Stiff equation sets

After the assembly and insertion of the boundai conditions the following set of global stiffness equations is derived... 

Families of finite elements and their corresponding shape functions, schemes for derivation of the elemental stiffness equations (i.e. the working equations) and updating of non-linear physical parameters in polymer processing flow simulations have been discussed in previous chapters. However, except for a brief explanation in the worked examples in Chapter 2, any detailed discussion of the numerical solution of the global set of algebraic equations has, so far, been avoided. We now turn our attention to this important topic. 

SOLVER Assembles elemental stiffness equations into a banded global matrix, imposes boundary conditions and solves the set of banded equations using the LU decomposition method (Gerald and Wheatley, 1984). SOLVER calls the following 4 subroutines. ... 

Equations (2.22) and (2.23) become indeterminate if ks = k. Special forms are needed for the analytical solution of a set of consecutive, first-order reactions whenever a rate constant is repeated. The derivation of the solution can be repeated for the special case or L Hospital s rule can be applied to the general solution. As a practical matter, identical rate constants are rare, except for multifunctional molecules where reactions at physically different but chemically similar sites can have the same rate constant. Polymerizations are an important example. Numerical solutions to the governing set of simultaneous ODEs have no difficulty with repeated rate constants, but such solutions can become computationally challenging when the rate constants differ greatly in magnitude. Table 2.1 provides a dramatic example of reactions that lead to stiff equations. A method for finding analytical approximations to stiff equations is described in the next section. 

Equation (8.22) for a(0) is also special because, due to symmetry, there is only one adjacent point, a(l). The overall set may be solved by any desired method. Euler s method is discussed below and is illustrated in Example 8.5. There are a great variety of commercial and freeware packages available for solving simultaneous ODEs. Most of them even work. Packages designed for stiff equations are best. The stiffness arises from the fact that VJJ) becomes very small near the tube waU. There are also software packages that will handle the discretization automatically. 

The entire equation set was solved numerically using Gear s method for stiff differential equations (Gear 1971). The initial mercury dose was administered at 100% methylmercury, administered as a bolus to the gut lumen compartment. The mass transport parameters listed in Table 2-6 were multiplied by the time-dependent compartment volumes to give the mass transport parameters used in the model equations. 

In above [K] is named as the stiffness matrix, whereas F denotes the load vector. This form of the equation set, where the unknown vector u can be explicitly found, is only handled for linear analysis. The nonlinearities associated with the nature of the manufacturing processes usually end up with a nonlinear set of equations with... 

For a system of S chemical species and R reactions c is the S vector of concentrations, k the R vector of time independent parameters (rate coefficients), and f the vector of the R rate expression functions. If the overall reaction is isothermal and takes place in a well-mixed vessel, equation (1) comprises a detailed chemical kinetic model (DCKM) of the reaction. The integration of the model equations can present difficulties because the rate coefficients may vary from one another by many orders of magnitude, and the differential equations are stiff. Numerical methods for the solution of stiff equations are discussed by Kee et al. [1]. Efficient solvers for stiff sets of equations have been developed and are available in various software packages. Some of these are described in Chapter 5. Additional information can be found in Refs. [2,3]. 

However, often the real problem is not with the numerical algorithm but with the engineer developing the equation set. If one is interested in the slower dynamic parts of the problem, a quasi-steady-state assumption should be made for the fast parts of the problem. On the other hand, if one is interested in the fast parts of the problem, the value of the slower parts essentially remains constant over these very short time periods. Therefore, stiff systems of equations should not arise in most properly formulated simulations that use order-of-magnitude scaling in model formation. 

The file ex43.m solves this set of differential equations using ode23s. The model differential equations are defined in m-file mode 143.m. The program ex43a. m solves the problem using the stiff equation solver ode 15. s. 

There is one more circumstance related to a variety of built-in integrators. It is the existence of so-called stiff ODE sets. The concept of stiffness may be illustrated by the example of the kinetic equation for a multi-step reaction ... 

The general approach to solving stiff equations is to use implicit methods. Historically, two chemical engineers, Curtis and Hirschfelder ([11]), proposed the first set of numerical formulas that are well-suited for stiff initial value problems by adopting ... 

A stiff differential equation set is a somewhat ideal equation for use with multiple ranges of time steps or in fact widi an enhanced logaridimically spacing. For... 

Figure 10.15 Error profile for stiff differential equation set with 400 solution points per decade in time using uniform logarithmically spaced or uniformly spaced points.

The other extreme is to evaluate the remaining terms at time n + 1)((50 the fully implicit or backward differencing approach. It leads to a set of algebraic equations from which the dependent variables at time (n -h 1)( 0 can be calculated. This approach is unconditionally stable (Richtmyer and Morton, 1967), and is the approach used here. We may of course also use other schemes in which intermediate weights are given to the forward and backward differences. These partially implicit schemes lead to improved accuracy. However, if attempts are made to use them on systems of stiff equations, the latter must be treated by asymptotic techniques. In chemical situations such techniques are equivalent to the use of the chemical quasi-steady-state or partial equilibrium assumptions at long times. They will be considered again in Section 9. 

With the introduction of Gear s algorithm (25) for integration of stiff differential equations, the complete set of continuity equations describing the evolution of radical and molecular species can be solved even with a personal computer. Many models incorporating radical reactions have been pubHshed. 

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