A predictor corrector method for a Finite Element 1143829/   Finite Element Method for

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  • IN DEGREE PROJECT MATHEMATICS,SECOND CYCLE, 30 CREDITS

    , STOCKHOLM SWEDEN 2017

    A predictor corrector method for a Finite Element Method for the variable density Navier-Stokes equations

    ISABEL HAASLER

    KTH ROYAL INSTITUTE OF TECHNOLOGYSCHOOL OF ENGINEERING SCIENCES

  • A predictor corrector method for a Finite Element Method for the variable density Navier-Stokes equations ISABEL HAASLER Degree Projects in Scientific Computing (30 ECTS credits) Degree Programme in Applied and Computational Mathematics (120 credits) KTH Royal Institute of Technology year 2017 Supervisor at KTH: Johan Hoffman, Johan Jansson Examiner at KTH: Michael Hanke

  • TRITA-MAT-E 2017:64 ISRN-KTH/MAT/E--17/64--SE Royal Institute of Technology School of Engineering Sciences KTH SCI SE-100 44 Stockholm, Sweden URL: www.kth.se/sci

  • Abstract

    Constructing Finite Element Methods for the incompressible Navier-Stokes equations that are computationally feasible requires the use ofpressure segregation methods such as the predictor corrector scheme.From an algebraic point of view these methods correspond to a pre-conditioned Richardson iteration on the pressure Schur complement.In this work the theory is extended to a variable density model. Wesuggest a modified preconditioner and investigate its performance ex-perimentally.

  • Sammanfattning

    En prediktorkorrigeringsmetod fr en Finita Elementmetoden frNavier-Stokes-ekvationer med variabel densitet

    Konstruktion av Finita Elementmetoder fr de inkompressibla Navier-Stokes ekvationer som r berkningsmssigt genomfrbara krver an-vndning av trycksegregeringsmetoder ssom prediktorkorrigeringss-chemat. Frn en algebraisk synvinkel motsvarar dessa metoder enfrkonditionerad Richardsoniteration p tryck-Schur-komplementet. Idetta arbete utvidgas teorin till en variabel-densitet-modell. Vi fres-lr en modifierad preconditioner och undersker prestanda experi-mentellt.

  • Acknowledgements

    First of all, I would like to express my gratitude to my supervisors Jo-han Hoffman and Johan Jansson for their guidance, advice and inputto my work, offering comments on the report and giving me the op-portunity to visit the research group at the Basque Institute of AppliedMathematics (BCAM).

    Moreover, I want to thank everyone who I got the chance to meet atBCAM for their guidance, support and hospitality, especially Daniel,Massimiliano, Magaly, Christina and Andrea. You made my stay inBilbao a fruitful, inspirational and enjoyable time.

    Special thanks to my fellow student and friend Georg Neumllerwhom I shared this experience the past few months with, going throughgood and challenging times together.

    Finally and most importantly, I want to thank my parents for theirinvaluable support and encouragement for everything I do.

  • Contents

    1 Introduction 1

    2 Background 42.1 A stabilized Finite Element Method for the Navier-Stokes

    equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52.1.1 The incompressible Navier-Stokes equations . . . 52.1.2 Weak form . . . . . . . . . . . . . . . . . . . . . . . 62.1.3 Finite element formulation . . . . . . . . . . . . . 72.1.4 The algebraic system . . . . . . . . . . . . . . . . . 8

    2.2 Pressure segregation methods . . . . . . . . . . . . . . . . 102.2.1 Incremental projection method . . . . . . . . . . . 112.2.2 Predictor corrector method . . . . . . . . . . . . . 122.2.3 Implementation of a predictor corrector method . 13

    2.3 Arbitrary Lagrangian-Eulerian methods . . . . . . . . . . 142.3.1 Lagrangian vs. Eulerian formulation . . . . . . . . 142.3.2 ALE formulation . . . . . . . . . . . . . . . . . . . 15

    2.4 Two-phase flow . . . . . . . . . . . . . . . . . . . . . . . . 162.4.1 Model for two-phase flow . . . . . . . . . . . . . . 162.4.2 An Overview of level set methods for two-phase

    flow . . . . . . . . . . . . . . . . . . . . . . . . . . . 172.5 The FEniCS project . . . . . . . . . . . . . . . . . . . . . . 19

    3 The variable density model 203.1 The variable density Navier-Stokes equations . . . . . . . 20

    3.1.1 Finite Element formulation for the variable den-sity model . . . . . . . . . . . . . . . . . . . . . . . 21

    3.1.2 A parameter free turbulence model . . . . . . . . 243.2 Test cases . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24

