Distributed PINN for Linear Elasticity - A Unified Approach for Smooth, Singular, Compressible and Incompressible Media

Over the last several decades, the Finite Element Method (FEM) has emerged as a numerical approach method of choice for the solution of problems in solid mechanics. Part of the reason for the success of FEM is that it provides a unified framework for discretizing even complex differential equations. However, despite this overall unification, FEM still requires specific variants or corrections depending on the problem at hand. For instance, problems with skewed meshes, discontinuity, singularity, incompressible media, etc. require the analyst to modify the discretization approach in order to preserve robustness. We speculate that local-polynomial bases such as those used in FEM do not sufficiently represent local physics and more "physics-informed" approaches may be more universal. Accordingly, in this paper, we evaluate the feasibility of one such approach - the recently developed Distributed Physics Informed Neural Network (DPINN) approach - to provide a truly unified framework for addressing problems in Solid Mechanics. The DPINN approach utilizes a piecewise-neural network representation for the underlying field, rather than the piece-polynomial representation that is common in FEM. We solve a series of problems in solid mechanics using either the single or domain-distributed version of DPINN and demonstrate that the approach is able to seamlessly solve varied problems with no special treatment required for volumetric locking or capturing discontinuities. Further, we also demonstrate that the DPINN approach, due to its meshless nature, is able to avoid the curse of dimensionality. We discuss the relative merits and demerits of the DPINN approach in comparison to FEM. We expect this work to be useful to researchers looking to develop unified computational frameworks for problems in solid mechanics.