
The finite element method (FEM) is a popular method for numerically solving
differential equation
In mathematics, a differential equation is an equation that relates one or more unknown functions and their derivatives. In applications, the functions generally represent physical quantities, the derivatives represent their rates of change, a ...
s arising in engineering and
mathematical modeling
A mathematical model is a description of a system using mathematical concepts and language. The process of developing a mathematical model is termed mathematical modeling. Mathematical models are used in the natural sciences (such as physics, ...
. Typical problem areas of interest include the traditional fields of
structural analysis
Structural analysis is a branch of Solid Mechanics which uses simplified models for solids like bars, beams and shells for engineering decision making. Its main objective is to determine the effect of loads on the physical structures and thei ...
,
heat transfer
Heat transfer is a discipline of thermal engineering that concerns the generation, use, conversion, and exchange of thermal energy (heat) between physical systems. Heat transfer is classified into various mechanisms, such as thermal conduction ...
,
fluid flow
In physics and engineering, fluid dynamics is a subdiscipline of fluid mechanics that describes the flow of fluids—liquids and gases. It has several subdisciplines, including '' aerodynamics'' (the study of air and other gases in motion) ...
, mass transport, and
electromagnetic potential.
The FEM is a general
numerical method
In numerical analysis, a numerical method is a mathematical tool designed to solve numerical problems. The implementation of a numerical method with an appropriate convergence check in a programming language is called a numerical algorithm.
Mathem ...
for solving
partial differential equations
In mathematics, a partial differential equation (PDE) is an equation which imposes relations between the various partial derivatives of a multivariable function.
The function is often thought of as an "unknown" to be solved for, similarly to ...
in two or three space variables (i.e., some
boundary value problem
In mathematics, in the field of differential equations, a boundary value problem is a differential equation together with a set of additional constraints, called the boundary conditions. A solution to a boundary value problem is a solution to ...
s). To solve a problem, the FEM subdivides a large system into smaller, simpler parts that are called finite elements. This is achieved by a particular space
discretization
In applied mathematics, discretization is the process of transferring continuous functions, models, variables, and equations into discrete counterparts. This process is usually carried out as a first step toward making them suitable for numeri ...
in the space dimensions, which is implemented by the construction of a
mesh
A mesh is a barrier made of connected strands of metal, fiber, or other flexible or ductile materials. A mesh is similar to a web or a net in that it has many attached or woven strands.
Types
* A plastic mesh may be extruded, oriented, e ...
of the object: the numerical domain for the solution, which has a finite number of points.
The finite element method formulation of a boundary value problem finally results in a system of
algebraic equation
In mathematics, an algebraic equation or polynomial equation is an equation of the form
:P = 0
where ''P'' is a polynomial with coefficients in some field, often the field of the rational numbers. For many authors, the term ''algebraic equation ...
s. The method approximates the unknown function over the domain.
The simple equations that model these finite elements are then assembled into a larger system of equations that models the entire problem. The FEM then approximates a solution by minimizing an associated error function via the
calculus of variations
The calculus of variations (or Variational Calculus) is a field of mathematical analysis that uses variations, which are small changes in functions
and functionals, to find maxima and minima of functionals: mappings from a set of functions t ...
.
Studying or
analyzing a phenomenon with FEM is often referred to as finite element analysis (FEA).
Basic concepts
The subdivision of a whole domain into simpler parts has several advantages:
* Accurate representation of complex geometry
* Inclusion of dissimilar material properties
* Easy representation of the total solution
* Capture of local effects.
Typical work out of the method involves:
# dividing the domain of the problem into a collection of subdomains, with each subdomain represented by a set of element equations to the original problem
# systematically recombining all sets of element equations into a global system of equations for the final calculation.
The global system of equations has known solution techniques, and can be calculated from the
initial value
In multivariable calculus, an initial value problem (IVP) is an ordinary differential equation together with an initial condition which specifies the value of the unknown function at a given point in the domain. Modeling a system in physics or o ...
s of the original problem to obtain a numerical answer.
In the first step above, the element equations are simple equations that locally approximate the original complex equations to be studied, where the original equations are often
partial differential equation
In mathematics, a partial differential equation (PDE) is an equation which imposes relations between the various partial derivatives of a multivariable function.
