Dear Prof. Wolfgang Bangerth,

> In that case, it is not enough to only pin the displacements at
> one vertex of your domain. That kills the translation mode. But you still 
> need
> to kill the rotation mode as well, which is one additional degree of 
> freedom
> in 2d (for a total of 3) and 3 more degrees of freedom in 3d (for a total 
> of 6).

Your comment is very exact and helpful! 

I found two ways to constrain several dofs (as you mentioned, at least 3 
dofs in 2D) for my problem: 1. I can use the "add_line" function, and do 
not use the "apply_boundary_values" function; or 2 . I can directly modify 
"std::map<types::global_dof_index, double> boundary_values" and use the 
"apply_boundary_values" function.

*I wonder if the above two ways are the same in dealii's internal 

Furthermore, it is amazing to receive your reply! I have seen most of your 
lectures on dealii on youtube. These lectures are very very helpful!

Thanks very much.


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