Finite Element Methods for Incompressible Flow Problems

Finite Element Methods for Incompressible Flow Problems
Author: Volker John
Publisher: Springer
Total Pages: 816
Release: 2016-10-27
Genre: Mathematics
ISBN: 3319457500

This book explores finite element methods for incompressible flow problems: Stokes equations, stationary Navier-Stokes equations and time-dependent Navier-Stokes equations. It focuses on numerical analysis, but also discusses the practical use of these methods and includes numerical illustrations. It also provides a comprehensive overview of analytical results for turbulence models. The proofs are presented step by step, allowing readers to more easily understand the analytical techniques.

Finite Element Methods for Viscous Incompressible Flows

Finite Element Methods for Viscous Incompressible Flows
Author: Max D. Gunzburger
Publisher: Elsevier
Total Pages: 292
Release: 2012-12-02
Genre: Technology & Engineering
ISBN: 0323139825

Finite Element Methods for Viscous Incompressible Flows examines mathematical aspects of finite element methods for the approximate solution of incompressible flow problems. The principal goal is to present some of the important mathematical results that are relevant to practical computations. In so doing, useful algorithms are also discussed. Although rigorous results are stated, no detailed proofs are supplied; rather, the intention is to present these results so that they can serve as a guide for the selection and, in certain respects, the implementation of algorithms.

On the Divergence Constraint in Mixed Finite Element Methods for Incompressible Flows

On the Divergence Constraint in Mixed Finite Element Methods for Incompressible Flows
Author: Volker John
Publisher:
Total Pages:
Release: 2015
Genre:
ISBN:

The divergence constraint of the incompressible Navier-Stokes equations is revisited in the mixed finite element framework. While many stable and convergent mixed elements have been developed throughout the past four decades, most classical methods relax the divergence constraint and only enforce the condition discretely. As a result, these methods introduce a pressure-dependent consistency error which can potentially pollute the computed velocity. These methods are not robust in the sense that a contribution from the right-hand side, which in fluences only the pressure in the continuous equations, impacts both velocity and pressure in the discrete equations. This paper reviews the theory and practical implications of relaxing the divergence constraint. Several approaches for improving the discrete mass balance or even for computing divergence-free solutions will be discussed: grad-div stabilization, higher order mixed methods derived on the basis of an exact de Rham complex, H(div)-conforming finite elements, and mixed methods with an appropriate reconstruction of the test functions. Numerical examples illustrate both the potential effects of using non-robust discretizations and the improvements obtained by utilizing pressure-robust discretizations.

Finite Element Methods in Incompressible, Adiabatic, and Compressible Flows

Finite Element Methods in Incompressible, Adiabatic, and Compressible Flows
Author: Mutsuto Kawahara
Publisher: Springer
Total Pages: 379
Release: 2016-04-04
Genre: Technology & Engineering
ISBN: 4431554505

This book focuses on the finite element method in fluid flows. It is targeted at researchers, from those just starting out up to practitioners with some experience. Part I is devoted to the beginners who are already familiar with elementary calculus. Precise concepts of the finite element method remitted in the field of analysis of fluid flow are stated, starting with spring structures, which are most suitable to show the concepts of superposition/assembling. Pipeline system and potential flow sections show the linear problem. The advection–diffusion section presents the time-dependent problem; mixed interpolation is explained using creeping flows, and elementary computer programs by FORTRAN are included. Part II provides information on recent computational methods and their applications to practical problems. Theories of Streamline-Upwind/Petrov–Galerkin (SUPG) formulation, characteristic formulation, and Arbitrary Lagrangian–Eulerian (ALE) formulation and others are presented with practical results solved by those methods.

Mathematical Aspects of Finite Element Methods for Incompressible Viscous Flows

Mathematical Aspects of Finite Element Methods for Incompressible Viscous Flows
Author: M. D. Gunzburger
Publisher:
Total Pages: 52
Release: 1986
Genre:
ISBN:

We survey some mathematical aspects of finite element methods for incompressible viscous flows, concentrating on the steady primitive variable formulation. We address the discretization of a weak formulation of the Navier Stokes equations; we then consider the div-stability condition, whose satisfaction insures the stability of the approximation. Specific choices of finite element spaces for the velocity and pressure are then discussed. Finally, the connection between different weak formulations and a variety of boundary conditions is explored.

Conforming Finite Element Methods for Incompressible and Nearly Incompressible Continua

Conforming Finite Element Methods for Incompressible and Nearly Incompressible Continua
Author: L. R. Scott
Publisher:
Total Pages: 44
Release: 1984
Genre:
ISBN:

Interest here is in finite element discretizations of problems involving an incompressibility condition. As model problems we consider the Stokes equations for the flow of a viscous, incompressible fluid and the equations of linear plane-strain elasticity for the deformation of an isotropic, nearly incompressible solid. In both cases the incompressibility condition takes the form of a divergence constraint. Although this is the most simple formulation, the proper understanding of how an approximate method satisfies the constraint represents an important step towards the understanding of more complicated situations, involving e.g. the Navier-Stokes equations or the equations of nonlinear elasticity. The finite element methods we study have the property that the approximations to the velocities, respectively to the displacements, are continuous; such methods are generally referred to as conforming.

Finite Element Techniques for Fluid Flow

Finite Element Techniques for Fluid Flow
Author: J. J. Connor
Publisher: Newnes
Total Pages: 321
Release: 2013-09-11
Genre: Technology & Engineering
ISBN: 1483161161

Finite Element Techniques for Fluid Flow describes the advances in the applications of finite element techniques to fluid mechanics. Topics covered range from weighted residual and variational methods to interpolation functions, inviscid fluids, and flow through porous media. The basic principles and governing equations of fluid mechanics as well as problems related to dispersion and shallow water circulation are also discussed. This text is comprised of nine chapters; the first of which explains some basic definitions and properties as well as the basic principles of weighted residual and variational methods. The reader is then introduced to the simple finite element concepts and models, and gradually to more complex applications. The chapters that follow focus on the governing equations of fluid flow, the solutions to potential type problems, and viscous flow problems in porous media. The solutions to more specialized problems are also presented. This book also considers how circulation problems can be tackled using finite elements, presents a solution to the mass transfer equation, and concludes with an explanation of how to solve general transient incompressible flows. This source will be of use to engineers, applied mathematicians, physicists, self-taught students, and research workers.