The function is often thought of as an "unknown" to be solved for, similarly to ...
s (PDE). To explain the approximation in this process, the finite element method is commonly introduced as a special case of
Galerkin method
In mathematics, in the area of numerical analysis, Galerkin methods, named after the Russian mathematician Boris Galerkin, convert a continuous operator problem, such as a differential equation, commonly in a weak formulation, to a discrete prob ...
. The process, in mathematical language, is to construct an integral of the
inner product
In mathematics, an inner product space (or, rarely, a Hausdorff pre-Hilbert space) is a real vector space or a complex vector space with an operation called an inner product. The inner product of two vectors in the space is a scalar, often ...
of the residual and the
weight function
A weight function is a mathematical device used when performing a sum, integral, or average to give some elements more "weight" or influence on the result than other elements in the same set. The result of this application of a weight function is ...
s and set the integral to zero. In simple terms, it is a procedure that minimizes the error of approximation by fitting trial functions into the PDE. The residual is the error caused by the trial functions, and the weight functions are
polynomial
In mathematics, a polynomial is an expression consisting of indeterminates (also called variables) and coefficients, that involves only the operations of addition, subtraction, multiplication, and positive-integer powers of variables. An ex ...
approximation functions that project the residual. The process eliminates all the spatial derivatives from the PDE, thus approximating the PDE locally with
* a set of
algebraic equations
In mathematics, an algebraic equation or polynomial equation is an equation of the form
:P = 0
where ''P'' is a polynomial with coefficients in some field, often the field of the rational numbers. For many authors, the term ''algebraic equation'' ...
for
steady state
In systems theory, a system or a process is in a steady state if the variables (called state variables) which define the behavior of the system or the process are unchanging in time. In continuous time, this means that for those properties ' ...
problems,
* a set of
ordinary differential equation
In mathematics, an ordinary differential equation (ODE) is a differential equation whose unknown(s) consists of one (or more) function(s) of one variable and involves the derivatives of those functions. The term ''ordinary'' is used in contras ...
s for
transient problems.
These equation sets are the element equations. They are
linear
Linearity is the property of a mathematical relationship ('' function'') that can be graphically represented as a straight line. Linearity is closely related to '' proportionality''. Examples in physics include rectilinear motion, the linear ...
if the underlying PDE is linear, and vice versa. Algebraic equation sets that arise in the steady-state problems are solved using
numerical linear algebra
Numerical linear algebra, sometimes called applied linear algebra, is the study of how matrix operations can be used to create computer algorithms which efficiently and accurately provide approximate answers to questions in continuous mathematics ...
methods, while
ordinary differential equation
In mathematics, an ordinary differential equation (ODE) is a differential equation whose unknown(s) consists of one (or more) function(s) of one variable and involves the derivatives of those functions. The term ''ordinary'' is used in contras ...
sets that arise in the transient problems are solved by numerical integration using standard techniques such as
Euler's method
In mathematics and computational science, the Euler method (also called forward Euler method) is a first-order numerical procedure for solving ordinary differential equations (ODEs) with a given initial value. It is the most basic explicit m ...
or the
Runge-Kutta method.
In step (2) above, a global system of equations is generated from the element equations through a transformation of coordinates from the subdomains' local nodes to the domain's global nodes. This spatial transformation includes appropriate
orientation adjustments as applied in relation to the reference
coordinate system. The process is often carried out by FEM software using
coordinate
In geometry, a coordinate system is a system that uses one or more numbers, or coordinates, to uniquely determine the position of the points or other geometric elements on a manifold such as Euclidean space. The order of the coordinates is si ...
data generated from the subdomains.
The practical application of FEM is known as ''finite element analysis'' (FEA). FEA as applied in
engineering
Engineering is the use of scientific method, scientific principles to design and build machines, structures, and other items, including bridges, tunnels, roads, vehicles, and buildings. The discipline of engineering encompasses a broad rang ...
is a computational tool for performing
engineering analysis Engineering analysis involves the application of scientific/mathematical analytic principles and processes to reveal the properties and state of a system, device or mechanism under study.
Engineering analysis is decompositional: it proceeds by se ...
. It includes the use of
mesh generation
Mesh generation is the practice of creating a polygon mesh, mesh, a subdivision of a continuous geometric space into discrete geometric and topological cells.
Often these cells form a simplicial complex.
