28th May, 2020

Universidade de Caxias do Sul (UCS)

Question

Asked 27th May, 2020

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 trying to implement a time-domain PML formulation for wave propagation, which can be found in this work

Here, the author has developed a set of equations to deal with the wave propagation in fluid and solid medium (please refer to eq. 5.9 of the work as I can't copy-paste them here). They are valid throughout the whole domain (PML and the physical domain), and within the physical domain the equations simplify to the general wave equation.

I'm using FEniCS as the Finite Element solver, and I need to provide the weak form of the PDE I'm willing to solve. Fortunately the weak form is in the work too (please see eqs. 5.10a-d).

I'm concerned only with the wave propagation in the fluid region, so I consider only eqs. 5.10a and 5.10c. Moreover, in eq. 5.10a, I believe the term with the integral over gamma vanishes as there is no fluid-solid interface.

The problem is I need to provide the Weak Form in only one equation. I'm confused with the term d phi/dx_{j} and I'm not sure how to rewrite eqs. 5.10a and 5.10c in the form a=L.

Can someone help me with this?

Many thanks!

Well, I have some experience with FEniCS. I think my question could be summarized into "how to write a linear system AX = b or a variational form a==L or F==0 out of this system of two equations?"

I think it is possible to rewrite the equations in a coordinate free form, this can help. But still, as far as I know, I have to put them all together to solve this using FEniCS.

Is it possible?

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These equations are written in index notation.

dphi_i/dx_j is the spatial derivative of the test function for the coordinate i wrt to coordinate j.

If you are new to FEniCS, then I suggest starting with weak formulation for the problem in a single variable to understand the syntax and then go to multiple variables. Refer to the FEniCS manual and tutorials on Poisson equation and elasticity.

1 Recommendation

You are predetermining (and limiting) the outcome by choosing to use FEniCS. You see this immediately, as you must solve a "weak form of the PDE" instead of the full governing equations. Many similar problems can be found in the open literature, most of which do not use this method. More often than not, previous researchers have successfully used the FDM, especially in the fields of aeronautics, electromagnetics, and oceanography. Implementation of the boundary conditions is significantly complicated by the approach you have chosen.

1 Recommendation

Well, I have some experience with FEniCS. I think my question could be summarized into "how to write a linear system AX = b or a variational form a==L or F==0 out of this system of two equations?"

I think it is possible to rewrite the equations in a coordinate free form, this can help. But still, as far as I know, I have to put them all together to solve this using FEniCS.

Is it possible?

1 Recommendation

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- Jan 2009

The use of a perfectly matched layer (PML) model is an efficient approach toward the bounded-domain modelling of wave propagation on unbounded domains. This paper formulates a three-dimensional PML for elastic waves by building upon previous work by the author and implements it in a displacement-based finite element setting. The novel contribution...

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- Oct 2022

The original B´erenger’s perfectly matched layer (PML) was quite effective in simulating wave propagation problem in unbounded domains. But its stability is very challenging to prove. Later, some equivalent PML models were developed by B´ecache and Joly [4] and their stabilities were established. Hence studying and developing efficicent numerical m...

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