Discussion Forum

Posted: 7 years ago Mar 19, 2018, 7:20 a.m. EDT

Dear Jan, The \nabla\cdot E equation is not explicitly included when solving the Maxwell's equation. However, it is done so implicitly, as \nabla\cdot(\nabla\times X) =0 for any vector quantity X. So, in your case, assuming harmonic time dependence and that your relative permability, relative permittivity, and the parameters g and a are homogeneous in space, applying the divergence operator to the second equation will result in \nabla\cdot E = 0, as the first term is the curl of the magnetic flux density and in the right hand side the magnetic flux density will be proportional to the curl of the electric field, from Faraday's law. You second equation seems to be a mixture of a transient equation and a time harmonic equation (the j\omega-factor in the right hand side). I think you might need to go back to the pure time-dependent equation, as your starting point. Once your are sure of your equations to solve, you can either solve the equations using the Electromagnetic Waves, Transient interface or using a number of Electromagnetic Waves, Frequency Domain interfaces from the RF Module. If you decide to use the transient interface, you treat the right hand side of your second equation as a external current density, call it J = (Jx, Jy, Jz). Then you can add an additional weak contribution for your domain and set the weak expression to \texrm{Jx*test(Ax)+Jy*test(Ay)+Jz*test(Az)}. The second_harmonic_generation model in the Application Libraries for both the RF and the Wave Optics Module demonstrates this approach. There the coupling is introduced using a Remanent electric displacement field in the Wave Equation, Electric node. If you decide to solve the problem using multiple time-harmonic Electromagnetic Waves, Frequency domain interfaces, you have to derive the equations for each harmonic component and the coupling terms between the different harmonic components. The general procdedure here is that you define your real fields as a summation over the different harmonics E=\sum_{k=-\infty}^{\infty}E_k\exp(ik\omega t). Here I have used the Comsol convention for time harmonic depependenc (\exp(i\omega t)). Furthermore, since the field is real, we also get that E_{-k} = E_k^*. Do the same also for B and a, although I guess that a is just a sine or cosine function. Once you have your expansions for E, B, and a, you insert them into your two curl equations. Then you gather all terms multiplying each exponential harmonic time dependence, \exp(ik\omega t). Each such factor will result in an equation to solve. The most interesting for you will be to derive the coupling terms from your second equation. Those terms you add to External current density nodes for each Electromagnetic Waves, Frequency domain interface. Those External current density terms will result in coupling to the nearest higher and lower harmonics. You specify the frequencies for the different harmonics in the Equation setting for each of the physics interfaces. This approach is demonstrated in the Wave Optics Module's Application Library model second_harmonic_generation_frequency_domain. In your case you must detemine how many harmonics you need to include. Furthermore, this approach doesnt't work well if you need to include a component for the DC field, as the RF Module's Electromagnetic waves, frequency domain interface is not applicable for DC fields, where the elecric and magnetic fields get uncoupled. Good luck with your interesting modeling project.