Dear Konrad,
I do not understand when a geometry gets complicated. Is a toroid inside a sphere, both centred at the origin, a complicated geometry? To start with, I will use a relatively simple mesh. I will not use local refinement, only global. I can do without refinement as well, i.e. making the mesh in gmsh. Yes, I would like to know more about orientations issues on complicated geometries. Could please tell me about it? I would very much prefer to use FE_Nedelec without hierarchical shape functions. The first order 3D FE_Nedelec will do nicely provided the orientation issues will be irrelevant to the mesh. John On Tuesday, April 20, 2021 at 9:43:31 PM UTC+2 Konrad Simon wrote: > Dear John, > > As Jean-Paul pointed out the entries of the solution vector for a Nedelec > element have a different meaning. The entries are weights that result from > edge moments (and in case of higher order also face moments and volume > moments), i.e, integrals on edges. Nedelec elements have a deep geometric > meaning: they discretize quantities that you can integrate (a physicist > would say "measure") along 1D-lines in 3D. > > Btw, if you are using Nedelec in 2d be aware that it has some difficulties > on complicated meshes. In 3D the lowest order is fine. NedelecSZ (for 2D > lowest order and 3D all orders), as Jean-Paul pointed out, are another > option. They are meant to by-pass certain mesh orientation issues on > complicated geometries (I can tell you a bit about that since currently I > am chasing some problems there). > > Best, > Konrad > > On Tuesday, April 20, 2021 at 9:23:23 PM UTC+2 Jean-Paul Pelteret wrote: > >> Hi John, >> >> Unfortunately, you’ve fallen into the trap of confusing what the entries >> in the solution vector mean for the different element types. For Nedelec >> elements, they really are coefficients of a polynomial function, so you >> can’t simply set each coefficient to 1 to visualise the shape function >> (like one might be inclined to try for Lagrange type polynomials). If you >> want a little piece of code that will plot the Nedelec shape functions for >> you, then take a look over here: >> https://github.com/dealii/dealii/issues/8634#issuecomment-595715677 >> >> BTW. In case you’re not aware, a newer (and probably more robust) >> implementation of a Nedelec element is the FE_NedelecSZ >> <https://dealii.org/current/doxygen/deal.II/classFE__NedelecSZ.html> that >> uses hierarchical shape functions to form the higher order bases. The PhD >> thesis of S. Zaglmayr that describes the element is mentioned in the class >> documentation. >> >> Best, >> Jean-Paul >> >> On 20. Apr 2021, at 15:47, John Smith <[email protected]> wrote: >> >> >> Dear Konrad, >> >> Thank you for your help! I have recently modified the step-3 tutorial >> program to save FE_Nedelec<2> fe(0) shape function into *.csv files. It is >> attached to this message. The result of this program is not what I would >> expect it to be. I have also attached a pdf file which compares my >> expectations (marked as “Expectations” in the pdf file) with the results of >> the modified step-3 program (marked as “Reality” in the pdf file). My >> expectations are based on the literature I have. My question is: why the >> shape functions of FE_Nedelec<2> fe(0) differ from my expectations? I guess >> my expectations are all wrong. I would also appreciate if you will share >> with me a reference to a paper or a book, so I can extend my modest library. >> >> Thanks! >> >> John >> >> P.S. I have sent you an e-mail as you have requested. Could you please >> check your spam folder? >> >> >> >> On Monday, April 19, 2021 at 8:47:29 AM UTC+2 Konrad Simon wrote: >> >>> Dear John, >>> >>> I had a look at the pdf you sent. I noticed some conceptual >>> inconsistencies that are important for the discretization. Let me start >>> like this: The curl-curl-problem that you are trying to solve is - >>> according to your description of the gauge - actually a curl-curl + >>> grad-div problem and hence a Laplace-problem that describes a >>> Helmholtz-decomposition. In other words in order to get a well-posed >>> simulation problem you need more boundary conditions. You already noticed >>> that since your curl-curl-matrix is singular (the kernel is quite >>> complicated and contains all longitudinal waves). >>> >>> You