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sculpturing from soft plastic materials, such as clay, wax, and plasticine, either by building up or carving out the mass. One of the basic preliminary operations in sculpturing, modeling is used to execute studies and models. It is also one of the first artistic techniques taught in kindergartens and schools. The word “modeling” is also used as a term for the indication of three-dimensionality in sculpture, as well as in painting and the graphic arts.
(or simulation), the study of entities by means of their models; the construction and study of models of objects and phenomena that actually exist (animate and inanimate systems, engineering structures, and physical, chemical, biological, and social processes) and of entities that are being built (to determine and refine the definition of their characteristics, expedite their methods of construction, and so on).
Modeling as a cognitive method is inseparable from the development of knowledge. It was conceived in ancient times, simultaneously with the beginnings of scientific knowledge, essentially as a type of reflection of reality. Its widespread use in a clearly expressed form, but without the use of the term itself, began during the Renaissance: Brunelleschi, Michelangelo, and other Italian architects and sculptors used models of the structures they designed; Galileo and Leonardo da Vinci not only used models in their theoretical works but also established the limitations on the use of the method. Newton was fully aware of the method, and by the 19th and 20th centuries it had become difficult to name a field of science or its applications where modeling was not very important. An exceptionally large methodological role was played in this regard by the works of such physicists and chemists as Kelvin (W. Thomson), J. C. Maxwell, F. A. Kekule, and A. M. Butlerov, for it was precisely these sciences that became the classical “proving ground” of modeling methods. The development of the first electronic computers (J. von Neumann, 1947) and the formulation of the fundamental principles of cybernetics (N. Wiener, 1948) brought out the truly universal significance of the new methods, in both abstract and applied fields of knowledge. Modeling has now assumed a general scientific nature and is used in studies of animate and inanimate nature and in the humanities and social sciences.
A single classification of the kinds of modeling is difficult in view of the ambiguity of the concept of the “model” in science and technology. Such a classification may be made on various bases: according to the nature of the models (that is, according to the medium of the model); according to the nature of the entities being modeled; according to the fields of application of the model (in technology, the physical sciences, and chemistry; modeling of animate processes; or modeling in psychology); and according to its levels (“depth”), beginning, for example, with the separation in physics of modeling at the microlevel (modeling at the levels of research relating to the elementary particles, atoms, and molecules). Any classification of modeling methods must of necessity be incomplete, especially since the terminology in this area rests not so much on “strict” rules as on linguistic, scientific, and practical traditions and even more often is defined within the limits of a specific context, outside of which it has no standard meaning (a typical example is the term “cybernetic modeling”).
Modeling that is performed on models that reproduce the basic geometric, physical, dynamic, and functional characteristics of the “original” is called object modeling. Such models are used to study processes that take place in the original—the object being studied or treated (for example, study of the properties of structural members, various mechanisms, and transportation equipment). If the model and the entity being modeled are of the same physical nature, then the modeling is called physical. A phenomenon (a system or process) may be investigated by means of experimental study of a phenomenon that is of a different physical nature but is described by the same mathematical relationships as the phenomenon being modeled. For example, mechanical and electrical oscillations are described by the same differential equations; consequently, electrical oscillations can be modeled by means of mechanical vibrations, and vice versa. Such “object-mathematical” modeling is used extensively to replace the study of certain phenomena by the study of others that are more suitable for laboratory investigation, particularly because they make possible the measurement of unknown quantities. Thus, electrical modeling makes possible the study of mechanical, hydrodynamic, and acoustic phenomena using electrical models. Electrical modeling is the basis of analog computers.
In symbolic modeling, the models are symbolic structures of any type: circuits, charts, diagrams, formulas, curves, and words and sentences in the alphabet of a natural or artificial language.
The most important kind of symbolic modeling is mathematical (logic-mathematical) modeling, which is achieved by means of the languages of mathematics and logic. Symbolic structures and their elements are always considered together with specific transformations and the operations on them that are performed by a human or a machine (transformations of mathematical, logical, and chemical formulas; transformations of the conditions of elements in a digital computer that correspond to the symbols of the machine language). The modern form of “material realization” of symbolic (mainly mathematical) modeling is simulation on general-purpose and specialized digital computers. Such machines are a kind of “blank” on which a description of any process or phenomenon may be recorded in the form of a program—that is, a system of rules encoded in machine language by which the machine can “reproduce” the course of the process being modeled.
