The application of the complex variable boundary element method to the solution of heat conduction problems in multiply ...

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Title:
The application of the complex variable boundary element method to the solution of heat conduction problems in multiply connected domains
Uncontrolled:
Multiply connected domains
Physical Description:
x, 192 leaves : ill. ; 28 cm.
Language:
English
Creator:
Kassab, Alain Jacques, 1958-
Publication Date:

Subjects

Subjects / Keywords:
Heat -- Conduction   ( lcsh )
Boundary element methods   ( lcsh )
Boundary value problems   ( lcsh )
Functions of complex variables   ( lcsh )
Domain structure -- Computer programs   ( lcsh )
Mechanical Engineering thesis Ph. D
Dissertations, Academic -- Mechanical Engineering -- UF
Genre:
bibliography   ( marcgt )
non-fiction   ( marcgt )

Notes

Thesis:
Thesis (Ph. D.)--University of Florida, 1989.
Bibliography:
Includes bibliographical references (leaves 100-104)
Statement of Responsibility:
by Alain Jacques Kassab.
General Note:
Typescript.
General Note:
Vita.

Record Information

Source Institution:
University of Florida
Rights Management:
All applicable rights reserved by the source institution and holding location.
Resource Identifier:
aleph - 001507891
oclc - 21663379
notis - AHC0774
System ID:
AA00002125:00001

Full Text












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 T-W 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,


never-ending


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-


Iiy-in~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


(6-1
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,


(4-18) ....


coding rules.


Coefficients


equations


(4-23)


and G.
I (4-24).''


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,


Zk-l;


zk .


Boundary


CVBEM


4-2


discretizat ion


in a doubly


Pictorial


used


connected


representation


to derive


domain.


limiting


process


Zo~4Zj


bz-O


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


5-7


Comparison


exact and
C'


CVBEM


generated


59


Page


methods ........









Figure


6-3


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


complex-variable


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


double-valued


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

steady-stat

dimensions.


equations


solution


subject


example,


analysis,

partial c

e heat


ordinary


appropriate


partial


boundary


unsteady-state


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


heat-conduction


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,


two-dimensional


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


heat-conduction


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


Semi-direct


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


semi-d i rect


boundary


1 n+nfl'rn 1


m + 1,h rn-


r'- ,, 1F -4- n r I


-,, a -.


. 1












numerical


modeling.


contributed


9-11


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

6,12-17


, 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.


24-29


More


recently,


RVBEM


have


been


used


conjunction


with


optimization


methods


design


thermal


systems.


30-32


only


recently


that


complex


variables


have


been












linear


trial


function


between


boundary


nodes


investigate


multizone,


anisotropic,


two-dimensional


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.


36-37


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.


40-42


Hromadka


successfully


tested


CVBEM


variety


steady


two-dimensional


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.


43-46


such


applications


CVBEM


can


accommodate


large-scale


inhomogeneous


subdomains


Lai47


applied


CVBEM


solution


density-stratified


inviscid


fluid-flow












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


pre-solution


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
E-zo


ZoEQ


(3-1)


which


relates


the


value


of a complex analytic


function


w at


point


inside


k-connected


Jordan


domain


complex


plane


integral


that


function


along


boundary


that


domain;


see


Figure


3-1.


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 two-dimensional


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.













(3-2)


which


field


relation


(3-3)


3-3),


n represents


outward


drawn


normal


to r,


tangential


coord i nate


along


again


positive


direction


defined


earlier.


Using


Fourier


heat


conduction


left-hand


side


(3-3)


enables


expressed


an


integral


heat


flux.


This


provides


link


between


stream


function


heat


flow.


Cauchy-Reimann


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


finite-length


segments denoted


The entire


boundary


union


these


segments,


i.e. ,


r.
r^U rJ


see


Figure


3-1.


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


3-2,


linear


basis


functions


Nj,l(s))


are


defined


on each


element


(3-4)


(3-5)


where


O

Using


these


basis


functions,


boundary


coordinate


is expressed


i(s)


parametrically


on


Nj ,2(s)


(3-6)


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 j-1 z j+1


z
j+2


ZN


gu re


.The


near


funct


ions


NJ


Nj,i











S-zi
Zj+l1


(3-7)


The


differential


coordinate


along


element


follows


from


(3-6)


(3-8)


j+-z.j)


similar


fashion,


complex


potential


expressed


parametrically


w[E(s)]


Nj ,(s)


(3-9)


Here,


using


specified


Hromadka s


complex


notation,


potential


.j=$(z.j)+i (zj)


nodal


point


Introducing


integral


(3-6),


formula,


(3-8),


equation


(3-9)


(3-1),


into


leads


Cauchy


first-order


approximation


to w(zo)


denoted


A o
W(Zo)


N fl N
2rit(zo) = [ i (
O1


S~f~


- zo


j+l-Z J


(3-10)


Then,


expanding


terms


integrand


rearranging,


can


expressed


sum


first-


j+i j


s>


(")"j+l
Zj+l]












where


(1-s)


Ij(zo)=


s--7


(3-12)


(zo-z .


