Nonlinear finite elements/Axial bar approximate solution

Approximate Solution: The Galerkin Approach

To find the finite element solution, we can either start with the strong form and derive the weak form, or we can start with a weak form derived from a variational principle.

Let us assume that the approximate solution is ${𝐮}_h(𝐱)$ and plug it into the ODE. We get

$AE\frac{d^2{𝐮}_h}{d{x}^2}+a𝐱=R_h(𝐱)$

where $R_h$ is the residual. We now try to minimize the residual in a weighted average sense

$\underset{0}{\overset{L}{\int }}{R}_h(𝐱)𝐰(𝐱)dx=0$

where $𝐰(𝐱)$ is a weighting function. Notice that this equation is similar to equation (5) (see 'Weak form: integral equation') with $𝐰$ in place of the variation $\delta 𝐮$. For the two equations to be equivalent, the weighting function must also be such that $𝐰(0)=0$.

Therefore the approximate weak form can be written as

$\underset{0}{\overset{L}{\int }}AE\frac{d{𝐮}_h}{dx}\frac{d𝐰}{dx}dx=\underset{0}{\overset{L}{\int }}𝐪𝐰dx+{𝑹𝐰|}_{x=L}.$

In Galerkin's method we assume that the approximate solution can be expressed as

${𝐮}_h(𝐱)=a_1{\phi }_1(𝐱)+a_2{\phi }_2(𝐱)+\dots +a_n{\phi }_n(𝐱)=\sum _{i=1}^n{a}_i{\phi }_i(𝐱).$

In the Bubnov-Galerkin method, the weighting function is chosen to be of the same form as the approximate solution (but with arbitrary coefficients),

$𝐰(𝐱)=b_1{\phi }_1(𝐱)+b_2{\phi }_2(𝐱)+\dots +b_n{\phi }_n(𝐱)=\sum _{j=1}^n{b}_j{\phi }_j(𝐱).$

If we plug the approximate solution and the weighting functions into the approximate weak form, we get

$\underset{0}{\overset{L}{\int }}AE\left(\sum _{i=1}^n{a}_i\frac{d{\phi }_i}{dx}\right)\left(\sum _{j=1}^n{b}_j\frac{d{\phi }_j}{dx}\right)dx=\underset{0}{\overset{L}{\int }}𝐪\left(\sum _{j=1}^n{b}_j{\phi }_j\right)dx+{𝑹\left(\sum _{j=1}^n{b}_j{\phi }_j\right)|}_{x=L}.$

This equation can be rewritten as

$\sum _{j=1}^n{b}_j\left[\underset{0}{\overset{L}{\int }}AE\left(\sum _{i=1}^n{a}_i\frac{d{\phi }_i}{dx}\frac{d{\phi }_j}{dx}\right)dx\right]=\sum _{j=1}^n{b}_j[\underset{0}{\overset{L}{\int }}𝐪{\phi }_{j}dx+{\left(𝑹{\phi }_j\right)|}_{x=L}].$

From the above, since $b_j$ is arbitrary, we have

$\underset{0}{\overset{L}{\int }}AE\left(\sum _{i=1}^n{a}_i\frac{d{\phi }_i}{dx}\frac{d{\phi }_j}{dx}\right)dx=\underset{0}{\overset{L}{\int }}𝐪{\phi }_{j}dx+{𝑹{\phi }_j|}_{x=L},j=1\dots n.$

After reorganizing, we get

$\sum _{i=1}^n\left[\underset{0}{\overset{L}{\int }}\frac{d{\phi }_j}{dx}AE\frac{d{\phi }_i}{dx}dx\right]a_i=\underset{0}{\overset{L}{\int }}{\phi }_j𝐪dx+{{\phi }_j𝑹|}_{x=L},j=1\dots n$

which is a system of $n$ equations that can be solved for the unknown coefficients $a_i$. Once we know the $a_i$s, we can use them to compute approximate solution. The above equation can be written in matrix form as

$𝐊𝐚=𝐟\leftrightarrow K_{ji}{a}_i=f_j$

where

$𝐊=\underset{0}{\overset{L}{\int }}{𝐁}^T𝐃𝐁dx\leftrightarrow K_{ji}=\underset{0}{\overset{L}{\int }}\frac{d{\phi }_j}{dx}AE\frac{d{\phi }_i}{dx}dx$

and

$f_j=\underset{0}{\overset{L}{\int }}{\phi }_j𝐪dx+{{\phi }_j𝑹|}_{x=L}.$

The problem with the general form of the Galerkin method is that the functions ${\phi }_i$ are difficult to determine for complex domains.