Minimum Unbalanced Displacement Norm

integrator MinUnbalDispNorm $dlambda1 < $Jd $minLambda $maxLambda >

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dlambda1	first load increment (pseudo-time step) at the first iteration in the next invocation of the analysis command.
Jd	factor relating first load increment at subsequent time steps (optional, default: 1.0)
$minLambda, $maxLambda	arguments used to bound the load increment (optional, default: $dLambda1 for both)

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Examples

integrator MinUnbalDispNorm 0.1;

{"integrator": ["MinUnbalDispNorm", 0.1]};

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Theory

Continuation

In this instance, the constraint equation involving \(\Delta \lambda_i^j\) is \[ \frac{\partial}{\partial \Delta \lambda_i^j}\left[\{\Delta u\}_i^{j} \cdot \{\Delta \delta\}_i^j\right]=0 \]

which guarantees a minimum value for the unbalanced displacement norm in each iteration. Expanding \(\{\Delta \delta\}_i^j\) as defined in equation (9) and evaluating equation (28) furnishes \[ \Delta \lambda_i^j=\frac{-\left\{\delta_1\right\}_i^{\mathrm{T}}\left\{\Delta \delta_{\mathrm{R}}\right\}_i^j}{\left\{\delta_1\right\}_i^{\mathrm{T}}\left\{\delta_{\mathrm{I}}\right\}_i} \]

Incrementation

The load increment at iteration \(i\), \(d\lambda_{1,i}\) , is related to the load increment at \((i-1)\), \(d\lambda_{1,i-1}\), and the number of iterations at \((i-1)\), \(J_{i-1}\), by the following:

\[ d\lambda_{1,i} = d\lambda_{1,i-1}\frac{J_d}{J_{i-1}} \]

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Code Developed by: fmk