Completions of nonlinear DAE flows based on index reduction techniques and their stabilization

Differential algebraic equations (DAEs) define a differential equation on a manifold. A number of ways have been developed to numerically solve some classes of DAEs. Motivated by problems in control theory, numerical simulation, and the use of general purpose modeling environments, recent research has considered the embedding of the DAE solutions of a general DAE into the solutions of an ODE where the added dynamics have special properties. This paper both provides new results on the linear time-varying case and considers the important nonlinear case.     

Convergence of multistep discretizations of DAEs

Standard ODE methods such as linear multistep methods encounter difficulties when applied to differential-algebraic equations (DAEs) of index greater than 1. In particular, previous results for index 2 DAEs have practically ruled out the use of all explicit methods and of implicit multistep methods other than backward difference formulas (BDFs) because of stability considerations. In this paper we embed known results for semi-explicit index 1 and 2 DAEs in a more comprehensive theory based on compound multistep and one-leg discretizations. This explains and characterizes the necessary requirements that a method must fulfill in order to be applicable to semi-explicit DAEs. Thus we conclude that the most useful discretizations are those that avoid discretization of the constraint. A freer use of e.g. explicit methods for the non-stiff differential part of the DAE is then possible.Dedicated to Germund Dahlquist on the occasion of his 70th birthdayThis author thanks the Centro de Estadística y Software Matemático de la Universidad Simón Bolivar (CESMa) for permitting her free use of its research facilities.Partial support by the Swedish Research Council for Engineering Sciences TFR under contract no. 222/91-405.     

Mixed symbolic–numerical computations with general DAEs I: System properties

Differential-algebraic equations (DAEs) arise in many ways in many types of problems. In this expository paper we discuss a variety of situations where we have found mixed symbolic-numerical calculations to be essential. The paper is designed to both familiarize the reader with several fundamental DAE ideas and to present some applications. The situations discussed include the analysis of DAEs, the solution of DAEs, and applications which include DAEs. Both successes and challenges will be presented. This revised version was published online in June 2006 with corrections to the Cover Date.     

Wavelet-based adaptive grids for multirate partial differential-algebraic equations

The mathematical model of electric circuits yields systems of differential-algebraic equations (DAEs). In radio frequency applications, a multivariate model of oscillatory signals transforms the DAEs into a system of multirate partial differential-algebraic equations (MPDAEs). Considering quasiperiodic signals, an approach based on a method of characteristics yields efficient numerical schemes for the MPDAEs in time domain. If additionally digital signal structures occur, an adaptive grid is required to achieve the efficiency of the technique. We present a strategy applying a wavelet transformation to construct a mesh for resolving steep gradients in respective signals. Consequently, we employ finite difference methods to determine an approximative solution of characteristic systems in according grid points. Numerical simulations demonstrate the performance of the adaptive grid generation, where radio frequency signals with digital structures are resolved.     

Global error estimation for index-1 and -2 DAEs

In this paper, we consider the extension of three classical ODE estimation techniques (Richardson extrapolation, Zadunaisky's technique and solving for the correction) to DAEs. Their convergence analysis is carried out for semi-explicit index-1 DAEs solved by a wide set of Runge-Kutta methods. Experimentation of the estimation techniques with RADAU5 is also presented: their behaviour for index-1 and -2 problems, and for variable step size integration is investigated. This revised version was published online in June 2006 with corrections to the Cover Date.     

Consistent initialization for DAEs in Hessenberg form

For nonlinear DAEs, we can hardly make a reasonable statement unless structural assumptions are given. Many results are restricted to explicit DAEs, often in Hessenberg form of order up to three. For the DAEs resulting from circuit simulation, different beneficial structures have been found and exploited for the computation of consistent initial values. In this paper, a class of DAEs in nonlinear Hessenberg form of arbitrary high order is defined and analyzed with regard to consistent initialization. For this class of DAEs, the hidden constraints can be systematically described and the consistent initialization can be determined step-by-step solving linear subproblems, an approach hitherto used for the DAEs resulting from circuit simulation. Finally, it is shown that the DAEs resulting from mechanical systems fulfill the defined structural assumptions. The algorithm is illustrated by several examples.     

Element-based index reduction in electrical circuit simulation

The numerical simulation of very large scale integrated circuits is an important tool in the development of new industrial circuits. In the course of the last years, this topic has received increasing attention. Common modeling approaches for circuits lead to differential-algebraic systems (DAEs). In circuit simulation, these DAEs are known to have index 2, given some topological properties of the network. This higher index leads to several undesirable effects in the numerical solution of the DAEs. Recent approaches try to lower the index of DAEs to improve the numerical behaviour. These methods usually involve costly algebraic transformations of the equations. Especially, for large scale circuit equations, these transformations become too costly to be efficient. We will present methods that change the topology of the network itself, while replacing certain elements in oder to obtain a network that leads to a DAE of index 1, while not altering the analytical solution of the DAE. This procedure can be performed prior to the actual numerical simulation. The decreasing of the index usually leads to significantly improved numerical behaviour. (© 2006 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Trajectory sensitivity analysis of hybrid systems with sliding motion

This paper concerns hybrid control systems exhibiting the sliding motion. It is assumed that the system’s motion on the switching surface is described by index-2 differential–algebraic equations (DAEs), which guarantee the accurate tracking of the sliding motion surface. For those systems the sensitivity analysis is performed with the help of solutions to system’s linearized equations. The paper states conditions under which the solutions to the linearized equations for original DAEs and the solutions to linearized equations for underlying ordinary differential equations (ODEs) exhibit similar properties. Due to the presence of sliding motion, we restrict the class of admissible control functions to piecewise differentiable functions. The presented sensitivity analysis might be useful in deriving the weak maximum principle for optimal control problems with hybrid systems exhibiting sliding motion and in establishing the global convergence of algorithms for solving those problems.     

