Sundials

Systems of differential-algebraic equations (DAEs) arise in many areas including chemical engineering, electrical circuit simulation, and robotics. Such systems are routinely generated by simulation and modeling environments, like MapleSim, Matlab/Simulink, and those based on the Modelica language. Before a simulation starts and a numerical solution method is applied, some kind of structural analysis (SA) is performed to determine the structure and the index of a DAE system. Structural analysis methods serve as a necessary preprocessing stage, and among them, Pantelides's graph-theory-based algorithm is widely used in industry. Recently, Pryce's Σ-method is becoming increasingly popular, owing to its straightforward approach and capability of analyzing high-order systems. Both methods are equivalent in the sense that (a) when one succeeds, producing a nonsingular Jacobian, the other also succeeds, and that (b) the two give the same structural index in the case of either success or failure. When SA succeeds, the structural results can be used to perform an index reduction process, or to devise a stage-by-stage solution scheme for computing derivatives or Taylor coefficients up to some order.Although such a success occurs on fairly many problems of interest, SA can fail on some simple, solvable DAEs with an identically singular Jacobian, and give incorrect structural information that usually includes the index. In this thesis, we focus on the iii Σ-method and investigate its failures. Aiming at making this SA more reliable, we develop two conversion methods for fixing SA failures. These methods reformulate a DAE on which the Σ-method fails into an equivalent problem on which SA is more likely to succeed with a nonsingular Jacobian. The implementation of our methods requires symbolic computations.We also combine our conversion methods with block triangularization of a DAE.Using a block triangular form of a Jacobian sparsity pattern, we identify which diagonal block(s) of the Jacobian is identically singular, and then perform a conversion on each singular block. This approach can reduce the computational cost and improve the efficiency of finding a suitable conversion for fixing SA's failures.

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“…To solve this DAE, a standard index-1 DAE solver can be used, for example, dassl [3], ida of sundials [21], or…”

Section: = F (M)mentioning

confidence: 99%

Conversion methods for improving structural analysis of differential-algebraic equation systems

2017

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“…f H 2 sh = nH 2 L, where nH 2 is H2 number density in a given cell and L is its physical size. The chemical network equations is solved by the CVODE package (Hindmarsh et al 2005). We assume that a gas has solar metallicity with the abundances given in Asplund et al (2005) We suppose that silicon is singly ionized and oxygen stays neutral.…”

Section: Modelmentioning

confidence: 99%

A Kennicutt–Schmidt relation at molecular cloud scales and beyond

Khoperskov

Vasiliev

2017

Monthly Notices of the Royal Astronomical Society

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Using N -body/gasdynamic simulations of a Milky Way-like galaxy we analyse a Kennicutt-Schmidt relation, Σ SFR ∝ Σ gas N , at different spatial scales. We simulate synthetic observations in CO lines and UV band. We adopt the star formation rate (SFR) defined in two ways: based on free fall collapse of a molecular cloud -Σ SFR, cl , and calculated by using a UV flux calibration -Σ SFR, UV . We study a KS relation for spatially smoothed maps with effective spatial resolution from molecular cloud scales to several hundred parsecs. We find that for spatially and kinematically resolved molecular clouds the Σ SFR, cl ∝ Σ gas N relation follows the power-law with index N ≈ 1.4. Using UV flux as SFR calibrator we confirm a systematic offset between the Σ SFR, UV and Σ gas distributions on scales compared to molecular cloud sizes. Degrading resolution of our simulated maps for surface densities of gas and star formation rates we establish that there is no relation Σ SFR, UV − Σ gas below the resolution ∼ 50 pc. We find a transition range around scales ∼ 50 − 120 pc, where the power-law index N increases from 0 to 1 − 1.8 and saturates for scales larger ∼ 120 pc. A value of the index saturated depends on a surface gas density threshold and it becomes steeper for higher Σ gas threshold. Averaging over scales with size of > ∼ 150 pc the power-law index N equals 1.3 − 1.4 for surface gas density threshold ∼ 5 M ⊙ pc −2 . At scales > ∼ 120 pc surface SFR densities determined by using CO data and UV flux, Σ SFR, UV /Σ SFR, cl , demonstrate a discrepancy about a factor of 3. We argue that this may be originated from overestimating (constant) values of conversion factor, star formation efficiency or UV calibration used in our analysis.

