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. To unsubscribe from this group and stop receiving emails from it, send an email to dealii+unsubscr...@googlegroups.com. To view this discussion on the web visit https://groups.google.com/d/msgid/dealii/8930320b-7558-4408-864d-01dea38c6263o%40googlegroups.com.