Usually the cells partition the geometric ...
techniques for dividing a
complex problem
Problem solving is the process of achieving a goal by overcoming obstacles, a frequent part of most activities. Problems in need of solutions range from simple personal tasks (e.g. how to turn on an appliance) to complex issues in business an ...
into small elements, as well as the use of software coded with a FEM algorithm. In applying FEA, the complex problem is usually a physical system with the underlying
physics
Physics is the natural science that studies matter, its fundamental constituents, its motion and behavior through space and time, and the related entities of energy and force. "Physical science is that department of knowledge which rel ...
such as the
Euler–Bernoulli beam equation, the
heat equation, or the
Navier-Stokes equations expressed in either PDE or
integral equation
In mathematics, integral equations are equations in which an unknown function appears under an integral sign. In mathematical notation, integral equations may thus be expressed as being of the form: f(x_1,x_2,x_3,...,x_n ; u(x_1,x_2,x_3,...,x_n ...
s, while the divided small elements of the complex problem represent different areas in the physical system.
FEA may be used for analyzing problems 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. FEA simulations provide a valuable resource as they remove multiple instances of creation and testing of hard prototypes for various high fidelity situations. 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 the cost of the simulation). Another example would be in
numerical weather prediction
Numerical weather prediction (NWP) uses mathematical models of the atmosphere and oceans to predict the weather based on current weather conditions. Though first attempted in the 1920s, it was not until the advent of computer simulation in th ...
, where it is more important to have accurate predictions over developing highly nonlinear phenomena (such as
tropical cyclone
A tropical cyclone is a rapidly rotating storm system characterized by a low-pressure center, a closed low-level atmospheric circulation, strong winds, and a spiral arrangement of thunderstorms that produce heavy rain and squalls. Dep ...
s in the atmosphere, or
eddies
In fluid dynamics, an eddy is the swirling of a fluid and the reverse current created when the fluid is in a turbulent flow regime. The moving fluid creates a space devoid of downstream-flowing fluid on the downstream side of the object. Fluid b ...
in the ocean) rather than relatively calm areas.
A clear, detailed and practical presentation of this approach can be found in ''The Finite Element Method for Engineers''.
History
While it is difficult to quote a date of the invention of the finite element method, the method originated from the need to solve complex
elasticity and
structural analysis
Structural analysis is a branch of Solid Mechanics which uses simplified models for solids like bars, beams and shells for engineering decision making. Its main objective is to determine the effect of loads on the physical structures and thei ...
problems in
civil
Civil may refer to:
*Civic virtue, or civility
*Civil action, or lawsuit
* Civil affairs
*Civil and political rights
*Civil disobedience
*Civil engineering
*Civil (journalism), a platform for independent journalism
*Civilian, someone not a membe ...
and
aeronautical engineering
Aerospace engineering is the primary field of engineering concerned with the development of aircraft and spacecraft. It has two major and overlapping branches: aeronautical engineering and astronautical engineering. Avionics engineering is sim ...
. Its development can be traced back to the work by
A. Hrennikoff and
R. Courant in the early 1940s. Another pioneer was
Ioannis Argyris. In the USSR, the introduction of the practical application of the method is usually connected with name of
Leonard Oganesyan. It was also independently rediscovered in China by
Feng Kang in the later 1950s and early 1960s, based on the computations of dam constructions, where it was called the ''finite difference method based on variation principle''. Although the approaches used by these pioneers are different, they share one essential characteristic:
mesh
A mesh is a barrier made of connected strands of metal, fiber, or other flexible or ductile materials. A mesh is similar to a web or a net in that it has many attached or woven strands.
Types
* A plastic mesh may be extruded, oriented, e ...
discretization
In applied mathematics, discretization is the process of transferring continuous functions, models, variables, and equations into discrete counterparts. This process is usually carried out as a first step toward making them suitable for numeri ...
of a continuous domain into a set of discrete sub-domains, usually called elements.
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 that arise from the problem of
torsion of a
cylinder
A cylinder (from ) has traditionally been a three-dimensional solid, one of the most basic of curvilinear geometric shapes. In elementary geometry, it is considered a prism with a circle as its base.
A cylinder may also be defined as an infi ...
. Courant's contribution was evolutionary, drawing on a large body of earlier results for PDEs developed by
Rayleigh Rayleigh may refer to:
Science
*Rayleigh scattering
*Rayleigh–Jeans law
*Rayleigh waves
*Rayleigh (unit), a unit of photon flux named after the 4th Baron Rayleigh
*Rayl, rayl or Rayleigh, two units of specific acoustic impedance and characte ...
