Set Of Differential Algebraic Equations Research Articles

A two-dimensional approach to multibody free dynamics in space environment

The equation of motion of a multibody system, described here as a chain of rigid bars and revolute joints orbiting around the Earth, is derived. For each bar two translational and one rotational equilibrium equations are written. The forces acting on each body are the gravitational forces and the reaction forces (unknown) acting on it's end joints. The complete set of equilibrium equations consists of N X differential equations, where N X is the order of the state vector. The total number of unknowns is N X + N R where N R =2 N J and N J is the number of joints. The N R additional equations, to make the system determinate, are provided by the nondifferential compatibility equations. The resulting system is a set of differential algebraic equations (DAE) for which the well-known method of reducing the system to ordinary differential equations (ODE) is applied. Since the internal forces are associated with the relative displacements between the bodies, which are small fractions of the distance of the multibody spacecraft from the center of the Earth, the task of obtaining these forces from inertial coordinates, from a numerical viewpoint, could be impossible. So the problem is reformulated in such a way that the equation of motion of the system, contains global quantities where no internal forces appear, and local equations where internal forces do appear. In the latter one, only quantities of the same order of the spacecraft dimensions are present. Numerical results complete the work.

A New Formulation of Multi-Zone Combustion Engine Models

Cylinder pressure simulation has grown to become an important tool when developing and evaluating new engine concepts and control strategies. A new formulation of zero-dimensional multi-zone models is developed and described. A general model structure is formulated that rely on a set of differential algebraic equations that are easy to solve. The selected formulation also results in models that are easy to scale, i.e. add new zones, and to increase complexity, which is a result of the selected structure. A number of important issues that can cause problems when simulating the model are treated. It is shown: a) How a new zone is initialized. b) How variables of varying magnitude can be scaled to avoid numerical difficulties. c) How numerical errors accumulated during the simulation can be reduced by using a set of consistency equations.

Computer Aided Generation of Approximate DAE Systems for Symbolic Analog Circuit Design

AbstractThe behavior of nonlinear analog circuits is described by a set of differential algebraic equations (DAE). We are developing simplification and approximation methods which can be applied to this DAE system in order to generate a reduced symbolic system, where a user given error bound is assured automatically. Either the reduced system can be analyzed further to achieve a better understanding of the circuit's behavior or it can be used as a behavioral model which can be solved numerically easier and faster. In the following we will focus on two aspects of the algorithm: simplification of subexpressions and monitoring of the index during simplification.

Printed circuit board

Liquefied gases in energy supply chain suffer the risks of disastrous accidents (e.g. boiling liquid expanding vapor explosion) normally resulted from a sudden release. The boiling response and its associated pressure transient under sudden release determine the severity of the accident, and are crucial for assessing the consequences of the sudden release. For predicting the pressure transient, a thermodynamic model of the liquefied gas under sudden release is presented in this paper. A three-stage modelling approach is proposed to describe the entire thermodynamic process after sudden release according to the venting and boiling behaviors of liquefied gas. A set of differential-algebraic equations are established based on mass and energy conservation for predicting the transient thermodynamic parameters. Comparisons of model predictions with experimental data show a consistent trend in pressure-time histories, with relative deviations of average pressures less than 5.78 %. The results show that the minimum pressure point has a time delay compared to the boiling inception point. The increase of release pressure and the decrease of vessel scale will strengthen the pressure rebound, and the increase of vent area will lead to higher depressurization and re-pressurization rate.

Detailed Kinetic Models in the Context of Reactor Analysis: Linking Mechanistic and Process Chemistry

Detailed kinetic models for the modeling of complex chemistries, including thermal cracking, catalytic reforming, catalytic cracking, and hydroprocessing, offer the compelling advantage chemical significance at the mechanistic level. They carry a considerable burden, however, in terms of species, reactions, and associated rate parameters. This, together with the batch and the plug flow reactor balances, requires solution of a large system of either stiff ordinary differential equations (ODE) or stiff differential algebraic equations (DAE), for both homogeneous and heterogeneous processes. It is often faster numerically to solve a stiff system of ODEs and, thus, it can be useful to convert a system of DAEs to ODEs for numerical solution schemes. For heterogeneous PFR systems, the reactor steady-state balances result in a set of DAEs, and it would therefore be desirable to construct the associated set of ODEs to minimize CPU demand. To this end, we propose that such a transformation can be achieved by makin...

