Hamed,

[...]
>
                                   
>                        4,5___________6,7
>                          |                      | 
>                          |                      |
>                          |                      | 
>                          |                      | 
>                        0,1__________ 2,3 
>
> when I just apply delta_x:
>
> 0 6: 1
>
> 0: 0.05
>
> 1 7: 1
>
> 2 6: 1
>
> 3 7: 1
>
> 4 6: 1
>
> 4: 0.05
>
> 5 7: 1
>
It seems that you are now constraining all the components, i.e. 
u_x(0,y)=u_x(L,y) and u_y(0,y)=u_y(L,y).
Is this really what you want to do? Otherwise, you should have a 
ComponentMask in your call to 
DoFTools::make_periodicity_constraints as well, i.e.

FEValuesExtractors::Scalar scalar_x(0)
DoFTools::make_periodicity_constraints(dof_handler,
                                             /*b_id*/ 0,
                                             /*b_id*/ 1,
                                             /*direction*/ 0,
                                             constraints,
                                             fe.component_mask(scalar_x));

and 

FEValuesExtractors::Scalar scalar_x(1)
DoFTools::make_periodicity_constraints(dof_handler,
                                             /*b_id*/ 2,
                                             /*b_id*/ 3,
                                             /*direction*/ 1,
                                             constraints,
                                             fe.component_mask(scalar_y));


> when I just have delta_y:
>
> 0 6: 1
>
> 1 7: 1
>
> 1: -0.1
>
> 2 6: 1
>
> 3 7: 1
>
> 3: -0.05
>
> 4 6: 1
>
> 5 7: 1
>  
>
 
>
As you can see, the inhomogenity applied on DoF 1 is ironically doubled and 
> that's why the corner element get distorted.
>
Again, I don't see an immediate reason why x- and y-direction don't behave 
similarly. Can you provide a minimal working code showing this?

Best,
Daniel

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