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Line Search

*Line Search, method [, default]
lambda_{min}, lambda_{max}, psi, n^{max}

In strongly nonlinear problems the Newton-Raphson solution technique, which is used in numgeo by default, may sometimes diverge during equilibrium iterations. To handle such difficulties, a line search algorithm is implemented in numgeo to enhance the convergence of the iterative method. The line search algorithm detects divergence and applies a scale factor \(\lambda\) to the computed solution correction (during the iterative refinement of one increment). The aim is to find a better configuration which would help to overcome divergence. By default, the line search algorithm is not enabled. The line search procedure can be activated by using the command above. For more detailed information on the implemented line search algorithm, the reader is referred to the numgeo Theory Manual.

  • method

    Set this keyword equal to root finding method to be used in the line search algorithm. This parameter is mandatory. As root finding methods a "Regula Falsi" and a "Secant" method are available. To choose which one to use, set method to Regula-Falsi, Secant or Linear respectively. In the subsequent line, the following parameter have to be specified:

    • \(\lambda_{min}\): is the lower bound for the Line Search scaling factor

    • \(\lambda_{max}\): is the upper bound for the Line Search scaling factor

    • \(\psi\): is the stop criterion. Line Search stops if the interval in the root finding process is less than \(\psi\) times the initial energy (see the Theory Manual)

    • \(n^{max}\): is the maximum number of line searches in each global iteration

    In case of the Linear line search approach, only \(\lambda_{min}\) and \(\lambda_{max}\) are required.

  • default

    Use this keyword to specify that the default values for the line search algorithm should be used. These are \(\lambda_{min}=0.25\), \(\lambda_{max}=1.25\), \(\psi = 0.8\) (\(\psi = 0.0\) in case of the Linear approach) and \(n^{max}=5\) (\(n^{max}=1\) in case of the Linear approach). Note that if default is used, no subsequent line is required.

Line search is not only useful in situations where equilibrium is not achieved due to divergence, but it can also increase the convergence rate for the problems with slow convergence.