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Grid generation method.pdf
Preface to the Third Edition
Contents
1 General Considerations
1.1 Introduction
1.2 General Concepts Related to Grids
1.2.1 Grid Cells
1.2.2 Requirements Imposed on Grids
1.2.3 Grid Classes
1.3 Methods for Grid Generation
1.3.1 Mapping Methods
1.3.2 Methods for Unstructured Grids
1.4 Big Codes
1.4.1 Interactive Systems
1.4.2 New Techniques
1.5 Comments
References
2 Coordinate Transformations
2.1 Introduction
2.2 General Notions and Relations
2.2.1 Jacobi Matrix
2.2.2 Tangential Vectors
2.2.3 Normal Vectors
2.2.4 Representation of Vectors Through the Base Vectors
2.2.5 Metric Tensors
2.2.6 Cross Product
2.3 Relations Concerning Second Derivatives
2.3.1 Christoffel Symbols
2.3.2 Differentiation of the Jacobian
2.3.3 Basic Identity
2.4 Conservation Laws
2.4.1 Scalar Conservation Laws
2.4.2 Vector Conservation Laws
2.5 Time-Dependent Transformations
2.5.1 Reformulation of Time-Dependent Transformations
2.5.2 Basic Relations
2.5.3 Equations in the Form of Scalar Conservation Laws
2.5.4 Equations in the Form of Vector Conservation Laws
2.6 Comments
References
3 Grid Quality Measures
3.1 Introduction
3.2 Curve Geometry
3.2.1 Basic Curve Vectors
3.2.2 Curvature
3.2.3 Torsion
3.3 Surface Geometry
3.3.1 Surface Base Vectors
3.3.2 Metric Tensors
3.3.3 Second Fundamental Form
3.3.4 Surface Curvatures
3.3.5 Curvatures of Discrete Surfaces
3.4 Metric-Tensor Invariants
3.4.1 Algebraic Expressions for the Invariants
3.4.2 Geometric Interpretation
3.5 Characteristics of Grid Lines
3.5.1 Sum of Squares of Cell Edge Lengths
3.5.2 Eccentricity
3.5.3 Curvature
3.5.4 Measure of Coordinate Line Torsion
3.6 Characteristics of Faces of Three-Dimensional Cells
3.6.1 Cell Face Skewness
3.6.2 Face Aspect-Ratio
3.6.3 Cell Face Area Squared
3.6.4 Cell Face Warping
3.7 Characteristics of Grid Cells
3.7.1 Cell Aspect-Ratio
3.7.2 Square of Cell Volume
3.7.3 Cell Area Squared
3.7.4 Cell Skewness
3.7.5 Characteristics of Nonorthogonality
3.7.6 Grid Density
3.7.7 Characteristics of Deviation from Conformality
3.7.8 Grid Eccentricity
3.7.9 Measures of Grid Warping and Grid Torsion
3.7.10 Quality Measures of Simplexes
3.8 Comments
References
4 Stretching Method
4.1 Introduction
4.2 Formulation of the Method
4.3 Theoretical Foundation
4.3.1 Model Problems
4.3.2 Basic Majorants
4.4 Basic Intermediate Transformations
4.4.1 Basic Local Stretching Functions
4.4.2 Basic Boundary Contraction Functions
4.4.3 Other Univariate Transformations
4.4.4 Construction of Basic Intermediate Transformations
4.4.5 Multidirectional Equidistribution
4.5 Comments
References
5 Algebraic Grid Generation
5.1 Introduction
5.2 Transfinite Interpolation
5.2.1 Unidirectional Interpolation
5.2.2 Tensor Product
5.2.3 Boolean Summation
5.3 Algebraic Coordinate Transformations
5.3.1 Formulation of Algebraic Coordinate Transformation
5.3.2 General Algebraic Transformations
5.4 Lagrange and Hermite Interpolations
5.4.1 Coordinate Transformations Based on Lagrange Interpolation
5.4.2 Transformations Based on Hermite Interpolation
5.5 Control Techniques
5.6 Transfinite Interpolation from Triangles and Tetrahedrons
5.7 Drag and Sweeping Methods
5.8 Comments
References
6 Grid Generation Through Differential Systems
6.1 Introduction
6.2 Elliptic Equations
6.2.1 Laplace Systems
6.2.2 Poisson Systems
6.2.3 Other Elliptic Equations
6.3 Biharmonic Equations
6.3.1 Formulation of the Approach
6.3.2 Transformed Equations
6.4 Orthogonal Systems
6.4.1 Derivation from the Condition of Orthogonality
6.4.2 Multidimensional Equations
6.5 Hyperbolic and Parabolic Systems
6.5.1 Specification of Aspect Ratio
6.5.2 Specification of Jacobian
6.5.3 Parabolic Equations
6.5.4 Hybrid Grid Generation Scheme
6.6 Grid Equations for Nonstationary Problems
6.6.1 Method of Lines
6.6.2 Moving-Grid Techniques
6.6.3 Time-Dependent Deformation Method
6.7 Comments
References
7 Variational Methods
