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We present an approach for determining the tractability index using truncated polynomial arithmetic. ... We realize the time differentiations using Algorithmic Differentiation techniques, specially by using the standard ADOL-C package with which calculating the derivatives becomes a simple shift and scaling ... Algorithm for the index determination The new algorithm we propose to compute the tractability index of DAEs introduces the following features: The approximations of the matrices A(t), B(t), and D(t) and ...doi:10.18452/2821 fatcat:zzbhq5zr6ncvpbnn2h3lhadjsa
“Reich et al. have presented a geometric theory and an algo- rithm for reducing DAE of high differential index to DAE of low differential index. ... A DAE has dif- ferential index r if a minimum of r +1 differentiations of it are required before no new constraints are obtained. ...
Several new algorithms are proposed that in effect transform DAEs (Differential Algebraic Equations) to a special index one form that can be simulated with standard DAE integrators. ... It is expected that these new algorithms will help to treat large Modelica models of any index in a better way as it is currently possible. ... Acknowledgements The authors would like to thank Martin Arnold from University Halle-Wittenberg for discussions to understand the fine details of multibody integrators. ...doi:10.3384/ecp17132565 dblp:conf/modelica/OtterE17 fatcat:kp5e46dhhbcyhn6ulqga5aszrq
We propose a new algorithm for index reduction of DAEs and establish the notion of differential algebraic elimination, which can provide the differential algebraic resultant of the enlarged system of original ... To make use of structure of DAEs, variable pencil technique is given to determine the termination of differentiation. ... Shizhong Zhao for his valuable discussions about removing the extraneous factors from resultant computations. ...arXiv:1504.04977v1 fatcat:y3lih6anvvcxpnbzq4i7cyhtti
A DAE has differential index r if a minimum of r + 1 differentiations of it are required before no new constraints are obtained. ... Reich et al. have presented a geometric theory and an algorithm for reducing DAE of high differential index to DAE of low differential index. ... Ascher's suggestions to explore linearized perturbation analysis of DAE. Also acknowledged are helpful discussions with E. Mansfield and E. Hubert about singular solutions. ...doi:10.1111/1467-9590.00159 fatcat:prkjm747vzcklpovunho4wswju
For large sets of DAEs occurring in chemical engineering practice, index analysis requires computational procedures. This paper presents a new approach for automatic index analysis. ... For the general nonlinear DAE, the novel approach gives a more precise picture in most cases. ... The new pattern after one structural differentiation step, entering a star in the derivative column for each nonzero variable, is given in eq 11. ...doi:10.1021/ie0341754 fatcat:2kvrivqt4feutmdbcpkez3lqp4
Motivated by the characteristics of DAE systems with large dimensions, high index or block structures, we first propose a modified Pantelides' algorithm (MPA) for any high order DAEs based on the Σ matrix ... A new generation of universal tools and languages for modeling and simulation multi-physical domain applications has emerged and became widely accepted; they generate large-scale systems of differential ... Acknowledgments: The authors wish to thank the referee for his or her very helpful comments and useful suggestions. Conflicts of Interest: The authors declare no conflict of interest. ...doi:10.3390/math8112057 fatcat:ndtnevquivg57nplkxwcrflvpu
Several techniques exist for index reduction and consistent initialization of higher index DAEs. ... Many such techniques change the original set of equations by differentiation, substitution, and/or introduction of new variables. This paper introduces substitute equations as a new language element. ... In literature, several algorithms can be found for index reduction. ...doi:10.1076/mcmd.184.108.40.20646 fatcat:33gcu3iilzdrzkvdjsher7bqay
We study linear Differential Algebraic Equations, DAE, with time varying coefficients. Such equations ... This allows us to propose a new generalization of the global index and a definition for the singularities of the initial system. ... This will lead us to generalize also the index and to a natural definition for the singularities of (1). We want to point out that our goal is not to propose a general theory for DAE. ...doi:10.1145/220346.220374 dblp:conf/issac/QuereV95 fatcat:rnvf2aqa3ffmxflh3bovit3sfe
As an important consequence we both establish a basis for the design of low-order integrators for high index DAEs and develop guidelines for the use of predictors in integrating general high-index DAEs ... DAEs arise in many applications and a variety of numerical methods have been developed for solving DAEs. ...
For the dynamic case a new algorithm is proposed. This algorithm is scalable for large problems and is a promising diagnosis tool to spread the usage of EO dynamic simulators. ... In the field of process simulation, a movement from modular oriented, which is currently the most widely used technique, to Equation-Oriented (EO) is clear. ... Historically, the analysis of such systems was limited to degrees of freedom and index analysis. A new algorithm for structural analysis of DAE systems was proposed. ...doi:10.3182/20070606-3-mx-2915.00096 fatcat:txwtsseapjfkpfauqrhvfyfie4
We study a deflation method to reduce and to solve linear dfferential-algebraic equations (DAEs). It consists to define a sequence of DAEs with index reduction of one unit by step. ... Moreover, we explain how this method extends in the case of linear time-varying DAEs. ... Conclusion In this paper, a deflation algorithm for DAEs has been studied. It highlights the key role of index notions, especially the Kronecker index and the differentiation index. ...arXiv:1109.3778v1 fatcat:e6665xgv7jbefdzmp6fjmkldgi
The dummy derivative method is a technique commonly used to solve DAEs that can benefit from such exploitation of underlying DAE structures and information found in the Signature Matrix method. ... It is useful to consider how existing methods using the Pantelides algorithm can benefit from such structural analysis. ... , and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made. ...doi:10.1007/s10543-016-0642-9 fatcat:c6ipkztnc5dmrorqb43cxxsx24
This article proposes modifications of the Parareal algorithm for its application to higher index differential algebraic equations (DAEs). ... For differential algebraic equations with a special structure as e.g. given in flux-charge modified nodal analysis, it is shown that the usage of the implicit Euler method as a time integrator suffices ... Acknowledgement We would like to thank Diana Estévez Schwarz and Lennart Jansen for the fruitful discussions as well as Pia Callmer for her assistance in implementing the algorithm. ...arXiv:2107.07755v1 fatcat:jjkhnjx5gfci3bpj6lgfai45au
Rigorous techniques for consistent initialization of differential-algebraic systems (DAEs) are in general difficult to apply. ... Determining the analogous system and smoothening adequately those perturbations are the key aspects for the consistent initialization of the DAEs involved. ... Acknowledgements The authors would like to thank CAPES for financial support and Professor Uri Ascher (University of British Columbia, Vancouver, Canada) for many stimulating discussions, particularly ...doi:10.1016/s0098-1354(01)00702-5 fatcat:croisulvlzec7dog7aaurfqy2u
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