Arc-Length Control

Arc-Length Control#

The ArcLength static integrator is particularly useful for tracing the response of a structure across limit points, where the solution may exhibit snap-through or snap-back behavior [2].

Model.integrator("ArcLength", s, alpha, det=False, exp=0.0, iterScale=1.0)
Parameters:

  • s (float) – the initial arc-Length

  • alpha (float) – a scaling factor on the reference loads.

  • det (bool) – if True, use the determinant to choose incrementation sign [1].

  • exp (float) – exponent parameter from [1]

  • iterScale – float, a scaling factor on the arc-length incrementation.

Examples#

The following snippets demonstrate the syntax for configuring the ArcLength integrator. A complete example that reproduces the work presented in [1] is available here.

import sees.openseespy as ops
model = ops.Model(ndm=2, ndf=3)

...

model.integrator("ArcLength", 1, det=True, exp=0.5)

integrator ArcLength 1 -det -exp 0.5

integrator ArcLength 1 -exp 0.5

Theory#

The arc-length control method is a continuation method that appends the following constraint equation to the static residual given in Static:

\[\Delta \boldsymbol{u}_{i+1} \cdot \Delta \boldsymbol{u}_{i+1} + \alpha^2 \Delta \lambda_{i+1}^2 = \Delta s^2\]

where

\[\begin{split}\begin{aligned} \Delta \boldsymbol{u}_{i+1}=\sum_{j=1}^i d \boldsymbol{u}_{(j)} =\Delta \boldsymbol{u}_{(i)} + d \boldsymbol{u} \\ \Delta \lambda_{i+1} =\sum_{j=1}^i d \lambda_{(j)}=\Delta \lambda_{(i)} + d \lambda \end{aligned}\end{split}\]

Recall the linearized static residual

\[\boldsymbol{K} d \boldsymbol{u} = d \lambda \, \boldsymbol{p}_{\mathrm{ref}} + \lambda_{(i)} \boldsymbol{p}_{\mathrm{ref}} - \boldsymbol{p}_{\sigma}(u_{(i)}) = d \lambda \, \boldsymbol{p}_{\mathrm{ref}} + g_{(i)}\]

and define \(d\hat{\boldsymbol{u}}\) and \(d\bar{\boldsymbol{u}}\) by

\[d \hat{\boldsymbol{u}} \triangleq \boldsymbol{K}^{-1}_{(i)}\boldsymbol{p}_{\mathrm{ref}} \qquad\text{ and }\qquad d \bar{\boldsymbol{u}} \triangleq \boldsymbol{K}^{-1}_{(i)} \boldsymbol{g}_{(i)}\]

so that

\[d \boldsymbol{u} = d \lambda \, d \hat{\boldsymbol{u}} + d \bar{\boldsymbol{u}}\]

Implementation#

The arc-length control method is implemented with a distinct increment and iteration phase.

Increment#

During load incrementation the iteration is \(i=1\) and the following assumption is taken:

\[d \boldsymbol{u}_{(1)} = d \lambda_{(1)} \, d \hat{\boldsymbol{u}}_{(1)} + \boldsymbol{0}\]

Thus the constraint equation simplifies to

\[d \lambda_{(1)} = \pm \sqrt{\frac{\Delta s^2}{d\hat{\boldsymbol{u}} \cdot d\hat{\boldsymbol{u}} + \alpha^2}}\]

where \(d \lambda\) from the previous time \((n-1)\) is used to determine the sign; if it was positive then the new \(d \lambda_{(1)}\) is assumed positive, otherwise negative.

Iterations#

During iterations (ie \(i>1\)) the constraint equation is expressed in terms of the linearization direction \(d\boldsymbol{u}\):

\[\left( \Delta \boldsymbol{u}_{(i)} + d\boldsymbol{u} \right) \cdot \left( \Delta \boldsymbol{u}_{(i)} + d \boldsymbol{u} \right) + \alpha^2 \left( \Delta \lambda_{(i)} + d\lambda \right)^2 = \Delta s^2\]

which expands to

\[\Delta u_{(i)} \cdot \Delta u_{(i)} + 2 \,d \boldsymbol{u} \cdot \Delta \boldsymbol{u}_{(i)} + d u \cdot du + \alpha^2 \, d {\lambda_{(i)}}^2 + 2 \alpha^2 d\lambda \, \Delta \lambda_{(i)} + \alpha^2 \, \Delta \lambda^2_{(i)} = \Delta s^2\]

assuming the constraint equation was solved at \(i-1\) then one has \(\Delta \boldsymbol{u}_{(i)} \cdot \Delta \boldsymbol{u}_{(i)} + \alpha^2 \Delta \lambda^2_{(i)} = \Delta s^2\), and the constraint for the current iteration simplifies to

\[d \boldsymbol{u} \cdot d \boldsymbol{u} + 2\, d\boldsymbol{u} \cdot \Delta \boldsymbol{u}_{(i)} + \alpha^2 d \lambda^2 + 2 \alpha^2 d\lambda \, \Delta \lambda_{(i)} = 0\]

Substituting the decomposed representation for \(d \boldsymbol{u}\) this furnishes a quadratic equation in \(d \lambda\):

\[ a \, d \lambda^2 + 2 b \, d \lambda + c =0\]

where we have defined the scalar constants

\[\begin{split}\begin{aligned} a &\triangleq d\hat{\boldsymbol{u}} \cdot d\hat{\boldsymbol{u}} + \alpha^2 \\ b &\triangleq d \hat{\boldsymbol{u}} \cdot \left( \Delta\boldsymbol{u}_{(i)} + d \bar{\boldsymbol{u}}\right) + \alpha^2 \Delta \lambda_{(i)} \\ c &\triangleq d \bar{\boldsymbol{u}} \cdot d \bar{\boldsymbol{u}} + \Delta \boldsymbol{u}_{(i)} \cdot d \bar{\boldsymbol{u}} \end{aligned}\end{split}\]

which can be solved for two roots. The root chosen is the one which will keep a positive angle between the incremental displacement before and after this step.

References#

  • Wempner, GA (1971) ‘Discrete approximations related to nonlinear theories of solids’

  • Crisfield, MA (1981) ‘A fast incremental/iterative solution procedure that handles “snap-through”’

  • Ramm E ‘Strategies for tracing nonlinear response near limit points’

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