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. 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