Elasticity/Rayleigh-Ritz method

The potential energy functional has the form

${\displaystyle \Pi [\mathbf {u} ]={\frac {1}{2}}\int _{\mathcal {B}}{\boldsymbol {\nabla }}\mathbf {u}$ :({\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 ${\displaystyle \Pi }$ over a restricted class of functions (the Rayleigh-Ritz method), by assuming that

${\displaystyle \mathbf {u} _{\text{approx}}=\mathbf {w} _{0}+\sum _{n=1}^{N}a_{n}\mathbf {w} _{n}}$

where ${\displaystyle \mathbf {w} _{n}}$ are functions that are chosen so that they vanish on ${\displaystyle \partial {\mathcal {B}}^{u}}$ and ${\displaystyle \mathbf {w} _{0}}$ is a function that approximates the boundary displacements on ${\displaystyle \partial {\mathcal {B}}^{u}}$. The constants ${\displaystyle a_{n}}$ are then chosen so that they make ${\displaystyle \Pi [\mathbf {u} _{\text{approx}}]}$ a minimum.

Suppose,

${\displaystyle \Pi [\mathbf {u} _{\text{approx}}]=\Pi _{\text{approx}}=\Pi [a_{1},a_{2},,a_{n}]}$

Then,

${\displaystyle \Pi _{\text{approx}}=A+{\frac {1}{2}}\sum _{m,n=1}^{N}B_{mn}~a_{m}~a_{n}+\sum _{n=1}^{N}D_{n}~a_{n}}$

where,

${\displaystyle {\begin{aligned}A&=\int _{\mathcal {B}}U(\mathbf {w} _{0})~dV-\int _{\mathcal {B}}\mathbf {f} \bullet \mathbf {w} _{0}~dV-\int _{\partial {\mathcal {B}}^{t}}{\widehat {\mathbf {t} }}\bullet \mathbf {w} _{0}~dA\\B_{mn}&=\int _{\mathcal {B}}{\boldsymbol {\nabla }}{\mathbf {w} _{m}}:({\text{C}}:{\boldsymbol {\nabla }}{\mathbf {w} _{n}})~dV\\D_{n}&=\int _{\mathcal {B}}{\boldsymbol {\nabla }}{\mathbf {w} _{0}}:({\text{C}}:{\boldsymbol {\nabla }}{\mathbf {w} _{n}})~dV-\int _{\mathcal {B}}\mathbf {f} \bullet \mathbf {w} _{n}~dV-\int _{\partial {\mathcal {B}}^{t}}{\widehat {\mathbf {t} }}\bullet \mathbf {w} _{n}~dA\end{aligned}}}$

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

${\displaystyle {\frac {\partial \Pi }{\partial a_{i}}}=0~~~(i=1,2,,n)}$

to get a set of ${\displaystyle N}$ equations which provide us with the values of ${\displaystyle a_{i}}$.

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.