(3-13)


integral


(3-12)


can


performed


to yield


Ij(zo)=(j +i
3 +1


J+--1
- *) + [7 j.0 +w .(l-7 j)]1n(
\ 3 /+ -7


upon


introducing


from


(3-13),


j+l (Zo-zj) -Wj(


-zj )


zo--Z


Sz.i+
I ^J


-zo
zo (3-14)


Now


fully


specified


each


node


then


equat ion;

b(zo)a

boundary


explicitly.


(3-11)


However r


conditions


boundary


(3-14)

regular

specify


are


sufficient


heat


conduction


any


integral


estimate


problems,


none


problem


thus


them


formed


using


equation


(3-11)


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.jl-Zj)


(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


(3-11),


nodal


point


approached


from


interior


are


contributed


two


adjacent


elements

arguments


approaches


logarithmic


term


(3-14)


vanishes


equals


unity,


thereby


leading


singularities


integrals


Fortunately,


Ik-1


these


singular


terms


can


manipulated


cancel


each


other


out.


With


this


mind,


contour


integral


equation


(3-11)


first


recast


Ij(zo) +Ik_1(zo) +Ik(zo))


(3-15)


S(zoI=kk (-
J=1
j k,k-l


Then


limit


applied


A \
w(zk)=


Sim
'7'/-& 97-.


I (zk)


or


-r 1


(zo)=












Finally,


(3-14)


introduced


into


(3-16),


after


much


manipulation




2ri 1


this


(zk)=


yields



Zk+1 -Zk
Wkn lzk-1-Zk


N
+

J k,k-1


j+1 (Zk-Zj) -j (Zk-Zj +1)
(z +1-z j


z +1 kz<-
z.-zk (3-17)
Zj k


Equation


(3-17)


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


(3-17)


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


(3-17)


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


(3-17)


clear


that


(3-17)


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)


(3-18)


where


refers


distance


refers


angle.


They


are


in turn


related


nodal-point


locations


d(zj+1 ,zk)


-xk)2+ (yj+l-k)21/2


(3-19)


d(zj,zk)


=[(x-xk)2+ (yj-yk)21/2


(3-20)


arg(z -zk)


(3-21)


Here


principal


value


(- r

taken;


see


Figure


3-1.


However,


confusion


may


arise


evaluating


logarithm


outside


summat ion


(3-17)


this


outside


logarithm,


0(zj +1,Zj ; zk)=arg(Zj +1-zk)


d(zj+l


=[(Xj+1












where


d(zk+l ,zk)


=[(Xk+l-k)2+ (Yk+ l-Yk)2]1/2


(3-23)


d(zk-l,Zk)


=[(xk-1


-Xk)2+ (k-l


(3-24)


and,


angle


re lated


interior


angle 0int


O(zk+1 ,Zk-l k)


= 2--0nt
lnt


(3-25)


illustrated


Figure


3-3.


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


k-1


3 x


figure


3-3.


Evaluat ion


angle


8(zk+l,Zk-1Zk).


















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,


54-56


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.


(N-M)

. 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


3-2)


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)


(4-1)


(4-2)


Here,


related


total


heat


flow


rate


per


unit


thermal


conductivity


(Q/k)


across


either


system


boundaries


or


_= (Zl)


_= (ZM+I)


rl













end-


starting-point


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


(3-10),


linear


CVBEM


approximation


Complex


so


QNM+1 I












2riw(zk)=


Sim
6x, 6y-O
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) (4-4)
J/


Equation


(4-4)


can


recast


2ri(zk) =


lim
6x, 6y-O\


lim
Zo Zk


2ri(zo)


(4-5)


Here


limit


inside


parentheses


recognized


that


time


2iriw(zk)


terms


previously


for


derived


j=k-1


(3-17)


fact


summation


this


(3-17)


can


recovered,


inner


limit


expressed


a summation


of Hj(zk)


N N
= Hj (zk)=
=1 i =1
Ji a=


j+l(zk-zj) -z (Zk-j+) Zj +l-Zk
z j+1- J ]1 z j- zk z
J+l J J


Notice


that


j=k-1


terms


have


now


been


included


summat ion


because


k-l+


Zk+l -
Hk=&kln ZkI-
k k zk-1-


Zk
zk


(4-7)


outer


limit


(4-5) is


applied


next,


notice


(4-6)


(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


(4-5)


(4-6)


write


2xri(z1) =


1rim
6x, 6y-&O


N
Hj(zl)


(4-7)


Since


zM=Zl+6x-i by


zM+ 1=ZN+6x- i 6y


Figure


4-2,


terms


j=1,M-1,


must


sorted


out


from


summation,


2rii (z1)


im y-
6x, 6y-a0


H1(Z1)+


HM-1(zl)+


HM(z) )+


HN(Z1)+


H(zl))


(4-8)


j#1,M-l,M,N


Then


applying


limit


as


indicated


using


(4-1)


relations zN+l=zl


WN+1 W,


there


is derived


/ z2-Z
2riu(z1)= 1ln ZM_--z1 +


ji1


Hj(z1)


,M,N


Z 1-N-l1 lM+1-Zl
zM+1 ZN-1 ZN-l -Z1j


(4-9)


l zM+1M-l-z
ilnzMl-1-Zl ]













P
H-i


P1


zi


ZM+1


^Sz





M-1
"---
L.\


Figure 4-2. Pictorial


representation of the


limiting


process


and bz-+O.


