75 Years of the FEM (Finite Element Method)

Finite Element Analysis is a simulation concept developed from the theoretical basis established by the Finite Element Method (FEM) — founded with a set of scientific papers in the 1940s.

Imagined as numerical techniques for finding approximate solutions to boundary value problems for partial differential equations, FEM is based on a problem domain’s subdivision into simpler parts called finite elements, and on the calculus of variational methods to minimize an associated error function.

The scientific pillars of the Finite Element Method came from the need to solve complex elasticity and structural analysis problems in civil and aeronautical engineering. The first development can be traced back to the work of A. Hrennikoff, 1941[1] and R. Courant, 1943[2]. Although these pioneers used different perspectives in their finite element approaches, they pointed out one common essential characteristic: mesh discretization of a continuous domain into a set of discrete sub-domains, usually called elements. Another fundamental mathematical contribution to the FEM is represented by the book “An Analysis of the Finite Element Method” by Gilbert Strang and George Fix, first published in 1973 [3]. After that, FEM has been generalized for the numerical modeling of physical systems in many engineering disciplines including electromagnetism, heat transfer, and fluid dynamics.

Benefits of FEM

Many specializations under Mechanical Engineering’s umbrella, such as aeronautical, biomechanical, and automotive industries, are commonly using integrated FEM in product design and development. Several modern FEM packages include specific components such as thermal, electromagnetic, fluid, and structural working environments. For example, in a structural simulation, the FEM helps in “producing stiffness and strength visualizations and also in minimizing weight, materials, and costs”.[4]

As its main capability, FEM offers a detailed visualization of bending and twisting places for a structure, indicating stresses and displacement distribution. Modern FEM applications software are offering a variety of simulation options for modeling and analysis.

The benefits of FEM consist of “increased accuracy, enhanced design and better insight into critical design parameters, virtual prototyping, fewer hardware prototypes, a faster and less expensive design cycle, increased productivity, and increased revenue” [5].

During the modern engineering history, FEM algorithms were embedded in many powerful design tools, contributing to standards of engineering and design process improvements.

Using FEM algorithms integrated in FEA applications, any engineering structure design can be developed, tested, and modified before the manufacturing of product prototypes.

Moreover, using the same digital format, any product can be optimized in any phase during the workflow process, from the engineering area to design and testing laboratory. FEA has also been proposed to be used in stochastic modelling for numerically solving probability models.

Main History Milestones:

1941-1942 – Hrennikoff and Courant developed mesh discretization methods for solving elasticity and structural analysis problems in civil and aeronautical engineering.

1956Ray W. Clough published the first paper on the Finite Element Method (FEM). “Finite elements” term coined in 1960 article. [6]

1959 – General Motors and IBM build computer system DAC-1 (Design Augmented by Computers) to facilitate cars design.

1960 – William Fetter from Boeing coins the term “computer graphic” for his human factors cockpit drawings. [7]

1965NASTRAN (NASA Structural Analysis) is developed as structural analysis solver tool.

1977 – FIESTA, the first professional FEM p-version code, initiated by Alberto Peano from ISME.

1982 – PROBE, developed by Barna Szabo and Kent Myers, 1st ‘industrial’ implementation of p-version FEA for research and aerospace applications.

1987 – MECHANICA, developed by RASNA Corp.

2001 – P-version FEM proven to be the most efficient for Plasticity by A. Duster. [8]

2006ASME guide for Verification and Validation in Computational Solid Mechanics is released.

2008 – NASA released a Standard for development of Models and Simulations.

2012 – Barna Szabo and Ricardo Actis introduce “Simulation governance: New technical requirements for software tools in computational solid mechanics”.[9]

2013 – SimScale officially released the world’s first cloud-based 3D simulation platform, offering a comprehensive set of linear and non-linear FEA capabilities for structural mechanics, fluid dynamics, and thermodynamics.


[1] Hrennikoff, A. (1941). “Solution of problems of elasticity by the framework method”. Journal of applied mechanics 8.4: 169–175.
[2] Courant, R. (1943). “Variational methods for the solution of problems of equilibrium and vibrations”. Bulletin of the American Mathematical Society 49: 1–23.
[3] Strang, G.; Fix, G. (1973). “An Analysis of the Finite Element Method”. Prentice Hall.
[4] Kiritsis, D.; Eemmanouilidis, Ch.; Koronios, A.; Mathew, J. (2009). “Engineering Asset Management” Proceedings of the 4th World Congress on Engineering Asset Management (WCEAM), 591-592.
[5] Hastings, J.K.; Judes, M.A.; Brauer, J.R. (1985) “Accuracy and Economy of Finite Element Magnetic Analysis”. 33rd Annual National Relay Conference
[6] Clough, R.W. (1960) “The Finite Element Method, in Plane Stress Analysis”, Proc. 2nd A.S.C.E. Conf: on Electronic Comp., Pittsburgh, PA
[7] Carlson, W. (2003) “A Critical History of Computer Graphics and Animation”. The Ohio State University
[8] Duster, A. (2001) “The p-version of the Finite Element Method compared to an adaptive h-version for the deformation theory of plasticity”. Computer Methods in Applied Mechanics and Engineering Journal Impact Factor & Information.
[9] Szabó, B.; Actis, R. (2011) “Simulation governance: New technical requirements for software tools in computational solid mechanics”. International Workshop on Verification and Validation in Computational Science University of Notre Dame, October 2011

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