Heat Transfer¶
Strong Form¶
The heat transfer module solves the heat equation
subject to the boundary conditions
where
Weak Form¶
We multiply this strong form of the PDE by an arbitrary function and integrate by parts to obtain the weak form
where
Discretization¶
After discretizing by the standard continuous Galerkin finite element method, i.e.
where \(\phi_i\) are nodal shape functions and \(u_i\) are degrees of freedom, we obtain the discrete equations in residual form
where
This system can then be solved using the selected nonlinear and ordinary differential equation solution methods. For example, if we use the backward Euler method we obtain
Given a known \(\mathbf{u}_n\), this is solved at each timestep \(\Delta t\) for \(\mathbf{u}_{n+1}\) using a nonlinear solver, typically Newton's method. To accomplish this, the above equation is linearized which yields
Note the above equation assumes the thermal capacitance and conductance are independent of temperature. The computed Newton iterations \(\mathbf{u}_{n+1}^{i+1} = \mathbf{u}_{n+1}^{i} + \Delta \mathbf{u}_{n+1}^{i+1}\) continue until the solution is converged.