Hi Sebastian, first of all, sorry for my late reply. Thank you very much for comment, it certainly raises some very interesting points. I think the only thing which is still left to be explained is the fact that a triangular grid, which yields the right isotropic result, becomes anisotropic just by rearranging the elastic properties to form squares-like clusters. I think this case does not correspond to any of your 4 points since I do this without h->inf. In fact, each of those clusters are just 2 FE.

Best, David. On Saturday, 11 July 2020 06:19:50 UTC+2, Sebastian Stark wrote: > > Hi David, > > I'm neither an expert, nor do I know the literature well, but looking on > your pictures, I think, the situations you are studying are geometrically > anisotropic. Just plot the distribution of angles the faces of your > inhomogenities make with the x-axis. For the quad-case, you'll get two > discrete peaks at 0 and 90 degree. for the triangular case, you get 0, 45 > and 90 degree. So, from this, the results do not seem surprising to me > (just consider the extreme case of cracks - if you have them only at 0 and > 90 degree oriented, this is unlikely to be isotropic). The fact that you > have randomly assigned elastic properties won't help to fix that. > > A few examples (in 2d): > > (1) equidistant circular inclusions in a matrix (matrix and inclusions two > different isotropic linearly elastic materials) -> this should be isotropic > if mesh size h->0 > > (2) equidistant square inclusions in a matrix, all aligned with x-axis > (matrix and inclusions two different isotropic linearly elastic materials) > -> probably anisotropic if mesh size h->0 > > (3) equidistant square inclusions in a matrix, random orientation of > inclusions (matrix and inclusions two different isotropic linearly elastic > materials) -> should be isotropic if mesh size h->0 and number of > orientations->infinity > > (4) equidistant square inclusions in a matrix, random orientation of > inclusions (random isotropic linearly elastic materials) -> should be > isotropic if mesh size h->0 and number of orientations->infinity > > Also to consider: In 3d, there are 21 elastic constants for a linearly > elastic material. In a mathematical 2d scenario, it should be 6. This > suggests that, in example (3), it is not strictly necessary to have random > orientation. Rather, a few (equally spaced) discrete orientations might be > good enough. If that's the case, how many does one need? I'm betting on 6, > not sure though. Alternatively, one could replace the square inclusions in > (2) by regular polygons and ask how many vertices the polygon needs for > isotropy. Again, I'm betting on 6. > > Related: Are there crystal structures with such a high degree of symmetry, > that they are elastically isotropic? For dielectric properties, a cubic > crystal is good enough already. But the dielectric tensor is rank 2 and the > elastic one rank 4. So you'll need much more symmetry in the crystal; and > considering that a crystal can have 6-fold rotational symmetry at most that > might be impossible. > > What I'm just noticing: Hexagonal crystals are elastically isotropic > perpendicular to the hexagonal axis. So, my bet on 6 might be good. And it > might explain your observation that the triangular elements are relatively > isotropic (though maybe not perfectly). > > I hope that gives you some input. If you have definitive answers to any of > the questions, I'm curious. > > Regards, > Sebastian > > > Am 11.07.20 um 00:12 schrieb David F: > > I have made a somewhat extensive study on his issue and prepared some > plots that will hopefully answer your questions, and also includes Bruno's > suggestion about distorting the mesh. The basic setup is: I sheared the > mesh along different orientations (see x-axis on the plots) and measured > the shear modulus (y-axis). I have repeated the random process of setting > the elastic properties many times to have good statistics (see errorbars on > the plots). Each element has an isotropic stiffness tensor with a Poisson > ratio of 1/3 and a shear modulus which is exponentially distributed with an > average of 10. I use linear shape functions unless otherwise stated. If the > picture are not big enough, you can find them in the links beneath them. > > > *1) I change the resolution. *By this I don't mean just a mesh with a > bigger number of elements, but importantly each inhomogeneities is > represented by a bigger number of elements. Therefore, we solve problems > with