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The finite element method (FEM) (its practical application often known as finite element analysis (FEA)) is a numerical technique for finding approximate solutions of partial differential equations (PDE) as well as integral equations. The solution approach is based either on eliminating the differential equation completely (steady state problems), or rendering the PDE into an approximating system of ordinary differential equations, which are then numerically integrated using standard techniques such as Euler's method, Runge-Kutta, etc.

In solving partial differential equations, the primary challenge is to create an equation that approximates the equation to be studied, but is numerically stable, meaning that errors in the input and intermediate calculations do not accumulate and cause the resulting output to be meaningless. There are many ways of doing this, all with advantages and disadvantages. The finite element method is a good choice for solving partial differential equations over complicated domains (like cars and oil pipelines), when the domain changes (as during a solid state reaction with a moving boundary), when the desired precision varies over the entire domain, or when the solution lacks smoothness. For instance, 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 its rear (thus reducing 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 cyclones in the atmosphere, or eddies in the ocean) rather than relatively calm areas.

History

The finite element method originated from the need for solving complex elasticity and structural analysis problems in civil and aeronautical engineering. Its development can be traced back to the work by Alexander Hrennikoff (1941) and Richard Courant (1942). While the approaches used by these pioneers are different, they share one essential characteristic: mesh discretization of a continuous domain into a set of discrete sub-domains, usually called elements. Starting in 1947, Olgierd Zienkiewicz from Imperial College gathered those methods together into what would be called the Finite Element Method, building the pioneering mathematical formalism of the method.

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. Courant's contribution was evolutionary, drawing on a large body of earlier results for PDEs developed by Rayleigh, Ritz, and Galerkin.

Development of the finite element method began in earnest in the middle to late 1950s for airframe and structural analysis and gathered momentum at the University of Stuttgart through the work of John Argyris and at Berkeley through the work of Ray W. Clough in the 1960s for use in civil engineering. By late 1950s, the key concepts of stiffness matrix and element assembly existed essentially in the form used today. NASA issued a request for proposals for the development of the finite element software NASTRAN in 1965. The method was again provided with a rigorous mathematical foundation in 1973 with the publication of Strang and Fix's An Analysis of The Finite Element Method, and has since been generalized into a branch of applied mathematics for numerical modeling of physical systems in a wide variety of engineering disciplines, e.g., electromagnetism and fluid dynamics.
Technical discussion

We will illustrate the finite element method using two sample problems from which the general method can be extrapolated. It is assumed that the reader is familiar with calculus and linear algebra.

P1 is a one-dimensional problem

$$\mbox{ P1 }:\begin{cases} u''(x)=f(x) \mbox{ in } (0,1), \\ u(0)=u(1)=0, \end{cases}$$

where f is given, u is an unknown function of x, and u'' is the second derivative of u with respect to x.

The two-dimensional sample problem is the Dirichlet problem

$$\mbox{P2 }:\begin{cases} u_{xx}(x,y)+u_{yy}(x,y)=f(x,y) & \mbox{ in } \Omega, \\ u=0 & \mbox{ on } \partial \Omega, \end{cases}$$

where $$\Omega$$ is a connected open region in the (x,y) plane whose boundary $$\partial \Omega$$ is "nice" (e.g., a smooth manifold or a polygon), and $$u_{xx}$$ and $$u_{yy}$$ denote the second derivatives with respect to x and y, respectively.

The problem P1 can be solved "directly" by computing antiderivatives. However, this method of solving the boundary value problem works only when there is only one spatial dimension and does not generalize to higher-dimensional problems or to problems like u+u''=f. For this reason, we will develop the finite element method for P1 and outline its generalization to P2.

Our explanation will proceed in two steps, which mirror two essential steps one must take to solve a boundary value problem (BVP) using the FEM.

In the first step, one rephrases the original BVP in its weak form. Little to no computation is usually required for this step. The transformation is done by hand on paper.
The second step is the discretization, where the weak form is discretized in a finite dimensional space.

