Abstract
The RifSimp package in Maple transforms a set of differential equations to Reduced Involutive Form. This paper describes the application of RifSimp to challenging real-world problems found in engineering design and modelling. RifSimp was applied to sets of equations arising in the dynamical studies of multibody systems. The equations were generated by the Maple package Dynaflex, which takes as input a graph-like description of a multibody mechanical system and generates a set of differential equations with algebraic constraints. Application of the standard RifSimp procedure to such Differential Algebraic Equations can require large amounts of computer memory and time, and can fail to finish its computations on larger problems.
We discuss the origin of these difficulties and propose an Implicit Reduced Involutive Form to assist in alleviating such problems. This form is related to RifSimp form by the symbolic inversion of a matrix. For many applications such as numerically integrating the multibody dynamical equations, the extra cost of symbolically inverting the matrix to obtain explicit RifSimp form can be impractical while Implicit Reduced Involutive Form is sufficient.
An approach to alleviating expression swell involving a hybrid analytic polynomial computation is discussed. This can avoid the excessive expression swell due to the usual method of transforming the entire input analytic differential system to polynomial form, by only applying this method in intermediate computations when it is required.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Preview
Unable to display preview. Download preview PDF.
References
Ascher, U., Petzold, L.: Computer Methods for Ordinary Differential Equations and Differential-Algebraic Equations. SIAM, Philadelphia (1998)
Buchberger, B., Collins, G.E.: Computer Algebra Symbolic and Algebraic Computation. Springer, Heidelberg (1983)
Rideau, P.: Computer Algegbra and Mechanics, The James Software. In: Computer Algebra in Industry I. John Wiley, Chichester (1993)
Corless, R.M., Jeffrey, D.J., Monagan, M.B., Pratibha: Two Perturbation Calculations in Fluid Mechanics Using Large-Expression Management. J. Symbolic Computation 11, 1–17 (1996)
Cox, D.A., Little, J.B., O’Shea, D.: Ideals, Varieties, and Algorithms. Springer, Heidelberg (1997)
Hubert, E.: Factorization free decomposition algorithms in differential algebra. J. Symbolic Computation 29, 641–662 (2000)
Mansfield, E.: A Simple Criterion for Involutivity. Journal of the London Mathematical Society 54, 323–345 (1996)
Pommaret, J.F.: Systems of Partial Differential Equations and Lie Pseudogroups. Gordon and Breach Science Publishers, Inc. (1978)
Wittkopf, A.D., Reid, G.J.: The Reduced Involutive Form Package. Maple Software Package. First distributed as part of Maple 7 (2001)
Reid, G.J., Lin, P., Wittkopf, A.D.: Differential-Elimination Completion Algorithms for DAE and PDAE. Studies in Applied Mathematics 106, 1–45 (2001)
Rudolf, C.: Road Vehicle Modeling Using Symbolic Multibody System Dynamics. Diploma Thesis, University of Waterloo in cooperation with University of Karlsruhe (2003)
Rust, C.J.: Rankings of Partial Derivatives for Elimination Algorithms and Formal Solvability of Analytic Partial Differential Equations. Ph.D. Thesis, University of Chicago (1998)
Seiler, W.M.: Analysis and application of the formal theory of partial differential equations. Ph.D. thesis, Lancaster University (1994)
Schiehlen, W.: Multibody Systems Handbook. Springer, Heidelberg (1990)
Shi, P., McPhee, J.: Symbolic Programming of a Graph-Theoretic Approach to Flexible Multibody Dynamics. Mechanics of Structures and Machines 30(1), 123–154 (2002)
Shi, P., McPhee, J.: DynaFlex User’s Guide. In: Systems Design Engineering, University of Waterloo (2002)
Shi, P., McPhee, J.: Dynamics of flexible multibody systems using virtual work and linear graph theory. Multibody System Dynamics 4(4), 355–381 (2000)
Shi, P., McPhee, J., Heppler, G.: A deformation field for Euler-Bernouli beams with application to flexible multibody dynamics. Multibody System Dynamics 4, 79–104 (2001)
Sommese, A.J., Verschelde, J., Wampler, C.W.: Numerical decomposition of the solution sets of polynomial systems into irreducible components. SIAM J. Numer. Anal. 38(6), 2022–2046 (2001)
Visconti, J.: Numerical Solution of Differential Algebraic Equations Global Error Estimation and Symbolic Index Reduction. Ph.D. Thesis. Laboratoire de Modélisation et Calcul. Grenoble (1999)
Wittkopf, A.D.: Algorithms and Implementations for Differential Elimination. Ph.D. Thesis. Simon Fraser University, Burnaby (2004)
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2005 Springer-Verlag Berlin Heidelberg
About this paper
Cite this paper
Zhou, W., Jeffrey, D.J., Reid, G.J., Schmitke, C., McPhee, J. (2005). Implicit Reduced Involutive Forms and Their Application to Engineering Multibody Systems. In: Li, H., Olver, P.J., Sommer, G. (eds) Computer Algebra and Geometric Algebra with Applications. IWMM GIAE 2004 2004. Lecture Notes in Computer Science, vol 3519. Springer, Berlin, Heidelberg. https://doi.org/10.1007/11499251_4
Download citation
DOI: https://doi.org/10.1007/11499251_4
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-540-26296-1
Online ISBN: 978-3-540-32119-4
eBook Packages: Computer ScienceComputer Science (R0)