries this geometry, such errors are not unexpected. These errors can reduced increasing number nodes. shown Figure 53, maximum error point effectively reduced less than when total number nodal points increased from Corresponding error also reduced from 1.6% (for 24 nodes) to 0.14% (for 72 nodes). Attention now directed irregular geometry tested Figure 54. Here a maximum error of 7% located point this takes place when temperature s are imposed outer boundary heat fluxes are imposed inner boundary. other nodal points other boundarycondition combinations, errors are less than Notice that N=50 for this example. Since errors can reduced increasing demonstrated Figure 53, CVBEM are still accurate solving this i rregulargeometry problem. 1.20 1.00 0.80 0.80 0.40 0.20 1.20 I I 1.20 1  . . 1.20 1.0 0 0.80 0.0 0.40 0.20 0.00 0.20 0.40 0.80 0.80 1.00 1.20 120 1.00 0.80 0.60 0.40 0.20 0.00 S0.20 0.40 0.50 0.80 1.00 , 1.20 1.20 Figure 57. Comparison isotherms of for exact nd CVBEM )=z pro generated blem. RVBEM the t cannot temperaturee used tc errors evaluate are stream compared functions, Figure. only Here, accuracy both methods appears to be dependent on the distance points measu red from Robincondition surface. The maximum error less than CVBEM when N=24; RVBEM appear more accurate (error less than 2.5% These errors tend converge rapidly when value doubled. depth Also measured form errors become less surface, dependent maximum on error being less than CVBEM, error evaluated .38% when N=24, drops down 0.83% N=48. The examples presented above have been solved using the impl i method. example Figure solved using three methods described Chapter comparison results. shown Figure 56, results than hybrid implicit method method appear slightly selected nodal better points. However, other points (not shown) results three methods are comparable. Inasmuch as the hybrid method requi res solution twice many equations compared with either implicit or explicit method, while hybrid results not show marked improvement salient feature found for implicit method. discussed previous chapter implicit method discards nodal equation for whose value iven. pointed out Hromadka, 49,50 once nodal value found implicit method, one can treat given unknown evaluate using (423) or (424). difference can then used gauge errors numerical solution. present check work, this results. method instance, been one used example successfully Figure (see data plotted diamonds) which flux specified outer boundary temperature specified inner method. boundary values solved A _ first are then using evaluated implicit reveal difference of 0(102) nodal points where 6 values were specified. Noticing this large difference, situation number rectified nodal either points increasing switching total explicit method solution. data explicit estimation Both plotted solution. also holds alternatives Figure noted work are that explicit satisfactori1 results such hybrid of the error methods. As mentioned earl ier 1 literature review, Hromadka has (44) Another concern arises positioning cut hence placement nodes M+1 boundaries, see Figure 43. The sensitivity method position cut investigated reso living problems Figure with cut positioned at addition, a test also mad e numbering nodal say, points following beginning defined inside contour boundary; integration that direction, nodes are numbered from inner boundary from M+1 outer boundary. Comparison of the results shows same level accuracy cases tested. solution methods thus appear insensitive cut position node numbering order. Another point interest related boundary discretization. Hromadka, 49,50 solution potential problems position cause in simply coefficient connected nodal matr domains points ices reported along become that boundary illconditioned. This occurs more frequently when geometry symmetric and same when