Finite element method

Lore & Background

The finite element method originated from the need to solve complex elasticity and structural analysis problems in civil and aeronautical engineering. Its development can be traced back to work by Alexander Hrennikoff and Richard Courant in the early 1940s. Another pioneer was Ioannis Argyris. In the USSR, the introduction of the practical application of FEM is usually connected with Alexander K. Oganesyan. It was also independently rediscovered in China by Feng Kang in the late 1950s and early 1960s, based on the computations of dam constructions, where it was called the 'finite difference method based on the variational principle'. Although the approaches used by these pioneers are different, they share one essential characteristic: the mesh discretization of a continuous domain into a set of discrete sub-domains, usually called elements.

Reader's Guide

The finite element method is significant because it provides a general numerical approach for solving partial differential equations in two- or three-space variables, including boundary value problems. Its ability to subdivide a large system into finite elements allows for accurate representation of complex geometry, inclusion of dissimilar material properties, easy representation of the total solution, and capture of local effects. The method gained momentum in the 1960s and 1970s due to developments by researchers at institutions such as the University of Stuttgart, University of California Berkeley, Swansea University, University of Paris 6, and Cornell University. A rigorous mathematical basis was provided in 1973 with a publication by Gilbert Strang and George Fix. FEM has since been generalized for numerical modeling in many engineering disciplines, including electromagnetism, heat transfer, and fluid dynamics. FEA simulations are valuable because they remove multiple instances of creating and testing complex prototypes for various high-fidelity situations, such as in frontal crash simulations or numerical weather prediction.

Did You Know?

• FEM subdivides a large system into smaller, simpler parts called finite elements.
• The method approximates the unknown function over the domain and minimizes an associated error function via the calculus of variations.
• FEM is commonly introduced as a special case of the Galerkin method.
• The practical application of FEM is known as finite element analysis (FEA).