REFERENCES

1. Abbott, M. B., "Computational Hydraulics: Elements of the Theory of Free Surface Flows," Pitman, London, 1979.

2. Basco, D. R., "Hydraul ics Instruction in the Computer Era-­Computational Hydraul ics," Proc. Civil Engineerir.g Education Spec. Conf., ASCE, Univ. Wiscons, Madison, 1979.

3. Brebbia, C. A., "The Boundary Element Method for Engineers," Pentech Press, London, 1978.

4. Brebbia, C. A., "Weighted Residual Classification of Approximate Methods," Applied Mathematical Modeling Journal, Vol. 2, No.3, September 1978.

5. Brebbia, C. A., "New Developments ih Boundary Element ~lethods," Proceedings of the Second International Seminar, University of Southampton, U.K., March 1980. Computational Mechanics Center publication.

6. Brebbia, C. A. et al., "The Boundary Element Method in Elasticity," Int. J. Mech. Sciences, Vol. 20, pp. 625-639, 1978.

7. Brebbia, C. A. and Chang, 0., "Boundary Elements Applied to Seepage Problems in Zoned Anisotropic Soils," Advances in Engineering Software Journal, Vol. 1, No.3, June 1979.

8. Brebbia, C. A. and Nakaguma, R., "Boundary Elements in Stress Analysis," Proceedings A.S.C.E., Eng. Mech. Division, EMI, February 1979.

9. Brebbia, C. A. and Walker, S., "General Formulation of Approximating Techniques in Engineering Science," Chapter in the Book "Systems Structures in Engineering," Tapir, Trondheim, Norway, 1978.

10. Brebbia, C. A. and ~~alker, S., "Boundary Elements An Introduction," Newnes-Butterworths, London. In Press.

11. Brevig, P., Greenhow, M., and Vinje, T., "Extreme Wave Forces on Submerged Wave Energy Devices," Applied Ocean Research, Vol. 4, No.4, 1982.

12. Butkov, E., "Mathematical Physics," Addison-Wesley, 1968.

13. Carslaw, H. S., and Jaeger, J. C., "Conducting of Heat in Solids," 2nd ed., Oxford Univ. Press (Clarendon), 1959.

14. Churchill, R. V., "Introduction to Complex Variables and Applications," McGraw-Hill, 1914.

15. Collatz, C.J., "The Numerical Treatment of Differential Equations," Springer-Verlag, 1966.

375

16. Delves, L. M., and Walsh, J., "Numerical Solution of Integral Equations," Clarendon Press, 1974.

17. Do1d, J. W., and Peregrine, D. H., "Steep Unsteady Water Waves: An Efficient Computational Scheme," School of Mathematics, University of Bristol Internal Rept. AM-16-04, 1984.

18. Donea, J., "On the Accuracy of Finite Element Solutions to the Transient Heat Conduction Equation," Int. J. Num. Meth. Engineering 8, pp. 103-110, 1974.

19. Fischer, H. B., List, E. J., Koh, R. C. Y., Imberger, J., and Brooks, N. H., "Mising in Inland and Coastal Waters," Academic Press, 1979.

20. Harr, M.E., "Groundwater and Seepage," McGraw-Hill, New York, 1962.

21. Hromadka II, T. V., and Guymon, G. L., "Application of a Boundary Integral Equation to Prediction of Freezing Fronts in Soils," Cold Regions Science & Technology, (6), 1982.

22. Hromadka II, T. V., and Guymon, G. L., "The Complex Variable Boundary Element Method: Development," Internat. Jour. Numerical Method in Engineering, April, 1983.

23. Hromadka II, T. V. and Guymon, G. L., "Subdomain Integration Model of Groundwater Flow," ASCE Journal of the Irrigation and Drainage Division, pp. 187-195, 1981.

24. Hromadka II, T. V., Guymon, G. L. and Pardoen, G., "Nodal Domain Integration Model of Unsaturated Two-Dimensional Soil-Water Flow: Development," Water Resources Research, (17), pp. 1425-1430, 1981.

25. Hromadka II, T. V. and Guymon, G. L., "Nodal Domain Integration Model of One-Dimensional Advection-Diffusion," Advances in Water Resources, (5), pp.9-16.