7.1 Introduction
7.2 Calculus of Variations
7.2.1 General Formulation
7.2.2 Euler--Lagrange Equations
7.2.3 Convexity Condition
7.2.4 Functionals Dependent on Metric Elements
7.2.5 Functionals Dependent on Tensor Invariants
7.3 Integral Grid Characteristics
7.3.1 Dimensionless Functionals
7.3.2 Dimensionally Heterogeneous Functionals
7.3.3 Functionals Dependent on Second Derivatives
7.4 Adaptation Functionals
7.4.1 One-Dimensional Functionals
7.4.2 Multidimensional Approaches
7.5 Functionals of Attraction
7.5.1 Lagrangian Coordinates
7.5.2 Attraction to a Vector Field
7.5.3 Jacobian-Weighted Functional
7.6 Energy Functionals of Harmonic Function Theory
7.6.1 General Formulation of Harmonic Maps
7.6.2 Application to Grid Generation
7.6.3 Relation to Other Functionals
7.7 Combinations of Functionals
7.7.1 Natural Boundary Conditions
7.8 Comments
References
8 Curve and Surface Grid Methods
8.1 Introduction
8.2 Grids on Curves
8.2.1 Formulation of Grids on Curves
8.2.2 Grid Methods
8.3 Formulation of Surface Grid Methods
8.3.1 Mapping Approach
8.3.2 Associated Metric Relations
8.4 Beltramian System
8.4.1 Beltramian Operator
8.4.2 Surface Grid System
8.5 Interpretations of the Beltramian System
8.5.1 Variational Formulation
8.5.2 Harmonic-Mapping Interpretation
8.5.3 Formulation Through Invariants
8.5.4 Formulation Through the Surface Christoffel Symbols
8.6 Control of Surface Grids
8.6.1 Control Functions
8.6.2 Monitor Approach
8.6.3 Control Through Variational Methods
8.6.4 Orthogonal Grid Generation
8.7 Hyperbolic Method
8.7.1 Hyperbolic Governing Equations
8.8 Comments
References
9 Comprehensive Method
9.1 Introduction
9.2 Hypersurface Geometry and Grid Formulation
9.2.1 Hypersurface Grid Formulation
9.2.2 Monitor Hypersurfaces
9.2.3 Metric Tensors
9.2.4 Relations Between Metric Elements
9.2.5 Christoffel Symbols
9.3 Functional of Smoothness
9.3.1 Formulation of the Functional
9.3.2 Geometric Interpretation
9.3.3 Euler--Lagrange Equations
9.3.4 Equivalent Forms
9.3.5 Inverted Beltrami Equations
9.4 Role of the Mean Curvature
9.4.1 Mean Curvature and Inverted Beltrami Grid Equations
9.4.2 Mean Curvature and Rate of Grid Clustering
9.4.3 Diffusion Functional
9.4.4 Dimensionless Functionals
9.5 Formulation of Comprehensive Grid Generator
9.5.1 Formulation of Control Metrics
9.5.2 Energy and Diffusion Functionals
9.5.3 Beltrami and Diffusion Equations
9.5.4 Inverted Beltrami and Diffusion Equations
9.5.5 Specification of Individual Control Metrics
9.5.6 Control Metrics for Generating Grids with Balanced Properties
9.6 Comments
References
10 Numerical Implementations of Comprehensive Grid Generators
10.1 One-Dimensional Equation
10.1.1 Numerical Algorithm
10.2 Multidimensional Finite Difference Algorithms
10.2.1 Parabolic Simulation
10.2.2 Two-Dimensional Equations
10.2.3 Three--Dimensional Problem
10.3 Spectral Element Algorithm
10.4 Finite Element Method
10.5 Inverse Matrix Method
10.6 Method of Minimization of Energy Functional
10.6.1 Generation of Fixed Grids
10.6.2 Adaptive Grid Generation
10.7 Parallel Mesh Generation
References
11 Control of Grid Properties
11.1 Grid Adaptation to Function Values
11.1.1 Control Operator
11.1.2 Grid Equations
11.2 Grid Generation with Node Clustering Near Isolated Points
11.3 Grids with Node Clustering Near Curves and Surfaces
11.4 Generation of Grids with Node Clustering in the Zones
11.4.1 Control Metric of a Monitor Surface
11.4.2 Spherical Control Metric
11.5 Application of Layer-Type Functions to Grid Codes
11.5.1 Specification of Basic Functions
11.5.2 Numerical Grids Aligned to Vector-Fields
11.5.3 Application to Grid Clustering
11.6 Generation of Multi-block Smooth Grids
11.6.1 Approaches to Smoothing Grids
11.6.2 Computation by Interpolation
References
12 Unstructured Methods
12.1 Introduction
12.2 Methods Based on the Delaunay Criterion
12.2.1 Dirichlet Tessellation
12.2.2 Incremental Techniques
