Table Of ContentLecture Notes
in Computational Science 14
and Engineering
Editors
M. Griebel, Bonn
D. E. Keyes, Norfolk
R. M. Nieminen, Espoo
D. Roose, Leuven
T. Schlick, New York
Springer-V erlag Berlin Heidelberg GmbH
Erik Dick
Kris Riemslagh
Jan Vierendeels
Editors
Multigrid Methods VI
Proceedings of the Sixth European Multigrid Conference
Held in Gent, Belgium, September 27-30, 1999
With 107 Figures
Springer
Editors
Erik Dick
Kris Riemslagh
Ian Vierendeels
Department of Flow, Heat and Combustion Mechanics
University of Gent
Sint -Pietersnieuwstraat 41
9000 Gent, Belgium
e-mail:
[email protected]
[email protected]
j an. [email protected]
Cataloging-in-Publication Data applied for
Die Deutsche Bibliothek -CIP-Einheitsaufnahme
Multigrid methods VI : proeeedings of the Sixth European Multigrid
Conferenee. held in Gent. Belgium. September 17 -30. 19991 Erik Dick
... (ed.). -Berlin; Heidelberg; New York; Bareelona; Hong Kong;
London ; Milan ; Paris; Singapore; Tokyo: Springer. 1000
(Lecture notes in eomputational ocienee and engineering ; 14)
ISBN 978-3-540-67157-2 ISBN 978-3-642-58312-4 (eBook)
DOI 10.10071978-3-642-58312-4
Front cover: paper by G. Cam~, G. Carte, H. Guillard, S. Lanteri
Mathematics Subject Classification (2000): 65M55 and 65N55
ISSN 1439-7358
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Preface
This volume contains 39 of the papers presented at the Sixth European Multi
grid Conference, held in Gent, September 27-30, 1999. There were 61 regis
tered participants from 12 different countries and 50 presentations were given.
Earlier European Multigrid Conferences were held in Cologne in 1981 and
1985, in Bonn in 1990, in Amsterdam in 1993 and in Stuttgart in 1996. The
next European Multigrid Conference is planned for 2002 in Heidelberg.
The Sixth European Multigrid Conference was organized by the Depart
ment of Flow, Heat and Combustion Mechanics from Ghent University. The
topics treated on the conference covered all aspects of Multigrid Methods: the
ory, analysis, computer implementation, applications in the fields of physics,
chemistry, fluid mechanics, structural mechanics, magnetism, ...
Gent, December 1999
Erik Dick, Kris Riemslagh, Jan Vierendeels
Table of Contents
Invited Contributions. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
Multigrid Strategies for CFD Problems on Non-Structured Meshes .... 1
G. Carre, G. Carte, H. Guillard, S. Lanteri
Detection of Strong Coupling in Algebraic Multigrid Solvers. . . . . . . . .. 11
T.F. Chan, P. Vanek
Steady Euler Solutions in O(N) Operations. . . . . . . . . . . . . . . . . . . . . . . .. 24
B. van Leer, D. Darmofal
The Acceleration of Multigrid Convergence by Recombination Techniques 34
C. W. Oosterlee, R. Wienands, T. Washio, F.J. Gaspar
Other Contributions ...................................... 44
Turbomachinery Computations on Parallel Computers Using a Multi-
grid Method. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 44
A. Arnone, P. Boncinelli, M. Marconcini
Approximate Schur Complement Multilevel Methods for General Sparse
Systems.. . . .. . ... . . .. . . .. . . .. .. .. . . . . . . .. .. . .. . . ... . .. .. .. . . .. 52
M. Benzi, M. DeLong
A Multigrid Approach to the Optimal Control of Solid Fuel Ignition
Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 59
A. Borzi, I<. Kunisch, M. Vanmaele
Convergent Adaptive Wavelet Methods for the Stokes Problem ....... 66
S. Dahlke, R. Hochmuth, I<. Urban
A Finite-ElementjEquivalent-Circuit Two-Level Method for Magnetic
Field Simulations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 73
H. De Gersem, S. Vandewalle, I<. Hameyer
Some Aspects of Multigrid for Mixed Discretizations ................ 80
J.E. Dendy, Jr., J.D. Moulton
Fixed and Adaptive Cache Aware Algorithms for Multigrid Methods 87
C. C. Douglas, J. Hu, W. Karl, M. Kowarschik, U. Rude, C. Weift
An Adaptive-Smoothing Multigrid Method for the Navier-Stokes Equa-
tions ............. . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . .. 94
D. Drikakis, O. Iliev, D. Vassileva
VIII Table of Contents
