APPENDIX A

INTEGRAL TRANSFORMS

In this appendix a brief review is given of some transform methods. These are techniques used to reduce a to an algebraic equation. The main transforms are the , the and the . These will be presented here, together with some of their main properties. Derivations of the theorems will be given in condensed form, or not at all. Complete derivations are given by Sneddon (1951) and Churchill (1972). Extensive tables of transforms have been published by the staff of the Bateman project (Bateman, 1954).

A.l Laplace transforms

The Laplace transform is particularly useful for problems in which the variables are defined in a semi-infinite domain, say for 0 < t < oo, where t may, for instance, be the time, and t = 0 indicates the initial value of time. The Laplace transform of a F(t) is defined as

f(s) = 100 F(t) exp( -st) dt, (A.1) where s is a parameter, which is assumed to be sufficiently large for the integral to exist. By the integration over the time domain, for various values of s, the function F(t) is transformed into a function f(s). For various functions the Laplace transform can be calculated, sometimes very easily, sometimes with considerable effort. Tables of such transforms are widely available (Churchill, 1972; Bateman, 1954). A short table is given in table A.l. The in this table can all be evaluated with little effort. The fundamental property of the Laplace transform appears when considering the transform of the time . Using partial integration this is found to be

oo dF(t) 1 -- exp(-st)dt = sf(s)- F(O). (A.2) 0 dt Thus differentiation with respect to time is transformed into multiplication by s, apart from the subtraction of the initial value F(O).

EXAMPLE In order to illustrate the application of the Laplace transform technique let us consider the differential equation

356 INTEGRAL TRANSFORMS 357

No. F(t) f(s) = Jt F(t) exp(-st)dt

1 1 1 - s 1 2 t s2 n! 3 tn 8n+l 1 4 exp(at) -- s-a a 5 sin( at) s2 + a2 s 6 cos( at) s2 + a2

Table A.l. Some Laplace transforms.

dF(t) 2F = 0 dt + , (A.3) with the initial condition F(O) = 5. Using the property (A.2) the differential equation (A.3) is transformed into the algebraic equation

(s+2)/-5=0, (A.4) the solution of which is 5 !=-. (A.5) s+2 Inverse transformation now gives, using transform no. 3 from table A.1,

F = 5 exp( -2t). (A.6)

Substitution into the original differential equation (A.3) will show that this is indeed the correct solution, satisfying the given initial condition. This example shows that the solution of the problem can be performed in a straightforward way. The main problem is the inverse transformation of the solution (A.5), which depends upon the availability of a sufficiently wide range of Laplace transforms. If the inverse transformation can not be found in a table of transforms it may be possible to use the general inverse transformation theorem (Churchill, 1972), but this requires considerable mathematical skill. 358 APPENDIX A

Heaviside's expansion theorem

A powerful inversion method is provided by the expansion theorem developed by Heaviside, one of the pioneers of the Laplace transform method. This applies to functions that can be written as a quotient of two polynomials,

!( ) = p(s) (A.7) s q(s)' where q(s) must be a polynomial of higher order than p(s). It is assumed that the function q(s) possesses single zeroes only, so that it may be written as q(s) = (s- sl)(s- s2) · · · (s- sn)· (A.8) One may now write

p(s) a1 a2 an f(s) = -() = --+ --+···+ --. (A.9) q S S - St S - S2 S - Sn The coefficient a; can be determined by multiplication of both sides of eq. (A.9) by (s- s;), and then passing into the limit s- s;. This gives 1. ( s - s;) p( s) a;= ,_,;tm q ( s ) . (A.lO) Because q(s;) = 0 the limit may be evaluated using L'Hopital's rule, giving p(s;) a; = --. (A.ll) q'(si) Inverse transformation of the expression (A.9) now gives, using formula no. 4 from table A.l, ~ p(s;) F(t) = L...J '(s·) exp(s;t). (A.l2) i=l q I This is Heaviside's expansion. It gives the inverse transform of the function (A.7). It has been derived here for the case of the quotient of two polynomials, but it can be used equally well for more general cases of the quotient of two functions, provided that the denominator is of higher order than the numerator, and that the denominator has zeroes of the first order only.

A.2 Fourier transforms

For certain partial differential equations the Fourier transform method can be used to derive solutions. These include problems of potential flow, and elasticity prob• lems, especially in the case of problems for infinite regions, semi-infinite regions, or infinite strips. The main principles of the method will be presented in this section. The main property of the Fourier transform can most easily be derived by first considering a Fourier expansion. For this purpose let there be given a function g( 8), which is periodic with a period 211", such that g( 8 + 211") = g( 8). This function can be written as INTEGRAL TRANSFORMS 359

1 00 g(O) = 2Ao + L {Ak cos(kO) + Bk sin(kO)}, (A.13) k=l where

Ak = -11+.,.. g(t) cos(kt) dt, (A.14) 7r _.,.. and 11+.,.. Bk = ; _.,.. g(t) sin(kt) dt, (A.15)

These formulas can be derived by multiplication of eq. (A.13) by cos(jO) or sin(jO), and then integrating the result from (} = -1r to (} = +1r. It will then appear that from the infinite series only one term is unequal to zero, namely for k = j. This leads to eqs. (A.14) and (A.15). For a function with period 21rl the Fourier expansion can be obtained from (A.13) by replacing(} with x/1, t by t/1 and then renaming g(x/1) as f(x). The result is

1 00 f(x) = 2Ao + L {Ak cos(kx/1) + Bk sin(kx/1)}, (A.16) k=l where now 11+'11'1 Ak = l f(t) cos(kt/1) dt, (A.17) 7r -'11'1 and 11+71'1 Bk = - 1 f(t) sin(kt/1) dt. (A.18) 7r -71'1

