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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 ПОСТРОЕНИЕ FE СЕТИ ПОСРЕДСТВОМ ТРЕУГОЛЬНЫХ ПОДРАЗБИЕНИЙ
Аннотация
В работе предложен новый алгоритм для построения треугольной сети для поверхности NURBS, заданной посредством контрольных точек. При этом используются две техники подразбиений - Modified Butterfly и Loop.
Ключевые слова: NURBS, контрольные точки, система автоматизированного проектирования, треугольная сеть, подразбиения.
Berzin D.V.
PhD, Associate Professor, Financial University under the Government of the Russian Federation, Moscow FINITE ELEMENT MESH AUTOMATIC GENERATION USING TRIANGULAR SUBDIVISION SCHEMES
Abstract
We suggest here a new algorithm for triangular finite element mesh generation for NURBS surface represented as a set of control points. We use a modern approach - subdivision techniques, which has many advantages. Two different subdivision schemes are presented here: Modified Butterfly and Loop ones.
Keywords: NURBS, control points, CAD system, FE mesh, subdivision.
1. The problem to solve
Suppose we are given a 2-dimensional surface S, defined by means of control points, for example, by data stored in IGES file type 126 [1]. Thus, we have a set of control points d weights w . , knot sequence (u n ,v k ), where i= -1, ... , L+1; j=-1, ... , M+1; n=0, ..., L; k=0, ... , M. And the corresponding NURBS surface has the following parametric form:
ZZ wsd„N(J(») Nj (v)
^ ' TZwNFwNJm" ’
i J
3 3
where N. (u), N . (v) are B-spline functions ([2], Ch.10,17).
i j
Triangular patches in CAD system development have certain advantages over quadrilateral ones ([2], Ch.24). For example, they do not suffer from some kinds of degeneracies and are thus better suited to describe complex geometries than are rectangular patches.
Our task is to construct a triangular finite element mesh satisfying the conditions ([3], [4], [5]):
Triangles should satisfy an aspect ratio, i.e. they must be close to regular triangles. Nodes of triangles must lie exactly on the given surface S. The distance d between triangle and surface should be less than number S , chosen by a user. User should be able to change the mesh adaptively (e.g., density of the mesh in some areas, the number S , and so on).
2. Solution of the problem
Without loss of generality, consider a bicubic B-spline surface S.
Step 1. Bringing a bicubic B-spline surface into a piecewise bicubic Bezier form. This is a standard CAGD procedure ([2], Ch.17), and can be realized by a subroutine, say, “Bezier”. Suppose, now we are given a rectangular net of points b 0 0 , ... , b 3L 3M . All of them lie on
S . Consider a planar rectangular domain R , spanned on points A ^, that there is a homeomorphism
g: R ^ S g( A nk ) = b 3n,3k .
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Remark. In general case there is a polyhedron K instead of rectangle R, K C R ([5], Ch.3).
Step 2. To construct an initial mesh, we pick out points from the set {b 3n3k }. We want the initial mesh to be close to the aspect
ratio demand. The subroutine, say, “Initial” uses diagonal transpose technique (e.g., see [4]): it starts from the rectangle A00AojA^Ajj ,
compares b 0 0 b 3 3 and b 0 3 b 3 0 , and chooses the shortest diagonal to divide a quadrilateral into 2 triangles. My colleague Nikita Kojekine
(see also [8] and [9]) wrote a computer program in C++ to get the results. The program divides each quadrilateral in the same way.
4. Subdivision process
Step 3a. For a subdivision process, we suggest an interpolating Modified Butterfly scheme ([6], Ch.4). We can suppose here that, loosely speaking, the subdivision surfaces f k (R) approach to a given surface S. Here fk ^ f , and f (R) is the subdivision surface. After
k k
k-th step of subdivision, we obtain a triangular net {B .. } C R, and a corresponding mesh {b .. }, where
У У
k k 0
f (Bk) = bk, g(Bj)e {b0. } k = (h1, ... .
A user can interactively choose a level of subdivision in different domains. Let the subroutine be called “Subdivision”.
Step 3b. Instead of Modified Butterfly scheme, sometimes it is advantageous to use other triangular schemes. One of most popular of them is (approximating) Loop scheme ([6], Ch.4). The explanation and results for the step 3 follows:
Program “Subdivision”. The program “subdivision surfaces” was created to demonstrate two different triangular subdivision schemes. One is interpolating Modified-Butterfly scheme, second is Loop scheme. The program was written in Microsoft Visual C++ using MFC (Microsoft Foundation Classes) technology, OpenGL and VTK (free-source visualization toolkit) by my colleague Nikita Kojekine. Program reads initial mesh files first. The very simple format was developed for them.
For example:
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Fig. 1 - Tetrahedron interpolated using Modified Butterfly Scheme, after 6 steps subdivisions
Fig. 2 - Same tetrahedron approximated using Loop scheme, after 6 steps of subdivision
The program is provided with more complex examples of meshes. For example, the ‘rose’ initial mesh and after subdivision:
Fig. 3 - Rose. Initial mesh on the left, and approximated image to the right. 3 steps of subdivision using Loop scheme
Another addition to "subdivision" program can be used to demonstrate the interpolation of colors using Modified Butterfly scheme too. Let us look at the same example with tetrahedron:
Fig. 4 - Tetrahedron after 5 subdivisions using Modified Butterfly subdivision scheme
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k k k
Step 4. After k-th level of subdivision we project (by a subroutine, say, “Projection”) a mesh {b .. } onto the surface S. Let a .. = P(b ..) ,
У У У
k
where P is a projection, a .. G S.