symmetri c problem doubly discret ization encountered connected domains. here these scheme solving problems, used. The problems diagnosing eigenvalue vector provided singular value on STT, backsubstitution . Alternatively, discretization scheme can altered create more staggered nodal distribution along boundary. condition number resulting coefficient matrix should reexamined prior final solution. examples above provide an exposition features CVBEM solution problems doubly connected domains. The methods have shown accurate even with boundary relatively converge coarse with discret izat ion refinement discretization scheme. next chapter, CVBEM will extended solution problems multiply connected domains. CHAPTER EXTENSION OF THE CVBEM TO MULTIPLY CONNECTED DOMAINS The analysis employed developing CVBEM for solution potential problems doubly connected domains can extended developing solution these problems multiply connected domains. first made triply connected domain shown Figure continuation doubly connected analysis. resulting analysis then generalized multiply connected domains with a connectivity higher than two . Some basic concepts useful for the doubly connected analysis are first reestablished for triply connected domains. General Concepts for Development the CVBEM in Multiply Connected Domain Comply Tripl ex Potential Connected at Nodal Points Domain Refer domain shown Figure 61. Here, nodal points are again sequentially numbered define continuous circuit around domain. positive direction about this circuit consistent with analysis simply doubly connected domains a it n.. e a , an nnn~r* F .J 1 __ _L A .. _ rf ~LL ~ nr Figure 61 Boundary discretizat ion used to derive CVBEM a triply connected domain. valued nodal point. similar fashion, complex potentials points M+1, MAB, MAB+1 are double valued follows that w(zM) zl) +i1o, M,1  o0 k' (61) w(ZME) =(ZMAB+1) i E' qME,MAB+1 (62) W(ZME+1 )=I (ZMAB) 11PE (63) Following (427), stream functions points can related those points ME+1 and M+1 ^(ZN) =(ZME+1) AT (ZMAB) = (ZM+1)AB' N ,ME+1 k MAB,M+1 (64) (65) These five equations will used later re late complex potentials doublevalued nodal points. Basis for the Formulation of CVBEM a Triply Connected Domain should noted that , Figure 61, boundary =J( E \II. AT  AB= point while that point inner boundary Figure loop 62. closing However, from stream above functions point points see MAB+1 are different; they are boundary loop closing from above. shown Figures 62, boundary over to The estabiis loop point first ME+1, concepts h the ec ses finally discussed luivalence above of nodal point departs then an will crosses upward useful points path. later derivation stream functions along lines cut. time being, basic equation will derived basis through analysis (46), presented Chapter contribution IV. the shown complex (44) potential A(zk) w(Zk) any boundary element comes from Hj(zk) term Hj(zk )=W Zk j+1 z]+i Z ln ZIZk ZZ k .ZkZj+ Zj+lZ W Zzj_ In ZiZ j+1 3/ j k (66) Here first term on righthand side accounts complex potential that located point zj+l while second term accounts complex potential located N M+I ME MAB+1 1 M *M l ME+1 MAB Figure 62. Formation of doublevalued stream functions at the cuts. f >lj Zn+l n1 n  n Figure 63 A nodal point shared two boundary elements. contribution complex potential by this point can be expressed ZkZnl In nznzn Zn1Zk ZkZn+1 nl Zn+lZk Zn+lzn ZnZk (67) noted that, nodal points where comply potentials potentials are d( remain ouble valued, unchanged, real only components the stream functions are double valued. Then, point used represent Figure any 62, such excess points stream , ZME, functions ZME+1, these points can derived sorting out stream function component from (67) writing AZknl (1 ZnZk  nZ p nZnl Zn1Zk/ ZkZn+ll f Zn+1Zk zn+lzn Znzk (68) where w(Zn) wO(zp) (69) denominator (69) complex