Construction of Customized Mass-Stiffness Pairs Using Templates
Publication: Journal of Aerospace Engineering
Volume 19, Issue 4
Abstract
This paper is a tutorial exposition of the template approach to the construction of customized mass-stiffness pairs for selected applications in structural dynamics. The exposition focuses on adjusting the mass matrix while a separately provided stiffness matrix is kept fixed. Two well known kinetic-energy discretization methods described in finite-element method (FEM) textbooks since the mid-1960s lead to diagonally lumped and consistent mass matrices, respectively. These two models are sufficient to cover many engineering applications. Occasionally, however, they fall short. The gap can be filled with a more general approach that relies on the use of templates. These are algebraic forms that carry free parameters. This approach is discussed in this paper using one-dimensional structural elements as examples. Templates have the virtue of producing a set of mass matrices that satisfy certain a priori constraint conditions such as symmetry, nonnegativity, invariance, and momentum conservation. In particular, the diagonally lumped and consistent versions can be obtained as instances. Thus those standard models are not excluded. Availability of free parameters, however, allows the mass matrix to be customized to special needs, such as high precision vibration frequencies or minimally dispersive wave propagation. An attractive feature of templates for FEM programming is that only one element implementation as module with free parameters is needed, and need not be recoded when the application problem class changes.
Get full access to this article
View all available purchase options and get full access to this article.
Acknowledgments
The pioneering contributions of Professor Noor to two of the tools heavily used in this paper: continuification of dynamic lattices as surveyed in Noor (1988) and the use of CAS in computational mechanics as discussed in Noor and Andersen (1979), are gratefully acknowledged. Preparation of this paper has been supported by the National Science Foundation under Grant No. NSFCMS–0219422.
References
Archer, J. S. (1963). “Consistent mass matrix for distributed mass systems.” J. Struct. Div., 89, 161–178.
Archer, J. S. (1965). “Consistent mass matrix formulation for structural analysis using finite element techniques.” AIAA J., 3, 1910–1918.
Belytschko, T., and Mullen, R. (1978). “On dispersive properties of finite element solutions.” Modern problems in elastic wave propagation, J. Miklowitz and J. D. Achenbach, eds., Wiley, New York, 67–82.
Born, M., and Huang, K. (1954). Dynamical theory of crystal lattices, Oxford, London.
Brillouin, L. (1946). Wave propagation in periodic structures, Dover, New York.
Duncan, W. J., and Collar, A. R. (1934). “A method for the solution of oscillations problems by matrices.” Philos. Mag., Series 7, 17, 865–885.
Duncan, W. J., and Collar, A. R. (1935). “Matrices applied to the motions of damped systems.” Philos. Mag., Series 7, 19, 197–214.
Felippa, C. A. (2001a). “Customizing high performance elements by Fourier methods.” Trends in computational mechanics, W. A. Wall et al., eds., CIMNE, Barcelona, Spain, 283–296.
Felippa, C. A. (2001b). “Customizing the mass and geometric stiffness of plane thin beam elements by Fourier methods.” Eng. Comput., 18, 286–303.
Felippa, C. A. (2001c). “A historical outline of matrix structural analysis: a play in three acts.” Comput. Struct., 79, 1313–1324.
Felippa, C. A. (2004). “A template tutorial.” Computational mechanics: Theory and practice, Chap. 3, K. M. Mathisen, T. Kvamsdal, and K. M. Okstad, eds., CIMNE, Barcelona, 29–68.
Flaggs, D. L. (1988). “Symbolic analysis of the finite element method in structural mechanics.” Ph.D. dissertation, Dept. of Aeronautics and Astronautics, Stanford University.
Frazer, R. A., Duncan, W. J., and Collar, A. R. (1938). Elementary matrices, and some applications to dynamics and differential equations, Cambridge University Press, Cambridge, U.K.
Hamming, R. W. (1973). Numerical methods for scientists and engineers, 2nd Ed., Dover, New York.
Hamming, R. W. (1998). Digital filters, 3rd Ed., Dover, New York.
Jones, H. (1960). The theory of Brillouin zones and electronic states in crystals, North-Holland, Amsterdam, The Netherlands.
Krieg, R. D., and Key, S. W. (1972). “Transient shell analysis by numerical time integration.” Advances in computational methods for structural mechanics and design, J. T. Oden, R. W. Clough, and Y. Yamamoto, eds., UAH, Huntsville, Ala., 237–258.
MacNeal, R. H., ed. (1970). “The NASTRAN theoretical manual.” NASA Rep. No. SP-221.
MacNeal, R. H. (1994). Finite elements: Their design and performance, Marcel Dekker, New York.
Melosh, R. J. (1962). “Development of the stiffness method to define bounds on the elastic behavior of structures.” Ph.D. dissertation, Dept. of Engineering Mechanics, Univ. of Washington, Seattle.
Melosh, R. J. (1963). “Bases for the derivation of matrices for the direct stiffness method.” AIAA J., 1, 1631–1637.
Noor, A. K. (1988). “Continuum modeling for repetitive lattice structures.” Appl. Mech. Rev., 41 (7), 285–296.
Noor, A. K., and Andersen, C. M. (1979). “Computerized symbolic manipulation in structural mechanics—Progress and potential.” Comput. Struct., 10, 95–118.
Park, K. C., and Flaggs, D. L. (1984). “An operational procedure for the symbolic analysis of the finite element method.” Comput. Methods Appl. Mech. Eng., 46, 65–81.
Przemieniecki, J. S. (1968). Theory of matrix structural analysis, McGraw-Hill, New York.
Warming, R. F., and Hyett, B. J. (1974). “The modified equation approach to the stability and accuracy analysis of finite difference methods.” J. Comput. Phys., 14, 159–179.
Ziman, J. M. (1967). Principles of the theory of solids, North-Holland, Amsterdam, The Netherlands.
Information & Authors
Information
Published In
Journal of Aerospace Engineering
Volume 19 • Issue 4 • October 2006
Pages: 241 - 258
Copyright
© 2006 ASCE.
History
Received: Oct 20, 2005
Accepted: Jan 6, 2006
Published online: Oct 1, 2006
Published in print: Oct 2006
Authors
Metrics & Citations
Metrics
Citations
Download citation
If you have the appropriate software installed, you can download article citation data to the citation manager of your choice. Simply select your manager software from the list below and click Download.