Combination and application of machine learning and computational mechanics

Abstract

With the steady development of computer science, machine learning and data science have made significant progress in recent decades. These techniques generally rely on a substantial amount of data samples to extract the abstract mapping hidden within the data. Hence, these technologies have gradually attracted the attention of researchers in the field of computational mechanics. Combining the recent studies of the authors and other researchers, this paper aims to interpret several forms of applications that integrate machine learning and data science with computational mechanics. In the first application, the core algorithm of the convolutional neural network is implemented to solve the linear elastic finite element problem. A standard finite element equation is transformed into an optimization problem in this method. The method is verified by a plane strain linear elastic finite element problem. The method demonstrates promising accuracy by comparing the results obtained by traditional finite element solver. However, some limitations of this method need to be addressed. First, though the optimization process can be accelerated by GPU, the efficiency of the proposed method is still lower than most mainstream numerical solvers. And, the framework of convolutional neural networks requires that the input layer data should be a constant matrix. This is a major challenge for solving nonlinear finite element equations whose stiffness matrices contain variables. These are the issues worth considerations in future studies. In the second application, a method is proposed to establish the implicit mapping between the effective mechanical property and the mesoscale structure of heterogeneous materials. Shale is employed in this paper as an example to illustrate the method. At the mesoscale, a shale sample is a complex heterogeneous composite that consists of multiple mineral constituents. The mechanical properties of each mineral constituent vary significantly, and mineral constituents are distributed in an utterly random manner within shale samples. Large quantities of shale samples are generated based on mesoscale scanning electron microscopy images using a stochastic reconstruction algorithm. Image processing techniques are employed to transform the shale sample images to finite element models. Finite element analysis is utilized to evaluate the effective mechanical properties of the shale samples. A convolutional neural network is trained based on the images of stochastic shale samples and their effective moduli. The trained network is validated to be able to predict the effective moduli of real shale samples accurately and efficiently. Not limited to shale, the proposed method can be further extended to predict effective mechanical properties of various heterogeneous materials. In the third application, the authors discuss a data-driven computational mechanics framework proposed by Kirchdoerfer and Ortiz. The most outstanding feature of the framework is that explicit material constitutive equations are no longer required. More specifically, experimental material response data are employed in the framework to replace constitutive equations. Combined with traditional compatibility and equilibrium equations, the framework is able to find the optimal stress-strain combination from a material response dataset to best fit the current element. With this framework, the errors and uncertainties induced by the empirical constitutive functions of traditional computational mechanics approaches can be avoided. The aforementioned applications are only the tip of an iceberg in the recent advancement of computational mechanics. Hence, researchers have reasons to believe that there would be more application scenarios that integrate data science and machine learning with computational mechanics in the future. Hopefully, computational mechanics methods with more robustness, efficiency, and fidelity will be developed.