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 1. Finite Element Software Free
 2. Finite Element Analysis Examples
 3. Finite Element Software
 4. Abaqus Finite Element Software
 5. Finite Element Software Engineer

Differential equations
Navier–Stokes differential equations used to simulate airflow around an
obstruction.
 * 
   Biology

Applied mathematicsSocial sciences

Classification

By variable type
 * Coupled / Decoupled
 * Homogeneous / Nonhomogeneous

Features

Solution
 * Lyapunov / Asymptotic / Exponential stability
 * Series / Integral solutions

 * Inspection
 * 
   Euler
 * Finite difference(Crank–Nicolson)
 * Finite element
 * Galerkin

The finite element method (FEM) is a numerical method for solving problems of
engineering and mathematical physics. Typical problem areas of interest include
structural analysis, heat transfer, fluid flow, mass transport, and
electromagnetic potential. The analytical solution of these problems generally
require the solution to boundary value problems for partial differential
equations. The finite element method formulation of the problem results in a
system of algebraic equations. The method approximates the unknown function over
the domain.[1] To solve the problem, it subdivides a large system into smaller,
simpler parts that are called finite elements. The simple equations that model
these finite elements are then assembled into a larger system of equations that
models the entire problem. FEM then uses variational methods from the calculus
of variations to approximate a solution by minimizing an associated error
function.
Studying or analyzing a phenomenon with FEM is often referred to as finite
element analysis (FEA).
 * 3Technical discussion
   * 3.3Weak formulation
 * 4Discretization
 * 5Various types of finite element methods

Every time you deliver a product to a customer, you are promising them that it
will work as advertised and make their life easier in some way. Using ANSYS
engineering simulation software to design your products ensures that you can
keep that promise, with every product and every order for every customer.


BASIC CONCEPTS[EDIT]

The subdivision of a whole domain into simpler parts has several advantages:[2]
 * Accurate representation of complex geometry
 * Inclusion of dissimilar material properties
 * Easy representation of the total solution
 * Capture of local effects.

A typical work out of the method involves (1) 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, followed by (2) 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 values 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 equations (PDE). To explain the
approximation in this process, FEM is commonly introduced as a special case of
Galerkin method. The process, in mathematical language, is to construct an
integral of the inner product of the residual and the weight functions 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
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 for steady state problems,
 * a set of ordinary differential equations for transient problems.

These equation sets are the element equations. They are 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 methods, while ordinary
differential equation sets that arise in the transient problems are solved by
numerical integration using standard techniques such as Euler's method or the
Runge-Kutta method.
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 data generated from the subdomains.
FEM is best understood from its practical application, known as finite element
analysis (FEA). FEA as applied in engineering is a computational tool for
performing engineering analysis. It includes the use of mesh generation
techniques for dividing a complex problem into small elements, as well as the
use of software program coded with FEM algorithm. In applying FEA, the complex
problem is usually a physical system with the underlying physics such as the
Euler-Bernoulli beam equation, the heat equation, or the Navier-Stokes equations
expressed in either PDE or integral equations, while the divided small elements
of the complex problem represent different areas in the physical system.


FINITE ELEMENT SOFTWARE FREE

FEA is a good choice 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.[3] 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.
FEM mesh created by an analyst prior to finding a solution to a magnetic problem
using FEM software. Colours indicate that the analyst has set material
properties for each zone, in this case a conducting wire coil in orange; a
ferromagnetic component (perhaps iron) in light blue; and air in grey. Although
the geometry may seem simple, it would be very challenging to calculate the
magnetic field for this setup without FEM software, using equations alone.
FEM solution to the problem at left, involving a cylindrically shaped magnetic
shield. The ferromagnetic cylindrical part is shielding the area inside the
cylinder by diverting the magnetic field created by the coil (rectangular area
on the right). The color represents the amplitude of the magnetic flux density,
as indicated by the scale in the inset legend, red being high amplitude. The
area inside the cylinder is low amplitude (dark blue, with widely spaced lines
of magnetic flux), which suggests that the shield is performing as it was
designed to.


