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.

${\begin{aligned}&&{\mathsf {The~boundary~value~problem~for~steady~heat~conduction}}\\&&\\&{\text{PDE:}}~~~&~~~-{\frac {1}{C_{v}~\rho }}{\boldsymbol {\nabla }}\bullet ({\boldsymbol {\kappa }}\bullet {\boldsymbol {\nabla T)}}=Q~~{\text{in}}~~\Omega \quad \\&{\text{BCs:}}~~~&~~~T={\overline {T}}(\mathbf {x} )~~{\text{on}}~~\Gamma _{T}~~{\text{and}}~~{\frac {\partial T}{\partial n}}=g(\mathbf {x} )~~{\text{on}}~~\Gamma _{q}\quad \\\end{aligned}}$

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

${\begin{aligned}&&{\mathsf {Poisson's~equation}}\\&&\\&{\text{PDE:}}~~~&~~~-k\nabla ^{2}T=Q~~{\text{in}}~~\Omega \quad \\&{\text{BCs:}}~~~&~~~T={\overline {T}}(\mathbf {x} )~~{\text{on}}~~\Gamma _{T}~~{\text{and}}~~{\frac {\partial T}{\partial n}}=g(\mathbf {x} )~~{\text{on}}~~\Gamma _{q}\quad \\\end{aligned}}$

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.

${\begin{aligned}&&{\mathsf {Laplace's~equation}}\\&&\\&{\text{PDE:}}~~~&~~~\nabla ^{2}T=0~~{\text{in}}~~\Omega \quad \\&{\text{BCs:}}~~~&~~~T={\overline {T}}(\mathbf {x} )~~{\text{on}}~~\Gamma _{T}~~{\text{and}}~~{\frac {\partial T}{\partial n}}=g(\mathbf {x} )~~{\text{on}}~~\Gamma _{q}\quad \\\end{aligned}}$

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:

${\begin{aligned}&&{\mathsf {The~boundary~value~problem~for~membrane~deformation}}\\&&\\&{\text{PDE:}}~~~&~~~-{\boldsymbol {\nabla }}\bullet (E~{\boldsymbol {\nabla \mathbf {u} )}}=Q~~{\text{in}}~~\Omega \quad \\&{\text{BCs:}}~~~&~~~\mathbf {u} ={\bar {\mathbf {u} }}(\mathbf {x} )~~{\text{on}}~~\Gamma _{u}~~{\text{and}}~~{\frac {\partial u}{\partial n}}=g(\mathbf {x} )~~{\text{on}}~~\Gamma _{t}\quad \\\end{aligned}}$

For a homogeneous membrane, we get

${\begin{aligned}&&{\mathsf {Poisson's~Equation}}\\&&\\&{\text{PDE:}}~~~&~~~-E\nabla ^{2}\mathbf {u} =Q~~{\text{in}}~~\Omega \quad \\&{\text{BCs:}}~~~&~~~\mathbf {u} ={\bar {\mathbf {u} }}(\mathbf {x} )~~{\text{on}}~~\Gamma _{u}~~{\text{and}}~~{\frac {\partial u}{\partial n}}=g(\mathbf {x} )~~{\text{on}}~~\Gamma _{t}\quad \\\end{aligned}}$

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

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Finite elements

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