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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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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”, starts from the rectangle A 00 A 01A 10 A n , consider six pairs of 3-dimensional triangles
corresponding to 8 points:
AAAAAAAA л 00 01,yi 0 2 10' л w'1 12’ л 2021 ’
and choose one pair which is more close to the aspect ratio condition. After that the subroutine goes to either a rectangle A oiA 02 A n A 12 or a rectangle A 10 A n A 20 A 21 ■ Thus, we obtain a net
B 00 , B 01, B 10- - - B lm, • where B ij e {A nk }, L1 ^L M 1 ^M
So, we have constructed a triangular mesh g(B ..) that satisfies conditions 1), 2), and conv (B ..) = R.
iJ iJ
Step 3. For a subdivision process, we suggest an interpolating Modified Butterfly scheme ([5], Ch.4). We can suppose here that, loosely speaking, the subdivision surfaces f k (R) approach to a given surface S. Here f k —— f , and f (R) is the subdivision surface. After k-th step
k k
of subdivision, we obtain a triangular net {B .. } CR, and a corresponding mesh {b .. }, where
4 4
f (B kij) = b kj- g(B j )e {b ij } k=0,1, ... .
A user can interactively choose a level of subdivision in different domains. Let the subroutine be called “Subdivision”.
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 ,,
4 ч 4
k
) , where P is a projection, a j £ S.
Step 5. Now one should verify a condition 3. We suggest here to use a distance d j between a barycenter of corresponding triangle and
S (instead of d), and verify a condition d j < £ /2. Let the subroutine be called “Distance”.
4. Conclusions and future work
In this paper we suggested a method for finite element mesh generation by a program consisting of 5 subroutines. However, there are some unsolved problems in this project. For example, what is the best algorithm in the step 2? How to show mathematically, that a difference between f and g is sufficient small in the step 3? How to connect nodes in the final mesh in the case of adaptive subdivision? These problems and others, together with computer implementation, are subjects of future research.
References
1. “Fuji technical research” company. Private communications, Tokyo, 2000.
2. Gerald Farin “Curves and surfaces for CAGD”. Academic press, 1993.
3. Ichiro Hagiwara. Private communications, Tokyo Institute of Technology, 2000.
4. K.-J. Bathe “Finite Element Procedures”. Prentice-Hall, 1996
5. “Subdivision for Modeling and Animation”. SIGGRAPH 99 Course Notes.
Берзин Д.В.
Кандидат физико-математических наук, доцент, Финансовый университет при Правительстве Российской Федерации,
Москва
КВАНТОВЫЕ ВЫЧИСЛЕНИЯ И АВТОМАТИЗИРОВАННЫЕ СИСТЕМЫ ПРОЕКТИРОВАНИЯ
Аннотация
В данной статье мы показываем, что применение квантовых вычислений в автоматизированных системах проектирования может существенно повысить их эффективность.
Ключевые слова: квантовые вычисления, системы автоматизированного проектирования.
Berzin D.V.
PhD, Associate Professor,
Financial University under the Government of the Russian Federation, Moscow QUANTUM COMPUTATIONS AND CAD
Abstract
In this paper we show how quantum computations can enhance CAD systems performance.
Keywords: quantum computations, CAD.
1. Motivation
Information processing (computing) is the dynamical evolution of a highly organized physical system produced by technology (computer) or nature (brain). The initial state of this system is (determined by) its input; its final state is the output. Physics describes nature in two complementary modes: classical and quantum. Up to the nineties, the basic mathematical models of computing, Turing machines, were classical objects, although the first suggestions for studying quantum models date back at least to 1980 ([Ma]). Roughly speaking, the motivation to study quantum computing comes from several sources: physics and technology, cognitive science, and mathematics. We will briefly discuss them in turn.
1) Physically, the quantum mode of description is more fundamental than the classical one. In the seventies and eighties it was remarked that, because of the superposition principle, it is computationally unfeasible to simulate quantum processes on classical computers ([Po], [Fe1]). Roughly speaking, quantizing a classical system with N states we obtain a quantum system whose state space is an (N-1)-dimensional complex projective space whose volume grows exponentially with N. One can argue that the main preoccupation of quantum chemistry is the struggle with resulting difficulties. Reversing this argument, one might expect that quantum computers, if they can be built at all, will be considerably more powerful than classical ones ([Fe1], [Ma]). Progress in the microfabrication techniques of modern computers has already led us to the level where quantum noise becomes an essential hindrance to the error-free functioning of microchips. It is only logical to start exploiting the essential quantum mechanical behavior of small objects in devising computers, instead of neutralizing it.
2) As another motivation, one can invoke highly speculative, but intriguing, conjectures that our brain is in fact a quantum computer (see a recent paper [Ha]). For example, the progress in writing efficient chess playing software (“Deep Blue”) shows that to simulate the
world championship level using only classical algorithms, one has to be able to analyze about 10 6 positions/sec and use about 1010 memory
-3
bytes. Since the characteristic time of neuronal processing is about 10 sec, it is very difficult to explain how the classical brain could possibly do the job and play chess as successfully as Karpov does. A less spectacular, but not less resource consuming task, is speech generation and perception, which is routinely done by billions of human brains, but still presents a challenge for modern computers using
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