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ФИЗИКО-МАТЕМАТИЧЕСКИЕ НАУКИ / PHYSICS AND MATHEMATICS
Берзин Д.В.
Кандидат физико-математических наук, доцент, Финансовый университет при Правительстве Российской Федерации, Москва О ГОМОТОПИЧЕСКОМ И КЛЕТОЧНОМ ПОДХОДАХ К ПРОСТРАНСТВЕННОМУ МОДЕЛИРОВАНИЮ
Аннотация
Существующее пространственное моделирование обычно исходит из геометрических свойств кривых и поверхностей. Но весьма немногие исследования затрагивали топологический аспект пространственного моделирования. В работах [1-9] было указано на важность такого аспекта. В настоящей работе мы предлагаем краткий обзор этих базовых идей и развиваем их. Ключевые слова: гомотопия, клеточные пространства, геометрическое моделирование.
Berzin D.V.
PhD, Associate Professor, Financial University under the Government of the Russian Federation, Moscow ON HOMOTOPY AND CELLULAR APPROACHES TO SHAPE MODELING
Abstract
The existing shape modeling usually starts from geometric properties of curves and surfaces. As for the topological level of shape modeling, limited researches in CAD have been conducted. In papers [1-9] the importance of homotopy and cellular topological approaches was emphasized. In this paper we suggest a summary of basic ideas, considered in papers [1-9], and develop these ideas.
Keywords: homotopy, cellular spaces, geometric modeling.
Introduction
In papers [1-9] two approaches to a spatial modeling were considered: homotopy and cellular. As a start point, the former one uses notions of homotopically equivalent maps from one topological space to another and homotopically equivalent topological spaces; the latter method uses notions of cellular spaces (i.e. CW-complexes) and so-called gluing map. Structure and properties of smooth manifolds give us a rich basis for computer-aided geometrical design (CAGD) and computer graphics (CG) methods. For example, Gaussian curvature and other invariants are of a great importance for CAGD [10,11]. It appears that sometimes a structure of a smooth manifold is not sufficient, so the notion of a cellular space is necessary. There are no direct practical applications of these approaches in papers [1-9]. However, the theoretical basis for future investigations seems to be rather interesting.
§1. Homotopy equivalence
We will start with a definition of a topological space. It is a basic one in topology. But this object is too general. Almost always mathematics deals with spaces with additional structures. Firstly, there are analytical structures: differential, Riemannian, symplectic, and so on. They are very natural. Secondly, combinatorial structures can be provided. One decomposes a space into similar parts and investigates how they are situated to each other. An important combinatorial structure is a cellular one.
Definition. A set X is called a topological space, if a set of sets { X c }aeI is determined, where X a are subsets of X (called
“open” sets), and I is an arbitrary set. In addition, three axioms must be held:
1) Empty set and the set X are open
2) Any union of open sets is open
3) Intersection of any two open sets is open.
Definition. Set VСX is called closed, if its complement to X is open.
Definition. Let X, Y be two topological spaces. Map f: X —— Y is called continuous, if for each open subset U C Y the inverse image f 1 (U) is open in X.
Definition. Let X, Y be two topological spaces. Map f : X —— Y is called homeomorphism, if it is a continuous one-to-one
correspondence and the inverse map f 1 is continuous too. Two spaces are homeomorphic, if there is a homeomorphism between them.
Often we need to restrict such wide classes of mathematical objects. For this aim additional separability axiom is used in the next definition.
Definition. Topological space X is Hausdorff space, if for each couple of points x,y E X there exist two corresponding open neighborhoods U, V in X, which intersection is empty: U О V = 0 , x E X,y E Y.
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Of course, our usual Euclidian space R is Hausdorff one.
Definition. Let f 0 : X —— Y and f ^ : X —— Y be two continuous maps between topological spaces X and Y. These maps are called
homotopically equivalent (or homotopic) if there exists a family ф. , for 0 < t < 1, of continuous (with respect to t and xE X simultaneously) maps:
Ф : X ^ Y,
and satisfying ф (x) = f 0 (x) , ф (x) = f ^ (x) . The family of maps
continuous map
F(x,t) : XX [0,1] ^ Y.
is called a homotopy between X and Y; it can also be regarded as a
In words, two maps are homotopic, if we can go from one to another by means of a continuous deformation with parameter tE [0,1]. Definition. Two topological spaces Xand Y are called homotopically equivalent if there are continuous maps f: X ^ Y and g : Y ^ X
such that the composition fg : Y —— Y is homotopic to the identical map id: Y —— Y and the composition gf: X ——X is homotopic to the identity map id: X ——X.
There are several well-known examples of homotopically equivalent (but not homeomorphic) spaces: Euclidian space R and a point; a Mobius strip (non-orientable surface) and a circle; a sphere with three holes and bouquet of two circles S 1 VS1 (i.e. two circles, intersecting at one common point); a torus with a hole and a bouquet of two circles; a circle and an annulus. We may choose f : S1 —— S 1 as
identical map, let h be a compression along radii, let g be a composition g=f identical maps.