12.2.3 Approaches for Insertion of New Points
12.2.4 Two-Dimensional Approaches
12.2.5 Constrained Form of Delaunay Triangulation
12.2.6 Point Insertion Strategies
12.2.7 Surface Delaunay Triangulation
12.2.8 Three-Dimensional Delaunay Triangulation
12.3 Advancing-Front Methods
12.3.1 Procedure of Advancing-Front Method
12.3.2 Strategies for Selecting Out-of-Front Vertices
12.3.3 Grid Adaptation
12.3.4 Advancing-Front Delaunay Triangulation
12.4 Meshing by Quadtree-Octree Decomposition
12.5 Three-Dimensional Prismatic Grid Generation
12.6 Comments
References
13 Applications of Adaptive Grids to Solution of Problems
13.1 Application to Unsteady Gas Dynamics Problems
13.1.1 Numerical Examples
13.2 Applications to Numerical Simulations of Tsunami Run-Up
13.2.1 Mathematical Model
13.2.2 Dynamically Adaptive Numerical Grid
13.2.3 Equations in Dynamic Curvilinear Coordinates
13.2.4 Numerical Algorithm
13.2.5 Some Results of Calculations
13.3 Application to Singularly-Perturbed Equations
13.3.1 Numerical Algorithm
13.4 Problem of Heat Transfer in Plasmas
13.4.1 The Tokamak Edge Region
13.4.2 Computations on Balanced Grids
13.5 Evaluations of Temperature-Profile Discrepancies
13.5.1 Mathematical Model for the Interaction of Heat Wave with Thermocouple
13.5.2 Generation of Adaptive Grid
13.5.3 Results of Numerical Experiments
13.6 Numerical Modeling of Nanopore Formation in Aluminium Oxide Films
13.6.1 Introduction
13.6.2 Mathematical Model
13.6.3 Numerical Approximation
13.6.4 Grid Generation
13.6.5 Numerical Experiments
13.7 Grids for Boundary Immersing Methods
13.7.1 Introduction
13.7.2 Formulation of the Method
13.7.3 Determination of Boundary Cells
13.7.4 Algorithm for Determining Interior Cells
13.7.5 Mesh Adaptation
References
Index
Scientific Computation
Vladimir D. Liseikin
Grid
Generation
Methods
Third Edition
Grid Generation Methods
Scientific Computation
Editorial Board
J.-J. Chattot, Davis, CA, USA
P. Colella, Berkeley, CA, USA
R. Glowinski, Houston, TX, USA
M.Y. Hussaini, Tallahassee, FL, USA
P. Joly, Le Chesnay, France
D.I. Meiron, Pasadena, CA, USA
O. Pironneau, Paris, France
A. Quarteroni, Lausanne, Switzerland
and Politecnico of Milan, Milan, Italy
M. Rappaz, Lausanne, Switzerland
R. Rosner, Chicago, IL, USA
P. Sagaut, Paris, France
J.H. Seinfeld, Pasadena, CA, USA
A. Szepessy, Stockholm, Sweden
M.F. Wheeler, Austin, TX, USA
More information about this series at http://www.springer.com/series/718
Vladimir D. Liseikin
Grid Generation Methods
Third Edition
123
Vladimir D. Liseikin
Institute of Computational Technologies SB
RAS
Novosibirsk
Russia
and
Novosibirsk State University
Novosibirsk
Russia
ISSN 1434-8322
Scientific Computation
ISBN 978-3-319-57845-3
DOI 10.1007/978-3-319-57846-0
ISSN 2198-2589
(electronic)
ISBN 978-3-319-57846-0
(eBook)
Library of Congress Control Number: 2017939626
1st edition: © Springer-Verlag Berlin Heidelberg 1999
2nd edition: © Springer Science+Business Media B.V. 2010
3rd edition: © Springer International Publishing AG 2017
This work is subject to copyright. All rights are reserved by the Publisher, whether the whole or part
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recitation, broadcasting, reproduction on microfilms or in any other physical way, and transmission
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The use of general descriptive names, registered names,
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The publisher, the authors and the editors are safe to assume that the advice and information in this
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Preface to the Third Edition
Grid generation codes represent an indispensable tool for solving field problems in
applied mathematics, mechanics, physics, and other areas of practical applications.
Despite the considerable success achieved in grid generation technologies, devel-
opment of more efficient and sophisticated algorithms and computer codes for
