Implementation of the Stress-Based Galerkin Finite Element Method for Two-Dimensional Elastodynamic Problems

This study considers a two-dimensional dynamic problem of linear elasticity formulated directly in terms of the stress tensor components. A coupled system of differential equations for two normal stress components and one shear stress component is derived from the equations of motion, kinematic relations, and Hooke’s law. Initial and traction boundary conditions are specified for the resulting system. Spatial discretization is performed using the Galerkin method with linear triangular finite elements. Element mass and stiffness matrices accounting for the coupling between the stress components are derived. Time integration is implemented using the Newmark method in a C++ program. An independent computational implementation is developed in FreeFEM++ using an implicit finite-difference approximation of the time derivatives. The sensitivity of the solution to the finite element size and time step is investigated. Successive refinement of the spatial and temporal discretizations leads to stabilization of the computed stresses. The method is validated using two benchmark problems. For a rectangular domain subjected to a time-dependent surface load, the stress histories computed in C++ and FreeFEM++ are in agreement. For transient wave propagation from a circular cavity, the results are compared with an analytical solution, the QL boundary element method, and the classical displacement-based finite element formulation. The minimum circumferential stress is −1.2250, compared with the analytical value of −1.2253, corresponding to a difference of approximately 0.025%. The direct stress formulation eliminates the need to recover stresses by differentiating the displacement field and yields a smoother time history for the discretization considered. The results confirm the applicability of the developed finite element scheme to dynamic elasticity problems formulated in terms of stresses.