Neural-net modeling for direct and inverse problems of shell theory

Effectiveness of the use of neural-net technology for the solving of shell theory problems is shown. Some results of neural-net interpolation and extrapolation for direct and inverse problems are discussed. Exact accuracy of neural-net solving opens wide latitude for shell constructions engineering design and optimization.


Introduction
Neural network technologies as a modern universal approximation tool are widely applied in both classic and new areas of science and technology [1,2]. Their use in the theory of shells, shells designing and research seems relevant and useful.
The joint use of solutions of direct and inverse problems can provide substantial assistance in the process of design engineering, which can be considered as finding the optimal form in the selection of certain parameters and characteristics.
The solution of the inverse problem is usually fraught with difficulties related to the ambiguity of the decision. We have to use different classical and non-classical mathematical methods for solving. For example, the definition of the inverse function or matrix inversion is required for solutions in linear problems without restrictions. For problems with constraints and nonlinear problems it is possible to apply the methods of mathematical programming.
In cases where the problem is poorly formalized or not formalized, we have to look for special innovative ways of solving. They include neural network approaches.
In terms of a neural network, inverse problem is an identification of dependencies between the input and output data, which are interchanged with respect to the direct problem. In this regard, the solution of direct and inverse problems can be carried out by a single algorithm. The neural network solution of the inverse problem will be simplified if we can build such a neural network, for which it would be possible to get a set of ambiguous solutions.
In this case the use of neural network technology, specifically neural network modeling, is convenient and efficient, it allows high-speed seek adoption of rational and optimal solutions.
Note that in this study, we have used standard neural network simulators, that were unfortunately not assumed to be applied to the problems of mechanics while being developed. It seems promising to develop specialized neural simulators, as well as the individual neural networks, which take into account the properties of the structure and, apparently, can significantly improve the results. However, it should be noted that this approach can restrict the class of problems.
As a neural simulator for most of the problems we used the program "Modely" written on the basis of interpolation of the solutions using Fourier series and minimization of the functional of squared deflection error using the Lagrange multipliers [3]. We consider the examples of solving direct and inverse problems [4] using neural networks.   Features of the neural network solving the problem. The best results that have been achieved are within 4% error and obtained by means of a neural network with 15 neurons and spectral density equal to 2.6. The process of network training, the selection of its architecture has a singularity. In the process of training the spectral density should be increased gradually, starting with 1.0 in increments of 0.1. Otherwise, training error increases greatly to 15% or higher. Changing the size of the network (number of neurons) and spectrum as downwards and upwards does not give positive results (Fig. 3).

Thin smooth shallow spherical shell on a square plan
If given moment M X , for which you want to determine the appropriate load bandwidth , is outside the training pattern interval (extrapolation problem -training pattern: M X = 4.2; 6.2; 8; 9.4; 10; tests: M X = 10.3; 11.5), the network training is very long and painstaking. It requires the use of step method neural network forecasting [3,5]. At each step of training the magnitude of the spectral density must be increased from 1, not more than 0.01 to 1.3. In this case, two-neuron network allows to achieve error on the test cases within 6%. For instance, for the training pattern of five examples (4-neuron network with the spectral density 2.5) the maximum error in the tests for deflections is 5.5 %. Reducing the pattern by a single example leads to an increase in errors to 8%.
The accuracy of neural network solutions can be significantly improved if instead of onestep (traditional) neural network extrapolation the step method of neural network forecasting is used [3,5].  5 show that with increasing pliability, quality change (even in sign) of moments epures near the contour and significant growth of deflections throughout the area occurs. The value  = 5 can be taken as conventional limit, below which a reinforcement shell contour can be called a very pliable. It may be recommended for design purposes to take  

5.
Inverse problem: what should be the parameter  (pliability of contour rib), to the deflections w/q and moments M X /q do not exceed the specified values The input parameters of the network -values of deflections w/q and the values of the bending moments M X /q at the nodes of the shell mesh.
The output network parameters -pliability ribs parameter .
Feature of the neural network solving the problem is the fact, that the inverse problem is solved much more accurate than direct both for the interpolation problem when the network functioning is carried out within the training pattern interval, and for extrapolation problem, when network functioning is carried out outside the training pattern interval. It confirms the need for reasonably select training examples, their number, as well as network architecture and its characteristics for each specific problem in view of its features.
In this problem, there is no need to use step method of neural forecasting as "traditional" (one-step) neural forecasting provides acceptable accuracy of the solution. However, the use of this method could significantly improve the results This problem can be considered as an optimization problem. Plotting of dependence of pliability parameter  on the value of the deflections and moments and their analysis can help in the selection of such pliability parameter of contour ribs, which would provide the desired (required) shell behavior, its stress-strain state (SSS).

Conclusion
Thus, the results of solving the problems under consideration show, that the use of neural network models allows to solve with sufficient accuracy (within 5% error) as a direct problem of calculating shells (to receive SSS parameters according to known geometric characteristics of shell and its load) and inverse problems (to select the design dimensions according to known moments and deflections). This opens up opportunities for the design of shell structures and their optimization. Note that the neural network solution of the inverse problem will be simplified, if we can build such neural network model, by which it would be possible to get a set of ambiguous solutions.
The examples discussed herein are illustrative a greater extent for the traditional method of using neural network technology (interpolation, extrapolation). It should be noted that the development and application of stepwise technology to these problems, set out in [3,5], using the completion of the training pattern by interpolation, additional training the neural network model and the step process, can significantly improve the results.
Neural network technologies have much more applications, and can include not only the static problem, but also the dynamics, to take into account the experimental data in order to bring the calculated results to the actual design.