Hi, Ray,

I recommend to use an approach proposed and described in detail in A. Srikantha Phani, J. Woodhouse, N. A. Fleck, Wave propagation in two-dimensional periodic lattices, J. Acoust. Soc. America 119, 1995 (2006); https://doi.org/10.1121/1.2179748

Hi,

a don't know the technical details concerning Abaqus software, but you should care about ill-posedness of your problem due to periodic boundary conditions. Although the periodic boundary conditions fulfill Hill-Mandel principle, your system is still free to translate-which means, you need somehow to add contraints to prevent rigid body motion. This can be done by constraining one of the model corner or add integral contraint in such way that the average of the periodic field is zero.

Hope it helps,

Petr

I do not know the softaware you mention. In general. we introduce periodic boundary conditions via the choice of the finite element space.

Dear Ray,

Abaqus does not have an automatic way of applying the PBCs. You might need to do them yourself through a python script or so. However, I have noticed a very similar question asked here on ResearchGate before and I think they might be able to help you. They developed their own Abaqus plug-in (EasyPBC) for this purpose. Check this link and contact Sadik Omairey if you are intersted:

https://www.researchgate.net/project/ABAQUS-Plugin-Tool-for-Periodic-RVE-Homogenisation-EasyPBC

Regards,

Mohamed

Hey, Ray,

I do not many operation details on Abaqus, but I have ever used COMSOL and Marc Mentat to implement these tied relations. The principles are all the same, so I would suggest you to read our lecture note "Notes_homog_Kouznetsova" (the previous attachment has been removed due to intellectual property rights of TU/e) to know more details on computational homogenization, of course also how to implement the periodic boundayr conditions.

Here to answer your questions:

1----------------------------------------------

In Chapter 4 of "Notes_homog_Kouznetsova", you can find the FEM-based implementations of periodic boundary conditions (I also attach my simple code "RelationDep" based on MATLAB). Note the implementation given in this lecture note is for

1) solid description;

2) quasi-static;

3) 2D case;

4) up-down and left-right boundaries simultaneously tied.

Since only up-down boundaries are tied in you case, I would suggest you to derive associated FEM-based implementation by yourself, by following detailed steps in this lecture note. It is very easy to achieve.

If you are considering the case under shell description, you can have a look at the FEM-based implementation in "E.W.C. Coenen, V.G. Kouznetsova, and M.G.D. Geers. Computational homogenization for heterogeneous thin sheets. International Journal for Numerical Methods in Engineering, 83(8-9):1180-1205, 2010."

If you are considering the dynamic case, you can have a look at the FEM-based implementation in "A. Sridhar, V.G. Kouznetsova, and M.G.D. Geers. Homogenization of locally resonant acoustic metamaterials towards an emergent enriched continuum. Computational Mechanics, 57(3):423-435, 2016."

2----------------------------------------------

Due to the irregular distribution of defects, you obviosuly need to take the RVE size into account such that to get sufficiently effective constitutive model and response. A convergence study is probably needed. You can look at a few similar discussions in "V.G. Kouznetsova, W.A.M. Brekelmans, and F.P.T. Baaijens. An approach to micromacro modeling of heterogeneous materials. Computational Mechanics, 27(1):37-48, 2001."

3-----------------------------------------------

Elaborate introductions on periodic boundary conditions and full prescribed boundary conditions (I think it is what you mean homogeneous boundary conditions) can be found in Chapter 4 of "Notes_homog_Kouznetsova".

Hope these information can help, and if you have more doubts, we can further discuss!

Lei

Hi, Ray,

I recommend to use an approach proposed and described in detail in A. Srikantha Phani, J. Woodhouse, N. A. Fleck, Wave propagation in two-dimensional periodic lattices, J. Acoust. Soc. America 119, 1995 (2006); https://doi.org/10.1121/1.2179748

Hi, Ray,

ABAQUS cannot deal with a model with complex boundaries directly. However, such the problem can be handled by individually combining real and imaginary parts through constraint conditions. You can find the solutions in Article Multiple flexural-wave attenuation zones of periodic slabs w...

Hello Ray,

It is a straight forward task if you are familiar with scripting in ABAQUS. Here is an easy-read on scripting: http://bertoldi.seas.harvard.edu/files/bertoldi/files/learnabaqusscriptinonehour.pdf

Now that you know scripting, you basically need to define the displacements nodes on periodic boundaries to be equal. Here is a simple example which might help you: https://imechanica.org/files/pbc.pdf