Finite Element Method with Adaptive Mesh Refinement for Solving Nonlinear PdesIn Engineering Design

Abstract

The Finite Element Method (FEM) has emerged as a cornerstone in solving Partial Differential Equations (PDEs), especially in engineering applications involving complex geometries and boundary conditions. However, when addressing nonlinear PDEs in real-world engineering design, challenges such as accuracy, convergence, and computational cost persist. Adaptive Mesh Refinement (AMR) enhances FEM by dynamically refining the computational mesh based on error estimations, providing localized accuracy improvements where needed. This paper investigates the integration of FEM with AMR for efficiently solving nonlinear PDEs in mechanical and structural engineering problems. The study elaborates on mathematical formulations, solution strategies, convergence criteria, and implementation in engineering design simulations. Real-world case studies and simulations demonstrate the effectiveness of this hybrid numerical approach in enhancing the performance and reliability of design analysis.

Keywords: Finite Element Method, Adaptive Mesh Refinement, Nonlinear PDEs, Engineering Design, Structural Simulation, Numerical Methods