Mathematically, the finite element method (FEM) is used for finding approximate solution of partial differential equations (PDE) as well as of integral equations such as the heat transport equation. The solution approach is based either on eliminating the differential equation completely (steady state problems), or rendering the PDE into an equivalent ordinary differential equation, which is then solved using standard techniques such as finite differences, etc. Euclid, Greek mathematician, 3rd century BC, as imagined by by Raphael in this detail from The School of Athens. ... In mathematics, a partial differential equation (PDE) is a relation involving an unknown function of several independent variables and its partial derivatives with respect to those variables. ... In mathematics, an integral equation is an equation in which an unknown function appears under an integral sign. ... The heat equation is an important partial differential equation which describes the variation of temperature in a given region over time. ... In mathematics, an ordinary differential equation (or ODE) is a relation that contains functions of only one independent variable, and one or more of its derivatives with respect to that variable. ... A finite difference is a mathematical expression of the form f(x + b) − f(x +a). ...

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 data 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 complex domains (like cars and oil pipelines), when the domain changes (as during a solid state reaction with a moving boundary), or when the desired precision varies over the entire domain. For instance, in simulating the weather pattern on Earth, it is more important to have accurate predictions over land than over the wide-open sea, a demand that is achievable using the finite element method. In mathematics, a partial differential equation (PDE) is a relation involving an unknown function of several independent variables and its partial derivatives with respect to those variables. ... In the mathematical subfield of numerical analysis, numerical stability is a property of numerical algorithms. ...

History

The finite-element method originated from the needs for solving complex elasticity, structural analysis problems in civil engineering 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 dramatically different, they share one essential characteristic: mesh discretization of a continuous domain into a set of discrete sub-domains. Hrennikoff's work discretizes the domain by using a lattice analogy while Courant's approach divides the domain into finite triangular subregions for solution of 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 picked up a lot of steam at Berkeley (see Early Finite Element Research at Berkeley) in the 1960s for use in civil engineering. The method was 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. Elasticity has meanings in two different fields: In physics and mechanical engineering, the theory of elasticity describes how a solid object moves and deforms in response to external stress. ... Structural analysis comprises the set of physical laws and mathematics required to study and predict the behavior of structures. ... The Falkirk Wheel in Scotland. ... Aerospace engineering is the branch of engineering concerning aircraft, spacecraft and related topics. ... Alexander Hrennikoff (1896–December 31, 1984) was a Russian-Canadian Civil Engineer, a founder of the Finite Element Method. ... Richard Courant (born January 8, 1888 at Lublinitz, today Poland, died January 27, 1972 at New York/USA) was a German and American mathematician. ... Mesh (cloth/textiles) Mesh - British musical band Mesh (mathematics) The Tyler mesh size is a scale of particle size in powders. ... Look up torsion in Wiktionary, the free dictionary. ... A right circular cylinder An elliptic cylinder In mathematics, a cylinder is a quadric surface, with the following equation in Cartesian coordinates: This equation is for an elliptic cylinder, a generalization of the ordinary, circular cylinder (a = b). ... John William Strutt, 3rd Baron Rayleigh (12 November 1842 – 30 June 1919) was an English physicist who (with William Ramsay) discovered the element argon, an achievement that earned him the Nobel Prize for Physics in 1904. ... Walter Ritz Walter Ritz (b. ... Boris Grigorievich Galerkin (4 March 1871 – 12 June 1945) was a Russian mathematician who developed the Galerkin Method, an important part of the finite element method. ... // Recovering from World War I and its aftermath, the economic miracle emerged in West Germany and Italy. ... Airframe is a novel by renowned author Michael Crichton first published in hardback edition in 1996 and as a paperback edition in 1997. ... Structural analysis comprises the set of physical laws and mathematics required to study and predict the behavior of structures. ... Sather tower (the Campanile) looking out over the San Francisco Bay and Mount Tamalpais. ... The 1960s decade refers to the years from January 1, 1960 to December 31, 1969, inclusive. ... The Falkirk Wheel in Scotland. ... 1973 (MCMLXXIII) was a common year starting on Monday. ... Gilbert Strang is an American mathematician who published (with George Fix) An Analysis of The Finite Element Method in 1973. ... Engineering is the design, analysis, and/or construction of works for practical purposes. ... Electromagnetism is the physics of the electromagnetic field; a field encompassing all of space which exerts a force on particles that possess the property of electric charge, and is in turn affected by the presence and motion of those particles. ... Fluid dynamics is the sub-discipline of fluid mechanics dealing with fluids (liquids and gases) in motion. ...