have of course the option to solve (0.0.2) with a Krylov-solver but >>> then you need to make sure that the right-hand side is in the orthogonal >>> complement of this kernel before each iteration which is quite difficult. I >>> would not recommend that option. >>> >>> Another point is that if you have a solution for your field A like in >>> (0.0.4) you can not have a similar representation for T. This is >>> mathematically not possible. >>> >>> Since you not care about the gauge let me tell you how I would tackle >>> this: The reason is - to come back to my original point - you are missing >>> a boundary condition that makes you gauge unique. Since you apply natural >>> boundary conditions (BCs) on A you must do so as well to determine div(A). >>> This second BC for A is applied to a different part of the system that is >>> usually neglected in the literature and A is partly determined from this >>> part (this part then describes the missing transversal waves). The >>> conditions you mention on curl(A) are some what contradictory to the space >>> in which A lives. Nedelec elements which you must use (you can not use FE_Q >>> to enforce the conditions) cannot generate your desired T_0. >>> >>> There are some principles when discretizing these problems (which are >>> not obvious) that you MUST adhere to (choice of finite elements, boundary >>> conditions, what is the exact system etc) if you want to get a stable >>> solution and these are only understood recently. I am solving very similar >>> problems (with Deal.II) in a fluid context and will be happy to share my >>> experiences with you. Just email me: [email protected] >>> >>> Best, >>> Konrad >>> >>> On Monday, April 19, 2021 at 5:51:17 AM UTC+2 Wolfgang Bangerth wrote: >>> >>>> On 4/18/21 12:24 PM, John Smith wrote: >>>> > >>>> > Thank you for your reply. It is a bit difficult to read formulas in >>>> text. So I >>>> > have put a few questions I have in a pdf file. Formulas are better >>>> there. It >>>> > is attached to this message. May I ask you to have a look at it? >>>> >>>> John: >>>> >>>> If I understand your first question right, then you are given a vector >>>> field J >>>> and you are looking for a vector field T so that >>>> curl T = J >>>> I don't really have anything to offer to this kind of question. There >>>> are many >>>> vector fields T that satisfy this because of the large null space of >>>> curl. You >>>> have a number of condition you would like to "approximate" but there >>>> are many >>>> ways to "approximate" something. In essence, you have two goals: To >>>> satisfy >>>> the equation above and to approximate something. You have to come up >>>> with a >>>> way to weigh these two goals. For example, you could look to minimize >>>> F(T) = ||curl T - J||^2 + alpha ||T-T_desired||^2 >>>> where T_desired is the right hand side of (0.0.5) and you have to >>>> determine >>>> what alpha should be. >>>> >>>> As for your other questions: (0.0.9) is indeed called "interpolation" >>>> >>>> The issue with the Nedelec element (as opposed to a Q(k)^dim) field is >>>> that >>>> for the Nedelec element, >>>> \vec phi(x_j) >>>> is not independent of >>>> \vec phi(x_k) >>>> and so you can't choose the matrix in (0.0.8) as the identity matrix. >>>> You >>>> realize this in (0.0.13). The point I was trying to make is that >>>> (0.0.13) >>>> cannot be exactly satisfied if T(x_j) is not a vector parallel to \hat >>>> e_j, >>>> which in general it will not be. You have to come up with a different >>>> notion >>>> of what it means to solve (0.0.13) because you cannot expect the left >>>> and >>>> right hand side to be equal. >>>> >>>> Best >>>> W> >>>> >>>> -- >>>> ------------------------------------------------------------------------ >>>> >>>> Wolfgang Bangerth email: [email protected] >>>> 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 [email protected]. >> To view this discussion on the web visit >> https://groups.google.com/d/msgid/dealii/348bd8f9-0891-42e5-b150-c3723eacb84bn%40googlegroups.com >> >> <https://groups.google.com/d/msgid/dealii/348bd8f9-0891-42e5-b150-c3723eacb84bn%40googlegroups.com?utm_medium=email&utm_source=footer> >> . >> <FE_Nedelec.pdf><step-3.cc> >> >> >> -- 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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