Introduction to finite elements/Steady state heat conduction

 Subject classification: this is a physics resource.

If the problem does not depend on time and the material is isotropic, we get the boundary value problem for steady state heat conduction.

{\displaystyle {\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

${\displaystyle k:={\frac {\kappa }{C_{v}~\rho }}}$

Then, the boundary value problem becomes

{\displaystyle {\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 ${\displaystyle \nabla ^{2}T}$ is the Laplacian

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

Laplace's equation

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

{\displaystyle {\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 ${\displaystyle \Omega }$. It is fixed along part of its boundary ${\displaystyle \Gamma _{u}}$. A transverse force ${\displaystyle \mathbf {f} }$ per unit area is applied. The final shape at equilibrium is nonplanar. The final displacement of a point ${\displaystyle \mathbf {x} }$ on the membrane is ${\displaystyle \mathbf {u} (\mathbf {x} )}$. There is no dependence on time.

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

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

{\displaystyle {\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

{\displaystyle {\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.