HISTORY[EDIT]

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 problems in civil and aeronautical engineering. Its
development can be traced back to the work by A. Hrennikoff[4] and R. Courant[5]
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.[6] In China, in the later 1950s and early 1960s,
based on the computations of dam constructions, K. Feng proposed a systematic
numerical method for solving partial differential equations. The method was
called the finite difference method based on variation principle, which was
another independent invention of the finite element method. Although the
approaches used by these pioneers are different, they share one essential
characteristic: meshdiscretization 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 orderelliptic 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.
The finite element method obtained its real impetus in the 1960s and 1970s by
the developments of J. H. Argyris with co-workers at the University of
Stuttgart, R. W. Clough with co-workers at UC Berkeley, O. C. Zienkiewicz with
co-workers Ernest Hinton, Bruce Irons[7] and others at the Swansea University,
Philippe G. Ciarlet at the University of Paris 6 and Richard Gallagher with
co-workers at Cornell University. Further impetus was provided in these years by
available open source finite element software programs. NASA sponsored the
original version of NASTRAN, and UC Berkeley made the finite element program SAP
IV[8] widely available. In Norway the ship classification society Det Norske
Veritas (now DNV GL) developed Sesam in 1969 for use in analysis of ships.[9] A
rigorous mathematical basis to the finite element method was provided in 1973
with the publication by Strang and Fix.[10] The method has since been
generalized for the numerical modeling of physical systems in a wide variety of
engineering disciplines, e.g., electromagnetism, heat transfer, and fluid
dynamics.[11][12]


TECHNICAL DISCUSSION[EDIT]


THE STRUCTURE OF FINITE ELEMENT METHODS[EDIT]

A finite element method is characterized by a variational formulation, a
discretization strategy, one or more solution algorithms and post-processing
procedures.
Examples of variational formulation are the Galerkin method, the discontinuous
Galerkin method, mixed methods, etc.
A discretization strategy is understood to mean a clearly defined set of
procedures that cover (a) the creation of finite element meshes, (b) the
definition of basis function on reference elements (also called shape functions)
and (c) the mapping of reference elements onto the elements of the mesh.
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.
There are various numerical solution algorithms that 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
action of the analyst. There are some very efficient postprocessors that provide
for the realization of superconvergence.


ILLUSTRATIVE PROBLEMS P1 AND P2[EDIT]

We will demonstrate 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
P1 :{u″(x)=f(x) in (0,1),u(0)=u(1)=0,{displaystyle {mbox{ P1
}}:{begin{cases}u'(x)=f(x){mbox{ in }}(0,1),u(0)=u(1)=0,end{cases}}}
where f{displaystyle f} is given, u{displaystyle u} is an unknown function of
x{displaystyle x}, and u″{displaystyle u'} is the second derivative of
u{displaystyle u} with respect to x{displaystyle x}.
P2 is a two-dimensional problem (Dirichlet problem)
P2 :{uxx(x,y)+uyy(x,y)=f(x,y) in Ω,u=0 on ∂Ω,{displaystyle {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 Ω{displaystyle Omega } is a connected open region in the
(x,y){displaystyle (x,y)} plane whose boundary ∂Ω{displaystyle partial Omega }
is nice (e.g., a smooth manifold or a polygon), and uxx{displaystyle u_{xx}} and
uyy{displaystyle u_{yy}} denote the second derivatives with respect to
x{displaystyle x} and y{displaystyle y}, respectively.
The problem P1 can be solved directly by computing antiderivatives. However,
this method of solving the boundary value problem (BVP) works only when there is
one spatial dimension and does not generalize to higher-dimensional problems or
to problems like u+u″=f{displaystyle 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.[13]


WEAK FORMULATION[EDIT]

The first step is to convert P1 and P2 into their equivalent weak formulations.

THE WEAK FORM OF P1[EDIT]

If u{displaystyle u} solves P1, then for any smooth function v{displaystyle v}
that satisfies the displacement boundary conditions, i.e. v=0{displaystyle v=0}
at x=0{displaystyle x=0} and x=1{displaystyle x=1}, we have
(1) ∫01f(x)v(x)dx=∫01u″(x)v(x)dx.{displaystyle int _{0}^{1}f(x)v(x),dx=int
_{0}^{1}u'(x)v(x),dx.}
Conversely, if u{displaystyle u} with u(0)=u(1)=0{displaystyle u(0)=u(1)=0}
satisfies (1) for every smooth function v(x){displaystyle v(x)} then one may
show that this u{displaystyle u} will solve P1. The proof is easier for twice
continuously differentiable u{displaystyle u} (mean value theorem), but may be
proved in a distributional sense as well.
We define a new operator or map ϕ(u,v){displaystyle phi (u,v)} by using
integration by parts on the right-hand-side of (1):
(2)∫01f(x)v(x)dx=∫01u″(x)v(x)dx=u′(x)v(x)|01−∫01u′(x)v′(x)dx=−∫01u′(x)v′(x)dx≡−ϕ(u,v),{displaystyle
{begin{aligned}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),dxequiv -phi (u,v),end{aligned}}}
where we have used the assumption that v(0)=v(1)=0{displaystyle v(0)=v(1)=0}.