,
Ritz
Ritz or The Ritz may refer to:
Facilities and structures Hotels
* The Ritz Hotel, London, a hotel in London, England
* Hôtel Ritz Paris, a hotel in Paris, France
* Hotel Ritz (Madrid), a hotel in Madrid, Spain
* Hotel Ritz (Lisbon), a hotel in ...
, and
Galerkin.
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
The University of Stuttgart (german: Universität Stuttgart) is a leading research university located in Stuttgart, Germany. It was founded in 1829 and is organized into 10 faculties. It is one of the oldest technical universities in Germany wi ...
,
R. W. Clough with co-workers at
UC Berkeley
The University of California, Berkeley (UC Berkeley, Berkeley, Cal, or California) is a public university, public land-grant university, land-grant research university in Berkeley, California. Established in 1868 as the University of Californi ...
,
O. C. Zienkiewicz with co-workers
Ernest Hinton,
Bruce Irons and others at
Swansea University
, former_names=University College of Swansea, University of Wales Swansea
, motto= cy, Gweddw crefft heb ei dawn
, mottoeng="Technical skill is bereft without culture"
, established=1920 – University College of Swansea 1996 – University of Wa ...
,
Philippe G. Ciarlet
Philippe G. Ciarlet (born 14 October 1938) is a French mathematician, known particularly for his work on mathematical analysis of the finite element method. He has contributed also to elasticity, to the theory of plates and shells and differentia ...
at the University of
Paris 6 and Richard Gallagher with co-workers at
Cornell University
Cornell University is a private statutory land-grant research university based in Ithaca, New York. It is a member of the Ivy League. Founded in 1865 by Ezra Cornell and Andrew Dickson White, Cornell was founded with the intention to ...
. Further impetus was provided in these years by available open source finite element programs. NASA sponsored the original version of
NASTRAN, and UC Berkeley made the finite element program SAP IV widely available. In Norway the ship classification society Det Norske Veritas (now
DNV GL
DNV (formerly DNV GL) is an international accredited registrar and classification society headquartered in Høvik, Norway. The company currently has about 12,000 employees and 350 offices operating in more than 100 countries, and provides ser ...
) developed
Sesam in 1969 for use in analysis of ships. A rigorous mathematical basis to the finite element method was provided in 1973 with the publication by
Strang and
Fix
Fix or FIX may refer to:
People with the name
* Fix (surname)
Arts, entertainment, and media Films
* ''Fix'' (film), a feature film by Tao Ruspoli Music
* ''Fix'' (album), 2015 album by Chris Lane
* "Fix" (Blackstreet song), 1997 song by Black ...
. The method has since been generalized for the
numerical modeling of physical systems in a wide variety of
engineering
Engineering is the use of scientific method, scientific principles to design and build machines, structures, and other items, including bridges, tunnels, roads, vehicles, and buildings. The discipline of engineering encompasses a broad rang ...
disciplines, e.g.,
electromagnetism
In physics, electromagnetism is an interaction that occurs between particles with electric charge. It is the second-strongest of the four fundamental interactions, after the strong force, and it is the dominant force in the interactions o ...
,
heat transfer
Heat transfer is a discipline of thermal engineering that concerns the generation, use, conversion, and exchange of thermal energy (heat) between physical systems. Heat transfer is classified into various mechanisms, such as thermal conduction ...
, and
fluid dynamics
In physics and engineering, fluid dynamics is a subdiscipline of fluid mechanics that describes the flow of fluids—liquids and gases. It has several subdisciplines, including '' aerodynamics'' (the study of air and other gases in motion) ...
.
Technical discussion
The structure of finite element methods
A finite element method is characterized by a
variational formulation, a discretization strategy, one or more solution algorithms, and post-processing procedures.
Examples of the variational formulation are the
Galerkin method
In mathematics, in the area of numerical analysis, Galerkin methods, named after the Russian mathematician Boris Galerkin, convert a continuous operator problem, such as a differential equation, commonly in a weak formulation, to a discrete prob ...
