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 <javascript:>. > 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. 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