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Extrapolation

*Extrapolation, <strategy>

Use this keyword to specify an extrapolation strategy of the initial estimate of the solution at the beginning of each increment (except for the first one). By default, a linear extrapolation is performed.

You can choose between the following <strategy>:

  • linear: Default, a linear extrapolation, i.e. \(\Delta d_{n+1} = \Delta d_n/\Delta t_n \Delta t_{n+1}\)

  • quadratic: quadratic extrapolation based on the last three known values of the solution \(\Delta d_{n+1}(d_{n-2},d_{n-1},d_n)\). For the first three increments of each new step, linear extrapolation is used.

  • coupled: Extrapolation of solution with time considering the different nature of solution variables - displacements are extrapolated linear in time and pressures (pw, pa) using a Backward Euler approach based on known velocities

  • constant: an unscaled approximation based on the last increment is assumed, i.e. \(\Delta d_{n+1} = \Delta d_n\).

  • none: no extrapolation is performed, the elements are called with \(\Delta d_{n+1} = 0\). This method usually slows down the convergence finding, but can help in case of strong convergence difficulties.

  • none-pw: no extrapolation is performed for the pore water pressure (\(\Delta d_{n+1} = 0\)). For the solid displacement, a linear extrapolation is applied (\(\Delta d_{n+1} = \Delta d_n/\Delta t_n \Delta t_{n+1}\)). This method may help in case of convergence finding in infiltration problems.

  • fos-linear: linear extrapolation of the solution increment with the factor of safety \(\text{FoS}\) instead of time, i.e. \(\Delta d_{n+1} = \Delta d_n / \Delta \text{FoS}_n \cdot \Delta \text{FoS}_{n+1}\). Intended for use in strength reduction analyses (*Reduction) where time has no physical meaning and the driving quantity is \(\text{FoS}\). For the first increment of each step, linear extrapolation in time is used as a fall-back.

  • fos-hyperbolic: hyperbolic extrapolation of the solution increment with the factor of safety \(\text{FoS}\). The last three converged values of \((\text{FoS}, \max|\Delta d|)\) are used to fit a hyperbolic model of the form \(\max|\Delta d| = A / (\text{FoS}_{\text{lim}} - \text{FoS})\), from which a scalar scaling factor \(\kappa\) is derived and applied to the previous solution increment: \(\Delta d_{n+1} = \kappa \Delta d_n\). Intended for use in strength reduction analyses near the critical \(\text{FoS}\), where the relation between \(\text{FoS}\) and displacements becomes strongly non-linear. Falls back to linear extrapolation when fewer than three converged increments are available or when the fit does not satisfy the safeguards on monotonicity, slope sign, asymptote location and scaling range.