# Finite element method

finite element analysisfinite elementfinite elementsFEMfinite element methodsFEAfinite-elementfinite element analysis (FEA)finite-element analysisfinite element modeling
The finite element method (FEM) is the most largely used method for solving problems of engineering and mathematical models.wikipedia
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### Mesh generation

grid generationmesh generatormeshing
It includes the use of mesh generation techniques for dividing a complex problem into small elements, as well as the use of software program coded with FEM algorithm.
Meshes are used for rendering to a computer screen and for physical simulation such as finite element analysis or computational fluid dynamics.

### Structural analysis

analysisherestructures
Typical problem areas of interest include the traditional fields of structural analysis, heat transfer, fluid flow, mass transport, and electromagnetic potential.
There are three approaches to the analysis: the mechanics of materials approach (also known as strength of materials), the elasticity theory approach (which is actually a special case of the more general field of continuum mechanics), and the finite element approach.

### Mathematical model

mathematical modelingmodelmathematical models
The finite element method (FEM) is the most largely used method for solving problems of engineering and mathematical models.
In engineering, physics models are often made by mathematical methods such as finite element analysis.

### Euler–Bernoulli beam theory

Euler–Bernoulli beam equationbeam theoryEuler-Bernoulli beam equation
In applying FEA, the complex problem is usually a physical system with the underlying physics such as the Euler-Bernoulli beam equation, the heat equation, or the Navier-Stokes equations expressed in either PDE or integral equations, while the divided small elements of the complex problem represent different areas in the physical system.
Additional analysis tools have been developed such as plate theory and finite element analysis, but the simplicity of beam theory makes it an important tool in the sciences, especially structural and mechanical engineering.

### Numerical linear algebra

computational linear algebrageneral matrix calculationlinear algebra
Algebraic equation sets that arise in the steady state problems are solved using numerical linear algebra methods, while ordinary differential equation sets that arise in the transient problems are solved by numerical integration using standard techniques such as Euler's method or the Runge-Kutta method.
Matrix methods are particularly used in finite difference methods, finite element methods, and the modeling of differential equations.

### Galerkin method

Galerkin approximationGalerkin's methodBubnov Galerkin
Examples of variational formulation are the Galerkin method, the discontinuous Galerkin method, mixed methods, etc.A discretization strategy is understood to mean a clearly defined set of procedures that cover (a) the creation of finite element meshes, (b) the definition of basis function on reference elements (also called shape functions) and (c) the mapping of reference elements onto the elements of the mesh.

### Nastran

NX NastranMSC NastranNASA Structural Analysis Computer Software
NASA sponsored the original version of NASTRAN, and UC Berkeley made the finite element program SAP IV widely available.
NASTRAN is a finite element analysis (FEA) program that was originally developed for NASA in the late 1960s under United States government funding for the aerospace industry.

### Bruce Irons (engineer)

Bruce IronsIrons, Bruce
The finite element method obtained its real impetus in the 1960s and 1970s by the developments of J. H. Argyris with co-workers at the University of Stuttgart, R. W. Clough with co-workers at UC Berkeley, O. C. Zienkiewicz with co-workers Ernest Hinton, Bruce Irons and others at Swansea University, Philippe G. Ciarlet at the University of Paris 6 and Richard Gallagher with co-workers at Cornell University.
Bruce Moncur Irons (6 October 1924 – 5 December 1983) was an engineer and mathematician, known for his fundamental contribution to the finite element method, including the patch test, the frontal solver and, along with Ian C. Taig, the isoparametric element concept.

### Numerical analysis

numerical methodsnumericalnumerical computation
It is in fact a particular numerical method for solving partial differential equations in two or three space variables (i.e. some boundary value problems).
This can be done by a finite element method, a finite difference method, or (particularly in engineering) a finite volume method.

### John Argyris

John H. ArgyrisIoannis ArgyrisJ. H. Argyris
The finite element method obtained its real impetus in the 1960s and 1970s by the developments of J. H. Argyris with co-workers at the University of Stuttgart, R. W. Clough with co-workers at UC Berkeley, O. C. Zienkiewicz with co-workers Ernest Hinton, Bruce Irons and others at Swansea University, Philippe G. Ciarlet at the University of Paris 6 and Richard Gallagher with co-workers at Cornell University. Another pioneer was Ioannis Argyris.
Johann Hadji Argyris FRS (Greek: Ιωάννης Χατζι Αργύρης; 19 August 1913 – 2 April 2004) was a Greek pioneer of computer applications in science and engineering, among the creators of the finite element method (FEM), and lately Professor at the University of Stuttgart and Director of the Institute for Statics and Dynamics of Aerospace Structures.

### Alexander Hrennikoff

A. Hrennikoff
Its development can be traced back to the work by A. Hrennikoff and R. Courant in the early 1940s.
Alexander Pavlovich Hrennikoff (Александр Павлович Хренников; 11 November 1896 — 31 December 1984) was a Russian-Canadian Structural Engineer, a founder of the Finite Element Method.