Lie Algebraic Control for the Stabilization of Nonlinear Multibody System Dynamics Simulation

It is well known that when equations of motion are formulated using Lagrange multipliers for multibody dynamic systems, one obtains a redundant set of differential algebraic equations. Numerical integration of these equations can lead to numerical difficulties associated with constraint violation drift. One approach that has been explored to alleviate this difficulty has been contraint stabilization methods. In this paper, a family of stabilization methods are considered as partial feedback linearizing controllers. Several stabilization methods including the range space method, null space method, Baumgarte's method, and the damping and stiffness penalty methods are examined. Each can be construed as a particular partial feedback linearizing controller. The paper closes by comparing several of these constraint stabilization methods to another method suggested by construction: the variable structure sliding (VSS) control. The VSS method is found to be the most efficient, stable, and robust in the presence of singularities.

PDEFIT: A Fortran code for data fitting in partial differential equations

PDEFIT is a computer program to estimate parameters in a system of one-dimensional differential equations and coupled ordinary differential equations. Equations without time-dependent derivatives are permitted. The model allows arbitrary transition conditions between separate integration areas for functions and derivatives, switching conditions and dynamic constraints. Proceeding from given experimental data, e.g. observation times and measurements, the distance of these measured data from the solution of a system of differential equations at designated spatial values is to be minimized in the L 2-L 1-, or L ∞-norm. The method of lines is used to discretize the partial differential equation with respect to polynomial approximation, difference formulae and several special upwind and related formulae for hyperbolic PDE's. The original system is transformed into a set of ordinary differential equations or, alternatively, into a set of differential algebraic equations, that is solved then by standard ODE or DAE solvers. We describe program organization and outline the usage of the corresponding FORTRAN code. A few examples are added to prove the feasibility of the proposed approach.

Globally Independent Coordinates for Real-Time Vehicle Simulation

Models of the dynamics of multibody systems generally result in a set of differential-algebraic equations (DAE). State-space methods for solving the DAE of motion are based on reduction of the DAE to ordinary differential equations (ODE), by means of local parameterizations of the constraint manifold that must be often modified during a simulation. In this paper it is shown that, for vehicle multibody systems, generalized coordinates that are dual to suspension and/or control forces in the model are independent for the entire range of motion of the system. Therefore, these additional coordinates, together with Cartesian coordinates describing the position and orientation of the chassis, form a set of globally independent coordinates. In addition to the immediate advantage of avoiding the computationally expensive redefinition of local parameterization in a state-space formulation, the existence of globally independent coordinates leads to efficient algorithms for recovery of dependent generalized coordinates. A topology based approach to identify efficient computational sequences is presented. Numerical examples with realistic vehicle handling models demonstrate the improved performance of the proposed approach, relative to the conventional Cartesian coordinate formulation, yielding real-time for vehicle simulation. [S1050-0472(00)00404-9]

Dynamic optimization of differential-algebraic systems through the dynamic simulator DYNSIM

A new structure of the simulator DYNSIM is proposed, where some of the design variables of the process are considered as control variables and are determined by applying the optimal control theory. An efficient algorithm to solve dynamic optimization problems for processes described by a set of differential-algebraic equations is developed and incorporated into DYNSIM.

The transition matrix for linear circuits

The state transition matrix /spl Phi/(t)=e/sup At/ plays an important role in the state variable analysis of linear time-invariant circuits. In this paper, we give a method to compute the equivalent matrix for a set of differential-algebraic equations. Specifically, the algorithm is illustrated for the modified nodal analysis (MNA). A benefit of using MNA formulation is that equation formulation is straightforward and computer-aided analysis of large circuits is simplified. Another benefit is that inconsistent initial conditions in the analysis of switched circuits is automatically and correctly handled by the algorithm. Comparison with the state variable approach is made, and application to the simulation of linear circuits is given. The method is based on circuit theory concepts accessible to all electrical engineers. In addition to the transition matrix, a numerical method to compute the zero state response of linear circuits for a restricted set of inputs is given. The transition matrix along with the zero state response results in a special algorithm that is used to compute the time response of lumped linear time-invariant circuits at equally spaced intervals of time. This method is compared with the solution of linear circuits by SPICE-like simulators.

