Initial solution estimate

To estimate the solution of the system of equations two approaches, numgeo uses a method based on the solutions obtained in previous increments. Only at the very first iteration of the first increment of each step, a so-called zero-call is performed. Details are given in the following two sections.

Reference Manual

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The very first iteration of the first increment of a step

At The very first iteration (\(i=0\)) of the first increment (\(k=1\)) of a step of the calculation no information from previous increments is available (we do not use information from previous steps). For the very first iteration (\(i=0\)) of each step the global linear system of equations \(K^{i=0}_{\alpha \beta} u^{i=0}_\beta = r^{i=0}_\alpha\) is build assuming \(u_\beta^{i=0}=[0,...,0]^T\). Before solving the system of equations the boundary conditions and external force components \(f_\alpha\) are added.

The solution obtained corresponds to the first correction \(c^{i=1}_\beta=u_\beta^{i=0}\) accounting for all boundary conditions and external forces. The now updated solution \(u^{i=1}_\beta = u_\beta^{i=0} + c_\beta ^{i=1}\) and \(\Delta u^{i=1}_\beta = c_\beta ^{i=1}\) can be seen as the initial estimation of the current increment. From here the solution method described in Newton-Raphson method is followed.

All subsequent increments

For all subsequent increments (\(k>1\)) the initial solution is based on an extrapolation of the solution of the previous increment (\(k^0=k-1\)) depending on the current solution procedure:

*Static, *Transient and *Reduction analyses: The user can choose between the following options.
- none: No extrapolation of the solution. The predicted solution increment is zero:
  
  \[ \Delta u^{i=1,k}_\beta = 0 \]
- none-pw: No extrapolation of the pore water pressure, linear extrapolation for all other degrees of freedom:
  
  \[ \Delta u^{i=1,k}_\beta = \begin{cases} 0 & \beta \in \text{pw} \\ \dfrac{\Delta t^k}{\Delta t^{k_0}} \Delta u^{k^0}_\beta & \text{otherwise} \end{cases} \]
- constant: The predicted solution increment is the converged increment of the previous increment:
  
  \[ \Delta u^{i=1,k}_\beta = \Delta u^{k^0}_\beta \]
- linear: Linear extrapolation based on the previous solution increment (default):
  
  \[ u^{i=1,k}_\beta = u_\beta^{k^0} + \dfrac{\Delta t^k}{\Delta t^{k_0}} \Delta u^{k^0}_\beta \]
  
  where \(u_\beta^{k^0}\) is the solution at the end of increment \(k_0\) (thus at the beginning of increment \(k\)) and \(\Delta u^{k^0}_\beta\) is the solution increment from the previous increment \(k_0\). \(\Delta t^k\) and \(\Delta t^{k_0}\) are the time steps of the current and previous increments, respectively.
- quadratic: Quadratic extrapolation based on the last three known values of the solution (at increments \(k-1\), \(k-2\) and \(k-3\)):
  
  \[ u^{i=1,k}_\beta = b_0 + b_1 (\Delta t^{k}-\Delta t^{k-1}) + b_2 (\Delta t^{k}-\Delta t^{k-1}) (\Delta t^{k}-\Delta t^{k-2}) \]
  
  where:
  
  \[ \begin{aligned} b_0 &= \Delta u^{k-1}_\beta, \\ b_1 &= \dfrac{\Delta u^{k-2}_\beta-\Delta u^{k-2}_\beta}{\Delta t^{k-2}-\Delta t^{k-1}}, \\ b_2 &= \left(\dfrac{\Delta u^{k-3}_\beta-\Delta u^{k-2}_\beta}{\Delta t^{k-3}-\Delta t^{k-2}} - \dfrac{\Delta u^{k-2}_\beta-\Delta u^{k-1}_\beta}{\Delta t^{k-2}-\Delta t^{k-1}} \right) \dfrac{1}{\Delta t^{k-3}-\Delta t^{k-1}} \end{aligned}\]
  
  Note that the first three increments of each new step, linear extrapolation is used.
- coupled: Special extrapolation of solution with time considering the different nature of solution variables
  - displacements \(u_i\): linear extrapolation in time
  - pressures \(p^w\) (water) and \(p^a\) (air): Backward Euler based on known velocities
- fos-linear: Linear extrapolation of the solution increment with the factor of safety \(\text{FoS}\) instead of time. Intended for use in strength reduction analyses, where time has no physical meaning and the driving quantity is \(\text{FoS}\):
  
  \[ \Delta u^{i=1,k}_\beta = \dfrac{\Delta \text{FoS}^k}{\Delta \text{FoS}^{k_0}} \Delta u^{k^0}_\beta \]
  
  where \(\Delta \text{FoS}^k\) is the target increment in \(\text{FoS}\) of the current increment \(k\) and \(\Delta \text{FoS}^{k_0}\) is the converged \(\text{FoS}\) increment from the previous increment \(k_0\). 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 \(\text{FoS}\). Intended for use in strength reduction analyses near the critical \(\text{FoS}\), where the relation between \(\text{FoS}\) and a scalar measure of the displacement \(s = \max |\Delta u|\) becomes strongly non-linear.
  