THE WEAK FORM OF P2[EDIT]

If we integrate by parts using a form of Green's identities, we see that if
u{displaystyle u} solves P2, then we may define ϕ(u,v){displaystyle phi (u,v)}
for any v{displaystyle v} by
∫Ωfvds=−∫Ω∇u⋅∇vds≡−ϕ(u,v),{displaystyle int _{Omega }fv,ds=-int _{Omega }nabla
ucdot nabla v,dsequiv -phi (u,v),}
where ∇{displaystyle nabla } denotes the gradient and ⋅{displaystyle cdot }
denotes the dot product in the two-dimensional plane. Once more ϕ{displaystyle
,!phi } can be turned into an inner product on a suitable space
H01(Ω){displaystyle H_{0}^{1}(Omega )} of once differentiable functions of
Ω{displaystyle Omega } that are zero on ∂Ω{displaystyle partial Omega }. We have
also assumed that v∈H01(Ω){displaystyle vin H_{0}^{1}(Omega )} (see Sobolev
spaces). Existence and uniqueness of the solution can also be shown.

A PROOF OUTLINE OF EXISTENCE AND UNIQUENESS OF THE SOLUTION[EDIT]

We can loosely think of H01(0,1){displaystyle H_{0}^{1}(0,1)} to be the
absolutely continuous functions of (0,1){displaystyle (0,1)} that are
0{displaystyle 0} at x=0{displaystyle x=0} and x=1{displaystyle x=1} (see
Sobolev spaces). Such functions are (weakly) once differentiable and it turns
out that the symmetric bilinear mapϕ{displaystyle !,phi } then defines an inner
product which turns H01(0,1){displaystyle H_{0}^{1}(0,1)} into a Hilbert space
(a detailed proof is nontrivial). On the other hand, the left-hand-side
∫01f(x)v(x)dx{displaystyle int _{0}^{1}f(x)v(x)dx} is also an inner product,
this time on the Lp spaceL2(0,1){displaystyle L^{2}(0,1)}. An application of the
Riesz representation theorem for Hilbert spaces shows that there is a unique
u{displaystyle u} solving (2) and therefore P1. This solution is a-priori only a
member of H01(0,1){displaystyle H_{0}^{1}(0,1)}, but using elliptic regularity,
will be smooth if f{displaystyle f} is.


DISCRETIZATION[EDIT]

A function in H01,{displaystyle H_{0}^{1},} with zero values at the endpoints
(blue), and a piecewise linear approximation (red)
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 u∈H01{displaystyle uin H_{0}^{1}} such that∀v∈H01,−ϕ(u,v)=∫fv{displaystyle
forall vin H_{0}^{1},;-phi (u,v)=int fv}
with a finite-dimensional version:
(3) Find u∈V{displaystyle uin V} such that∀v∈V,−ϕ(u,v)=∫fv{displaystyle forall
vin V,;-phi (u,v)=int fv}
where V{displaystyle V} is a finite-dimensional subspace of H01{displaystyle
H_{0}^{1}}. There are many possible choices for V{displaystyle V} (one
possibility leads to the spectral method). However, for the finite element
method we take V{displaystyle V} to be a space of piecewise polynomial
functions.


