Machine learning has achieved remarkable breakthroughs in different scientific disciplines, including the image recognition, cognitive science, genomics, drug discovery, and material designing. Recently, we start to witness a machine-learning revolution in many branches of engineering sciences, such as structural mechanics. One of the most fundamental challenges of applying machine learning algorithms is that the accuracy of the models largely depends on the size and quality of the data that drive the models. In the cases where the accessibility to a large collected dataset is limited, the accuracy of the trained model can hardly be guaranteed. To overcome this issue, most recently, the concept of physics-guided machine learning method is proposed that incorporates the already-known physics laws into the training process to get a reasonable prediction with a small size dataset. A successful application of this method is solving partial differential equations (PDE). In this study, we apply the physics-guided machine learning method to investigate the finite elastic deformation of plates and shells that is govern by the F√∂ppl‚Äìvon K√°rm√°n equations, which in the field of structural mechanics are notoriously difficult to solve by conventional theoretical or numerical methods. We construct a neural network that takes the spatial coordinates as the input and the displacement field as the output to approximate the exact solution. We then formulate the loss function of the neural network with two different approaches. The first approach is to directly introduce the governing equations and boundary conditions into the loss function. The other approach takes the advantage of the minimal total potential energy principle in structural mechanics, where we treat the total potential energy as the loss. Three different loading cases (i.e. the in-plane central hole tension of square plate, out-of-plane compression of fixed-edge square plate, and in-plane buckling of square plate) are studied to validate the machine learning model. A set of data points from the boundary and solution domain are used for training the model. The predictions of the trained model are compared with the finite element simulation results. Despite the existence of the geometry and loading nonlinearities introduced in the central-hole tension case and the difficulties of solving the high order coupled PDEs in the out-plane compression case, the physics-guided machine learning model turns out to be capable of well characterizing elastic deformation of plates. Compared with directly introducing the PDEs and boundary conditions, taking the total potential energy as the loss has a better performance in terms of training accuracy and efficiency.

A deep learning framework of elastic plates and shells

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