Dear Martin, I would like to start by thanking you for all of your help. Here is a link to my fork: https://github.com/AlexanderCicchino/dealii It is not fully working yet, I need your help for the "evaluate" as you mentioned earlier. I have been trying multiple different ways but cannot seem to get it to work properly. First, I have a test setup in: tests/mappings/mapping_q_generic_GCL_curvilinear.cc where currently I am outputting everything, this will change in the future and I will later on add a more complicated curvilinear mesh to test. But, importantly, the current implementation fails the test. Also, I made changes to source/fe/mapping_q_generic.cc as you suggested in the function mentioned above. The last part, which is including the evaluate is in:
https://github.com/AlexanderCicchino/dealii/blob/master/source/fe/mapping_q_generic.cc line 1622. We have X_l*grad(X_m) evaluated at each quadrature point, now we need to somehow have the gradient of that evaluated at the quadrature points. I assumed after line 1622 that that gradient is written in "grad_Xl_grad_Xm" which is gradient( X_l * gradient(X_m) ) then I loop cyclically through it for the conservative curl form. Please let me know how you suggest I should proceed/setup the evaluate call at line 1624. Also, I noticed that the conservative curl form from Kopriva is not well posed for 2D. In the past, for 2D we would extend the grid by unit 1 in the z direction to properly evaluate the metric terms since, for example we need: d/(d \zeta) ( z* dy / (d\eta) ). Any suggestions on how to implement the 2D version in mapping Q generic? Thank you, Alex On Wednesday, June 24, 2020 at 9:40:55 AM UTC-4, Martin Kronbichler wrote: > > Dear Alex, > > Great! I would suggest to start by simply adding new code to the > maybe_update_q_points_Jacobians_... function with the option to turn it off > or on. Depending on how the final implementation will look like we might > want to move that to a separate place, but I think it will be less > repetitive if we use a single place. > > Best, > Martin > On 22.06.20 19:59, Alexander Cicchino wrote: > > Dear Martin, > > Thank you very much! I have been working on making the test case not > depend on our in house flowsolver's functions. > I think that implementing Eq. 36 the "conservative curl" form would be > sufficient. > Yes this procedure sounds perfect to me, and I agree with the dimension of > the object described. I have been going through the source code that you > sent to familiarize myself with the objects. Should I be adding to the > function maybe_update_q_points_Jacobians_and_grads_tensor or should I > create a new function for it? > > Thank you, > Alex > > On Friday, June 19, 2020 at 5:09:14 AM UTC-4, Martin Kronbichler wrote: >> >> Dear Alex, >> >> Great! The first thing we need to know is the equation. I had a quick >> look in the paper by Kopriva and I think we want to use either equation >> (36) or (37), depending on whether we consider the conservative or >> invariant curl form, respectively. In either case, it appears that we need >> to do this in a two-step procedure. The first step is to compute X_l and >> \nabla_\xi X_m, which in deal.II speak are the "q_points" and "Jacobians". >> The implementation in mapping_q_generic.cc is a bit involved because we >> have a slow algorithm (working for arbitrary quadrature rules) and a fast >> one for tensor product quadrature rules. Let us consider the fast one >> because I think we have most ingredients available, whereas we would need >> to fill additional fields for the slow algorithm. The source code for those >> parts is here: >> >> >> https://github.com/dealii/dealii/blob/9e05a87db802ecd073bf7567d77f3491170d84b4/source/fe/mapping_q_generic.cc#L1463-L1592 >> >> I skipped the part on the Hessians (second derivative of transformation) >> because we won't need them. The important parts here are the extractions of >> the positions in line 1554 and the extraction of the gradients >> (contravariant transformations) in line 1575. Those two parts will be the >> starting point for the second phase we need to do in addition: According to >> the algorithm by Kopriva, we need to define this as the curl of the >> discrete interpolation of X_l \nabla_\xi X_m. To get the curl, we need >> another round through the SelectEvaluator::evaluate() call in that function >> to get the reference-cell gradient of that object, from which we can then >> collect the entries of the curl. To call into