Differential algebraic equation

The extension of the approach to constrained multibody systems and differential-algebraic equations affects the formulation of the multiple shooting method and the computation of sensitivity matrices. The former requires a more sophisticated treatment because variations of initial values and parameters may no longer be consistent with the algebraic equations. The latter can be done efficiently by exploiting the fact that the number of degrees of freedom of the system is reduced due to the presence of constraints. 

We first consider semi-explicit index 1 DAEs of the form 

As discussed in Sec. 5.1.2 a solution exists only for consistent initial values. However, if the initial values at the shooting nodes are degrees of freedom for 

To overcome this problem it has been proposed in [BES87] to solve in each subinterval [Tj Tj i] the m — 1 consistent extended initial value problems 

In order to ensure a continuous solution it has to be required additionally to the matching conditions (7.2.9) for y that the solution of the boundary value problem fulfills the original algebraic equations. Thus the pointwise constraints 

However, as discussed in any text on linear algebra, it is inefficient to actually compute the matrix inverse. Rather, the A matrix is decomposed into the product of upper and lower triangular matrices, 

Standard software for these operations is readily available. 

Many chemical-kinetics problems, such as the homogeneous mass-action kinetics problems discussed in Section 16.1, are easily posed as a system of standard-form ordinary differential equations (ODE), 

There are many high-quality, well-documented, software packages available to solve stiff problems in this form. However, one often encounters chemically reacting flow problems that are not easily posed as standard-form ODEs. In these cases problems can often be posed easily in a more general form, called differential-algebraic equations (DAE), 

The vector function g can be, and generally is, nonlinear. It may also have components that do not involve y (i.e., algebraic constraints). 

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Equations (4) and (9) along with (8) and (7) form the a set of the differential-algebraic equations dependent on X which describes the behaviour of the NA water. system, namely the conformational transitions in... 

Differential-algebraic equations can be written in the general notation... 

The Galerkin formulation results in a large set of differential/algebraic equations that are most easily expressed in the form... 

This equation must be solved for yn +l. The Newton-Raphson method can be used, and if convergence is not achieved within a few iterations, the time step can be reduced and the step repeated. In actuality, the higher-order backward-difference Gear methods are used in DASSL [Ascher, U. M., and L. R. Petzold, Computer Methods for Ordinary Differential Equations and Differential-Algebraic Equations, SIAM, Philadelphia (1998) and Brenan, K. E., S. L. Campbell, and L. R. Petzold, Numerical Solution of Initial-Value Problems in Differential-Algebraic Equations, North Holland Elsevier (1989)]. 

Off-line analysis, controller design, and optimization are now performed in the area of dynamics. The largest dynamic simulation has been about 100,000 differential algebraic equations (DAEs) for analysis of control systems. Simulations formulated with process models having over 10,000 DAEs are considered frequently. Also, detailed training simulators have models with over 10,000 DAEs. On-line model predictive control (MPC) and nonlinear MPC using first-principle models are seeing a number of industrial applications, particularly in polymeric reactions and processes. At this point, systems with over 100 DAEs have been implemented for on-line dynamic optimization and control. 

By discretizing the differential algebraic equations model using some standard discretization techniques (Liebman et al., 1992 Alburquerque and Biegler, 1996) to convert the differential constraints to algebraic constraints, the NDDR problem can be solved as the following NLP problem ... 

The application of simultaneous optimization to reactor-based flowsheets leads us to consider the more general problem of differentiable/algebraic optimization problems. Again, the optimization problem needs to be reconsidered and reformulated to allow the application of efficient nonlinear programming algorithms. As with flowsheet optimization, older conventional approaches require the repeated execution of the differential/algebraic equation (DAE) model. Instead, we briefly describe these conventional methods and then consider the application and advantages of a simultaneous approach. Here, similar benefits are realized with these problems as with flowsheet optimization. 

Ascher, U. M., and Petzold, L. R., Projected implicit Runge-Kutta methods for differential algebraic equations, SIAM J. Num. Anal. 28(4), 1097 (1991). 

Gritsis, D., Pantelides, C. C., and Sargent, R. W. H., The dynamic simulation of transient systems described by index two differential-algebraic equations, Proc. Third International PSE Symposium, Sydney, Australia, p. 132 (1988). 

Morison, K., Optimal control of processes described by systems of differential-algebraic equations, Ph.D. thesis. University of London (1984). 

Pelzold, L. R., Differential/algebraic equations are not ODEs, SIAM Journal on Scientific and Statistical Computing, No. 3, pp. 367-385 (1982). 

Renfro, J. G, Morshedi, A. M., and Asbjornsen, O. A., Simultaneous optimization and solution of systems described by differential/algebraic equations, Comp, and Chem. Eng. 11(5), 503-... 

