Finite-element collocation method

The second approach, applied by Margolis (1978) and by Heimerl and Coffee (1980) to ozone decomposition flames, employs a method-of-lines technique. In combination with finite-element collocation methods this technique provides a general approach to the numerical solution of partial differential equations. Taking Eqs. (4.12) and (4.13) as the working examples 

An important feature of the method of lines is selection of the basis functions i (co), which determines the precision of (spatial) curve fitting. The piecewise polynomials known as B splines meet the requirements. Curve fitting by means of spline functions entails division of the solution space into subintervals by means of a series of points called knots. Knots may be either single or multiple, a multiple knot being formed by the coincidence of two or more such points. They are numbered in nondecreasing order of location Si, S2. 5i,. A normalized B spline of order k takes nonzero values only over a range of k subintervals between knots, and, for example, Bij (co), the ith normalized B spline of order k for the knot sequence s, is zero outside the interval + nonnegative at = s, and w = Si + j, and strictly 

A spline function of order k with knot sequence s is then defined as any linear combination of splines of order k for the particular knot sequence. 

The7 th derivative of a B spline function of order fc, with coefficients a,-, is given (de Boer, 1978) by 

We now return to Eqs. (6.1) and (6.2). If the right-hand sides of these equations are taken to be B spline functions of order k, then it can be shown (Margolis, 1978) that the spatial truncation error, that is, the amount by which the approximate solution fails to solve the partial differential equations (4.12) and (4.13) at time t, is 0((5 ). Next, if the number of continuity conditions to be satisfied at each interior (i = 2. .. /) breakpoint is v, then only the first m = kl — v l — 1) knots of the sequence form the origins of fresh polynomial pieces. It follows that the number of pp coefficients and hence the 

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The answer to this difficulty lies in the use of piecewise approximants, such as cubic splines, which are in general use in the mathematics literature (11). Carey and Finlayson (12) have introduced a finite-element collocation method along these lines, which uses polynomial approximants on sub-intervals of the domain, and apply continuity conditions at the break-points to smooth the solution. It would seem more straight-forward, however, to use piecewise polynomials which do not require explicit continuity... 

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). 

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. 

Different techniques are commonly used to solve the diffusion equation (Carslaw and Jaeger, 1959). Analytic solutions can be found by variable separation, Fourier transforms or more conveniently Laplace transforms and other special techniques such as point sources or Green functions. Numerical solutions are calculated for the cases which have no simple analytic solution by finite differences (Mitchell, 1969 Fletcher, 1991), which is the simplest technique to implement, but also finite elements, particularly useful for complicated geometry (Zienkiewicz, 1977), and collocation methods (Finlayson, 1972). 

Instead, the simultaneous method can be extended to select adaptively a sufficient number of finite elements. Here, we note that even if we set any element length to zero, the collocation equations and the continuity equations are still satisfied. Thus, any number of zero length (or dummy) elements can be added without changing the control or state profiles, or the solution to the NLP. Vasantharajan and Biegler (1990) take advantage of this important property and propose an adaptive element addition approach embedded within the simultaneous solution strategy. 

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). 

The aim of this part of the book is to present the main and current numerical techniques that are used in polymer processesing. This chapter presents basic principles, such as error, interpolation and numerical integration, that serve as a foundation to numerical techniques, such as finite differences, finite elements, boundary elements, and radial basis functions collocation methods. 

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]. 

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 one of the first articles on this subject [8], the general analytical solution of Eq. (3) was derived. This general solution is easy to find, but it contains infinite series and (integration) constants that depend on the boundary conditions. Those were determined for the central cells of square and triangular arrays, using the boundary collocation method [8]. More recent publications on this subject are based mostly on complete numerical solution using finite-element methods. 

The approach proposed in this paper belongs to this class and uses the orthogonal collocation method on finite elements [1, 2, 3] to convert the DAEs... 

FD methods are point approximations, because they focus on discrete points. In contrast, finite element methods focus on the concentration profile inside one grid element. As an example of these segment methods, orthogonal collocation on finite elements (OCFE) is briefly discussed below. 

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