- Finite element method
The finite element method (FEM) (sometimes referred to as finite element analysis) is a numerical technique for finding approximate solutions of
partial differential equations (PDE) as well as of integral equations. The solution approach is based either on eliminating the differential equation completely (steady state problems), or rendering the PDE into an approximating system of ordinary differential equations, which are then solved using standard techniques such as Euler's method, Runge-Kutta, etc.
partial differential equations, the primary challenge is to create an equation that approximates the equation to be studied, but is numerically stable, meaning that errors in the input data and intermediate calculations do not accumulate and cause the resulting output to be meaningless. There are many ways of doing this, all with advantages and disadvantages. The Finite Element Method is a good choice for solving partial differential equations over complex domains (like cars and oil pipelines), when the domain changes (as during a solid state reaction with a moving boundary), when the desired precision varies over the entire domain, or when the solution lacks smoothness. For instance, in a frontal crash simulation it is possible to increase prediction accuracy in "important" areas like the front of the car and reduce it in its rear (thus reducing cost of the simulation); Another example would be the simulation of the weather pattern on Earth, where it is more important to have accurate predictions over land than over the wide-open sea.
The finite-element method [cite book|last=Ciarlet| first=Phillippe G. | title=The Finite Element Method for Elliptic Problems | publisher=North-Holland | location=Amsterdam | year=1978] originated from the need for solving complex elasticity and
structural analysisproblems in civil and aeronautical engineering. Its development can be traced back to the work by Alexander Hrennikoff(1941) and Richard Courant(1942). While the approaches used by these pioneers are dramatically different, they share one essential characteristic: mesh discretization of a continuous domain into a set of discrete sub-domains, usually called elements.citation
author = Waterman, Pamela J.
title = Meshing: the Critical Bridge
journal = Desktop Engineering Magazine
date = Aug. 1, 2008
url = http://www.deskeng.com/articles/aaakfj.htm]
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 [cite journal|authorlink=Richard Courant|first=R. L.|last=Courant|title=Variational Methods for the Solution of Problems of Equilibrium and Vibration|journal=Bulletin of the American Mathematical Society |volume=49|year=1943|pages=1-23|url=http://projecteuclid.org/DPubS?verb=Display&version=1.0&service=UI&handle=euclid.bams/1183504922&page=record] . 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
airframeand structural analysisand gathered momentum at the University of Stuttgartthrough the work of John Argyrisand at Berkeley through the work of Ray W. Cloughin the 1960s for use in civil engineering. [ cite web|url=http://www.edwilson.org/History/fe-history.pdf |title=Early Finite Element Research at Berkeley |accessdate=2007-10-25 |last=Clough |first=Ray W. |coauthors=Edward L. Wilson |format=PDF ] By late 1950s, the key concepts of stiffness matrix and element assembly existed essentially in the form used todaycite journal | last =Turner | first = M.J. | coauthors = R.W. Clough, H.C. Martin, and L.C. Topp | title =Stiffness and Deflection Analysis of Complex Structures | journal =Journal of the Aeronautical Sciences | volume =23 | pages = 805–82 | year =1956 ] and NASA issued request for proposals for the development of the finite element software NASTRANin 1965. 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" [cite book | first=Gilbert | last=Strang | coauthors=George Fix | title=An Analysis of the Finite Element Method | year=1973 | publisher= Prentice-Hall | location=Englewood Cliffs] , and has since been generalized into a branch of applied mathematics for numerical modeling of physical systems in a wide variety of engineeringdisciplines, e.g., electromagnetismand fluid dynamics.
A variety of specializations under the umbrella of the mechanical engineering discipline (such as aeronautical, biomechanical, and automotive industries) commonly use integrated FEM in design and development of their products. Several modern FEM packages include specific components such as thermal, electromagnetic, fluid, and structural working environments. In a structural simulation, FEM helps tremendously in producing stiffness and strength visualizations and also in minimizing weight, materials, and costs. FEM allows detailed visualization of where structures bend or twist, and indicates the distribution of stresses and displacements. FEM software provides a wide range of simulation options for controlling the complexity of both modeling and analysis of a system. Similarly, the desired level of accuracy required and associated computational time requirements can be managed simultaneously to address most engineering applications. FEM allows entire designs to be constructed, refined, and optimized before the design is manufactured. This powerful design tool has significantly improved both the standard of engineering designs and the methodology of the design process in many industrial applications.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.cite web|title=Vodafone McLaren-Mercedes: Feature - Stress to impress|author=McLaren-Mercedes|year=2006|url=http://www.mclaren.com/features/technical/stress_to_impress.php|accessdate=2006-10-03] 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.
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
calculusand linear algebra.
P1 is a one-dimensional problem:where is given and is an unknown function of , and is the second derivative of with respect to .
The two-dimensional sample problem is the
where is a connected open region in the plane whose boundary is "nice" (e.g., a
smooth manifoldor a polygon), and and denote the second derivatives with respect to and , respectively.
The problem P1 can be solved "directly" by computing
antiderivatives. However, this method of solving the boundary value problemworks only when there is only one spatial dimension and does not generalize to higher-dimensional problems or to 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, or
variationalform. 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
The first step is to convert P1 and P2 into their
variationalequivalents. If solves P1, then for any smooth function that satisfies the displacement boundary conditions, i.e. at and ,we have
Conversely, if for a given , (1) holds for every smooth function then one may show that this will solve P1. (The proof is nontrivial and uses
integration by partson the right-hand-side of (1), we obtain
where we have used the assumption that .
A proof outline of existence and uniqueness of the solution
We can loosely think of to be the
absolutely continuousfunctions of that are at and (see Sobolev spaces). Such function are (weakly) "once differentiable" and it turns out that the symmetric bilinear mapthen defines an inner productwhich turns into a Hilbert space(a detailed proof is nontrivial.) On the other hand, the left-hand-side is also an inner product, this time on the Lp space. An application of the Riesz representation theoremfor Hilbert spaces shows that there is a unique solving (2) and therefore P1.
The variational form of P2
If we integrate by parts using a form of
Green's theorem, we see that if solves P2, then for any ,
where denotes the
gradientand denotes the dot productin 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 spaces). Existence and uniqueness of the solution can also be shown.
The basic idea is to replace the infinite dimensional linear problem::Find such that:with a finite dimensional version:
:(3) Find such that:
where is a finite dimensional 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 linear functions.
For problem P1, we take the interval , choose values
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