In structural engineering, the analysis of complex systems often involves the consideration of various factors such as material properties, loading conditions, and geometric configurations. Discuss the fundamental principles underlying the finite element method $($FEM$)$ as a numerical technique for solving these intricate structural problems. Explore the mathematical basis of FEM, including the formulation of stiffness matrices, shape functions, and the process of discretization. Elaborate on the advantages and limitations of FEM compared to other analytical methods such as the finite difference method and the boundary element method. Additionally, delve into the concept of convergence in FEM analysis and its significance in ensuring the accuracy of computational results. How does FEM facilitate the analysis of nonlinear behavior in structures, and what are the key challenges associated with nonlinear finite element analysis? Lastly, discuss recent advancements in FEM technology and their potential implications for the future of structural engineering design and analysis.