finite element method


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finite element method

[¦fī‚nīt ′el·ə·mənt ‚meth·əd]
(engineering)
An approximation method for studying continuous physical systems, used in structural mechanics, electrical field theory, and fluid mechanics; the system is broken into discrete elements interconnected at discrete node points.

Finite element method

A numerical analysis technique for obtaining approximate solutions to many types of engineering problems. The need for numerical methods arises from the fact that for most practical engineering problems analytical solutions do not exist. While the governing equations and boundary conditions can usually be written for these problems, difficulties introduced by either irregular geometry or other discontinuities render the problems intractable analytically. To obtain a solution, the engineer must make simplifying assumptions, reducing the problem to one that can be solved, or a numerical procedure must be used. In an analytic solution, the unknown quantity is given by a mathematical function valid at an infinite number of locations in the region under study, while numerical methods provide approximate values of the unknown quantity only at discrete points in the region. In the finite element method, the region of interest is divided up into numerous connected subregions or elements within which approximate functions (usually polynomials) are used to represent the unknown quantity.

The physical concept on which the finite element method is based has its origins in the theory of structures. The idea of building up a structure by fitting together a number of structural elements (see illustration) was used in the early truss and framework analysis approaches employed in the design of bridges and buildings in the early 1900s. By knowing the characteristics of individual structural elements and combining them, the governing equations for the entire structure could be obtained. This process produces a set of simultaneous algebraic equations. The limitation on the number of equations that could be solved posed a severe restriction on the analysis. The introduction of the digital computer has made possible the solution of the large-order systems of equations.

Structures modeled by fitting together structural elements: ( a ) truss structure; ( b ) two- dimensional planar structureenlarge picture
Structures modeled by fitting together structural elements: (a) truss structure; (b) two- dimensional planar structure

The finite element method is one of the most powerful approaches for approximate solutions to a wide range of problems in mathematical physics. The method has achieved acceptance in nearly every branch of engineering and is the preferred approach in structural mechanics and heat transfer. Its application has extended to soil mechanics, heat transfer, fluid flow, magnetic field calculations, and other areas.

References in periodicals archive ?
King, "A finite element method for interface problems in domains with smooth boundaries and interfaces," Advances in Computational Mathematics, vol.
Finite element method for a nonlocal problem of Kirchhoff type.
combined the strain smoothing technique [28] used in mesh-free methods [29] into the FEM and formulated a cell/element-based smoothed finite element method (S-FEM) [6].
As it can be seen in Figure 2 on the graphs, the spectral element method on the coarsest mesh just on the 4th order demonstrates the accuracy of the order of 1%, whereas the finite element method on a mesh of 849 thousand elements still gives an error of more than 10% for the finite elements of the first order and about 2% for the finite elements of the second order.
The light diffusion equation in a frequency domain is numerically calculated by using the finite element method (FEM) [3, 6].
Recently, extended finite element method, collocation boundary element, cell-based smoothed finite element method, and so forth were used to solve the problem of fracture in piezoelectric materials.
Although only the simple axial element has been used, the procedure described is common to the finite element method for all element and analysis.
Holberg C et al attempted to create an anisotropic finite element method model of mandibular bone and orthodontic bracket.
Finally we carry out some numerical experiments to verify that the mixed finite element method can effectively exclude all the spurious eigenvalues, including the nonphysical zero eigenvalues.
applied the hierarchical stochastic finite element method for structural analysis [15].
In this paper, a new expanded mixed finite element method is proposed and studied for Sobolev equation with convection-term.
Finite element method based software package COMSOL Multiphysics and its Conjugate Heat Transfer interface was selected to perform the modelling.

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