I _


Zo-+Z


6y


8x


M+l












where


natural


logarithm


term


inside


bracket


comes


from


HN-1 (Zl)


term


summation.


This


related


term .


WM+1


is noted


(4-2),


that


because


thus


producing


way


this


(4-9)


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
k-l,k


be deduced




Hj(zk)

,M,N


+. i l M+nl-(-Zk
1 zk


(4-10)


Equation


(4-10)


can


used


for


any


point


boundaries


domain


Figure


long


following


rules


are


adopted:


When


k=l,


k-l=M-1


when


k=M+1,


k-l=N-1.


(ii)


Singularities


logarithmic


terms


can


cancelled


out


limiting procedure.


For


example,


when


k=l,


term


brackets


(4-10)


becomes


zl-1ZN-1 i M+l-Z)+ zl:N-l zl-zl 1


ilnzM+1-z~
191 24-2


w(zl)


Zk-ZN-1 ZM+1-Zkh_ Zk--ZM1nl ( Z1-Zk
-M+1 --ZN-1 N-1 k 1 M-l M-l kj












(iii)Rule


above


also applies


summation.


example,


when


k=1,


k-l=M-1;


then,


HM-l


should


excluded


from


summation


(4-9)


(iv)


summation


terms


Hj(zk)


(4-10)


, (zM)


is set


equal


to ((zl)


W(zM+1)


set


equal


O(zN)


for the


reasons


previously


explained;


see


paragraph


following


(4-9).


Since


Z(zM)


related


to 3(zl)


&(ZM+1)


related


to c(zN)


equations


(4-1)


(4-2),


AM
w(ZM)


i(zN)


equations are


redundant.


CVBEM


developed


above


are


strictly


valid


doubly


connected


domain


shown


Figure


4-1.


order


develop


connected


general

domai n


method

analyzed


that


earlier,


reducible


new


numbering


simply

system


introduced


cut


as


shown


effectively


Figure


eliminated


4-3.


the


new


doubly


system,


connected


domain


The


point


now


takes


place


point


M-l,


point


takes


place


point


N-1


(Figure


4-1).


With


these


changes,


the


new


nodal


equation


can


then


be derived


2ri0 (zk)
an^


{Jzk) z k+


-Zk
- Zk/


























FN r

+1


r-M+1



['-I .


Figure 4-3. 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 Zl-Zk


Zk-ZN i ZM+1lZk
ZM+l-Z ZN--Zk
M1C+1 N7 ZN --k


for


(4-12)


special


-- (ln(


---- [I(


ZM+1-z1
M 1


Z-ZNN InzM+1-z1
M+1 N ZN-Z1


-z


--Z


(4-13)


-z


AM+1=


_( ZM--ZN ln(ZMlN
_M+l-N I N-


-z


ZM+1 -ZM Z -Z M1 i ZN-ZM+
Z1-ZM ZM--ZM+1-- I Z-M+l


,/ZN- -ZMM I1-
[Z-zM nzM-I


)+ln(


(4-14)


(4-15)


(4-16)


Notice


that


equations


(4-11)


(4-16)


are


val id


doubly


connected


domain


new


numbering


system


Figure


4-3.


new


numbering


system


when


k=1,


k-l=M


when


k=M+1


k-l=N.


informative


compare


equations


(4-11)


3-17).


Clearly


right-hand


side


exactly


Zk--ZM nfzl
Zl-zM In zM


M+1-ZN >
zl-zN 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


(4-11)


thus


reducible


simply


connected


domain


equation


(3-17)


is noted


that,


development


follow


remainder


this


chapter,


node


numbering


will


refer


new


system


Figure


4-3,


unless


noted


otherwise


now


estimated


possible


temperature


to derive


stream


equations


function,


relating


(zkk)


specified


temperature


stream


function,


first


term


right-hand


side


(4-17),


associated


with


single


valued


nodal


equations,


expressed


27ri [(zk)]S=


w(Zk)1ln


d(Zk+l,Zk)
d(zkl ,Zk)


+iG(zk+lZk-1


;zk)


N
+k

j j+lk


L[j+1(zk-zj)


-W(zk- Zj+l)](C+iD)


(4-18)


Here,


have


been


given 1


, (3-23)


3-25).


