Possible new definition of the friction bound

[mariusnevland]

The friction bound is currently defined as

$$b=-F\lambda_n \ \ \ \ \ \ \ (1)$$

in PorePy.

However, it is defined to be

$$b=-F(\lambda_n+\tilde{c}([[\mathbf{u}]]_n-g)) \ \ \ \ \ \ \ (2)$$

in the paper by Stefansson et al (A fully coupled numerical model of thermo-hydro-mechanical processes and fracture contact mechanics in porous media, p. 11) and in the article by Hüeber et al (A primal-dual active set algorithm for three-dimensional contact problems with coulomb friction, p.20-21).

At the moment, (2) is used in the definition of the normal fracture deformation equation in PorePy, while (1) is used for the tangential fracture deformation equation.

Should we change the friction bound from (1) to (2) in the tangential fracture deformation equation? On one hand, the two definitions should give the same solution, as (2) reduces to (1) in the case of contact. Moreover, an open cell is defined to be when $b \leq 0$, which both (1) and (2) will satisfy when $\lambda_n=0$ and $[[\mathbf{u}_n-g]]>0$. However, I believe definition (2) could result in easier numerical treatment, as it might make it easier to detect an open cell. Currently in PorePy, a cell is regarded as open if $b<10^{-5}$. If (2) is used, this tolerance could possibly be lowered, as the presence of the $\tilde{c}([[\mathbf{u}]]_n-g)$ term will reduce $b$ even further.