33 research outputs found
Numerical simulation of two-dimensional Bingham fluid flow by semismooth Newton methods
AbstractThis paper is devoted to the numerical simulation of two-dimensional stationary Bingham fluid flow by semismooth Newton methods. We analyze the modeling variational inequality of the second kind, considering both Dirichlet and stress-free boundary conditions. A family of Tikhonov regularized problems is proposed and the convergence of the regularized solutions to the original one is verified. By using Fenchel’s duality, optimality systems which characterize the original and regularized solutions are obtained. The regularized optimality systems are discretized using a finite element method with (cross-grid P1)–Q0 elements for the velocity and pressure, respectively. A semismooth Newton algorithm is proposed in order to solve the discretized optimality systems. Using an additional relaxation, a descent direction is constructed from each semismooth Newton iteration. Local superlinear convergence of the method is also proved. Finally, we perform numerical experiments in order to investigate the behavior and efficiency of the method
A Multigrid Optimization Algorithm for the Numerical Solution of Quasilinear Variational Inequalities Involving the -Laplacian
In this paper we propose a multigrid optimization algorithm (MG/OPT) for the
numerical solution of a class of quasilinear variational inequalities of the
second kind. This approach is enabled by the fact that the solution of the
variational inequality is given by the minimizer of a nonsmooth energy
functional, involving the -Laplace operator. We propose a Huber
regularization of the functional and a finite element discretization for the
problem. Further, we analyze the regularity of the discretized energy
functional, and we are able to prove that its Jacobian is slantly
differentiable. This regularity property is useful to analyze the convergence
of the MG/OPT algorithm. In fact, we demostrate that the algorithm is globally
convergent by using a mean value theorem for semismooth functions. Finally, we
apply the MG/OPT algorithm to the numerical simulation of the viscoplastic flow
of Bingham, Casson and Herschel-Bulkley fluids in a pipe. Several experiments
are carried out to show the efficiency of the proposed algorithm when solving
this kind of fluid mechanics problems
A semismooth Newton method for implicitly constituted non-Newtonian fluids and its application to the numerical approximation of Bingham flow
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.Comment: 25 page
A Dual-Mixed Approximation for a Huber Regularization of Generalized -Stokes Viscoplastic Flow Problems
In this paper, we propose a dual-mixed formulation for stationary
viscoplastic flows with yield, such as the Bingham or the Herschel-Bulkley
flow. The approach is based on a Huber regularization of the viscosity term and
a two-fold saddle point nonlinear operator equation for the resulting weak
formulation. We provide the uniqueness of solutions for the continuous
formulation and propose a discrete scheme based on Arnold-Falk-Winther finite
elements. The discretization scheme yields a system of slantly differentiable
nonlinear equations, for which a semismooth Newton algorithm is proposed and
implemented. Local superlinear convergence of the method is also proved.
Finally, we perform several numerical experiments in two and three dimensions
to investigate the behavior and efficiency of the method
A BDF2-Semismooth Newton Algorithm for the Numerical Solution of the Bingham Flow with Temperature Dependent Parameters
This paper is devoted to the numerical solution of the non-isothermal
instationary Bingham flow with temperature dependent parameters by semismooth
Newton methods. We discuss the main theoretical aspects regarding this problem.
Mainly, we focus on existence of solutions and a multiplier formulation which
leads us to a coupled system of PDEs involving a Navier-Stokes type equation
and a parabolic energy PDE. Further, we propose a Huber regularization for this
coupled system of partial differential equations, and we briefly discuss the
well posedness of these regularized problems. A detailed finite element
discretization, based on the so called (cross-grid ) -
elements, is proposed for the space variable, involving weighted
stiffness and mass matrices. After discretization in space, a second order BDF
method is used as a time advancing technique, leading, in each time iteration,
to a nonsmooth system of equations, which is suitable to be solved by a
semismooth Newton algorithm. Therefore, we propose and discuss the main
properties of a SSN algorithm, including the convergence properties. The paper
finishes with two computational experiment that exhibit the main properties of
the numerical approach
A damped Newton algorithm for computing viscoplastic fluid flows
International audienceFor the first time, a Newton method is proposed for the unregularized viscoplastic fluid flow problem. It leads to a superlinear convergence for Herschel-Bulkley fluids when 0<n<1, where n is the power law index. Performances are enhanced by using the inexact variant of the Newton method and, for solving the Jacobian system, by using an efficient preconditioner based on the regularized problem. A demonstration is provided by computing a viscoplastic flow in a pipe with a square cross section. Comparisons with the augmented Lagrangian algorithm show a dramatic reduction of the required computing time while this new algorithm provides an equivalent accuracy for the prediction of the yield surfaces