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.

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

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

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

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.

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

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 $\mathrm{\Omega }$. It is fixed along part of its boundary ${\mathrm{\Gamma }}_u$. 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.

File:MembraneProb.png 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:

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

For a homogeneous membrane, we get

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

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