Civil Engineering Reference
In-Depth Information
problem is expected to be quadratic, such as the large rotation of structures.
If parabolic extrapolation is used in a step, it begins after the second incre-
ment of the step, that is, the first increment employs no extrapolation, and
the second increment employs linear extrapolation. Consequently, slower
convergence rates may occur during the first two increments of the succeed-
ing steps in a multistep analysis. Nonlinear problems are commonly solved
using Newton's method, and linear problems are commonly solved using
the stiffness method. Details of the aforementioned solution methods are
outside the scope of this topic; however, the methods are presented in details
in [1.12-1.18].
Most general-purpose computer programs adopt a
convergence criterion
for
the solution of nonlinear problems automatically. Convergence criterion is
the method used by software to govern the balance equations during the iter-
ative solution. The iterative solution is commonly used to solve the equations
of nonlinear problems for unknowns, which are the degrees of freedom at the
nodes of the finite element model. Most general-purpose computer programs
have control parameters designed to provide reasonably optimal solution of
complex problems involving combinations of nonlinearities as well as efficient
solution of simpler nonlinear cases. However, the most important consider-
ation in the choice of the control parameters is that any solution accepted as
“converged” is a close approximation to the exact solution of the nonlinear
equations. Modelers can reset many solution control parameters related to the
tolerances used for equilibrium equations. If less strict convergence criterion is
used, results may be accepted as converged when they are not sufficiently close
to the exact solution of the nonlinear equations. Caution should be considered
when resetting solution control parameters. The lack of convergence is often
due to modeling issues, which should be resolved before changing the accu-
racy controls. The solution can be terminated if the balance equations failed to
converge. It should be noted that linear cases do not require more than one
equilibrium iteration per increment, which is easy to converge. Each incre-
ment of a nonlinear solution will usually be solved by multiple equilibrium
iterations. The number of iterations may become excessive, in which case,
the increment size should be reduced and the increment will be attempted
again. On the other hand, if successive increments are solved with a minimum
number of iterations, the increment size may be increased. Modelers can spec-
ify a number of time incrementation control parameters. Most general-
purpose computer programs may have trouble with the element calculations
because of excessive distortion in large-displacement problems or because of
very large plastic-strain increments. If
this occurs and automatic time
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