Elasticity/Rayleigh-Ritz method

The Rayleigh-Ritz method

The potential energy functional has the form

$\mathrm{\Pi }[𝐮]=\frac{1}{2}\underset{ℬ}{\int }\mathrm{\nabla }𝐮:(\text{C}:\mathrm{\nabla }𝐮)dV-\underset{ℬ}{\int }𝐟\bullet 𝐮dV-\underset{\partial ℬ}{\int }\widehat{𝐭}\bullet 𝐮dV$

The standard method of finding an approximate solution to the mixed boundary value problem is to minimize $\mathrm{\Pi }$ over a restricted class of functions (the Rayleigh-Ritz method), by assuming that

${𝐮}_{\text{approx}}={𝐰}_0+\sum _{n=1}^N{a}_n{𝐰}_n$

where ${𝐰}_n$ are functions that are chosen so that they vanish on $\partial {ℬ}^u$ and ${𝐰}_0$ is a function that approximates the boundary displacements on $\partial {ℬ}^u$. The constants $a_n$ are then chosen so that they make $\mathrm{\Pi }[{𝐮}_{\text{approx}}]$ a minimum.

Suppose,

$\mathrm{\Pi }[{𝐮}_{\text{approx}}]={\mathrm{\Pi }}_{\text{approx}}=\mathrm{\Pi }[a_1,a_2,,a_n]$

Then,

${\mathrm{\Pi }}_{\text{approx}}=A+\frac{1}{2}\sum _{m,n=1}^N{B}_{mn}{a}_m{a}_n+\sum _{n=1}^N{D}_n{a}_n$

where,

$\begin{array}{cc}A& =\underset{ℬ}{\int }U({𝐰}_0)dV-\underset{ℬ}{\int }𝐟\bullet {𝐰}_{0}dV-\underset{\partial {ℬ}^t}{\int }\widehat{𝐭}\bullet {𝐰}_{0}dA\\ B_{mn}& =\underset{ℬ}{\int }\mathrm{\nabla }{𝐰}_m:(\text{C}:\mathrm{\nabla }{𝐰}_n)dV\\ D_n& =\underset{ℬ}{\int }\mathrm{\nabla }{𝐰}_0:(\text{C}:\mathrm{\nabla }{𝐰}_n)dV-\underset{ℬ}{\int }𝐟\bullet {𝐰}_{n}dV-\underset{\partial {ℬ}^t}{\int }\widehat{𝐭}\bullet {𝐰}_{n}dA\end{array}$

To minimize ${\mathrm{\Pi }}_{\text{approx}}$ we use the relations

$\frac{\partial \mathrm{\Pi }}{\partial a_i}=0(i=1,2,,n)$

to get a set of $N$ equations which provide us with the values of $a_i$.

This is the approach taken for the displacement-based finite element method. If, instead, we choose to start with the complementary energy functional, we arrive at the stress-based finite element method.

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