System Of Nonlinear Differential-algebraic Equations Research Articles

Order Reduction of General Nonlinear DAE Systems by Automatic Tearing

The object-oriented approach to modelling has recently made possible to build models of large-scale real systems. However, the resulting system of equations is generally a nonlinear DAE (Differential Algebraic Equations) system of large dimension, which must be reduced in some way to make it tractable for numerical solutions. A way to do this is automatic symbolic tearing, which aims at splitting the DAE system into two parts: a core consisting of a reduced implicit DAE system and a set of explicit assignments. The problem is here dealt with by a graph theoretic approach, first proving the NP-completeness in the general case, then formulating the problem with reference to a bipartite graph and finally defining an efficient and flexible algorithm for automatic tearing. It is also shown how the proposed algorithm can easily incorporate both general and domain-specific heuristic rules, and can also be used to deal with equations in vector form. The application to serial multibody systems is considered as a significant example.

Convergence analysis of waveform relaxation for nonlinear differential-algebraic equations of index one

We give a new and simple convergence theorem on the waveform relaxation (WR) solution for a system of nonlinear differential-algebraic equations of index one. We show that if the norms of certain matrices derived from the Jacobians of the system functions are less than one, then the WR solution converges. The new sufficient condition includes previously reported conditions as special cases. Examples are given to confirm the theoretical analysis.

Monotone Waveform Relaxation for Systems of Nonlinear Differential-Algebraic Equations

We present a monotone waveform relaxation algorithm which produces very tight upper and lower bounds of the transient response of a class of systems described by nonlinear differential-algebraic equations (DAEs) that satisfy certain Lipschitz conditions. The choice of initial iteration is critical and we give two methods of finding it. We show that the class of systems in which monotone convergence of waveform relaxation is possible is actually larger than previously reported. Numerical experiments are given to confirm the monotonicity of convergence of the algorithm.

THE ROLE OF MEMBRANE ELECTROSTATICS IN THE REGULATION OF CELL VOLUME AND ION CONCENTRATIONS

We present a model to study how membrane surface negative charges can affect the electro-osmotic regulation properties of a cell. This model is based on the cellular analog proposed by Jakobsson, which includes passive and active ion transports; we further introduce the effect of membrane surface charges, using a generalized formulation of the Gouy–Chapman theory. We derive a system of nonlinear differential-algebraic equations (DAEs) which describes the dynamics of the cellular analog. The system admits a unique asymptotically stable stationary state, in which the Na-pump rate, which is crucial for electro-osmotic regulation, is inversely related to the Ca2+level in the extracellular milieu; numerical integration shows that this apparent inhibition of the Na-pump by external Ca2+results from a decrease in the electrostatic field produced by surface charges at the external side of the membrane. Furthermore, the degree of stability of the stationary state dramatically depends on the amount of negative charges on the membrane; a maximal stability is obtained for densities around - e /500 Å2, where the Na-pump is maximally activated by an increase in the Na content of the cytoplasm.

A direct translator from neutral model format to the SPARK simulation environment

The neutral model format (NMF) provides generic, machine-readable descriptions of equation-based component models of a physical system in a building description. It is intended to allow model interchange between different simulation environments. The simulation problem analysis and research kernel (SPARK) is a simulation environment that generates source and executable code for solving non-linear differential-algebraic equation systems. It is intended to automate the generation of modular, easy to maintain simulation programs. The goal is to generate simulation code automatically from NMF models transparently to the user. A translator achieving this goal is described here, which transforms NMF syntax into SPARK syntax using LEX and YACC for parsing and numerical inversion techniques generating SPARK code. Thus the practicality of NMF as a common description language for building simulation is demonstrated for the case of a simulation environment developed independently from NMF. The purpose of this paper is to give an account of the experience of building a translator from a general model representation format (NMF) to a particular simulation environment. We first present the NMF format, then the SPARK building simulation environment. Then the 1st translator from NMF to SPARK via a computer algebra system is presented, along with examples. That translator being too circuitous, a newer and direct translator is then presented, along with practical examples. Thus are demonstrated the usefulness of the NMF concept and the methodology for building a translator from it to an independently developed simulation environment for building science.

Optimal Estimation of States in Differential-Algebraic Equation Systems

Based on a minimum least squares criterion and without any assumption on process model error and measuring device noise, an optimal estimation algorithm has been developed for nonlinear differential-algebraic equation systems. The optimal estimator has been applied to an industrial scale fixed-bed reactor with decaying catalysts where phthalic anhydride is produced by the partial oxidation of o-xylene. Although a simplified reactor model was utilised by the estimator, using three real-time temperature measurements, the catalyst activity profile was accurately inferred and the dynamic behaviour of the reactor was excellently tracked. Computational experience indicated that the estimator could be implemented on-line in analogous way.

Algorithm 731: A moving-grid interface for systems of one-dimensional time-dependent partial differential equations

In the last decade, several numerical techniques have been developed to solve time-dependent partial differential equations (PDEs) in one dimension having solutions with steep gradients in space and in time. One of these techniques, a moving-grid method based on a Lagrangian description of the PDE and a smoothed-equidistribution principle to define the grid positions at each time level, has been coupled with a spatial discretization method that automatically discreizes the spatial part of the user-defined PDE following the method of lines approach. We supply two FORTRAN subroutines, CWRESU and CWRESX, which compute the residuals of the differential algebraic equations (DAE) system obtained from semidiscretizing, respectively, the PDE and the set of moving-grid equations. These routines are combined in an enveloping routine SKMRES, which delivers the residuals of the complete DAE system. To solve this stiff, nonlinear DAE system, a robust and efficient time-integrator must be applied, for example, a BDF method such as implemented in the DAE solvers SPRINT [Berzins and Furzeland 1985; 1986; Berzins et al. 1989] and DASSL [Brenan et al. 1989; Petzold 1983]. Some numerical examples are shown to illustrate the simple and effective use of this software interface.