26. Hromadka II, T. V. and Guymon, G. L., "Application of a Boundary Integral Equation to Prediction of Freezing Fronts in Soils," Cold Regions Science & Technology, (6), pp.115-121.

27. Hromadka II, T. V. and Guymon, G. L., "A Complex Variable Boundary Element Method: Development," International Journal of Numerical Methods in Engineering, April, 1984.

28. Hromadka II, T. V., "Determining Relative Error Bound!> for the CVBEM," Engineering Analysis. Vol. 2, No.2, pgs. 75-80, 1985.

29. Hromadka II, T. V., "Linking the Complex Variable Boundary Element Method to the Analytic Function Method," Numerical Heat Transfer, Vol. 7. No.2. pp. 235-240.

376

30. Hromadka II, T. V., "Nodal Domain Integration Model of Two­Dimensional Advection-Diffusion Processes," Advances in Water Resources, vol. 7, pp. 76-80, 1983.

31. Hromadka II, T. V., "Complex Polynomial Approximation of the Laplace Equation," ASCE Hydraulics Division, vol. 110, No.3, 1984.

32. Hromadka II, T. V., Guymon, G. L. and Yen, C. C., "The Complex Variable Boundary Element Method: Applications," Numerical Methods in Engineering, Vol. 21, pp. 1013-1025, 1985.

33. Hromadka II, T. V. and Durbin, T. J., "Adjusting the Nodal Point Distribution in Domain Groundwater Flow Models," Proceedings, Fifth International Conference on FEM in Water Resources, University of Vermont, 1984.

34. Hromadka II, T. V., "The Complex Variable Boundary Element Method: Development of Approximative Boundaries," Engineering Analysis, Vol. I, No.4, pp. 218-222, 1984.

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.

37. Hromadka II; T. V., "Locating CVBEM Collocation Points for Steady StatcHeat Transfer Problems," Engineering Analysis, Vol. 2, No.2, pp. 100-106, 1985.

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.

42. Hromadka II, T. V., "Complex Variable Boundary Elements in Computational Mechanics, Topics in Boundary Element Research," Vol. 3, Springer­Verlag, 1986 (in-press).

377

43. Hromadka II, T. V., "Developing Accurate Solutions of Potential Problems Using an Approximate Boundary and a Boundary Element Method," Boston, BETECH Conf., Springer-Verlag, 1986.

44. Hromadka II, T. V., Yen, C. C., and Lai, Chintu, "Solving Time­Dependent Slow Moving Interface Problems in Heat Transfer Using the CVBEM," Proceedings, Int. Conf. on Computational Mechanics (ICCM86 Tokyo), Japan Society of Mechanical Engineers (JSME), May 25-29, 1986, published by Springer-Verlag.

45. Yen, C. C., Hromadka II, T. V., and Lai, Chintu, "The Complex Variable Boundary Element Method in Groundwater Contaminant Transport," Proceedings, Int. Conf. on Computational Mechanics, (ICCM86 Tokyo), JSME, May 25-29, 1986, Springer-Verlag publishers.

46. Yen, C. C. and Hromadka II, T. V., "A Complex Polynomial Approxi­mation of Two-Dimensional Potential Flow Problems Using Generalized Fourier Series," Microsoftware for Engineers, Vol. 2, No.2, pp. 75-82, 1986.

47. Hunt, B., and Isaacs, L. T., "Integral Equation Formulation for Ground-Water Flow," ASCE Hyd. Jour. Div. HYlO, Oct. 1981.

48. Kaplan, W., "Advanced Calculus," Addison-Wesley, 1952.

49. Lai, C., "Numerical Modeling of Unsteady Open Channel Flows," in "Advances in Hydroscience," (V. T. Chow and B. C. Yen, Ed.), Vol. 14, Academic Press, pp. 161-333, 1986.

50. Lai, C., Schaffranek, R. W., and Baltzer, R. A., "An Operational System for Implementing Simulation Models, A Case Study," Seminar on Compo Hydraul., Proc. 26th Annual Hydraul. Spec. Conf., ASCE, Univ. of Maryland, College Park, pp. 415-454, 1978.

51. Lai, C., "Analysis of Stratified Flow by the Complex-Variable Boundary-Element Method," Proc., International Conf. on Computational Mechanics, Tokyo, Japan, 1986.

