Isogeometric approximations of Biot's model for poromechanical continua: inf-sup stability and spurious oscillations

Abstract:

Finite element discretizations of Biot's model for poroelastic continua are nowadays widely used. By virtue of its ease of implementation, the deformation and pressure spaces are commonly discretized using identical finite element spaces. In combination with an implicit time integration procedure, this approach yields meaningful results in a wide range of applications, but it has been established that it is prone to spurious oscillatory solutions when the time step size is not sufficiently large in relation to the mesh size. Stabilization techniques have been proposed to tackle this problem. However, they can be unattractive for practical purposes, since they generally imply an increase in computation cost and can be difficult to implement in existing software frameworks. Therefore, evaluation of alternative discretizations that resolve the unphysical pressure oscillations is of interest. In this contribution, we study the suitability of various isogeometric finite element discretizations to ameliorate such oscillatory effects. In particular, we analyze the discrete inf-sup stability of various families of inf-sup stable discretizations, namely Taylor-Hood (TH), Raviart-Thomas (RT) and Nédélec (N) elements. Our study establishes a link between the discrete inf-sup constant and the observed pressure oscillations for various benchmark problems. The presented numerical analysis conveys that the considered compatible element spaces yield stability constants independent of the time step size, this in contrast to the equal order discretization. As a result, the considered isogeometric finite elements ameliorate the stability problems observed using finite elements with identical …

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