REFERENCES Abbott, M. B., Computational Hydraulics: Elements of the …978-1-4612-4660... ·...
Transcript of REFERENCES Abbott, M. B., Computational Hydraulics: Elements of the …978-1-4612-4660... ·...
REFERENCES
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35. Hromadka II, T. V. and Pardoen, G., "Appl ication of the CVBEM to Nonuniform St. Venant Torsion," Computer Methods in Applied Mechanics & Engineering, 53, pp. 149-161, Elsevier Publishers, 1985.
36. Hromadka II, T. V., "Predicting Two-Dimensional Steady-State Freezing Fronts using the CVBEM," ASME Journal of Heat Transfer, Vol. 108, No.1, pp. 235-237, Feb., 1986.
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38. Hromadka II, T. V. and Durbin, T. J., "Modeling Steady-State, Advective Contaminant Transport by the CVBEM," Engineering Analysis, Vol. 3, No.1, pp. 9-15, 1986.
39. Hromadka II, T. V., "Computer Interation and the CVBEM in Engineering Design," Engineering Analysis, Vol. 2, No.3, pp. 163-167, 1985.
40. Hromadka II, T. V., "Variable Trial Functions and the CVBEM," Numerical Methods for POE," John Wiley & Sons, Inc., Vol. 1, pp. 259-278, 1985.
41. Hromadka, II, T. V., "Analyzing Numerical Errors in Domain Heat Transport Models using the CVBEM," Proceedings, Fourth International Symposium on Offshore Mechanics and Artic Engineering, February 17, 1985, Dallas, Texas.
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LIST OF SYMBOLS
Note: Other symbols, mostly local, minor, or for purely mathematical
substitution, are defined in the text.
A
A
B -" B
C
c
D
D
V
E(j)
e
e .. lJ
e<p (rJ, eljJ(1;;) ..)
F
area, surface area, cross-sectional area
complex constant
complex constant
body force vector
a constant, complex constant
mass concentration
a (m x 2m) matrix system associated with ~
a (m x 2m) matrix system associated with ¢ specific heat
diffusion coefficient
complex constant
domain
minlzl
modulus of elasticity, Young's modulus
boundary error (at j) of Hk approximation function
strain (normal)
s tra in tensor
boundary error function
force vector
gravitational potential
linear global trial function
k-degree global trial function
acceleration of gravity
total head, height, depth
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h
M
M ~
M
m
N ~
n
n*
p
q ~
q
r
r
S
S
s,s*
height, elevation
transport coefficient, conductivity
permeability of the medium, conductivity
coefficient of permeability, hydraulic conductivity, seepage coefficient, permeability
thermal conductivity
length, length dimension
volumetric latent heat of fusion
length
direction cosine
a maximum value
mass, mass dimension
momentum vector
torque
direction cosine
linear basis function; See (3.6), (3.7) for definitions
unit outward vector
distance measured along equipotential line
pressure intensity
porosity
discharge, rate of flow
rate of flow per unit width, two-dimensional rate of flc
solute mass flux vector
heat flux, flux
position vector
polar-cylindrical coordinate
retardation factor
line segment, straight line segment, length of the line
strength of the source (negative for the sink)
distance measured along stream line
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T
T
T
T(z)
t
u
u ~
u
v
v
v
w
w
x x
y
y
Z
z
z
temperature
time dimension
transmissivity
Schwartz-Christoffel transformation
time
(x) component of velocity
displacement
displacement vector
magnitude of velocity vector • velocity vector, V(u,v,w)
volume
specific discharge, Darcian velocity, discharge velocity
pore velocity, effective velocity, average linear velocity
product of error e and departure d,--a measure of computational accuracy used in one of the CVBEM accuracy analysis methods
(y) component of velocity
diSplacement
(z) component of velocity
displacement, longitudinal displacement of a torsion bar
(x) component of body force
Cartesian coordinate, distance along the x-axis
(y) component of body force
Cartesian coordinate, distance along the y-axis