-1
h. Then fg and gf are homotopy equivalent to corresponding
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Deformations of real objects often can be considered as homotopy deformations.
§2. Cellular spaces
Now we are ready to give a strict definition of CW-complex.
Definition. Cellular space (or CW-complex, which is the same) is Hausdorff topological space, represented as a union of nonintersecting sets (called cells)
X= U( Ueq)
qe J ie [q
where q is a dimension of a cell eq ; i is its number; Iq is some set, corresponding to dimension q ; J is a subset of the set of integer non-negative numbers.
In addition, the following conditions must be satisfied:
1. Each cell of zero dimension is a point of X. For every positive q the closure eq of each cell eq is the image of a closed q -
dimensional ball Bq under some continuous map (called characteristic map)
X: B ^ eq,
where eq С X , and the restriction of this map to the open ball int Bq is a homeomorphism. Here ball Bq is an ordinary ball in q-
dimensional Euclidian space R q :
Bq = {||w|| < 1; w e R q }
1. (C-axiom - “closure finite”).
A boundary of each cell (with dimension more or equal than 1) is contained in a union of a finite number of cells of less dimensions.
2. (W-axiom - “weak topology”).
A subset K of X is closed if and only if the intersection K О eq is closed for every cell eq of space X.
It is easy to understand that for finite CW-complexes (i.e. consisting of finite number of cells) C-axiom and W-axiom are held automatically.
Consider an important example of a cellular subspace of X - its skeleton
skpX = U ( U ei) ,
qeJ ,q< p ie I q
where p is an integer non-negative number.
Thus, from the above definitions we can obtain the next procedure of cellular spaces constructing. Consider a restriction Fq of
the characteristic map %q to the boundary Sq 1 of the closed q-dimensional ball Bq, and after that consider continuous map from a union of spheres into a skeleton:
F-- U ST' ^ X .
ie J 1 q
so that Fq is a restriction of Fq to the sphere Sq 1. Thus,
Л, X = Skq -, XU F,
(U Bq)
ie]
1 q
Definition. Continuous map Fq is called gluing map (or attaching map).
Indeed, gluing map identifies each point x from Sq 1 with its image F q (x) e skq ^ X .
So, we have a sequence
sk0X c skxx c ... c skqX c ... с X .
which is called a “filtration ”.
Roughly say, if we add a differential structure to a cellular space, we will obtain a smooth manifold. Differential geometry, an important branch of mathematics, describes various properties of manifolds [17,18]. Two-dimensional smooth manifold is called a surface. In [13-16] so-called homotopy model based on Morse theory [19] for smooth manifolds was constructed. The scheme
VRML data —^ Homotopy data —^ VRML data
is used there. Homotopy data consist of information about critical points of Morse function chosen as a height function (Reeb graph), cross sections, and other. Each critical point of Morse function has its index, that is an integer non-negative number. For example, for peak points index X =2, for saddle points X =1, and for pit points X =0.
References
1. T. L. Kunii “Valid Computational Shape Modeling: Design and Implementation.” Preprint of International Journal of Shape Modeling, 1999.
2. T. L. Kunii “Invariants of Cyberworlds”, 1999.
3. T. L. Kunii “A Cellular Model”, 1999.
4. T. L. Kunii “Homotopy Modeling as World Modeling.” Proceedings of Computer Graphics International ’99, pp.130-141.
5. T. L. Kunii “Technological Impact of Modern Abstract Mathematics” Proceedings of Third Asian Technology Conference in Mathematics, 1998, pp.13-23.
6. T. L. Kunii, Y. Saito, M. Shiine “A Graphics Compiler for a 3-Dimensional Captured Image Database and Captured Image Reusability.” Proceedings of CAPTECH’98.
7. T. L. Kunii “Graphics with Shape Property Inheritance.” Proceedings of PG’98.
8. T. L. Kunii “The 3-rd Industrial Revolution through Integrated Intelligent Processing Systems.” Proceedings of ICIPS’97,
pp.1-6
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9. T. L. Kunii, T. Maeda “On the Silhouette Cartoon Animation”
10. A. T. Fomenko, T. L. Kunii “Topological Modeling for Visualization”, Springer, 1998.
11. G. Farin “Curves and surfaces for CAGD”. Academic Press, 1992.
12. A. T. Fomenko, D. B. Fuchs “Course of Homotopic Topology.” Kluwer Academic Publishers.
13. Y. Shinagawa, T. L. Kunii, Y. L. Kergosien “Surface Coding Based on Morse Theory.” IEEE Computer Graphics & Applications, 1991.
14. Y. Shinagawa, T. L. Kunii “Constructing a Reeb Graph Automatically from Cross Sections.” IEEE Computer Graphics & Applications, 1991.
15. M. Hilaga “Enhanced homotopy model for orientable closed surface”. Master thesis, Department of Information Science, Faculty of Science, University of Tokyo, 1999.
