Conference article

Ali Baharev
Faculty of Mathematics, University of Vienna, Oskar-Morgenstern-Platz 1, 1090 Vienna, Austria

Arnold Neumaier
Faculty of Mathematics, University of Vienna, Oskar-Morgenstern-Platz 1, 1090 Vienna, Austria

Hermann Schichl
Faculty of Mathematics, University of Vienna, Oskar-Morgenstern-Platz 1, 1090 Vienna, Austria

Download articlehttp://dx.doi.org/10.3384/ecp17132353

Published in: Proceedings of the 12th International Modelica Conference, Prague, Czech Republic, May 15-17, 2017

Linköping Electronic Conference Proceedings 132:39, p. 353-362

Published: 2017-07-04

ISBN: 978-91-7685-575-1

ISSN: 1650-3686 (print), 1650-3740 (online)

State-of-the-art Modelica implementations may fail in various ways when tearing is turned on: Completely incorrect results are returned without a warning, or the software fails with an obscure error message, or it hangs for several minutes although the problem is solvable in milliseconds without tearing. We give three detailed examples and an in-depth discussion why such failures are inherent in tearing and cannot be fixed within the traditional approach.

Without compromising the advantages of tearing, these issues are resolved for the first time with staircase sampling. This is a non-tearing method capable of robustly finding all well-separated solutions of sparse systems of nonlinear equations without any initial guesses. Its robustness is demonstrated on the steady-state simulation of a particularly challenging distillation column. This column has three solutions, one of which is missed by most methods, including problem-specific tearing methods. All three solutions are found with staircase sampling.

B. Bachmann, P. Aronßon, and P. Fritzson. Robust initialization of differential algebraic equations. In 1st International Workshop on Equation-Based Object-Oriented Modeling Languages and Tools (Berlin; Germany; July 30; 2007), Linköping Electronic Conference Proceedings, pages 151–163. Linköping University Electronic Press; Linköpings universitet, 2007.

A. Baharev, H. Schichl, and A. Neumaier. Decomposition methods for solving nonlinear systems of equations. Submitted, 2017a. URL http://reliablecomputing.eu/baharev_tearing_survey.pdf.

A. Baharev, H. Schichl, and A. Neumaier. Ordering matrices to bordered lower triangular form with
minimal border width. Submitted, 2017b. URL http://reliablecomputing.eu/baharev_tearing_exact_algorithm.pdf.

Ali Baharev and Arnold Neumaier. A globally convergent method for finding all steady-state solutions of distillation columns. AIChE J., 60:410–414, 2014.

Ali Baharev, Ferenc Domes, and Arnold Neumaier. A robust approach for finding all well-separated solutions of sparse systems of nonlinear equations. Numerical Algorithms, pages 1–27, 2016. doi:10.1007/s11075-016-0249-x. URL https://doi.org/10.1007/s11075-016-0249-x.

N. Bekiaris, G. A. Meski, C. M. Radu, and M. Morari. Multiple steady states in homogeneous azeotropic distillation. Ind. Eng. Chem. Res., 32:2023–2038, 1993.

J. F. Boston and S. L. Sullivan. A new class of solution methods for multicomponent, multistage separation processes. Can. J. Chem. Eng., 52:52–63, 1974.

Emanuele Carpanzano. Order reduction of general nonlinear DAE systems by automatic tearing. Mathematical and Computer Modelling of Dynamical Systems, 6(2):145–168, 2000.

François E Cellier and Ernesto Kofman. Continuous system simulation. Springer Science & Business Media, 2006.

R. de P. Soares and A. R. Secchi. EMSO: A new environment for modelling, simulation and optimisation. In Computer Aided Chemical Engineering, volume 14, pages 947–952. Elsevier, 2003.

E. J. Doedel, X. J. Wang, and T. F. Fairgrieve. AUTO94: Software for continuation and bifurcation problems in ordinary differential equations. Technical Report CRPC-95-1, Center for Research on Parallel Computing, California Institute of Technology, Pasadena CA 91125, 1995.

M. F. Doherty, Z. T. Fidkowski, M. F. Malone, and R. Taylor. Perry’s Chemical Engineers’ Handbook, chapter 13, page 33.

McGraw-Hill Professional, 8th edition, 2008. I. S. Duff, A. M. Erisman, and J. K. Reid. Direct Methods for Sparse Matrices. Clarendon Press, Oxford, 1986.

Dymola User Manual. Volume 2. Dymola 2017 FD01, Dassault Systèmes AB, 2016.

H. Elmqvist and M. Otter. Methods for tearing systems of equations in object-oriented modeling. In Proceedings ESM’94, European Simulation Multiconference, Barcelona, Spain, June 1–3, pages 326–332, 1994.

Hilding Elmqvist. A Structured Model Language for Large Continuous Systems. PhD thesis, Department of Automatic Control, Lund University, Sweden, May 1978.

Robert Fourer, David M. Gay, and Brian Wilson Kernighan. AMPL: A Modeling Language for Mathematical Programming. Brooks/Cole USA, 2003.

Peter Fritzson. Principles of Object-Oriented Modeling and Simulation with Modelica 2.1. Wiley-IEEE Press, 2004.

