Explain the principles behind the Finite Element Method $($FEM$)$ in civil engineering and its significance in structural analysis. Discuss the fundamental assumptions and mathematical formulations involved in FEM, elucidating how it breaks down complex structures into simpler elements for computational analysis. Explore the role of shape functions, interpolation techniques, and discretization in FEM, highlighting their contributions to accurate solutions. Delve into the concept of equilibrium and compatibility within FEM, elucidating how these principles ensure structural integrity and stability. Furthermore, analyze the advantages and limitations of FEM compared to other analytical methods, such as the finite difference method or the boundary element method, and how it has revolutionized the field of structural engineering by enabling the analysis of intricate structures with unprecedented accuracy and efficiency.