16. S. Noda “Development of CG systems fundamental technology using homotopy modeling”. Master thesis, Department of Mechanical Engineering and Science, Faculty of Engineering, Tokyo Institute of Technology, 2000 (In Japanese).
17. A. B. Dubrovin, A.T. Fomenko, S. P. Novikov “Modern Geometry. Methods and Applications.” Springer-Verlag. Part 1, 1984. Part 2, 1985. Part 3, 1990.
18. S. Kobayashi, K. Nomizu “Foundations of Differential Geometry”. John Wiley & Sons, 1996
19. J. Milnor “Morse Theory.” Princeton University Press, 1963.
Берзин Д.В.
Кандидат физико-математических наук, доцент, Финансовый университет при Правительстве Российской Федерации, Москва
ПОСТРОЕНИЕ FE СЕТИ МЕТОДОМ ПОДРАЗБИЕНИЙ
Аннотация
В данной работе мы применяем Modified Butterfly интерполяцию к построению треугольной FE сети.
Ключевые слова: система автоматизированного проектирования, сеть, метод конечных элементов
Berzin D.V.
PhD, Associate Professor, Financial University under the Government of the Russian Federation, Moscow FINITE ELEMENT MESH GENERATION USING SUBDIVISION TECHNIQUE
Abstract
We describe here our results on application of Modified Butterfly interpolation subdivision scheme to triangular finite element mesh generation. The surface of an initial object is represented by a scanned data or in a NURBS form.
Keywords: CAD, finite element method, triangular mesh.
1. Introduction
Subdivision method [1] is a way to describe a surface of an object using a polygonal model, i.e. a mesh, by a sequence of successive refinements. There are many research results on mesh generation for finite element analysis. Nevertheless, as we know, there are no profound researches have been conducted in the application of subdivision technique to this problem. We choose a subdivision as a basic approach for our FE mesh generation, because it has certain advantages over other modeling methods (for instance, modeling with B-spline or NURBS). Firstly, subdivision is easy to implement and has low computational cost. Secondly, subdivision can handle arbitrary topology quite well without losing efficiency; this is one of its key advantages. Thirdly, subdivision allows more flexible controls of the shape and size of features than is possible with splines. In addition to choosing locations of control points, one can manipulate the coefficients of subdivision to achieve effects such as sharp creases or control the behavior of the boundary curves. Fourthly, there is a good elaborated adaptive subdivision technique, which enable to provide more dense mesh in the desired areas [2].
In this work, we suggest methods for triangular mesh generation applying progressive subdivision technique and avoiding triangles with too high aspect ratio. Using C++ programming language, my colleague Nikita Kojekine wrote the computer programs, which are implemented to construct the mesh.
2. Initial data
Suppose we have initial data that describes the surface of an object. We consider two cases: either the surface is given in NURBS form or we have a set of unorganized points on it (scanned data). As we know, both of cases occur rather often in practical applications [3].
Firstly, suppose we are given a 2-dimensional surface S, defined by means of control points. Without lost of generality, consider a
cubic case. Thus, we have a set of control points d .. , weights w ■■
iJ 4
knot sequence (u ,v k ), where i= -1, ... , L+1; j=-1, ... , M+1; n=0,
..., L; k=0, ... , M. The corresponding NURBS surface has the following parametric form:
S(u,v)
EEVA» N 3 (v)
l j_____
XX^N» N 3(v)
3 3
where N(u), N . (v) are B-spline functions [4].
In the case of scattered data, we just have a set of unorganized points, more-or-less uniformly distributed on the surface.
The number r=max (AB, BC, AC)/min (AB, BC, AC) is called the aspect ratio of a triangle ABC. The goal is to construct a triangular mesh in a way that the number of triangles with high aspect ratio would be as less as possible.
3. Initial mesh generation for NURBS
If we are given a NURBS surface, we construct an initial mesh for further subdivision process in the two following steps.
Step 1. Bringing a NURBS into a piecewise rational Bezier form is a standard CAGD procedure [4]. Schematically, the procedure can
be depicted as follows: {d ^ } —— {b k n }, where {b k n
} is a set of Bezier control points, k=0, ...,3L, n=0, ...,3M. Thus, now we are given a
rectangular net of Bezier points {b 3[ 3m
l=0,...,L
; each of them lie on S. Consider a planar rectangular domain R , spanned on points A
nk ’
that there is a homeomorphism g: R —— S, g(A nk ) = b 3n3k . Note, that in general case there is a polyhedron K instead of rectangle R, K 4
e R , see [1].
Step 2. It is needed the initial mesh to be close to the aspect ratio demand. We suggest here using the diagonal transpose technique (e.g., [5]). Starting from the rectangle A 00 A 01A 10 A n, we compare corresponding diagonals b 00 b 33 and b 03 b 30 , and chooses the shortest diagonal to divide a quadrilateral into 2 triangles. The program divides each quadrilateral in the same way. Thus, we have constructed a triangular mesh g(A nk ) = b 3n3k , where nodes b 3n 3k lie exactly on the surface.
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