A
#(k) '








35
Table 4-1.

Expressions for coefficients C, D, A, B, and F


in Equation


[A(xj+1-X) +B(yj+1


[B(xj+1


-xj) -A(yj+1


(4-18)


-Yj)]/F
-y )]/F
J
j~ ]/


where


d(zj+1 ,zk)
d(zj,zk)


o (zj+1,Zj;zk)

(xj+1-Xj)2+ (Yj+1-Yj)2












Next


term Ak


is decomposed


Ak Ak


+iAk


(4-19)


components


can


derived


expanding


(4-12)


(4-16)


into


their


real


imaginary


components


with


help


explicit


expressions


for


(4-19),


giving


= [Ek-CN, k (xk-XN) +DN k (Yk-N) +CM, k (xk-XM) -DM,k (Yk-M)


(4-20)


= [Fk-CN,k (Yk-YN)-DN,k(xk-XN)+CM,k (Yk--YM)+DM,k(xk-XM)]


(4-21)


where


coefficients


D's,


are


listed


Table


4-2.


Coding


rules


for


subscripts


encountered


general


expressions


C's


are


given


table.


Provisions


are


also


made


table


for


modifications


required


to evaluate


nodal


equation


(4-17)


can


now


be written


Ai[(z (
2xi[4 (zk) + i (zk) D


+iAk)


(4-22)


as


Atk


A k


as


C's,


, AM,


M+1'


2ri[ (zk)]S+i (A k









Tab 1


Equat ions


4-2.


coding rules.


Equat ion


=[Ek-CN, k (k-N) +DN,k Yk-N) +CM, k (Xk-M) -DM,k Yk


(4-20)


=[Fk-CN, k Yk


-YN)-DN, k (xk


-xN


) +CM k (Yk-YM) +DM, k (Xk-XM)


(4-21)


where


(ZM+1


,zk)


,zk)


(ZM+1


,zl;zk)


k (Xf -e)

k(xf-xe)


e)l /


-A,k (Yf-


d(zf,zk)
d(ze,zu)


,Ze;


= arg(zf-zk)


arg(ze-zk)


Xf -Xe


(Yf--Ye)


Coding


rules


subscripts


Ce,k


through


equations


given


above:


e=M


f=1l


k= [


A^k


%k


-YM)


^k


k(Yf-Y










Table


4-2


(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


(4-23)


4(zk)=B *k


N
A*ak+E
j=1


[G3, j+G4,j j+l


-G1


(4-24)


,* jj-G 2, jj +1 &5+ 2-k


where,


expressions


A's


,Gi, j


are


given


explic


(4-23)


itly


Tables


(4-24)


4-3.


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


4-3.


Coefficients


equations


(4-23)


G for the
and (4-24).


nodal


d(zk+l


,zk)


d(zk-l,k)
Zk+l Zk-l; zk)


G2 j

3,j


-Y j+l) -C(k-j+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


4-1.


-yj)


[D(Yk












values


boundary.


However,


when


a Neumann


condition


imposed


boundary,


Cauchy-Reimann


conditions


can


used


to derive


j+1=


(4-25)


Consistent

function


with

the


the

heat


linear


flux


approximation


approximated


the complex

linear over


element


equat ion


(4-25)


becomes


=- J1
*sb. =0.- ,
-J+1 'J O0


[N.j, (


) .j+N.j
3 3 J


-zj)ds
3_


(4-26)


Evaluating the


integral


equation


(4-26)


summing over


elements


gives


a relation


between


j +1 and


.j+3 =ij+E
i-l


(4-27)


where


(4-28)


(As)j+i-=d(zj+iz j+i 1)=- Z+-Z.+i


(4-29)


(kqa,j+i-1 2Lkk/J+i k +-1


k(0)


() +l(zj +l


k a, j+1-1(+1-1]












problem


with


heat


flux


spec ified


along a


boundary,


equation


(4-27)


can


used


prepare


a data set


for


direct


input


into


nodal


equation


(4-23)


(4-24),


depending


method


assembly


equations


chosen


solving


problem.


Robin


convective)


condition


imposed


boundary


is modeled


by the


relation


- 4o0


= 0


(4-30)


Here,


convective


coefficient


(which


may


also


account

ambient


for

source


similar


radiation


or


that


sink


linearized),


temperature.


leading


equation


Following


procedure


(4-27),


stream


functions


are


related


Robin


condition


following expression:


1 1

_- h 1
(4-31)


Here


(h ,


(4-32)


on


j1 h- 1Lh4 i]
j -ti-1 2 +1 ,k j+i-


f( h












again


inner


relation


M>j+l>1


boundary.


between


boundary;


outer


boundary,


Equation (4-31)


stream


however,


N>j +>M+ 1


provides


function


because


on


additional


temperature


complexity,


not


possible


use


as


direct


input


nodal


equations


(4-23)


(4-24).


use


solution


will


discussed


next


section.


