Stiffness, mathematical

E. Hairer and G. Wanner. Solving Ordinary Differential Equations II. Stiff and Differential-Algebraic Problems, volume 14 of Springer Series in Computational Mathematics. Springer-Verlag, New York, New York, second edition, 1996. 

Naturally, fibers and whiskers are of little use unless they are bonded together to take the form of a structural element that can carry loads. The binder material is usually called a matrix (not to be confused with the mathematical concept of a matrix). The purpose of the matrix is manifold support of the fibers or whiskers, protection of the fibers or whiskers, stress transfer between broken fibers or whiskers, etc. Typically, the matrix is of considerably lower density, stiffness, and strength than the fibers or whiskers. However, the combination of fibers or whiskers and a matrix can have very high strength and stiffness, yet still have low density. Matrix materials can be polymers, metals, ceramics, or carbon. The cost of each matrix escalates in that order as does the temperature resistance. 

Thus, the Tsai-Wu tensor failure criterion is obviously of more general character than the Tsai-Hill or Hoffman failure criteria. Specific advantages of the Tsai-Wu failure criterion include (1) invariance under rotation or redefinition of coordinates (2) transformation via known tensor-transformation laws (so data interpretation is eased) and (3) symmetry properties similar to those of the stiffnesses and compliances. Accordingly, the mathematical operations with this tensor failure criterion are well-known and relatively straightforward. 

Mathematical models of the reaction system were developed which enabled prediction of the molecular weight distribution (MWD). Direct and indirect methods were used, but only distributions obtained from moments are described here. Due to the stiffness of the model equations an improved numerical integrator was developed, in order to solve the equations in a reasonable time scale. 

The rheological characteristics of AB cements are complex. Mostly, the unset cement paste behaves as a plastic or plastoelastic body, rather than as a Newtonian or viscoelastic substance. In other words, it does not flow unless the applied stress exceeds a certain value known as the yield point. Below the yield point a plastoelastic body behaves as an elastic solid and above the yield point it behaves as a viscoelastic one (Andrade, 1947). This makes a mathematical treatment complicated, and although the theories of viscoelasticity are well developed, as are those of an ideal plastic (Bingham body), plastoelasticity has received much less attention. In many AB cements, yield stress appears to be more important than viscosity in determining the stiffness of a paste. 

In a transported PDF simulation, the chemical source term, (6.249), is integrated over and over again with each new set of initial conditions. For fixed inlet flow conditions, it is often the case that, for most of the time, the initial conditions that occur in a particular simulation occupy only a small sub-volume of composition space. This is especially true with fast chemical kinetics, where many of the reactions attain a quasi-steady state within the small time step At. Since solving the stiff ODE system is computationally expensive, this observation suggests that it would be more efficient first to solve the chemical source term for a set of representative initial conditions in composition space,156 and then to store the results in a pre-computed chemical lookup table. This operation can be described mathematically by a non-linear reaction map ... 

You may wonder why we would ever be satisfied with anything less than a very accurate integration. The ODEs that make up the mathematical models of most practical chemical engineering systems usually represent a mixture of fast dynamics and slow dynamics. For example, in a distillation column the liquid flow or hydraulic dynamic response occurs fairly rapidly, of the order of a few seconds per tray. The composition dynamics, the rate of change of hquid mole fractions on the trays, are usually much slower—minutes or even hours for columns with many trays. Systems with this mixture of fast and slow ODEs are called stiff systems. 

It is instructive to study a much simpler mathematical equation that exhibits the essential features of boundary-layer behavior. There is a certain analogy between stiffness in initial-value problems and boundary-layer behavior in steady boundary-value problems. Stiffness occurs when a system of differential equations represents coupled phenomena with vastly different characteristic time scales. In the case of boundary layers, the governing equations involve multiple physical phenomena that occur on vastly different length scales. Consider, for example, the following contrived second-order, linear, boundary-value problem ... 

Next, we have to solve for the Yj s from the species continuity equations, Equation (32). Unfortunately, these equations cannot be integrated by a similar simple point iteration scheme as they are mathematically "stiff"16 and iterative approaches are unstable. To solve these simultaneous equations, we turn to a perturbation analysis developed by Newman17 where the equations are linearized about an initial guess, and the resulting linear equations are solved numerically. The solution is then used as the next guess, and the linear equations are resolved. The procedure is repeated until the solution no longer changes. 

