Conversion methods for improving structural analysis of differential-algebraic equation systems
- 169 Downloads
Structural analysis (SA) of a system of differential-algebraic equations (DAEs) is used to determine its index and which equations to be differentiated and how many times. Both Pantelides’s algorithm and Pryce’s \(\varSigma \)-method are equivalent: if one of them finds correct structural information, the other does also. Nonsingularity of the Jacobian produced by SA indicates success, which occurs on many problems of interest. However, these methods can fail on simple, solvable DAEs and give incorrect structural information including the index. This article investigates \(\varSigma \)-method’s failures and presents two conversion methods for fixing them. Under certain conditions, both methods reformulate a DAE system on which the \(\varSigma \)-method fails into a locally equivalent problem on which SA is more likely to succeed. Aiming at achieving global equivalence between the original DAE system and the converted one, we provide a rationale for choosing a conversion from the applicable ones.
KeywordsDifferential-algebraic equations Structural analysis Modeling Symbolic computation
Mathematics Subject Classification34A09 65L80 41A58 68W30
The authors acknowledge with thanks the financial support for this research: GT was supported in part by the McMaster Centre for Software Certification through the Ontario Research Fund, Canada, NSN was supported in part by the Natural Sciences and Engineering Research Council of Canada, and JDP was supported in part by the Leverhulme Trust, the UK. The authors thank the anonymous reviewers for providing valuable suggestions on improving this article.
- 7.Griewank, A.: On automatic differentiation. In: Iri, M., Tanabe, K. (eds.) Mathematical Programming: Recent Developments and Applications, pp. 83–108. Kluwer Academic Publishers, Dordrecht (1989)Google Scholar
- 10.MapleSim: Technological superiority in multi-domain physical modeling and simulation (2012). http://www.maplesoft.com/view.aspx?sf=7032
- 12.Mazzia, F., Iavernaro, F.: Test Set for Initial Value Problem Solvers. Tech. Rep. 40, Department of Mathematics, University of Bari (2003). http://pitagora.dm.uniba.it/~testset/
- 13.Nedialkov, N., Pryce, J.D.: DAETS User Guide. Tech. Rep. CAS 08-08-NN, Department of Computing and Software, McMaster University, Hamilton (2013). pp 68, DAETS is available at http://www.cas.mcmaster.ca/~nedialk/daets
- 23.Scholz, L., Steinbrecher, A.: Regularization of DAEs based on the signature method. BIT Numer. Math. 56(1), 319–340 (2016)Google Scholar
- 24.Scholz, L., Steinbrecher, A.: Structural-algebraic regularization for coupled systems of DAEs. BIT Numer. Math. 56(2), 777–804 (2016)Google Scholar
- 25.Sjölund, M., Fritzson, P.: Debugging symbolic transformations in equation systems. In: Proceedings of Equation-Based Object-Oriented Modeling Languages and Tools (EOOLT), pp. 67–74 (2011)Google Scholar
- 26.Tan, G.: Conversion methods for improving structural analysis of differential-algebraic equation systems. Ph.D. thesis, School of Computational Science and Engineering, McMaster University, 1280 Main St W, Hamilton, Ontario, L8S 1A8, Canada, pp. 168 (2016)Google Scholar
- 27.Tan, G., Nedialkov, N.S., Pryce, J.D.: Conversion methods, block triangularization, and structural analysis of differential-algebraic equation systems. Tech. Rep. CAS 16-04-NN, Department of Computing and Software, McMaster University, Hamilton, pp. 25 (2016). http://www.cas.mcmaster.ca/cas/0reports/CAS-16-04-NN.pdf
- 28.The MathWorks, Inc.: Matlab Symbolic Math Toolbox (2016). http://www.mathworks.com/products/symbolic/