Chain-based representations for solid and physical modeling

IEEE Transactions on Automation Science and Engineering

Volumn 6, Issue 3, 2009, Pages 454-467

  Dicarlo, Antonio ^a   Milicchio, Franco ^a   Paoluzzi, Alberto ^a   Shapiro, Vadim ^b  

^a ROMA TRE UNIVERSITY   (Italy)

^b UNIVERSITY OF WISCONSIN MADISON   (United States)

Author keywords

Algorithms; Computational geometry; Finite element methods; Geometric modeling; Mesh generation; Sparse matrices; Spatial data structures; Topology

Indexed keywords

BIDIAGONAL; CELL COMPLEXES; CHAIN COMPLEXES; CODIMENSION; COMPUTATIONAL DOMAINS; COMPUTATIONAL SCIENCE AND ENGINEERINGS; EULER OPERATORS; FIELD COMPUTATION; FINITE-ELEMENT METHODS; FIRST-PRINCIPLES; GEOMETRIC MODELING; LOCAL MESH REFINEMENT; MATRIX; MESH REFINEMENT; ORIENTABILITY; PHYSICAL MODELING; SPARSE MATRICES; SPATIAL DATA STRUCTURES;

CELL MEMBRANES; COMPUTATION THEORY; DATA STRUCTURES; FINITE ELEMENT METHOD; GEOMETRY; MATHEMATICAL OPERATORS; MATRIX ALGEBRA; MODELS; TOPOLOGY;

MESH GENERATION;

EID: 68149169180     PISSN: 15455955     EISSN: None     Source Type: Journal    
DOI: 10.1109/TASE.2009.2021342     Document Type: Conference Paper

References (37)

