Application of deep learning for improving the efficiency of solving contact mechanics problems by the finite element method

Abstract

The finite element method (FEM) is one of the most widely used numerical techniques for modeling problems in contact mechanics. Despite its universality, FEM has well-known limitations, particularly when solving nonlinear, multiscale, or singular problems that require highly refined meshes and therefore lead to a sharp increase in computational cost. Recent advances in machine learning, especially deep learning, have demonstrated significant poten-tial to improve numerical simulations. Physics-Informed Neural Networks (PINNs), convolu-tional neural networks (CNNs), and neural operators have been successfully applied in com-putational mechanics to accelerate calculations, predict stress and displacement fields, and adaptively refine meshes. These developments indicate that hybrid approaches combining FEM with neural networks can overcome the shortcomings of classical methods.
The purpose of this research is to integrate FEM with deep learning tools in order to increase accuracy and efficiency in solving contact mechanics problems. A hybrid approach is proposed in which coarse-mesh FEM results are supplemented with error information and refined using a feedforward neural network. The input data include stress states and error values in the vicinity of a target point, while the reference data are obtained from fine-mesh FEM simulations. This makes it possible to apply the trained model flexibly to new problems without the need for remeshing or direct use of geometric and boundary conditions.
The method was tested on two-dimensional stress analysis problems using quadrilateral elements. Datasets of training and verification examples were generated, where each example included the difference between stresses from coarse and fine meshes as well as local error information and feedforward neural network with five hidden layers of eighty neurons each was investigated. The results show that the neural network significantly reduces error compared to coarse-mesh FEM, producing stresses much closer to those obtained with fine meshes while requiring considerably fewer computational resources.
The study demonstrates that integrating FEM with deep learning provides an effective balance between accuracy and efficiency. The hybrid approach allows for reliable stress pre-diction with reduced computational cost, which is especially valuable for multiscale problems and cases requiring repeated analyses such as optimization and computational design. Alt-hough the method currently provides refined solutions primarily at selected points and requires large training datasets, its adaptability and ability to accelerate calculations make it a promising tool for future applications in engineering and scientific simulations.