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Springer - Computational Differential Geometry Approach to Grid Generation - 2nd Edition - 2007

V.D. Liseikin
A Computational
Differential
Geometry Approach
to Grid Generation
Second Edition
With 81 Figures
Including 3 Color Figures

Preface to the Second Edition
This second edition of A Computational Differential Geometry Approach to
Grid Generation is significantly expanded by new material that centers on
the recent advances in grid generation technology based on the numerical solution
of Beltrami and diffusion equations in monitor metrics. It gives a more
detailed and practice-oriented description of the monitor metrics for providing
the generation of adaptive, field-aligned, and balanced numerical grids.
New finite-difference codes are described for generating both structured and
unstructured surface and domain grids. Numerous applications of the codes
for the generation of numerical grids with individual and balanced properties
in surfaces and domains, in particular, in the tokamak-edge region are
demonstrated. The new edition also boasts examples of the implementations
of the grid generation codes in the codes for the numerical investigations of
gas-dynamics and magnetized plasmas problems.
Grid technology, which has had a significant impact on the efficiency of
numerical codes, remains a rapidly advancing field of computational physics
and applied mathematics. New achievements are being added by the creation
of more sophisticated techniques, modification of the available methods, and
implementation of more subtle tools as well as the results of the theories of
differential equations, calculus of variations, and Riemannian geometry in the
formulation of grid models and analysis of grid properties.
The development of comprehensive differential and variational grid generation
techniques reviewed in the monographs of J.F. Thompson, Z.U.A.Warsi,
and C.W. Mastin, P. Knupp, and S. Steinberg, and V.D. Liseikin has been
largely based on a popular concept in accordance with which a grid model
realizing the required grid properties should be formulated through a linear
combination of basic and control grid operators with weights. A typical,
basic grid operator is the operator responsible for the well-posedness of the
grid model and construction of unfolding grids, e.g., the Laplace equations
(generalized Laplace equations referred also to as second-order Beltrami equations)
or the function of grid smoothness, which produces fixed non-folding
grids while grid clustering is controlled by source terms in differential grid
formulations or by an adaptation function in variational models. However,
such a formulation does not obey the fundamental invariance laws with respect
to parameterizations of physical geometries and frequently results in
VI Preface to the Second Edition
cumbersome governing grid equations. Besides this, the choice of the weight
and control functions for providing well-posedness, grid non-degeneracy, and
adaptation is largely based on unreliable theoretical assumptions borrowed
from one-dimensional models.
The current book revises this popular concept and pursues a more updated
and somewhat revolutionary one based on the general fact that an
arbitrary one-to-one, smooth multidimensional coordinate transformation deriving
a numerical grid in a domain or on a surface is realized by a solution of
a system of the Beltrami equations in a suitable monitor metric specified in
the physical geometry. The system can be interpreted as the multidimensional
equidistribution principle in which the monitor metric tensor is an extension
of a scalar-valued weight function. With this interpretation for a mathematical
model for generating grids in domains or on surfaces, one need only choose
the Beltrami equations, without any complementary control operators that
worsen the model, while the required grid properties are realized through the
specification of suitable metric tensors.
Thus the single Beltrami mathematical model provides a real foundation
for the solution of the challenging problem of the development of comprehensive
grid generators. Consequently the efforts of research should be directed
towards implementing this model into grid technology by developing
approaches for formulating metrics in physical geometries and establishing
necessary relations between them and the required grid properties for the
purpose of setting up an adequate control of the grid quality by the choice
of the suitable metric.
One natural approach for formulating metric tensors and corresponding
tensor-valued weight functions is based on the notion of a monitor surface
over the physical geometry that undergoes a gridding process. The monitor
surface is defined as the graph of some (in general vector-valued) function
that takes into account the behavior of the physical solution. This monitor
surface, having an inherent metric tensor that can be considered as the very
tensor-valued weight function, is suitable for generating adaptive grids with
the use of a smoothness functional (which is the functional of energy) whose
Euler–Lagrange equations are, in fact, equivalent to the Beltrami equations
in the metric of the monitor surface. The resulting grid derived by this metric
tends to cluster its nodes in the zones of the large gradient of the function. The
approach for formulating the adaptive metric is readily extended to define
more general monitor metrics in domains or on surfaces, thus turning them
into Riemannian manifolds whose implementation in grid technology allows
one to generate grids satisfying the most broad mesh quality requirements.