  Near a global failure mechanism the relation between \(\text{FoS}\) and \(s\) is approximately hyperbolic:
  
  \[ s(\text{FoS}) = \dfrac{A}{\text{FoS}_{\text{lim}} - \text{FoS}} \]
  
  which linearizes as
  
  \[ \dfrac{1}{s} = a_0 + a_1 \, \text{FoS} \qquad \text{with} \qquad a_1 = -\dfrac{1}{A} < 0. \]
  
  A least-squares fit of this linear model is performed on the last three converged points \((\text{FoS}^{k-1}, s^{k-1})\), \((\text{FoS}^{k-2}, s^{k-2})\), \((\text{FoS}^{k-3}, s^{k-3})\). The predicted displacement measure at the current target factor of safety \(\text{FoS}^{k}\) is then:
  
  \[ s^{k} = \dfrac{1}{a_0 + a_1 \, \text{FoS}^{k}} \]
  
  and the scalar scaling factor
  
  \[ \kappa = \dfrac{s^{k}}{s^{k-1}} \]
  
  is applied to the previous converged solution increment:
  
  \[ \Delta u^{i=1,k}_\beta = \kappa \, \Delta u^{k^0}_\beta \]
  
  The direction of the predicted increment is thus preserved from the previous converged step (as in linear and fos-linear); only the magnitude is adjusted to reflect the non-linear softening of the mechanism near failure.
  Fall-back to linear
  
  The hyperbolic extrapolation requires at least three converged increments and makes several assumptions that may not hold early in the analysis. The implementation falls back to linear extrapolation in time when any of the following conditions is met:
  - fewer than three converged increments are available
  - the three displacement samples are not monotonically increasing with \(\text{FoS}\)
  - the fitted slope \(a_1\) is not strictly negative (data not on a hyperbolic trajectory)
  - the predicted denominator \(a_0 + a_1 \, \text{FoS}^k\) is non-positive (the target \(\text{FoS}\) is at or past the predicted \(\text{FoS}_{\text{lim}}\))
  - the scaling factor \(\kappa\) is outside a safety range \([\kappa_{\min}, \kappa_{\max}]\)
*Dynamic analysis
- Initial velocity and initial acceleration
  
  \[ \begin{aligned} u^{i=0,k}_\beta &= u_\beta^{i,k^0} + \Delta u^{i,k^0}_\beta & \\ \dot{u}^{i=0,k}_\beta &= 0 \\ \ddot{u}^{i=0,k}_\beta&= 0 \end{aligned} \]
- Normal time with Hilber-Hughes-Taylor (HHT) time integration scheme:
  
  \[ \begin{aligned} u^{i=0,k}_\beta &= u_\beta^{i,k^0} + \Delta u^{i,k^0}_\beta & \\ \dot{u}^{i=0,k}_\beta &= \dot{u}^{i,k^0}_\beta + \Delta t \left((1-\gamma) \ddot{u}^{i,k^0}_\beta + \gamma \ddot{u}^{i=0,k}_\beta \right) \\ \ddot{u}^{i=0,k}_\beta&= \left((\Delta u_\beta^{i=0,k}-\Delta t \dot{u}^{i,k^0})/\Delta t^2 - (0.5-\beta)/\ddot{u}^{i,k^0}_\beta\right) / \beta \end{aligned} \]
- Normal time with Backwards-Euler (BWE) time integration scheme:
  
  \[ \begin{aligned} u^{i=0,k}_\beta &= u_\beta^{i,k^0} + \Delta u^{i,k^0}_\beta \\ \dot{u}^{i=0,k}_\beta &= \dfrac{\Delta u^{i=0,k}_\beta}{\Delta t} \\ \ddot{u}^{i=0,k}_\beta &= \dfrac{\dot{u}^{i=0,k}_\beta - \dot{u}^{i,k^0}_\beta}{\Delta t} \end{aligned} \]

From here the solution method described in Newton-Raphson method is followed.

Cutbacks

If the previous increment was not successful and a cutback in time (or factor of safety) is requested, the predicted solution increment of the current increment is reset to zero:

\[ \Delta u^{i=1,k}_\beta = 0 \]

That is, no extrapolation is performed and the iteration restarts from the last converged state with the reduced time or factor-of-safety increment.