Abstract
We propose a semismooth Newton method for non-Newtonian models of incompressible flow where the constitutive relation between the shear stress and the symmetric velocity gradient is given implicitly; this class of constitutive relations captures for instance the models of Bingham and Herschel–Bulkley. The proposed method avoids the use of variational inequalities and is based on a particularly simple regularisation for which the (weak) convergence of the approximate stresses is known to hold. The system is analysed at the function space level and results in mesh-independent behaviour of the nonlinear iterations.
Highlights
IntroductionAfter introducing the relevant weak and finite element formulations, we will proceed in Section 3 to write the problem (1.1a) with the regularised constitutive relation (1.6) in terms of a function F between appropriate Banach spaces and formulate a semismooth Newton algorithm that, with appropriate assumptions, can be shown to fall within the framework of semismooth Newton methods in function spaces of [49]; in particular, we will see that some expressions describing the Bingham constitutive relation are more advantageous
Let Ω ⊂ Rd be a Lipschitz polygonal/polyhedral domain for d ∈ {2, 3} and consider the following system for the velocity u : Ω → Rd, shear stress tensor S : Ω → Rdsy×md and pressure p : Ω → R of an incompressible fluid: αu − div S + div(u ⊗ u) + ∇p = f div u = 0 u=0 in Ω, in Ω, on ∂Ω, (1.1a) where f : Ω → Rd is given, and α ≥ 0 is e.g. a parameter that arises from an implicit time discretisation (α = 0 for the steady problem)
In the finite element formulation of the problem we look for functions (Sn, un, pn) ∈ Σnsym × V n × M0n such that:
Summary
After introducing the relevant weak and finite element formulations, we will proceed in Section 3 to write the problem (1.1a) with the regularised constitutive relation (1.6) in terms of a function F between appropriate Banach spaces and formulate a semismooth Newton algorithm that, with appropriate assumptions, can be shown to fall within the framework of semismooth Newton methods in function spaces of [49]; in particular, we will see that some expressions describing the Bingham constitutive relation are more advantageous. The analysis from [41] is closely tied to the particular structure of the Herschel–Bulkley relation, which is a constraint not present here: the framework employed here captures a wider variety of models; for instance, given the implicit nature of the constitutive relation, one could very consider stress-dependent viscosities or even swap the roles of the shear stress S and the symmetric velocity gradient D(u), and obtain models describing inviscid (i.e. Euler) fluids for low values of |D(u)|, and viscous otherwise (see [8] for a more detailed description of related models)
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