Index Reduction in Differential-Algebraic Equations Using Dummy Derivatives

Mattsson, Sven Erik; Söderlind, Gustaf

doi:10.1137/0914043

Cited by 259 publications

(219 citation statements)

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“…If a DAE is of index ≥ 2, then the DAE is of high index and we need a high-index DAE solver, such as radau5 for DAEs of index ≤ 3 [20], or daets for DAEs of any index [39]. Index reduction methods [24,29] and regularization techniques [25,55] exist, and can be used to reduce a high-index DAE to a DAE of index-1 or a regularized (and thus regular) DAE, respectively. Recent works by Pryce and McKenzie focus on the dummy derivatives (DDs) index reduction method [31,32,34,35,44,47].…”

Section: = F (M)mentioning

confidence: 99%

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

2017

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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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Section: = F (M)mentioning

confidence: 99%

“…Another research direction is combining the dummy derivative index reduction method [29] with our conversion methods. When the conditions for applying them are violated, appending some differentiated equations and replacing some "genuine" derivatives by dummy derivatives (which are algebraic) may make a conversion possible.…”

Section: Chapter 8 Conclusionmentioning

confidence: 99%

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

2017

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“…To deal with high-index DAEs, we utilize JModelica.org's index reduction, which is based on the method of dummy derivatives [49] and reduces the index of the DAE to at most 1. The method of dummy derivatives prevents the numerical drift of straightforward reduction by differentiation and is applicable to a wide class of DAEs.…”

Section: Differential-algebraic Equationsmentioning

confidence: 99%

Symbolic elimination in dynamic optimization based on block-triangular ordering

Magnusson

Åkesson

2017

Optimization Methods and Software

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We consider dynamic optimization problems for systems described by differential-algebraic equations (DAEs). Such problems are usually solved by discretizing the full DAE. We propose techniques to symbolically eliminate many of the algebraic variables in a preprocessing step before discretization. These techniques are inspired by the causalization and tearing techniques often used when solving DAE initial value problems. Since sparsity is crucial for some dynamic optimization methods, we also propose a novel approach to preserving sparsity during this procedure.The proposed methods have been implemented in the open-source JModelica.org platform. We evaluate the performance of the methods on a suite of optimal control problems solved using direct collocation. We consider both computational time and probability of solving the problem in a timely manner. We demonstrate that the proposed methods often are an order of magnitude faster than the standard way of discretizing the full DAE, and also significantly increase probability of successful convergence.

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“…(1)) are defined through the FixedTranslation and FixedRotation models. The linear system of equations under consideration is found as a strong connected component through Tarjan's algorithm [8] after index reduction has been applied [7]. Moreover, most Modelica compilers apply symbolic simplifications to the equations of motion.…”

Section: Equation Graphmentioning

confidence: 99%