1. B. G. Baumgart, Winged-Edge Polyhedron Representation Artificial Intelligence Lab., Stanford Univ., Stanford, CA, Tech. Rep. CS-TR-72-320, 1972.
2. L. Guibas and J. Stolfi, "Primitives for the manipulation of general subdivisions and the computation of voronoi," ACM Trans. Graphics, vol. 4, no. 2, pp. 74-123, 1985.
3. M. Mäntylä, Introduction to Solid Modeling. New York: Freeman, 1988.
4. E. Brisson, "Representing geometric structures in d dimensions: Topology and order," Discrete Comput. Geometry, vol. 9, no. 1, pp. 387-426, 1993.
5. A. Requicha, Mathematical Models of Rigid Solid Objects Production Automation Project, Univ. Rochester, Rochester, NY, Tech. Memo 28, 1977.
6. A. Requicha, "Representations for rigid solids: Theory, methods, and systems," Comput. Surv., vol. 12, no. 4, pp. 437-464, 1980.
7. C. Silva, Alternative definitions of faces in boundary representatives of solid objects Production Automation Project, Univ. Rochester, Rochester, NY, 1981.
8. M. A. O'Connor and J. R. Rossignac, "SGC: A dimension-independent model for pointsets with internal structures and incomplete boundaries," in Proc. Geometric Modeling for Product Engineering., Sep. 1989, pp. 145-180.
9. T. Woo, "A combinatorial analysis of boundary data structure schemata," IEEE Comput. Graph. Applicat., vol. 5, no. 3, pp. 19-27, 1985.
10. H. Masuda, "Topological operators and Boolean operations for complex-based nonmanifold geometric models," Comput.-Aided Des., vol. 25, no. 2, pp. 119-129, 1993.
11. Y. Yamaguchi and F. Kimura, "Nonmanifold topology based on coupling entities," IEEE Comput. Graph. Applicat., vol. 15, no. 1, pp. 42-50, 1995.
12. C. Eastman and K. Weiler, Geometric Modeling Using the Euler Operators. Pittsburgh, PA: Inst. Physical Planning, Carnegie-Mellon Univ., 1979.
13. J. R. Munkres, Elements of Algebraic Topology. Reading, MA: Addison-Wesley, 1984.
14. A. Hatcher, Algebraic Topology. Cambridge, U.K.: Cambridge Univ. Press, 2002.
15. C.Hoffmann, Geometric and Solid Modeling: An Introduction. San Francisco, CA: Morgan Kaufmann, 1989.
16. F. H. Branin, "The algebraic-topological basis for network analogies and the vector calculus," in Proc. Symp. Generalized Networks, 1966, vol. 16, pp. 453-491.
17. E. Tonti, On the formal structure of physical theories Istituto di Matematica del Politecnico di Milano, Milan, Italy, 1975.
18. R. Palmer and V. Shapiro, "Chain models of physical behavior for engineering analysis and design," Res.Eng.Des., vol. 5, no. 3, pp. 161-184, 1993.
19. G. Strang, "A framework for equilibrium equations," SIAM Rev., vol. 30, no. 2, pp. 283-297, 1988.
20. C. Mattiussi, "An analysis of finite volume, finite element, and finite difference methods using some concepts from algebraic topology," J. Comput. Phys., vol. 133, no. 2, pp. 289-309, 1997.
21. E. Tonti, "A direct discrete formulation of field laws: The cell method," Comput. Model. Eng. Sci., vol. 2, no. 2, pp. 237-258, 2001.
22. J. Hyman and M. Shashkov, "Natural discretizations for the divergence, gradient and curl on logically rectangular grids," Int. J. Comput. Math. Applicat., vol. 33, no. 4, pp. 81-104, 1997.
23. F. Milicchio, A. DiCarlo, A. Paoluzzi, and V. Shapiro, "A codimension-zero approach to discretizing and solving field problems," Adv. Eng. Inform., vol. 22, no. 2, pp. 172-185, 2008.
24. F. Milicchio, "Towards topological unification of finite computational methods," Ph.D. dissertation, Univ. Roma Tre, Rome, Italy, 2007.
25. A. Paoluzzi, V. Pascucci, and M. Vicentino, "Geometricprogramming: A programming approach to geometric design," ACM Trans. Graph., vol. 14, no. 3, pp. 266-306, 1995.
26. G. Scorzelli, A. Paoluzzi, and V. Pascucci, "Parallel solid modeling using BSP dataflow," J. Comput. Geom. Applicat., 2007.
27. A. Paoluzzi, V. Pascucci, and G. Scorzelli, G. Elber, N. Patrikalakis, and P. Brunet, Eds., "Progressive dimension-independent boolean operations," in Proc. 9thACMSymp. Solid Modeling Applicat., New York, 2004, pp. 203-211.
28. A. DiCarlo, F. Milicchio, A. Paoluzzi, and V. Shapiro, "Solid and physical modeling with chain complexes," in Proc. 2007 ACM Symp., Solid and Physical Modeling, SPM'07, Beijing, China, 2007, pp. 73-84.
29. C. L. Bajaj and V. Pascucci, "Splitting a complex of convex polytopes in any dimension," in Proc. 12th Annu. Symp. Computat. Geometry, SCG'96, Philadelphia, PA, 1996, pp. 88-97.
30. T. Davis, "Direct methods for sparse linear systems," in Fundamentals of Algorithms. Philadelphia, PA: SIAM, 2006.
31. V. Ramaswamy and V. Shapiro, "Combinatorial laws for physically meaningful design," J. Comput. Inform. Sci. Eng., vol. 4, no. 1, pp. 3-10, 2004.
32. H. Whitney, Geometric Integration Theory. Princeton, NJ: Princeton Univ. Press, 1957.
33. J. A. Chard and V. Shapiro, "A multivector data structure for differential forms and equations," IMACS J. Math. Comput. Simul., vol. 54, no. 1, pp. 33-64, 2000.
34. M. Nakahara, Geometry, Topology and Physics. New York: Inst. Physics, 1990.
35. R. Bott and L. Tu, Differential Forms in Algebraic Topology, 3rd ed. New York: Springer, 1995.
36. D. Arnold, R. Falk, and R. Winther, "Finite element exterior calculus, homological techniques, and applications," Acta Numerica, vol. 15, pp. 1-155, 2006.
37. A. Bossavit, "Whitney forms: A class of finite elements for three-dimensional computations in electromagnetism," Proc. Inst. Elect. Eng., vol. 135, no. 8, pp. 493-500, 1988.