### Philippe G. Ciarlet

CiarletPhilippe CiarletCiarlet, Philippe G.
The finite element method obtained its real impetus in the 1960s and 1970s by the developments of J. H. Argyris with co-workers at the University of Stuttgart, R. W. Clough with co-workers at UC Berkeley, O. C. Zienkiewicz with co-workers Ernest Hinton, Bruce Irons and others at Swansea University, Philippe G. Ciarlet at the University of Paris 6 and Richard Gallagher with co-workers at Cornell University.
Philippe G. Ciarlet (born 1938, Paris) is a French mathematician, known particularly for his work on mathematical analysis of the finite element method.

### Gilbert Strang

Strang, GilbertStrang
A rigorous mathematical basis to the finite element method was provided in 1973 with the publication by Strang and Fix.
William Gilbert Strang (born November 27, 1934 ), usually known as simply Gilbert Strang or Gil Strang, is an American mathematician, with contributions to finite element theory, the calculus of variations, wavelet analysis and linear algebra.

### George Fix

FixFix, George
A rigorous mathematical basis to the finite element method was provided in 1973 with the publication by Strang and Fix.
George J. Fix (10 May 1939 – 10 March 2002) was an American mathematician who collaborated on several seminal papers and books in the field of finite element method.

### SESAM (FEM)

Sesam
In Norway the ship classification society Det Norske Veritas (now DNV GL) developed Sesam in 1969 for use in analysis of ships.
It is based on the displacement formulation of the Finite Element Method.

### Torsion (mechanics)

torsiontorsionaltorsionally
Hrennikoff's work discretizes the domain by using a lattice analogy, while Courant's approach divides the domain into finite triangular subregions to solve second order elliptic partial differential equations (PDEs) that arise from the problem of torsion of a cylinder.

### Ray William Clough

Ray W. CloughRay CloughR. W. Clough
The finite element method obtained its real impetus in the 1960s and 1970s by the developments of J. H. Argyris with co-workers at the University of Stuttgart, R. W. Clough with co-workers at UC Berkeley, O. C. Zienkiewicz with co-workers Ernest Hinton, Bruce Irons and others at Swansea University, Philippe G. Ciarlet at the University of Paris 6 and Richard Gallagher with co-workers at Cornell University.
Ray William Clough, (July 23, 1920 – October 8, 2016), was Byron L. and Elvira E. Nishkian Professor of structural engineering in the department of civil engineering at the University of California, Berkeley and one of the founders of the finite element method (FEM).

### P-FEM

p-versionp-refinements
Examples of discretization strategies are the h-version, p-version, hp-version, x-FEM, isogeometric analysis, etc.
p-FEM or the p-version of the finite element method is a numerical method for solving partial differential equations.

### Isogeometric analysis

iso-geometric analysis (IGA)
Examples of discretization strategies are the h-version, p-version, hp-version, x-FEM, isogeometric analysis, etc.
Isogeometric analysis is a computational approach that offers the possibility of integrating finite element analysis (FEA) into conventional NURBS-based CAD design tools.

### Extended finite element method

eXtended FEMeXtented Finite Elements (XFEM)x-FEM
Examples of discretization strategies are the h-version, p-version, hp-version, x-FEM, isogeometric analysis, etc.
It extends the classical finite element method (FEM) approach by enriching the solution space for solutions to differential equations with discontinuous functions.

### Partial differential equation

partial differential equationsPDEPDEs
It is in fact a particular numerical method for solving partial differential equations in two or three space variables (i.e. some boundary value problems).
Alternatives are numerical analysis techniques from simple finite difference schemes to the more mature multigrid and finite element methods.

### Hp-FEM

Fichera corner problemhigh-order finite element methodshp
Examples of discretization strategies are the h-version, p-version, hp-version, x-FEM, isogeometric analysis, etc.
hp-FEM is a general version of the finite element method (FEM), a numerical method for solving partial differential equations based on piecewise-polynomial approximations that employs elements of variable size

### Richard Courant

R. CourantCourantCourant, Richard
Its development can be traced back to the work by A. Hrennikoff and R. Courant in the early 1940s.
Courant's name is also attached to the finite element method, with his numerical treatment of the plain torsion problem for multiply-connected domains, published in 1943.

### Analysis

analysesanalyzinganalytical
Studying or analyzing a phenomenon with FEM is often referred to as finite element analysis (FEA).

### Engineering

engineerengineersengineered
FEA as applied in engineering is a computational tool for performing engineering analysis.
CAD together with digital mockup (DMU) and CAE software such as finite element method analysis or analytic element method allows engineers to create models of designs that can be analyzed without having to make expensive and time-consuming physical prototypes.