MinUnbalDispNorm#

Model.integrator("MinUnbalDispNorm", dlam0[, jd, min, max])

Use the minimum unbalanced displacement norm continuation method [1]

Parameters:

dlam0 (float) – First load increment (pseudo-time step) at the first iteration in the next invocation of the analysis command.
jd (float) – Factor relating first load increment at subsequent time steps. (optional, default: 1.0)
min (float) – arguments used to bound the load increment (optional, default: dlam0)
max (float) – arguments used to bound the load increment (optional, default: dlam0)

integrator MinUnbalDispNorm $dlambda11 <$Jd $minLambda $maxLambda>

Argument	Type	Description
$dlambda11	float	First load increment (pseudo-time step) at the first iteration in the next invocation of the analysis command.
$Jd	float	Factor relating first load increment at subsequent time steps. (optional, default: 1.0)
$minLambda	float	arguments used to bound the load increment (optional, default: $dLambda11)
$maxLambda	float	arguments used to bound the load increment (optional, default: $dLambda11)

Theory#

Continuation#

In this method, the constraint equation governing the evolution of $\Delta \lambda$ is

\[\frac{\partial}{\partial \Delta \lambda}\left. d \boldsymbol{u} \cdot d \boldsymbol{u}\right.=0\]

which guarantees a minimum value for the unbalanced displacement norm in each iteration. Evaluating the constraint equation furnishes

\[d \lambda = -\frac{d\hat{\boldsymbol{u}} \cdot d\bar{\boldsymbol{u}}}{d\hat{\boldsymbol{u}} \cdot d\hat{\boldsymbol{u}}}\]

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:

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

MinUnbalDispNorm

Contents

MinUnbalDispNorm#

Theory#

Continuation#

Incrementation#

References#