Chapter 1
Computational Electromagnetics - Introductory Aspects
This introductory section deals with the character of a physical model and the corresponding mathematical method to solve the problem of interest. The models are characterized to be simplified imaginary simulations of the real-world systems one attempts to understand. However, models include only those properties and relationships required to understand aspects of real systems that are of interest at the given moment, i.e. those aspects of real systems one knows, or those one is aware of after all. The rest of the information about a real system is simply neglected. Furthermore, this introductory section discusses the fundamental framework to describe electromagnetic phenomena - Maxwell's equations, wave equations, and conservation laws.
1.1 The Character of Physical Models Representing Natural Phenomena
1.1.1 Scientific Method, a Definition, History, Development . ?
Scientists create tools, that's what they do...
C.P. Snow
Science could be considered as an entire set of facts, definitions, theorems, techniques, and relationships, and is tested on phenomena in the real, objective, and external world and, itself, has many elements of imagination, logic, creativity, judgment, metaphor, and instrumentations.
The essence of science is definitely more in research methods and specific way of reasoning, and less in particular facts and results.
Scientific insight starts with observing a certain phenomenon, and then organizing the collected observations in a sort of hypothesis that is tested on additional observations, and if necessary, modified. Then, predictions based on these modified hypotheses are carried out, and some experiments are performed to test the predictions. When the range of predictions provided by the hypothesis is considered to be satisfactory for the scientific community, the hypothesis is referred to as a scientific theory or natural law.
This rather successful methodology, more than four centuries old, is called science or scientific method.
Scientific method was born in the beginning of the seventeenth century with Galileo having abandoned Aristotle's theory of motion. It was Galileo who came up with the principle of the relativity of motion and with the statement that only change in motion required force.
At the same time, a separation of science from philosophy began in the form of shift from consideration of the nature of phenomenon (essence) to explanation of the behavior of a phenomenon. Namely, the Aristotelian essentialistic approach to the explanation of natural phenomena was replaced by the mathematical predictive approach. Instead of asking the question why scientists started to ask how [1]. As once Kelvin pointed out - to know something about phenomena means to measure them and express them in terms of numbers.
What is considered to be one of the crucial issues in the analysis of a natural phenomenon is related to the development and application of a physical model enabling one to predict the behavior of a system with a certain level of accuracy.
One of the crucial aspects of the scientific method and related technological progress is definitely the physical model of a natural phenomenon of interest.
1.1.2 Physical Model and the Mathematical Method to Solve the Problem - The Essence of Scientific Theories
Therefore, the goal of the scientific method is to establish the model of a physical phenomenon and to develop related mathematical methods for the analysis of the given problem.
Various theoretical and experimental procedures are used while developing a model. Models are simplified imaginary simulations of the natural systems one attempts to understand and include only those properties and relationships required to understand aspects of real systems that are currently of interest, i.e. those aspects one knows, or, those one is aware of after all. The rest of the details about a real system are simply neglected from a model.
The concept of physical model represents the essence of reductionistic approach within the scientific method. How much the model of a given physical phenomenon is satisfactory depends on what is required from the model. In the language of mathematics, almost all problems arising in electromagnetics can be formulated in terms of differential, integral, or variational equations.
Generally, there are two basic approaches to solving problems in electromagnetics - the differential (field) approach and the integral (source) approach.
The field approach deals with a solution of a corresponding differential equation with associated initial and boundary conditions, specified at a boundary of a computational domain. Solving some differential equation type one obtains the spatial and temporal distribution of the corresponding field or potential.
Historically, this approach has been developed by Boskovic, Faraday, Maxwell, and others, and is generally very useful for handling the problems with closed domains and clearly specified boundary conditions, the so-called interior field problems.
The source or integral approach is based on the solution of a corresponding integral equation that yields the distribution of electromagnetic filed sources in terms of charge or current distribution, respectively.
In the past, this approach was promoted by Franklin, Cavendish, and Ampere, among others, and is convenient for the treatment of the exterior (unbounded) field problems.
Thus, a classical boundary-value problem can be formulated in terms of the operator equation:
1.1 on the domain O with conditions
1.2 prescribed on the boundary G.
L is the linear differential operator, u is the solution of the problem, and p is the excitation function representing the known sources inside the domain. Note that u usually represents potentials (such as scalar potential ?) or fields (such as electric field E).
The character of the differential approach is depicted in Figure 1.1 [2].
Figure 1.1 Differential approach concept.
Methods for the solution of the interior field problem are generally referred to as differential methods or field methods.
Essentially, a differential approach isolates the calculation domain from the rest of the world. The interaction of the domain of interest with the rest is expressed (de facto replaced) by a set of prescribed boundary conditions.
If instead of differential operator L one considers an integral operator g, then unknowns are related to field sources (charge or current densities, respectively), distributed along the boundary G´. Namely, it can be written as
1.3 where h denotes the excitation function.
Figure 1.2 illustrates the character of the integral approach [2].
Figure 1.2 Integral approach concept.
Solution methods for the exterior field problem are generally referred to as integral methods or source methods. In this case, the domain of interest is unbounded (infinite). However, the source distribution represents all that exists, i.e. all interactions coming from the outer world are neglected. For example, when a basic electromagnetic model is developed for a dipole antenna in engineering electromagnetics the antenna is assumed to be insulated in free space [2].
Finally, for dynamic phenomena the initial condition of a physical system has to be considered. Basically, any law of nature represents physical states in a mathematical form (written in terms of differential, integral, or variational equations). Thus, a prescribed initial condition (behavior of the considered physical quantity at t = 0) by definition implies that nothing exists earlier than t = 0. This is also considered to be the origin of time asymmetry in physical laws.
Generally, techniques for the solution of operator equations can be referred to as analytical, numerical, or hybrid methods. Analytical solution methods provide exact solutions but are, on the other hand, limited to a narrow range of applications, mostly related to canonical problems.
Unfortunately, there are not many realistic scenarios in physics or practical engineering problems that can be worked out using these techniques.
Numerical techniques are applicable to almost all scientific engineering problems, but the main drawbacks are related to the limits governed by the approximation contained in the model itself.
Moreover, the criteria for accuracy, stability, and convergence are not always straightforward and clear to the researcher in a particular area [2].
1.1.3 Philosophical Aspects Behind Scientific Theories
One of the crucial questions in the philosophy of science is how physical models really work, or, more generally, how scientific theories are developed or "upgraded."
Looking back into the history of physics, the development of Maxwell's kinetic theory of gases and electromagnetic field theory was not motivated by experimental findings that were not compatible with the existing paradigm (in the sense of Kuhn [3]), as was...