(z) component of body force
Cartesian coordinate, distance along the z-axis
cylindrical coordinate
angle
real part of complex constant a + i8
383
a{z)
S
r " r
Y
Y, Yxy' Yyz
6
e:
e:, e:x' e:y , e:z
1;;
T)
T)
8
e
e
-&-
-&-
A
~
~
\)
F;,
-F;,
F;,k
F;,U
p
linear trial function
imaginary part of complex constant a + is
boundary, true (or exact) boundary
approximative boundary {associated with the ~(z) solutio
speci fi c wei ght
shearing strain
a (small) positive number
a (small) positive number
unit elongation
(z) component of vorticity
(y) component of vorticity
known nodal value (~ or $)
= 'aa = '11 + '22 + '33 = ax + 0y + °z
angle, angle of twist per unit length
polar coordinate
spherical coordinate
= e = e + e + e = e: + e: + e: volume expansion (l(l 11 22 33 X Y z'
::: (1+v)tl-2V) , Lame's constant
dynamic viscosity
modulus of elasticity; Lame's constant
Poisson's ratio
(x) component of vorticity
unknown value (~ or ~) at the specified nodal point
known nodal value
unknown nodal value
density
384
W
w
-w
nonnal stress
stress
stress tensor
potential function, state function
velocity potential. head
stress function
spherical coordinate
stream function
warping function
flux function
domain; control volume
= w{z), an analytic function defined in domain V
warping (of cross section)
vorticity vector
Wij rotation tensor
w{Zj) =Wj =CP{Zj) +iljJ{zj) =CPj +iljJ
w{Zj) =Wj =~(Zj) +i~{Zj) =~j +i~j
~(Zj) =~j =~(Zj) +i~{Zj) =~j +i~
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exact nodal value of w{z)
specified nodal point value (cf. Definitions 3, 3a)
approximation function nodal values (cf. Defintion 7)
Abbreviation
AFM
BEM
BIEM
CVBEM
FDM
FEM
MOe
POE
SOR
1-,2-,3-D
analytical function method
boundary element method
boundary integral equation method
complex variable boundary element method
finite difference method
finite element method
method of characteristics
partial differential equation
successive overrelaxation
one-, two-, three-dimensional
386
Subscripts
x, y, z
1, 2, 3
i. j, K, £, Ct
R, I
j
k, u
f, t
P
d
X-, y-, z- component of a quantity
x, y, z
1 or 2 or 3
real, imaginary part
1, 2,···,m, nodal pOint number
known, unknown
frozen, thawed
pore
discharge, Darcian
387
AC B
A:JB .. ... A • B
v
MATHEMATICAL NOTATIONS
inclusion, A is contained in B
inclusion, A includes B
scalar product of vectors
vector product of vectors
m x 2m matrices associated with real coefficients, imaginary coefficients
domain
sUbstantial differentiation operator
V:: {(x,y): x>O, y?O} (The domain) V is composed of elements x,y, whose values are in x>O, y~O
HI approximation function
Hk approximation function
defined in (3.10), Definition 7
defined in (3.17), Definition 9
I
1m z
inf x ... .. .. i, j, k
max (a,b,c, ... )
min (a,b,c, ... )
N(P)
n!
P
R(w,P)
Re z
sup x
z
identity matrix
imaginary part of z
= ;:r unit imaginary number
greatest lower bound of x
rectangular cartesian base vectors
the maximum among a, b, c, .•.
the minimum among a, b, c, .•.
norm of partition P
factorial
(cf. p. 56)
set of all simple polygonal domain
Riemann sum for w relative to partition P (cf. p. 63)
real part of z
least upper bound of x
complex variable z = x + iy
388
point Z approaches Zo from the interior (of n)
another boundary slightly interior of boundary r
r:: {z: Z ( t) = ~ ( t) + i lJ! ( t), t 1 ~ t ~ t2 }
l:!.
l:!. .. II
.. 'i;2 - II . O •. lJ
£
W
{w} =
U
n m U j=l
.... II
w1
w2
wn
(The curve) r is composed of elements z, which are determined by z(t) = ~(t) + ilJ!(t), with t lying in
tl ~ t ~ t2
operator, in l:!.~, l:!.=1 if ~=~, l:!.=i if ~=lJ!
increment of quantity, e.g. l:!.z =Zj+l - Zj
vector differential operator
Laplacian operator
Kronecker delta, = 0 if itj, =1 if i =j
element of, member of
such that
complex variable function, w = f(z) = ~(x,y) + ilJ!(x,y)
vector, column matrix
union; A UB, union of A and B
intersection; A(/B, intersection of A and B
union of elements from 1 to m
389