Constraint logic programming for structure-based reasoning about dynamic physical systems

The paper describes a constraint logic programming approach for reasoning about dynamic physical systems based on structure. The approach takes a bond graph model of a system and computes a causal graph representation of its causal structure. Causal graphs can be used for causal explanations and for explaining the effect of modeling abstractions on causal structure. The paper shows how causal graphs can be used to compute a simulation model of a system in the form of a set of differential algebraic equations. The topological properties of the causal graph determine whether a simulation model is regular i.e. conforming to the criterion of ‘real-time representation’ or causality. In case the simulation model is not regular, a method is given for identifying all causal problems of a bond graph model.

An efficient method of simulating the detailed electromechanical response of industrial power systems

A new method is presented for simulating the large-disturbance, electromechanical response of mine and other industrial power systems. With this method, detailed waveforms that include the transient components of machine currents and bus voltages can be obtained rapidly. This approach may be a useful supplement to traditional steady-state analysis techniques when considering circuit breaker requirements in light of a variety of system transients including the following: starting of large motors or groups of motors, reclosure of motors following temporary isolation or during bus transfers, and rapid changes in the mechanical loading of motors. The short simulation times associated with this method are achieved by modeling the power system with line charging capacitance neglected. This eliminates high frequency modes, which often lead to long computation times in other detailed simulations, but retains the machine stator transients that affect significantly the instantaneous currents and voltages throughout the system. With shunt capacitance neglected, however, the system model is represented by a set of differential-algebraic equations (DAEs). The crux of the new method is a systematic procedure for establishing a readily simulated state-space model from these DAEs.

Parallel‐in‐time method based on shifted‐picard iterations for power system ty‐ansient stability analysis

AbstractThe transient stability analysis is the most time‐consuming computer simulation in power system studies. For this reason, in recent years, parallel processing has been applied for implementing on‐line time‐domain simulations of power system transient behaviour. As is well known, the problem of power system transient simulation consists, basically, in the solution of an initial value problem for a mixed set of Ordinary Differential Equations (ODE) and algebraic equations. The objective of this paper is to present an algorithm based on Shifted‐Picard dynamic iterations, which solves the above mentioned set of non‐linear Differential Algebraic Equations (DAE) by the iterative solution of a linear set of DAE. This feature is important because the time behaviour of the linear set of differential equations can be obtained by the solution of an integral. In this case, a multiprocessor parallel‐in‐time implementation of such algorithms is possible because each processor is devoted to the evaluation of the required variables relative to each time step. Moreover, parallelism‐in‐time allows multigrid techniques to be implemented in a natural way. Test results on realistic‐sized power systems are presented.

Parallel-in-time implementation of transient stability simulations on a transputer network

The most time consuming computer simulation in power system studies is the transient stability analysis. Parallel processing has been applied for time domain simulations of power system transient behavior. In this paper, a parallel implementation of an algorithm based on Shifted-Picard dynamic iterations is presented. The main idea is that a set of nonlinear Differential Algebraic Equations (DAEs), which describes the system, can be solved by the iterative solution of a linear set of DAEs. The time behavior of the linear set of differential equations can be obtained by the evaluation of the convolution integral. In the parallel-in-time implementation of the proposed algorithm, each processor is devoted to the evaluation of the complete set of variables relative to each time step. The quadrature formula, adopted for the integral evaluation, can be easily parallelized by using a number of processors equal to the number of time steps. The algorithm, implemented on a transputer network with 32 Inmos T800/20 adopting a uni-directional ring topology, has been tested on standard power systems.< <ETX xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">&gt;</ETX>

On the connection between nonlinear differential-algebraic equations and singularly perturbed control systems in nonstandard form

Considers a class of control systems represented by nonlinear differential equations depending on a small parameter. The systems are not in the standard singularly perturbed form, and therefore, one of the challenges is to show that the control systems do represent singularly perturbed two-time-scale systems. Assumptions are introduced which guarantee that an equivalent representation for the systems can be obtained in the standard singularly perturbed form, thereby justifying the two-time-scale property. The equations for the slow dynamics are characterized by a set of differential-algebraic equations which have been studied previosly in the literature. The fast dynamics are characterized by differential equations. Both the slow and the fast dynamics are easily derived and are defined in terms of variables that define the original control system. Control design for the class of systems being considered is studied using the composite control approach.< <ETX xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">&gt;</ETX>

