Engineering problems nonlinear models

One main aim of this book is to teach current and future engineers how to use MAT-LAB as a black box for algorithms and codes in the most efficient and smooth manner in order to solve important chemical and biological engineering problems through realistic nonlinear models. 

Models linear in 6, with unconstrained parameters, can be fitted directly by solving Eq. (6.3-5). Efficient algorithms and software for such problems are available [Lawson and Hanson (1974, 1995) Dongarra. Bunch, Moler, and Stewart (1979) Anderson et ah. (1992)], and will not be elaborated here. We will focus on nonlinear models with bounded parameters, which are common in chemical kinetics and chemical reaction engineering. 

Tien attempting to gel an analytical solution to a physical problem, there is always the tendency to oversimplify the problem to make the mathematical model sufficiently simple to warrant an analytical solution. Therefore, it is common practice to ignore any effects that cause mathematical complications such as nonlincarities in the differential equation or the boundary conditions. So it comes as no surprise that nonlinearities such as temperature dependence of tliernial conductivity and tlie radiation boundary conditions aie seldom considered in analytical solutions. A maihematical model intended for a numerical solution is likely to represent the actual problem belter. Therefore, the numerical solution of engineering problems has now become the norm rather than the exception even when analytical solutions are available. 

Engineers develop mathematical models to describe processes of interest to them. For example, the process of converting a reactant A to a product B in a batch chemical reactor can be described by a first order, ordinary differential equation with a known initial condition. This type of model is often referred to as an initial value problem (IVP), because the initial conditions of the dependent variables must be known to determine how the dependent variables change with time. In this chapter, we will describe how one can obtain analytical and numerical solutions for linear IVPs and numerical solutions for nonlinear IVPs. 

Nonlinear optimization is one of the crucial topics in the numerical treatment of chemical engineering problems. Numerical optimization deals with the problems of solving systems of nonlinear equations or minimizing nonlinear functionals (with respect to side conditions). In this article we present a new method for unconstrained minimization which is suitable as well in large scale as in bad conditioned problems. The method is based on a true multi-dimensional modeling of the objective function in each iteration step. The scheme allows the incorporation of more given or known information into the search than in common line search methods. 

Multiple approaches for the quantification of model-form uncertainties are presented, discussed, and applied to a nonlinear transient concrete creep engineering problem. This entry presents the application bounds of each approach such as requirements on the availability of experimental data and whether or not the approach can account for probabilistic model predictions - predictions including the quantification of parametric and/or predictive uncertainties. These approaches include the traditional adjustment factor approaches, probabilistic adjustment factor approach (an adaptation of the prior), and the Bayesian model averaging (BMA) approach. 

A common NLR problem in chemical and biomolecular engineering involves finding model coefficients (parameters) for models in which the parameters occur nonlinearly. A typical problem is solved in Example 9.5. 

Chapters 6, 7, 8 and 9 provide important applications of nonlinear models to example engineering problems. Topics covered include data fitting to nonlinear models using least squares techniques, various statistical methods used in the analysis of data and parameter estimation for nonlinear engineering models. These important topics build upon the basic nonlinear analysis techniques of the preceding chapters. These topics are not extensively covered by most introductory books on numerical methods. 

As a matter of definition a transcendental function is a function for which the value of the function can not be obtained by a finite number of additions, subtractions, multiplications or divisions. Exponential, trigonometric, logarithmic and hyperbolic functions are all examples of transcendental functions. Such functions play extremely important roles in engineering problems and are the source of many of the nonlinear equations of interest in this book. For engineering models an important feature of transcendental functions is that their argument must be a dimensionless mathematical variable. 

The unknown model parameters will be obtained by minimizing a suitable objective function. The objective function is a measure of the discrepancy or the departure of the data from the model i.e., the lack of fit (Bard, 1974 Seinfeld and Lapidus, 1974). Thus, our problem can also be viewed as an optimization problem and one can in principle employ a variety of solution methods available for such problems (Edgar and Himmelblau, 1988 Gill et al. 1981 Reklaitis, 1983 Scales, 1985). Finally it should be noted that engineers use the term parameter estimation whereas statisticians use such terms as nonlinear or linear regression analysis to describe the subject presented in this book. 

---