Assembly


Boundary


Element


Equations


close


examination


(4-23)


(4-24)


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


left-hand


Then,


estimated


side


equations


that


also


appear


depend


prescribed


same


point


right-hand


side


these


equations.


This


a distinct


feature


CVBEM


that


leads


diffe rent


solution


methods


will


discussed


later.


also


noted


that


coefficients


G's


(4-23)


(4-24)


are


solely


dependent


nodal


on












whose


boundaries


are


fully


specified,


coefficients


can


be evaluated.


final


point


related


those


source


terms,


2rA
2Ak,


2r-Ak


(4-23)


(4-24).


They


are


course,


result


perturbat ion


term


(4-17).


shown


(4-20)


(4-21),


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 ,


(4-33)


STk


2rtk


(4-34)


possible


combine


(4-23)


(4-24)


matrix


equation


CR, CR,


C C
I, CI


ST,


A


A


(4-35)


Here


subscripts


R and


for


partitioned


matrices


as












equation.


matrix


desirable


These


formulation


that


part it ioned


given


matrices


are


equation


elements


each


order


(4-35)


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


(4-35)


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


(4-3)


(4-27),


related


heat


flux


boundaries.


the


solution


heat


conduction


problems


with


prescribed


Dirichlet


Neumann


are


conditions,


given


either


temperature


boundaries.


heat-flux


heat


fluxes


values


are,


nevertheless,


related


(4-27).


However,


Robin


condition


imposed


on


boundary,


none


temperatures


or


heat


fluxes


are


given,


yet


they


are


still


related


(4-31).


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+k-th


equation


(4-35).


meantime,


since


k-th


given,


equation


longer


deleted


from


unknown;


(4-35).


this


permits


implicit


equation


for


node


then


obtained


rearranging


nodal


equation


(4-24)


N

j=1
j ,j+l1k


[G3, j 4j+G4, j+1


-G1


Sj -G2 j]+
,) 3 02,1 3+11+


(4-36)


This


equation


can


then


used


together


with


other


equations

unknown *1


solve


appears


unknown


both


sides


Inasmuch


(4-35),


this


as


method


called


implicit.


The


method


explicit


that,


solution


since


method


differs


given,


from


can


implicit


set


equal


k-th


equation


(4-35).


other


hand,


A* k


= (B*


2i ^,













equation


node


can


obtained


from


the


nodal


equation


(4-23)


(1-B*) k


N
=A*gk+E
j l=1
j ,j+l1k


[G1, j+G2, j+1


+G3,j j+G4, j +1]


2-A k
2r.k


(4-37)


This


equation


can


then


used


together


with


other


equations


solve


unknown


Notice


that


this


time


unknown


appears


only


left-hand


side


equation


(4-35),


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


(4-31)


supply


additional


equations.


discussed


equation


(4-31)


complicated


because


contains


both


unknowns


direct


input


into


equations


(4-23)


thus


impossible.


must


then


append


(4-31)


(4-35)


raise


; number


equations












(4-35).


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


(4-35)


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


4-4.


addition,


implicit


explicit


formulations


are


implemented


FORTRAN


program


provided


Appendix










Table


4-4.


Implicit,


explicit,


hybrid


solution


methods.


Implicit


Method:


1S
sol\


(4-24)


specified
le for Ck


zk, then
loading the


set tk=
following


. Also
equation


the matrix equations


N

j kk-1


[G3, jj+G4, j+1


-G1, j-G2, j+1 A
1Ck


A(Z
w(zk) k + Ck


Upon


solution


(4-23) 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 kk-l


[G1 jj+G2, jj+G3,+G4, j +G, jj+1 k


Upon


solution


the matrix


equations,


set


w(zk) =k+ k'


Explicit


Method:


(4-23) to sol
in the matrix


spec ified
ve for k
equations:


at
by


z then
loading


set
the


following


Also


use


equation


N
(1-B*) k=A*k +

j kk-l


[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
(4-24) 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


4-4


(Continued)


Hybrid


Method:


S. Also
loading
equations


is
use
the


specified
(4-23) and
following


z then
(4-24) t4


two


equat ions


Ck tand
r k a
n the


A
set_ =
d k by
matrix


k= A*#+B*k+
j kk-1


Gl ,I j +G2, j3+l


.iS + G4
, J4+


O= (B*


-1) k-A* + +
Jkk-1


[G3, j +G4, j J+1


1 J J-2,


jS -+l] + ah,


Upon


solution


matrix equations,


set


w(zk)


p. If
ok=ad"
1 oad 1 ng


specified


use


(4-23)


following


z ,
(4- 4)


two


then


set


to solve


equations


set


for
in


matrix


equations:


,jj++G2


+ k-E
jk, k-1


N
k =B k-A k+ t
j Sk ,k-l


+G3 j S+G4


, J++1-1 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


louble-valued


are


known


derived


potentials


boundary


focusing


nodes


nodes


attention


, zo)
w{Zo)


only


Figure


4-1.


