Finite-element orthogonal collocation

There are really many different variants of this method based on the selection of the support points and the elements to discretize the interval. Some of them have special names that highlight their approach. For instance, if the points are selected as the roots of an orthogonal polynomial and if the elements have only one point in common, the method is said to be a finite-element orthogonal collocation. On the other hand, if each element consists of three points and the adjacent elements share two points, the method is said a finite-difference method. In some cases, when the elements have common points, the single residual is not zeroed, but the sum of residuals is calculated in the same point using all the elements that are sharing it. The aim of these variants is to find a well-conditioned system of equations with a structure that makes its solution particularly efficient when the number of variables is rather large. 

Boundary value methods provide a description of the solution either by providing values at specific locations or by an expansion in a series of functions. Thus, the key issues are the method of representing the solution, the number of points or terms in the series, and how the approximation converges to the exact answer, i.e., how the error changes with the number of points or number of terms in the series. These issues are discussed for each of the methods finite difference, orthogonal collocation, and Galerkin finite element methods. 

Parabolic Equations m One Dimension By combining the techniques apphed to initial value problems and boundary value problems it is possible to easily solve parabolic equations in one dimension. The method is often called the method of lines. It is illustrated here using the finite difference method, but the Galerldn finite element method and the orthogonal collocation method can also be combined with initial value methods in similar ways. The analysis is done by example. 

Other methods can be used in space, such as the finite element method, the orthogonal collocation method, or the method of orthogonal collocation on finite elements (see Ref. 106). Spectral methods employ Chebyshev polynomials and the Fast Fourier Transform and are quite useful for nyperbohc or parabohc problems on rec tangular domains (Ref. 125). 

The simultaneous solution strategy offers several advantages over the sequential approach. A wide range of constraints may be easily incorporated and the solution of the optimization problem provides useful sensitivity information at little additional cost. On the other hand, the sequential approach is straightforward to implement and also has the advantage of well-developed error control. Error control for numerical integrators (used in the sequential approach) is relatively mature when compared, for example, to that of orthogonal collocation on finite elements (a possible technique for a simultaneous approach). 

Using piecewise constant control profiles and orthogonal collocation on finite elements, this approach was further developed by Renfro (Renfro, 1986 Renfro et al, 1987) to deal with much larger problems. More recent simultaneous applications that involve SQP, orthogonal collocation, and piecewise constant control profiles have been presented by Patwardhan et al (1988) for online control, and by Eaton and Rawlings (1988) for optimization of batch crystallizers. These studies have shown that simultaneous approaches can be applied successfully to small-scale applications with complex constraints. 

Note that state variable profiles are one order higher than the controls because they have explicit interpolation coefficients defined at the beginning of each element. With this representation of Z(t) and U(t), we can extend this approach to piecewise polynomials and apply orthogonal collocation on NE finite elements (of length Aoc,). This leads to the following nonlinear algebraic equations ... 

One of the most populax numerical methods for this class of problems is the method of weighted residuals (MWR) (7,8). For a complete discussion of these schemes several good numerical analysis texts are available (9,10,11). Orthogonal collocation on finite elements was used in this work to solve the model as detailed by Witkowski (12). 

Fig. 13. Orthogonal collocation on finite elements global indexing system.

Fig. 15. Comparison of profiles computed by orthogonal collocation and orthogonal collocation on finite elements.

Regardless of whether orthogonal collocation or orthogonal collocation on finite elements is used for the discretization, the resulting linear state-space... 

Among the variety of methods which have been proposed for simulation of packed bed dynamics three techniques have been used with success (1) Crank-Nicholson technique [10], (2) transformation to integral equation [11], (3) orthogonal collocation on finite elements [12]. In the following computation, we have used the Crank-Nicholson method with the nonequidistant space steps in the Eigenberger and Butt version [10]. 

Orthogonal collocation on two finite elements is used in the radial direction, as in the steady-state model (1), with Jacobi and shifted Legendre polynomials as the approximating functions on the inner and outer elements, respectively. Exponential collocation is used in the infinite time domain (4, 5). The approximating functions in time have the form... 

In the last section we considered explicit expressions which predict the concentrations in elements at (t + At) from information at time t. An error is introduced due to asymmetry in relation to the simulation time. For this reason implicit methods, which predict what will be the next value and use this in the calculation, were developed. The version most used is the Crank-Nicholson method. Orthogonal collocation, which involves the resolution of a set of simultaneous differential equations, has also been employed. Accuracy is better, but computation time is greater, and the necessity of specifying the conditions can be difficult for a complex electrode mechanism. In this case the finite difference method is preferable7. 

This is one of the variants of the finite element methods. The essence of orthogonal collocation (OC) is that a set of orthogonal polynomials is fitted to the unknown function, such that at every node point, there is an exact fit. The points are called collocation points, and the set of polynomials is chosen suitably, usually as Jacobi polynomials. The optimal choice of collocation points is to make them the roots of the polynomials. There are tables of such roots, and thus point placements, in Appendix A. The notable things here are the small number of points used (normally, about 10 or so will do), their... 

A general computational scheme using orthogonal collocation on finite elements has been developed for calculation of rates of mass transfer accompanied by single or multistep reactions. The method can be used to predict enhancement in absorption or desorption rates for a wide class of industrially important situations. 

Approximate vs. Numerical Solution. The accuracy of the approximate reaction factor expression has been tested over wide ranges of parameter values by comparison with numerical solutions of the film-theory model. The methods of orthogonal collocation and orthogonal collocation on finite elements (7,8) were used to obtain the numerical solutions (details are given by Shaikh and Varma (j>)). Comparisons indicate that deviations in the approximate factor are within few percents (< 5%). It should be mentioned that for relatively high values of Hatta number (M >20), the asymptotic form of Equation 7 was used in those comparisons. 

In this woric, discretisation of both space and time derivatives was executed, based on either central finite difference (CFD) or orthogonal collocation cm finite elements (OCFE) discretisation in the spatial domain and backward finite difference (BFD) discretisation in the time domain. 

Here, the last two equations define the flow rate and the mean residence time, respectively. This formulation is an optimal control problem, where the control profiles are q a), f(a), and r(a). The solution to this problem will give us a lower bound on the objective function for the nonisothermal reactor network along with the optimal temperature and mixing profiles. Similar to the isothermal formulation (P3), we discretize (P6) based on orthogonal collocation (Cuthrell and Biegler, 1987) on finite elements, as the differential equations can no longer be solved offline. This type of discretization leads to a reactor network more... 

The solution to (P12) gives us the optimal separation profile as a function of age within the reactor. However, except in the case of reactive phase equilibrium, the assumption of a continuous separation profile is not really required. Furthermore, a continuous separation profile may not be implementable in practice. To address this, we take advantage of the structure of a discretization procedure for the differential equation system. In this case, we choose orthogonal collocation on finite elements to discretize the above model. This results... 

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