Tjoa, I.-B., and Biegler, L. T., Simultaneous solution and optimization strategies for parameter estimation of differential-algebraic equations systems, I EC Research, 30, 376 (1991). 

The system of equations in the Von Mises form leads to a coupled system of nonlinear differential-algebraic equations. The transport equations (Eqs. 7.59 and 7.62) have parabolic characteristics, with the axial coordinate z being the timelike direction. The other three equations (Eqs. 7.60, 7.61, and 7.63) are viewed as algebraic constraints—in the sense that they have no timelike derivatives. 

An alternative to the standard-form representation is the differential-algebraic equation (DAE) representation, which is stated in a general form as g(r, y, y). The lower portion of Fig. 7.3 illustrates how the heat equation is cast into the DAE form. The boundary conditions can now appear as algebraic constraints (i.e., they have no time derivatives). For a problems as simple as the heat equations, this residual representation of the boundary conditions is not necessary. However, recall that implicit boundary-condition specification is an important aspect of solving boundary-layer equations. 

Following the very brief introduction to the method of lines and differential-algebraic equations, we return to solving the boundary-layer problem for nonreacting flow in a channel (Section 7.4). From the DAE-form discretization illustrated in Fig. 7.4, there are several important things to note. The residual vector F is structured as a two-dimensional matrix (e.g., Fuj represents the residual of the momentum equation at mesh point j). This organizational structure helps with the eventual software implementation. In the Fuj residual note that there are two timelike derivatives, u and p (the prime indicates the timelike z derivative). As anticipated from the earlier discussion, all the boundary conditions are handled as constraints and one is implicit. That is, the Fpj residual does not involve p itself. 

An important issue in the boundary-layer problem, and in differential-algebraic equations generally, is the specification of consistent initial conditions. We think first of the physical problem (not in Von Mises form), since the inlet profiles of u, v, and T, as well as pressure p, must be specified. However, all the initial conditions are not independent, as they would be for a system of standard-form ordinary differential equations. So assuming that the axial velocity u and temperature T profiles are specified, the radial velocity must be required to satisfy certain constraints. 

At the channel inlet, the radial-velocity v profile cannot be specified independently of the axial-velocity and temperature profiles. Explain why this is so. Explain the implications concerning consistent initial conditions from the differential-algebraic-equation perspective. Develop an algorithm to determine the consistent initial v profile, given the u profile. 

In a highly simplified form, a governing system of differential-algebraic equations can be written as... 

Because the problem is represented by differential-algebraic equations all the initial conditions are not independent. Determine the surface composition that is consistent with the gas-phase initial condition of A = 1 and B = C = 0. 

Taken together, the system of equations represents a set of stiff ordinary differential equations, which can be solved numerically. Because more than one dependent-variable derivative can appear in a single equation (e.g., the momentum equation has velocity and pressure derivatives), it is usually more convenient to use differential-algebraic equation (DAE) software (e.g., Dassl) for the solution rather than standard-form ODE software. 

The plug-flow problem may be formulated with a variable cross-sectional area and heterogeneous chemistry on the channel walls. Although the cross-sectional area varies, we make a quasi-one-dimensional assumption in which the flow can still be represented with only one velocity component u. It is implicitly assumed that the area variation is sufficiently small and smooth that the one-dimensional approximation is valid. Otherwise a two- or three-dimensional analysis is needed. Including the surface chemistry causes the system of equations to change from an ordinary-differential equation system to a differential-algebraic equation system. 

The site-fraction constraint (Eq. 16.64) means that all the s in Eq. 16.63 are not independent. Therefore only Ks — 1 of Eq. 16.63 are solved. Solving the plug-flow problem requires satisfying the algebraic constraints represented by Eqs. 16.63 and 16.64 at every point along the channel surface. The coupled problem is posed naturally as a system of differential-algebraic equations. 

A brief explanation of differential-algebraic equations (DAE) facilitates a further mathematical discussion of the stagnation-flow equations. In general, DAEs are stated as a vector residual equation, where w is the dependent-variable vector and the prime denotes a time derivative. For the discussion here, it is convenient to consider a restricted class of DAEs called semi-explicit nonlinear DAEs, which are represented as... 

Dassl, solves stiff systems of differential-algebraic equations (DAE) using backward differentiation techniques [13,46]. The solution of coupled parabolic partial differential equations (PDE) by techniques like the method of lines is often formulated as a system of DAEs. It automatically controls integration errors and stability by varying time steps and method order. 

U.M. Ascher and L.R. Petzold. Computer Methods for Ordinary Differential Equations and Differential-Algebraic Equations. SIAM, Philadelphia, PA, 1998. 

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