Hu–Washizu principle

In continuum mechanics, and in particular in finite element analysis, the Hu–Washizu principle is a variational principle which says that the action

${\displaystyle \int _{V^{e}}\left[{\frac {1}{2}}\varepsilon ^{T}C\varepsilon -\sigma ^{T}\varepsilon +\sigma ^{T}(\nabla u)-{\bar {p}}^{T}u\right]dV-\int _{S_{\sigma }^{e}}{\bar {T}}^{T}u\ dS}$

is stationary, where ${\displaystyle C}$ is the elastic stiffness tensor. The Hu–Washizu principle is used to develop mixed finite element methods.^[1] The principle is named after Hu Haichang and Kyūichirō Washizu.

The Euler–Lagrange equations of the Hu–Washizu functional are the following equations:

${\displaystyle {\begin{cases}\nabla \cdot {\boldsymbol {\sigma }}+\mathbf {f} =\mathbf {0} &{\text{(Equilibrium)}}\\{\boldsymbol {\sigma }}={\frac {\partial W}{\partial {\boldsymbol {\varepsilon }}}}&{\text{(Constitutive law)}}\\{\boldsymbol {\varepsilon }}=\nabla ^{s}\mathbf {u} &{\text{(Compatibility)}}\end{cases}}}$

with appropriate boundary conditions

${\displaystyle {\boldsymbol {\sigma }}\cdot \mathbf {n} =\mathbf {t} \quad {\text{on}}\quad \partial \Omega _{t}}$.