Smart finite elements: An application of machine learning to reduced-order modeling of multi-scale problems

Abstract

To design structures using state-of-the-art materials like composites and metamaterials, we need predictive tools that are capable of taking into account the phenomena occurring at different length scales. However, the upscaling of nonlinear mesoscale behavior to perform system-level predictions is intractable when using conventional modeling techniques. Other methods like multiscale finite elements are capable of solving arbitrary problems, but they tend to be computationally expensive because they rely on detailed models of the element's internal displacement field. We propose a method that utilizes machine learning to generate a direct relationship between the element's state and its forces, skipping altogether the complex and unnecessary task of finding its internal displacements. To generate our model, we choose an existing finite element formulation, extract data from an instance of that element, and feed that data to the machine learning algorithm. The result is an approximated model of the element that can be used in the same context. Unlike most data-driven techniques applied to individual elements, our method is not tied to any particular machine learning algorithm, and it does not impose any restriction on the solver of choice. In addition, we guarantee that our elements are physically accurate by enforcing frame indifference and conservation of linear and angular momentum. Our results indicate that this can considerably reduce the error of the method and the computational cost of producing and solving the model.