EXAMPLE As an example consider the block function defined by

!( ) _ { 0, if lxl > 7rl/2, (A.19) x - 1, if lxl < 7rl/2. For this case the coefficients Ak and Bk can easily be calculated, using the expres• sions (A.17) and (A.I8). The factors Bk are all zero, which is a consequence of the fact that the function f(x) is even, f(-x) = f(x). The factors Ak are equal to zero when k is even, and the uneven terms are proportional to I/k. The series (A.I6) finally can be written as I 2 { x I 3x 1 5x 1 ( 7x ) } f(x) = 2" +; cos( l)- acos( T) + S cos( T)- 7 cos T + .... (A.20) 360 APPENDIX A f(x) /[\

2 ···················:···················:···················:···················:·················· ~ ~ 1 1 ~ ~ ~ ~ ·· · · · · · ·· ··· ·······I·················· ·I·················· ·I·················· ~ ·· ··· ··· ·· ·· · · · · · · ~ ~ l 1 f..- ...... ~-· ...... : ...... -~····· ... ·Jro....-+--1114

...... : ...... : ...... : ...... ~ ~ 1 ~

~ l l X OL-~~-=--~--~=--~---=0 1 2 3 ~~~------~--~ 4 5,.. 11"1 Figure A.l. , 40 terms.

The first term of this series represents the average value of the function, the second term causes the main fluctuation, and the remaining terms together modify this first sinusoidal fluctuation into the block function. Figure A.1 shows the approximation of the series (A.20) by its first 40 terms. It appears that the approximation is reasonably good, except very close to the discontinuities. The approximation becomes better, of course, when more terms are taken into account.

FROM FOURIER SERIES TO FOURIER TRANSFORM Substitution of (A.17) and (A.18) into (A.16) gives 1 J+..-1 f(x) = -2 I f(t) dt + 7r -..-1

00 1 J+..-1 {; f(t) 1rl -..-I f(t) cos[k(t- x)f~ dt. (A.21)

The interval can be made very large by writing 1/1 = aa. Then the formula (A.21) becomes aa J+..-/Aa f(x) =- f(t) dt + 27r -..-/A a oo a a !+..-/A a {; f(t) 7 -..-/Aa f(t) cos[kaa(t- x)] dt. (A.22)

If aa ~ 0 this reduces to

f(x) = -11oo da J+oo f(t) cos[a(x- t)] dt. (A.23) 7r 0 -oo INTEGRAL TRANSFORMS 361

This can also be written as l(x) = 100 {A(o:) cos(o:x) + B(o:) sin(o:x)} do:, (A.24) where A(o:) = ;11 -oo00 l(t) cos(o:t) dt, (A.25) and

00 B(o:) = ;11 -oo l(t) sin(o:t) dt. (A.26)

It can be seen from (A.25) that A(o:) is an even function, A(-o:) = A(o:), and from (A.26) it can be seen that B(o:) is uneven, B(-o:) = -B(o:). Therefore, if G(o:) =A+ iB, it follows that

00 I: G(o:) do:= 21 { A(o:) cos(o:x) + B(o:) sin(o:x)} do:, (A.27) and thus eq. (A.24) may also be written as

l(x) = 211 -oo00 G(o:) exp(-io:x)do:, (A.28) where G(o:) =;11 -oo00 l(t) exp(io:t)dt, (A.29)

Finally, by writing F(o:) = ~G(o:) the factor ~can be eliminated from the expres• sion (A.28),

l(x) =I: F(o:) exp(-io:x)do:, (A.30) where now

00 F(o:) = 217r 1-oo l(t) exp(io:t) dt, (A.31)

This is the basic formula of the Fourier transform method. The function F(o:) is called the Fourier transform of l(x). It may be noted that the asymmetry of the formulas is often eliminated by writing a factor 1/$ in each of the two integrals. The main property of the Fourier transform appears when considering the Fourier transform of the a?- I/ dx 2. This is found to be, using partial integration,

1 1oo a?- I 2 - -d2 exp(io:t) dt = -o: F(o:), (A.32) 27r _00 t 362 APPENDIX A if it is assumed that f(x) and its derivative dffdx tend towards zero for a---+ -oo and a ---+ oo. Thus the second derivative is transformed. into multiplication by -a2. When it is known that the function f( x) is even, f( -x) = f( x), one may write f(x) = 100 Fc(a) cos(ax)da, (A.33) where now Fe( a)=-21 00 f(t) cos( at) dt. (A.34) 7r 0 The function Fe( a) is called the Fourier cosine-transform of f(x). For uneven functions, f( -x) =-f(x), the Fourier -transform may be used,

f(x) = 100 F.(a) sin( ax) da, (A.35) where

00 sin( at) dt. (A.36) F8 (a) = -21 f(t) 7r 0 Both for the Fourier cosine transform and for the Fourier sine transform vari• ous examples are given in the tables published by Churchill (1972) and Bateman (1954).

EXAMPLE As an example consider the problem of potential flow in a half plane y > 0, see figure A.2. The differential equation is

--~: ~:::~:::~:::~:::~:::~:::~:::~:::~:::~:::~:::~:::~:::~:::~:::~:::~:::r.::~:::~:::~:::~:::~:::~:::~:::~:::·~:::~:::~:::~:::~:::~:::~:::~:::~:::-: ~X ...... ' ......

y

Figure A.2. Half plane.

