Finite element collocation

Numerical integration techniques are necessary in modeling and simulation of batch and bio processing. In this chapter we described error and stability criteria for numerical techniques. Various numerical techniques for solution of stiff and non-stiff problems are discussed. These methods include one-step and multi-step explicit methods for non-stiff and implicit methods for stiff systems, and orthogonal collocation method for ordinary as well as partial differential equations. These methods are an integral part of some of the packages like MATLAB. However, it is important to know the theory so that appropriate method for simulation can be chosen. 

Many batch reactors deal with constant-density reaction systems. This includes most liquid-phase reactions as well as all gas phase reactions occuring in a constant-volume bomb. This is a constant volume batch reactor and is the focus of this chapter. Much of this chapter is derived from Levenspiel[24]. 

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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... 

Bhatia, S.K. (1998). Determination of pore size distributions by regularization and finite element collocation. Chem. Eng. Sci., 53, 3239—49. 

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... 

In a first discretization step, we apply a suitable spatial discretization to Schrodinger s equation, e.g., based on pseudospectral collocation [15] or finite element schemes. Prom now on, we consider tjj, T, V and H as denoting the corresponding vector and matrix representations, respectively. The total... 

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. 

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). 

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). 

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 ... 

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. 

Fig. 15. ODE solver for state differential equations using collocation on finite elements with information processed from element to element.

As with finite differences, the finite-element approach can be recast, using vectors and matrices, in the form Au = b, with A and b known and u to be determined. There are two basic approaches. In the first case, referred to as collocation, substimtion of Eq. (15.6) in (15.1) leads to... 

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... 

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