Everything about Fem totally explained
The
finite element method (FEM) is used for finding approximate solutions of
partial differential equations (PDE) as well as of
integral equations such as the
heat transport equation. The solution approach is based either on eliminating the differential equation completely (steady state problems), or rendering the PDE into an approximating system of
ordinary differential equations, which are then solved using standard techniques such as
Euler's method,
Runge-Kutta, etc.
In solving
partial differential equations, the primary challenge is to create an equation that approximates the equation to be studied, but is
numerically stable, meaning that errors in the input data and intermediate calculations don't accumulate and cause the resulting output to be meaningless. There are many ways of doing this, all with advantages and disadvantages. The Finite Element Method is a good choice for solving partial differential equations over complex domains (like cars and oil pipelines), when the domain changes (as during a solid state reaction with a moving boundary), when the desired precision varies over the entire domain, or when the solution lacks smoothness. For instance, in simulating the weather pattern on Earth, it's more important to have accurate predictions over land than over the wide-open sea, a demand that's achievable using the finite element method.
History
The finite-element method originated from the needs for solving complex
elasticity,
structural analysis problems in
civil engineering and
aeronautical engineering. Its development can be traced back to the work by
Alexander Hrennikoff (1941) and
Richard Courant (1942). While the approaches used by these pioneers are dramatically different, they share one essential characteristic:
mesh discretization of a continuous domain into a set of discrete sub-domains. Hrennikoff's work discretizes the domain by using a lattice analogy while Courant's approach divides the domain into finite triangular subregions for solution of second order elliptic partial differential equations (PDEs) that arise from the problem of
torsion of a cylinder. Courant's contribution was evolutionary, drawing on a large body of earlier results for PDEs developed by
Rayleigh,
Ritz, and
Galerkin. Development of the finite element method began in earnest in the middle to late
1950s for
airframe and
structural analysis and gathered momentum at the
University of Stuttgart through the work of
John Argyris and at
Berkeley through the work of
Ray W. Clough in the
1960s for use in
civil engineering. The method was provided with a rigorous mathematical foundation in
1973 with the publication of
Strang and
Fix's
An Analysis of The Finite Element Method, and has since been generalized into a branch of applied mathematics for numerical modeling of physical systems in a wide variety of
engineering disciplines, for example,
electromagnetism and
fluid dynamics.
The development of the
finite element method in structural mechanics is often based on an energy principle, for example, the
virtual work principle or the
minimum total potential energy principle, which provides a general, intuitive and physical basis that has a great appeal to structural engineers.
Technical discussion
We will illustrate the finite element method using two sample problems from which the general method can be extrapolated. It is assumed that the reader is familiar with
calculus and
linear algebra.
P1 is a
one-dimensional problem
» .
Comparison to the finite difference method
The
finite difference method (FDM) is an alternative way for solving PDEs. The differences between FEM and FDM are:
The finite difference method is an approximation to the differential equation; the finite element method is an approximation to its solution.
The most attractive feature of the FEM is its ability to handle complex geometries (and boundaries) with relative ease. While FDM in its basic form is restricted to handle rectangular shapes and simple alterations thereof, the handling of geometries in FEM is theoretically straightforward.
The most attractive feature of finite differences is that it can be very easy to implement.
There are several ways one could consider the FDM a special case of the FEM approach. One might choose basis functions as either piecewise constant functions or Dirac delta functions. In both approaches, the approximations are defined on the entire domain, but need not be continuous. Alternatively, one might define the function on a discrete domain, with the result that the continuous differential operator no longer makes sense, however this approach isn't FEM.
There are reasons to consider the mathematical foundation of the finite element approximation more sound, for instance, because the quality of the approximation between grid points is poor in FDM.
The quality of a FEM approximation is often higher than in the corresponding FDM approach, but this is extremely problem dependent and several examples to the contrary can be provided.
Generally, FEM is the method of choice in all types of analysis in structural mechanics (for example solving for deformation and stresses in solid bodies or dynamics of structures) while computational fluid dynamics (CFD) tends to use FDM or other methods (for example, finite volume method). CFD problems usually require discretization of the problem into a large number of cells/gridpoints (millions and more), therefore cost of the solution favors simpler, lower order approximation within each cell. This is especially true for 'external flow' problems, like air flow around the car or airplane, or weather simulation in a large area.
There are many finite element software packages, some free and some proprietary.
Some examples of FEM software available on the market
ABAQUS: American software
CosmosWorks : Franco-American software from SolidWorks, which belong to Dassault CosmosWorks
NISA: Indian software NISA
ANSYS: American software
CAST3M: French software CASTEM
SYSTUS: French software
SYSWELD: French software
Code Aster: French software Aster
Nastran: American software
PERMAS: German software PERMAS
SAMCEF: Belgian softwareSAMCEF
Morfeo: Belgian softwareMorfeo
JMAG: Japanese software
freeFEM: a GPL-licensed software freefem.org
CalculiX: Open-Source-FEM, uses a partially compatible ABAQUS file format
Some examples of explicit software:
ABAQUS: American software
EuroPlexus: French software EuroPlexus
LS DYNA: American software
PAM: French software PAM
Radioss: French softwareFurther Information
Get more info on 'Fem'.
|
External Link Exchanges
Do you know how hard it is to get a link from a large encyclopaedia? Well we're different and will prove it. To get a link from us just add the following HTML to your site on a relevant page:
<a href="http://finite_element_method.totallyexplained.com">Finite element method Totally Explained</a>
Then simply click through this link from your web page. Our crawlers will verify your link, extract the title of your web page and instantly add a link back to it. If you like you can remove the words Totally Explained and embed the link in article text.
As long as your link remains in place, we'll keep our link to you right here. Please play fair - our crawlers are watching. Your site must be closely related to this one's topic. Any kind of spamming, dubious practises or removing the link will result in your link from us being dropped and, potentially, your whole site being banned. |
