Finite volume-based supervised machine learning models for linear elastostatics

This article proposes two approaches for combining finite volume and machine learning techniques to solve linear elastostatic problems. The first approach adopts a classical supervised machine learning model and generates the training dataset by finite volume-based solvers. The second approach applies a physics-informed model to enforce the governing equations without requiring a priori ground-truth data; as a result, all training cases are solved within the training process. Although the methods presented apply to a wide range of computational problems, this study is limited to linear elastostatics to demonstrate the concept. To develop a physics-informed approach consistent with a finite volume discretisation, we create symbolic Gauss-based gradient and divergence operators as a function of the displacement field. This allows for a finite volume-based residual of the momentum equation to be used as the loss of the network within the training process. For both approaches, the trained models can be used as surrogates or initialisers for classical solvers. The results for three problems are presented: a plate with a hole, a curved plate, and a cantilever beam. It is demonstrated that both approaches can be used as a surrogate or initialiser with an acceptable level of accuracy; however, the classical supervised approach requires much less computational effort than the physics-informed approach. In particular, employing the classical supervised model as an initialiser for the solution of 500 configurations from the cantilever beam case can reduce the overall computational time by up to 461%.