Dear Ross,
I am using dealii to solve the curl-curl equation:
curl(mu^(-1)curl(E)) + (-omega^2*epsilon+j*omega*sigma)*E=0,
with boundary conditions: n x E = n X F ,
(Dirichlet)
or n X (curl(E)) = n X (curl(F))
(Neumann)
where F is a known exact solution.
Based on your code, I implemented my own exact solution and the assembly of
system of equations. My exact solution (including real and imaginary parts)
is as follows:
E_re = [0, sin(pi*x)/a*cos(k_z10_re*z)*exp(k_z10_im*z), 0],
E_im = [0, -sin(pi*x)/a*sin(k_z10_re*z)*exp(k_z10_im*z), 0]
And the exact solution for your case (if I take p=[0 1 0] and k=[0 0 0.1])
is given by:
E_re = [0, cos(0.1*z), 0],
E_im = [0, sin(0.1*z), 0]
The domain I want to solve this problem is a 3D hyperrectangle with width
= 19.05e-3, height = 9.524e-3, and length = 15.24e-3. I've tried to solve
the above equation with the two exact solutions. When I used the build-in
mesh generator in dealii (i.e., GridGenerator::hyper_rectangle
(triangulation, p1, p2);) to generate the mesh. The problem can be solved
correctly for both cases:
my exact solution, built-in mesh generator:
<https://lh3.googleusercontent.com/-OBpgHd_d1zc/Wjn4QmGmkYI/AAAAAAAAAIQ/jJ5M9dvfZMwveEkHfkTlyBARSuYERbEGQCLcBGAs/s1600/Picture1.png>
your exact solution, built-in mesh generator:
Cycle 0:Number of degrees of freedom: 3888
Done
cycle cells dofs L2 Error H(curl) Error
0 64 3888 9.73783895e-12 1.03298196e-11
When I use the mesh generated from Gmsh software (after conversion into
hexes), if I provide your exact solution, the problem can still be solved
correctly, as shown bellow:
Cycle 0:number of cells:244
Number of degrees of freedom: 13176
Done
cycle cells dofs L2 Error H(curl) Error
0 244 13176 1.36020912e-07 8.58355818e-05
However, when I switch to my exact solution, the results are totally wrong.
The electric field obtained has totally wrong direction (seems to be
arbitrary but the correct one should be along y-axis). The results are as
follows
<https://lh3.googleusercontent.com/-4-YDq6oGyq8/Wjn5pzuHvqI/AAAAAAAAAIg/X_p-s_5mmV8MhxyXB35InyJdzYKFU25wwCLcBGAs/s1600/Picture2.png>
<https://lh3.googleusercontent.com/-eGwOwq4oVpU/Wjn5-INq1CI/AAAAAAAAAIk/BXQVHh9_eZMxMD4x3U0pEOqFtDCOasbrgCLcBGAs/s1600/Picture3.png>
My question is like this, based on your experience of using FE_Nedelec, can
we use FE_Nedelec on a mesh imported from other mesh generators like Gmsh
instead of the one generated from dealii itself? Does dealii of the current
version has a good support for this kind of mesh? FYI, the following is the
mesh generated from Gmsh (after transformation).
<https://lh3.googleusercontent.com/-as7ovfmYh_c/WjrIv6gI89I/AAAAAAAAAI4/uZRGx-1lTjkc31XFhpRR64mwD_TUyk28ACLcBGAs/s1600/mesh.png>
Any suggestions and help is much appreciated
Thanks in advance.
Jianan Zhang
在 2014年7月15日星期二 UTC-4下午12:58:46，Ross Kynch写道：
>
> Hi all,
>
> I'm solving a complex-valued vector-wave equation (the curl-curl
> formulation of the Maxwell equations):
>
> curl(curl(E)) + kappa*E=0
>
> with BCs given by
> n x E = g on \Gamma_D (Dirichlet)
> n x curl(E) = h on \Gamma_N (Neumann)
>
> note, I've set mu =1 for this problem.
>
> the wave propagation solution is given by:
> E = p*exp(i*k*x), x, k and p in R^3
>
> with p orthogonal to k, |k| < 1, |p|=1.
>
> I've chosen k=(-0.1 -0.1, 0.2) and p=(1/sqrt(3))*[1 1 1].
>
> I've implemented this for both Neumann and Dirichet BCs using Nedelec
> elements and splitting the system into real/imaginary parts. I am seeing
> the correct rates of convergence in the L2 and H(curl) norms of the error
> for the Neumann version with both h and p refinement.
>
> However, when using Dirichlet BCs, after p=0, something seems to go wrong.
> Am I enforcing the boundary conditions correctly for the
> project_boundary_values_curl_conforming routine? - I have a function with
> dim+dim components, with the first dim(3) being the real part and the
> latter dim being the imaginary part - is this the correct way to do this?
>
> I've included an example with Dirichlet BCs, although if you uncomment the
> 666-678 in the code then the boundaries will be changed to Neumann. It
> takes the polynomial order as input, so run it with "./wave_propagation -p
> 2" for order 2 elements, etc.
>
> For comparison, here are the results when I've run the code p=0,1,2 -
> using a direct solver for now, so going above p=2 causes memory problems,
> also runtime increases due to the slow generation of the Nedelec elements
> at higher orders.
>
> Thanks,
>
> Ross
>
> Dirichlet:
> ./wave_propagation -p 0
> cycle cells dofs L2 Error H(curl) Error
> 0 8 108 1.15904018e-01 1.24176508e-01
> 1 64 600 5.77896943e-02 6.19578648e-02
> 2 512 3888 2.88743606e-02 3.09624645e-02
> ./wave_propagation -p 1
> cycle cells dofs L2 Error H(curl) Error
> 0 8 600 8.35079734e-01 2.59980668e+00
> 1 64 3888 7.93792725e-01 3.36290438e+00
> 2 512 27744 7.77603612e-01 4.42928910e+00
> ./wave_propagation -p 2
> cycle cells dofs L2 Error H(curl) Error
> 0 8 1764 8.20717491e-01 2.50234579e+00
> 1 64 12168 7.87196814e-01 3.17447088e+00
> 2 512 90000 7.74601489e-01 4.11893779e+00
>
> Neumann:
> ./wave_propagation -p 0
> 0 8 108 1.15471804e-01 1.23771671e-01
> 1 64 600 5.77352583e-02 6.19069021e-02
> 2 512 3888 2.88675425e-02 3.09560822e-02
> ./wave_propagation -p 1
> cycle cells dofs L2 Error H(curl) Error
> 0 8 600 2.58268149e-03 2.79282760e-03
> 1 64 3888 6.45549584e-04 6.98219104e-04
> 2 512 27744 1.61377755e-04 1.74553607e-04
> ./wave_propagation -p 2
> cycle cells dofs L2 Error H(curl) Error
> 0 8 1764 4.17828432e-05 4.53286783e-05
> 1 64 12168 5.22313050e-06 5.66700077e-06
> 2 512 90000 6.52897524e-07 7.08401532e-07
>
>
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