G. H. Golub and C. F. van Loan. Matrix Computations. The Johns Hopkins University Press, Baltimore, USA, 3rd edition, 1996.

gPROMS. Process Systems Enterprise Limited, gPROMS. https://www.psenterprise.com, 2017. [Online; accessed 21-Jan-2017].

Prem K. Gupta, Arthur W. Westerberg, John E. Hendry, and Richard R. Hughes. Assigning output variables to equations using linear programming. AIChE Journal, 20(2):397–399, 1974.

T. E. Güttinger and M. Morari. Comments on “multiple steady states in homogeneous azeotropic distillation”. Ind. Eng. Chem. Res., 35:2816–2816, 1996.

T. E. Güttinger, C. Dorn, and M. Morari. Experimental study of multiple steady states in homogeneous azeotropic distillation. Ind. Eng. Chem. Res., 36:794–802, 1997.

HSL. A collection of Fortran codes for large scale scientific computation., 2017. URL http://www.hsl.rl.ac.uk.

IEEE 754. IEEE standard for floating-point arithmetic. IEEE Std 754-2008, pages 1–70, Aug 2008.
doi: https://doi.org/10.1109/IEEESTD.2008.4610935.

A. Kannan, M. R. Joshi, G. R. Reddy, and D. M. Shah. Multiplesteady-states identification in homogeneous azeotropic distillation using a process simulator. Ind. Eng. Chem. Res., 44: 4386–4399, 2005.

A. Kröner, W. Marquardt, and E.D. Gilles. Getting around consistent initialization of DAE systems? Computers & Chemical Engineering, 21(2):145–158, 1997.

F. Magnusson and J. Åkesson. Symbolic elimination in dynamic optimization based on block-triangular ordering. Optimization Methods and Software, 2017. doi: https://doi.org/10.1080/10556788.2016.1270944. Published online: 17 Jan 2017.

S. Mattsson, H. Elmqvist, and M. Otter. Physical system modeling with Modelica. Control. Eng. Pract., 6:501–510, 1998.

S. E. Mattsson, M. Otter, and H. Elmqvist. Modelica hybrid modeling and efficient simulation. In Decision and Control, 1999. Proceedings of the 38th IEEE Conference on, volume 4, pages 3502–3507, 1999.

L. M. Naphthali and D. P. Sandholm. Multicomponent separation calculations by linearization. AIChE J., 17:148–153, 1971.

L. A. Ochel and B. Bachmann. Initialization of equationbased hybrid models within OpenModelica. In 5th International Workshop on Equation-Based Object-Oriented Modeling Languages and Tools (University of Nottingham; Nottingham, UK; April 19, 2013), Linköping Electronic Conference Proceedings, pages 97–103. Linköping University Electronic Press; Linköpings universitet, 2013.

Online Supplement, 2017. URL https://github.com/baharev/failure-modes-of-tearing.

C. C. Pantelides. The consistent initialization of differentialalgebraic systems. SIAM Journal on Scientific and Statistical Computing, 9(2):213–231, 1988.

P. C. Piela, T. G. Epperly, K. M. Westerberg, and A. W. Westerberg. ASCEND: An object-oriented computer environment for modeling and analysis: The modeling language. Computers & Chemical Engineering, 15(1):53–72, 1991.

M. Sielemann and G. Schmitz. A quantitative metric for robustness of nonlinear algebraic equation solvers. Mathematics and Computers in Simulation, 81(12):2673–2687, 2011.

M. Sielemann, F. Casella, and M. Otter. Robustness of declarative modeling languages: Improvements via probability-one homotopy. Simulation Modelling Practice and Theory, 38: 38–57, 2013.

P. Täuber, L. Ochel, W. Braun, and B. Bachmann. Practical realization and adaptation of Cellier’s tearing method. In Proceedings of the 6th International Workshop on Equation-Based Object-Oriented Modeling Languages and Tools, pages 11–19, New York, NY, USA, 2014. ACM.

M. Tiller. Introduction to physical modeling with Modelica. Springer Science & Business Media, 2001.

J. Unger, A. Kröner, and W. Marquardt. Structural analysis of differential-algebraic equation systems — theory and applications. Computers & Chemical Engineering, 19(8):867–882, 1995.

A. Vadapalli and J. D. Seader. A generalized framework for computing bifurcation diagrams using process simulation programs. Comput. Chem. Eng., 25:445–464, 2001.

R.C. Vieira and E.C. Biscaia Jr. Direct methods for consistent initialization of DAE systems. Computers & Chemical Engineering, 25(9–10):1299–1311, 2001.

A. Wächter and L. T. Biegler. On the implementation of an interior-point filter line-search algorithm for large-scale nonlinear programming. Mathematical Programming, 106:25–57, 2006.

A. W. Westerberg and F. C. Edie. Computer-aided design, Part 1 Enhancing Convergence Properties by the Choice of Output Variable Assignments in the Solution of Sparse Equation Sets. The Chemical Engineering Journal, 2:9–16, 1971a.

A. W. Westerberg and F. C. Edie. Computer-Aided Design, Part 2 An approach to convergence and tearing in the solution of sparse equation sets. Chem. Eng. J., 2(1):17–25, 1971b.