(A.37) INTEGRAL TRANSFORMS 363 with the boundary condition

_ 0 . f _ { 0, if lxl > a, (A.38) y- · - p, if lxl < a. Because the boundary condition (A.38) is symmetric with respect to the y-axis, it can be expected that the solution will be even, and therefore the Fourier cosine transform (A.34) may be used. The transformed problem is, using (A.32),

2 dF -a F + dy2 = 0. (A.39) The solution of this ordinary differential equation which vanishes at infinity is F = A(a)exp(-ay). (A.40) From this it follows that the value at the surface y = 0 is y = 0 : F = A( a). (A.41) The transformed boundary condition is, with (A.38) and (A.34),

y = 0 : F = 2p sin( a a) . (A.42) 7r a From eqs. (A.41) and (A.42) the integration factor A( a) can be determined,

A( a)= 2p sin(aa). (A.43) 7r a The final solution of the transformed problem is 2p sin(aa) ( ) F =- exp -ay. (A.44) 7r a The solution of the original problem can now be obtained by the inverse transform, (A.33),

00 f =-2p 1 sin(aa) cos( ax) exp ( -ay ) d a. (A.45) 7r o a Although this integral has been obtained as a Fourier integral, it can actually most easily be found in a table of Laplace transforms, because of the function exp( -ay) in the integral. In such tables the following integral may be found 00 sin(at) a 1 -- exp( -st) dt =arctan(-). (A.46) 0 t s Using this result, and some trigonometric relations to bring the integrand of (A.45) into the correct form to apply (A.46), the final solution of the problem considered here is found to be f = ~ arctan(a + x) +~arctan( a-x). (A.47) 7r y 7r y It can easily be verified that this solution satisfies the differential equation (A.37) and the boundary condition (A.38). Thus the expression (A.47) is indeed the solution of the problem. 364 APPENDIX A

A.3 Hankel transforms

For problems with radial symmetry a useful solution method is provided by the Hankel transform. This transform is defined by

F(~) = 100 f(r) r Jo(~r) dr. (A.48) The inverse transform is

(A.49)

For a derivation of this relation the reader is referred to the literature, see e.g. Sneddon (1951). The main property of the Hankel transform is that it transforms the often appearing in radially symmetric problems into a simple multiplication. Thus

00 d2 f 1 df . 1 { -d2 + --d } rJo(~r)dr = -e F(~). (A.50) 0 r r r This property can be derived by using partial integration, and noting that the Bessel function w = J0 (~r) satisfies the differential equation d2 w 1 dw 2 (A.51) -dr 2 +--d r r +~ w=O. Thus the combination d2 f fdr2 + (1/r) df fdr is transformed into multiplication of the Hankel transform F(e) by -e. This means that a differential equation in which this combination of appears may be transformed into an alge• braic equation. In many cases this algebraic equation is relatively simple to solve, but the problem then remains to find the inverse transform. For the inverse trans• formation tables of transforms may be consulted, but if the tables do not give the inverse transform, it may be a formidable mathematical problem to derive it, as an illustration of Timman's principle of conservation of misery. APPENDIX B

SOLUTION OF LINEAR EQUATIONS

The development of a numerical model for the solution of an engineering problem often leads to a set of linear algebraic equations. The solution of such a system of equations is a standard problem in linear algebra. In many cases, especially when using the or the , the most effective procedures to solve the system of linear equations use the particular structure of the system resulting from the physical nature of the model. In order to provide some basic theory for the various methods used in this book some of the methods used are briefly presented in this appendix, together with elementary listings in Turbo Pascal. Versions of the programs in BASIC have been given earlier by Bear & Verruijt (1987). Comprehensive presentations of solution methods can be found in textbooks on numerical analysis (for instance Ralston & Rabinowitz, 1978). Listings of computer programs are given by Press et a/. (1986).

B.l Reference problem

The solution methods will be presented on the basis of a simple example, see figure B.l. The example concerns a linear system, consisting of 12 linear resis-

3 6 9 r t--

5 2--; t-- 8

~~ 1 4 7

Figure B.l. Reference problem. tors, connecting 9 nodal points. Through the resistors a fluid (or electricity, or

365 366 APPENDIX B heat) may flow, depending upon the potential difference between the two ends, by equations of the form

I= F1- F2. (B.1) R The system of equations is obtained by requiring that the total flow into each node is zero. The system can be written as

n L: I{i Fi = Qi, (B.2) i=l where Qi is the volume of fluid supplied to the system at node i, and where n is the number of nodes. It is assumed that for all nodes Qi = 0. For a physical problem the system of equations must not only describe the physical processes in the interior of the body, but must also take into account the boundary conditions. It is assumed that these are that F1 = 9 and F9 = 0. If all resistors in the network are equal, the solution of the problem then is F2 = F4 = 4,

Fs = Fs = 2. It can easily be verified that now the total fluxes into each node are zero.

B.2 Gauss-Seidel iteration

A very simple method to solve the system of equations (B.2) in an iterative way is to successively update all values Fi using the algorithm

n Fi = Fi + R(Qi- L:PiiFi)/Pii· (B.3) i=l If R = 1 this means that the error in the i-th equation is calculated (the term between parentheses), and then the value of Fi is updated such that the equation is satisfied. The process must be executed for all variables, and then repeated a number of times, until all errors are practically zero. Experience shows that the number of iterations needed to obtain sufficient accuracy usually is of the order of magnitude of the number of equations (n). By taking the value of the overrelaxation factor R somewhat greater than 1 convergence is often considerably faster. The value of this factor must be smaller than 2 (1 ~ R < 2). The method has been found to converge reasonably fast if the main diagonal of the matrix is dominant. This is the case for problems of potential flow, such as groundwater flow problems. In its standard form the Gauss-Seidel algorithm requires storage of a n X n matrix, and about n3 multiplications. In real physical problems it appears that SOLUTION OF LINEAR EQUATIONS 367 a large proportion of the matrix coefficients is zero, so that valuable computer storage and time would be wasted to store zeros, and to execute multiplications involving the number zero. This can be avoided by restructuring the matrix, storing only the non-zero coefficients, using some sort of bookkeeping system to keep track of their actual position in the original matrix. An effective method is to define a pointer matrix, which indicates the position of the non-zero coefficients in the system matrix. This matrix is given the dimensions n x 10, where the number 10 has been chosen in the expectation that the maximum number of non-zero coefficients in any row of the system matrix will be less than 10. In many methods from , in which the system matrix is composed on the basis ofa finite number of discrete contributions in the neighborhood of each point only, the maximum number of non-zero coefficients is indeed of the order of magnitude of say 6 or 8. In the case of the reference problem shown in figure B.1 the pointer matrix is as follows.