Speedup and synchronisation overhead analysis of Gauss-Seidel type algorithms on a Sequent balance machine

The parallelisation and implementation of Gauss-Seidel (G-S) algorithms for power flow analysis have been investigated previously. Numerous runs to demonstrate the speedup have been illustrated on a Sequent Balance shared-memory multi-instruction, multidata access (SM MIMD) machine. The authors extend the idea and investigate-the effects of acceleration factors. It is shown on systems ranging from teens to thousands that when the acceleration factors are used, the implementation using colour-by-colour synchronisation is more reliable and has better convergence rate, even though it takes a longer time to synchronise. The authors also analyse the dependence of synchronisation overhead in terms of system sizes, network connection and number of processors. Comparisons between G-S and fast decoupled load-flow algorithms are also made. The implications on nCUBE implementations are also discussed. It is also shown that the idea of parallel G-S algorithm can be easily extended to solve the transient stability problem which involves a set of algebraic differential equations.

Non-standard singularly perturbed control systems and differential-algebraic equations

We consider a class of linear control systems represented by equations depending on a small parameter but which are not in the standard singularly perturbed form. One of the challenging system theoretic problem is to obtain an equivalent representation for the control system which, if possible, is in the standard singularly perturbed form. Assumptions are introduced which guarantee that an equivalent representation can be obtained which is in the standard singularly perturbed form, thereby justifying the two time scale property. The equations for the slow dynamics are characterized by a set of differential-algebraic equations which are easily derived from the original equations by setting the parameter to zero; the original assumptions guarantee the existence of solutions of the obtained differential-algebraic equations. The fast dynamics are characterized by different equations in terms of matrices that define the original control system. Control design for the system being considered is studied using the composite control approach. The problem of contract force and position regulation in a robot with its end effector in contact with a stiff surface is considered as an example

Algebraic theory of differential inequalities

We obtain differential-algebraic analogues of some basic theorems of real algebra and semialgebraic geometry. Proofs are based on: a differential version of the real spectrum of a differential ring containing Q; an Artin-Schreier theory for such rings; the model theory of ordered differential fields. Results include: an algebraic characterization of the differential inequalities which are consequences of a given finite set of algebraic differential equations and inequalities; a differential counterpart of the Hormander-Lojasiewicz inequality.

Non-linear dynamic phenomena in the behaviour of a railway wheelset model

A dicone moving on a pair of cylindrical rails can be considered as a simplified model of a railway wheelset. Taking into account the non-linear friction laws of rolling contact, the equations of motion for this non-linear mechanical system result in a set of differential-algebraic equations. Previous simulations performed with the differential-algebraic solver DASSL, [2], and experiments, [7], indicated non-linear phenomena such as limit-cycles, bifurcations as well as chaotic behaviour. In this paper the non-linear phenomena are investigated in more detail with the aid of special in-house software and the path-following algorithm PATH [10]. We apply Poincare sections and Poincare maps to describe the structure of periodic, quasiperiodic and chaotic motions. The analyses show that part of the chaotic behaviour of the non-linear system can be fully understood as a non-linear iterative process. The resulting stretching and folding processes are illustrated by series of Poincare sections.

Finite element methods for unsteady solidification problems arising in prediction of morphological structure

Galerkin finite element methods are presented for calculation of the dynamic transitions between planar and deep two-dimensional cellular interface morphologies in directional solidification of a binary alloy from models that include solute transport, the phase diagram, and the interfacial free energy between melt and crystals. The unknown melt-solid interface shape is accounted for in the finite element formulation by mapping the equations to a fixed domain. Novel nonorthogonal transformations are introduced combining cylindrical and Cartesian coordinate interface representations for approximating the deep cellular interfaces that evolve from a planar solidification front. The algorithm for time integration combines a fully implicit Adams-Moulton algorithm with the Isotherm-Newton method for solving the nonlinear set of differential-algebraic equations that result from the spatial discretization of the moving-boundary problem. The fully implicit scheme is found to be more accurate and efficient than an explicit predictor-corrector algorithm. Sample calculations show the connectivity between families of shapes with resonant spatial wavelengths.