Scalable Efficient Methods for Incompressible Fluid-dynamics in Engineering Problems Open Access

Villa, Umberto Emanuele (2012)

Permanent URL:


Accurate and effective methods for the numerical solution of incompressible fluid dynamics is an old but still important challenging problem, as more and more complex problems in engineering biology, ecology, medicine, sport are tackled with computational methods.

In this thesis, we investigate efficient solvers for two important models that govern the motion of a fluid, the incompressible Navier-Stokes and the Brinkman equations. The former describes the motion of an incompressible fluid in either an open or closed domain. The latter is used for describing the dynamics in a matrix of an inhomogeneous porous media, alternating bubbles and open channels.

For the solution of the unsteady Navier-Stokes Equations, we move from the pressure correction algebraic factorization formerly proposed by Saleri, Veneziani (2005), and we introduce the incremental formulation of pressure corrected schemes. These schemes feature an intrinsic hierarchical nature, such that an accurate approximation of the pressure Schur complement is obtained by computing intermediate low-order guesses. When used as a splitting method instead of a preconditioner, the difference between the pressure at two successive correction steps provides a natural a-posteriori estimator with no additional computational cost. We consider the basics settings of the method and its more stable variants; we also discuss implementation details that make the method competitive for real interest problems.

For the solution of the Brinkman Equations, we follow the approach presented in Mardal, Winther (2011) to precondition symmetric saddle point problems in a Hilbert settings. More specifically, we first present a novel mixed formulation of the Brinkman problem, with improved stability properties, in which we introduce the flow's vorticity as additional unknown. Based on stability analysis of the problem in the H(curl)-H(div)-L2 norms, we derive a scalable block diagonal preconditioner which is optimal in the constant coefficient case.

Algorithms and preconditioners analysed in this thesis have been implemented in a parallel C++ code, using the finite element libraries LifeV and MFEM, and the linear algebra libraries Trilinos and HYPRE.
We emphasize the performance of the proposed algorithms in solving problems of practical interest, involving complex geometries and realistic flow conditions. Numerical experiments in 2D and 3D confirm the effectiveness of our approach showing good efficiency and parallel scalability properties of the solvers proposed.

Table of Contents

1 Introduction...1

1.1 The incompressible Navier-Stokes equations...2
1.2 The Brinkman Equations...6
1.3 Thesis outline...7

2 Discretization of the unsteady Navier-Stokes equations...13

2.1 Governing equations...14
2.2 Weak formulation and Galerkin Projection...17
2.3 Space discretization of the generalized Oseen Problem...21
2.4 Time discretization...26

2.4.1 Treatment of the non-linear convective term...30

2.5 A note on mass lumping for high order finite element...32

2.5.1 Mass lumping and orthogonal finite element basis...33
2.5.2 Accuracy of mass-lumped finite elements...35

3 Algebraic splittings and block preconditioners...41

3.1 Velocity-pressure splittings methods...42

3.1.1 Incremental formulation of splitting methods...50

3.2 The high order Yosida splitting...51
3.3 Algorithmic form of High Order Yosida schemes...54
3.4 Analysis of the pressure corrected splittings...55

3.4.1 Non-singularity and consistency...56
3.4.2 Stability analysis...58
3.4.3 Convergence analysis...62

3.5 Algebraic splitting as preconditioners...65

3.5.1 Block preconditioners and approximated Schur Complement operators...66
3.5.2 Spectral properties of algebraic splitting preconditioners...68
3.5.3 Comparison with the Cahouet-Chabard preconditioner...71
3.5.4 Comparison with the Least Squares Commutator preconditioner...72

4 Time Adaptivity...75

4.1 Time adaptivity for computational fluid-dynamics...75
4.2 Analysis of the incremental formulation of High Order Yosida schemes...77
4.3 Adaptation rule...82
4.4 A posteriori error estimators for the Navier-Stokes problem...84

4.4.1 Algebraic splitting based estimators...87
4.4.2 Preconditioned unsplit solvers estimators...88

4.5 Numerical results...90

4.5.1 Preliminary 2D results...90
4.5.2 3D Womersley test case...93
4.5.3 Sensitivity with respect to the mesh size...93
4.5.4 An adaptive 3D blood flow simulation...95

5 Implementation...101

5.1 Libraries and Software...104

5.1.1 LifeV...104
5.1.2 Trilinos...105
5.1.3 SuiteSparseQR...109

5.2 On the numerical solution of the discrete Laplacian with direct methods...112

5.2.1 Parallel performance results...113
5.2.2 The effect of the ordering strategy...115

5.3 Management of Block Operators in LifeV/Trilinos...117

5.3.1 Overview of the block linear algebra module...119

5.4 Scalability Results...123

5.4.1 Weak scalability test...126
5.4.2 Strong scalability test...131

6 The Brinkman Problem...139

6.1 Mixed formulation of the Brinkman Problem...141

6.1.1 Functional spaces and orthogonal decompositions...142
6.1.2 Weak formulation...144

6.2 Well-posedness of the mixed variational formulation...146
6.3 Discretization...153

6.3.1 Analysis of the discrete problem...154

6.4 Discretization error numerical results...157

6.4.1 Discretization error for constant coefficients...157
6.4.2 Discretization error for non-constant smooth coefficients...160
6.4.3 Discretization error for coefficients with jumps...160

6.5 Preconditioning...162

6.5.1 Augmented Lagragian formulation...167

6.6 Scalability results...169

6.6.1 Software and implementation details...170
6.6.2 Constant coefficient weak scalability test...171
6.6.3 The case of non-constant smooth coefficients...173
6.6.4 The case of coefficients with discontinuities...175

7 Conclusion...179

About this Dissertation

Rights statement
  • Permission granted by the author to include this thesis or dissertation in this repository. All rights reserved by the author. Please contact the author for information regarding the reproduction and use of this thesis or dissertation.
  • English
Research field
Committee Chair / Thesis Advisor
Committee Members
Last modified

Primary PDF

Supplemental Files