Numerical Treatment of Second Order Differential-Algebraic Systems

We consider the numerical treatment of systems of second order differential-algebraic equations (DAEs). The classical approach of transforming a second order system to first order by introducing new variables can lead to difficulties such as an increase in the index or the loss of structure. We show how we can compute an equivalent strangeness-free second order system using the derivative array approach and we present Runge-Kutta methods for the direct numerical solution of second order DAEs. (© 2006 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

A Structure Preserving Index Reduction Method For MNA

Transient analysis in industrial chip design leads to very large systems of differential-algebraic equations (DAEs). The numerical solution of these DAEs strongly depends on the so called index of the DAE. In general, the higher the index of the DAE is, the more sensitive the numerical solution will be to errors in the computation. So, it is advisable to use mathematical models with small index or to reduce the index. This paper presents an index reduction method that uses information based on the topology of the circuit. In addition, we show that the presented method retains structural properties of the DAE. (© 2006 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

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)     

Constraint preserving integrators for general nonlinear higher index DAEs

Summary. In the last few years there has been considerable research on numerical methods for differential algebraic equations (DAEs) where is identically singular. The index provides one measure of the singularity of a DAE. Most of the numerical analysis literature on DAEs to date has dealt with DAEs with indices no larger than three. Even in this case, the systems were often assumed to have a special structure. Recently a numerical method was proposed that could, in principle, be used to integrate general unstructured higher index solvable DAEs. However, that method did not preserve constraints. This paper will discuss a modification of that approach which can be used to design constraint preserving integrators for general nonlinear higher index DAEs. Received August 25, 1993 / Revised version received April 7, 1994     

The Index of an Infinite Dimensional Implicit System

The idea of the index of a differential algebraic equation (DAE) (or implicit differential equation) has played a fundamental role in both the analysis of DAEs and the development of numerical algorithms for DAEs. DAEs frequently arise as partial discretizations of partial differential equations (PDEs). In order to relate properties of the PDE to those of the resulting DAE it is necessary to have a concept of the index of a possibly constrained PDE. Using the finite dimensional theory as motivation, this paper will examine what one appropriate analogue is for infinite dimensional systems. A general definition approach will be given motivated by the desire to consider numerical methods. Specific examples illustrating several kinds of behavior will be considered in some detail. It is seen that our definition differs from purely algebraic definitions. Numerical solutions, and simulation difficulties, can be misinterpreted if this index information is missing.     

A new approach to the incorporation of servo constraints in multibody dynamics

A new index reduction approach is developed to solve the servo constraint problems [2] in the inverse dynamics simulation of underactuated mechanical systems. The servo constraint problem of underactuated systems is governed by differential algebraic equations (DAEs) with high index. The underlying equations of motion contain both holonomic constraints and servo constraints in which desired outputs (specified in time) are described in terms of state variables. The realization of servo constraints with the use of control forces can range from orthogonal to tangential [3]. Since the (differentiation) index of the DAEs is often higher than three for underactuated systems, in which the number of degrees of freedom is greater than the control outputs/inputs, we propose a new index reduction method [1] which makes possible the stable numerical integration of the DAEs. We apply the proposed method to differentially flat systems, such as cranes [1,4,5], and non-flat underactuated systems. (© 2016 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Beyond conventional Runge–Kutta methods in numerical integration of ODEs and DAEs by use of structures and local models

There are two parts in this paper. In the first part we consider an overdetermined system of differential-algebraic equations (DAEs). We are particularly concerned with Hamiltonian and Lagrangian systems with holonomic constraints. The main motivation is in finding methods based on Gauss coefficients, preserving not only the constraints, symmetry, symplecticness, and variational nature of trajectories of holonomically constrained Hamiltonian and Lagrangian systems, but also having optimal order of convergence. The new class of (s,s)$(s,s)$-Gauss–Lobatto specialized partitioned additive Runge–Kutta (SPARK) methods uses greatly the structure of the DAEs and possesses all desired properties. In the second part we propose a unified approach for the solution of ordinary differential equations (ODEs) mixing analytical solutions and numerical approximations. The basic idea is to consider local models which can be solved efficiently, for example analytically, and to incorporate their solution into a global procedure based on standard numerical integration methods for the correction. In order to preserve also symmetry we define the new class of symmetrized Runge–Kutta methods with local model (SRKLM).     

