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Finite Element Analysis
In the field of solid mechanics, finite element analysis is the process of reducing materials with infinite degrees of freedom to a set of finite beam, bar, truss, or frame elements with limited degrees of freedom in order to perform structural analyses that will predict how the material deforms as well as the stresses, strains, and reaction forces that result from the deformations.

Modeling Deformations with Partial Differential Equations
For axial elongation of a bar, the displacement u can be modeled by

$$\frac{\partial}{\partial x}(B\frac{\partial u}{\partial x})+C*u-f=0, 0<x<L$$

where $$x$$ being the abscissa along the central axis of the bar and $$L$$ being the total length of the bar. The variables $$B, C$$ and $$f$$ depend on the application of the differential equation. In the case of solid mechanics $$ B=E(x)*A(x)$$ the elastic modulus multiplied by the cross sectional area of the bar, and $$ f=f(x)$$ is the axial force applied to the bar, while $$ C=0 $$

Approximation Functions
In order to analyze a bar with the preceding differential equation, we must use a method of approximating the solution that allows for generalized applications. In reality, the axial displacement $$u(x)$$ is infinitely variable for $$0<x<L$$ so we choose to approximate it with a polynomial of the form

$$ \displaystyle u = \sum_{j=1}^n  u_j \psi_j(x)$$

which can be any order desired, depending on the accuracy sought for a given solution.

Weak Formulation
The weak formulation is then constructed to facilitate the introduction of a weight function that will allow a bilinear, symmetric, functional to be obtained. The purpose of this is to reduce the differential equation to a set of integrable polynomial expressions that can then be evaluated and manipulated with principles from linear algebra.

$$0= \int\limits_{x_0}^{x_1} w(x) \bigg[ \frac{\partial}{\partial x}(E(x)A(x)\frac{\partial u}{\partial x})-f \bigg]\, dx$$

This expression assures that the residual error in our approximate solution is defined to be zero, in a weighted integral sense.Integration by parts distributes the differentiation between the assumed displacement function and the weight function

$$0= \int\limits_{x_0}^{x_1} \bigg[ \frac{\partial w}{\partial x}\frac{\partial u}{\partial x}EA-wf \bigg]\, dx- \bigg[w(x)EA\frac{\partial u}{\partial x}\bigg]$$

which can then be evaluated for the functions of x, with $$w(x)=\psi_i(x)$$

$$0= \sum_{j=1}^n u_j \int\limits_{x_0}^{x_1} \bigg[ \frac{\partial \psi_i}{\partial x}\frac{\partial \psi_j}{\partial x}EA-\psi_if \bigg]\, dx- \bigg[\psi_i(x)EA\frac{\partial u}{\partial x}\bigg]~where~ i,j=1,2,\cdots,n$$

The indices i and j indicate where the function will be evaluated. For an element with two nodes, the $$u_j$$'s are the magnitudes of the nodal displacements and the $$\psi_j(x)$$ functions are evaluated at the nodes, giving an interpolation of the nodal values down the length of the bar.

Developing Interpolation Functions
In order to properly distribute the nodal displacements to the center section of the bar element, the interpolation functions must be constructed such that the value of the nodal displacement at a given node does not contribute to the others, and furthermore, the interpolation functions must not alter the magnitudes of the nodal displacements when evaluated at the nodes. This implies that the interpolation functions should sum to unity everywhere throughout the bar and all but one should reduce to zero at the nodes.

$$\sum_{j=1}^{n}\psi_j(x)=1$$

$$\psi_j(x_i)=\begin{cases} 0, & \mbox{if }i \neq j \\ 1, & \mbox{if }i=j \end{cases}$$

From these properties, the following interpolation functions are derived:

$$\displaystyle \psi_1=1-x/L$$

$$\displaystyle \psi_2=x/L$$

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