The development of the finite element method in structural mechanics is often based on an energy principle, e.g., the virtual work principle or the minimum total potential energy principle, which provides a general, intuitive and physical basis that has a great appeal to structural engineers. Finite element method (FEM) is a powerful technique originally developed for numerical solution of complex problems in structural mechanics, and it remains the method of choice for complex systems. ... A force F, which may be real (actual) or imaginary (fictitious), acting on a particle is said to do virtual work when the particle is imagined to undergo a real or imaginary displacement component D in the direction of the force. ... As one of the energy principles in structural mechanics, the minimum total potential energy principle asserts that a structure or body shall deform or displace to a position that minimizes the total potential energy. ...

Technical discussion

We will illustrate the finite element method using two sample problems from which the general method can be extrapolated. We assume that the reader is familiar with calculus and linear algebra. We will use the one-dimensional Calculus (from Latin, counting stone) is a major area in mathematics. ... Linear algebra is the branch of mathematics concerned with the study of vectors, vector spaces (also called linear spaces), linear maps (also called linear transformations), and systems of linear equations. ...

where f is given and 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 In mathematics, Dirichlet problems are a class of partial differential equation (PDE) problems which ask you to solve for the values of a function in a region given the value of the function on the boundary of that region. ...

where Ω is a connected open region in the (x,y) plane whose boundary 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. In mathematics, a manifold M is a type of space, characterized in one of two equivalent ways: near every point of the space, we have a coordinate system; or near every point, the environment is like that in Euclidean space of a given dimension. ... Look up polygon in Wiktionary, the free dictionary. ...

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. In calculus, an antiderivative, primitive or indefinite integral of a function f is a function F whose derivative is equal to f, i. ... Shows a region where a differential equation is valid and the associated boundary values In mathematics, in the field of differential equations, a boundary value problem is a differential equation together with a set of additional restraints, called the boundary conditions. ...

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, or variational 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. Calculus of variations is a field of mathematics which deals with functions of functions, as opposed to ordinary calculus which deals with functions of numbers. ... A BlueGene supercomputer cabinet. ...

Variational formulation

The first step is to convert P1 and P2 into their variational equivalents. If u solves P1, then for any smooth function v we have Calculus of variations is a field of mathematics which deals with functions of functions, as opposed to ordinary calculus which deals with functions of numbers. ...

(1)

Conversely, if for a given u, (1) holds for every smooth function v(x) then one may show that this u will solve P1. (The proof is nontrivial and uses Sobolev spaces.) In mathematics, a Sobolev space is a normed space of functions obtained by imposing on a function f and its weak derivatives up to some order k the condition of finite Lp norm, for given p ≥ 1. ...

By using integration by parts on the right-hand-side of (1), we obtain In calculus, and more generally in mathematical analysis, integration by parts is a rule that transforms the integral of products of functions into other, possibly simpler, integrals. ...

(2)

where we have made the additional assumption that v(0) = v(1) = 0.

A proof outline of existence and uniqueness of the solution

We can define to be the functions of (0,1) of bounded variation that are 0 at x = 0 and x = 1. Such function are "once differentiable" and it turns out that the symmetric bilinear map φ then defines an inner product which turns into a Hilbert space (a detailed proof is nontrivial.) On the other hand, the left-hand-side is also an inner product, this time on the Lp space L²(0,1). An application of the Riesz representation theorem for Hilbert spaces shows that there is a unique u solving (2) and therefore P1. In mathematics, given f, a real-valued function on the interval [a, b] on the real line, the total variation of f on that interval is the supremum running over all partitions P = { x1, ..., xn } of the interval [a, b]. In effect, the total variation is the vertical component of... In mathematics, a bilinear operator is a generalized multiplication which satisfies the distributive law. ... In mathematics, an inner product space is a vector space with additional structure, an inner product (also called a scalar product), which allows us to introduce geometrical notions such as angles and lengths of vectors. ... The mathematical concept of a Hilbert space (named after the German mathematician David Hilbert) generalizes the notion of Euclidean space in a way that extends methods of vector algebra from the plane and three-dimensional space to spaces of functions. ... In mathematics, the Lp and spaces are spaces of p-power integrable functions, and corresponding sequence spaces. ... There are several well-known theorems in functional analysis known as the Riesz representation theorem. ...