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“…Sundials' CVODE routine [52] was employed using backward differentiation with absolute and relative tolerances both set to 10 −12 . Equations (40) and (41)- (43) were solved using Muller's method with an absolute tolerance of 10 −5 .…”

Section: Numerical Implementationmentioning

confidence: 99%

Modeling of RAFT Copolymerization with Crosslinking of Styrene/Divinylbenzene in Supercritical Carbon Dioxide

López‐Domínguez

Hernández‐Ortiz

Vivaldo‐Lima

2017

Macro Theory & Simulations

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medium for radical polymerizations since it is nontoxic and has a readily accessible critical point, T c = 31 °C and P c = 74 bar (mild conditions compared to water whose critical conditions are T c = 374 °C and P c = 221 bar). [19] Furthermore, side reactions in CO 2 , such as chain transfer to solvent and formation of complexes with monomer and polymer, are not expected. [20] Reversible deactivation radical homoand copolymerizations in scCO 2 have been reported for RAFT, [21][22][23][24][25][26] ATRP, [27] and NMRP. [28,29] In these reports the polymerizations have been claimed to proceed in a controlled manner with low molar mass dispersities (Ð < 1.5). Modeling of (co)polymerization kinetics and molar mass development in scCO 2 for linear systems or conventional (uncontrolled) nonlinear ones has already been addressed in the literature. Since the polymer is often insoluble in scCO 2 , the polymerization proceeds in dispersion or precipitation in two reaction sites, the CO 2 -rich and the polymer-rich phases. [30] Comprehensive models allow computation of the full molar mass distribution (for linear cases) by using phase equilibrium, a kinetic description of the reaction and mass transport calculations. [31][32][33][34] On the other hand, simplified models, such as the method of moments, and simple equilibrium calculations have been also used to describe the behavior of these simpler systems. [35][36][37] To the best of our knowledge, the modeling of copolymerization kinetics, molar mass development, and gel fraction evolution for vinyl/divinyl monomers in the presence of RDRP controllers, in scCO 2 , has not been addressed or has been addressed assuming a single phase situation. [38] In this contribution, we use a multifunctional polymer molecule approach developed in our group, [39] along with a simple partition of the main components in the scCO 2 -rich and polymer-rich phases, to model the RAFT copolymerization with crosslinking of vinyl/divinyl monomers in scCO 2 . The resulting model represents a RAFT dispersion copolymerization (with crosslinking) process and it could very well describe a process where scCO 2 is replaced by water or any other appropriate solvent as the continuous phase. The model is validated by comparing calculated and experimental profiles of conversion versus time, number and weight average molar masses versus conversion, gel fraction and swelling index versus conversion and copolymer composition versus conversion for the RAFT copolymerization of styrene and divinylbenzene in scCO 2 . The experimental data used for model validation purposes was generated in our laboratory. [23] RAFT Copolymerization A kinetic model for the reversible addition-fragmentation chain transfer radical copolymerization of vinyl/divinyl monomers in supercritical carbon dioxide, using a multifunctional approach for polymer network formation, is presented. The process is assumed to proceed as a dispersion polymerization in three stages, with two phases: CO 2 -and polymer-rich phases. A simple model for partit...

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Sundials

Cited by 2,148 publications

References 30 publications

Conversion methods for improving structural analysis of differential-algebraic equation systems

Conversion methods for improving structural analysis of differential-algebraic equation systems

A Kennicutt–Schmidt relation at molecular cloud scales and beyond

Modeling of RAFT Copolymerization with Crosslinking of Styrene/Divinylbenzene in Supercritical Carbon Dioxide

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