Operations with symbols are always associated to some extent with the concept of symbolic structures and their transformations: the formulas, mathematical equations, and other expressions of the specific type used in constructing a model of the scientific language are interpreted (represented) using the concepts of the objective area to which the original belongs. Therefore, the actual construction of symbolic models or their fragments can be replaced by a mental picture of the symbols and (or) the operations on them. This type of symbolic modeling is sometimes known as conceptual modeling. Incidentally, this term is often used to designate “intuitive” modeling, which does not use any firmly established systems of symbols but rather proceeds on the level of “model notions.” Such modeling is a necessary condition of any cognitive process in its initial stage.
In terms of the aspect of an entity that is being modeled, it is appropriate to draw a distinction between modeling of the structure of an object and modeling of its behavior (the operations of the processes that take place in it). This distinction is highly relative in chemistry or physics, but it takes on a precise meaning in the life sciences, where the distinction of the structure and functions of living systems is a fundamental methodological priciple of investigation, and in cybernetics, where the emphasis is on modeling of the functions of the systems under study. In cybernetic modeling the systems are usually abstracted from the structure by treating the latter as a “black box,” the description (model) of which is built up in terms of the relationships between the states of its “inputs” and “outputs” (the inputs correspond to the external actions on the system under study, and the outputs are its reactions to them—that is, its behavior).
Stochastic modeling, which is based on determination of the probability of one or another event, is used for a number of complex phenomena, such as turbulence or pulsations in regions of flow separation. Such models define a certain average, total result rather than indicate the whole course of individual random processes in a given phenomenon.
The concept of modeling is an epistemological category that describes one of the most important means of cognition. The feasibility of modeling—that is, of transferring to the original the results obtained through the construction and study of models—is based on the fact that a model reflects (reproduces) in a certain sense any of the original’s characteristics; such an indication (and the associated idea of similarity) is explicitly or implicitly based on the precise concepts of isomorphism or homomorphism (or their generalizations) between the entity under study and some other “original” entity and frequently is achieved by preliminary theoretical or experimental study of both. Consequently, established theories of the phenomena under investigation, or at least sufficiently well-grounded theories and hypotheses indicating the maximum permissible simplifications in the construction of a model, are necessary for successful modeling. The results of modeling increase substantially if a theory that refines the similarity concept used in the modeling procedure may be used during the construction of the model and the transfer of its results to the original. Such a theory, which is based on the use of the dimensionality concept for physical quantities, has been well developed for phenomena of the same physical nature. However, no analogous theory has yet been developed for the modeling of complex systems and processes, such as those studied in cybernetics; this has brought about the intensive development of the theory of complex systems—the general theory of the construction of models for complicated dynamic systems of animate nature, technology, and the socio-economic sphere.
Modeling is always used in conjunction with other general and specialized scientific methods. Above all, it is closely associated with experimentation. The study of any phenomenon by means of its model (by object modeling, symbolic modeling, or digital computer simulation) can be regarded as a special kind of experiment—a “model experiment” that differs from the usual (“direct”) experiment in that the cognition process includes an “intermediate link,” the model, which is simultaneously the means and the objective of the experimental investigation that replaces the entity under study. A model experiment makes possible the study of entities for which a direct experiment would be difficult, economically unprofitable, or generally impossible for one or another reason (the modeling of unique structures, such as hydraulic-engineering works; complex industrial structures, economic systems, social phenomena; processes taking place in space; or conflicts and military actions).
The study of symbolic models (particularly mathematical models) can also be treated like certain experiments (“experiments on paper” and intellectual experiments). This becomes particularly obvious in light of the possibility of performing such experiments using computer technology. One kind of model experiment is the cybernetic-model experiment, during which the “actual” experimental operation with the entity under study is replaced by an algorithm (program) of its operation, which proves to be a unique model of the entity’s behavior. By entering this algorithm into a digital computer and, so to speak, “playing it back,” information is obtained on the behavior of the original in a specified medium and on its functional relationships with a changing “environment.”
It is thus possible, first of all, to distinguish between “material” (object) and “ideal” modeling. The former may be treated as “experimental,” and the latter, as “theoretical,” although such a contrast is, of course, highly arbitrary, not only because of the interdependence and mutual effect of these kinds of modeling but also because of the existence of such “hybrid” forms as the “imaginary experiment.” As was indicated above, “material” modeling is subdivided into physical and object-mathematical modeling, and a particular case of the latter is analog modeling (simulation). Further, “ideal” modeling can take place both on the level of the most general “model notions,” which may not have been fully recognized and defined, and on the level of fairly detailed symbolic systems; in the former case one speaks of imaginary (intuitive) modeling, and in the latter, of symbolic modeling (the most important and commonplace kind is logic-mathematical modeling). Finally, digital computer simulation (often called cybernetic simulation) is object-mathematical in form but symbolic in content.