Multigrid on Massively Parallel Architectures ....................... 101
R.D. Falgout, J.E. Jones
Experiences with Multigrid-Prolongation for Two-Equation 'furbu-
lence Models in Flows with High Reynolds Numbers ................. 108
J. Fassbender
Additive Aspects of Hybrid Multigrid/Domain Decomposition Solution
of Fluid Flow Problems on Parallel Computers ...................... 115
L. Fournier, S. Lanteri
Biharmonic and Bi-Helmholtz Type Scattered Data Interpolation using
Quadtrees and Multigrid Technique ............................... 122
C. Gaspar
Algebraic Multigrid for Solving Electromechanical Problems .......... 129
M. K altenbacher, S. Reitzinger
Multigrid Methods for Mortar Finite Elements ...................... 136
R.H. Krause, B.I. Wohlmuth
The Sparse-Grid Combination Technique Applied to Time-Dependent
Advection Problems ............................................. 143
B. Lastdrager, B. Koren, J. Verwer
Multigrid Acceleration in a 3D Navier-Stokes Solver Using Unstruc-
tured Hexahedral Meshes with Adaptation ......................... 150
B. Leonard, A. Patel, C. Hirsch
Analysis of Several Multigrid Implicit Algorithms for the Solution of
the Euler Equations on Unstructured Meshes ....................... 157
I. Lepot, P. Geuzaine, F. Meers, J.A. Essers, J.M. Vaassen
Efficient Large-Eddy Simulations of Compressible Flows
Using Multigrid ................................................. 164
B. Lessani, S. Smirnov, C. La cor, T. Baelmans, J. Meyers
Application of Multigrid in Two-Equation 'furbulence Modelling ...... 171
B. Merci, J. Steelant, E. Dick
Evaluation of Multigrid as a Solver for Stress Analysis Problems in
Semiconductor Process Simulation ................................ 179
S. Mijalkovic
Multilevel Techniques for the Solution of the Inverse Problem of Elec-
trocardiography ................................................. 186
M. Mohr, U. Rude
Table of Contents IX
Robust Multigrid Algorithms for 3D Elliptic Equations on Structured
Grids .......................................................... 193
R.S. Montero, M. Prieto, I.M. Llorente, F. Tirado
A Multigrid Newton-Krylov Solver for Non-linear Systems ........... 200
V.A. Mousseau, D.A. Knoll, w.J. Rider
Estimation of the Condition Number of Additive Preconditioners on
Tensor Product Grids ........................................... 207
C. Pflaum
Multigrid Algorithms for the Tracking of Electron Beams ............ 214
C. Poplatl, U. van Rienen
Stable Semi-Iterative Smoother for Cascadic and Multigrid Algorithms 221
V. V. Shaidurov, G. Timmermann
Scalability of Parallel Multigrid Adaption .......................... 228
J. Stiller, W.E. Nagel, U. Fladrich
From Molecular Dynamics to Continuum Models .................... 235
S. Ta'asan
Multigrid With an Immersed Interface ............................. 242
J.M. Thijssen
Analysis of a Non-Standard Multigrid Preconditioner by Spectral Por-
trait Computation .............................................. 249
X. Vasseur
Treatment of All Speed Flows and High Aspect Ratios in CFD Appli-
cations ........................................................ 256
J. Vierendeels, K Riemslagh, B. Merci, E. Dick
On the Algebraic Construction of Multilevel Transfer Operators (for
Convection-Diffusion-Reaction Equations) .......................... 264
C. Wagner
Upwind Prolongations for a Highly-Unstructured Euler Solver ........ 271
K WarendorJ, U. Kuster, R. Ruhle
Fourier Analysis for Krylov Subspace Acceleration of Multigrid with
Application to 3D Anisotropic Problems ........................... 278
R. Wienands, C. W. Oosterlee
Parallel Adaptively Refined Sparse Grids ........................... 285
C. Zumbusch
Author Index .............................................. 293
Multigrid Strategies for CFD Problems on
Meshes
~on-Structured
Gilles Carre , Gilles Carte , Herve Guillard , and Stephane Lanteri
INRIA Centre de Sophia Antipolis
BP 93,06902 Sophia Antipolis Cedex
France
Abstract. For solving a problem on a fine mesh, the multigrid technology requires
the definition of coarse levels, coarse grid operators and inter-grid transfer operators.
