Nonlinear finite elements/Steady state heat conduction

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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.

Figure 1. The membrane problem.

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.