When submitted to periodic sinusoidal heat loads at a given frequency, the temperature response of a body can, in some cases, be assumed periodic, sinusoidal, and of same frequency around an equilibrium temperature. If the temperature changes are small enough or if the thermal properties are constant, this time-dependent periodic problem may be replaced by an equivalent linear steady-state problem in the frequency domain, which is far less computationally expensive. The Heat Transfer interfaces support such frequency domain studies via the Thermal Perturbation, Frequency Domain study sequence.

where T0 is the equilibrium temperature that verifies the steady-state heat transfer equation and may come from the solution of a previous study, T′ is the complex amplitude of the harmonic perturbation around T0, and ω is the angular frequency, related to the ordinary frequency, f, according to

From the temperature decomposition in Equation 4-29, and according to the heat transfer equation in Equation 4-14, heat transfer by conduction in solids is then governed by:

where Q′ejωt is the harmonic perturbation in domain around an average heat source, Q. Removing the terms of the steady-state heat transfer equation satisfied by T0, and simplifying by ejωt, this reduces to:

which is the governing equation for T′. For constant material properties, the linearized form reads:

Here, ρ0, Cp, 0, and k0 denote the density, heat capacity at constant pressure, and thermal conductivity, evaluated at T0, that is: ρ(T0), Cp(T0), and k(T0), respectively.

When the linearized heat transfer equation, such as Equation 4-30 or Equation 4-31, can still describe the model accurately, the problem becomes steady-state in the frequency domain, therefore computationally less expensive than a time-dependent simulation. An automatic linearization process is performed by COMSOL Multiphysics so that no additional action is needed from the user to get these equations, even in the presence of temperature-dependent coefficients, in domains and boundaries. Only the expressions of the material properties and other parameters, as functions of the temperature, are required as for usual nonlinear modeling.