1 2 4 0 0 0 0 0 0 3 2 1 3 5 0 0 0 0 0 4 3 2 6 0 0 0 0 0 0 3 4 1 5 7 0 0 0 0 0 4 5 2 4 6 8 0 0 0 0 5 6 3 5 9 0 0 0 0 0 4 7 4 8 0 0 0 0 0 0 3 8 7 5 9 0 0 0 0 0 4 9 6 8 0 0 0 0 0 0 3

The first row of this pointer matrix indicates that the only nodes that are connected to point 1 are nodes 2 and 4, so that in the equation describing continuity of flow in node 1, only the values F1, F2 and F4 will appear. The last column of the matrix is used to indicate the actual number of non-zero coefficients in each row. The coefficients of the matrix P now are stored in the following way.

Pu p12 p14 0 0 0 0 0 0 Ql p22 p21 p23 p25 0 0 0 0 0 Q2 p33 p32 p36 0 0 0 0 0 0 Q3 p44 p41 p45 p47 0 0 0 0 0 Q4 Pss Ps2 Ps4 Pss Pss 0 0 0 0 Qs Pss Ps3 Pss Psg 0 0 0 0 0 Qs Fr7 p74 P1s 0 0 0 0 0 0 Q7 Pss Ps1 Pss Psg 0 0 0 0 0 Qs Pgg p96 Pgs 0 0 0 0 0 0 Qg

A program, in Turbo Pascal, that reads the input data from a datafile "exam- ple.dat", creates the pointer matrix, sets up the system of equations, and then solves the system of equations by Gauss-Seidel iteration, is reproduced below. 368 APPENDIX B

program gs; const nn=300;mmF250;vv=10; var x,y,£,q:array[1 .. nn] o£ real;ip:array[1 .. nn] o£ integer; r:array[1 .. -:1 o£ real;np:array[1 .. -,1. .2] o£ integer; pt:array[1. .nn,1. .1111] o£ integer;p:array[1. .nn,1 .. vv] o£ real; n,m,nit:integer;rx:real;data:text; procedure input; var i,j: integer; begin assign(data,'example.dat');reset(data);readln(data,n,m,nit,rx); £or i:=1 to n do readln(data,x[i] ,y[i] ,ip[i] ,f[i] ,q[i]); for j:=1 tom do readln(data,np[j,1],np[j,2],r[j]); close(data); end; procedure pointer; var i,j,k,1,kk,1l,ia,ii,kb:integer; begin for i:=1 ton do for j:=1 to 1111 do pt[i,j]:=O; for i:=1 ton do begin pt[i,1] :=i;pt[i,1111]:=1;end; for j:=1 tom do begin for k:=1 to 2 do begin kk:=np[j,k];for 1:=1 to 2 do begin 11:=np[j,1];ia:=O; for ii:=1 to pt[kk,1111] do if pt[kk,ii]=11 then ia:=1; i£ ia=O then begin kb:=pt[kk,llll]+1;pt[kk,vv]:=kb;pt[kk,kb]:=11; end; end; end; end; end; procedure matrix; var i,j,k,1,h:integer;a:rea1; begin for i:=1 ton do for j:=1 to 1111 do p[i,j]:=O; for i:=1 ton do p[i,llv] :=q[i]; for j:=1 tom do begin a:=1/r[j]; for i:=1 to 2 do begin k:=np[j,i];1:=np[j,3-i];p[k,1] :=p[k,1]+a; for h:=2 to pt[k,vv] do if (pt[k,h]=1) then p[k,h] :=p[k,h]-a; end; end; end; procedure so1vegs; SOLUTION OF LINEAR EQUATIONS 369

var it,i,j:integer;a:real; begin ~or it:=1 to nit do begin ~or i:=1 ton do i~ ip[i]

Program GS.

The constant nn denotes the maximum number of nodes, mm denotes the maximum number of elements, and wv denotes the width ofthe pointer matrix and the system matrix. In the program the data defining the problem are read from a dataset named "example.dat". For the problem illustrated in figure B.l this dataset is as follows.

9 12 20 1.6 0.0 0.0 1 6.0 0.0 0.0 1.0 -1 0.0 0.0 0.0 2.0 -1 0.0 0.0 1.0 0.0 -1 0.0 0.0 1.0 1.0 -1 0.0 0.0 1.0 2.0 -1 0.0 0.0 2.0 0.0 -1 0.0 0.0 2.0 0.0 -1 0.0 0.0 2.0 2.0 1 0.0 0.0 1 2 1.0 2 3 1.0 4 6 1.0 6 6 1.0 7 8 1.0 8 9 1.0 1 4 1.0 4 7 1.0 2 s 1.0 s 8 1.0 370 APPENDIX B

3 6 1.0 6 9 1.0

Dataset exa.mple.da.t.

The first line of this dataset specifies that the number of nodes is 9, the number of elements is 12, the number of iterations should be 20, and the overrelaxation factor is 1.5. The next 9 lines give the data of the nodes, with the numbers in the third column indicating whether the values of Fi are given (if ip[i] > 0) or unknown (if ip[i] < 0). The final 12 lines give the data of the elements: the two node numbers, and the resistance. It may be noted that in the program the boundary conditions are taken into account by simply skipping the updating algorithm if ip[i] > 0. When running the program the output on the screen is as follows.

i X y f 1 0.000 0.000 6.000 2 0.000 1.000 4.000 3 0.000 2.000 3.000 4 1.000 0.000 4.000 5 1.000 1.000 3.000 6 1.000 2.000 2.000 7 2.000 0.000 3.000 8 2.000 1.000 2.000 9 2.000 2.000 0.000

Output.

