Re: [deal.II] Mesh-induced elastic anisotropy and distorting the quad. points as a way to palliate it

```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 ```

Re: [deal.II] Mesh-induced elastic anisotropy and distorting the quad. points as a way to palliate it

```
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
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
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.

summary.png

*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, ```

Re: [deal.II] Mesh-induced elastic anisotropy and distorting the quad. points as a way to palliate it

```
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: bange...@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
---
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 view this discussion on the web visit

```

Re: [deal.II] Mesh-induced elastic anisotropy and distorting the quad. points as a way to palliate it

```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.

On Thursday, 9 July 2020 05:03:45 UTC+2, Wolfgang Bangerth wrote:
>
> On 7/2/20 10:06 PM, David F wrote:
> >
> > *_Q2_:* why the system behaves as anisotropic if its local inhomogeneous
> > elastic properties are isotropic? If you have any comment or suggestion
> > the problem of mesh-induced elastic anistropy in FEM, I would like to
> know it.
>
> I don't know how exactly you choose your coefficient, but if you alternate
> layers of isotropic materials, then you get an anisotropic material. Think
> about layering styrofoam plates with steel plates -- the resulting stack
> of
> layers is essentially incompressible under loads from the side (because
> the
> steel plates provide the stiffness), but is quite compressible if you load
> it
> from top and bottom (because the styrofoam layers will simply collapse).
>
> 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
---
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 view this discussion on the web visit

```

Re: [deal.II] Mesh-induced elastic anisotropy and distorting the quad. points as a way to palliate it

```
On 7/2/20 10:06 PM, David F wrote:

*_Q2_:* why the system behaves as anisotropic if its local inhomogeneous
elastic properties are isotropic? If you have any comment or suggestion about
the problem of mesh-induced elastic anistropy in FEM, I would like to know it.

I don't know how exactly you choose your coefficient, but if you alternate
layers of isotropic materials, then you get an anisotropic material. Think
about layering styrofoam plates with steel plates -- the resulting stack of
layers is essentially incompressible under loads from the side (because the
steel plates provide the stiffness), but is quite compressible if you load it
from top and bottom (because the styrofoam layers will simply collapse).

Best
W.

--

Wolfgang Bangerth  email: bange...@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
---
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 view this discussion on the web visit

```

[deal.II] Mesh-induced elastic anisotropy and distorting the quad. points as a way to palliate it

```Hello everyone,

I'm trying to solve a 2D solid mechanics homogenization problem, in which I
have element-wise constant elastic properties, which are inhomogeneous and
isotropic from element to element (i.e., I am assembling the system using
the same 4-rank stiffness tensor for all the quadrature points of a certain
element, but that tensor is different for each element). For this system, I
would like to compute its effective elastic properties, which I do by
standard homogenization approaches. The system should behave as an
isotropic solid. However, I observe significant anisotropy (and clearly not
due to random fluctuations that might arise because the element-to-element
inhomogeneous properties are randomly distributed). I attribute this to a
mesh dependency of the solution, since I have solved the same problem with
a unstructured triangular mesh with another FEM package and I don't observe
this issue. I believe the structured quadrilateral mesh induces some
artificial elastic anisotropy, which is not there in the case of the
unstructured triangular mesh due to its topological disorder.

I've thought of a way that might palliate this issue, which is to set
different elastic properties at the quadrature points themselves (i.e., the
properties are no longer element-wise constant). This seems to work to some
extent since the system becomes less anisotropic, however it is not good
enough.

*Q1:* is there a preferred way in dealII in which I could randomly distort
a bit the location of the quadrature points? I think this extra distortion
might help get rid of the mesh artifacts. Is is possible to do it with the
in-built Lagrange linear FE or another type of FE is more suitable within
dealII for this task? Basically I have no idea where to start from to do
something like this, so any suggestion is welcome.

*Q2:* why the system behaves as anisotropic if its local inhomogeneous
elastic properties are isotropic? If you have any comment or suggestion
about the problem of mesh-induced elastic anistropy in FEM, I would like to
know it.

David.

--
The deal.II project is located at http://www.dealii.org/
For mailing list/forum options, see