A Finite Difference Scheme For Solving Partial Differential Equations In Heat Conduction Problems In Mechanical Engineering

Abstract

When it comes to mechanical engineering, the majority of thermo physical processes are governed by partial differential equations (PDEs), notably those that are associated with transient and steady-state heat conduction respectively. Due to the fact that analytical solutions are only applicable to issues that have simplified geometries and boundary conditions, finite-difference approximations have emerged as a viable option for obtaining approximate solutions for areas that are more intricate. For the two-dimensional heat conduction problem, the current study presents an ordered finite-difference (FDM) scheme that has been customized to function properly. The approach is excellent for engineering applications because it has numerical stability, consistency, and convergence. It accomplishes these goals by using central difference approximations in space and an implicit backward Euler method in time. In this paper, we offer a rigorous mathematical derivation of the strategy, which is reinforced by stability analysis using the von Neumann methodology. For the purpose of solving typical heat transfer issues, numerical calculations are carried out. These problems include transient heat conduction in a rectangular plate and steady-state distribution in a finned surface. When compared with benchmarked analytical and semi-analytical solutions, comparison analysis reveals root-mean-square errors that are less than 1.5%, which is evidence of the model's impressive level of accuracy. Additionally, the method that was developed is used to simulate actual thermal conditions using experimental datasets that are already in existence. This demonstrates the program's practical applicability in the simulation of component design and production. The findings provide evidence that the approach is robust, computationally efficient, and amenable to use in the context of more complex mechanical systems. The purpose of this effort is to establish a replicable model that is compliant with the existing production requirements for thermal condition-based simulation-driven design.