FOR PROBLEM P1[EDIT]

We take the interval (0,1){displaystyle (0,1)}, choose n{displaystyle n} values
of x{displaystyle x} with 0=x0<x1<⋯<xn<xn+1=1{displaystyle 0=x_{0}<x_{1}<cdots
<x_{n}<x_{n+1}=1} and we define V{displaystyle V} by:
V={v:[0,1]→R:v is continuous, v|[xk,xk+1] is linear for k=0,…,n, and
v(0)=v(1)=0}{displaystyle V={v:[0,1]rightarrow mathbb {R} ;:v{mbox{ is
continuous, }}v|_{[x_{k},x_{k+1}]}{mbox{ is linear for }}k=0,dots ,n{mbox{, and
}}v(0)=v(1)=0}}
where we define x0=0{displaystyle x_{0}=0} and xn+1=1{displaystyle x_{n+1}=1}.
Observe that functions in V{displaystyle V} are not differentiable according to
the elementary definition of calculus. Indeed, if v∈V{displaystyle vin V} then
the derivative is typically not defined at any x=xk{displaystyle x=x_{k}},
k=1,…,n{displaystyle k=1,ldots ,n}. However, the derivative exists at every
other value of x{displaystyle x} and one can use this derivative for the purpose
of integration by parts.
A piecewise linear function in two dimensions


FOR PROBLEM P2[EDIT]

We need V{displaystyle V} to be a set of functions of Ω{displaystyle Omega }. In
the figure on the right, we have illustrated a triangulation of a 15 sided
polygonal region Ω{displaystyle 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{displaystyle V} 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 h>0{displaystyle 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{displaystyle V} must
also change with h{displaystyle h}. For this reason, one often reads
Vh{displaystyle V_{h}} instead of V{displaystyle V} in the literature. Since we
do not perform such an analysis, we will not use this notation.


CHOOSING A BASIS[EDIT]

16 scaled and shifted triangular basis functions (colors) used to reconstruct a
zeroeth order Bessel function J0 (black).
The linear combination of basis functions (yellow) reproduces J0 (blue) to any
desired accuracy.
To complete the discretization, we must select a basis of V{displaystyle V}. In
the one-dimensional case, for each control point xk{displaystyle x_{k}} we will
choose the piecewise linear function vk{displaystyle v_{k}} in V{displaystyle V}
whose value is 1{displaystyle 1} at xk{displaystyle x_{k}} and zero at every
xj,j≠k{displaystyle x_{j},;jneq k}, i.e.,
vk(x)={x−xk−1xk−xk−1 if x∈[xk−1,xk],xk+1−xxk+1−xk if x∈[xk,xk+1],0
otherwise,{displaystyle v_{k}(x)={begin{cases}{x-x_{k-1} over
x_{k},-x_{k-1}}&{mbox{ if }}xin [x_{k-1},x_{k}],{x_{k+1},-x over
x_{k+1},-x_{k}}&{mbox{ if }}xin [x_{k},x_{k+1}],0&{mbox{
otherwise}},end{cases}}}
for k=1,…,n{displaystyle k=1,dots ,n}; this basis is a shifted and scaled tent
function. For the two-dimensional case, we choose again one basis function
vk{displaystyle v_{k}} per vertex xk{displaystyle x_{k}} of the triangulation of
the planar region Ω{displaystyle Omega }. The function vk{displaystyle v_{k}} is
the unique function of V{displaystyle V} whose value is 1{displaystyle 1} at
xk{displaystyle x_{k}} and zero at every xj,j≠k{displaystyle x_{j},;jneq 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 thehp-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:
 * moving nodes (r-adaptivity)
 * refining (and unrefining) elements (h-adaptivity)
 * changing order of base functions (p-adaptivity)
 * combinations of the above (hp-adaptivity).


SMALL SUPPORT OF THE BASIS[EDIT]

Solving the two-dimensional problem uxx+uyy=−4{displaystyle 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 matrixL of the discretized linear system
(c) The computed solution, u(x,y)=1−x2−y2.{displaystyle u(x,y)=1-x^{2}-y^{2}.}
The primary advantage of this choice of basis is that the inner products
⟨vj,vk⟩=∫01vjvkdx{displaystyle langle v_{j},v_{k}rangle =int
_{0}^{1}v_{j}v_{k},dx}
and
ϕ(vj,vk)=∫01vj′vk′dx{displaystyle phi (v_{j},v_{k})=int _{0}^{1}v_{j}'v_{k}',dx}
will be zero for almost all j,k{displaystyle j,k}.(The matrix containing
⟨vj,vk⟩{displaystyle langle v_{j},v_{k}rangle } in the (j,k){displaystyle (j,k)}
location is known as the Gramian matrix.)In the one dimensional case, the
support of vk{displaystyle v_{k}} is the interval [xk−1,xk+1]{displaystyle
[x_{k-1},x_{k+1}]}. Hence, the integrands of ⟨vj,vk⟩{displaystyle langle
v_{j},v_{k}rangle } and ϕ(vj,vk){displaystyle phi (v_{j},v_{k})} are identically
zero whenever |j−k|>1{displaystyle |j-k|>1}.
Similarly, in the planar case, if xj{displaystyle x_{j}} and xk{displaystyle
x_{k}} do not share an edge of the triangulation, then the integrals
∫Ωvjvkds{displaystyle int _{Omega }v_{j}v_{k},ds}
and
∫Ω∇vj⋅∇vkds{displaystyle int _{Omega }nabla v_{j}cdot nabla v_{k},ds}
are both zero.