Cauchy


integral


evaluated


limit


small


gap


goes


zero


A(zo)


lim
Ex by-0


P--Zo


(4-38)


Using


linear


elements,


(4-38)


can


integrated


to yield


A(zo)


lim
6x, byO0


N
1 V'
27rni I
j=1


Hj(zo)
J


(4-39)


account


for


double-valued


complex


potential


points


Figure


4-1,


HM-,


are


sorted


out


from


summation,


they


are


evaluated


complex


indicated


potential


limit.


(zo)


The


CVBEM


the


approximation


numbering


system


of Figure 4-1


then


A /
W;(zo)


N

j=1
J M,N


H (zo)+2-(-zM_ 1 zM_ zo)


_ Zo-zN-1
- z -z


i ZM+1l-zo
z_-zo


(4-40)


as


HM+1'


HN-1'


+ In(ZM+1-Zo
2 z-zo


I














2iriw(zo) =


Hj(zo)


i A


(4-41)


where,


A =(zo--zM 1n(Z)zo+ 1 /zM+l-zo
zIZM/ n zM-zo/ +n zl-zo )


/ Zo-ZN ZM+1-zo
-ZM+1-ZN)\ ZN-o )


(4-42)


Notice


here


that


zo r'.


With


derivation


of the


boundary-


interior-node


nodal-equation


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


f|I I [


S | r1 I j


I










Exact


solution:


(z)=z


-2


-1


2



1



-1



-2


Equipotential


lines


Figure


5-1. 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


(4-3)


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


5-2.


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


M-12
N-24
Ro-1.0
R1-0.6


Exaot solution
o0 (z)- z
+ J (z)- z2
o (s)- ez


0-speaified 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


(4-3)


Yet,


like


previous


example,


taken


to be


unknown


computed


the


solution.


different


set


Dirichlet


conditions


imposed


boundaries


annulus


shown


Figure


5-3.


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


5-4.


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


5-5


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


i-umbe
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


5-6.


Here


w(z)=z2


test


case


Figure


5-2


re-studied


effects


solution


methods


used.


The


cross


plotted


for


implicit


method


Figure


is plotted


as square


Figure


5-6.




























a a ag


act Solution


0(r#)-
0l-#*


U-12


Ina(r/Ro)-(1/Bi)
ln(B/Na)-(1/B1)


e,-100 'C


a1-10


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)


(5-1)


where


w(z)


exact


solution,


A \
w(z)


represents


CVBEM


solution.


computations


were


performed


double


precision


Microsoft


elimination


8-Mhz


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


5-1.


addition,


total


heat


flow


rates


across


boundaries


are


computed


'=7.099x10-16


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


first-degree


piecewise


linear


analytic

boundary


function


domain.


For


the al


nnulus


shown in


Figure


5-2,


the


first


case ,










Table


5-1.


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 0.3706E-14
2 0.1776E-14 0. 1332E-14
3 0.6661E-15 0.1110E-14
4 0.4441E-15 0.1332E-14
5 0.1776E-14 0.8882E-15
6 0.1110E-15 0.1457E-13
7 0.6328E-14 0.2220E-15
8 0.8882E-15 0.1776E-14
9 0.7772E-15 0.1887E-14
10 0.6217E-14 0.1110E-15


Definition of the errors eC and e, above:

e I=CVBEM tEXACTI
e' = CVBEM -DEXACT1












w(z)=z2


problem are


plotted


Figure


5-7.


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


5-3,


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


5-4.


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


boundary-condition


combinations,


errors


are


less


than


Notice


that


N=50


for


this


example.


Since


errors


can


reduced


increasing


demonstrated


Figure


5-3,


CVBEM


are


still


accurate


solving this


i rregular-geometry


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


S-0.20


-0.40


-0.50


-0.80


-1.00


-, -1.20
1.20


Figure


5-7.


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


Robin-condition


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


5-6,


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


(4-23)


or


(4-24).


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


(4-4)












Another


concern


arises


positioning


cut


hence


placement


nodes


M+1


boundaries,


see


Figure


4-3.


The


sensitivity


method


position


cut


investigated


re-so


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


ill-conditioned.


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


6-1.


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


6-1 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'


(6-1)


w(ZME) =(ZMAB+1)- i E'


qME,MAB+1


(6-2)


W(ZME+1


)=I


(ZMAB) 11PE


(6-3)


Following


(4-27),


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


(6-4)


(6-5)


These


five


equations


will


used


later


re late


complex


potentials


double-valued


nodal


points.