, 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. Examples of discretization strategies are the h-version,
p-version,
hp-version,
x-FEM,
isogeometric analysis, etc. Each discretization strategy has certain advantages and disadvantages. A reasonable criterion in selecting a discretization strategy is to realize nearly optimal performance for the broadest set of mathematical models in a particular model class.
Various numerical solution algorithms can be classified into two broad categories; direct and iterative solvers. These algorithms are designed to exploit the sparsity of matrices that depend on the choices of variational formulation and discretization strategy.
Postprocessing procedures are designed for the extraction of the data of interest from a finite element solution. In order to meet the requirements of solution verification, postprocessors need to provide for ''a posteriori'' error estimation in terms of the quantities of interest. When the errors of approximation are larger than what is considered acceptable then the discretization has to be changed either by an automated adaptive process or by the action of the analyst. There are some very efficient postprocessors that provide for the realization of
superconvergence.
Illustrative problems P1 and P2
The following two problems demonstrate the finite element method.
P1 is a one-dimensional problem
:
where
is given,
is an unknown function of
, and
is the second derivative of
with respect to
.
P2 is a two-dimensional problem (
Dirichlet problem
In mathematics, a Dirichlet problem is the problem of finding a function which solves a specified partial differential equation (PDE) in the interior of a given region that takes prescribed values on the boundary of the region.
The Dirichlet pro ...
)
:
where
is a connected open region in the
plane whose boundary
is nice (e.g., a
smooth manifold
In mathematics, a differentiable manifold (also differential manifold) is a type of manifold that is locally similar enough to a vector space to allow one to apply calculus. Any manifold can be described by a collection of charts (atlas). One m ...
or a
polygon
In geometry, a polygon () is a plane figure that is described by a finite number of straight line segments connected to form a closed '' polygonal chain'' (or ''polygonal circuit''). The bounded plane region, the bounding circuit, or the two t ...
), and
and
denote the second derivatives with respect to
and
, respectively.
The problem P1 can be solved directly by computing
antiderivative
In calculus, an antiderivative, inverse derivative, primitive function, primitive integral or indefinite integral of a function is a differentiable function whose derivative is equal to the original function . This can be stated symbolically ...
s. However, this method of solving the
boundary value problem
In mathematics, in the field of differential equations, a boundary value problem is a differential equation together with a set of additional constraints, called the boundary conditions. A solution to a boundary value problem is a solution to ...
(BVP) works only when there is one spatial dimension and does not generalize to higher-dimensional problems or problems like
. 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 formulation Weak formulations are important tools for the analysis of mathematical equations that permit the transfer of concepts of linear algebra to solve problems in other fields such as partial differential equations. In a weak formulation, equations or ...
s.
The weak form of P1
If
solves P1, then for any smooth function
that satisfies the displacement boundary conditions, i.e.
at
and
, we have
Conversely, if
with
satisfies (1) for every smooth function
then one may show that this
will solve P1. The proof is easier for twice continuously differentiable
(
mean value theorem
In mathematics, the mean value theorem (or Lagrange theorem) states, roughly, that for a given planar arc between two endpoints, there is at least one point at which the tangent to the arc is parallel to the secant through its endpoints. It ...
), but may be proved in a
distributional sense as well.
We define a new operator or map
by using
integration by parts
In calculus, and more generally in mathematical analysis, integration by parts or partial integration is a process that finds the integral of a product of functions in terms of the integral of the product of their derivative and antiderivative. ...
on the right-hand-side of (1):
where we have used the assumption that
.
The weak form of P2
If we integrate by parts using a form of
Green's identities
In mathematics, Green's identities are a set of three identities in vector calculus relating the bulk with the boundary of a region on which differential operators act. They are named after the mathematician George Green, who discovered Green's ...
, we see that if
solves P2, then we may define
for any
by
:
where
denotes the
gradient
In vector calculus, the gradient of a scalar-valued differentiable function of several variables is the vector field (or vector-valued function) \nabla f whose value at a point p is the "direction and rate of fastest increase". If the gr ...
and
denotes the
dot product
In mathematics, the dot product or scalar productThe term ''scalar product'' means literally "product with a scalar as a result". It is also used sometimes for other symmetric bilinear forms, for example in a pseudo-Euclidean space. is an alg ...
in the two-dimensional plane. Once more
can be turned into an inner product on a suitable space
of once differentiable functions of
that are zero on
. We have also assumed that
(see
Sobolev space
In mathematics, a Sobolev space is a vector space of functions equipped with a norm that is a combination of ''Lp''-norms of the function together with its derivatives up to a given order. The derivatives are understood in a suitable weak sense ...
s). Existence and uniqueness of the solution can also be shown.
