An Augmented Lagrangian Preconditioner for Implicitly Constituted Non-Newtonian Incompressible Flow

Mixed Kirchhoff stress–displacement–pressure formulations for incompressible hyperelasticity

“…Strikingly, the continuous augmentation instead adds a term to the displacement-displacement block. This structural distinction does not occur in other problems, such as in stress-velocity-pressure formulations for non-Newtonian flow [29]; here it is caused by the fact that the pressure appears in the equation for the stress (2.9a), rather than the equation for the displacement (2.9b). Nevertheless, the continuous augmentation also successfully controls the Schur complement of the linearised systems [55, Lemma 3].…”

Implicit Type Constitutive Relations for Elastic Solids and Their Use in the Development of Mathematical Models for Viscoelastic Fluids

“…Readers interested in analytical and semianalytical solutions to boundary value problems for fluids described by constitutive relations of the type T δ = g(D) or k(T δ , D) = O are referred to Málek et al [8], Le Roux and Rajagopal [7], Srinivasan and Karra [22], Narayan and Rajagopal [23], Fusi and Farina [24], Fusi [25], Housiadas et al [26], Gomez-Constante and Rajagopal [27] and Fetecau and Bridges [28], to name a few. The numerical solution of the corresponding governing equations is investigated in Janečka et al [29], while a rigorous numerical analysis for various models that fall into this class is discussed in Diening et al [30], Stebel [31], Hirn et al [32], Süli and Tscherpel [33], Farrell et al [34], Farrell and Gazca-Orozco [35]. A rigorous mathematical theory for some of the aforementioned models is developed in Bulíček et al [21,36] and Maringová and Žabenský [37]; see also Blechta et al [38] for a recent review.…”

“…Either (38) or (35) can serve as a starting point for the specification of constitutive relations. Once we have postulated a formula for Gibbs free energy, which is tantamount to the specification of the energy storage mechanism in the material of interest, we can proceed with the specification of the entropy production mechanisms.…”

“…Finally, let us remark that once the specific Gibbs free energy is given, one can then exploit the relation between the entropy and the specific Gibbs free energy (see (24a)) and convert (35) or (38), respectively, to an evolution equation for the temperature:…”

A semismooth Newton method for implicitly constituted non-Newtonian fluids and its application to the numerical approximation of Bingham flow

“…Regarding the numerical analysis, in [18] and [44] the authors established (weak) convergence of the finite element approximations to a weak solution of the system for the steady and unsteady problems, respectively. This was later extended to formulations including the shear stress S S S in [21], and an augmented Lagrangian preconditioner was proposed in [20].…”