Geometric Conservation Law

[Alexander Cicchino]

Hello,

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.

Thank you,

Alex

[Wolfgang Bangerth]

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.

--
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Wolfgang Bangerth email: bang...@colostate.edu
www: http://www.math.colostate.edu/~bangerth/

[Alexander Cicchino]

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

[Martin Kronbichler]

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

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[Alexander Cicchino]

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

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[Martin Kronbichler]

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

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[Alexander Cicchino]

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

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[Martin Kronbichler]

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

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[Alexander Cicchino]

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

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[Alexander Cicchino]

Dear Martin,

As an update, I figured out a way to have it work in 3D without needing to use another Evaluate call. The only issue is that I need the quadrature in the reference element since I construct an FE_DQGArbitraryNodes to evaluate the derivative (I do this since the metric Jacobian is in a sense "collocated" on the quadrature points so the gradient needed is just the derivative of the basis functions).

I have it passing collocated and uncollocated curvilinear test cases in 3D now perfectly, but I currently construct the FE_DGQArbitraryNodes in mapping_q_generic.cc by explicitly constructing a Quadrature like the testcase (example QGauss if the testcase uses QGauss).

I couldn't find a way to extract the reference element quadrature from the MappingQGeneric InternalData, do you have any suggestions?

Thank you,

Alex

[Alexander Cicchino]

Dear Martin or whom it may concern,

I have solved the previous problem and am confirming that the conservative curl form has now been implemented and passes 2 complicated tests for GCL on symmetric and non-symmetric curvilinear grids for different polynomial degrees. Turns out that we did not need to use the "evaluate" call. The changes are found in:

https://github.com/AlexanderCicchino/dealii/blob/master/source/fe/mapping_q_generic.cc

lines 1599 - 1697 (line 1602 to switch it on or off)

and the 2 tests are /tests/mappings/mapping_q_generic_GCL_curvilinear.cc for symmetric curv. grid and /tests/mappings/mapping_q_generic_GCL_curvilinear_nonsym.cc for non-sym. curv grid.

How should I proceed with the pull request?

Also, for additional work, are there any suggestions on how to proceed with 2D. Additionally, are there any suggestions for templating the mapping for ease automatically differentiating the metric terms?

Thank you for all the help,

Alex

[alexande...@gmail.com]

Dear all,

To complete this topic for both MappingQGeneric and MappingFEField, I am wondering if it is possible to extract the quadrature in the reference element in the function maybe_update_Jacobians and maybe_compute_q_points? These are needed to ensure that the normals are consistent at each surface quadrature node with the metric terms that satisfy GCL.

I was able to get the reference quadrature nodes in maybe_update_q_points_Jacobians_and_grads_tensor in fe/mapping_q_generic.cc by using data.shape_info.data[0].quadrature, but when it is not a tensor product basis (in the 2 functions listed above for mapping_q_generic.cc) I noticed that data.shape_info is empty (and there is no data.shape_info in mapping_fe_field.cc). Ideally, is it possible to fill data.shape_info in mapping_q_generic.cc or are there any other suggestions on how to extract the quadrature on the reference element for both mapping_q_generic.cc and mapping_fe_field.cc (just the 1D representation is sufficient)?

Thank you,

Alex