A proof outline of existence and uniqueness of the solution
We can loosely think of
to be the
absolutely continuous
In calculus, absolute continuity is a smoothness property of functions that is stronger than continuity and uniform continuity. The notion of absolute continuity allows one to obtain generalizations of the relationship between the two central ope ...
functions of
that are
at
and
(see
Sobolev spaces
In mathematics, a Sobolev space is a vector space of functions equipped with a norm that is a combination of ''Lp''-norms of the function together with its derivatives up to a given order. The derivatives are understood in a suitable weak sense ...
). Such functions are (weakly) once differentiable and it turns out that the symmetric
bilinear map
In mathematics, a bilinear map is a function combining elements of two vector spaces to yield an element of a third vector space, and is linear in each of its arguments. Matrix multiplication is an example.
Definition
Vector spaces
Let V, ...
then defines an
inner product
In mathematics, an inner product space (or, rarely, a Hausdorff pre-Hilbert space) is a real vector space or a complex vector space with an operation called an inner product. The inner product of two vectors in the space is a scalar, often ...
which turns
into a
Hilbert space
In mathematics, Hilbert spaces (named after David Hilbert) allow generalizing the methods of linear algebra and calculus from (finite-dimensional) Euclidean vector spaces to spaces that may be infinite-dimensional. Hilbert spaces arise natu ...
(a detailed proof is nontrivial). On the other hand, the left-hand-side
is also an inner product, this time on the
Lp space
In mathematics, the spaces are function spaces defined using a natural generalization of the -norm for finite-dimensional vector spaces. They are sometimes called Lebesgue spaces, named after Henri Lebesgue , although according to the Bourb ...
. An application of the
Riesz representation theorem
:''This article describes a theorem concerning the dual of a Hilbert space. For the theorems relating linear functionals to measures, see Riesz–Markov–Kakutani representation theorem.''
The Riesz representation theorem, sometimes called the ...
for Hilbert spaces shows that there is a unique
solving (2) and therefore P1. This solution is a-priori only a member of
, but using elliptic operator, elliptic regularity, will be smooth if
is.
Discretization

P1 and P2 are ready to be discretized which leads to a common sub-problem (3). The basic idea is to replace the infinite-dimensional linear problem:
:Find
such that
:
with a finite-dimensional version:
where
is a finite-dimensional Linear subspace, subspace of
. There are many possible choices for
(one possibility leads to the spectral method). However, for the finite element method we take
to be a space of piecewise polynomial functions.
For problem P1
We take the interval
, choose
values of
with
and we define
by:
:
where we define
and
. Observe that functions in
are not differentiable according to the elementary definition of calculus. Indeed, if
then the derivative is typically not defined at any
,
. However, the derivative exists at every other value of
and one can use this derivative for the purpose of
integration by parts
In calculus, and more generally in mathematical analysis, integration by parts or partial integration is a process that finds the integral of a product of functions in terms of the integral of the product of their derivative and antiderivative. ...
.
For problem P2
We need
to be a set of functions of
. In the figure on the right, we have illustrated a Polygon triangulation, triangulation of a 15 sided
polygon
In geometry, a polygon () is a plane figure that is described by a finite number of straight line segments connected to form a closed '' polygonal chain'' (or ''polygonal circuit''). The bounded plane region, the bounding circuit, or the two t ...
al region
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
would consist of functions that are linear on each triangle of the chosen triangulation.
One hopes that as the underlying triangular mesh 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. To measure this mesh fineness, the triangulation is indexed by a real-valued parameter
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
must also change with
. For this reason, one often reads
instead of
in the literature. 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 (linear algebra), basis of
. In the one-dimensional case, for each control point
we will choose the piecewise linear function
in
whose value is
at
and zero at every
, i.e.,
:
for
; this basis is a shifted and scaled tent function. For the two-dimensional case, we choose again one basis function
per vertex
of the triangulation of the planar region
. The function
is the unique function of
whose value is
at
and zero at every
.
Depending on the author, the word "element" in the "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". The 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 element method, 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 an 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 unrefined) elements (h-adaptivity)
* changing order of base functions (p-adaptivity)
* combinations of the above (hp-FEM, 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
.