Basis


for the


Formulation


of CVBEM


a Triply


Connected


Domain


should


noted


that ,


Figure


6-1,


boundary


=J(


E-


\II.
AT -



AB=












point


while


that


point


inner


boundary


Figure


loop


6-2.


closing


However,


from


stream


above


functions


point


points


see


MAB+1


are


different;


they


are


boundary


loop


closing


from above.


shown


Figures


6-2,


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

(4-6),


presented


Chapter


contribution


IV.

the


shown


complex


(4-4)


potential


A(zk)
w(Zk)


any


boundary


element


comes


from


Hj(zk)


term


Hj(zk


)=-W


Zk-
j+1 z]+i


Z ln


ZI-Zk
ZZ k


.Zk-Zj+ Zj+l-Z
-W Z-zj_ In Zi--Z
j+1 3/ j k


(6-6)


Here


first


term on


right-hand


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


6-2.


Formation of double-valued stream functions


at the cuts.


f--


---->lj

























Zn+l


n1


n -
n-


Figure


6-3


A nodal


point


shared


two


boundary


elements.













contribution


complex


potential


by this


point


can


be expressed


Zk-Zn-l In
n-zn-zn-


Zn-1-Zk


Zk-Zn+1 nl Zn+l-Zk
Zn+l-zn Zn-Zk


(6-7)


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


6-2,


such


excess


points


stream


, ZME,


functions


ZME+1,

these


points


can


derived


sorting


out


stream


function


component


from


(6-7)


writing


AZkn-l (1 Zn-Zk -
nZ p n-Zn-l Zn-1Zk/


Zk-Zn+ll f Zn+1-Zk
zn+l-zn Zn--zk


(6-8)


where


w(Zn)


-wO(zp)


(6-9)


denominator


(6-9)


complex


number


i=J-l.


Here,


for


sake


generality,


first


subscript


is used


represent


specific


point


whose


excess


stream


n-p












(6-9),


as defined


'.
i-p

(6-1)


can


through


expressed


terms


those


(6-5)


Equations


(6-8)


(6-9)


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_ -Z-11 ZM-Zk
MM-l-1 M-l


Zk-ZM+1 IM+1-Zk
-M+1-M I Zk


(6-10)


Z-Z N ZN-Zk
AN=-N-(M+1) ZN ZN-1 zN-l-Zk


Zk N+1
ZN+1 ZN


ZN+-Zk
N k


(6-11)


Then


since


ZM+1-Zk
A +AN= o In M- Zk
z1-zk


_k- N-1 I Z I M+1--Zk
ZM+1-ZN-1 ZN-1-zk


+zk-ZM-1 In z-zk
Z18


- N-(M+1)M-1 o


(6-12)












(4-10)


This


term


accounts


for


double-valued


stream


function


equation


component


(4-10),


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


6-1,

ME+1,


they


must


accounted


for.


Applying


(6-8)


these


points and


using the


relation


that


(6-13)


ME-(MAB+1)- (ME+1)-MAB E


gives


_A +A In- ZMAB-Zk zk-ZME+2 in ZMAB-Zk
ME ME+1 WE ZMAB+ Zk ZMAB-ZME+2 ZME+2- k


Zk-ZME-1 ZMEAB+1-zk
ZMAB+1-ZME-1 / ZME-1-zk


(6-14)


now


necessary


return


concepts


established


earlier


this


chapter.


previously


mentioned,


double-valued


stream


functions


only


appear












(6-12)


these


(6-14)


equations


must


are


related ;


equivalent


indeed,


eac


points


other


following


manner:


Nodal


Points


(6-12)


Corresponding


Points


(6-14)


MAB


MAB+1


ME+2


ME-1


Establishing


these


extension


relations


analysis


for


ows


a doubly


for


connected


direct


domain


triply


connected


domain.


particular,


nodal


equation


(4-10)


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+1-zk
1 k


N

j=1
J fk-


Zk-_ZN-
ZM+1 Z


Hj(zk)


l,k,
MA ,M
1 zM+1
In zN
<-1 ^N-1


Zk-ZM-1 In
ZlZ-M-1l


1- zk
ZM-l-Zk


-i E[ln(


ZMAB-zk
ZMAB+1-Zk


Zk-zME+2 I ZMAB -Zk
ZMAB-ZME+2/ ZME+2 Zk


2ri i(


-Zk
-Zk












rules


following


(4-10)


are


accordingly


amended


include


following


relat ions:


In addition


when


k=MAB+1


k-1=ME-1


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


6-4.


Points


are


renumbered


according


following


scheme:


Points


in Old


System


Corresponding


Points


in New


System


M-1

MAB


MAB+1


ME-1


ME+2

N-I


LT+1


, U(


























































Figure


6-4. 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)


(6-16)


Here


subscript


refers


triply


connected.