MATRIX FORM OF THE PROBLEM[EDIT]

If we write u(x)=∑k=1nukvk(x){displaystyle u(x)=sum _{k=1}^{n}u_{k}v_{k}(x)} and
f(x)=∑k=1nfkvk(x){displaystyle f(x)=sum _{k=1}^{n}f_{k}v_{k}(x)} then problem
(3), taking v(x)=vj(x){displaystyle v(x)=v_{j}(x)} for j=1,…,n{displaystyle
j=1,dots ,n}, becomes
−∑k=1nukϕ(vk,vj)=∑k=1nfk∫vkvjdx{displaystyle -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.{displaystyle
j=1,dots ,n.} (4)
If we denote by u{displaystyle mathbf {u} } and f{displaystyle mathbf {f} } the
column vectors (u1,…,un)t{displaystyle (u_{1},dots ,u_{n})^{t}} and
(f1,…,fn)t{displaystyle (f_{1},dots ,f_{n})^{t}}, and if we let
L=(Lij){displaystyle L=(L_{ij})}
and
M=(Mij){displaystyle M=(M_{ij})}
be matrices whose entries are
Lij=ϕ(vi,vj){displaystyle L_{ij}=phi (v_{i},v_{j})}
and
Mij=∫vivjdx{displaystyle M_{ij}=int v_{i}v_{j}dx}
then we may rephrase (4) as
−Lu=Mf.{displaystyle -Lmathbf {u} =Mmathbf {f} .} (5)
It is not necessary to assume f(x)=∑k=1nfkvk(x){displaystyle f(x)=sum
_{k=1}^{n}f_{k}v_{k}(x)}. For a general function f(x){displaystyle f(x)},
problem (3) with v(x)=vj(x){displaystyle v(x)=v_{j}(x)} for j=1,…,n{displaystyle
j=1,dots ,n} becomes actually simpler, since no matrix M{displaystyle M} is
used,
−Lu=b{displaystyle -Lmathbf {u} =mathbf {b} }, (6)
where b=(b1,…,bn)t{displaystyle mathbf {b} =(b_{1},dots ,b_{n})^{t}} and
bj=∫fvjdx{displaystyle b_{j}=int fv_{j}dx} for j=1,…,n{displaystyle j=1,dots
,n}.
As we have discussed before, most of the entries of L{displaystyle L} and
M{displaystyle M} are zero because the basis functions vk{displaystyle v_{k}}
have small support. So we now have to solve a linear system in the unknown
u{displaystyle mathbf {u} } where most of the entries of the matrix
L{displaystyle 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{displaystyle 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{displaystyle L} is usually referred to as the stiffness matrix,
while the matrix M{displaystyle M} is dubbed the mass matrix.


GENERAL FORM OF THE FINITE ELEMENT METHOD[EDIT]

In general, the finite element method is characterized by the following process.
 * One chooses a grid for Ω{displaystyle 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
uxxxx+uyyyy=f{displaystyle u_{xxxx}+u_{yyyy}=f}, one may use piecewise quadratic
basis functions that are C1{displaystyle C^{1}}.
Another consideration is the relation of the finite-dimensional space
V{displaystyle V} to its infinite-dimensional counterpart, in the examples above
H01{displaystyle H_{0}^{1}}. A conforming element method is one in which the
space V{displaystyle 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 H01{displaystyle
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{displaystyle h} is
bounded above by Chp{displaystyle Ch^{p}}, for some C<∞{displaystyle C<infty }
and p>0{displaystyle 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{displaystyle d} method will have an error of order p=d+1{displaystyle
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 Rn{displaystyle mathbb {R} ^{n}}.