evaluate one more time, we >> also need a new data.shape_info object that does the collocation evaluation >> of derivatives. That should only be two lines that I can show you how and >> where to add, so let us not worry about that part now. What is important to >> understand first (in terms of index notation vs tensor notation) is the >> dimension of the object. I believe that X_l \nabla_\xi X_m is a rank-two >> tensor, so it has dim*dim components, and we compute the gradient that >> gives us a dim * dim * dim tensor. Taking the curl in the derivative and >> inner tensor dimension space, we get rid of one component and up with a dim >> * dim tensor as expected. The final step we need to do is to divide by the >> determinant of the Jacobian (contravariant vectors), because the inverse >> Jacobian in deal.II does not yet pre-multiply with the determinant. >> >> Does that procedure sound reasonable to you? If yes, we could start >> putting together the ingredients. It would be good to have a mesh (the >> curvilinear case you were mentioning) where we can test those formulas. >> >> Best, >> Martin >> On 17.06.20 18:37, Alexander Cicchino wrote: >> >> Dear Martin, >> >> Thank you for your response. Yes I agree that only some local >> computations are necessary to implement the identities. >> Yes I would be interested in this feature and trying to implement it. Do >> you have any suggestions on where I should start and overall practices I >> should follow? >> >> Thank you, >> Alex >> >> On Wednesday, June 17, 2020 at 1:19:29 AM UTC-4, Martin Kronbichler >> wrote: >>> >>> Dear Alex, >>> >>> This has been on my list of things to implement and verify with deal.II >>> over a range of examples for quite a while, so I'm glad you bringing the >>> topic up. It is definitely true that our way to define Jacobians does not >>> take those identities into account, but I believe we should add support for >>> them. The nice thing is that only some local computations are necessary, so >>> having the option to use it in the polynomial mapping classes would be >>> great. If you would be interested in this feature and trying to implement >>> things, I'd be happy to guide you to the right places in the code. >>> >>> Best, >>> Martin >>> On 17.06.20 06:02, Alexander Cicchino wrote: >>> >>> Thank you for responding Wolfgang Bangerth. >>> >>> The GCL condition comes from the discretized scheme satisfying >>> free-stream preservation. I will demonstrate this for 2D below, (can be >>> interpreted for spectral, DG, finite difference, finite volume etc): >>> Consider the conservation law: \frac{\partial W}{\partial t} + >>> \frac{\partial F}{\partial x} +\frac{\partial G}{\partial y} =0 >>> Transforming this to the reference computational space (x,y)->(\xi, >>> \eta): >>> J*\frac{\partial W}{\partial t} + J*\frac{ \partial \xi}{\partial x} * >>> \frac{\partial F}{\partial \xi} + J * \frac{ \partial \eta}{\partial x}* >>> \frac{\partial F}{\partial \eta} + J * \frac{ \partial \xi}{\partial y} * >>> \frac{\partial G}{\partial \xi} + J*\frac{ \partial \eta}{\partial >>> y}*\frac{\partial G}{\partial \eta} >>> Putting this in conservative form results in: >>> J\frac{\partial W}{\partial t} + \frac{\partial}{\partial \xi} ( >>> J*F*\frac{\partial \xi}{\partial x} +J*G*\frac{\partial \xi}{\partial y} ) >>> + \frac{\partial}{\partial \eta} ( J*F*\frac{\partial \eta}{\partial x} >>> +J*G*\frac{\partial \eta}{\partial y} ) - F*( GCL in x) - G*(GCL in y) =0 >>> >>> where GCL in x = \frac{\partial }{\partial \xi} ( det(J)* \frac{\partial >>> \xi >>> }{\partial x}) + \frac{\partial }{\partial \eta}( det(J)* >>> \frac{\partial >>> \eta}{\partial x} ) >>> similarly for y. >>> >>> So for the conservative numerical scheme to satisfy free stream >>> preservation, the GCL conditions must go to zero. >>> For linear grids, there are no issues with the classical definition for >>> the inverse of the Jacobian, but what Kopriva had shown (before him Thomas >>> and Lombard), was that the metric Jacobian has to be calculated in either a >>> "conservative curl form" or an "invariant curl form" since it reduces the >>> GCL condition to the divergence of a curl, which is always discretely >>> satisfied. In the paper by Kopriva, he shows this, an example in 3D: >>> Analytically >>> J*\frac{\partial \xi}{\partial x} = \frac{\partial z}{\partial \zeta} * >>> \frac{\partial y}{\partial \eta} - \frac{\partial z}{\partial \eta} * >>> \frac{\partial y}{\partial \zeta} >>> >>> but the primer