Dear Howe,

Regarding 1., the main question is whether you added the 'mapping' variable as the first arguments to all FEValues and FEFaceValues constructors, to calls to VectorTools::interpolate* and VectorTools::integrate_difference calls and the like, i.e., all routines that internally construct an FEValues or FEFaceValues object. There is defaults for many of those data structures using linear mappings, MappingQ1, but you don't want to use them but rather the high order description.


Regarding 2, I guess you are using a curved boundary description as explained in step-1? You need to tell the triangulation to use a curved description.

Regarding 3, your mesh looks quite good. You would probably want to use a volume manifold on the whole circular part of the domain, though. And ideally a TransfiniteInterpolationManifold on the regions where you go from the circle to the straight ends, but the latter is not yet available in any of the releases yet, only the github code, and it is not critical and you should get better results up to degree 3 at least.

Best,
Martin


On 09.08.2017 10:06, Howe wrote:
Dear Martin,

Thanks for your rapid response.
1.
The MappingQ is set to be the same as the order of velocity, as is shown in the following code snippet:

        /template <int dim>
        //NS<dim>::NS (ParameterHandler &prm)
        //:
        //   parameters (&prm),
        //   degree (prm.get_integer("pressure degree")),
        //   fe( FE_Q<dim>(QGaussLobatto<1>(degree+2)), dim,
        // FE_Q<dim>(QGaussLobatto<1>(degree+1)), 1),
        //   fe_scalar (FE_Q<dim>(QGaussLobatto<1>(degree+2))),
        //   dof_handler (triangulation),
        //   dof_handler_scalar (triangulation),
        //mapping (degree+2),
        //   computing_timer (std::cout,
        //           TimerOutput::summary,
        //           TimerOutput::wall_times)/


I am not quite sure whether the computation of the lift/drag in my code is right, and my implementation is almost the same as the one in this post: https://groups.google.com/d/msg/dealii/rS6soTb69ig/C4QchAyEGwAJ
The only change is the first line :

        /QGauss<dim-1> face_quadrature_formula(degree+2);/


2. I am using deal.II 8.4.0 now, and I think i am not using manifold description in my code

3. The mesh is shown as follows:

<https://lh3.googleusercontent.com/-Vs0s6So28js/WYrCbyok9mI/AAAAAAAABSc/rfs-1D1TA8c71IQZpOEyOoU3C1idNr1DQCLcBGAs/s1600/Image.png>


Best,

Howe



在 2017年8月9日星期三 UTC+8下午3:25:20,Martin Kronbichler写道:

    Dear Howe,

    How did you run your simulation? From your picture, it appears
    that a higher order method is worse at higher degrees than a lower
    order method, which does not match with my experience. If that
    were the case, nobody would use high orders. However, you need to
    bring many pieces in place to really get to the benefit of the
    high order method for somewhat more complicated examples such as
    the flow around a cylinder. Here is a list of things to look at:

    - Do you use a high-order polynomial mapping MappingQ of the same
    or higher degree as the interpolation space? Do you use this
    mapping in all routines that evaluate quantities, such as the
    usual assembly, the computation of the lift/drag, and so on?
    - Do you use a manifold description that extends into the domain?
    (Look into TransfiniteInterpolationManifold.) Without, you will
    not get more than third order convergence.
    - Do you have a good mesh around the area of interest? Flows
    around cylinders tend to be really really sensitive to the mesh
    quality around the cylinder.

    For the Navier-Stokes equations around the cylinder, if everything
    is done right one gets significantly improved results in terms of
    accuracy over the number of degrees of freedom up to degree (6,5)
    (velocity,pressure). Beyond that picture is less clear. At least
    with the meshes that we tried in our group it was not worth to go
    beyond. You can have a look a our results in section 5.4 and Figs.
    9 and 10 of this preprint:
    https://arxiv.org/pdf/1706.09252.pdf
    <https://arxiv.org/pdf/1706.09252.pdf>

    Best,
    Martin


    On 09.08.2017 09:01, Howe wrote:
    Dear Jaekwang

    Have you solved this problem? If yes, Could pls share your
    solution with us?
    I am simulating a steady state flow over a cylinder, and the
    drag/lift coefficient shows an unexpected trend of change as i
    increase the discretization order and refine the mesh.

    
<https://lh3.googleusercontent.com/-Gz5932Zt6e0/WYqvbO3P-4I/AAAAAAAABSM/6hBHn1FO5W0P7X3SPfQ4iyRaldDYzOt3QCLcBGAs/s1600/Image.png>

    As is shown in the figure, the Cd increased as the cells
    increased for all the discretization orders, however, for a fixed
    cells, the Cd decreased as the discretization order increased.

    In my opinion, to increase the order and refine the mesh should
    both make the approximation more close to the exact solution,
    thus should have the same trend of change.


    在 2016年9月18日星期日 UTC+8下午11:57:16,Jaekwang Kim写道:


        Hello, I am a starter of dealii and am learning a lot these
        days with the help of video lectures and tutorial examples.

        I modified step-22 code (stokes flow code) into my own
        problem, the flow around sphere.

        and I intend to evaluate the drag force (which is
        analytically given by stokes equation)

        My code reached quite close to the value since the absolute
        error  : abs(drag_calculated-drag_exact)/drag_exact is around
        10^(-3)

        However, I expected that if I input higher 'degree' I will
        receive more accurate result, but it didn't

        Obviously Q2 is better than Q1. and Q3 is better than Q2. But
        Q4 or Q4 is not better than Q2 or Q3?

        Is there any reason on this?

        (To be specific, if i say degree 2 , that mean I use (2+1)
        for velocity, (2) for pressure, and (2+2) for Gauss integral....


        Thank you

        Jaekwang Kim

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