VARIOUS TYPES OF FINITE ELEMENT METHODS[EDIT]


AEM[EDIT]

The Applied Element Method, or AEM combines features of both FEM and Discrete
element method, or (DEM).


GENERALIZED FINITE ELEMENT METHOD[EDIT]

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.[14]


FINITE ELEMENT ANALYSIS EXAMPLES


MIXED FINITE ELEMENT METHOD[EDIT]

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.


HP-FEM[EDIT]

The hp-FEM combines adaptively, elements with variable size h and polynomial
degree p in order to achieve exceptionally fast, exponential convergence
rates.[15]


HPK-FEM[EDIT]

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[EDIT]

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 is
able to 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 remesh 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 Radioss, 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)[EDIT]

The introduction of the scaled boundary finite element method (SBFEM) came from
Song and Wolf (1997).[16] 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[EDIT]

The S-FEM, Smoothed Finite Element Methods, are 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.


FINITE ELEMENT SOFTWARE


SPECTRAL ELEMENT METHOD[EDIT]

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 Lagragian
interpolants and used only with certain quadrature rules.[17]


MESHFREE METHODS[EDIT]


DISCONTINUOUS GALERKIN METHODS[EDIT]


FINITE ELEMENT LIMIT ANALYSIS[EDIT]


STRETCHED GRID METHOD[EDIT]


LOUBIGNAC ITERATION[EDIT]

Loubignac iteration is an iterative method in finite element methods.



LINK WITH THE GRADIENT DISCRETISATION METHOD[EDIT]

Some types of finite element methods (conforming, nonconforming, mixed finite
element methods) are particular cases of the gradient discretisation 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[EDIT]

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.[18]
 * 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 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[EDIT]

Visualization of how a car deforms in an asymmetrical crash using finite element
analysis.[1]
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.[19]
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 discretised
problem. However, there is a value at which the results converge and further
mesh refinement does not increase accuracy.[20]
Finite Element Model of a human knee joint.[21]
This powerful design tool has significantly improved both the standard of
engineering designs and the methodology of the design process in many industrial
applications.[22] The introduction of FEM has substantially decreased the time
to take products from concept to the production line.[22] It is primarily
through improved initial prototype designs using FEM that testing and
development have been accelerated.[23] 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.[22]
In the 1990s FEA was proposed for use in stochastic modelling for numerically
solving probability models[24] and later for reliability assessment.[25] The
stochastic finite element method has since been applied to many branches of
engineering,[26] often being applied to characterise variability in material
properties.[27]


SEE ALSO[EDIT]


REFERENCES[EDIT]