These are indeed the correct results.

B.3 Conjugate method

Another iterative method is the method of conjugate (Reid, 1971), which will always converge in just n iterations, where n is the number of unknowns, provided that the system matrix is positive definite. This is usually the case for which the equations can be established on the basis of a minimum principle, such as problems of potential flow and elasticity problems. The method starts with an estimate Fi of the solution. This estimate will give rise to residuals ui when substituted into the system of equations,

n ui = Qi- L:PijFj. (B.4) i=l A second vector Vi is introduced, which is initially set equal to Ui, (B.5) The actual iterative process now consists of a repeated execution of the following operations, SOLUTION OF LINEAR EQUATIONS 371

n Uo = L:uiui, (B.6) i=l

n wi = L:;l-J, (B.7) j=l

n Vo = LViWi, (B.8) i=l A= Uo/Vo, (B.9) Fi=Fi+AVi, (B.lO)

ui = ui -Awi, (B.ll)

n Wo = L:uiui,· (B.12) i=l B = Wo/Uo, (B.13) (B.l4)

The process is repeated until the value of Uo (which is a measure for the error) is smaller than some given small value. A Thrbo Pascal procedure that executes the calculations is reproduced below. procedure solvecg; var u,v,v:array[l .. nn] of real; it,i,j,iv:integer;ee,uu,cc,vv,vv,aa,bb:real; begin ee:=O.OOOOOl;ee:=ee•ee;it:=l; for i:=l to n do begin iv:=pt[i,vv];u[i]:=O;if ip[i]ee) do begin for i:=l to n do begin v[i]:=O;iv:=pt[i,vv];for j:=l to iv do begin 372 APPENDIX B

v[i] :=v[i]+p[i,j]•v[pt[i,j]]; end; end; vv:=O;for i:=l ton do vv:=vv+v[i]•v[i]; aa:=uu/vv;for i:=l to n do if ip[i]

Procedure solvecg.

This procedure may be substituted for the procedure solvegs in the program given above. The reader may verify that the program will also give the correct output data. It may be noted that the number of iterations and the relaxation factor, which are read in the first line of the procedure input, are no longer used. The conjugate gradient method is especially valuable for large systems, be• cause it usually needs much less iterations than the number of nodes. Another advantage is that its storage requirements are small, because it can be formulated using a pointer matrix. Actually, storage requirements can be further reduced by making use of the symmetry of the system matrix, and by assembling the non-zero coefficients in a pointer vector, omitting all zeros.

B.4 Elimination using wave front method

Although iterative methods of solution of systems of linear equations are often very effective, they are not always applicable, especially if the system matrix is not positive definite. Therefore direct methods of solution are also frequently used. The principle of such methods is simply that equation number 1 is used to express the F1 into all other ones, and then this relation is used to eliminate F1 from all other equations. The remaining system of n- 1 equations then contains only n- 1 variables, and the process of elimination may be repeated until finally only 1 equation with 1 unknown variable remains. For sparse systems of equations, with many zero coefficients, special procedures have been developed, in which the sparsity of the system matrix is maintained as well as possible. Examples are the method of LDU-decomposition, which is restricted to symmetric matrices, or the wave front method (Irons, 1970), which has no such restiction. In this method the variables are stored using a pointer matrix, and during elimination this pointer matrix is updated, because non-zero coefficients are created. A Thrbo Pascal procedure that executes the calculations is reproduced below. SOLUTION OF LINEAR EQUATIONS 373 procedure boundary; var i,j:integer; begin for i:=1 to n do begin if ip [i] >O then begin p[i,1]:=1;p[i,vv] :=f[i];for j:=2 to vv-1 do p[i,j]:=O; end; end; end; procedure solvevf; var i,j,k,l,kc,jj,jk,jl,ii,ij,ik:integer;a,c:real; begin for i:=1 to n do begin kc:=pt[i,vv];c:=1/p[i,1];p[i,vv] :=c•p[i,vv]; for ii:=1 to kc do p[i,ii]:=c•p[i,ii]; if kc>1 then begin for j:=2 to kc do begin jj :=pt [i,j] ;l:=pt [jj ,vv]; for ik:=2 to 1 do if pt[jj,ik]=i then jk:=ik; a:=p[jj ,jk] ;p[jj ,vv] :=p[jj ,vv]-a•p[i,vv]; pt[jj,jk] :=pt[jj,1] ;pt[jj,1] :=0; p[jj,jk] :=p[jj,1] ;p[jj,1] :=0; 1:=1-1;pt[jj,vv]:=1; for ii:=2 to kc do begin ij:=-1; for ik:=1 to 1 do if pt[jj,ik]=pt[i,ii] then ij:=ik; i:f ij=-1 then begin 1:=1+1;if 1=vv then begin vriteln('Vidth o:f matrix p too small');halt; end; ij :=1;pt [jj ,vv] :=1;pt[jj ,ij] :=pt [i,ii]; end; p[jj,ij] :=p[jj,ij]-a•p[i,ii]; end; end; end; end; :for i:=1 to n do begin j:=n-i+1;1:=pt[j,vv];i:f 1>1 then :for k:=2 to 1 do p[j,vv]:=p[j,vv]-p[j,k]•p[pt[j,k],vv]; end; :for i:=1 ton do :f[i] :=p[i,vv]; end;

Procedure solvewf. 374 APPENDIX B

This procedure may be substituted for the procedure solvegs in the program given above. In order to take into account the boundary conditions the system matrix must be modified before the elimination process starts. This is done in the pro• cedure boundary, which must be called before calling solvewf. It can be verified that a program using the procedure solvewf will also give the correct output data. This procedure is used in several programs in this book. REFERENCES

