Mathematical and numerical study of symmetry and positivity of the tensor-valued spring constant defined from P1-FEM for two- and three-dimensional linear elasticity

In this study, we consider a spring-block system that approximates a $d$-dimensional linear elastic body, where $d=2$ or $d=3$. We derive a $d\times d$ matrix as the spring constant using the P1 finite element method with a triangular mesh for the linear elasticity equations. We mathematically analyze the symmetry and positive-definiteness of the spring constant. Even if we assume full symmetry of the elasticity tensor, the symmetry of the matrix obtained as the spring constant is not trivial. However, we have succeeded in proving this in a unified manner for both 2D and 3D cases. This is an alternative proof for the 2D case in Notsu-Kimura (2014) and is a new result for the 3D case. We provide a necessary and sufficient condition for the spring constant to be positive-definite in the case of an isotropic elasticity tensor, along with a sufficient condition in terms of mesh regularity and the Poisson ratio. These theoretical results are supported by several numerical experiments. The positive-definiteness of the spring constant derived from the finite element method plays a vital role in fracture simulations of elastic bodies using the spring-block system.