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/
>>
>> --
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