 1.  ^Daryl L. Logan (2011). A first course in the finite element method.
     Cengage Learning. ISBN978-0495668251.
 2.  ^Reddy, J. N. (2006). An Introduction to the Finite Element Method (Third
     ed.). McGraw-Hill. ISBN9780071267618.
 3.  ^'Finite Elements Analysis (FEA)'. www.manortool.com. Retrieved 2017-07-28.
 4.  ^Hrennikoff, Alexander (1941). 'Solution of problems of elasticity by the
     framework method'. Journal of Applied Mechanics. 8.4: 169–175.
 5.  ^Courant, R. (1943). 'Variational methods for the solution of problems of
     equilibrium and vibrations'. Bulletin of the American Mathematical Society.
     49: 1–23. doi:10.1090/s0002-9904-1943-07818-4.
 6.  ^'СПб ЭМИ РАН'. emi.nw.ru. Retrieved 17 March 2018.
 7.  ^Hinton, Ernest; Irons, Bruce (July 1968). 'Least squares smoothing of
     experimental data using finite elements'. Strain. 4 (3): 24–27.
     doi:10.1111/j.1475-1305.1968.tb01368.x.
 8.  ^'SAP-IV Software and Manuals'. NISEE e-Library, The Earthquake Engineering
     Online Archive.
 9.  ^Gard Paulsen; Håkon With Andersen; John Petter Collett; Iver Tangen
     Stensrud (2014). Building Trust, The history of DNV 1864-2014. Lysaker,
     Norway: Dinamo Forlag A/S. pp. 121, 436. ISBN978-82-8071-256-1.
 10. ^Strang, Gilbert; Fix, George (1973). An Analysis of The Finite Element
     Method. Prentice Hall. ISBN978-0-13-032946-2.
 11. ^Olek C Zienkiewicz; Robert L Taylor; J.Z. Zhu (31 August 2013). The Finite
     Element Method: Its Basis and Fundamentals. Butterworth-Heinemann.
     ISBN978-0-08-095135-5.
 12. ^Bathe, K.J. (2006). Finite Element Procedures. Cambridge, MA: Klaus-Jürgen
     Bathe. ISBN978-0979004902.
 13. ^Smith, I.M.; Griffiths, D.V.; Margetts, L. (2014). Programming the Finite
     Element Method (Fifth ed.). Wiley. ISBN978-1-119-97334-8.
 14. ^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.
 15. ^P. Solin, K. Segeth, I. Dolezel: Higher-Order Finite Element Methods,
     Chapman & Hall/CRC Press, 2003
 16. ^Song, Chongmin; Wolf, John P. (5 August 1997). 'The scaled boundary
     finite-element method - alias consistent infinitesimal finite-element cell
     method - for elastodynamics'. Computer Methods in Applied Mechanics and
     Engineering. 147 (3–4): 329–355. Bibcode:1997CMAME.147..329S.
     doi:10.1016/S0045-7825(97)00021-2.
 17. ^'Spectral Element Methods'. State Key Laboratory of Scientific and
     Engineering Computing. Retrieved 2017-07-28.
 18. ^'What's The Difference Between FEM, FDM, and FVM?'. Machine Design.
     2016-04-18. Retrieved 2017-07-28.
 19. ^Kiritsis, D.; Eemmanouilidis, Ch.; Koronios, A.; Mathew, J. (2009).
     'Engineering Asset Management'. Proceedings of the 4th World Congress on
     Engineering Asset Management (WCEAM): 591–592.
 20. ^'Finite Element Analysis: How to create a great model'. Coventive
     Composites. 2019-03-18. Retrieved 2019-04-05.
 21. ^Naghibi Beidokhti, Hamid; Janssen, Dennis; Khoshgoftar, Mehdi; Sprengers,
     Andre; Perdahcioglu, Emin Semih; Boogaard, Ton Van den; Verdonschot, Nico
     (2016). 'A comparison between dynamic implicit and explicit finite element
     simulations of the native knee joint'(PDF). Medical Engineering & Physics.
     38 (10): 1123–1130. doi:10.1016/j.medengphy.2016.06.001. PMID27349493.
 22. ^ abcHastings, J. K., Juds, M. A., Brauer, J. R., Accuracy and Economy of
     Finite Element Magnetic Analysis, 33rd Annual National Relay Conference,
     April 1985.
 23. ^McLaren-Mercedes (2006). 'McLaren Mercedes: Feature - Stress to impress'.
     Archived from the original on 2006-10-30. Retrieved 2006-10-03.
 24. ^Peng Long; Wang Jinliang; Zhu Qiding (19 May 1995). 'Methods with high
     accuracy for finite element probability computing'. Journal of
     Computational and Applied Mathematics. 59 (2): 181–189.
     doi:10.1016/0377-0427(94)00027-X.
 25. ^Haldar, Achintya; Mahadevan, Sankaran (2000). Reliability Assessment Using
     Stochastic Finite Element Analysis. John Wiley & Sons. ISBN978-0471369615.
 26. ^Arregui Mena, J.D.; Margetts, L.; et al. (2014). 'Practical Application of
     the Stochastic Finite Element Method'. Archives of Computational Methods in
     Engineering. 23 (1): 171–190. doi:10.1007/s11831-014-9139-3.
 27. ^Arregui Mena, J.D.; et al. (2018). 'Characterisation of the spatial
     variability of material properties of Gilsocarbon and NBG-18 using random
     fields'. Journal of Nuclear Materials. 511: 91–108.
     doi:10.1016/j.jnucmat.2018.09.008.


FURTHER READING[EDIT]

 * 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).


EXTERNAL LINKS[EDIT]


ABAQUS FINITE ELEMENT SOFTWARE

Wikimedia Commons has media related to Finite element modelling.

 * IFER – Internet Finite Element Resources – describes and provides access to
   finite element analysis software via the Internet


FINITE ELEMENT SOFTWARE ENGINEER

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