Finite Element Method

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	CATEGORIES ABOUT FINITE ELEMENT METHOD
	numerical differential equations
	partial differential equations
	continuum mechanics
	finite element methodnumerical differential equations
	partial differential equations
	continuum mechanics
	finite element method
	structural analysis
	numerical analysis

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Mathematically , the finite element method (FEM) is used for finding approximate solution of Partial Differential Equation s (PDE) as well as of Integral Equation s such as the Heat Transport Equation . The solution approach is based either on eliminating the differential equation completely (steady state problems), or rendering the PDE into an equivalent Ordinary Differential Equation , which is then solved using standard techniques such as Finite Difference s, etc. In solving Partial Differential Equation s, the primary challenge is to create an equation that approximates the equation to be studied, but is Numerically Stable , meaning that errors in the input data and intermediate calculations do not accumulate and cause the resulting output to be meaningless. There are many ways of doing this, all with advantages and disadvantages. The Finite Element Method is a good choice for solving partial differential equations over complex domains (like cars and oil pipelines), when the domain changes (as during a solid state reaction with a moving boundary), or when the desired precision varies over the entire domain. For instance, in simulating the weather pattern on Earth, it is more important to have accurate predictions over land than over the wide-open sea, a demand that is achievable using the finite element method. HISTORY The finite-element method originated from the needs for solving complex in the 1960s for use in Civil Engineering . The method was provided with a rigorous mathematical foundation in 1973 with the publication of Strang and Fix's ''An Analysis of The Finite Element Method'', and has since been generalized into a branch of applied mathematics for numerical modeling of physical systems in a wide variety of Engineering disciplines, e.g., Electromagnetism and Fluid Dynamics . The development of the Finite Element Method In Structural Mechanics is often based on an energy principle, e.g., the Virtual Work principle or the Minimum Total Potential Energy Principle , which provides a general, intuitive and physical basis that has a great appeal to structural engineers. TECHNICAL DISCUSSION We will illustrate the finite element method using two sample problems from which the general method can be extrapolated. We assume that the reader is familiar with Calculus and Linear Algebra . We will use the one-dimensional :$\backslash mbox\{P1\; \}:\backslash begin\{cases\}$ u''=f \mbox{ in } (0,1), \ u(0)=u(1)=0, \end{cases} where $f$ is given and $u$ is an unknown function of $x$, and $u\text{'}\text{'}$ is the second derivative of $u$ with respect to $x$. The two-dimensional sample problem is the Dirichlet Problem :$\backslash mbox\{P2\; \}:\backslash begin\{cases\}$ u_{xx}+u_{yy}=f & \mbox{ in } \Omega, \ u=0 & \mbox{ on } \partial \Omega, \end{cases} where $\backslash Omega$ is a connected open region in the $(x,y)$ plane whose boundary $\backslash partial\; \backslash Omega$ is "nice" (e.g., a Smooth Manifold or a Polygon ), and $u\_\{xx\}$ and $u\_\{yy\}$ denote the second derivatives with respect to $x$ and $y$, respectively. The problem P1 can be solved "directly" by computing Antiderivative s. However, this method of solving the Boundary Value Problem works only when there is only one spatial dimension and does not generalize to higher-dimensional problems or to problems like $u+u\text{'}\text{'}=f$. 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, or Variational form. Little to no computation is usually required for this step, the transformation is done by hand on paper. The second step is the discretization, where the weak form is discretized in a finite dimensional space. After this second step, we have concrete formulae for a large but finite dimensional linear problem whose solution will approximately solve the original BVP. This finite dimensional problem is then implemented on a Computer . VARIATIONAL FORMULATION The first step is to convert P1 and P2 into their Variational equivalents. If $u$ solves P1, then for any smooth function $v$ that satisfies the displacement boundary conditions, i.e. $v=0$ at $x=0$ and $x=1$,we have (1) $\backslash int\_0^1\; f(x)v(x)\; \backslash ,\; dx\; =\; \backslash int\_0^1\; u\text{'}\text{'}(x)v(x)\; \backslash ,\; dx.$ Conversely, if for a given $u$, (1) holds for every smooth function $v(x)$ then one may show that this $u$ will solve P1. (The proof is nontrivial and uses Sobolev Space s.) By using Integration By Parts on the right-hand-side of (1), we obtain (2)$$ \begin{align} \int_0^1 f(x)v(x) \, dx & = \int_0^1 u''(x)v(x) \, dx \
	:<math>\begin{matrix} V	\{v:[0,1] ightarrow \Bbb R\: v\mbox{ is continuous, }v_{[x_k,x_{k+1}]} \mbox{ is linear for }\

{x_{k+1}-x \over x_{k+1}-x_k} & \mbox{ if } x \in {Link without Title} , \
0 & \mbox{ otherwise},\end{cases}

for $k=1,...,n$. For the two-dimensional case, we choose again one basis function $v\_k$ per vertex $x\_k$ of the triangulation of the planar region $\backslash Omega$. The function $v\_k$ is the unique function of $V$ whose value is $1$ at $x\_k$ and zero at every $x\_j,\backslash ;j$
eq 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, in which case he might describe his elements as being curvilinear. On the other hand, some authors replace "piecewise linear" by "piecewise quadratic" or even "piecewise polynomial". The author might then say "higher order element" instead of "higher degree polynomial." Finite element method is not restricted to triangles (or tetrahedra in 3-d, or higher order simplexes in multidimensional spaces), but can be defined on quadrilateral subdomains (hexahedra, prisms, or pyramids in 3-d, and so on). Higher order shapes (curvilinear elements) can be defined with polynomial and even non-polynomial shapes (e.g. ellipse or circle).

Methods that use higher degree piecewise polynomial basis functions are often called Spectral Element Method s, especially if the degree of the polynomials increases as the triangulation size $h$ goes to zero.

More advanced implementations (adaptive finite element methods) utilize a method to assess the quality of the results (based on error estimation theory) and modify the mesh during the solution aiming to achieve approximate solution within some bounds from the 'exact' solution of the continuum problem. Mesh adaptivity may utilize various techniques, the most popular are:

• moving nodes (r-adaptivity)


• refining (and unrefining) elements (h-adaptivity)


• changing order of base functions (p-adaptivity)


• combinations of the above (e.g. hp-adaptivity)

Small support of the basis


''L'' of the discretized linear system.]]

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

:$\backslash langle\; v\_j,v\_k\; angle=\backslash int\_0^1\; v\_j\; v\_k\backslash ,dx$

and

:$\backslash phi(v\_j,v\_k)=\backslash int\_0^1\; v\_j\text{'}\; v\_k\text{'}\backslash ,dx$