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FPL stabilisation:

*Material, ...
*FPL-stabilisation
alpha_fpl

Use this command to control the stabilisation parameter \(\alpha\) in the stabilized u-p elements for saturated soils and partially saturated soils.

For these elements the conversation of mass for the free water is enhanced by adding a stabilisation term \(f^{wf,\text{stab}}_K\). The stabilisation term used in numgeo is a modified version of the Fluid-Pressure-Laplacian (FPL) stabilisation 1:

\(f^{wf,\text{stab}}_K = \int_\mathfrak{B} \dfrac{\partial N_I^p}{\partial x_i} \kappa^{FPL} \dfrac{\partial N_I^p}{\partial x_i} \dot{p}^{wf}_K \mbox{d} v,\)

wherein \(\kappa^{FPL}\) is the stabilisation factor and is implemented in numgeo based on the FPL for unsaturated materials 2:

\(\kappa^{FPL} = \alpha^{FPL} \left(\dfrac{\left(S^w l_e\right)^2}{4 M} + \dfrac{l_e^2 c}{6} - \dfrac{\Delta t k}{\rho^w g}\right).\)

Therein are:

  • \(S^w\) the degree of saturation,

  • \(\Delta t\) the time increment,

  • \(l_e\) the characteristic element length (approximated as the longest distance of any two nodes of a finite element),

  • \(M\) the constrained modulus of the material (approximated from the current material tangent stiffness \(\frac{\partial \sigma_{ij}}{\partial \varepsilon_{kl}}\) assuming incremental linear elasticity),

  • \(c\) the storage capacity assuming incompressibility of the solid particles: \(c=\frac{\varphi^{wf}}{\bar{K}^{wf}} + \frac{\partial \varphi^w}{\partial \bar{p}^{w}}\) and

  • \(k\) the representative hydraulic conductivity 3 \(k = \frac{p^w_{,i}}{\parallel p^w_{,i}\parallel} \frac{k^{wf} K_{ij}^{ref}}{\mu^{wf}} \frac{p^w_{,j}}{\parallel p^w_{,j}\parallel}.\)

  • \(\alpha^{FPL}\) controls the amount of numerical diffusion and is typically in the range of \(0 \leq \alpha^{FPL} \leq 1\). If the keyword *FPL-stabilisation is not defined, a default value of \(\alpha^{FPL}=0.5\) is used.

  1. F. Brezzi and J. Pitkäranta. On the Stabilization of Finite Element Approximations of the Stokes Equations. In Wolfgang Hackbusch, editor, Efficient Solutions of Elliptic Systems, pages 11–19. Vieweg+Teubner Verlag, 1984. URL: http://link.springer.com/10.1007/978-3-663-14169-3_2 (visited on 2024-03-15), doi:10.1007/978-3-663-14169-3_2. ↩

  2. Andrzej Truty and Thomas Zimmermann. Stabilized mixed finite element formulations for materially nonlinear partially saturated two-phase media. Computer Methods in Applied Mechanics and Engineering, 195(13-16):1517–1546, 02 2006. URL: https://linkinghub.elsevier.com/retrieve/pii/S0045782505002938 (visited on 2024-01-04), doi:10.1016/j.cma.2005.05.044. ↩

  3. A. Truty. A Galerkin/least‐squares finite element formulation for consolidation. International Journal for Numerical Methods in Engineering, 52(8):763–786, 11 2001. URL: https://onlinelibrary.wiley.com/doi/10.1002/nme.224, doi:10.1002/nme.224. ↩