Projected explicit and implicit Taylor series methods for DAEs

The recently developed new algorithm for computing consistent initial values and Taylor coefficients for DAEs using projector-based constrained optimization opens new possibilities to apply Taylor series integration methods. In this paper, we show how corresponding projected explicit and implicit Taylor series methods can be adapted to DAEs of arbitrary index. Owing to our formulation as a projected optimization problem constrained by the derivative array, no explicit description of the inherent dynamics is necessary, and various Taylor integration schemes can be defined in a general framework. In particular, we address higher-order Padé methods that stand out due to their stability. We further discuss several aspects of our prototype implemented in Python using Automatic Differentiation. The methods have been successfully tested on examples arising from multibody systems simulation and a higher-index DAE benchmark arising from servo-constraint problems.Peer Reviewe

The properties of differential-algebraic equations representing optimal control problems

This paper outlines a procedure for transforming a general optimal control problem to a system of Differential-Algebraic Equations (DAEs). The Kuhn-Tucker conditions consist of differential equations, complementarity conditions and corresponding inequalities. These latter are converted to equalities by the addition of a new variable combining the slack variable and the corresponding Lagrange multipliers. The sign of this variable indicates whether the constraint is active or not. The concept of the tractability index is introduced as a general purpose tool for determining the index of a system of DAEs by checking for the nonsingularity of the elements of the matrix chain. This is helpful in determining the well-conditioning of the problem, and an appropriate method for solving it numerically. In the examples used here, the solution of all the differential equations could be performed analytically. The given examples are tested by the numerical determination of the tractability index chain, and the results confirm the previously known properties of the examples

Systems of Differential Algebraic Equations in Computational Electromagnetics

Starting from space-discretisation of Maxwell's equations, various classical formulations are proposed for the simulation of electromagnetic fields. They differ in the phenomena considered as well as in the variables chosen for discretisation. This contribution presents a literature survey of the most common approximations and formulations with a focus on their structural properties. The differential-algebraic character is discussed and quantified by the differential index concept

A Projector Based Representation of the Strangeness Index Concept

The strangeness index concept is generalized and represented by a matrix chain similar to the structure of the tractability index. The properties of the related projectors are proven. A decoupling of the DAE and a representation of a solution is given

A Shooting Method for Fully Implicit Index-2 Differential-Algebraic Equations

A shooting method for two-point-boundary value problems for fully implicit index-1 and -2 differential-algebraic equations is presented. A combination of the shooting equations with a method of the calculation of consistent initial values leads to a system of nonlinear algebraic equations with nonsingular Jacobian. Examples are given

A new algorithm for the index determination in DAEs by Taylor series using Algorithmic Differentiation

We present an approach for determining the tractability index using truncated polynomial arithmetic. In particular, computing the index this way generates a sequence of matrices that contains itself derivatives. We realize the time differentiations using Algorithmic Differentiation techniques, specially by using the standard ADOL-C package with which calculating the derivatives becomes a simple shift and scaling of coefficients. We present the theory supporting the procedure we propose, as well as the implementation issues behind it to provide a convenient interface to the standard ADOL-C functionality. We give also examples of academic and practical problems and report several experimental results we have obtained with them

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 staring 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

Index determination of DAEs

The index definition of DAEs with properly stated leading term bases on a matrix sequence with suitably chosen projectors. A way of realization of this matrix sequence is presented, it includes the calculation of suitable projectors using generalized inverses of the sequence matrices