M. Abramowitz and I.A. Stegun, Handbook of Mathematical Functions, National Bureau of Standards, Washington, 1964. American Petroleum Institute, API Recommended Practice for Planning, Design and Constructing Fixed Offshore Platforms, API, Dallas, 1981. C.A. Appel and T.E. Reilly, Selected reports that include computer programs by the USGS for simulation of ground-water flow and quality, Water Resources Investigations Report 87-4271, USGS, Denver, CO, 1988. P.K. Banerjee ·and R. Butterfield, Boundary element methods in geomechanics, in : Finite Elements in Geomechanics (G. Gudehus, editor), Wiley, London, 529-570, 1977. P.K. Banerjee and T.G. Davies, The behaviour of axially and laterally loaded single piles embedded in non-homogeneous soils, Geotechnique, 28, 309-326, 1978. R.A. Barron, Consolidation of fine-grained soils by drain wells, Transactions ASCE, 113, 718-754, 1948. H. Bateman, Tables of Integral Transforms, 2 vols, McGraw-Hill, New York, 1954. K.J. Bathe, Finite Element Procedures in Engineering Analysis, Prentice-Hall, Englewood Cliffs, 1982. J. Bear, Dynamics of Fluids in Porous Media, American Elsevier, New York, 1972. J. Bear and A. Verruijt, Modeling Groundwater Flow and Pollution, Reidel, Dor• drecht, 1987. M.A. Biot, General theory of three-dimensional consolidation, J. Appl. Phys., 12, 155-164, 1941. M.A. Biot, Theory of propagation of elastic waves in a fluid-saturated porous solid, J. Acoust. Soc. Amer., 28, 168-191, 1956. A.W. Bishop, The use of the slip circle in the stability analysis of earth slopes, Geotechnique, 5, 1955. L. Bjerrum, Geotechnical problems involved in foundations of structures in the North Sea, Geotechnique, 23, 319-359, 1973. H. Blum, Einspannungsverhiiltnisse bei Bohlwerken, Wilhelm Ernst & Sohn, Berlin, 1931. J .R. Booker and J .C. Small, An investigation of the stability of numerical solu• tions of Biot's equations of consolidation, Int. J. of Solids and Structures, 11, 907-917, 1974. J .E. Bowles, Analytical and Computer Methods in Foundation Engineering, McGraw-Hill, New York, 1974. J .E. Bowles, Foundation Analysis and Design, 4th ed., McGraw-Hill, New York, 1988. J.D. Bredehoeft and G.F. Pinder, Mass transport in flowing groundwater, Water Resources Research, 9, 194-210, 1973.

375 376 REFERENCES

J. Brinch Hansen, A revised and extended formula for bearing capacity, Bulletin of the Danish Geotechnical Institute, 28, 5-11, 1970. M.A. Celia et al., Computational Methods in Water Resources, 2 vols., Elsevier, Amsterdam, 1988. R.V. Churchill, Operational Mathematics, 2nd ed., McGraw-Hill, New York, 1958. C. W. Cryer, A comparison of the three-dimensional consolidation theories of Biot and Terzaghi, Quart. J. Mech. and Appl. Math., 16, 401-412, 1963. R. De Borst and P.A. Vermeer, Possibilities and limitations of finite elements for limit analysis, Geotechnique, 34, 199-210, 1984. C.S. Desai and J.F. Abel, Introduction to the Finite Element Method, Van Nos• trand Reinhold, New York, 1972. C.S. Desai and J.T. Christian, Numerical Methods in Geotechnical Engineering, McGraw-Hill, New York, 1977. W. Feller, An Introduction to Probability Theory and its Applications, 2, Wiley, New York, 1966. W. Fellenius, Erdstatische Berechnungen, Ernst, Berlin, 1927. L. Fox, Numerical Solution of Ordinary and Partial Differential Equations, Per• gamon Press, New York, 1962. A.O. Garder, D.W. Peaceman and A.L. Pozzi, Numerical calculation of mul• tidimensional miscible displacement by the method of characteristics, Soc. Petroleum Engineers Journal, 4, 26-36, 1964. A.E. Green and W. Zerna, Theoretical Elasticity, Clarendon Press, Oxford, 1954. J.G.M. van der Grinten, An Experimental Study of Shock-induced Wave Propa- gation in Porous Media, Ph.D. Thesis, Eindhoven, 1987. W. Grobner and N. Hofreiter, Integraltafel, Springer, Wien, 1961. G. Gudehus, Finite Elements in Geomechanics, Wiley, London, 1977. P. van der Heijde et al., Groundwater Management : The Use of Numerical Models, 2nd ed., AGU, Washington, D.C., 1985. M. Hetenyi, Beams on Elastic Foundation, University of Michigan Press, Ann Arbor, 1946. S.P. Huyakorn and K. Nilkuha, Solution of transient transport equation using an upstream finite element scheme, Applied Mathematical Modeling, 3, 7-17, 1979. I. Javandel, C. Doughty, and C.F. Tsang, Groundwater Transport: Handbook of Mathematical Models, AGU, Washington, D.C., 1984. G. de Josselin de Jong, Wat gebeurt er in de grand tijdens het heien?, De Inge• nieur, 68, B77-B88, 1956. G. de Josselin de Jong, Consolidatie in drie dimensies, LGM-Mededelingen, 7, 25-73, 1963. W. Kinzelbach, Groundwater Modelling, Elsevier, Amsterdam, 1986. D.E. Knuth, The TF;Xbook, Addison-Wesley, Reading, 1986. L.F. Konikow and J.D. Bredehoeft, Modeling flow and chemical quality changes in an irrigated stream-aquifer system, Water Resources Research, 10, 546- 562, 1974. REFERENCES 377