Modeling must presuppose the use of abstraction and idealization. By depicting the properties of an original that are essential (from the viewpoint of the purposes of an investigation) and avoiding the nonessential properties, the model appears as a specific mode of realization of an abstraction—that is, as a certain abstract, idealized entity. In addition, the entire process of transferring knowledge from the model to the original depends to a large extent on the character and levels of the abstractions and idealizations that form the basis of the modeling. Of particular importance is the separation of the three levels of abstraction on which modeling can be accomplished: the level of potential practicability (when the transfer mentioned above presupposes abstraction from the boundedness of cognitive-practical human activity in space and time), the level of “real” practicability (when the transfer is treated as a process that can actually be accomplished, although perhaps only in some future period of human experience), and the level of practical convenience (when the transfer is not only realizable but also is desirable in order to achieve some specific cognitive or practical tasks).
It must be remembered, however, that on all levels the modeling of a given original cannot provide complete knowledge about it at any of its stages. This aspect of modeling is particularly important when the object being modeled is a complicated system whose behavior depends on a substantial number of inter-related factors of unlike nature. During cognition such systems are represented by various models that are more or less justifiable; here some of the models can be related to one another, whereas others may differ profoundly. Consequently, the problem of comparison or evaluation of the adequacy of different models for the same phenomenon arises, necessitating the formulation of accurately defined criteria of comparison. If the criteria are based on experimental data, an added difficulty then arises because a good agreement of the conclusions derived from a model and the data from observation and experiment is not as yet a unique confirmation of the model’s correctness, since it is possible to construct other models of a given phenomenon that will also be supported by empirical facts. From this arises the creation of mutually complementary or even conflicting models; the conflicts may be “removed” as a science develops (and then, during modeling, may reappear on a deeper level). For example, at a certain stage in the development of theoretical physics, physical processes were modeled on a “classical” level, using models assuming an incompatibility between the particle and wave concepts; this incompatibility was removed by the creation of quantum mechanics, based on the thesis of a wave-particle dualism contained in the very nature of matter.
Modeling of various forms of brain activity is another example of this type of model. The models created for the intellect and the psychic functions—for example, in the form of heuristic programs for a digital computer—indicate that thinking as an information process may be modeled in different aspects (deductive, or formal-logical; inductive; neutrological; or heuristic) that can be “correlated” by means of further logical, psychological, physiological, evolutionary-genetic, and cybernetic modeling research.
Modeling deeply permeates theoretical thinking. In addition, the development of any science may be treated as a whole in a very general but completely reasonable sense as “theoretical modeling.” An important cognitive function of modeling is to serve as the impetus and source for new theories. A theory frequently comes into being initially in the form of a model, providing an approximate, simplified explanation of a phenomenon, and emerges as a primary working hypothesis, which can grow into a “pretheory”—the predecessor of the developed theory. Moreover, new ideas and types of experiment present themselves and previously unknown facts are discovered in the course of modeling. Such an “interweaving” of theoretical and experimental modeling is quite typical in the development of physical theories (for example, kinetic molecular theory or the theory of nuclear forces).
Modeling is not only one of the methods of depicting the phenomena and processes of the real world but also—despite its relativity, as mentioned above—is an objective practical criterion for checking the validity of our knowledge, either directly or by establishing its relation to another theory expressed as a model whose adequacy is considered for all practical purposes to be proved. When used in organic harmony with other methods of cognition, modeling emerges as a process of extension of cognition, which moves from relatively information-poor models to more complete models that more fully reveal the essence of the real phenomena being studied.
Various types of modeling are usually used in modeling more or less complicated systems (for examples, see , below: Modeling of power systems; Modeling of reaction vessels).
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Modelirovanie kak metod nauchnogo issledovaniia. Moscow, 1965.
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In physical modeling the study of a specific power system is replaced by the study of a similar power system of a different size (with respect to power, voltage, frequency, length of the transmission line, and scale) but of the same physical nature with respect to the elements of the model that are most important for the conditions of the given problem. Physical models of power systems that include electric machines that represent, on a reduced scale of power (down to 1/10,000 or 1/20,000) and voltage (1/1000), a real power system with its regulation, protective, and other equipment are commonly in use in the USSR and abroad. Physical models are used for studies of entire power systems as a whole and of power lines (usually higher-frequency models) and regulation and protective equipment.
Physical modeling of power systems is used mainly for the study and verification of fundamental theoretical conclusions, refinement of equivalent circuits and design formulas, and monitoring of the operation of apparatus, installations, new protective circuits, and means of power transmission, and also for determining the general characteristics of electromagnetic, electromechanical, and wave processes in systems that are operating under unusual conditions or for which there is no accurate mathematical description.