doesn't satisfy free-stream preservation while the latter >>> ("conservative curl form") does. >>> >>> I will put together a unit test for a curvilinear grid. >>> >>> Thank you, >>> Alex >>> >>> On Tuesday, June 16, 2020 at 10:24:59 PM UTC-4, Wolfgang Bangerth wrote: >>>> >>>> >>>> Alexander, >>>> >>>> > I am wondering if anybody has also found that the inverse of the >>>> Jacobian from >>>> > FE Values, with MappingQGeneric does not satisfy the Geometric >>>> Conservation >>>> > Law (GCL), in the sense of: >>>> > >>>> > Kopriva, David A. "Metric identities and the discontinuous spectral >>>> element >>>> > method on curvilinear meshes." /Journal of Scientific Computing/ 26.3 >>>> (2006): 301. >>>> > >>>> > on curvilinear elements/manifolds in 3D. >>>> > That is: >>>> > \frac{\partial }{\partial \hat{x}_1} *det(J)* \frac{\partial >>>> \hat{x}_1 >>>> > }{\partial x_1} + \frac{\partial }{\partial \hat{x}_2} *det(J)* >>>> \frac{\partial >>>> > \hat{x}_2}{\partial x} + \frac{\partial }{\partial \hat{x}_3} * >>>> > det(J)*\frac{\partial \hat{x}_3 }{\partial x_1} != 0 (GCL says it >>>> should =0, >>>> > similarly for x_2 and x_3) >>>> > >>>> > If so or if not, also, has anybody found a remedy to have the inverse >>>> of the >>>> > Jacobian from FE Values with MappingQGeneric to satisfy the GCL. >>>> >>>> I'm not sure any of us have ever thought about it. (I haven't -- but I >>>> really >>>> shouldn't speak for anyone else.) Can you explain what this equality >>>> represents? Why should it hold? >>>> >>>> I'm also unsure whether we've ever checked whether it holds (exactly or >>>> approximately). Can you create a small test program that illustrates >>>> the >>>> behavior you are seeing? >>>> >>>> Best >>>> W. >>>> >>>> -- >>>> ------------------------------------------------------------------------ >>>> >>>> Wolfgang Bangerth email: bang...@colostate.edu >>>> www: >>>> http://www.math.colostate.edu/~bangerth/ >>>> >>>> -- >>> The deal.II project is located at http://www.dealii.org/ >>> For mailing list/forum options, see >>> https://groups.google.com/d/forum/dealii?hl=en >>> --- >>> You received this message because you are subscribed to the Google >>> Groups "deal.II User Group" group. >>> To unsubscribe from this group and stop receiving emails from it, send >>> an email to dea...@googlegroups.com. >>> To view this discussion on the web visit >>> https://groups.google.com/d/msgid/dealii/4f313231-dbb3-445f-923c-9eaff17ab783o%40googlegroups.com >>> >>> <https://groups.google.com/d/msgid/dealii/4f313231-dbb3-445f-923c-9eaff17ab783o%40googlegroups.com?utm_medium=email&utm_source=footer> >>> . >>> >>> -- >> The deal.II project is located at http://www.dealii.org/ >> For mailing list/forum options, see >> https://groups.google.com/d/forum/dealii?hl=en >> --- >> You received this message because you are subscribed to the Google Groups >> "deal.II User Group" group. >> To unsubscribe from this group and stop receiving emails from it, send an >> email to dea...@googlegroups.com. >> To view this discussion on the web visit >> https://groups.google.com/d/msgid/dealii/b764b4b7-02d2-4139-95d9-68c30ad4f2a9o%40googlegroups.com >> >> <https://groups.google.com/d/msgid/dealii/b764b4b7-02d2-4139-95d9-68c30ad4f2a9o%40googlegroups.com?utm_medium=email&utm_source=footer> >> . >> >> -- > The deal.II project is located at http://www.dealii.org/ > For mailing list/forum options, see > https://groups.google.com/d/forum/dealii?hl=en > --- > You received this message because you are subscribed to the Google Groups > "deal.II User Group" group. > To unsubscribe from this group and stop receiving emails from it, send an > email to dea...@googlegroups.com <javascript:>. > To view this discussion on the web visit > https://groups.google.com/d/msgid/dealii/cf020345-b304-45d2-a228-4081da2d4effo%40googlegroups.com > > <https://groups.google.com/d/msgid/dealii/cf020345-b304-45d2-a228-4081da2d4effo%40googlegroups.com?utm_medium=email&utm_source=footer> > . > > -- The deal.II project is located at http://www.dealii.org/ For mailing list/forum options, see https://groups.google.com/d/forum/dealii?hl=en --- You received this message because you are subscribed to the Google Groups "deal.II User Group" group. To unsubscribe from this group and stop receiving emails from it, send an email to dealii+unsubscr...@googlegroups.com. To view this discussion on the web visit https://groups.google.com/d/msgid/dealii/2ba99ff8-ef0d-463d-bdf0-5f6b328da005o%40googlegroups.com.