Step 5. Now one should verify a condition 3). We suggest here to use a distance d x between a barycenter of corresponding triangle and
S (instead of d), and verify a condition dx<£ /2. Let the subroutine be called “Distance”. For the mesh to be conforming, we can use a
method from [7] for dividing big triangles into several smaller ones.
References
1. “Fuji technical research” company. Private communications, Tokyo, 2000.
2. Gerald Farm “Curves and surfaces for CAGD”. Academic press, 1993.
3. Ichiro Hagiwara. Private communications, Tokyo Institute of Technology, 2000.
4. Ho-Le K. “Finite element mesh generation methods: review and classification”. Computer-Aided Design, 20:27-38, 1988
5. K.-J. Bathe “Finite Element Procedures”. Prentice-Hall, 1996
6. “Subdivision for Modeling and Animation”. SIGGRAPH 99 Course Notes.
7. Rivara M.C. “Algorithms for refining triangular grids suitable for adaptive and multi-grid techniques”. Int. J. Numer. Meth. Eng. Vol 20 (1984) pp. 745-756.
8. Dmitry Berzin "Finite element mesh generation using subdivision technique" // Research Journal of International Studies, №8 (27) 2014, p. 6
9. Dmitry Berzin "Finite element automatic mesh generation using Modified Butterfly subdivision scheme" // Research Journal of International Studies, №8 (27) 2014, p. 8
Берзин Д.В.
Кандидат физико-математических наук, доцент, Финансовый университет при Правительстве Российской Федерации, Москва О ТОПОЛОГИЧЕСКИХ МЕТОДАХ В ПРОСТРАНСТВЕННОМ МОДЕЛИРОВАНИИ
Аннотация
Существует множество хороших работ, касающихся геометрических методов пространственного моделирования. Но очень немногие исследования затрагивали топологический аспект. В настоящей работе мы предлагаем краткий обзор базовых топологических концепций в пространственном моделировании.
Ключевые слова: симплициальные комплексы, гомотопия, клеточные пространства, гомеоморфизм.
Berzin D.V.
PhD, Associate Professor, Financial University under the Government of the Russian Federation, Moscow ON TOPOLOGICAL METHODS IN SHAPE MODELING
Abstract
There are many excellent research results on geometrical shape modeling. As for the topological level of the modeling, limited researches have been conducted. In this paper we overview some basic concepts of topology, which can be applied to shape modeling.
Keywords: simplicial complex, cell space, homeomorphism
1. Introduction
The notions of topological space and homeomorphism are ones of fundamental in mathematics. Roughly speaking, homeomorphisms describe objects deformations, and the concept of homeomorphism is useful for discovering important properties of objects, that are not changing under such deformations. These properties are called topological, in contrast to metrical ones, which are connected with distances between points, angles between lines and so on. For example, cube and tetrahedron are different from metrical point of view, but they are homeomorphic. Subtle metric properties are not important for many problems, and it is needed to reveal “rough” topological properties. Topology (in particular, homotopy topology) is an important branch of mathematics (e.g., see [1], [2]). Its basic concepts, useful for shape modeling, are homeomorphism, homotopy, simplicial complex, cell space.
It is known, that the concept of manifold is a fundamental in geometry. Structure and properties of smooth manifolds give us a good basis for computer-aided geometrical design (CAGD) and computer graphics (CG) methods. For example, Gaussian curvature and other invariants are of a great importance in geometric modeling. It appears that sometimes a structure of a smooth manifold is not sufficient, so the notion of simplicial complex or cellular space is necessary. Simplicial complex can be regarded as a triangulated object (manifold or nonmanifold). A cellular space is an object, constructed from some primitives - cells, and can be regarded as generalization of the notion “smooth manifold”.
Triangulations of subdivided manifolds (and non-manifolds) are used extensively in solid modeling. Paoluzzi et al. [3] provide an overview of related work and analyze the benefits of representing such triangulations using simplicial complexes. Bertolotto et al. [4] present hierarchical simplicial representations for subdivided objects, but these do not support changes of topological type. Polyhedra can also be represented using more general representations. The simplicial set representation of Lang and Lienhardt [5] generalizes simplicial complexes to allow incomplete and degenerate simplices. Cell complexes (i.e. cell spaces), formed by subdividing manifolds into non-simplicial cells, can be represented using radial edge structure [6] or the cell tuple structure [7]. Kunii et al. [8-14] and [23] used so-called cellular approach for shape modeling. The authors give a definition of homotopy and cellular space, and examples of cellular decompositions of geometric objects together with corresponding attaching maps as well. The progressive simplicial complex (PSC) representation for geometrical objects was described in [15]. The PSC representation expresses an arbitrary triangulated model M (e.g. any dimension, non-orientable, nonmanifold) as the result of successive refinements applied to the base model M1 that always consists of a single vertex. Thus both geometric and topological complexities are recovered progressively. Combinatorial and topological properties of meshed solids were also used in [16]. In [17] the authors use the following modeling pipeline: state space - configuration space - image space. The state space is represented by a data structure that is topologically general and computationally practical: the simplicial complex. Topological approaches are used also in [18]. The authors proposed a novel technique, called Topology Matching, in which similarity between polyhedral models is quickly, accurately, and automatically calculated by comparing Multiresolutional Reeb Graphs (MRGs). By the way, Prof. Kunii and his followers use widely Reeb Graph representation of objects in their research [19-22].
2. Homotopy equivalence
We will start with a definition of a topological space. This concept 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. Important combinatorial structures are simplicial and cellular ones.
A set X is called a topological space, if a set of sets {X a}aGi is determined, where X a are subsets of X (called “open” sets), and I is an arbitrary set. In addition, three axioms hold:
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