The


term


2ri [A(zk)] S


addition,


recognized


that


braced


equation


term


been


(4-11)


changed


above


Equations


have


been


given


as


(4-12)


through


(4-16)


new


term Ek


appearing


(6-16)


is defined


"k;kfL,L+1,
LN,LT+1


zk-z LN zL+ -zk zL-Zk
- zL+1 ZLN Z LN- k Z L+1-Zk


Zk-ZLT+1 n
zL-z LT+1


zL-Zk
ZLT+1-Zk]


for


special


k subscripts


of 2,


= ln ZL-ZL+1
[ln LN-ZL+1


-[in ZL-ZLN
ZL+1-ZLN


SL+1-zLT+1 i
ZL-ZLT+1In


_ ZLN-LT+1 1
ZL-ZLT+1 n


ZL-ZL+1
ZLT+1-ZL+l1


ZL ZLN]
zLT+1 ZLN


(6-18)


(6-19)


zL-ZLN nZL+1-ZL zLT+1-ZL
zLL+1-zLN N ZN--Z 1 zIr.I-z, ]


(6-20)


(6-17)


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


(4-12)


(6-17),


(4-13)


(6-18),


(4-14)


(6-19),


(4-15)


(6-20),


(4-16)


(6-21)


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


(4-19)


through


(4-24)


can


then


revised


to develop


computer


codes


for the


evaluation


term


(6-16).


Development


of Computer Codes


for


Evaluating Terms


(6-16)


There


are


three


terms


right-hand


side


(6-16).


The


first


term


can


coded


using


equation


(4-18)


which


constants C and


The


(6-16)


imaginary


components


D have


been


have


been given


defined


been


derived


Table


(4-19)


as


4-1.


real


(4-20)


i$E k












term


new


can


evaluated


revising


previous


coding


First


following


(4-19)


(4-21),


is decomposed


real


imaginary


components


(6-22)


where


=Uk-KLT+l


,k(Xk-XLT+1)+HLT+1,k Yk-YLT+1)


+KLN,k(Xk


=Vk-KLT+l


-XLN) -HLN,k Yk-YLN)


-YLT+1)-HLT+1


(6-23)


k (Xk-XLT+1)


+KLN, k (Yk-YLN) +HLN, k (Xk-XLN)


(6-24)


Here


coefficients


are


given


together


with


their


coding


rules


Table


6-1.


Notice


that


expressions


given


this


table


are


deduced


from


Table


4-2,


in which


nodal-point


locations


are


changed


according to


list


equivalence


points


along


cuts


given


earl ier.


Next


(4-22)


is written


+i~
I~k


-k =-k
-k-k


k (Yk


9k










Tabi


Equat ions


6-1.


coding rules.


Equation


=-Uk-KLT+l k (Xk-XN) +HLT+l, k (Yk-YN)


+KLN, k (Xk -M) -HLN,k (Yk


(6-23)


k =Vk-KLT+1, k (Yk -YN) -HLT+, k (Xk-XN)


+KLN,k (Yk


-YM) +HLN, k (k-M)]


(6-24)


where


(ZL+1,


SZL+1;zk)

k (xf -xe)
-4

k(x -xe)
k (Xf-Xe) -
d(zf ,Zk)
d(ze, z,)


Ak
e,k


e,k (Yf-Ye


e,k (Yf


-Ye)


= arg(zf -zk)


arg(ze-zk)


(Yf--Ye)


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 6-1


(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
j-1
j j+1


-2
2 k


,jj+G4


(6-26)


,j+1 j+l


(6-27)


-E-
2r thk


,j.j+l] +A1


Here


first


three


terms


right-hand


sides


come


from


2,ri [i (zk)]


coefficients


these


S term i

equations


(6-16).


can


Hence,


found


using


expressions given


Table


4-3.


Finally


a matrix


equation


can


constructed


(4-35).


this


equation,


right-hand


side


terms


are


re-


defined


for the


triply


connected


domain


ST4k


6-28)


-A*~


-G ,j-G2
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


(4-36)


(4-37)


those


equations


Table


4-4


as follows:


2rA
2x9k


2,bk


2- (\OA

(6-30)


derivation


nodal


equations


boundaries


now


complete.


Development


of Equations


for


Interior


Points


derive


equations


interior


points


Zo6E1


use


again


mad e


equations


(4-38)


through


(4-42).


Referring


numbering


system


given


Figure


6-1,


double-valued


stream


functions


points


ZM+r,


ZMAB'


ZMAB+1


can


derived


sorting


out


HM-l


HM+l

from


, HMAB,


HMAB+1


, HME-1,


summation


HME,


(4-39).


HME+1'


Following


HN1,


the


terms


same


limiting


procedure


leading


(4-41) ,


interior


equation


derived


for


triply


connected


domain


new


numbering system


(6-31)


- 2 OA

E Ek)


-E ,k)


2 i[ (zo)] T=2ri[ (z ")]q+i(WoA--FE)












2ii [t (zo)]S=


Hj(zo)


(6-32)


oLN n ZL+1-Zo ZL-Zo
- ZL+-ZLN zLN-Zo zL+1- Z


zo-zLT+1 in ZL-o ZO
zLzLT+1 zLT+10 o


(6-33)


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


6-5.


With


exact


solution


taken


w(z


this


example


analogous


previously


solved


conduction


problem


hollow


square


region


Figure


5-1.


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,