Elasticity/Rayleigh-Ritz method

The Rayleigh-Ritz method
The potential energy functional has the form

\Pi[\mathbf{u}] = \frac{1}{2}\int_{\mathcal B} \boldsymbol{\nabla}\mathbf{u}:(\text{C}:\boldsymbol{\nabla}\mathbf{u})~dV - \int_{\mathcal B} \mathbf{f}\bullet\mathbf{u}~dV - \int_{\partial{\mathcal B}} \widehat{\mathbf{t}}\bullet\mathbf{u}~dV $$

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

\mathbf{u}_{\text{approx}} = \mathbf{w}_0 + \sum_{n=1}^{N} a_n \mathbf{w}_n $$ where $$\mathbf{w}_n$$ are functions that are chosen so that they vanish on $$\partial{\mathcal B}^{u}$$ and $$\mathbf{w}_0$$ is a function that approximates the boundary displacements on $$\partial{\mathcal B}^{u}$$. The constants $$a_n$$ are then chosen so that they make $$\Pi[\mathbf{u}_{\text{approx}}]$$ a minimum.

Suppose,

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

\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{align}

A & = \int_{\mathcal B} U(\mathbf{w}_0)~dV - \int_{\mathcal B} \mathbf{f}\bullet\mathbf{w}_0~dV - \int_{\partial{\mathcal B}^{t}} \widehat{\mathbf{t}}\bullet\mathbf{w}_0~dA \\ B_{mn} & = \int_{\mathcal B} \boldsymbol{\nabla}{\mathbf{w}_m}:(\text{C}:\boldsymbol{\nabla}{\mathbf{w}_n})~dV\\ D_n & = \int_{\mathcal B} \boldsymbol{\nabla}{\mathbf{w}_0}:(\text{C}:\boldsymbol{\nabla}{\mathbf{w}_n})~dV - \int_{\mathcal B} \mathbf{f}\bullet\mathbf{w}_n~dV - \int_{\partial{\mathcal B}^{t}} \widehat{\mathbf{t}}\bullet\mathbf{w}_n~dA \end{align}$$ To minimize $$\Pi_{\text{approx}}$$ we use the relations

\frac{\partial \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.