Nonlinear finite elements/Steady state heat conduction

Subject classification: this is a physics resource.

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.

$\begin{array}{ccc}& & 𝖳𝗁𝖾𝖻𝗈𝗎𝗇𝖽𝖺𝗋𝗒𝗏𝖺𝗅𝗎𝖾𝗉𝗋𝗈𝖻𝗅𝖾𝗆𝖿𝗈𝗋𝗌𝗍𝖾𝖺𝖽𝗒𝗁𝖾𝖺𝗍𝖼𝗈𝗇𝖽𝗎𝖼𝗍𝗂𝗈𝗇\\ & & \\ & \text{PDE:}& -\frac{1}{C_v\rho }\mathrm{\nabla }\bullet (𝜿\bullet \mathrm{\nabla }𝑻\mathit{)}=Q\text{in}\Omega \\ & \text{BCs:}& T=\bar{T}(𝐱)\text{on}{\Gamma }_T\text{and}\frac{\partial T}{\partial n}=g(𝐱)\text{on}{\Gamma }_q\\ \end{array}$

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

$\begin{array}{ccc}& & 𝖯𝗈𝗂𝗌𝗌𝗈{𝗇}^{\prime }𝗌𝖾𝗊𝗎𝖺𝗍𝗂𝗈𝗇\\ & & \\ & \text{PDE:}& -k{\mathrm{\nabla }}^{2}T=Q\text{in}\Omega \\ & \text{BCs:}& T=\bar{T}(𝐱)\text{on}{\Gamma }_T\text{and}\frac{\partial T}{\partial n}=g(𝐱)\text{on}{\Gamma }_q\\ \end{array}$

where ${\mathrm{\nabla }}^{2}T$ is the Laplacian

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

Laplace's equation

Finally, if there is no internal source of heat, the value of $Q$ is zero, and we get Laplace's equation.

$\begin{array}{ccc}& & 𝖫𝖺𝗉𝗅𝖺𝖼{𝖾}^{\prime }𝗌𝖾𝗊𝗎𝖺𝗍𝗂𝗈𝗇\\ & & \\ & \text{PDE:}& {\mathrm{\nabla }}^{2}T=0\text{in}\Omega \\ & \text{BCs:}& T=\bar{T}(𝐱)\text{on}{\Gamma }_T\text{and}\frac{\partial T}{\partial n}=g(𝐱)\text{on}{\Gamma }_q\\ \end{array}$

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 $𝐟$ per unit area is applied. The final shape at equilibrium is nonplanar. The final displacement of a point $𝐱$ on the membrane is $𝐮(𝐱)$. There is no dependence on time.

The goal is to find the displacement $𝐮(𝐱)$ 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 ($\mathrm{\nabla }𝐮$).

If the membrane if inhomogeneous, the boundary value problem is:

$\begin{array}{ccc}& & 𝖳𝗁𝖾𝖻𝗈𝗎𝗇𝖽𝖺𝗋𝗒𝗏𝖺𝗅𝗎𝖾𝗉𝗋𝗈𝖻𝗅𝖾𝗆𝖿𝗈𝗋𝗆𝖾𝗆𝖻𝗋𝖺𝗇𝖾𝖽𝖾𝖿𝗈𝗋𝗆𝖺𝗍𝗂𝗈𝗇\\ & & \\ & \text{PDE:}& -\mathrm{\nabla }\bullet (E\mathrm{\nabla }𝐮\mathit{)}=Q\text{in}\Omega \\ & \text{BCs:}& 𝐮=\overline{𝐮}(𝐱)\text{on}{\Gamma }_u\text{and}\frac{\partial u}{\partial n}=g(𝐱)\text{on}{\Gamma }_t\\ \end{array}$

For a homogeneous membrane, we get

$\begin{array}{ccc}& & 𝖯𝗈𝗂𝗌𝗌𝗈{𝗇}^{\prime }𝗌𝖤𝗊𝗎𝖺𝗍𝗂𝗈𝗇\\ & & \\ & \text{PDE:}& -E{\mathrm{\nabla }}^2𝐮=Q\text{in}\Omega \\ & \text{BCs:}& 𝐮=\overline{𝐮}(𝐱)\text{on}{\Gamma }_u\text{and}\frac{\partial u}{\partial n}=g(𝐱)\text{on}{\Gamma }_t\\ \end{array}$

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

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