L.F. Konikow and J.D. Bredehoeft, Computer model of two-dimensional solute transport and dispersion in ground water, Techniques of Water Resources Investigations of the USGS, Book 7, Chapter C2, 1978. A.P. Kooijman, A Numerical Model for Laterally Loaded Piles and Pile Groups, Ph.D. Thesis, Delft, 1989. T.W. Lambe and R.V. Whitman, Soil Mechanics, Wiley, New York, 1969. L. Lamport, D.TEX: A Document Preparation System, Addison-Wesley, Reading, 1986. E.C. Leong and M.F. Randolph, Finite element analyses of soil plug response, Int. J. Numer. and Anal. Methods Geomech., 15, 121-141, 1991. J.A. Liggett and P.L.F. Liu, The Boundary Method for Porous Media Flow, Allen & Unwin, London, 1983. O.S. Madsen, Wave-induced pore pressures and effective stresses in a porous bed, Geotechnique, 28, 377-393, 1978. J. Mandel, Consolidation des sols, Geotechnique, 7, 287-299, 1953. M.G. McDonald and A.W. Harbaugh, A modular three-dimensional finite differ• ence groundwater flow model, Techniques of Water Resources Investigations of the USGS, Book 6, Chapter A1, 1988. K.L. Meijer and A.G. van Os, Pore pressures near moving underwater slope, J. Geotechn. Eng. Div., ASCE, 102, no. GT4, 361-372, 1976. T.N. Narasimhan and P.A. Witherspoon, An integrated finite difference method for analyzing fluid flow in porous media, Water Resources Research, 12, 57-64, 1976. A.G. van Os, Snelle deformatie van korrelvormig materiaal onder water, Poly• technisch Tijdschrift, 32, 461-467, 1977. A.G. van Os and W. van Leussen, Basic research on cutting forces in saturated sand, Journal of Geotechnical Engineering, 113, 1501-1516, 1987. G.F. Pinder and H.H. Cooper Jr., A numerical technique for calculating the transient position of the saltwater front, Water Resources Research, 6, 875- 882, 1970. G.F. Pinder and W.S. Gray, Finite Element Simulation in Surface and Subsurface Hydrology, Academic Press, New York, 1977. H.G. Poulos, Behaviour of laterally loaded piles, J. Soil Mech. Found. Div., ASCE, 97, no. SM5, 711-751, 1971. H.G. Poulos, Marine Geotechnics, Unwin Hyman, London, 1988. L. Prandtl, Uber die Harte plastischer Korper, Nachr. Kgl. Ges. Wiss. Gottin• gen, Math.-phys. Klasse, 1920. T.A. Prickett and C.G. Lonnquist, Selected digital computer techniques for groundwater resource evaluation, Illinois Water Survey Pub/., 55, 1-62, 1971. P.A.C. Raats, Transport in structured porous media, Flow and Transport in Porous Media (A. Verruijt and F.B.J. Barends, editors), Balkema, Rotter• dam, 221-226, 1981. M.F. Randolph, The response of flexible piles to lateral loading, Geotechnique, 31, 247-259, 1981. 378 REFERENCES

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A. Verruijt, The influence of soil properties on the behaviour of offshore struc• tures, Behaviour of Offshore Structures (J.A. Battjes, editor), Elsevier, Am• sterdam, 7-19, 1985. A. Verruijt and A.P. Kooijman, Laterally loaded piles in a layered elastic medium, Geotechnique, 39, 39-46, 1989. A. Verruijt, Finite element modeling of transport in porous media, Appl. Scien• tific Res., 48, 129-139, 1991. A. Verruijt and W. Swidzinski, Advective transport in a multilayered system of aquifers, Transport in Porous Media, 12, 31-42, 1993. H. Vidal, The principle of reinforced earth, Highway Research Record, 282, 1-16, 1969. H.F. Wang and M.P. Anderson, Introduction to Groundwater Modeling, Freeman, New York, 1982. M.J. Wichura, The PJCI'EX Manual, 1-800-USA-BOOKS, Lawrence, NY, 1987. C.R. Wylie, Advanced Engineering Mathematics, 2nd ed., McGraw-Hill, New York, 1960. T. Yamamoto, H.L. Koning, J.B. Sellmeijer and E. van Hijum, On the response of a poro-elastic bed to water waves, J. Fluid Mech., 87, 193-206, 1978. G.T. Yeh, An orthogonal-upstream finite element approach to modeling aquifer contaminant transport, Water Resources Research, 22, 952-964, 1986. W.W.G. Yeh, Y.S. Yoon and K.S. Lee, Aquifer parameter identification with kriging and optimum parameterization, Water Resources Research, 19, 225- 233, 1983. O.C. Zienkiewicz, The Finite Element Method, 3d ed., McGraw-Hill, London, 1977. W. Zijl, Finite element methods based on a transport velocity representation for groundwater motion, Water Resources Research, 20, 137-145, 1984. INDEX

