11,058 research outputs found
Mathematical and numerical analysis of a simplified time-dependent viscoelastic flow
A time-dependent model corresponding to an Oldroyd-B viscoelastic fluid is considered, the convective terms being disregarded. Global existence in time is proved in Banach spaces provided the data are small enough, using the implicit function theorem and a maximum regularity property for a three fields Stokes problem. A finite element discretization in space is then proposed. Existence of the numerical solution is proved for small data, so as a priori error estimates, using again an implicit function theore
Energy-stable linear schemes for polymer-solvent phase field models
We present new linear energy-stable numerical schemes for numerical
simulation of complex polymer-solvent mixtures. The mathematical model proposed
by Zhou, Zhang and E (Physical Review E 73, 2006) consists of the Cahn-Hilliard
equation which describes dynamics of the interface that separates polymer and
solvent and the Oldroyd-B equations for the hydrodynamics of polymeric
mixtures. The model is thermodynamically consistent and dissipates free energy.
Our main goal in this paper is to derive numerical schemes for the
polymer-solvent mixture model that are energy dissipative and efficient in
time. To this end we will propose several problem-suited time discretizations
yielding linear schemes and discuss their properties
Numerical analysis of the Oseen-type Peterlin viscoelastic model by the stabilized Lagrange-Galerkin method, Part II: A linear scheme
This is the second part of our error analysis of the stabilized
Lagrange-Galerkin scheme applied to the Oseen-type Peterlin viscoelastic model.
Our scheme is a combination of the method of characteristics and
Brezzi-Pitk\"aranta's stabilization method for the conforming linear elements,
which leads to an efficient computation with a small number of degrees of
freedom especially in three space dimensions. In this paper, Part II, we apply
a semi-implicit time discretization which yields the linear scheme. We
concentrate on the diffusive viscoelastic model, i.e. in the constitutive
equation for time evolution of the conformation tensor a diffusive effect is
included. Under mild stability conditions we obtain error estimates with the
optimal convergence order for the velocity, pressure and conformation tensor in
two and three space dimensions. The theoretical convergence orders are
confirmed by numerical experiments.Comment: See arXiv:1603.01339 for Part I: a nonlinear schem
Numerical analysis of the Oseen-type Peterlin viscoelastic model by the stabilized Lagrange-Galerkin method, Part I: A nonlinear scheme
We present a nonlinear stabilized Lagrange-Galerkin scheme for the Oseen-type
Peterlin viscoelastic model. Our scheme is a combination of the method of
characteristics and Brezzi-Pitk\"aranta's stabilization method for the
conforming linear elements, which yields an efficient computation with a small
number of degrees of freedom. We prove error estimates with the optimal
convergence order without any relation between the time increment and the mesh
size. The result is valid for both the diffusive and non-diffusive models for
the conformation tensor in two space dimensions. We introduce an additional
term that yields a suitable structural property and allows us to obtain
required energy estimate. The theoretical convergence orders are confirmed by
numerical experiments. In a forthcoming paper, Part II, a linear scheme is
proposed and the corresponding error estimates are proved in two and three
space dimensions for the diffusive model.Comment: See arXiv:1603.01074 for Part II: a linear schem
Theoretical and numerical comparison of hyperelastic and hypoelastic formulations for Eulerian non-linear elastoplasticity
The aim of this paper is to compare a hyperelastic with a hypoelastic model
describing the Eulerian dynamics of solids in the context of non-linear
elastoplastic deformations. Specifically, we consider the well-known
hypoelastic Wilkins model, which is compared against a hyperelastic model based
on the work of Godunov and Romenski. First, we discuss some general conceptual
differences between the two approaches. Second, a detailed study of both models
is proposed, where differences are made evident at the aid of deriving a
hypoelastic-type model corresponding to the hyperelastic model and a particular
equation of state used in this paper. Third, using the same high order ADER
Finite Volume and Discontinuous Galerkin methods on fixed and moving
unstructured meshes for both models, a wide range of numerical benchmark test
problems has been solved. The numerical solutions obtained for the two
different models are directly compared with each other. For small elastic
deformations, the two models produce very similar solutions that are close to
each other. However, if large elastic or elastoplastic deformations occur, the
solutions present larger differences.Comment: 14 figure
Non-Newtonian Rheology in Blood Circulation
Blood is a complex suspension that demonstrates several non-Newtonian
rheological characteristics such as deformation-rate dependency,
viscoelasticity and yield stress. In this paper we outline some issues related
to the non-Newtonian effects in blood circulation system and present modeling
approaches based mostly on the past work in this field.Comment: 26 pages, 5 figures, 2 table
Finite element approximation of the viscoelastic flow problem: a non-residual based stabilized formulation
In this paper, a three-field finite element stabilized formulation for the incompressible viscoelastic fluid flow problem is tested numerically. Starting from a residual based formulation, a non-residual based one is designed, the benefits of which are highlighted in this work. Both formulations allow one to deal with the convective nature of the problem and to use equal interpolation for the problem unknowns View the MathML sources-u-p (deviatoric stress, velocity and pressure). Additionally, some results from the numerical analysis of the formulation are stated. Numerical examples are presented to show the robustness of the method, which include the classical 4: 1 planar contraction problem and the flow over a confined cylinder case, as well as a two-fluid formulation for the planar jet buckling problem.Peer ReviewedPostprint (author's final draft
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