2.2.6 `Softening' behaviour and displacement control
2.2.8 General remarks ± incremental and rate methods
The various iterative methods described provide an essential tool-kit for the solution of non-linear problems in which ®nite element discretization has been used. The precise choice of the optimal methodology is problem dependent and although many comparative solution cost studies have been published10;15;25 the dierences are often marginal. There is little doubt, however, that exact Newton±Raphson processes (with line search) should be used when convergence is dicult to achieve.
Also the advantage of symmetric update matrices in the quasi-Newton procedures frequently make these a very economical candidate. When non-symmetric tangent moduli exist it may be better to consider one of the non-symmetric updates, for example, a Broyden method.11;26
We have not discussed in the preceding direct iterative methods such as the various conjugate direction methods27ÿ31or dynamic relaxation methods in which an explicit dynamic transient analysis (see Chapter 18 of Volume 1) is carried out to achieve a steady-state solution.32;33These forms are often characterized by:
1. a diagonal or very sparse form of the matrix used in computing trial increments da (and hence very low cost of an iteration) and
2. a signi®cant number of total iterations and hence evaluations of the residual . 34 Solution of non-linear algebraic equations
These opposing trends imply that such methods oer the potential to solve large problems eciently. However, to date such general solution procedures are eective only in certain problems.34
One ®nal remark concerns the size of increments f or to be adopted. First, it is clear that small increments reduce the total number of iterations required per computational step, and in many applications automatic guidance on the size of the increment to preserve a (nearly) constant number of iterations is needed. Here such processes as the use of the `current stiness parameter' introduced by Bergan20 can be eective.
Second, if the behaviour is path dependent (e.g. as in plasticity-type constitutive laws) the use of small increments is desirable to preserve accuracy in solution changes.
In this context, we have already emphasized the need for calculating such changes by using always the accumulated ain change and not in adding changes arising from each iterative dain step in an increment.
Third, if only a single Newton±Raphson iteration is used in each increment of
then the procedure is equivalent to the solution of a standard rate problem incrementally by direct forward integration. Here we note that if Eq. (2.3) is rewritten
as P a f0 2:48
we can, on dierentiation with respect to obtain dP
da da
d f0 2:49
f (λ)
a
∆f4
∆f3
∆f2
∆f1
Possible divergence
Fig. 2.8 Direct integration procedure.
Iterative techniques 35
and write this as
da
d Kÿ1T f0 2:50
Incrementally, this may be written in an explicit form by using an Euler method as
an Kÿ1Tnf0 2:51
This direct integration is illustrated in Fig. 2.8 and can frequently be divergent as well as being only conditionally stable as a result of the Euler explicit method used.
Obviously, other methods can be used to improve accuracy and stability. These include Euler implicit schemes and Runge±Kutta procedures.
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