From a mathematical point of view, we can see that Equation (5.10) is in a (nonstandard) singularly perturbed form. This suggests that the integrated processes under consideration will feature a dynamic behavior with at least two distinct time scales. Drawing on the developments in Chapters 2, 3, and 4, the following section demonstrates that these systems evolve in effect over three distinct time scales and proposes a method for deriving reduced-order, non-stiff models for the dynamics in each time scale. 

Egly et al. (1979), Cuille and Reklaitis (1986), Mujtaba (1989), Reuter et al. (1989), Albet et al. (1991), Basualdo and Ruiz (1995) and Wajge and Reklaitis (1999) considered the development of mathematical models to simulate BREAD processes. In most cases, the model was posed as a system of Differential and Algebraic Equations (DAEs) and a stiff solution method was employed for integration. 

The mathematical models of the reacting polydispersed particles usually have stiff ordinary differential equations. Stiffness arises from the effect of particle sizes on the thermal transients of the particles and from the strong temperature dependence of the reactions like combustion and devolatilization. The computation time for the numerical solution using commercially available stiff ODE solvers may take excessive time for some systems. A model that uses K discrete size cuts and N gas-solid reactions will have K(N + 1) differential equations. As an alternative to the numerical solution of these equations an iterative finite difference method was developed and tested on the pyrolysis model of polydispersed coal particles in a transport reactor. The resulting 160 differential equations were solved in less than 30 seconds on a CDC Cyber 73. This is compared to more than 10 hours on the same machine using a commercially available stiff solver which is based on Gear s method. 

Mathematical models that contain ordinary differential equations face an inherent computational difficulty associated with the stiffness of the equations. Stiffness of ordinary differential equations depends on the relative magnitudes of the response modes or the characteristic time constants of the system being modeled. In solid fuel conversion problems where particles of varying sizes are considered the differential equations for the thermal transients of the particles are usually stiff. Estab-... 

Over the past ten years the numerical simulation of the behavior of complex reaction systems has become a fairly routine procedure, and has been widely used in many areas of chemistry, [l] The most intensive application has been in environmental, atmospheric, and combustion science, where mechanisms often consisting of several hundred reactions are involved. Both deterministic (numerical solution of mass-action differential equations) and stochastic (Monte-Carlo) methods have been used. The former approach is by far the most popular, having been made possible by the development of efficient algorithms for the solution of the "stiff" ODE problem. Edelson has briefly reviewed these developments in a symposium volume which includes several papers on the mathematical techniques and their application. [2]... 

It is important to note that in using computer-aided models for batch distillation, the various assumptions of the model can have a significant impact on the accuracy of the results e.g., see the discussion of the effects of holdup above. Uncertainties in the physical and chemical parameters in the models can be addressed most effectively by a combination of sensitivity calculations using simulation tools, along with comparison to data. The mathematical treatment of stiffness in the model equations can also be very important, and there is often a substantial advantage in using simulation tools that take special account of this stiffness. (See the 7th edition of Perry s Chemical Engineers Handbook for a more detailed discussion of this aspect). 

Operation of a batch distillation is an rmsteady state process whose mathematical formulation is in terms of differential equations since the compositions in the stiff and of the holdups on individual trays change with time. This problem and methods of solution are treated at length in the literature, for instance, by Holland and Liapis (Computer Methods for Solving Dynamic Separation Problems, 1983, pp. 177-213). In the present section, a simplified analysis will be made of batch distillation of binary mixtures in colunms with negligible holdup on the trays. Two principal modes of operating batch distillation columns may be employed ... 

In some problems, certain parameters vary quickly and others vary more slowly. For example, in packed-bed chemical reactors, the concentration can change quickly in time, but the temperature will not change rapidly because of the large heat capacity of the solid. Such problems are called stiff [Finlayson, 1997 (p. 3-56), 1990 (Vol. BI, p. 1-60-61)]. The mathematical definition is that the eigenvalues of the Jacobian are widely separated, certainly by factors of thousands, perhaps by factors of millions. It... 

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