Numerical Methods for Stiff Index-3 DAEs

Space semidiscretization of PDAEs, i.e. coupled systems of PDEs and algebraic equations, give raise to stiff DAEs and thus the standard theory of numerical methods for DAEs is not valid. As the study of numerical methods for stiff ODEs is done in terms of logarithmic norms, it seems natural to use also logarithmic norms for stiff DAEs. In this paper we show how the standard conditions imposed on the PDAE and the semidiscretized problem are formally the same if they are expressed in terms of logarithmic norms. To study the mathematical problem and their numerical approximations, this link between the standard conditions and logarithmic norms allow us to use for stiff DAEs techniques similar to the ones used for stiff ODEs. The analysis is done for problems which appear in the context of elastic multibody systems, but once the tools, i.e., logarithmic norms, are developed, they can also be used for the analysis of other PDAEs/DAEs.     

Detecting structures in differential algebraic equations: Computational aspects

Differential algebraic equations (DAEs) are often automatically generated, in particular, by coupling different tools. These DAEs are unstructured in the sense that they do not reveal their mathematical structure a priori. In view of a reliable treatment of those DAEs, their mathematical structure should be uncovered and monitored also by computational methods. We discuss several computational aspects of the tractability index concept.     

Iterative Solution of SPARK Methods Applied to DAEs

In this article a broad class of systems of implicit differential–algebraic equations (DAEs) is considered, including the equations of mechanical systems with holonomic and nonholonomic constraints. Solutions to these DAEs can be approximated numerically by applying a class of super partitioned additive Runge–Kutta (SPARK) methods. Several properties of the SPARK coefficients, satisfied by the family of Lobatto IIIA-B-C-C^*-D coefficients, are crucial to deal properly with the presence of constraints and algebraic variables. A main difficulty for an efficient implementation of these methods lies in the numerical solution of the resulting systems of nonlinear equations. Inexact modified Newton iterations can be used to solve these systems. Linear systems of the modified Newton method can be solved approximately with a preconditioned linear iterative method. Preconditioners can be obtained after certain transformations to the systems of nonlinear and linear equations. These transformations rely heavily on specific properties of the SPARK coefficients. A new truly parallelizable preconditioner is presented.     

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…     

Some remarks on solvability and various indices for implicit differential equations

It has been shown [17,18,21] that the notion of index for DAEs (Differential Algebraic Equations), or more generally implicit differential equations, could be interpreted in the framework of the formal theory of PDEs. Such an approach has at least two decisive advantages: on the one hand, its definition is not restricted to a “state-space” formulation (order one systems), so that it may be computed on “natural” model equations coming from physics (which can be, for example, second or fourth order in mechanics, second order in electricity, etc.) and there is no need to destroy this natural way through a first order rewriting. On the other hand, this formal framework allows for a straightforward generalization of the index to the case of PDEs (either “ordinary” or “algebraic”). In the present work, we analyze several notions of index that appeared in the literature and give a simple interpretation of each of them in the same general framework and exhibit the links they have with each other, from the formal point of view. Namely, we shall revisit the notions of differential, perturbation, local, global indices and try to give some clarification on the solvability of DAEs, with examples on time-varying implicit linear DAEs. No algorithmic results will be given here (see [34,35] for computational issues) but it has to be said that the complexity of computing the index, whatever approach is taken, is that of differential elimination, which makes it a difficult problem. We show that in fact one essential concept for our approach is that of formal integrability for usual DAEs and that of involution for PDEs. We concentrate here on the first, for the sake of simplicity. Last, because of the huge amount of work on DAEs in the past two decades, we shall mainly mention the most recent results. This revised version was published online in June 2006 with corrections to the Cover Date.     

