Graph theory, irreducibility, and structural analysis of differential-algebraic equation systems

The $\Sigma$-method for structural analysis of a differential-algebraic equation (DAE) system produces offset vectors from which the sparsity pattern of a system Jacobian is derived. This pattern implies a block-triangular form (BTF) of the DAE that can be exploited to speed up numerical solution. The paper compares this fine BTF with the usually coarser BTF derived from the sparsity pattern of the \sigmx. It defines a Fine-Block Graph with weighted edges, which gives insight into the relation between coarse and fine blocks, and the permitted ordering of blocks to achieve BTF. It also illuminates the structure of the set of normalised offset vectors of the DAE, e.g.\ this set is finite if and only if there is just one coarse block

How AD Can Help Solve Differential-Algebraic Equations

A characteristic feature of differential-algebraic equations is that one needs to find derivatives of some of their equations with respect to time, as part of so called index reduction or regularisation, to prepare them for numerical solution. This is often done with the help of a computer algebra system. We show in two significant cases that it can be done efficiently by pure algorithmic differentiation. The first is the Dummy Derivatives method, here we give a mainly theoretical description, with tutorial examples. The second is the solution of a mechanical system directly from its Lagrangian formulation. Here we outline the theory and show several non-trivial examples of using the "Lagrangian facility" of the Nedialkov-Pryce initial-value solver DAETS, namely: a spring-mass-multipendulum system, a prescribed-trajectory control problem, and long-time integration of a model of the outer planets of the solar system, taken from the DETEST testing package for ODE solvers

Towards Integrating Hybrid DAEs with a High-Index DAE Solver

J.D. Pryce and N.S. Nedialkov have developed a Taylor series method and a C++ package, DaeTs, for solving numerically an initial-value problem differential-algebraic equation (DAE) that can be of high index, high order, nonlinear, and fully implicit. Numerical results have shown this method to be efficient and very accurate, and particularly suitable for problems that are of too high an index for present DAE solvers. However, DaeTs cannot be applied to systems of DAEs that change at points it time, also called hybrid or multi-mode DAEs. This paper presents methods for extending Daets with the capability to integrate hybrid DAEs. Methods for event location and consistent initializations are given. Daets is applied to simulate a model of a parallel robot: a hybrid system of index-3 DAEs with closed-loop control

WFPC2 Observations of Massive and Compact Young Star Clusters in M31

We present color magnitude diagrams of four blue massive and compact star clusters in M31: G38, G44, G94, and G293. The diagrams of the four clusters reveal a well-populated upper main sequence and various numbers of supergiants. The U-B and B-V colors of the upper main sequence stars are used to determine reddening estimates of the different lines of sight in the M31 disk. Reddening values range from E(B-V) = 0.20 +/- 0.10 to 0.31 +/- 0.11. We statistically remove field stars on the basis of completeness, magnitude and color. Isochrone fits to the field-subtracted, reddening-corrected diagrams yield age estimates ranging from 63 +/- 15 Myr to 160 +/- 60 Myr. Implications for the recent evolution of the disk near NGC 206 are discussed.Comment: 17 pages, Latex, ApJ, in Pres

Towards Integrating Hybrid DAEs with a High-Index DAE Solver

J.D. Pryce and N.S. Nedialkov have developed a Taylor series method and a C++ package, DaeTs, for solving numerically an initial-value problem differential-algebraic equation (DAE) that can be of high index, high order, nonlinear, and fully implicit. Numerical results have shown this method to be efficient and very accurate, and particularly suitable for problems that are of too high an index for present DAE solvers. However, DaeTs cannot be applied to systems of DAEs that change at points it time, also called hybrid or multi-mode DAEs. This paper presents methods for extending Daets with the capability to integrate hybrid DAEs. Methods for event location and consistent initializations are given. Daets is applied to simulate a model of a parallel robot: a hybrid system of index-3 DAEs with closed-loop control

Multibody dynamics in natural coordinates through automatic differentiation and high-index DAE solving

The Natural Coordinates (NCs) method for Lagrangian modelling and simulation of multibody systems is valued for giving simple, sparse models. We describe our version of it and compare with the classical approach of Jal´on and Bayo (JBNCs). Our NCs use the high-index differential-algebraic equation solver Daets. Algorithmic differentiation, not symbolic algebra, forms the equations of motion from the Lagrangian. We obtain significantly smaller equation systems than JBNCs, at the cost of a non-constant mass matrix for fully 3D models—a minor downside in the Daets context. Examples in 2D and 3D are presented, with numerical results