generating grids still remains an important problem. Serious difficulties arise in grid
generation in domains with complicated boundary geometries, specifically, with
discretely defined boundary segments and in the case when grids have to be adapted
to solution singularities, such as boundary and interior layers, shocks, detonation
waves, combustion fronts, high-speed jets, and phase transition zones. A promising
tool to deal with the numerical problems having such singularities is adaptive grid
generation technology. With increasing complexity of the physical problems, there
is an increased need for more reliable, robust, and fully automated grid generation
codes which enable one to generate suitable meshes in a uniform “black box” mode,
without human interaction. The development of such grid systems is a challenging
problem in computational physics and applied mathematics.
Grid technology still remains a rapidly advancing field of computational and
applied mathematics. New achievements are being added by the creation of more
sophisticated techniques, modification of the available methods, and implementa-
tion of more subtle tools as well as the results of the classical theories of differential
equations, calculus of variations, and Riemannian geometry in the formulation of
grid models and analysis of grid properties. Therefore, there is a clear need of
students, researchers, and practitioners in the field of applied mathematics and
industry for the creation of new books and/or updated editions of the available
books which will complement the existing ones, providing a description of current
developments relating to grid methods, grid codes, and their applications to the
solving of actual problems.
This third edition of the monograph “Grid Generation Methods” is significantly
expanded with new material that discusses recent advances in grid generation
technology. It includes a description of updated grid generation methods, which
were partly presented in the former monograph of the author, as well as new
adaptive approaches for structured and unstructured grids and numerical algorithms
v
vi
Preface to the Third Edition
for their generation. Special attention is paid to a review of those promising
approaches and methods which have been developed recently and/or have not been
sufficiently covered in other monographs. In particular, this book includes an
application for generating grids for immersed boundary methods. It also describes a
stretching method adjusted to the numerical solution of singularly perturbed
equations having large-scale solution variations, e.g.,
those modeling high-
Reynolds-number flows. A number of functionals related to conformality, orthog-
onality, energy, and alignment are described. This book includes differential and
variational techniques for generating uniform, conformal, and harmonic coordinate
transformations on hypersurfaces for the development of a comprehensive approach
to the construction of both fixed and adaptive grids in the interior and on the
boundary of domains in a unified manner. The monograph is also concerned with
the description of all essential grid quality measures such as skewness, curvature,
torsion, angle and length values, and conformality. It gives a more detailed and
practice-oriented description of control metrics for providing the generation of
adaptive, field-aligned, and balanced numerical grids by means of the numerical
solution of inverted Beltrami and diffusion equations in the control metrics. Some
numerical algorithms are described for generating surface and domain grids. One
more new feature of this book is the implementation of adaptive grid technology to
the numerical solution of problems in mechanics, physics, fluids, plasmas, and
nanotechnologies. Emphasis is placed on mathematical formulations, explanations,
and examples of various aspects of grid generation and their applications.