Examples of analog simulation of power systems are DC or AC calculating boards, otherwise known as design models, in which a set of resistances and reactances represent an electrical network, whereas the power supplies—the generators (stations) in the power system—are replaced by adjustable transformers (an AC model) or DC sources such as storage batteries (a DC model). The actual physical processes that take place in the system being studied are not reproduced in such a model. The impedances and electromotive forces that constitute the equivalent circuit of the system under study according to the design equations adopted may be changed manually or automatically, thereby depicting the actual variations that take place in the system. With certain assumptions, the values of the voltages, currents, and powers measured on such a model (the equivalent circuit) describe the actual process in the power system.
In the simulation of power systems on analog computers (for example, the MN-7, MN-14, and MPT-10), some processes are also reproduced that are of a different nature from the processes in a power system but are described formally by the same differential equations as for the power system.
Varieties of analog models include analog-physical models and digital-analog or hybrid models, which combine in one unit both analog and physical models, an analog model and digital computer elements, or a specialized digital computer. Specialized analog models exist that can operate both in real time and on a changed time scale and are used for rapid prediction of processes, which is essential in controlling power systems.
Analog modeling is used for computations with equivalent circuits that require no check of their physical adequacy in terms of the real system but do require the study of the effect of substantial variations in individual parameters of the elements and the initial conditions of the processes.
Mathematical modeling of power systems is achieved in practice by compilation of a system of equations that is adapted for solution on a digital computer and is introduced in the form of algorithms or programs with which numerical characteristics of the processes that take place in the power system under study are obtained on the computer in the form of curves or tables.
Mathematical modeling of power systems is used extensively for design and operational calculations performed with specific parameters that are changed during the study of competing variants; this is of particular importance in technical-economic analysis, optimization, and distribution of currents, powers, and voltages in complicated power systems. The absence of physical visualization in the results obtained makes particularly acute the question of conformity between the calculations and reality (that is, the question of approval of the programs compiled). The most appropriate algorithmic language for writing the programs that guide power-system calculations on a digital computer is FORTRAN, which is universally used in power-engineering practice.
REFERENCESTetel’baum, I. M. Elektricheskoe modelirovanie. Moscow, 1959.
Azar’ev, D. I. Matematicheskoe modelirovanie elektricheskikh sistem. Moscow-Leningrad, 1962.
Gorushkin, V. I. Vypolnenie energeticheskikh raschetov s pomoshch’iu vychislitel’nykh mashin. Moscow, 1962.
Voprosy teorii i primeneniia matematicheskogo modelirovaniia. Moscow, 1965.
Primenenie analogovykh vychislitel’nykh mashin v energeticheskikh sistemakh, 2nd ed. Moscow, 1970.
Mathematical modeling has made possible the study of a reaction vessel as a whole and the accomplishment of a scale transition. The process in a reaction vessel is made up of a large number of chemical and physical interactions on various structural levels (the molecule, the macroregion, an element of the vessel, and the vessel). A multistage mathematical model of the reaction vessel is constructed according to the structural levels of the process. The kinetic model, whose equations describe the reaction rate as a function of the concentration of the reagents, the temperature, and the pressure over their entire region of variation, encompassing the practical operating conditions of a process, corresponds to the first level (the chemical transformation itself). The characteristics of subsequent structural levels depend on the type of reactor. For example, in a reactor with a fixed catalyst layer the second level is the process taking place in a single grain of the catalyst, when mass and heat transfer in the porous grain are important. Each succeeding structural level contains all the preceding levels as components: for example, the mathematical description of the process in a single grain of catalyst includes both transfer and kinetic equations. The model for the third level includes, in addition, equations for mass, heat, and momentum transfer in the catalyst layer. Models for vessels of other types (fluidized-bed vessels; vessels of the column type, with a suspended catalyst) also have a hierarchical structure.
Mathematical modeling is used to select the optimum conditions for the conduct of the process; to determine the required amount of catalyst, the dimensions and shape of the reaction vessel, the parametric sensitivity of the process to the initial and boundary conditions, and the transitional modes; and to study the stability of the process. In a number of cases theoretical optimization is performed first—the optimum conditions for achieving the highest output of useful product are determined, regardless of whether they can be realized, and then, in the second stage, an engineering solution is chosen that provides the closest approach to the theoretical optimum mode, taking into account economic and other factors. To achieve the modes that were found, as well as normal operation of the reaction vessel, uniform distribution of the reactive mixture over the cross section of the vessel and complete mixing ef flows of unlike emnpo sition and temperature must be provided. These problems are solved by using physical (aerohydrodynamic) modeling of the vessel design selected.
M. G. SLIN’KO