Finite Element Analysis of Nonlinear Partial Differential Equations in Multiphysics Engineering Applications

Abstract

The accurate numerical solution of nonlinear partial differential equations (PDEs) is fundamental to modeling and analyzing complex engineering systems involving interacting physical phenomena. Multiphysics engineering applications, such as thermo-mechanical analysis, fluid–structure interaction, and coupled electromagnetic–thermal systems, often exhibit strong nonlinear behavior that poses significant challenges for conventional analytical and numerical methods. This study presents a comprehensive finite element analysis (FEA) framework for solving nonlinear PDEs arising in coupled multiphysics environments. The proposed methodology employs the finite element method (FEM) with weak-form discretization, adaptive meshing, and an iterative Newton–Raphson solution strategy to efficiently resolve nonlinearities while maintaining numerical stability and convergence. A generalized computational framework is developed to model the interaction among multiple physical fields under realistic boundary and loading conditions. The performance of the proposed approach is evaluated through representative engineering case studies, demonstrating its capability to accurately predict temperature distributions, stress fields, deformation characteristics, and coupled physical responses. The numerical results are validated through comparisons with benchmark solutions and published literature, showing high accuracy, improved convergence behavior, and computational efficiency. Furthermore, sensitivity analyses are conducted to investigate the influence of mesh density, material properties, and nonlinear coupling parameters on solution accuracy. The findings demonstrate that the proposed finite element framework provides a robust and versatile tool for the numerical simulation of complex multiphysics systems, offering enhanced predictive capability for engineering design, optimization, and decision-making. The study contributes to the advancement of computational engineering by providing an efficient and scalable numerical methodology for solving nonlinear multiphysics problems, with potential applications in aerospace, automotive, civil, biomedical, and energy engineering