In order to control the required grid properties by the monitor metrics,
one needs a knowledge of geometric characteristics of the monitor geometries
and their relations to the resulting grid behavior. This knowledge can be attained
with the aid of the theory of multidimensional differential geometry of
Riemannian manifolds adjusted to the features of grid technology. The theory
Preface to the Second Edition VII
of multidimensional differential geometry is really one of the most promising
branches of the pure mathematical field of science, capable of pushing grid
technology to a more advanced level in its development. Indeed, many notions
and characteristics of common surfaces, such as metric tensors, their invariants,
first and second fundamental forms, curvatures and torsions of lines, the
mean and Gauss curvatures, and Christoffel symbols, have already been used
by many authors as natural elements in defining grid quality measures and
formulating appropriate variational and differential grid techniques in a unified
manner regardless of the geometry of the physical domains and surfaces.
A theory of more general geometric objects, such as regular multidimensional
surfaces and Riemannian manifolds implemented for generating grids
with necessary properties, is expected to become a highly beneficial tool for
boosting grid technology. The known relations and techniques of differential
geometry also present an efficient means for transforming and modernizing
the physical and grid equations into a suitable form. It is presumable that
the science of differential geometry will play in numerical grid technology
the same role played by the science of matrices in the theory of difference
approximations of boundary value problems.
Therefore, there is a need for a monograph that is essentially aimed at
providing deep and balanced insight into the fields of grid science, multidimensional
geometry adjusted to grid technology, and up-to-date achievements
of the applications of geometric tools to the creation of advanced grid techniques.
With this background the reader will be able to formulate and develop
well-posed grid models and algorithms and analyze grid properties with geometry
related tools, thus taking part in the solution of the very challenging
problem of the development of advanced comprehensive grid generators.
This monograph gives an account of the geometrization of popular comprehensive
grid methods and presents an important extension to the methods
related to the application of the technique of Riemannian manifolds to the
formulation of grid equations by developing some procedures for the construction
of monitor metric tensors. Contrary to classical geometric studies,
which center on geometric features and characteristics of specified Riemannian
manifolds, the problem of finding appropriate monitor metrics for producing
grid systems with the required properties is somewhat an inverse
problem of the creation of Riemannian manifolds with desirable geometric
characteristics. In accordance with the concept of the inverse problem, the
author of the monograph discusses rather thoroughly some new techniques
aimed at the construction of special monitor metrics in physical geometries.
The techniques are designed by generalizing the projection approach in which
the monitor metric in an n-dimensional physical geometry is borrowed from
a natural metric of the n-dimensional surface derived by a height monitor
function over the geometry. This technology allows the required metric to
be defined through the original metric of the physical geometry and certain
vector-valued functions.
VIII Preface to the Second Edition
The book establishes and reviews some of the relations of the Riemannian
geometry for the purpose of obtaining new equations with implemented metric
characteristics aimed at facilitating the control of the generation of grids
with the required properties. Taking advantage of the relations established,
the author has converted the equations into a compact form convenient for
numerical treatment via the available algorithms.
The technique of multidimensional differential geometry is also applied
to study the qualitative effect of a general class of monitor metrics on the
resulting mesh. For this purpose a new characteristic of grid clustering is
formulated. Certain relations between this measurement and some geometric
characteristics of grid hypersurfaces and the monitor functions forming the
monitor metrics are established. The well-known results for grids generated
by inverted Laplace equations about node-clustering near concave boundary
segments of domains and node-rarefaction near convex boundary segments
are, using these relations, extended to arbitrary boundary segments and to
more general Beltrami equations in monitor metrics. On the basis of the
established formulas, the monitor functions are readily estimated in the inverted
diffusion and Beltrami grid equations to provide grid clustering or, if it
is reasonable, grid rarefaction near arbitrary segments of physical geometries.
Some relations of the mean curvature of the monitor surfaces to the Beltrami
equations for grid generation are exhibited. The book also includes
a chapter devoted to the implementation of the comprehensive grid equations
and the energy functional into numerical codes and to the application
of the codes to the numerical solution of some gas-dynamics and plasmarelated
problems.
Since grid technology has widespread applications to nearly all field problems,
this monograph will be useful for a broad range of readers, including
teachers, students, and researchers as well as practitioners in applied mathematics,
mechanics, biology, medicine, and physics interested in the numerical
analysis of multidimensional field problems with complicated geometries and
complex solutions.
