Stable Numerical Schemes for Fluids, Structures and their Interactions

By Cornel Marius Murea

This book presents numerical algorithms for solving incompressible fluids, elastic structures and fluid-structure interactions. It collects some of the fundamental finite element methods as well as new approaches.For Stokes and Navier-Stokes equations, the mixed finite element method is employed. An arbitrary Lagrangian Eulerian framework is used for fluids in a moving domain. Schemes for linear and St Venant-Kirchhoff non-linear dynamic elasticity are presented. For fluid-structure interaction, two schemes are analyzed: the first is fully implicit and the second is semi-implicit, where the fluid domain is computed explicitly and consequently the computational time is considerably reduced.The stability of the schemes is proven in this self-contained book. Every chapter is supplied with numerical tests for the reader. These are aimed at Masters students in Mathematics or Mechanical Engineering.

• Presents a self-contained monograph of schemes for fluid and elastic structures, including their interactions
• Provides a numerical analysis of schemes for Stokes and Navier-Stokes equations
• Covers dynamic linear and non-linear elasticity and fluid-structure interaction

• Language: English
• Publisher: Elsevier Science
• Release date: Sep 1, 2017
• ISBN: 9780081023808

Cornel Marius Murea

Cornel Marius Murea is a lecturer at the University of Haute Alsace, France. His research area includes numerical analysis and scientific computing of fluid-structure interaction problems.

Book preview

Stable Numerical Schemes for Fluids, Structures and their Interactions

Cornel Marius Murea

Series Editor

Roger Prud'homme

Table of Contents

Cover

Title page

Dedication

Copyright

Preface

1: Mixed Finite Elements for the Stokes Equations

Abstract

1.1 Stokes equations

1.2 Existence and uniqueness for an abstract mixed problem

1.3 Existence and uniqueness for the Stokes equations

1.4 Finite element approximation of an abstract mixed problem

1.5 Mixed finite elements for the Stokes equations

1.6 Boundary conditions for the Stokes equations

1.7 Numerical tests

2: Numerical Schemes for the Navier-Stokes Equations

Abstract

2.1 Navier-Stokes equations

2.2 Implicit Euler scheme, semi-implicit treatment of the nonlinear term

2.3 Implicit Euler scheme and implicit treatment of the nonlinear term

2.4 Implicit Euler scheme and explicit treatment of the nonlinear term

2.5 c˜Some properties of the trilinear forms c and c˜

2.6 Numerical tests: flow around a disc

3: The ALE Method for Navier-Stokes Equations in a Moving Domain

Abstract

3.1 Lagrangian and Eulerian coordinates

3.2 Arbitrary Lagrangian Eulerian (ALE) coordinates

3.3 Weak non-conservative and conservative formulations

3.4 Discretization of order one in time for the weak conservative formulation

3.5 Stabilized discretization of order one in time for the non-conservative formulation

3.6 Finite element discretization in space

3.7 Numerical tests

4: Linear Elastodynamics

Abstract

4.1 Two-dimensional linear elasticity

4.2 Abstract formulation

4.3 Backward Euler scheme

4.4 Implicit centered scheme

4.5 Mid-point scheme

4.6 Numerical tests

5: Nonlinear Elastodynamics

Abstract

5.1 Total Lagrangian formulation

5.2 The St Venant-Kirchhoff model

5.3 Total Lagrangian backward Euler scheme

5.4 Total Lagrangian mid-point scheme

5.5 Numerical tests

6: Numerical Schemes for the Fluid-Structure Interaction

Abstract

6.1 Non-conservative and conservative weak formulations

6.2 Discretization in time of the conservative form: implicit domain calculation

6.3 Discretization in time of the non-conservative form: explicit domain calculation

6.4 Coupling strategies

6.5 The constants in the Poincaré, Korn and trace inequalities

6.6 Numerical tests

Appendix: Functional Analysis Tools

A.1 Sobolev spaces

A.2 Closed, surjective operators

Bibliography

Index

Dedication

To my daughter Marie-Laetitia, with love, tenderness and affection

Copyright

First published 2017 in Great Britain and the United States by ISTE Press Ltd and Elsevier Ltd

