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Typically, nonlinear equations of this type are solved using a Newton“
Raphson iterative process whereby given a solution estimate xk at iteration
k, a new value xk+1 = xk + u is obtained in terms of an increment u by
establishing the linear approximation,
R(xk+1 ) ≈ R(xk ) + DR(xk )[u] = 0 (1.34)
This directional derivative is evaluated with the help of the chain rule as,

d
DR(xk )[u] = R(xk + u)
d =0

d R1 (x1 + u1 , x2 + u2 )
=
R2 (x1 + u1 , x2 + u2 )
d =0

= Ku (1.35)
where the tangent matrix K is,

‚Ri
K(xk ) = [Kij (xk )]; Kij (xk ) = (1.36)
‚xj xk

If we substitute Equation (1.35) for the directional derivative into (1.34), we
obtain a linear set of equations for u to be solved at each Newton“Raphson
iteration as,
K(xk )u = ’R(xk ); xk+1 = xk + u (1.37a,b)
For equations with a single unknown x, such as Equation (1.11) for the
truss example seen in Section 1.3.2 where R(x) = T (x) ’ F , the above
Newton“Raphson process becomes,

R(xk )
u=’ ; xk+1 = xk + u (1.38a,b)
K(xk )
This is illustrated in Figure 1.12.
18 INTRODUCTION



T R

K


R(xk )
0
F x
u


x
xk+1 xk

FIGURE 1.12 Newton“Raphson iteration.


In practice the external load F is applied in a series of increments as,
l
F= ∆Fi (1.39)
i=1
and the resulting Newton“Raphson algorithm is given in Box 1.1 where bold-
face items generalize the above procedure in terms of column and square
matrices. Note that this algorithm re¬‚ects the fact that in a general FE
program, internal forces and the tangent matrix are more conveniently eval-
uated at the same time. A simple FORTRAN program for solving the one-
degree-of-freedom truss example is given in Box 1.2. This program stops
once the sti¬ness becomes singular, that is, at the limit point p. A tech-
nique to overcome this de¬ciency is dealt with in Section 7.5.3. The way
in which the Newton“Raphson process converges toward the ¬nal solution
is depicted in Figure 1.13 for the particular choice of input variables shown
in Box 1.2. Note that only six iterations are needed to converge to values
within machine precision. We can contrast this type of quadratic rate of
convergence with a linear convergence rate, which, for instance, would re-
sult from a modi¬ed Newton“Raphson scheme based, per load increment,
on using the same initial sti¬ness throughout the iteration process.
19
1.4 DIRECTIONAL DERIVATIVE AND LINEARIZATION




BOX 1.1: NEWTON“RAPHSON ALGORITHM
INPUT geometry, material properties, and solution parameters
r
INITIALIZE F = 0, x = X (initial geometry), R = 0
r
FIND initial K (typically (1.13))
r
LOOP over load increments
r
r FIND ∆F (establish the load increment)
r SET F = F + ∆F
r SET R = R ’ ∆F
r DO WHILE ( R / F > tolerance )
SOLVE Ku = ’R (typically (1.38a))
r
UPDATE x = x + u (typically (1.38b))
r
FIND T (typically (1.12)) and K (typically (1.13))
r
FIND R = T ’ F (typically (1.11))
r
r ENDDO
r ENDLOOP




1
Residual norm




Modified Newton’Raphson


Newton’Raphson
’10
1e



2 4 6 8 10 12 14 16 18 20 Iterations

FIGURE 1.13 Convergence rate.
20 INTRODUCTION




BOX 1.2: SIMPLE TRUSS PROGRAM


c------------------------------------------------------
program truss
c------------------------------------------------------
c
c Newton-Raphson solver for 1 D.O.F. truss example
c
c Input:
c
c d ---> horizontal span
c x ---> initial height
c area ---> initial area
c e ---> Young modulus
c nincr ---> number of load increments
c fincr ---> force increment
c cnorm ---> residual force convergence norm
c miter ---> maximum number of Newton-Raphson
c iterations
c
c------------------------------------------------------
c
implicit double precision (a-h,o-z)
double precision l,lzero
c
data d,x,area,e,f,
resid /2500.,2500.,100.,5.e5,0.,0./
data nincr,fincr,cnorm,miter /1,1.5e7,1.e-20,20/
c
c initialize geometry data and stiffness
c
lzero=sqrt(x**2+d**2)
vol=area*lzero
stiff=(area/lzero)*e*(x/lzero)*(x/lzero)
c
c starts load increments
c
do incrm=1,nincr
f=f+fincr
resid=resid-fincr
21
1.4 DIRECTIONAL DERIVATIVE AND LINEARIZATION




BOX 1.3:


c
c Newton-Raphson iteration
c
rnorm=cnorm*2
niter=0
do while((rnorm.gt.cnorm).and.(niter.lt.miter))
niter=niter+1
c
c find geometry increment
c
u=-resid/stiff
x=x+u
l=sqrt(x*x+d*d)
area=vol/l
c
c find stresses and residual forces
c
stress=e*log(l/lzero)
t=stress*area*x/l
resid=t-f
rnorm=abs(resid/f)
print 100, incrm,niter,rnorm,x,f
c
c find stiffness and check stability
c
stiff=(area/l)*(e-2.*stress)*(x/l)*(x/l)
+(stress*area/l)
if(abs(stiff).lt.1.e-20) then
print *, ™ near zero stiffness - stop™
stop
endif
enddo
enddo
stop
100 format(™ increment=™,i3,™ iteration=™,i3,™
residual=™,g10.3,
& ™ x=™,g10.4,™ f=™,g10.4)
end
CHAPTER TWO

MATHEMATICAL PRELIMINARIES




2.1 INTRODUCTION

In order to make this text su¬ciently self-contained, it is necessary to include
this chapter dealing with the mathematical tools that are needed to achieve
a complete understanding of the topics discussed in the remaining chapters.
Vector and tensor algebra is discussed, as is the important concept of the
general directional derivative associated with the linearization of various
nonlinear quantities that will appear throughout the text.
Readers, especially with engineering backgrounds, are often tempted to
skip these mathematical preliminaries and move on directly to the main
text. This temptation need not be resisted, as most readers will be able to
follow most of the concepts presented even when they are unable to under-
stand the details of the accompanying mathematical derivations. It is only
when one needs to understand such derivations that this chapter may need
to be consulted in detail. In this way, this chapter should, perhaps, be ap-
proached like an instruction manual, only to be referred to when absolutely
necessary. The subjects have been presented without the excessive rigors of
mathematical language and with a number of examples that should make
the text more bearable.


2.2 VECTOR AND TENSOR ALGEBRA

Most quantities used in nonlinear continuum mechanics can only be de-
scribed in terms of vectors or tensors. The purpose of this section, however,
is not so much to give a rigorous mathematical description of tensor alge-
bra, which can be found elsewhere, but to introduce some basic concepts

1
2 MATHEMATICAL PRELIMINARIES



x3

v3



e3

v

e2

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