After this second step, we have concrete formulae for a large but finite dimensional linear problem whose solution will approximately solve the original BVP. This finite dimensional problem is then implemented on a computer.
Weak formulation

The first step is to convert P1 and P2 into their equivalent weak formulations. If u solves P1, then for any smooth function v that satisfies the displacement boundary conditions, i.e. v=0 at x=0 and x=1, we have

(1)$$\int_0^1 f(x)v(x) \, dx = \int_0^1 u''(x)v(x) \, dx.$$

Conversely, if u with u(0)=u(1)=0 satisfies (1) for every smooth function v(x) then one may show that this u will solve P1. The proof is easier for twice continuously differentiable u (mean value theorem), but may be proved in a distributional sense as well.

By using integration by parts on the right-hand-side of (1), we obtain

(2)
\begin{align} \int_0^1 f(x)v(x) \, dx & = \int_0^1 u''(x)v(x) \, dx \\ & = u'(x)v(x)|_0^1-\int_0^1 u'(x)v'(x) \, dx \\ & = -\int_0^1 u'(x)v'(x) \, dx = -\phi (u,v). \end{align}

where we have used the assumption that v(0)=v(1)=0.
A proof outline of existence and uniqueness of the solution

We can loosely think of$$H_0^1(0,1$$ ) to be the absolutely continuous functions of (0,1) that are 0 at x=0 and x=1 (see Sobolev spaces). Such functions are (weakly) "once differentiable" and it turns out that the symmetric bilinear map \!\,\phi then defines an inner product which turns $$H_0^1(0,1)$$ into a Hilbert space (a detailed proof is nontrivial). On the other hand, the left-hand-side $$\int_0^1 f(x)v(x)dx$$ is also an inner product, this time on the Lp space $$L^2(0,1)$$ . An application of the Riesz representation theorem for Hilbert spaces shows that there is a unique u solving (2) and therefore P1. This solution is a-priori only a member of $$H_0^1(0,1)$$ , but using elliptic regularity, will be smooth if f is.
The weak form of P2

If we integrate by parts using a form of Green's identities, we see that if u solves P2, then for any v,

$$\int_{\Omega} fv\,ds = -\int_{\Omega} \nabla u \cdot \nabla v \, ds = -\phi(u,v),$$

where $$\nabla$$ denotes the gradient and$$\cdot denotes the dot product in the two-dimensional plane. Once more\( \,\!\phi$$ can be turned into an inner product on a suitable space $$H_0^1(\Omega)$$ of "once differentiable" functions of $$\Omega$$ that are zero on \partial $$\Omega$$ . We have also assumed that $$v \in H_0^1(\Omega)$$ (see Sobolev spaces). Existence and uniqueness of the solution can also be shown.
Discretization
A function in H_0^1, with zero values at the endpoints (blue), and a piecewise linear approximation (red).

The basic idea is to replace the infinite dimensional linear problem:

Find $$u \in H_0^1$$ such that
$$\forall v \in H_0^1, \; -\phi(u,v)=\int fv$$

with a finite dimensional version:

(3) Find$$u \in V$$ such that
$$\forall v \in V, \; -\phi(u,v)=\int fv$$

where V is a finite dimensional subspace of $$H_0^1.$$ There are many possible choices for V (one possibility leads to the spectral method). However, for the finite element method we take V to be a space of piecewise polynomial functions.

For problem P1, we take the interval (0,1), choose n values of x with $$0=x_0<x_1<...<x_n<x_{n+1}=1$$ and we define V by

$$\begin{matrix} V=\{v:[0,1] \rightarrow \Bbb R\;: v\mbox{ is continuous, }v|_{[x_k,x_{k+1}]} \mbox{ is linear for }\\ k=0,...,n \mbox{, and } v(0)=v(1)=0 \} \end{matrix}$$

where we define $$x_0=0$$ and$$x_{n+1}=1$$ . Observe that functions in V are not differentiable according to the elementary definition of calculus. Indeed, if v \in V then the derivative is typically not defined at any $$x=x_k, k=1,...,n$$ . However, the derivative exists at every other value of x and one can use this derivative for the purpose of integration by parts.
A piecewise linear function in two dimensions.