(The matrix containing
in the
location is known as the Gramian matrix.)
In the one dimensional case, the support (mathematics), support of
is the interval
. Hence, the integrands of
and ''
'' are identically zero whenever
.
Similarly, in the planar case, if
and
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), taking
for
, becomes
If we denote by
and
the column vectors
and
, and if we let
:
and
:
be matrices whose entries are
:
and
:
then we may rephrase (4) as
It is not necessary to assume
. For a general function
, problem (3) with
for
becomes actually simpler, since no matrix
is used,
where
and
for
.
As we have discussed before, most of the entries of
and
are zero because the basis functions
have small support. So we now have to solve a linear system in the unknown
where most of the entries of the matrix
, which we need to invert, are zero.
Such matrices are known as sparse matrix, sparse matrices, and there are efficient solvers for such problems (much more efficient than actually inverting the matrix.) In addition,
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
is usually referred to as the stiffness matrix, while the matrix
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
. 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.
Separate consideration is the smoothness of the basis functions. For second-order elliptic boundary value problems, piecewise polynomial basis function that is 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
, one may use piecewise quadratic basis functions that are Smooth function#Order of continuity,
.
Another consideration is the relation of the finite-dimensional space
to its infinite-dimensional counterpart, in the examples above
. A conforming element method is one in which space
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
.
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
is bounded above by
, for some
and
, then one has an order ''p'' method. Under certain hypotheses (for instance, if the domain is convex), a piecewise polynomial of order
method will have an error of order
.
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 (spectral element method, SFEM). These are not to be confused with spectral methods.
For vector partial differential equations, the basis functions may take values in
.
Various types of finite element methods
AEM
The Applied Element Method or AEM combines features of both FEM and Discrete element method, or (DEM).
A-FEM
The Augmented-Finite Element Method is introduced by Yang and Lui whose goal was to model the weak and strong discontinuities without the need of extra DoFs as in PuM stated.
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.
Mixed finite element method
The mixed finite element method is a type of finite element method in which extra independent variables are introduced as nodal variables during the discretization of a partial differential equation problem.
Variable – polynomial
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)'' to achieve best convergence rates.
XFEM
The extended finite element method (XFEM) is a numerical technique based on the generalized finite element method (GFEM) and the partition of unity method (PUM). It extends the classical finite element method by enriching the solution space for solutions to differential equations with discontinuous functions. Extended finite element methods enrich the approximation space so that it can naturally reproduce the challenging feature associated with the problem of interest: the discontinuity, singularity, boundary layer, etc. It was shown that for some problems, such an embedding of the problem's feature into the approximation space can significantly improve convergence rates and accuracy. Moreover, treating problems with discontinuities with XFEMs suppresses the need to mesh and re-mesh the discontinuity surfaces, thus alleviating the computational costs and projection errors associated with conventional finite element methods, at the cost of restricting the discontinuities to mesh edges.
Several research codes implement this technique to various degrees:
1. GetFEM++
2. xfem++
3. openxfem++
XFEM has also been implemented in codes like Altair Radios, ASTER, Morfeo, and Abaqus. It is increasingly being adopted by other commercial finite element software, with a few plugins and actual core implementations available (ANSYS, SAMCEF, OOFELIE, etc.).
Scaled boundary finite element method (SBFEM)
The introduction of the scaled boundary finite element method (SBFEM) came from Song and Wolf (1997). The SBFEM has been one of the most profitable contributions in the area of numerical analysis of fracture mechanics problems. It is a semi-analytical fundamental-solutionless method which combines the advantages of both the finite element formulations and procedures and the boundary element discretization. However, unlike the boundary element method, no fundamental differential solution is required.
S-FEM
The S-FEM, Smoothed Finite Element Methods, is a particular class of numerical simulation algorithms for the simulation of physical phenomena. It was developed by combining meshfree methods with the finite element method.
Spectral element method
Spectral element methods combine the geometric flexibility of finite elements and the acute accuracy of spectral methods. Spectral methods are the approximate solution of weak form partial equations that are based on high-order Lagrangian interpolants and used only with certain quadrature rules.
Meshfree methods
Discontinuous Galerkin methods
Finite element limit analysis
Stretched grid method
Loubignac iteration
Loubignac iteration is an iterative method in finite element methods.