exactly the same physical domain but with different resolution. In the > legend, n means the resolution of the inhomogeneities. For n=1 each > inhomogeneity is described by 1 FE. For n=2, by 2^2, and for n=4 by 4^2. We > can see that for shearing with angle 0 (see pictures on the bottom for > clarity) the shear modulus is minimum, while it is maximum for 45 degrees, > when the principal axes are aligned with the mesh. The magnitude of the > anisotropy is the difference between the maximum and the minimum. The > difference decreases by increasing the resolution, but actually the > relative difference is very similar, and it seems that by just increasing > the resolution this problem won't go away. Finally, I have distorted the > mesh, which doesn't change the behavior at all. > > [image: summary.png] > > Link to the picture <https://ibb.co/bFps0Vs> > > > > > > *2) I change the order of the shape functions.* I use the original set > up, in which each inhomogeneity is represented by 1 element. We see that > increasing the shape function order has a somewhat similar effect as > increasing the resolution of the inhomogeneity (expected, since in both > cases we are increasing the number of dofs of each inhomogeneity). > Therefore, increasing the order of the shape functions doesn't seem to be > enough to fix the issue. > > [image: plots_shape_functions.png] > > Link to the picture <https://ibb.co/0X2W5M2> > > *3) I try different types of mesh.* In this case, I compare the solution > provided by dealII (i.e. quadrilateral mesh) with the solution obtained > with a triangular mesh using the python FENICS package. Lastly, I solve > using the triangular mesh but allocating the inhomogeneities in such a way > that even if the mesh is triangular, the structure of the inhomogeneities > looks quadrilateral. It seems that a triangular mesh, even it is > structured, is able to provide an isotropic solution. Interestingly, *the > same* mesh fails and behaves like a quadrilateral mesh if the structure > depicted by the elastic properties look like those of the quadrilateral > mesh. > > [image: summary3.png] > > Link to the picture <https://ibb.co/PF5z8t3> > > > > In summary, form all the tings that I tried, the only solution is to use a > triangular mesh. Very interestingly, by arranging the elastic properties in > a quad-like manner we can force the triangular mesh to give the wrong > result of a quadrilateral mesh. Therefore, do you think it is possible to > do the opposite, i.e., to make dealII's quadrilateral mesh "behave more > triangular-like" by playing with elastic properties, quadrature points etc.? > > Do you know the reason of the anisotropy in the quadrilateral mesh, or if > there is some literature about this? I couldn't find anything. > > > Best, > David. > > > On Friday, 10 July 2020 17:38:58 UTC+2, Wolfgang Bangerth wrote: >> >> On 7/10/20 9:15 AM, David F wrote: >> > I have a 2D system for which I create the stiffness tensor of an >> isotropic >> > material, but for each finite element I create it with a different >> shear >> > modulus. The shear modulus is random for each element (I use an >> exponential >> > distribution, but any distribution leads to the same behavior as long >> as the >> > std is high), with no structure such as layers or anything else. In >> this case, >> > the system should clearly be macroscopically isotropic (up to >> statistical >> > fluctuations due to the random properties) for symmetry reasons. >> >> At least in the limit h->0 I agree. For finite mesh sizes, I would expect >> that >> the material has a degree of anisotropy that goes to zero as you make the >> mesh >> smaller. It is true that the axes of anisotropy should be oriented in >> random >> ways for different realizations of the same experiment on the same mesh. >> When >> you do your computations, have you checked (for different realizations of >> the >> random process): >> (i) whether the orientation of anisotropy is always the same, and always >> related to the principal directions of the mesh? >> (ii) how the magnitude of anisotropy behaves as you refine the mesh? >> >> 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 <javascript:>. > To view this discussion on the web visit > https://groups.google.com/d/msgid/dealii/759917d0-227a-456e-bfac-00eab3ed71cdo%40googlegroups.com > > <https://groups.google.com/d/msgid/dealii/759917d0-227a-456e-bfac-00eab3ed71cdo%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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