Finite element method with energy minimization (FEMEM) in structural analysis: State-of-the-art
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The Finite Element Method (FEM) is a powerful and widely adopted numerical technique in structural engineering for the analysis of structures and systems. Although FEM is highly effective for linear problems and can address certain classes of nonlinear problems through appropriate iterative schemes, there exist structural analysis cases that cannot be satisfactorily solved using conventional FEM formulations. Such cases include problems with non-unique equilibrium states, nonlinear or missing boundary conditions, and structures with ill-conditioned flexibility matrices, for example truss systems following the failure of one or more members. To overcome these limitations, an emerging approach—the Finite Element Method with Energy Minimization (FEMEM)—has been introduced. FEMEM is based on the same fundamental principles of structural mechanics as FEM; however, it differs in its solution strategy. Instead of solving matrix equations, FEMEM formulates the problem as the minimization of a functional, namely the total potential energy of the structure. In this paper, the fundamental concepts of FEMEM are first introduced, followed by a comparative discussion with classical FEM to highlight the advantages and limitations of the method. Subsequently, representative applications of FEMEM are presented for various structural systems, ranging from trusses and truss-like structures to planar and three-dimensional structural models. Finally, directions for future research are discussed, with emphasis on extending the applicability of FEMEM to more complex structural systems and a wider range of nonlinearities, including material, geometric, and constraint-related effects.
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