The variational form of P2

If we integrate by parts using a form of Green's theorem, we see that if u solves P2, then for any v, In physics and mathematics, Greens theorem gives the relationship between a line integral around a simple closed curve C and a double integral over the plane region D bounded by C. Greens theorem was named after British scientist George Green and is a special two-dimensional case of...

where denotes the gradient and denotes the dot product in the two-dimensional plane. Once more φ can be turned into an inner product on a suitable space of "once differentiable" functions of Ω that are zero on . We have also assumed that . The space can no longer be defined in terms of functions of bounded variation, but see Sobolev spaces. Existence and uniqueness of the solution can also be shown.. For other uses, see Gradient (disambiguation). ... In mathematics, the dot product, also known as the scalar product, is a binary operation which takes two vectors over the real numbers R and returns a real-valued scalar quantity. ... In mathematics, a Sobolev space is a normed space of functions obtained by imposing on a function f and its weak derivatives up to some order k the condition of finite Lp norm, for given p ≥ 1. ...

Discretization

A function in H¹₀, with zero values at the endpoints (blue), and a piecewise linear approximation (red).

A piecewise linear function in two dimensions.

Basis functions v_k (blue) and a linear combination of them, which is piecewise linear (red).

The basic idea is to replace the infinite dimensional linear problem: Image File history File links Download high resolution version (1002x651, 42 KB) Licensing I, the creator of this work, hereby release it into the public domain. ... Image File history File links Download high resolution version (1002x651, 42 KB) Licensing I, the creator of this work, hereby release it into the public domain. ... Image File history File links Finite_element_triangulation. ... Image File history File links Finite_element_triangulation. ... Image File history File links Download high resolution version (1002x651, 54 KB) Licensing I, the creator of this work, hereby release it into the public domain. ... Image File history File links Download high resolution version (1002x651, 54 KB) Licensing I, the creator of this work, hereby release it into the public domain. ...

Find such that

with a finite dimensional version:

(3) Find such that

where V is a finite dimensional subspace of . 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 linear functions. The concept of a linear subspace (or vector subspace) is important in linear algebra and related fields of mathematics. ... In applied mathematics, Spectral methods are algorithms to solve certain kinds of partial differential equations numerically using some sort of Fast Fourier Transform. ...

For problem P1, we take the interval (0,1), choose n x values 0 = x₀ < x₁ < ... < x_n < x_{n + 1} = 1 and we define V by

where we define x₀ = 0 and x_{n + 1} = 1. Observe that functions in V are not differentiable according to the elementary definition of calculus. Indeed, if 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. In calculus, and more generally in mathematical analysis, integration by parts is a rule that transforms the integral of products of functions into other, possibly simpler, integrals. ...

For problem P2, we need V to be a set of functions of Ω. In the figure on the right, we have illustrated a triangulation of a 15 sided polygonal region Ω 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

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 , i.e., In linear algebra, a basis is a set of vectors that, in a linear combination, can represent every vector in a given vector space, and such that no element of the set can be represented as a linear combination of the others. ...

for k = 1,...,n. For the two-dimensional case, we choose again one basis function v_k per vertex x_k of the triangulation of the planar region Ω. The function v_k is the unique function of V whose value is 1 at x_k and zero at every .

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, in which case he might describe his 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).

Methods that use higher degree piecewise polynomial basis functions are often called spectral element methods, especially if the degree of the polynomials increases as the triangulation size h goes to zero. In mathematics, the spectral element method is a high order finite element method. ...

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:

• moving nodes (r-adaptivity)
• refining (and unrefining) elements (h-adaptivity)
• changing order of base functions (p-adaptivity)
• combinations of the above (e.g. hp-adaptivity)

Small support of the basis

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

and

will be zero for almost all j,k. In the one dimensional case, the support of v_k is the interval [x_{k − 1},x_{k + 1}]. Hence, the integrands of and φ(v_j,v_k) are identically zero whenever | j − k | > 1. In mathematics, the support of a real-valued function f on a set X is sometimes defined as the subset of X on which f is nonzero. ...

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

and

are both zero.

Matrix form of the problem

If we write and then problem (3) becomes

(4) for j = 1,...,n.

