Finite Element Analysis for Laminated Plates with Holes or Patches

Question

As the title says, I am trying to code in FEM a plate structure that either has a hole in one of the layers or one of the layers is made of patches of plates, rather than one whole plate. However, while I have a slight idea of how to implement this, I'm really not very sure and I would like some questions answered before I start trudging through my code haphazardly.

From what I understand of FEM, I understand that in order to represent a hole or a patch (which is basically a plate with a hole AROUND it) is to change the connectivity matrix for a given layer. However, I don't quite get how I would go about doing that. For example, if I had a 6x6 element plate structure and one of the layers of that plate was to have a 2x2 element hole in the middle, do I simply remove those elements from my connectivity vector? I just have a feeling that would cause some of my matrix dimensions to not match and throw the whole thing into chaos.

Additionally, what if the location/dimensions of my hole or plate didn't align perfectly with the coordinates of my elements? how do I implement that? Would I have to write a completely new coordinate system for my nodes?

I'm sorry if my questions seem either too obvious or too vague. I just wanted to know if I'm heading in the right direction before investing in any major change to my code. If any of you could point me in the right direction or provide some good practice/sample pieces of code that would be great. I'm learning FEM basically through trial and error.

Answer 1

The strength of the FE method is its ability to represent complex geometries. That way is it preferred method by engineers. Thus representation of holes or other geometrical features is for it natural.

In general, to solve your problem, you have to have a global enumeration of nodes. Then you iterate over finite elements, calculate local stiffness matrix and assemble it global stiffness matrix, by copying block of DOFs adjacent to nodes, one by one, into the global matrix using global nodal indexes to find the right place. In your case, you have 6-DOFs at the node, i.e. three displacements and three rotations. You can find many books about that, for example, Zienkewicz and Taylor book or look at internet resources https://www.colorado.edu/engineering/cas/courses.d/AFEM.d/

However, you raise two issues which are problematic, i.e. laminate plate (1) and non-conforming meshes (2).

1) For Kirchoff-Love or Reissner-Mindlin you can consider laminate plate by modifying the stiffness equation and appropriate integration through the thickness. I refer you to a vast number of papers in this area. However, you can not stack Kirchoff-Love or Reissner-Mindlin FE element on top of each other; you will not capture adequately bending stiffness. If you like that, you need to modify formulation. I strongly believe the best solution is with using solid-shell elements.

2) You consider the case when nodes on subsequent layers do not much, in that case, you have non-conforming mesh. In principle, you can not do that, unless you have a special methodology to resolve those type of problems. You need to look at the mortar method or Nitsche method which both which provide a solution for non-conforming meshes.