Nonlinear finite elements/Steady state heat conduction

Steady state heat conduction
If the problem does not depend on time and the material is isotropic, we get the boundary value problem for steady state heat conduction. $$ { \begin{align} & &\mathsf{ The~ boundary~ value~ problem ~for~ steady~ heat ~conduction}\\ & & \\ & \text{PDE:} &- \frac{1}{C_v~\rho} \boldsymbol{\nabla} \bullet (\boldsymbol{\kappa}\bullet\boldsymbol{\nabla T)} = Q \text{in}\Omega\quad\\ & \text{BCs:} & T = \overline{T}(\mathbf{x})\text{on}\Gamma_T \text{and} \frac{\partial T}{\partial n} = g(\mathbf{x})\text{on}\Gamma_q\quad\\ \end{align} } $$

Poisson's equation
If the material is homogeneous the density, heat capacity, and the thermal conductivity are constant. Define the  thermal diffusivity as

k := \frac{\kappa}{C_v~\rho} $$ Then, the boundary value problem becomes $$ { \begin{align} & &\mathsf{ Poisson's~ equation}\\ & & \\ & \text{PDE:} &- k \nabla^2 T = Q \text{in}\Omega\quad\\ & \text{BCs:} & T = \overline{T}(\mathbf{x})\text{on}\Gamma_T \text{and} \frac{\partial T}{\partial n} = g(\mathbf{x})\text{on}\Gamma_q\quad\\ \end{align} } $$ where $$\nabla^2 T$$ is the  Laplacian

\nabla^2 T := \boldsymbol{\nabla} \bullet \boldsymbol{\nabla T} $$

Laplace's equation
Finally, if there is no internal source of heat, the value of $$Q$$ is zero, and we get Laplace's equation. $$ { \begin{align} & &\mathsf{ Laplace's~ equation}\\ & & \\ & \text{PDE:} &\nabla^2 T = 0 \text{in}\Omega\quad\\ & \text{BCs:} & T = \overline{T}(\mathbf{x})\text{on}\Gamma_T \text{and} \frac{\partial T}{\partial n} = g(\mathbf{x})\text{on}\Gamma_q\quad\\ \end{align} } $$

The Analogous Membrane Problem
The thin elastic membrane problem is another similar problem. See Figure 1 for the geometry of the membrane.

The membrane is thin and elastic. It is initially planar and occupies the 2D domain $$\Omega$$. It is fixed along part of its boundary $$\Gamma_u$$. A transverse force $$\mathbf{f}$$ per unit area is applied. The final shape at equilibrium is nonplanar. The final displacement of a point $$\mathbf{x}$$ on the membrane is $$\mathbf{u}(\mathbf{x})$$. There is no dependence on time.

The goal is to find the displacement $$\mathbf{u}(\mathbf{x})$$ at equilibrium.

It turns out that the equations for this problem are the same as those for the heat conduction problem - with the following changes:
 * The time derivatives vanish.
 * The balance of energy is replaced by the balance of forces.
 * The constitutive equation is replaced by a relation that states that the vertical force depends on the displacement gradient ($$\boldsymbol{\nabla} \mathbf{u}$$).

If the membrane if inhomogeneous, the boundary value problem is: $$ { \begin{align} & &\mathsf{ The~boundary~ value~ problem~ for~ membrane~ deformation}\\ & & \\ & \text{PDE:} &- \boldsymbol{\nabla} \bullet (E~\boldsymbol{\nabla \mathbf{u})} = Q \text{in}\Omega\quad\\ & \text{BCs:} & \mathbf{u} = \bar{\mathbf{u}}(\mathbf{x})\text{on}\Gamma_u \text{and} \frac{\partial u}{\partial n} = g(\mathbf{x})\text{on}\Gamma_t\quad\\ \end{align} } $$

For a homogeneous membrane, we get $$ { \begin{align} & &\mathsf{ Poisson's~ Equation}\\ & & \\ & \text{PDE:} &- E \nabla^2 \mathbf{u} = Q \text{in}\Omega\quad\\ & \text{BCs:} & \mathbf{u} = \bar{\mathbf{u}}(\mathbf{x})\text{on}\Gamma_u \text{and} \frac{\partial u}{\partial n} = g(\mathbf{x})\text{on}\Gamma_t\quad\\ \end{align} } $$

Note that the membrane problem can be formulated in terms of a problem of minimization of potential energy.