active earth pressure 69, 125 CONSOL2D (Program) 304 adsorption 337 contraction 199 advection 308, 310 Courant number 345 Airy functions 87, 105 CPT 5, 6, 91 ALP (Program) 103 creep 203 analytical element method 315 critical time step 190 ANISEL (Program) 263 cyclic loading 35, 141, 199 aquifer 219 aquitard 220 damping 141, 150, 341 axially loaded piles 78 Darcy's law 11, 219, 314 deformations of unstable slope 213 beam theory 53 degree of consolidation 18 bearing capacity 6, 89, 93, 195 diffusivity 309, 339 bending moment 54 dilatancy 199 Bernoulli's assumption 123 DISP (Program) 352 Biot's theory 8, 203 dispersion 308, 309, 310, 337, 354 Bishop's method 204 dispersivity 310 Bjerrum's relation 42, 200 distribution coefficient 338 Blum 68, 77 theorem 222, 248 Blum's approximation 126 drained deformations 26 Boussinesq 2 Dupuit 219, 326 Brinch Hansen's formula 6, 90, 195 dynamic effects 93 bulk modulus 25 dynamic pile loading test 91 central limit theorem 348 effective stress 12, 25 characteristic frequency 173 eigen frequency 173, 17 4 characteristics 175 eigen functions 173 cohesion 5, 126 elastic foundation 54 cohesive material 81 elastic modulus 2 compatibility 25 elastic plug model 106 compressibility 4, 280 elasticity 244 compression modulus 25, 244 ELASTO (Program) 255 compression waves 167 elasto-plastic response 97, 125 cone penetration 91 elasto-plasticity 165, 266 cone penetration test 5 equilibrium 244, 291 conjugate gradients 370 equilibrium adsorption 342 conjugated gradients 239 expansion theorem 358 conservation of mass 220, 314 explicit procedure 20 consolidation 8, 202, 291 consolidation coefficient 14, 36 FEMFLUX (Program) 323 CONSOLlD (Program) 287 FEM3D (Program) 333

381 382 REFERENCES finite differences 19, 342 LLP (Program) 140 finite elements 222, 244, 280, 291 long waves 39 finite pile 169, 171 longitudinal dispersivity 311 fluid compressibility 9, 36 FLUX (Program) 317 Mandel-Cryer effect 30, 32 Fourier series 358 mass balance 308 Fourier transform 358 MULAT 325 friction 89, 91, 183, 191 FRICTION (Program) 192 N avier equations 245 friction angle 5, 126 neutral earth pressure 69 frictional material 85 non-steady flow 234 normal distribution 348 246 NSF (Program) 239 Gauss-Seidel method 227, 366 NUMCONS (Program) 22 gravity foundations 194 numerical plug model 110 groundwater flow 219 GWF (Program) 231 oedometer test 5 offshore platforms 194 Hankel transform 364 one-dimensional consolidation 12, 280 heat transfer 339 overrelaxation factor 228 Heaviside 358 Hooke's law 25, 244, 292 partitioning coefficient 309 hydraulic conductivity 11, 220, 314 passive earth pressure 69, 125 Peclet number 309 IMPACT (Program) 191 permeability 11, 220, 280 impedance 176, 189 PILAT (Program) 163 inclination factors 196 pile driving 167 incompressible fluid 40 pile group 165 infiltration 220 plastic material 125 infinite pile 168 plastic plug model 109 Integral transforms 356 plastic potential 268 Koppejan 206 PLASTO (Program) 275 plate loading test 5 Lame constants 25, 244, 267, 292 plug 92, 106 Laplace transform 168, 356 PLUG (Program) 120 lateral earth pressure 86 point resistance 80, 89 laterally loaded piles 122 pointer matrix 228, 367 layered system 151 Poisson's ratio 2 LDU-decomposition 251 pollution transport 308, 337 leakage 220 pore pressures 197, 198, 200 leap frog 188 porosity 328 Linear equations 365 Prandtl 90, 195 linear equilibrium isotherm 309 preshearing 48 linear material 78, 122 liquefaction 48 quake 82, 113 REFERENCES 383

radial consolidation 31 uncoupled consolidation 29 radiation condition 187 undrained deformations 27 radiation damping 187 undrained shear strength 5, 7 references 375 reflection 178 221, 281, 294 reflection coefficient 180 viscosity 11, 149, 150 reservoir deformations 203 viscous damping 148 resistance 220 volumetric weight 11 resonance 175 retardation 310, 337, 340, 342 water waves 35 retardation factor 313 wave equation 168 wave front method 258, 372 safety factor 204 waves in piles 167 SEABED (Program) 50 weighted residuals 246 171 WINKLER (Program) 64 settlements 18, 202 yield condition 267 shakedown 104, 143 shape factors 90, 196 shear modulus 25, 244 shear strength 5 sheet pile wall 67 short waves 40 SLOPE (Program) 211 slope stability 204 SORPTION (Program) 345 SPWALL (Program) 75 stability 345 stability factor 204 static pile loading test 91 stiffness 2 storage equation 10, 36, 198 stroke 70, 75, 126 subgrade modulus 2, 55, 79, 123, 183

Terzaghi's formula 4 Terzaghi's principle 12, 25, 292 Terzaghi's problem 13, 305 three-dimensional consolidation 23, 291 three-dimensional transport 326 transmission 178 transmission coefficient 180 transmissivity 220, 234 transport 308 transport equation 311 transverse dispersivity 311 Theory and Applications of Transport in Porous Media

Series Editor: Jacob Bear, Technion -Israel Institute of Technology, Haifa, Israel

1. H. I. Ene and D. Polisevski: Thermal Flow in Porous Media. 1987 ISBN 90-277-2225-0 2. J. Bear and A. Verruijt: Modeling Groundwater Flow and Pollution. With Computer Programms for Sample Cases. 1987 ISBN 1-55608-014-X; Ph 1-55608-015-8 3. G. I. Barenhlatt, V. M. Entov and V. M. Ryzhik: Theory of Fluid Flows Through Natural Rocks. 1990 ISBN 0-7923-0167-6 4. J. Bear andY. Bachmat: Introduction to Modeling of Transport Phenomena in Porous Media. 1990 ISBN 0-7923-0557-4; Ph (1991) 0-7923-1106-X 5. J. Bear and J-M. Buchlin (eds.): Modelling and Applications of Transport Phenomena in Porous Media. 1991 ISBN 0-7923-1443-3 6. Ne-Zheng Sun: Inverse Problems in Groundwater Modeling. 1994 ISBN 0-7923-2987-2 7. A. Verruijt: Computational Geomechanics. 1995 ISBN 0-7923-3407-8

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