Apart from any fair dealing for the purposes of research or private study, or criticism or review, as permitted under the Copyright, Designs and Patents Act 1988, this publication may only be reproduced, stored or transmitted, in any form or by any means, with the prior permission in writing of the publishers, or in the case of reprographic reproduction in accordance with the terms and licenses issued by the CLA. Enquiries concerning reproduction outside these terms should be sent to the publishers at the undermentioned address:

ISTE Press Ltd

27-37 St George’s Road

London SW19 4EU

UK

www.iste.co.uk

Elsevier Ltd

The Boulevard, Langford Lane

Kidlington, Oxford, OX5 1GB

UK

www.elsevier.com

Notices

Knowledge and best practice in this field are constantly changing. As new research and experience broaden our understanding, changes in research methods, professional practices, or medical treatment may become necessary.

Practitioners and researchers must always rely on their own experience and knowledge in evaluating and using any information, methods, compounds, or experiments described herein. In using such information or methods they should be mindful of their own safety and the safety of others, including parties for whom they have a professional responsibility.

To the fullest extent of the law, neither the Publisher nor the authors, contributors, or editors, assume any liability for any injury and/or damage to persons or property as a matter of products liability, negligence or otherwise, or from any use or operation of any methods, products, instructions, or ideas contained in the material herein.

For information on all our publications visit our website at http://store.elsevier.com/

© ISTE Press Ltd 2017

The rights of Cornel Marius Murea to be identified as the author of this work have been asserted by him in accordance with the Copyright, Designs and Patents Act 1988.

British Library Cataloguing-in-Publication Data

A CIP record for this book is available from the British Library

Library of Congress Cataloging in Publication Data

A catalog record for this book is available from the Library of Congress

ISBN 978-1-78548-273-1

Printed and bound in the UK and US

Preface

Cornel Marius Murea June 2017

This book presents stable numerical schemes for incompressible fluids, linear and nonlinear elastic structures, and fluid-structure interactions. It reviews a selection of fundamental methods as well as some new approaches.

For fluids governed by the Stokes equations, we present the mixed finite element method, and for the Navier-Stokes equations, three discretization schemes of order one in time will be analyzed. In cases where the domain occupied by the fluid is moving, the ALE (Arbitrary Lagrangian-Eulerian) formalism will be used, and we demonstrate the stability of conservative and non-conservative schemes. Unconditionally stable schemes are analyzed for linear and nonlinear elastic structures. For the fluid-structure interaction, we first examine an implicit scheme, and we then present a semi-implicit scheme where the fluid domain is calculated explicitly, but the velocity and the pressure of the fluid and the structure displacement are evaluated implicitly.

We give a detailed demonstration of stability for each scheme presented. Each chapter finishes with numerical tests that confirm the theoretical results.

The book is aimed at second-year Master’s degree students in mathematics, as well as students at engineering schools. It is based on two courses that I have taught at the University of Strasbourg for second-year Master’s degree students of mathematics.

I thank Professor Frédéric Hecht for FreeFem++, an extremely powerful and user-friendly software that I used to carry out the numerical tests.

I would like to express my gratitude to the production team at ISTE and to Professor Roger Prud’homme, the co-ordinator of the collection, for publishing this work and giving me the opportunity to fulfill a deeply held ambition.

I would like to express my thanks to the University of Upper Alsace for the six month Leave for Research or Thematic Conversions, which allowed me to finish this project.

I thank my wife for her daily help and for giving me the time to write this book. I dedicate this work to my daughter for the constant happiness she brings me.

I thank my mother for her kindness and generosity, and my sister for her support and encouragement.

1

Mixed Finite Elements for the Stokes Equations

Abstract

be an open, non-empty set. We denote its border with ∂is denoted by x = (x1, x.

Keywords

Abstract mixed problem; Boundary conditions; Finite element approximation; Homogeneous Dirichlet condition; Mixed finite elements; Neumann condition; Numerical tests; Slip condition; Stokes Equations

1.1 Stokes equations

be an open, non-empty set. We denote its border with ∂is denoted by x = (x1, xare denoted by

are denoted by

The scalar product of two vectors v, w are two tensors, we note

, we denote the (linearized) strain rate tensor by

, such that

   [1.1]

   [1.2]

   [1.3]

where

– μ > 0 is the dynamic viscosity,

–