CC BY
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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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4. Initial mesh generation for the scattered data
We use three main methods for initial mesh generation from a set of unorganized points. First is based on the famous Delaunay triangulation technique [6] (Prog.2), which is optimal in a certain meaning and can be used for surfaces with simple geometry, especially flat ones. Second method is based on Hoppe's surface reconstruction [7]. Undesirable holes are sealed up with a specially designed dynamic programming algorithm. Third method is based on a surface reconstruction CSRBF technique [8]. Like in the second method, we often need to pick out nodes for a successive coarser triangulation of the polygonal model. In the Fig. 1 (left), we show a surface of an engineering device reconstructed from a scanned data by the method based on Hoppe’s technique. After that (Fig. 1, right) we stop up holes. Red contours denote boundaries of the holes.
Fig. 1. Left: detail, reconstructed from a scanned data; right: after that holes are sealed up
5. Subdivision process and the final stage of mesh generation
For a subdivision process, we suggest an interpolating Modified Butterfly scheme (see [9] for details). In the case of a given NURBS surface S, we may suppose here that the subdivision surfaces f k (R) approach to S, where k is step of subdivision. Here f k —— f , and f (R)
k k
is the limit subdivision surface. After k-th step of subdivision, we obtain a triangular net {A } CR, and a corresponding mesh {b ..},
ч 4
wlieref (B j ) = b j, g(B у )G {b 0 } .
Here are some examples of triangular mesh generation.
Fig. 2. Delaunay triangulation; after first step of subdivision; after second step
In the Fig. 2 we show an example of Delaunay triangulation of scattered points with successive subdivisions. The points lie on the surface of a plate detail with a hole and a groove. In the figure 3 we depict an initial mesh for a NURBS 3-dimentional surface using the transpose diagonal technique. After that, we apply the Modified Butterfly subdivision scheme twice, as it was shown in Fig. 2.
Finally, for the mesh to be conforming (in the case of adaptive subdivision), we can use a method [10] for dividing big triangles into several smaller ones.
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In the case of NURBS surface, after subdivision process we may need to project a mesh onto the given surface S. The projection algorithm follows. According to subdivision scheme, each generating node b is a some barycentric combination of initial nodes: b= b
3l ,3m
(u ■ ,v .), where (X, =1. Evaluate a point on the surface at (u,v). The resulting pointp(u,v) is a projection of the point b.
6. Conclusion and future work
In this article, we have proposed new triangular mesh generation techniques, which use the interpolation subdivision scheme. The mesh satisfies the aspect ratio demand; hence, it is suitable for FEM analysis. However, a lot of work should be done in the area of subdivision methods application to mesh generation. It is desirable to reveal a class of cases, where subdivision is more advantageous for such a generation. Hence, a detailed comparison with other existing methods is necessary. The development of adaptive subdivision technique for mesh generation and further computer programming are the subject of a future research as well.
References
1. Denis Zorin et al. “Subdivision for Modeling and Animation”. SIGGRAPH’2000 Course Notes.
2. Labsik U, Kobbelt L., Schneider R., Seidel H.-P. “Progressive transmission of subdivision surfaces” // Computational Geometry 15 (2000) 25-39
3. “Fuji Technical Research” company. Private communications, Tokyo, 2000
4. Gerald Farin “Curves and surfaces for computer aided geometric design”. Academic Press, 1993
5. Ho-Le K. “Finite element mesh generation methods: review and classification”. Computer-Aided Design, 20:27-38, 1988
6. M. de Berg et al. “Computational geometry. Algorithms and applications”. Springer-Verlag, 2000
7. Hugues Hoppe “Surface reconstruction from unorganized points”. PhD thesis, University of Washington, 1994
8. Nikita Kojekine, Vladimir Savchenko, Dmitry Berzin, Ichiro Hagiwara “Software tools for compactly supported radial basis functions”. Proceedings of Fourth IASTED International Conference “Computer Graphics and Imaging”, August 13-16, 2001, Honolulu, Hawaii, USA
9. Zorin D., Schroder P., Sweldens W. “Interpolating subdivision for meshes with arbitrary topology”. Computer Graphics Proceedings (SIGGRAPH’96), 189-192
10. 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.
Берзин Д.В.
Кандидат физико-математических наук, доцент, Финансовый университет при Правительстве Российской Федерации, Москва АВТОМАТИЧЕСКОЕ ПОСТРОЕНИЕ FE СЕТИ ПОСРЕДСТВОМ МЕТОДА MODIFIED BUTTERFLY
ПОДРАЗБИЕНИЙ
Аннотация
В работе предложен новый алгоритм для построения треугольной сети для поверхности NURBS, заданной посредством контрольных точек. При этом используется современный и эффективный метод - техника подразбиений.
Ключевые слова: система автоматизированного проектирования, треугольная сеть, NURBS, подразбиения.
Berzin DV
PhD, Associate Professor, Financial University under the Government of the Russian Federation, Moscow FINITE ELEMENT MESH AUTOMATIC GENERATION USING MODIFIED BUTTERFLY SUBDIVISION SCHEME
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 technique, which has many advantages.
Keywords: CAD system, FE mesh, subdivision.
1. Given CAD system
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 , weights w j , 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:
s(u,v)
N 3 (v)
l J___________
XX^NKu) N 3 (v)
l J
where N3 (u), N 3 (v) are B-spline functions ([2], Ch.10,17).
1 J
2. Formulation of the problem
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]):
1) Triangles should satisfy an aspect ratio, i.e. they must be close to regular triangles.
2) Nodes of triangles must lie exactly on the given surface S.
3) The distance d between triangle and surface should be less than number £ , chosen by a user.
4) User should be able to change the mesh adaptively (e.g., density of the mesh in some areas, the number £ , and so on).
3. 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 (fig.3). 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 00 , ... , b3L3M . All 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 3n,3k.
Remark. In general case there is a polyhedron K instead of rectangle R, K C R
4
([5], Ch.3).
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