For problem P2, we need V to be a set of functions of \Omega. In the figure on the right, we have illustrated a triangulation of a 15 sided polygonal region \Omega in the plane (below), and a piecewise linear function (above, in color) of this polygon which is linear on each triangle of the triangulation; the space V would consist of functions that are linear on each triangle of the chosen triangulation.

One often reads $$V_h$$ instead of V in the literature. The reason is that one hopes that as the underlying triangular grid becomes finer and finer, the solution of the discrete problem (3) will in some sense converge to the solution of the original boundary value problem P2. The triangulation is then indexed by a real valued parameter h>0 which one takes to be very small. This parameter will be related to the size of the largest or average triangle in the triangulation. As we refine the triangulation, the space of piecewise linear functions V must also change with h, hence the notation $$V_h$$ . Since we do not perform such an analysis, we will not use this notation.
Choosing a basis
Basis functions vk (blue) and a linear combination of them, which is piecewise linear (red).

To complete the discretization, we must select a basis of V. In the one-dimensional case, for each control point x_k we will choose the piecewise linear function $$v_k$$ in V whose value is 1 at $$x_k and zero at every \( x_j,\;j \neq k$$ , i.e.,

$$v_{k}(x)=\begin{cases} {x-x_{k-1} \over x_k\,-x_{k-1}} & \mbox{ if } x \in [x_{k-1},x_k], \\ {x_{k+1}\,-x \over x_{k+1}\,-x_k} & \mbox{ if } x \in [x_k,x_{k+1}], \\ 0 & \mbox{ otherwise},\end{cases}$$

for k=1,...,n; this basis is a shifted and scaled tent function. For the two-dimensional case, we choose again one basis function $$v_k$$ per vertex x_k of the triangulation of the planar region $$\Omega$$ . The function $$v_k$$ is the unique function of V whose value is 1 at $$x_k$$ and zero at every $$x_j,\;j \neq k.$$

Depending on the author, the word "element" in "finite element method" refers either to the triangles in the domain, the piecewise linear basis function, or both. So for instance, an author interested in curved domains might replace the triangles with curved primitives, and so might describe the elements as being curvilinear. On the other hand, some authors replace "piecewise linear" by "piecewise quadratic" or even "piecewise polynomial". The author might then say "higher order element" instead of "higher degree polynomial". Finite element method is not restricted to triangles (or tetrahedra in 3-d, or higher order simplexes in multidimensional spaces), but can be defined on quadrilateral subdomains (hexahedra, prisms, or pyramids in 3-d, and so on). Higher order shapes (curvilinear elements) can be defined with polynomial and even non-polynomial shapes (e.g. ellipse or circle).

Examples of methods that use higher degree piecewise polynomial basis functions are the hp-FEM and spectral FEM.

More advanced implementations (adaptive finite element methods) utilize a method to assess the quality of the results (based on error estimation theory) and modify the mesh during the solution aiming to achieve approximate solution within some bounds from the 'exact' solution of the continuum problem. Mesh adaptivity may utilize various techniques, the most popular are:

changing order of base functions (p-adaptivity)

Small support of the basis
Solving the two-dimensional problem $$u_{xx}+u_{yy}=-4$$ in the disk centered at the origin and radius 1, with zero boundary conditions.
(a) The triangulation.
(b) The sparse matrix L of the discretized linear system.
(c) The computed solution, $$u(x, y)=1-x^2-y^2.$$

The primary advantage of this choice of basis is that the inner products

$$\langle v_j,v_k \rangle=\int_0^1 v_j v_k\,dx$$

and

$$\phi(v_j,v_k)=\int_0^1 v_j' v_k'\,dx$$

will be zero for almost all j,k. (The matrix containing $$\langle v_j,v_k \rangle$$ in the (j,k) location is known as the Gramian matrix.) In the one dimensional case, the support of $$v_k$$ is the interval $$[x_{k-1},x_{k+1}]$$ . Hence, the integrands of $$\langle \( v_j,v_k \rangle$$ and $$\phi(v_j,v_k)$$ are identically zero whenever |j-k|>1.

