Mathematical modeling basic principles

The beauty of finite-element modelling is that it is very flexible. The system of interest may be continuous, as in a fluid, or it may comprise separate, discrete components, such as the pieces of metal in this example. The basic principle of finite-element modelling, to simulate the operation of a system by deriving equations only on a local scale, mimics the physical reality by which interactions within most systems are the result of a large number of localised interactions between adjacent elements. These interactions are often bi-directional, in that the behaviour of each element is also affected by the system of which it forms a part. The finite-element method is particularly powerful because with the appropriate choice of elements it is easy to accurately model complex interactions in very large systems because the physical behaviour of each element has a simple mathematical description. 

In recent years some theoretical results have seemed to defeat the basic principle of induction that no mathematical proofs on the validity of the model can be derived. More specifically, the universal approximation property has been proved for different sets of basis functions (Homik et al, 1989, for sigmoids Hartman et al, 1990, for Gaussians) in order to justify the bias of NN developers to these types of basis functions. This property basically establishes that, for every function, there exists a NN model that exhibits arbitrarily small generalization error. This property, however, should not be erroneously interpreted as a guarantee for small generalization error. Even though there might exist a NN that could... 

There are two basic classes of mathematical models (see Fig. 5.3-18) (1) purely empirical models, and (2) models based on physicochemical principles. 

As this chapter aims at explaining the basics, operational principles, advantages and pitfalls of vibrational spectroscopic sensors, some topics have been simplified or omitted altogether, especially when involving abstract theoretical or complex mathematical models. The same applies to methods having no direct impact on sensor applications. For a deeper introduction into theory, instrumentation and related experimental methods, comprehensive surveys can be found in any good textbook on vibrational spectroscopy or instrumental analytical chemistry1"4. 

Classical mechanics which correctly describes the behaviour of macroscopic particles like bullets or space craft is not derived from more basic principles. It derives from the three laws of motion proposed by Newton. The only justification for this model is the fact that a logical mathematical development of a mechanical system, based on these laws, is fully consistent... 

First principle mathematical models These models solve the basic conservation equations for mass and momentum in their form as... 

Simplified mathematical models These models typically begin with the basic conservation equations of the first principle models but make simplifying assumptions (typically related to similarity theory) to reduce the problem to the solution of (simultaneous) ordinary differential equations. In the verification process, such models must also address the relevant physical phenomenon as well as be validated for the application being considered. Such models are typically easily solved on a computer with typically less user interaction than required for the solution of PDEs. Simplified mathematical models may also be used as screening tools to identify the most important release scenarios however, other modeling approaches should be considered only if they address and have been validated for the important aspects of the scenario under consideration. 

Successful approaches to designing an extraction process begin with an appreciation of the fundamentals (basic phase equilibrium and mass-transfer principles) and generally rely on both experimental studies and mathematical models or simulations to define the commercial technology. Small-scale e3q)eriments using representative feed usually are needed to accurately quantify physical properties and phase equilibrium. Additionally, it is common practice in industry to perform... 

First principle mathematical models These models solve the basic conservation equations for mass and momentum in their form as partial differential equations (PDEs) along with some method of turbulence closure and appropriate initial and boundary conditions. Such models have become more common with the steady increase in computing power and sophistication of numerical algorithms. However, there are many potential problems that must be addressed. In the verification process, the PDEs being solved must adequately represent the physics of the dispersion process especially for processes such as ground-to-cloud heat transfer, phase changes for condensed phases, and chemical reactions. Also, turbulence closure methods (and associated boundary and initial conditions) must be appropriate for the dis-... 

The study of tumor growth forms the foundation for many of the basic principles of modem cancer chemotherapy. The growth of most tumors is illustrated by the Gompertzian tumor growth curve (Fig. 124-5). Gompertz was a German insurance actuary who described the relationship between age and expected death. This mathematical model also approximates tumor-cell proliferation. In the... 

Once the continuum hypothesis has been adopted, the usual macroscopic laws of classical continuum physics are invoked to provide a mathematical description of fluid motion and/or heat transfer in nonisothermal systems - namely, conservation of mass, conservation of linear and angular momentum (the basic principles of Newtonian mechanics), and conservation of energy (the first law of thermodynamics). Although the second law of thermodynamics does not contribute directly to the derivation of the governing equations, we shall see that it does provide constraints on the allowable forms for the so-called constitutive models that relate the velocity gradients in the fluid to the short-range forces that act across surfaces within the fluid. 

These simplifying assumptions are made primarily on the basis of an intuitive assessment of the important features of the systems being considered and the experience gained by previous attempts to address these and related problems in natural bodies of water. The basic principle to be applied to this conceptual model, which can then be translated into mathematical terms, is that of conservation of mass. 

The basic principles of modeling the physical, chemical and biological processes that determine pesticide fate in unsaturated soil are reviewed. The mathematical approaches taken to integrate diffusion, convection, sorption, degradation and volatilization are presented. Deterministic and stochastic models formulated to describe these processes in a soil-water pesticide system are contrasted and evaluated. The use of pesticide models for research or management purposes dictates the degree of resolution with thich these processes are modeled. 

This short review of zeolite and zeotype synthesis is written for those who are relatively new to the field. It aims to present an overall introduction to some fundamental aspects of the subject and to indicate where further information can be found. An account of experimental practice is followed by a summary of mathematical modelling procedures. Observations from crystallisation studies then introduce basic principles of the synthesis process. 

Here the basic concepts of system theory and principles for mathematical modelling are presented for the development of diffusion reaction models for fixed bed catalytic reactors. 

Basic Principles of Mathematical Modelling for Industrial Fixed Bed Catalytic Reactors... 

BASIC PRINCIPLES OF MATHEMATICAL MODELLING FOR INDUSTRIAL FIXED BED CATALYTIC REACTORS... 

Part II (Chapters 4 and 5) introduces the reader to the modeling requirements for process control. It demonstrates how we can construct useful models, starting from basic principles, and determines the scope and difficulties of mathematical modeling for process control purposes. 

Turbulent diffusion is concerned with the behavior of individual particles that are supposed to faithfully follow the airflow or, in principle, are simply marked minute elements of the air itself. Because of the inherently random character of atmospheric motions, one can never predict with certainty the distribution of concentration of marked particles emitted from a source. Although the basic equations describing turbulent diffusion are available, there does not exist a single mathematical model that can be used as a practical means of computing atmospheric concentrations over all ranges of conditions. 

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