52. Lai, C.,and Hromadka II, T. V., "Modeling Hydraulic Problems Using the CVBEM and the Microcomputer," Proc., 1985 ASCE Hydraulics Div. Special ty Conf., Orlando, Florida, 1985.

53. Lai, C., and Hromadka II, T. V., "Modeling Complex Two-Dimensional Flows by the Complex-Variable Boundary-Element Method," Proc., International Symp. on Refined Flow Modeling and Turbulence Measurements, Univ. of Iowa, Iowa City, Iowa, 1985.

54. Lai, C., and Hromadka II, T. V., "Some Advances in CVBEM Modeling of Two-Dimensional Potential Flow," Advancement in Aerodynamics, Fluid Mechanics and Hydraulics, Proc., ASCE Specialty Conf., Minneapolis, Minn., 1986.

378

55. Lau, P. and Brebbia, C. A., "The Cell Collocation Method," Int. J. Mech. Sci., 20, pp. 83-95, 1978.

56. Liggett, J. A., and Cunge, J. A., "Numerical Methods of Solution of the Unsteady Flow Equations," in "Unsteady Flow in Open Channels" (K. Mahmood and V. Yevjevich, Ed.), Chap. 4, Vol. I, Water Resources Publications, Ft. Collins, Colo., 1975.

57. Liggett, J. A. and Liu, P. L-F., "The Boundary Integral Equation Method for Porous Media Flow," George Allen and Unwin, London, 1983.

58. Remson,!., Hornberger, G. M., and Mo1z, F. J., "Numerical Methods in Subsurface Hydrology," Wiley-Interscience, 1971.

59. Richardson, J. G., "Flow through Porous Media," in "Handbook of Fluid Dynamics," (V. L. Streeter, Ed.), Sec. 16, McGraw-Hill, 1961.

60. Richtmyer, R. D., and Morton, K. W., "Difference Methods for Initial­Value Problems," Interscience Publishers (Wiley), 1967.

61. Schultz, W. W., "Integral Equation Algorithm for Breaking Waves," Proc., 11th IMCS World Congress, Oslo, Norway, 1985.

62. Soko1nikoff, 1. S., "Mathematical Theory of Elasticity," McGrall­Hill, 1956.

63. Streeter, V. L., "Fl ui d Dynami cs," McGraw-Hi 11, 1948.

64. Telles, J. C. and Brebbia, C. A., "On the Application of the Boundary Element Method to Plasticity," Applied Mathematical Modeling, 1980.

65. Timoshenko, S., and Goodier, J. N., "Theory of Elasticity," McGraw-Hill, 1951.

66. Van Der Veer, P., "Calculation Method for Two-Dimensional Ground­water Flow," Rijkswaterstaat Communications, no. 28/1978, Govern­ment Publishing Office, The Hague, The Netherlands, 1978.

67. Vinje, T. and Brevig, P., "Numerical Calculations of Forces from Break Waves," Hydrodynamics in Ocean Engineering, Norwegian Inst. Tech., pp. 547-5fi5, 1981.

68. Wrobel, L. and Brebbia, C. A., "Boundary Elements for Fluid Flow," Advances in Water Resources Journal, Vol. 2, No.2, June 1979.

69. Wrobel, L. and Brebbia, C. A., "The Boundary Element Method for Steady State and Transient Heat Conduction," Proceeding International Conference on Numerical Methods in Thermal Problems, Pineridge Press, Swansea, Wales, July 1979.

379

70. Wrobel. L. C. and Brebbia. C. A .• "Boundary Elements in Thermal Problems." Chapter in the Book "Recent Numerical Advances in Thermal Problems." Edited by ~wis et al. To be published by John Wiley in July 1980.

71. Yin. C. S .• "Ideal-fluid Flow." in "Handbook of Fluid Dynamics," (V. L. Streeter. Ed.). Sec. 4. McGraw-Hill. 1961.

72. Zienkiewicz, O. C. and Cheung, Y. K •• "Finite Elements in the Solution of Field Problems." The Engineer 220. pp. 507-510. 1965.

73. Zienkiewicz, O. C .• "The Finite Element Method in Engineering Science," McGraw-Hill. 1971.

380

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

381

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

382

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~

385

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