Theory of formal integrability and DAEs: effective computations

This paper deals with the computation of the formally integrable systems underlying a given quasi-linear polynomial DAE. We use as stopping condition the criterium of differential stability, which happens to be equivalent to the formal integrability in dimension 1. A symbolic method is developed to compute effectively a finite collection of so-called triangular stable DAEs, whose solutions are precisely all the solutions of the initial system. Besides, this algorithm enables to determine the generic points of a triangular DAE, by checking the non-nullity of a single polynomial. This revised version was published online in June 2006 with corrections to the Cover Date.     

A Simple Structural Analysis Method for DAEs

We describe a straightforward method for analysing the structure of a differential-algebraic system. It generalizes the method of Pantelides, but is more directly informative and applies to DAEs with derivatives of any order. It naturally leads to a numerical method for the initial value problem that combines projection and index reduction. We illustrate the method by examples, and justify it with proofs. We prove that it succeeds on a fairly wide class of systems encountered in practice, and show its relation to the Pantelides method and to the Campbell-Gear derivative-array equations.This revised version was published online in October 2005 with corrections to the Cover Date.     

计算微分代数系统稳定域的迭代Lyapunov函数方法

用迭代Lyapunov函数方法对微分代数系统稳定域进行了研究,根据所研究的微分代数系统形式,构造一个Lyapunov函数,然后对这个Lyapunov函数进行逐次迭代,给出了微分代数系统稳定域逐次扩大的迭代算法,数值实验表明迭代Lyapunov函数方法应用于微分代数系统稳定域的估计比单个Lyapunov函数具有良好的优越性。     

Existence and uniqueness of optimal solutions for multirate partial differential algebraic equations

The numerical simulation of electric circuits including multirate signals can be done by a model based on partial differential algebraic equations. In the case of frequency modulated signals, a local frequency function appears as a degree of freedom in the model. Thus the determination of a solution with a minimum amount of variation is feasible, which allows for resolving on relatively coarse grids. We prove the existence and uniqueness of the optimal solutions in the case of initial-boundary value problems as well as biperiodic boundary value problems. The minimisation problems are also investigated and interpreted in the context of optimal control. Furthermore, we construct a method of characteristics for the computation of optimal solutions in biperiodic problems. Numerical simulations of test examples are presented.     

THE EXISTENCE AND UNIQUENESS OF THE SOLUTIONS OF DAES

1Intr0ducti0nDifferential-algebraicequations(DAEs)areveryusefu1inwidefields(cf.[1]).Bydifferential-algebraicequations,wemeanthoseequati0nswhosepartsof"derivative"cann0tbeexpressedexplicitly.Forexample,weconsidertheimplicitdifferentialequationwithmappingFsm00thssufficient1y.Itisusuallyreferredt0adifferential-algebraicequation(DAE)whentherank0fD.F(t,x,p)islessthann,wheretheremightbesomepurea1gebraic,whichwecallc0nstraintequations.TheDAEs,inparticular,theexistenceanduniquenessofitssolutions…     

Mixed symbolic–numerical computations with general DAEs II: An applications case study

A variety of theorems and properties of nonlinear DAEs were discussed in part I. This paper illustrates many of these ideas within the context of analyzing a specific nonlinear system that exhibits a variety of interesting features. This revised version was published online in June 2006 with corrections to the Cover Date.     

微分代数方程去奇异化分析

利用去奇异化方法讨论了拟线性微分代数方程在奇点邻域内光滑解的性质.通过尺度参数的微分同胚变换,将拟线性微分代数方程转化为相应的常微分方程,从而构造出在孤立奇点邻域内的初始微分代数方程的光滑解,给出解存在的充分条件,并进一步讨论了解的性质.     

SOlvaBILITY OF HIGHER INDEX TIME-VARYING LINEAR DIFFERENTIAL-ALGEBRAIC EQUATIONS

1 IntroductionNormal differential-algebraic equatiOns (DAEs) are siugular ordiuary differe11tial equations(ODEs)f(x,(t),x(f),f) = 0, (1.1)wllere the partial Jacobian f;(y, x, f) E L(n") is everywliere singular but has constant rank.Such systelns are of special interest in view of various applicatiOns, e.g. electrical networks,constrailled lllecl1anica1 systenis of rigid bodies, coutrol theory, singular perturbatio11 and dis-cretization of partia1 differential equations, etc. (cf [1,2,3]).I…     

ON A REGULARIZATION OF INDEX 2 DIFFERENTIAL-ALGEBRAIC EQUATIONS WITH PROPERLY STATED LEADING TERM

In this article, linear regular index 2 DAEs A(t)[D(t)x(t)]′+B(t)x(t)=q(t) are considered. Using a decoupling technique, initial condition and boundary condition are properly formulated. Regular index 1 DAEs are obtained by a regularization method. We study the behavior of the solution of the regularization system via asymptotic expansions. The error analysis between the solutions of the DAEs and its regularization system is given.     

热弹性动力学耦合问题的微分代数方法

对一般的热机械问题提出了一种有效的数值方法,并对二维的热弹性问题进行了测试．该方法的基本思路是将描述热机械耦合问题的偏微分方程进行降阶,使之成为一组微分代数方程,应力应变关系被写成代数方程．所得到的微分代数系统采用全隐式的向后差分公式进行求解．对该方法进行了详细的说明．为了验证该方法的有效性,将其应用于一个动态非耦合的热弹性问题的求解和一个耦合的二维热弹性问题的求解．