Similarly, in the planar case, if $$x_j$$ and $$x_k$$ do not share an edge of the triangulation, then the integrals

$$\int_{\Omega} v_j v_k\,ds$$

and

$$\int_{\Omega} \nabla v_j \cdot \nabla v_k\,ds$$

are both zero.
Matrix form of the problem

If we write $$u(x)=\sum_{k=1}^n u_k v_k(x) and f(x)=\sum_{k=1}^n f_k v_k(x) then problem (3), taking \( v(x)=v_j(x) for j=1,...,n$$ , becomes

$$-\sum_{k=1}^n u_k \phi (v_k,v_j) = \sum_{k=1}^n f_k \int v_k v_j dx for j=1,...,n. (4)$$

If we denote by $$\mathbf{u} and\( \mathbf{f} the column vectors \( (u_1,...,u_n)^t$$ and $$(f_1,...,f_n)^t$$ , and if we let

$$L=(L_{ij})$$

and

$$M=(M_{ij})$$

be matrices whose entries are

$$L_{ij}=\phi (v_i,v_j)$$

and

$$M_{ij}=\int v_i v_j dx$$

then we may rephrase (4) as

$$-L \mathbf{u} = M \mathbf{f}. (5)$$

It is not, in fact, necessary to assume $$f(x)=\sum_{k=1}^n f_k v_k(x)$$ . For a general function f(x), problem (3) with $$v(x)=v_j(x) for j=1,...,n becomes actually simpler, since no matrix M is used, \( -L \mathbf{u} = \mathbf{b}$$ , (6)

where $$\mathbf{b}=(b_1,...,b_n)^t and b_j=\int f v_j dx for j=1,...,n.$$

As we have discussed before, most of the entries of L and M are zero because the basis functions v_k have small support. So we now have to solve a linear system in the unknown \mathbf{u} where most of the entries of the matrix L, which we need to invert, are zero.

Such matrices are known as sparse matrices, and there are efficient solvers for such problems (much more efficient than actually inverting the matrix.) In addition, L is symmetric and positive definite, so a technique such as the conjugate gradient method is favored. For problems that are not too large, sparse LU decompositions and Cholesky decompositions still work well. For instance, Matlab's backslash operator (which uses sparse LU, sparse Cholesky, and other factorization methods) can be sufficient for meshes with a hundred thousand vertices.

The matrix L is usually referred to as the stiffness matrix, while the matrix M is dubbed the mass matrix.
General form of the finite element method

In general, the finite element method is characterized by the following process.

One chooses a grid for \Omega. In the preceding treatment, the grid consisted of triangles, but one can also use squares or curvilinear polygons.
Then, one chooses basis functions. In our discussion, we used piecewise linear basis functions, but it is also common to use piecewise polynomial basis functions.

A separate consideration is the smoothness of the basis functions. For second order elliptic boundary value problems, piecewise polynomial basis function that are merely continuous suffice (i.e., the derivatives are discontinuous.) For higher order partial differential equations, one must use smoother basis functions. For instance, for a fourth order problem such as $$u_{xxxx}+u_{yyyy}=f$$ , one may use piecewise quadratic basis functions that are $$C^1.$$

Another consideration is the relation of the finite dimensional space V to its infinite dimensional counterpart, in the examples above $$H_0^1$$ . A conforming element method is one in which the space V is a subspace of the element space for the continuous problem. The example above is such a method. If this condition is not satisfied, we obtain a nonconforming element method, an example of which is the space of piecewise linear functions over the mesh which are continuous at each edge midpoint. Since these functions are in general discontinuous along the edges, this finite dimensional space is not a subspace of the original $$H_0^1.$$

Typically, one has an algorithm for taking a given mesh and subdividing it. If the main method for increasing precision is to subdivide the mesh, one has an h-method (h is customarily the diameter of the largest element in the mesh.) In this manner, if one shows that the error with a grid h is bounded above by $$Ch^p$$ , for some $$C<\infty$$ and p>0, then one has an order p method. Under certain hypotheses (for instance, if the domain is convex), a piecewise polynomial of order d method will have an error of order p=d+1.

