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:

|{ΔU}(i)| < εd |t+Δt{U}(i)|

where |{α}| denotes the Euclidean norm of {α}, and εd 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 εf of the applied load increment, i.e.,

|t+Δt{R} - t+Δt{F}(i)| < εf |t+Δt{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:

({ΔU}(i))T (t+Δt{R} - t+Δt{F}(i-1)) < εe ({ΔU}(1))T (t+Δt{R} - t{F})

where εe 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.