A codimension-zero approach to discretizing and solving field problems

Advanced Engineering Informatics

Volumn 22, Issue 2, 2008, Pages 172-185

  Milicchio, F ^a   DiCarlo, A ^a   Paoluzzi, A ^a   Shapiro, V ^b  

^a ROMA TRE UNIVERSITY   (Italy)

^b UNIVERSITY OF WISCONSIN MADISON   (United States)

Author keywords

Algebraic topology; Finite methods; Solid and physical modeling

Indexed keywords

COMPUTATIONAL COMPLEXITY; GRAPH THEORY; SYSTEMS ANALYSIS;

ALGEBRAIC TOPOLOGY; FINITE METHODS;

PROBLEM SOLVING;

EID: 40749148075     PISSN: 14740346     EISSN: None     Source Type: Journal    
DOI: 10.1016/j.aei.2007.08.013     Document Type: Article

References (29)

1. Lodish H., Berk A., Zipursky L.S., Matsudaira P., Baltimore D., and Darnell J. Molecular Cell Biology. fourth ed. (2000), W.H. Freeman, New York, NY, USA
2. Boal D. Mechanics of the Cell. third ed. (2005), Cambridge University Press, Cambridge, UK
3. Maxwell J.C. On the mathematical classification of physical quantities. In: Niven W. (Ed). The Scientific Papers of James Clerk Maxwell (1890), Cambridge University Press 257-266
4. Kron G. Numerical solution of ordinary and partial differential equations by means of equivalent circuits. J. Appl. Phys. 126 (1945) 172-186
5. Roth J.P. An application of algebraic topology to numerical analysis: On the existence of a solution to the network problem. Proc. Nat. Acad. Sci. 41 (1955) 518-521
6. F.H. Branin, The algebraic-topological basis for network analogies and the vector calculus, in: Proceedings of the Symposium on Generalized Networks, vol. 16, Polytechnic Institute of Brooklyn, 1966, pp. 453-491.
7. E. Tonti, On the formal structure of physical theories, Technical report, Istituto di Matematica del Politecnico di Milano, 1975.
8. Palmer R., and Shapiro V. Chain models of physical behavior for engineering analysis and design. Res. Eng. Des. 5 3 (1993) 161-184
9. Bamberg P., and Sternberg S. A Course in Mathematics for Students of Physics (1988), Cambridge University Press, Cambridge, England
10. Bamberg P., and Sternberg S. A Course in Mathematics for Students of Physics (1990), Cambridge University Press, Cambridge, England
11. Palmer R.S. Chain models and finite elements analysis: an executable chains formulation of plane stress. Comput. Aided Geom. Des. 12 7 (1995) 733-770
12. Egli R., and Stewart N.F. A framework for system specification using chains on cell complexes. Comput. Aided Des. 32 9 (2000) 447-459
13. Egli R., and Stewart N.F. A framework for system specification using chains on cell complexes. Math. Comput. Simulat. 66 (2004) 449-468
14. Schwalm W., Moritz B., Giona M., and Schwalm M. Vector difference calculus for physical lattice models. Phys. Rev. E 59 (1999) 1217-1233
15. Strang G. A framework for equilibrium equations. SIAM Rev. 30 2 (1988) 283-297
16. Mattiussi C. An analysis of finite volume, finite element, and finite difference methods using some concepts from algebraic topology. J. Comput. Phys. 133 (1997) 289-309
17. Tonti E. A direct discrete formulation of field laws: The cell method. Comput. Model. Eng. Sci. 2 2 (2001) 237-258
18. Hyman J., and Shashkov M. Natural discretizations for the divergence, gradient and curl on logically rectangular grids. Int. J. Comput. Math. Appl. 33 4 (1997) 81-104
19. Chard J.A., and Shapiro V. A multivector data structure for differential forms and equations. IMACS Trans. J., Math. Comput. Simulat. 54 1 (2000) 33-64
20. Ramaswamy V., and Shapiro V. Combinatorial laws for physically meaningful design. J. Comput. Informat. Sci. Eng. 4 3 (2004) 3-10
21. Paoluzzi A., Pascucci V., and Vicentino M. Geometric programming: a programming approach to geometric design. ACM Trans. Graph. 14 3 (1995) 266-306
22. G. Scorzelli, A. Paoluzzi, V. Pascucci, Parallel solid modeling using BSP dataflow, J. Comput. Geom. Appl., in press.
23. Paoluzzi A., Pascucci V., and Scorzelli G. Progressive dimension-independent boolean operations. In: Elber G., Patrikalakis N., and Brunet P. (Eds). Proceeding of the Ninth ACM Symposium on Solid Modeling and Applications (2004), ACM Press 203-212
24. Munkres J.R. Elements Of Algebraic Topology (1984), Addison-Wesley, Reading MA
25. Franz W. Algebraic Topology (1968), Frederick Ungar Publishing Co., New York, NY
26. Skiena S. Implementing Discrete Mathematics: Combinatorics and Graph Theory with Mathematica (1990), Addison-Wesley, Reading, MA
27. Bajaj C.L., and Pascucci V. Splitting a complex of convex polytopes in any dimension. SCG'96: Proceedings of the twelfth annual symposium on Computational geometry (1996), ACM Press, Philadelphia, PA 88-97
28. In: McCormick S.F. (Ed). Multigrid Methods. Lecture Notes in Pure and Applied Mathematics vol. 110 (1988), Dekker, New York, NY
29. Shapira Y. Matrix-Based Multigrid: Theory and Applications. Numerical Methods and Algorithms vol. 2 (2003), Kluwer Academic Publishers, Boston, MA