Explore the theoretical foundations of finite element analysis $($FEA$)$ in structural engineering and elucidate its significance in predicting the behavior of complex structures under various loading conditions. Delve into the mathematical formulations and computational procedures involved in FEA, emphasizing its applications in simulating stress, deformation, and failure patterns in engineering structures. Discuss the advantages, limitations$,$ and potential advancements of FEA methodology without referring to specific software packages or case studies.