Teatreros a prueba de patada, toletazo, auditoría y juicio

boundary generalised co-ordinates Generation of subscripts for generalised co-ordinates Formulation of matrices of coefficients

LINT, QUDT, CUBT, QURT

Flow Chart Elucidating The

Definition

Of

The Global Functions

diagram r e la tin g the various m atrices.

The th ird stage i s concerned with s a tis fy in g the boundary con d ition s o f equations ( 5*66) and ( 5*72) and d efin in g the i n i t i a l con d ition s. Av consequence o f f u lly s a tis fy in g the boundary

con d ition s in the manner described i s that the f ir s t o f the *3n* equations i s rendered t r iv ia l. However, there remain only *3n—1* glob al unknowns and these may be determined from the sim ultaneous so lu tio n o f the remaining se t o f *3n -l* n on -lin ear algebraic

equations. Thus in th is stage o f the program valu es are assigned to the known global parameters through the ap p lication o f equations

(5*5^) a&d (5*72) and the i n i t i a l con d ition s are s e t.

The fourth stage o f the program e n ta ils the p resen tation o f the global equations in the manner required by the so lu tio n ro u tin e. To th is end, the remaining global unknowns are made equivalent to *x^f where

i = l,3 n - l (5 .8 2 )

Thus the parameter fx^f appearing in the n on-linear so lu tio n rou tin e described in the follow in g chapter, rep resen ts the glob al unknowns, *U (i)*, which are not elim inated through the ap p lication o f boundary con d ition s.

The equations are defined as the fu nctions * f^ (x )' through the m u ltip lica tio n o f the m atrices o f c o e ffic ie n ts by the appropriate column vectors o f gen eralised co-ord in ates. The b asic p r in c ip le o f the operation i s illu s tr a te d in the flow diagram in Figure (37)* The actual programming employed i s rather more complex. SUrther d e ta ils o f the method are provided by the ’comment* statem ents included in the program l i s t i n g given in Appendix I I .

5.8 Summary

In th is chapter the c la s s ic a l d iffe r e n tia l continuous f ie ld and boundary expressions have been converted in to a glob al set o f d iscre te algebraic fu nction s by th e ap p lication o f the

(59) C ollocation U n ite Element, or Bypar F in ite D ifferen ce, method • Consider now the solu tion o f these equations.

CHAPTER 6

SOLUTION OP DISCRETE GLOBAL SYSTEM 6.1 Introduction

The global equations provided by the co-ordinate

transform ation o f the lo c a l f ie ld and boundary analogues c o n sist o f a se t o f n on -lin ear sim ultaneous algebraic equations in clu d in g terms o f up to the fourth order# The algorithm chosen for th eir solu tion i s a hybrid o f the Newton-Raphson and the Steepest Descent methods* developed by Pow ell(?3)(74)^ Software d e ta ils being a v a ila b l© ^ ^ , the present study w ill content i t s e l f w ith a broad o u tlin e o f the method and a d escrip tion o f some o f i t s more important features#

6# 2 The Newton-Raphson Method#

The underlying p r in c ip le s o f the Newton-Raphson method may be appreciated most e a s ily by considering i t s ap p lication to a sin g le v a ria b le fu nction , * f(x )* . This i s a sp ecia l case o f the method which, i s equivalent to Newton1 s technique for fin d in g the root o f * f ( x ) 1#

The rela tio n sh ip ; (h)

f ' ( x ^ ) = ...

i s se lf-e v id e n t from Figure (38)# Re-arranging the terms g iv e s the

w ell known Newton-Raphson formula^ 5

x (* » l) . x (k) . _ £ ^ . . . ( 6. 2 )

f » U w )

A solu tion may bo gainod by an ite r a tio n procedure in which the ( v )

estim ate, fi ' 71, o f the solu tion i s replaced by ;

x(k+l) = x(k) +

h (k)

where j f ( x ^ ) 4- f ’ ( x ^ ) S^k ) = 0 . . . . ( 6, 4) The procecure converges to the root o f the equation and the so lu tio n i s obtained when the updated estim ate agrees with the previous

f(x)

CkK

f(X )

The N e w to n - R a p h s o n Iteration