Finite element method: Difference between revisions

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| year = 2005

| publisher = SIAM

| doi = 10.1137/040606041030601077

| hdl = 20.500.11850/147656

| doi-broken-date = 2024-11-02

| hdl-access = free

}}</ref> To solve a problem, FEM subdivides a large system into smaller, simpler parts called '''finite elements'''. This is achieved by a particular space [[discretization]] in the space dimensions, which is implemented by the construction of a [[Types of mesh|mesh]] of the object: the numerical ___domain for the solution that has a finite number of points. FEM formulation of a boundary value problem finally results in a system of [[algebraic equation]]s. The method approximates the unknown function over the ___domain.<ref>{{cite book

| title = A first course in the finite element method

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The practical application of FEM is known as finite element analysis (FEA). FEA, as applied in [[engineering]], is a computational tool for performing [[engineering analysis]]. It includes the use of [[mesh generation]] techniques for dividing a [[complex system|complex problem]] into smaller elements, as well as the use of software coded with a FEM algorithm. When applying FEA, the complex problem is usually a physical system with the underlying [[physics]], such as the [[Euler–Bernoulli beam theory|Euler–Bernoulli beam equation]], the [[heat equation]], or the [[Navier-Stokes equations|Navier]][[Euler–Bernoulli beam theory|–]]Stokes equations, expressed in either PDEs or [[integral equation]]s, while the divided, smaller elements of the complex problem represent different areas in the physical system.

FEA may be used for analyzing problems over complicated domains (e.g., cars and oil pipelines) when the ___domain changes (e.g., during a solid-state reaction with a moving boundary), when the desired precision varies over the entire ___domain, or when the solution lacks smoothness. FEA simulations provide a valuable resource, as they remove multiple instances of creating and testing complex prototypes for various high-fidelity situations.<ref>{{Cite journal |date=2022 |title=Editorial Board |url=https://doi.org/10.1016/s0168-874x(22)00118-4 |journal=Finite Elements in Analysis and Design |volume=211 |pagesarticle-number=103845 |doi=10.1016/s0168-874x(22)00118-4 |issn=0168-874X|url-access=subscription }}</ref> For example, in a frontal crash simulation, it is possible to increase prediction accuracy in important areas, like the front of the car, and reduce it in the rear of the car, thus reducing the cost of the simulation. Another example would be in [[numerical weather prediction]], where it is more important to have accurate predictions over developing highly nonlinear phenomena, such as [[tropical cyclone]]s in the atmosphere or [[Eddy (fluid dynamics)|eddies]] in the ocean, rather than relatively calm areas.

A clear, detailed, and practical presentation of this approach can be found in the textbook ''The Finite Element Method for Engineers''.<ref>{{Cite book |last=Huebner |first=Kenneth H. |title=The Finite Element Method for Engineers |publisher=Wiley |year=2001 |isbn=978-0-471-37078-9}}</ref>

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===Generalized finite element method===

The generalized finite element method (GFEM) uses local spaces consisting of functions, not necessarily polynomials, that reflect the available information on the unknown solution and thus ensure good local approximation. Then a [[partition of unity]] is used to “bond”"bond" these spaces together to form the approximating subspace. The effectiveness of GFEM has been shown when applied to problems with domains having complicated boundaries, problems with micro-scales, and problems with boundary layers.<ref>{{cite journal | first1=Ivo | last1=Babuška | author1-link=Ivo Babuška | first2=Uday | last2=Banerjee | first3=John E. | last3=Osborn | author3-link=John E. Osborn (mathematician) | title=Generalized Finite Element Methods: Main Ideas, Results, and Perspective | journal=[[International Journal of Computational Methods]] | date=June 2004 | issue=1 | pages= 67–103 | doi=10.1142/S0219876204000083 | volume=1}}</ref>

===Mixed finite element method===

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The [[finite difference method]] (FDM) is an alternative way of approximating solutions of PDEs. The differences between FEM and FDM are:

* The most attractive feature of the FEM is its ability to handle complicated geometries (and boundaries) with relative ease. While FDM in its basic form is restricted to handle rectangular shapes and simple alterations thereof, the handling of geometries in FEM is theoretically straightforward.<ref name=":0" /><ref name=":1">{{Cite journal |last=Topper |first=Jürgen |date=January 2005 |title=Option pricing with finite elements |url=http://dx.doi.org/10.1002/wilm.42820050119 |journal=Wilmott |volume=2005 |issue=1 |pages=84–90 |doi=10.1002/wilm.42820050119 |doi-broken-date=3 December 2024 |issn=1540-6962|url-access=subscription }}</ref>

* FDM is not usually used for irregular CAD geometries but more often for rectangular or block-shaped models.<ref>{{Cite news|url=http://www.machinedesign.com/fea-and-simulation/what-s-difference-between-fem-fdm-and-fvm|title=What's The Difference Between FEM, FDM, and FVM?|date=2016-04-18|work=Machine Design|access-date=2017-07-28|archive-date=2017-07-28|archive-url=https://web.archive.org/web/20170728024918/http://www.machinedesign.com/fea-and-simulation/what-s-difference-between-fem-fdm-and-fvm|url-status=live}}</ref>

* FEM generally allows for more flexible mesh adaptivity than FDM.<ref name=":1" />