Semiempirical model of the real membrane bending

Abstract

The tasks of constructing mathematical models from heterogeneous data, including differential equations, boundary and initial conditions, observational data and other information about the modeled object, are of great practical importance, in particular, in the construction of digital counterparts of complex technical objects. Especially relevant is the search for methods for constructing the above mathematical models in a situation where the physical model and, consequently, the differential equation is known with insufficient precision for modeling purposes. We have developed new methods for constructing mathematical models of the type mentioned above and check them on a model problem with real measurements. In this paper, we consider the solution of the problem of modeling the deflection of a loaded circular membrane, in the center of which the weight of a given mass is located. The accuracy of the models expressing the dependence of the deflection of the membrane from the distance to the center is compared. We constructed the first model on the basis of an analytical solution of the equation of equilibrium conditions. The second model was obtained with the help of the original modification of the refined Euler method. When constructing the second model, it is necessary to select the same number of coefficients as in the construction of the first model. We built the third model in the form of an output of a neural network. The coefficients of the models were selected from the data obtained experimentally. The resulting approximate accuracy models outperform the model based on the exact solution. The neural network model turned out to be the most accurate, but it requires the selection of a larger number of coefficients.