The book is divided into two parts. Part I of the book gives a geometric
background needed for the development of grid generators. The grid equations,
codes, and applications are described in Part II.
Part I of the monograph includes Chaps. 1–4. Chapter 1 gives a general introduction
to the subject of numerical grids and methods of their generation.
Chapters 2–4 introduce the reader to multidimensional differential geometry
for the purpose of better understanding those of its techniques that are suitable
for the implementation into advanced grid generation technologies. The
geometric implementation in grid technology pursued in the book assumes the
development of robust techniques for producing appropriate monitor metrics
over both physical domains and surfaces thus converting them into Riemannian
manifolds. The metrics should guarantee generation of grids with the
necessary properties through popular mathematical models.
Preface to the Second Edition IX
Part II of the book is devoted to the implementation of geometric tools
into the development of grid techniques and codes. It contains Chaps. 5–7.
Chapter 5 deals with fundamental elliptic grid models formulated through
the operators of Beltrami and diffusion and establishes compact formulas of
monitor metrics. Two-dimensional Beltrami equations in the natural metric of
a physical surface were originally proposed byWarsi for generating fixed grids
on the surface. The ordinary Laplace equations that are the Beltrami equations
in the Euclidean metric were applied to generate fixed grids in domains
by Crowley and Winslow. One justification of the Beltramian operator is
demonstrated in Chap. 5 by the proof of the statement that an arbitrary nondegenerate
smooth transformation of a physical domain or surface is realized
as a solution of the Dirichlet boundary value problem for the system of Beltrami
grid equations in some appropriate metric. The chapter also discusses
some variational and harmonic interpretations of the Beltrami equations, in
particular, a variational approach for generating harmonic maps through the
minimization of energy functionals, which was suggested by Dvinsky.
With the help of the geometric relations, established in Chap. 4, the grid
equations introduced in Chap. 5 are transformed in Chap. 6 to equations in
invariant forms with respect to independent logical variables. Special monitor
metrics over two-dimensional surfaces are designed that result in simpler
transformed equations, even in comparison with the equations that have
been used for generating fixed grids. The chapter also establishes relations
between the monitor functions and geometric characteristics of the Riemannian
manifolds produced and the coordinate lines and surfaces generated by
a corresponding mathematical model, for the purpose of realization of grid
control through a suitable specification of the monitor functions.
Chapter 7 gives a description of some computational codes for generating
grids with the numerical solution in the logical domain of the elliptic
equations obtained in Chap. 6 by changing mutually dependent and independent
variables in the original Beltrami and diffusion equations. Some
numerical aspects related to the development of grid generation codes are
reviewed, in particular, the application of layer-type functions to formulating
monitor metrics and description of two techniques for generating smooth
block-structured grids. Numerical results related to the application of the
grid technology advocated in the book to some gas-dynamics and plasma
problems are also exhibited in this chapter.
The book ends with a list of references.
Acknowledgement
The author is very grateful for helpful suggestions in geometry, algebra, and
numerical techniques made by his colleagues, Professors Borisov, Churkin,
Glasser, Kuzminov, Sharafutdinov, and Shvedov.
The author is also much obliged to the researchers who responded to his
requests and sent files of their papers and of pictures for figures, namely
X Preface to the Second Edition
B.S. Azarenok (Figs. 7.10–7.14), A.A. Charakhch’yan (Figs. 7.8–7.9), and
A.H. Glasser (Figs. 5.11–5.12). Figures 7.10–7.14 were published in Azarenok
(2000, 2002), 7.8 in Charakhch’yan and Ivanenko (1997), 7.9 in Lomonosov,
Frolova and Charakhch’yan (1997), and 5.10–5.11 in Glasser et al. (2005).
The work over the book was partly supported by the US Civilian Research
& Development Foundation (CRDF): Award NO RU-M1-2579-NO-04.
In particular, the efforts related to the development of grid generation codes,
computing figures of grids, and preparing the text of the book in Latex code,
made by I. Kitaeva, Yu. Likhanova, and D. Patrakhin, whom the author
thanks very much, were remunerated by payments from the CRDF grant.
Figures 5.6, 7.29 and 7.37 were published in Glasser, Liseikin and Kitaeva
(2005), 5.4, 5.8, 7.32 and 7.34 in Glasser et al. (2005).
Novosibirsk, March 2006 Vladimir D. Liseikin

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