Termination Schemes
For an incremental procedure based on iterative methods to be effective,
practical termination schemes should be provided. At the end of each iteration,
convergence should be evaluated within realistic tolerances. Very loose
tolerances will lead to inaccurate results, while very strict tolerances
can needlessly increase the computational cost. A bad divergence check
can end the iterative process when the solution has not diverged or allow
the process to continue searching for unrealizable solution.
A number of procedures have been introduced as convergence criteria
for terminating an iterative process. Three convergence criteria will
be discussed below:
Displacement Convergence
This criterion is based on the displacement increments during iterations.
It is given by:
|{DU}(i)| <
ed |t+Dt{U}(i)|
where |{a}| denotes the Euclidean norm of {a}, and ed is the displacement tolerance.
Force Convergence
This criterion is based on the out-of-balance (residual) loads during
iterations. It requires that the norm of the residual load vector to be
within a tolerance ef of the applied load increment, i.e.,
|t+Dt{R}
- t+Dt{F}(i)|
< ef |t+Dt{R}
- t{F}|
Energy Convergence
In this criterion, the increment in the internal energy during each
iteration, which is the work done by the residual forces through the incremental
displacements, is compared with the initial energy increment. Convergence
is assumed realized when the following condition is satisfied:
({DU}(i))T
(t+Dt{R}
- t+Dt{F}(i-1))
< ee ({DU}(1))T
(t+Dt{R}
- t{F})
where ee is the energy tolerance.
In addition, a number of schemes are used as divergence criteria. One
of these schemes is based on the divergence of the residual loads. Another
is based on the divergence of the incremental energy.
Related Topics
Numerical
Procedures
Incremental
Control Techniques
Performing
Nonlinear Static Analysis
Setting
the Properties of the Nonlinear Analysis
Setting
the Result Options of Nonlinear Analysis