This book will introduce a reader to structured and unstructured grid methods, as
well as automated technologies for the generation of adaptive grids for the
numerical solution of applied problems with complicated domain segments and
complicated solution structures. These technologies are based on advanced alge-
braic, elliptic, variational, Delaunay, advancing-front, and quad–octree methods, as
well as on the methods of finite differences and volumes. The technologies are
indispensable for the numerical solution of differential equations, modeling various
complex physical processes in energetics, ecology, industry, as well as the medical
sphere. Furthermore, this book includes chapters devoted to the implementation of
comprehensive grid methods into numerical codes and to the application of the
codes to the numerical solution of a range of mechanical, fluid, and plasma-related
problems. The new and fast-developing computational tools discussed throughout
this book enable a detailed analysis of real-world problems that simply lie beyond
the reach of traditional methods.
The major area of attention of this book is grid-mapping techniques. In addition,
however,
the author has also included an elementary introduction to basic
unstructured approaches to mesh generation. A more detailed description of
unstructured mesh techniques and corresponding aspects related to parallel pro-
cessing, mesh quality enhancement, and mesh modification and optimization can be
found in the books of the leading experts on these technologies: Computational
Grids: Adaptation and Solution Strategies by G.F. Carey (1997), Delaunay
Triangulation and Meshing by P.-L. George and H. Borouchak (1998), Mesh
Generation Application to Finite Elements by P.J. Frey and P.-L. George (2008),
Preface to the Third Edition
vii
and Finite Element Mesh Generation by D.S.H. Lo (2015). These books, though,
do not give a detailed introduction to advanced mapping approaches developed in
recent years. Thus, the current monograph and these books complement each other,
presenting a comprehensive description of all the popular grid generation approa-
ches. As grid generation methodology is well on its way to becoming a formal
subject in university curricula, the books mentioned and the current book taken
together will provide materials fully sufficient to support a one-year university
course related to structured and unstructured mesh technologies.
Since grid technology has a widespread application across nearly all field
problems, this new edition of the monograph will be of significant interest to a
broad range of readers: teachers, students, researchers, and practitioners in applied
mathematics, mechanics, physics, and other areas of application. In addition, it
could be used as a textbook for advanced undergraduates or for first-year post-
graduate students or as a tutorial for mathematicians, engineers, and scientists who
are engaged in the computation of equations in multidimensional domains with
complicated boundary geometries.
Chapter 1 of this book provides a general introduction to the subject of grids. It
gives an outline of structured, unstructured, hybrid, overlapping, and composite
grids. The chapter delineates some of the basic classes of methods, in particular
manual or semiautomatic methods, mapping methods, and unstructured methods.
The chapter also includes a description of various types of grid topology and
touches on certain issues of comprehensive grid codes.
Chapter 2 deals with several mathematical relations that are necessary only for
the generation of grids by means of the mapping approach and which are connected
with and derived from the metric tensors of coordinate transformations. As an
example of an application of these relations, the chapter presents a technique aimed
at obtaining conservation-law equations in new fixed or time-dependent coordi-
nates. In the procedures described, the deduction of the expressions for the trans-
formed equations is based only on the formula for the differentiation of the Jacobian
of the coordinate transformations.
Very important issues of grid generation, connected with a description of grid
quality measures in forms suitable for formulating grid techniques and efficiently
analyzing the necessary mesh properties, are discussed in Chap. 3. The definitions
of the grid quality measures are based on the metric tensors and on the relations
between the metric elements considered in Chap. 2. Special attention is paid to the
invariants of the metric tensors, which are the basic elements for the definition of
many important grid quality measures. Clear algebraic and geometric interpreta-
tions of the invariants are presented.
Chapter 4 describes a stretching method based on the application of special
nonuniform stretching coordinates in the regions of large variation of the solution.
The use of stretching coordinates is extremely effective for the numerical solution
of problems with boundary and interior layers. The chapter acquaints the reader
with various types of singularity arising in solutions to equations with a small
parameter affecting the higher derivatives. The solutions of these equations undergo
large variations in very small boundary and interior zones, called boundary or
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