If we denote by and the column vectors (u₁,...,u_n)^t and (f₁,...,f_n)^t, and if let L = (L_ij) and M = (M_ij) be matrices whose entries are L_ij = φ(v_i,v_j) and then we may rephrase (4) as

(5) .

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 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 is based on sparse LU) can be sufficient for meshes with a hundred thousand vertices. In the mathematical subfield of numerical analysis a sparse matrix is a matrix populated primarily with zeros. ... In mathematics, the conjugate gradient method is an algorithm for the numerical solution of particular systems of linear equations, namely those whose matrix is symmetric and positive definite. ... In linear algebra, the LU decomposition is a matrix decomposition which writes a matrix as the product of a lower and upper triangular matrix. ... In mathematics, the Cholesky decomposition, named after André-Louis Cholesky, is a matrix decomposition of a symmetric positive-definite matrix into a lower triangular matrix and the transpose of the lower triangular matrix. ... MATLAB is a numerical computing environment and programming language. ...

The matrix L is usually referred to as the stiffness matrix, while the matrix M is dubbed the mass matrix.

Comparison to the finite difference method

The finite difference method (FDM) is an alternative way for solving PDEs. The differences between FEM and FDM are: In mathematics, more precisely in numerical analysis, finite differences play an important role, they are one of the simplest ways of approximating a differential operator, and are extensively used in solving differential equations. ...

• The finite difference method is an approximation to the differential equation; the finite element method is an approximation to its solution.

• The most attractive feature of the FEM is its ability to handle complex 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. One might choose basis functions as either piecewise constant functions or Dirac delta functions. In both approaches, the approximations are defined on the entire domain, but need not be continuous. Alternatively, one might define the function on a discrete domain, with the result that the continuous differential operator no longer makes sense, however this approach is not FEM.

• 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 (e.g., finite volume method). 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 in a large area. In mathematics, a function f(x) of a real number variable x is defined piecewise, if f(x) is given by different expressions on various intervals. ... In mathematics a constant function is a function whose values do not vary and thus are constant. ... The Dirac delta function, often referred to as the unit impulse function and introduced by the British theoretical physicist Paul Dirac, can usually be informally thought of as a function δ(x) that has the value of infinity for x = 0, the value zero elsewhere. ... A computer simulation of high velocity air flow around the Space Shuttle during re-entry. ... The finite volume method is a method for representing and evaluating partial differential equations as algebraic equations. ...

There are many finite element software packages, some free and some proprietary. This article is about free software as defined by the sociopolitical free software movement; for information on software distributed without charge, see freeware. ... Proprietary software is software with restrictions on using, copying and modifying as enforced by the proprietor. ...

See also

• Galerkin method
• Finite element method in structural mechanics
• Finite element analysis
• Rayleigh-Ritz method
• Discrete element method
• Spectral method
• Meshfree methods
• Boundary element method
• Electromagnetic modeling

The Galerkin method is an approach invented by the Russian mathematician Boris Galerkin. ... Finite element method (FEM) is a powerful technique originally developed for numerical solution of complex problems in structural mechanics, and it remains the method of choice for complex systems. ... Visualization of how a car deforms in an asymmetrical crash using finite element analysis. ... In engineering, the Rayleigh-Ritz method is a widely used, classical method for the calculation of the natural vibration frequency of a structure in the second or higher order. ... The term discrete element method (DEM) is a family of numerical methods for computing the motion of a large number of particles like molecules or grains of sand. ... In applied mathematics, Spectral methods are algorithms to solve certain kinds of partial differential equations numerically using some sort of Fast Fourier Transform. ... Meshfree methods are a particular class of numerical simulation algorithms for the simulation of physical phenomena. ... The boundary element method is a numerical computational method of solving linear partial differential equations which have been formulated as integral equations (i. ... Computational electromagnetics, computational electrodynamics or electromagnetic modeling refers to the process of modeling the interaction of electromagnetic fields with physical objects and the environment. ...

External links

• IFER -- Internet Finite Element Resources - an annotated list of FEA links and programs
• Workshop "The Finite Element Method in Biomedical Engineering, Biomechanics and Related Fields"
• Finite Element Analysis Resources- Finite Element news, articles and tips
• Finite Element Books- books bibliography
• Mathematics of the Finite Element Method
• Finite Elements Lecture Notes - Lecture notes of the Introduction into Finite Elements course at the Delft University of Technology.
• Finite Element Methods for Partial Differential Equations