Smart finite elements: A novel machine learning application

Abstract

Many multiscale finite element formulations can become computationally expensive because they rely on detailed models of the element’s internal displacement field. This issue is exacerbated in the presence of nonlinear problems, where numerical iterations are generally needed. We propose a method that utilizes machine learning to generate a direct relationship between the element state and its forces, which avoids the complex task of finding the internal displacement field and eliminates the need for numerical iterations. 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.