    3.2.1 2D test case . . . . . . . . . . . . . . . . . . . . . . 25

  • CONTENTS

    3.2.2 3D test case . . . . . . . . . . . . . . . . . . . . . . 25

    4 A pressure segregation method for the variable density Navier-Stokes equations 294.1 Algebraic approach to predictor corrector methods . . . . 30

    4.1.1 The discretized system . . . . . . . . . . . . . . . . 304.1.2 Derivation of the Schur system . . . . . . . . . . . 314.1.3 Preconditioned Richardson iteration . . . . . . . . 324.1.4 A preconditioner leading to a predictor corrector

    scheme . . . . . . . . . . . . . . . . . . . . . . . . . 334.1.5 Predictor corrector scheme . . . . . . . . . . . . . 36

    4.2 Extension of the preconditioner to the variable densitymodel . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37

    5 Results of numerical experimentation 395.1 Maximal time step size k . . . . . . . . . . . . . . . . . . . 405.2 Convergence of Fixed point iteration . . . . . . . . . . . . 42

    5.2.1 Convergence without Schur preconditioning term 425.2.2 Convergence with Schur preconditioning term . . 44

    5.3 Accumulated number of fixed point iterations . . . . . . 475.4 Comparison with 3D simulations . . . . . . . . . . . . . . 52

    6 Discussion 586.1 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . 586.2 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . 596.3 Open questions and future work . . . . . . . . . . . . . . 59

    Bibliography I

    List of Figures V

    List of Tables VII

  • Chapter 1

    Introduction

    The Navier-Stokes equations are one of the foundation for engineeringapplications concerned with the description of fluids. Among othersin mechanical, chemical and biomedical engineering the simulationand analysis of the dynamics of fluids is a fundamental topic.

    As the Navier-Stokes equations are analytically solvable only un-der some special conditions and computational possibilities advancequickly, the development of efficient numerical solving strategies is ofgreat interest and a particularly active research area. One field in Com-putational Fluid Dynamics (CFD) is concerned with the developmentof Finite Element Methods (FEMs) for the Navier-Stokes equations.

    This approach is particularly challenging when the Reynolds num-ber is large, as vortices appear at a large variety of scales and resolvingall of them requires a massive amount of computational power thatexceeds even the capability of supercomputing clusters. A residual-based numerical stabilization of the FEM has been shown to serve as aparameter free turbulence model [14]. With this approach small scalevortices do not need to be resolved in full detail to extract certain aver-aged information about the fluid mechanics. Hence, FEM simulationsare viable even for high Reynolds numbers if computations are paral-lelized.

    The FEM for the Navier-Stokes equations results in large algebraicsystems, which should be solved with iterative methods, as these re-quire less memory and are more easily parallelizable. Unfortunately,it is troublesome to find well-scaling preconditioners for the mono-lithic problem [7]. For incompressible flow it is efficient to decouplethe velocity and pressure equations of the Navier-Stokes system and

    1

  • 2 CHAPTER 1. INTRODUCTION

    solve them separately. In order for the solutions to converge to themonolithic solution, the decoupled equations are solved iteratively inthe form of a fixed-point method. With a suitable preconditioner thisapproach has proven to be sufficiently efficient.

    However, there have not yet been many attempts to extend thismethod to the variable density incompressible Navier-Stokes equa-tions. A variable density model can be used to model the interactionof two fluids. Such a two-phase model is for example required for ap-plications in marine engineering. The efficient numerical modeling offloating structures with breaking waves, at high Reynolds numbers, isan ongoing challenge.

    The objective of this master thesis is to extend the iterative solv-ing strategy to an FEM for the variable density incompressible Navier-Stokes equations. In particular, we aim to find an effective precondi-tioner.

    This work is outlined as follows. Chapter 2 gives an introduction tothe mathematical background for the following work. After present-ing the Navier-Stokes equations a stabilized FEM is derived. Pressuresegregation methods are introduced and two schemes presented. Wedescribe two approaches to model interfaces, the interface tracking Ar-bitrary Lagrangian-Eulerian (ALE) method and the interface capturinglevel set method. A short introduction to the FEniCS project whichforms the computational framework for this master thesis is given.

    Our model for the variable density incompressible Navier-Stokesequations is portrayed in Chapter 3. A stabilized Finite Element Methodis developed and explained. Also, the test cases we base our numericalinvestigation on are presented.

    In Chapter 4 we follow an algebraic approach to predictor correc-tor methods. This allows us to derive a suitable preconditioner for theFEM with a constant but arbitrarily chosen density. With some addi-tional considerations and heuristics we extend the derived method tosuggest a preconditioner for the variable density model.

    The performance of the proposed pr