Poets know – everything is like everything else. Modeling is based on the same premise. Modeling is the construction and study of models. A model, in turn, is a system, by studying which information about another system is obtained.
At first glance, this seems nonsense. Is it possible, looking at one object, to get an idea of another object. Where is that sea, and where is that cottage?
Meanwhile, in order to look at ourselves, we use a mirror. We identify our reflection in the mirror glass with ourselves. Although our reflection differs from the original in some respects. For example, the right and left in the mirror change places. But we almost automatically make allowances for this distinction, which is not essential in this case, and use the mirror to our advantage and greater convenience. All the boys come away from the mirror clean and combed. And the girls in general are beautiful!
A model, metaphorically speaking, is such a mirror attached to the object under study.
By creating a model, we decide which properties of the system under study are important to us and which are secondary. For example, when studying aircraft wings in a wind tunnel, what is important to us is their shape and the material of which they are made. The color of the wings in this case is unimportant. Although, when calculating the visibility of an airplane, the color of its planes is probably the most important information.
Having determined the main and non-main properties of the system or object being modeled, we establish certain relationships between the properties of the system and its model. For example, if the size of the house model is half the size of the real house, the volume, and hence the weight of the model will be eight times less than the real house.
Then we start investigating the model and determine the various relationships between the parameters we are interested in. For example, at what velocity of the airflow will the wing vibrations start. This is a formulation of the problem of flutter, the oscillations of the aircraft that unexpectedly occur at certain values of the airflow velocity flowing around the wing. Without solving this problem, airplanes would not be able to fly at high speeds. To solve it, we had to observe the destruction of a large number of wing models in the wind tunnel. Here we immediately see the advantage of modeling. We do not test the strength of an expensive airplane, but a cheap model, recalculating the properties of the model into the properties of the simulated real airplane. Saves money, and most importantly, test pilots don’t have to risk their lives.
Another area of application of models is resistance of materials and structural mechanics. How strong should the steel be for the bridge? How thick should the supporting columns be to prevent the building from collapsing? Can a skyscraper be built out of brick? Here, the model of a real material is a specimen that is tested on special test rigs. The strength characteristics obtained from the test results are recalculated into the strength characteristics of real parts of machines or buildings.
And when “populating” a new building, you can’t do without simulation, either. In order to optimally arrange the furniture in the rooms, no one drags heavy tables and bulky refrigerators back and forth. All items are modeled by small rectangles of paper that move across the surface of a sheet of paper with a plan of the room shown on it.
And in medicine, we do not do without modeling. No human being is exactly like any other. At the same time, all human beings have enough similarities, both in “detail” and in “function. A medical doctor studies anatomy from a single skeleton, and sometimes even from a model skeleton, and understands how all humans are arranged. A psychologist studies how a particular person reacts to certain stimuli and then draws general conclusions about the behavior of all people.
There are two kinds of modeling – mathematical and physical. In mathematical modeling, we study systems of relations describing processes occurring in a simulated object. Relationships may be described by equations, often rather complicated, which are derived on the basis of a theoretical model of the process or system under study. But mathematical models can also be probabilistic. In such models, changes in input parameters determine the behavior of output parameters not rigidly, but with a certain degree of probability.
A mathematical model is always a compromise between the real complexity of the system under study and the simplicity required to describe it. There are not always “qualitative” theories that can accurately calculate what happens, for example, when there is a voltage drop in a large power grid. Even the behavior of the flow of water flushed down the toilet bowl depending on its shape is a serious theoretical problem.
In physical modeling, we study the properties of models that are physically similar to the originals. For example, in car crash tests, a lot of crashing cars simulates the behavior of any car that will eventually be released on the road.
The physical models are investigated on real-world setups or test benches. The test results are translated into real-world results using calculations based on a special mathematical apparatus called similarity theory. An example of testing physical models is the already described testing of aircraft models in a wind tunnel. Or the calculation of a hydroelectric dam. The disadvantage of physical modeling is the relative labor intensity of creating and testing models and less versatility of the physical modeling method.
But in any case, physical and mathematical modeling, complementing each other, allow to change our world in the desired direction.