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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.

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 a linear extrapolation of the solution of the previous increment (\(k^0=k-1\)) depending on the current solution procedure:

  • *Static and *Transient analysis: The user can choose between the following three options.

    • 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

  • *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

      \[ \begin{aligned} u^{i=0,k}_\beta &= u_\beta^{i,k^0} + \Delta u^{i,k^0}_\beta & \\ \dot{u}^{i=0,k}_\beta &= \dot{u}^{i,k^0}_\beta + \Delta t \left((1-\gamma) \ddot{u}^{i,k^0}_\beta + \gamma \ddot{u}^{i=0,k}_\beta \right) \\ \ddot{u}^{i=0,k}_\beta&= \left((\Delta u_\beta^{i=0,k}-\Delta t \dot{u}^{i,k^0})/\Delta t^2 - (0.5-\beta)/\ddot{u}^{i,k^0}_\beta\right) / \beta \end{aligned} \]

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