For non-structured 3-D meshes in CFD, two major MG techniques have emerged
in the last years. The first one relies on the use of non-nested triangulations while
the second technique is associated to finite volume discretization and agglomera
tion/aggregation techniques. In this paper, we first present some automatic ways
to coarsen 3-D meshes and show that these geometrical methods result in efficient
multigrid algorithms. Then, we briefly describe the volume agglomeration method
and shows an example of its application in a 3-D industrial CFD code.
1 Introduction
In contrast to structured grids, the use of unstructured meshes offers some
major advantages in the mesh generation process. Moreover, it allows a larger
flexibility in adapting the mesh to complex geometries and solutions. As a
consequence, this type of data representation is now a common tool in Com
putational Fluid Dynamics. However, this approach also requires sophisti
cated discretization methods and solution algorithms. As a matter of fact,
classical discretization schemes and solvers using the regularity of the mesh
may fail or become less efficient on non-structured meshes. This is in par
ticular the case for Multigrid methods that were originally formulated for
structured grids. To run efficiently on non-structured meshes, the solution
algorithms have to be adapted to the irregularity of the mesh. In structured
MG algorithms, the building blocks of the methods are the inter-grid transfer
and coarse grid operators. Unstructured multigrid algorithms add the addi
tional difficulty of defining the coarse levels. This paper will present some
approaches to solving this difficulty in the context of industrial three dimen
sional problems.
In the past ten years, two main MG techniques have appeared in CFD on
non-structured meshes. The first one, already applied in structural mechan
ics, relies on the use of non-nested triangulations. The second technique is
specific to CFD and is associated to finite volume discretization and agglom
eration/ aggregation techniques. For the first class of methods that explicitly
E. Dick et al. (eds.), Multigrid Methods VI
© Springer-Verlag Berlin Heidelberg 2000
2 G. Carre et al.
define a hierarchy of grids, an important practical problem is to construct
a sequence of grids with different resolutions. A coarse to fine path can be
used i.e fine levels are generated by successive refinements starting from a
given coarse mesh. While this approach possesses the advantages of simplic
ity, it requires that the coarse level is sufficiently large to adequately represent
the geometry and that the successive refinements do not introduce artificial
features in the geometry (for instance sharp edges). Moreover, in three di
mensions, successive refinements of tetraedra results in a loss of the quality
of the mesh in such a way that the overall quality of the fine mesh can be
rather poor. These reasons explain why an efficient mutigrid strategy cannot
be entirely based on successive refinements and that these methods have to
be complemented by other techniques to generate the coarse meshes.
An alternative is to generate all the mesh levels independently. However,
this method places an excessive burden on the mesh generation step with
the result that industrial users are not able to use multigrid in a production
setting. Thus for practical purpose, one has to derive strategies for the auto
matic construction of coarse grids from a given non-structured one. We will
discuss several ways to perform this task. While the problem in two dimen
sion appears to be relatively simple, this is not the case in a three dimensional
setting and practical algorithms have appeared only very recently. We will
review some of them with a particular emphasis on the node nested strategy
first advocated in [4]. Then we will present one example of coarse mesh gen
eration and numerical computation of three dimensional flows using this last
method.
As an alternative to pure geometrical methods, one can consider the vol
ume agglomeration MG technique of Lallemand & Dervieux [10] based on
finite volume discretisations. This method appears as one of the most suc
cessful in CFD and for hyperbolic problems compares very favorably to ge
ometrical methods. We will discuss its implementation in a 3-D industrial
code and consider the extension of the Volume Agglomeration MG technique
to second-order discretisations or elliptic equations. This important topic is
still under development and have not received a definite answer [8], [5], [6].
2 Geometrical methods
In this section, we consider methods that explicitely define a hierarchy of grids
and first discuss the usefulness in CFD of methods based on mesh refinement
algorithms.
2.1 Nested mesh approach
n =
Let be a bounded domain of JRn(n 2,3), and consider a coarse triangu
lation ~ of this domain defined by the set of nodes N1. Associated to this