number i=Jl. Here, for sake generality, first subscript is used represent specific point whose excess stream np (69), as defined '. ip (61) can through expressed terms those (65) Equations (68) (69) can now used to derive excess stream functions along lines cut. They are first applied finding stream functions points M and along cut N/1. Along this cut, AM=*M_ Z11 ZMZk MMl1 Ml ZkZM+1 IM+1Zk M+1M I Zk (610) ZZ N ZNZk AN=N(M+1) ZN ZN1 zNlZk Zk N+1 ZN+1 ZN ZN+Zk N k (611) Then since ZM+1Zk A +AN= o In M Zk z1zk _k N1 I Z I M+1Zk ZM+1ZN1 ZN1zk +zkZM1 In zzk Z18  N(M+1)M1 o (612) (410) This term accounts for doublevalued stream function equation component (410), doubly correspondingly, connected accounts nodal for double value triply connected domain. The above analysis thus consistent with reducible doubly connected domain formulation developed Chapter For excess the t stream riply connected functions also domain appear shown Figure points 61, ME+1, they must accounted for. Applying (68) these points and using the relation that (613) ME(MAB+1) (ME+1)MAB E gives _A +A In ZMABZk zkZME+2 in ZMABZk ME ME+1 WE ZMAB+ Zk ZMABZME+2 ZME+2 k ZkZME1 ZMEAB+1zk ZMAB+1ZME1 / ZME1zk (614) now necessary return concepts established earlier this chapter. previously mentioned, doublevalued stream functions only appear (612) these (614) equations must are related ; equivalent indeed, eac points other following manner: Nodal Points (612) Corresponding Points (614) MAB MAB+1 ME+2 ME1 Establishing these extension relations analysis for ows a doubly for connected direct domain triply connected domain. particular, nodal equation (410) can modified include an extra term that accounts for double valued stream functions along cut ME/(ME+1) z =w(zk)ln z +iSo[ln( Zk Zk M+1zk 1 k N j=1 J fk Zk_ZN ZM+1 Z Hj(zk) l,k, MA ,M 1 zM+1 In zN <1 ^N1 ZkZM1 In ZlZM1l 1 zk ZMlZk i E[ln( ZMABzk ZMAB+1Zk ZkzME+2 I ZMAB Zk ZMABZME+2/ ZME+2 Zk 2ri i( Zk Zk rules following (410) are accordingly amended include following relat ions: In addition when k=MAB+1 k1=ME1 when k=MAB k+1=ME+2. (iv) addition ZMAB+1) set equal to (ZME) w(ZME+1) is set equal to w( ZMAB) are formulations for "(ZME) zME+1) redundant. The other rules remain unchanged. derivation developed repetition Chapter only that follows, will changes equations extended, will rules avoid highlighted. point departure, old node numbering system given Figure changed new system shown Figure 64. Points are renumbered according following scheme: Points in Old System Corresponding Points in New System M1 MAB MAB+1 ME1 ME+2 NI LT+1 , U( Figure 64. New numbering disc ret i zat ion system boundary for the CVBEM in a triply connected domain. Development S . a Triply Connected Domain new numbering system, complex potential point can evaluated using 2ri [t(zk)]T=2ri [t(zk)] S+i ($oAk E k) (616) Here subscript refers triply connected. The term 2ri [A(zk)] S addition, recognized that braced equation term been (411) changed above Equations have been given as (412) through (416) new term Ek appearing (616) is defined "k;kfL,L+1, LN,LT+1 zkz LN zL+ zk zLZk  zL+1 ZLN Z LN k Z L+1Zk ZkZLT+1 n zLz LT+1 zLZk ZLT+1Zk] for special k subscripts of 2, = ln ZLZL+1 [ln LNZL+1 [in ZLZLN ZL+1ZLN SL+1zLT+1 i ZLZLT+1In _ ZLNLT+1 1 ZLZLT+1 n ZLZL+1 ZLT+1ZL+l1 ZL ZLN] zLT+1 ZLN (618) (619) zLZLN nZL+1ZL zLT+1ZL zLL+1zLN N ZNZ 1 zIr.Iz, ] (620) (617) of Nodal Eaquations for CVBEM L+1 derivation excess stream functions above, points along cuts are shown equivalent each other. This also applies new system close relationship thus expected between equations. Indeed, careful comparison (412) (617), (413) (618), (414) (619), (415) (620), (416) (621) reveal s that following points are equivalent to each other: Points Along Cut (M+1)/1 Points Along Cut (L+1)/1 LT+1 cross link thus established between points equations. Equations (419) through (424) can then revised to develop computer codes for the evaluation term (616). Development of Computer Codes for Evaluating Terms (616) There are three terms righthand side (616). The first term can coded using equation (418) which constants C and The (616) imaginary components D have been have been given defined been derived Table (419) as 41. real (420) i$E k term new can evaluated revising previous coding First following (419) (421), is decomposed real imaginary components (622) where =UkKLT+l ,k(XkXLT+1)+HLT+1,k YkYLT+1) +KLN,k(Xk =VkKLT+l XLN) HLN,k YkYLN) YLT+1)HLT+1 (623) k (XkXLT+1) +KLN, k (YkYLN) +HLN, k (XkXLN) (624) Here coefficients are given together with their coding rules Table 61. Notice that expressions given this table are deduced from Table 42, in which nodalpoint locations are changed according to list equivalence points along cuts given earl ier. Next (422) is written +i~ I~k k =k kk k (Yk 9k Tabi Equat ions 61. coding rules. Equation =UkKLT+l k (XkXN) +HLT+l, k (YkYN) +KLN, k (Xk M) HLN,k (Yk (623) k =VkKLT+1, k (Yk YN) HLT+, k (XkXN) +KLN,k (Yk YM) +HLN, k (kM)] (624) where (ZL+1, SZL+1;zk) k (xf xe) 4 k(x xe) k (XfXe)  d(zf ,Zk) d(ze, z,) Ak e,k e,k (YfYe e,k (Yf Ye) = arg(zf zk) arg(zezk) (YfYe) Coding rules for subscripts e and f in the Ke,k through Fe equations given above: e=LN, f=L+1. Zk) ,Ze Xe YM)] ) / Table 61 (Continued). Modifications through e equations ~L+1' LN 'L LT+1: k=L+1 KLN (ii) L+1= k=LN: Uk=ln HLN,L+1= LN LN = d(zL, d(ZLN ,ZL+l) ,ZL+l) , Vk=O(ZL,ZLN;ZL+1), HLN,LN (iii)For k=L: Uk=ln d( zLT+l, ZL) d(ZL+l, ZL) , Vk=(ZLT+1,ZL+1 ;zL)' KLT+1, L HLT+1, L=0 (iv) k=LT+1 LT+1, LT+1 HLT+1 ,LT+1 Then, nodal equations for a triply connected domain are read ly obtained A N S(Zk) =A* k+B* k+ j1k J ,j+1$k , jj+G2, j+l +G3 ,j.+G4 b(zk) =B*4k jj+l]+ 27 N  C j1 j j+1 2 2 k ,jj+G4 (626) ,j+1 j+l (627) E 2r thk ,j.j+l] +A1 Here first three terms righthand sides come from 2,ri [i (zk)] coefficients these S term i equations (616). can Hence, found using expressions given Table 43. Finally a matrix equation can constructed (435). this equation, righthand side terms are re defined for the triply connected domain ST4k 628) A*~ G ,jG2 1,a jj G2 =12 (WoAk E k) Once again, three solution methods can used to solve matrix equation described Chapter this effort, changes must be made (436) (437) those equations Table 44 as follows: 2rA 2x9k 2,bk 2 (\OA (630) derivation nodal equations boundaries now complete. Development of Equations for Interior Points derive equations interior points Zo6E1 use again mad e equations (438) through (442). Referring numbering system given Figure 61, doublevalued stream functions points ZM+r, ZMAB' ZMAB+1 can derived sorting out HMl HM+l from , HMAB, HMAB+1 , HME1, summation HME, (439). HME+1' Following HN1, the terms same limiting procedure leading (441) , interior equation derived for triply connected domain new numbering system (631)  2 OA E Ek) E ,k) 2 i[ (zo)] T=2ri[ (z ")]q+i(WoAFE) 2ii [t (zo)]S= Hj(zo) (632) oLN n ZL+1Zo ZLZo  ZL+ZLN zLNZo zL+1 Z zozLT+1 in ZLo ZO zLzLT+1 zLT+10 o (633) The above equation concludes derivation the CVBEM equations for points inside a triply connected domain. Numeri 1 Application of Triply Connected the CVBEM Domain The domains problem development now the verified rectangular CVBEM for solving region triply heat illustrated connected conduction Figure 65. With exact solution taken w(z this example analogous previously solved conduction problem hollow square region Figure 51. Here again, horizontal boundaries are insulated; vertical boundaries are imposed with uniform temperatures, heat flows steadily from the right left. numerical solution this problem, boundaries are discretized using sixteen nodes, Dirichlet conditions are specified at all nodal points. total rate heat flow across outer boundary )=z, 