Mathematics of Computation

			Journals Home Search Author Info Subscribe Tech Support My Subscriptions Help
ISSN 1088-6842(e) ISSN 0025-5718(p)

Algebraic algorithms for the analysis of mechanical trusses

Author(s): I. Babuska; S. A. Sauter.
Journal: Math. Comp. 73 (2004), 1601-1622.
MSC (2000): Primary 65T50, 06B10, 35J55
Posted: April 27, 2004
MathSciNet review: 2059728
Retrieve article in: PDF
This article is available free of charge

Abstract | References | Similar articles | Additional information

Abstract: Infinite periodic lattices can be used as models for analyzing and understanding various properties of mechanical truss constructions with periodic structures. For infinite lattices, the problems of connectivity and stability are nontrivial from the mathematical point of view and have not been addressed adequately in the literature. In this paper, we will present a set of algebraic algorithms, which are based on ideal theory, to solve such problems.

For the understanding of the notion ``complicated three-dimensional lattices'', it is essential to have this paper with colored figures.

References:

1.: L. Asimov and B. Roth.
The Rigidity of Graphs.
Trans. Am. Math. Soc., 245:279-289, 1978. MR 80i:57004a
2.: I. Babuska and S. A. Sauter. Efficient Solution of Lattice Equations by the Recovery Method. Part 1: Scalar Elliptic Problems. TICAM REPORT 02-39, University of Texas at Austin, 2002.
3.: W. Brown.
Matrices over Commutative Rings.
Marcel Dekker, New York, 1992. MR 93k:15028
4.: G. Constantinides and A. C. Payatakes.
A Three Dimensional Network Model for Consolidated Porous Media. Basic Studies.
Chem. Eng. Comm., 81:55-81, 1980.
5.: H. Crapo and W. Whiteley.
Encyclopedia of Mathematics and Its Applications, chapter ``The Geometry of Rigid Structures''.
Cambridge Univ. Press, 1989.
6.: V. Deshpande, N. Fleck, and M. Ashby.
Effective properties of the octet truss material.
Technical report, Cambridge Univ., Dept. Eng., 2000.
7.: I. Fatt.
The Network Model of Porous Media.
Trans. Am. Inst. Mem. Metall Pet. Eng., 207:144-181, 1956.
8.: L. Gibson and M. Ashby.
Cellular Solids, Structures and Properties.
Pergamon Press, Exeter, 1989.
9.: M. L. Glasser and J. Boersma.
Exact Values for the Cubic Lattice Green Functions.
J. Phys., 133:5017-5023, 2000. MR 2001i:82022
10.: J. C. Hansen, S. Chien, R. Skalog, and A. Hoger.
A Classic Network Model Based on the Structure of the Red Blood Cell Membrane.
Biophy. J., 70:146-166, 1996.
11.: G. S. Joyce.
On the Simple Cubic Lattice Green Functions.
Proc. Roy. Soc., London Ser. A, 445:463-477, 1994. MR 95j:33041
12.: G. Laman.
On Graphs and Rigidity of Plane Skeletal Structures.
Journal of Engineering Math., 4:331-340, 1970. MR 42:4430
13.: A. Maradudin, E. Montroll, G. Weiss, R. Herman, and H. Milnes.
Green's functions for monoatomic simple cubic lattices.
Acad. Roy. Belg. Cl. Sci. Mem.Coll. in-4deg, 14(2):176, 1960. MR 22:7440
14.: P. G. Martinson.
Fast Multiscale Methods for Lattice Equations.
PhD thesis, University of Texas at Austin, 2002.
15.: A. Noor.
Continuum modelling for repetitive lattice structures.
Appl. Mech. Rev., 41(7):285-296, 1988.
16.: M. Ostoja-Starzewski.
Lattice Models in Micromechanics.
App. Mech. Review, 55(1):35-60, 2002.
17.: M. Ostoja-Starzewski, P. Y. Shang, and K. Alzebdoh.
Spring Network Models in Elasticity and Fractures of Composites and Polycristals.
Comp. Math. Sci., 7:89-93, 1996.
18.: G. I. Pshenichnov.
Theory of Lattice Plates and Shells.
World Scientific, Singapore, 1993. MR 97i:73001
19.: R. Zurmühl and S. Falk.
Matrizen und Ihre Anwendungen.
Springer Verlag, Berlin, Heidelberg, 5 edition, 1984. MR 86a:15002

	Publications Meetings The Profession Membership Programs Math Samplings Policy & Advocacy In the News About the AMS
\|