A Numerical Study of Convergence Speed in Solving Stress Field by Control Volume Based Finite Difference Method

Abstract

The study presents an efficient implementation of the control volume-based finite difference method (CVFDM) integrated with a line-by-line solver for stress and strain analysis. The Navier's equation was discretized for each element, yielding fifteen displacement unknowns represented in a single equation. For this study, a three-unknown formulation per element was adopted. A line-by-line solver employing the TriDiagonal Matrix Algorithm (TDMA) was utilized to solve the equations. Dynamic memory allocation for updating displacements at previous element rows, enhancing convergence speed. Variables were solved and stored contiguously along a row in each time step, the iteration continued until the desired accuracy was achieved, eliminating the need for redundant boundary condition updates and reducing overall simulation time. A finite difference method (FDM)-based stress analysis application was developed based on the novel approach proposed in this work. Numerical simulations of three problems conducted using this application demonstrate a high level of agreement with theoretical solutions. The modified CVFDM with line-by-line solver proves to be an efficient and robust approach for stress and strain analysis, providing accurate and reliable results.