Crystal plasticity finite element method (CPFEM)
Crystal plasticity finite element method (CPFEM) is an advanced numerical tool developed by Franz Roters. Metals can be regarded as crystal aggregates and it behave anisotropy under deformation, for example, abnormal stress and strain localization. CPFEM based on slip (shear strain rate) can calculate dislocation, crystal orientation and other texture information to consider crystal anisotropy during the routine. Now it has been applied in the numerical study of material deformation, surface roughness, fractures and so on.
Virtual element method (VEM)
The virtual element method (VEM), introduced by Beirão da Veiga et al. (2013) as an extension of mimesis (mathematics), mimetic finite difference method, finite difference (MFD) methods, is a generalisation of the standard finite element method for arbitrary element geometries. This allows admission of general polygons (or polyhedra in 3D) that are highly irregular and non-convex in shape. The name ''virtual'' derives from the fact that knowledge of the local shape function basis is not required, and is in fact never explicitly calculated.
Link with the gradient discretization method
Some types of finite element methods (conforming, nonconforming, mixed finite element methods) are particular cases of the gradient discretization method (GDM). Hence the convergence properties of the GDM, which are established for a series of problems (linear and non-linear elliptic problems, linear, nonlinear, and degenerate parabolic problems), hold as well for these particular finite element methods.
Comparison to the finite difference method
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.
* FDM is not usually used for irregular CAD geometries but more often rectangular or block shaped models.
* FEM generally allows for more flexible mesh adaptivity than FDM.[
* The most attractive feature of finite differences is that it is 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 Discretization, 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) tend 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 the cost of the solution favors simpler, lower-order approximation within each cell. This is especially true for 'external flow' problems, like airflow around the car or airplane, or weather simulation.
]
Application
A variety of specializations under the umbrella of the mechanical engineering discipline (such as aeronautical, biomechanical, and automotive industries) commonly use integrated FEM in the 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. The mesh is an integral part of the model and it must be controlled carefully to give the best results. Generally the higher the number of elements in a mesh, the more accurate the solution of the discretized problem. However, there is a value at which the results converge and further mesh refinement does not increase accuracy.
This powerful design tool has significantly improved both the standard of engineering designs and the methodology of the design process in many industrial applications.[Hastings, J. K., Juds, M. A., Brauer, J. R., ''Accuracy and Economy of Finite Element Magnetic Analysis'', 33rd Annual National Relay Conference, April 1985.] 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.[
In the 1990s FEM was proposed for use in stochastic modelling for numerically solving probability models and later for reliability assessment.]
See also
*Applied element method
*Boundary element method
*Céa's lemma
*Computer experiment
*Direct stiffness method
*Discontinuity layout optimization
*Discrete element method
*Finite difference method
*Finite element machine
*Finite element method in structural mechanics
*Finite volume method
*Finite volume method for unsteady flow
*Infinite element method
*Interval finite element
*Isogeometric analysis
*Lattice Boltzmann methods
*List of finite element software packages
*Meshfree methods
*Movable cellular automaton
*Multidisciplinary design optimization
*Multiphysics
*Patch test (finite elements), Patch test
*Rayleigh–Ritz method
*Space mapping
*STRAND7
*Tessellation (computer graphics)
*Weakened weak form
References
Further reading
*G. Allaire and A. Craig:
Numerical Analysis and Optimization: An Introduction to Mathematical Modelling and Numerical Simulation
'.
*K. J. Bathe: ''Numerical methods in finite element analysis'', Prentice-Hall (1976).
*Thomas J.R. Hughes: ''The Finite Element Method: Linear Static and Dynamic Finite Element Analysis,'' Prentice-Hall (1987).
*J. Chaskalovic: ''Finite Elements Methods for Engineering Sciences'', Springer Verlag, (2008).
*Endre Süli
''Finite Element Methods for Partial Differential Equations''
*O. C. Zienkiewicz, R. L. Taylor, J. Z. Zhu :
The Finite Element Method: Its Basis and Fundamentals
', Butterworth-Heinemann (2005).
*N. Ottosen, H. Petersson : ''Introduction to the Finite Element Method, '' Prentice-Hall (1992).
*Zohdi, T. I. (2018) A finite element primer for beginners-extended version including sample tests and projects. Second Edition https://link.springer.com/book/10.1007/978-3-319-70428-9
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