If instead of making h smaller, one increases the degree of the polynomials used in the basis function, one has a p-method. If one combines these two refinement types, one obtains an hp-method (hp-FEM). In the hp-FEM, the polynomial degrees can vary from element to element. High order methods with large uniform p are called spectral finite element methods (SFEM). These are not to be confused with spectral methods.

For vector partial differential equations, the basis functions may take values in \mathbb{R}^n.

area.
Various types of finite element methods
AEM

The Applied Element Method, or AEM combines features of both FEM and Discrete element method, or (DEM).
Main article: Applied element method
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” 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.
hp-FEM

The hp-FEM combines adaptively elements with variable size h and polynomial degree p in order to achieve exceptionally fast, exponential convergence rates.
hpk-FEM

The hpk-FEM combines adaptively elements with variable size h, polynomial degree of the local approximations p and global differentiability of the local approximations (k-1) in order to achieve best convergence rates.
XFEM
Main article: Extended finite element method
S-FEM
Main article: Smoothed finite element method
Spectral methods
Main article: Spectral method
Meshfree methods
Main article: Meshfree methods
Discontinuous Galerkin methods
Main article: Discontinuous Galerkin method
Finite element limit analysis
Main article: Finite element limit analysis
Stretched grid method
Main article: Stretched grid method
Comparison to the finite difference method
Question book-new.svg This unreferenced section requires citations to ensure verifiability.

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.

The most attractive feature of finite differences is that it can be very easy to implement.

There are several ways one could consider the FDM a special case of the FEM approach. E.g., first order FEM is identical to FDM for Poisson's equation, if the problem is discretized by a regular rectangular mesh with each rectangle divided into two triangles.

There are reasons to consider the mathematical foundation of the finite element approximation more sound, for instance, because the quality of the approximation between grid points is poor in FDM.

The quality of a FEM approximation is often higher than in the corresponding FDM approach, but this is extremely problem-dependent and several examples to the contrary can be provided.

Generally, FEM is the method of choice in all types of analysis in structural mechanics (i.e. solving for deformation and stresses in solid bodies or dynamics of structures) while computational fluid dynamics (CFD) tends to use FDM or other methods like finite volume method (FVM). CFD problems usually require discretization of the problem into a large number of cells/gridpoints (millions and more), therefore cost of the solution favors simpler, lower order approximation within each cell. This is especially true for 'external flow' problems, like air flow around the car or airplane, or weather simulation.
Application
Visualization of how a car deforms in an asymmetrical crash using finite element analysis.

A variety of specializations under the umbrella of the mechanical engineering discipline (such as aeronautical, biomechanical, and automotive industries) commonly use integrated FEM in design and development of their products. Several modern FEM packages include specific components such as thermal, electromagnetic, fluid, and structural working environments. In a structural simulation, FEM helps tremendously in producing stiffness and strength visualizations and also in minimizing weight, materials, and costs.

FEM allows detailed visualization of where structures bend or twist, and indicates the distribution of stresses and displacements. FEM software provides a wide range of simulation options for controlling the complexity of both modeling and analysis of a system. Similarly, the desired level of accuracy required and associated computational time requirements can be managed simultaneously to address most engineering applications. FEM allows entire designs to be constructed, refined, and optimized before the design is manufactured.

This powerful design tool has significantly improved both the standard of engineering designs and the methodology of the design process in many industrial applications. The introduction of FEM has substantially decreased the time to take products from concept to the production line. It is primarily through improved initial prototype designs using FEM that testing and development have been accelerated. In summary, benefits of FEM include 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.