Numerical integration of positive linear differential-algebraic systems

In the simulation of dynamical processes in economy, social sciences, biology or chemistry, the analyzed values often represent non-negative quantities like the amount of goods or individuals or the density of a chemical or biological species. Such systems are typically described by positive ordinary differential equations (ODEs) that have a non-negative solution for every non-negative initial value. Besides positivity, these processes often are subject to algebraic constraints that result from conservation laws, limitation of resources, or balance conditions and thus the models are differential-algebraic equations (DAEs). In this work, we present conditions under which both these properties, the positivity as well as the algebraic constraints, are preserved in the numerical simulation by Runge–Kutta or multistep discretization methods. Using a decomposition approach, we separate the dynamic and the algebraic equations of a given linear, positive DAE to give positivity preserving conditions for each part separately. For the dynamic part, we generalize the results for positive ODEs to DAEs using the solution representation via Drazin inverses. For the algebraic part, we use the consistency conditions of the discretization method to derive conditions under which this part of the approximation overestimates the exact solution and thus is non-negative. We analyze these conditions for some common Runge–Kutta and multistep methods and observe that for index-1 systems and stiffly accurate Runge–Kutta methods, positivity is conditionally preserved under similar conditions as for ODEs. For higher index problems, however, none of the common methods is suitable.     

Optimal Control of Differential–Algebraic Equations of Higher Index,Part 2: Necessary Optimality Conditions

This paper deals with optimal control problems described by higher index DAEs. We introduce a class of problems which can be transformed to index one control problems. For these problems we show in the accompanying paper that, if the solutions to the adjoint equations are well–defined, then the first-order approximations to the functionals defining the problem can be expressed in terms of the adjoint variables. In this paper we show that the solutions to the adjoint equations are essentially bounded measurable functions. Then, based on the first order approximations, we derive the necessary optimality conditions for the considered class of control problems. These conditions do not require the transformation of the DAEs to index-one system; however, higher-index DAEs and their associated adjoint equations have to be solved.     

Some software aspects of DAE simulation & optimization software

Considerable numerical software has been written for simulation and optimization of dynamical systems. From the beginning of their development, differential algebraic equations (DAEs) have often been proposed as a way to make modeling easier. The modeler need only write down equations relating the variables in the model. However, much DAE software requires at least as much user numerical and mathematical expertise as explicit methods. An important aspect of our research has been working toward helping the idea of DAEs achieve its promise in modeling and simulation by both pushing the software to handle more general problems and to also allow for less user expertise. Some recent examples are presented where this research impacts on software and their underlying algorithms. Space necessitates we assume the reader has a rough idea of what a DAE is. The examples are implicit Scicos, and optimization of DAE models. (© 2008 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Variational splines for the numerical solution of singular differential equations

The numerical parametrization method (PM), originally created for optimal control problems, is specificated for classical calculus of variation problems that arise in connection with singular implicit (IDEs) and differential-algebraic equations (DAEs). The PM for IDEs is based on representation of the required solution as a spline with moving knots and on minimization of the discrepancy functional with respect to the spline parameters. Such splines are named variational splines. For DAEs only finite entering functions can be represented by splines, and the functional under minimization is the discrepancy of the algebraic subsystem. The first and the second derivatives of the functionals are calculated in two ways – for DAEs with the help of adjoint variables, and for IDE directly. The PM does not use the notion of differentiation index, and it is applicable to any singular equation having a solution. (© 2008 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Solution of Index 2 Implicit Differential-Algebraic Equations by Lobatto Runge-Kutta Methods

We consider the numerical solution of systems of index 2 implicit differential-algebraic equations (DAEs) by a class of super partitioned additive Runge–Kutta (SPARK) methods. The families of Lobatto IIIA-B-C-C^*-D methods are included. We show super-convergence of optimal order 2s–2 for the s-stage Lobatto families provided the constraints are treated in a particular way which strongly relies on specific properties of the SPARK coefficients. Moreover, reversibility properties of the flow can still be preserved provided certain SPARK coefficients are symmetric.     

The computation of consistent initial values for nonlinear index-2 differential–algebraic equations

The computation of consistent initial values for differential–algebraic equations (DAEs) is essential for starting a numerical integration. Based on the tractability index concept a method is proposed to filter those equations of a system of index-2 DAEs, whose differentiation leads to an index reduction. The considered equation class covers Hessenberg-systems and the equations arising from the simulation of electrical networks by means of Modified Nodal Analysis (MNA). The index reduction provides a method for the computation of the consistent initial values. The realized algorithm is described and illustrated by examples.