Scalable Efficient Methods for Incompressible Fluid-dynamics in Engineering Problems Open Access
Villa, Umberto Emanuele (2012)
Abstract
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
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