FEA has also been proposed to use in stochastic modelling, for numerically solving probability models. See the references list.  

Applied element method
Boundary element method
Direct stiffness method
Discontinuity layout optimization
Discrete element method
Finite element machine
Finite element method in structural mechanics
Galerkin method
Interval finite element
Isogeometric analysis
List of finite element software packages
Movable Cellular Automata
Multidisciplinary design optimization
Multiphysics
Patch test
Rayleigh-Ritz method
Weakened weak form v
t

Numerical partial differential equations
Finite difference methods
Heat Equation and related: FTCS scheme · Crank–Nicolson method Hyperbolic: Lax–Friedrichs method · Lax–Wendroff method · MacCormack method · Upwind scheme · Other: Alternating direction implicit method · Finite-difference time-domain method
Finite volume methods
Godunov's scheme · High-resolution scheme · MUSCL scheme · AUSM · Riemann solver
Finite element methods
hp-FEM · Extended finite element method · Discontinuous Galerkin method · Spectral element method · Meshfree methods · Mortar methods
Other methods
Spectral method · Pseudospectral method · Method of lines · Multigrid methods · Collocation method · Level set method · Boundary element method · Immersed boundary method · Analytic element method · Particle-in-cell · Isogeometric analysis
Domain decomposition methods
Schur complement method · Fictitious domain method · Schwarz alternating method · Additive Schwarz method · Abstract additive Schwarz method · Neumann–Dirichlet method · Neumann–Neumann methods · Poincaré–Steklov operator · Balancing domain decomposition · BDDC · FETI · FETI-DP
References

^ Giuseppe Pelosi (2007). "The finite-element method, Part I: R. L. Courant: Historical Corner". doi:10.1109/MAP.2007.376627.
^ E. Stein (2009), Olgierd C. Zienkiewicz, a pioneer in the development of the finite element method in engineering science. Steel Construction, 2 (4), 264-272.
^ Matrix Analysis Of Framed Structures, 3rd Edition by Jr. William Weaver, James M. Gere, 3rd Edition, Springer-Verlag New York, LLC, ISBN 978-0-412-07861-3, First edition 1966
^ Strang, Gilbert; Fix, George (1973). An Analysis of The Finite Element Method. Prentice Hall. ISBN 0130329460.
^ Roberto Coccioli, Tatsuo Itoh, Giuseppe Pelosi, Peter P. Silvester (1996). "Finite-element methods in microwaves: a selected bibliography". doi:10.1109/74.556518.
^ Ronald L. Ferrari (2007). "The Finite-Element Method, Part 2: P. P. Silvester, an Innovator in Electromagnetic Numerical Modeling". doi:10.1109/MAP.2007.4293978.
^ Babuška, Ivo; Banerjee, Uday; Osborn, John E. (June 2004). "Generalized Finite Element Methods: Main Ideas, Results, and Perspective". International Journal of Computational Methods 1 (1): 67–103. doi:10.1142/S0219876204000083.
^ P. Solin, K. Segeth, I. Dolezel: Higher-Order Finite Element Methods, Chapman & Hall/CRC Press, 2003
^ a b c Hastings, J. K., Juds, M. A., Brauer, J. R., Accuracy and Economy of Finite Element Magnetic Analysis, 33rd Annual National Relay Conference, April 1985.
^ McLaren-Mercedes (2006). "Vodafone McLaren-Mercedes: Feature - Stress to impress". Archived from the original on 2006-10-30. Retrieved 2006-10-03.
^ "Methods with high accuracy for finite element probability computing" by Peng Long, Wang Jinliang and Zhu Qiding, in Journal of Computational and Applied Mathematics 59 (1995) 181-189
^ Achintya Haldar and Sankaran mahadan: "Reliability Assessment Using Stochastic Finite Element Analysis", John Wiley & sons.