Hello!
I apologize for the long question, but I’m a bit lost so I’ll try to be as precise as I can.
I am following the implementation of a time-domain PML formulation for wave propagation, which can be found in this work. The author has developed a set of equations for the wave propagation in fluid (\Omega_F) and solid (\Omega_S) medium. It reads, for \Omega_F
(eq. 5.9)
where a_j, b and c are functions either of \alpha or \beta (from the PML formulation). The equations are valid throughout the fluid domain \Omega_F, and within the physical domain a_j, b and c equal 0 so we have the classical wave equation, as shown in the figure.
\partial^2 p / \partial t ^2 = v^2 \nabla ^2p
\partial v_j / \partial t = - (1/ \rho) \partial p/ \partial x_j
The author has also given the weak form as follows
(eq. 5.10a)
(eq. 5.10c)
where the second term of the first equation (the integral over \Gamma) concerns the fluid-solid interface (nonexistent in this scenario).\phi and \psi_j are test functions.
Now I wish to translate this to FEniCS. For sake of simplicity, I’m concerned only with \Omega_F and will use a point source in its physical domain. So I believe the second term of eq. 5.10a vanishes and F = 0. \alpha and \beta will be set depending if we’re in the PML or in the physical domain.
I have read the Mixed formulation for Poisson equation example, and, according to what I understood, I’ll have to put eqs. 5.10a and 5.10c together. But I’m confused about how to deal with the j components and with \partial \phi / \partial x_j.
It would be very helpful if somebody could give me some advice on how to approach this problem. Thanks in advance!
