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Whenever I use a mesh that's more complex than the cube, (even the monkey), I often end up with results like this:

Original mesh for fracturing: Original mesh with particles for fracturing:

Cell Fracture settings:

Cell Fracture settings:

Results with weird extra geometry: Results with weird extra geometry:

Same thing in a different angle:

Same thing different angle:

Working on Blender 3.0.0 on Windows 10. Any help you guys could give would be amazing. If there's any more information I need to provide, please just let me know. Thanks.

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I gues this happens because of Non Manifold geometry. Cell Fracture is a boolean based addon. So to make it work properly you need to close all the gaps in Suzanne.
To search and fix the problem follow next steps:

  1. Select the mesh and press TAB to enter the Edit Mode
  2. Press LMB anywhere in viewport to reset selection
  3. Make sure that you are in Vertex or Edge Selection Mode
  4. In Viewport Header go to Select > Select All by Trait > Non Manifold to inspect the model and find the gaps
  5. Press Z > Wireframe to switch to Wireframe Shading. Now you can see the gaps clearly. These gaps mean that Suzanne consists of three Non Manifold objects the head and two eyes. And the result whitch we'll get after Cell Fracture(a lot of boolean operations) most likely will be poor. If you take a look at your screenshots - all problems are located at those gaps place
  6. Hover the mouse cursor over the eyes one at a time and press L to use Select Linked
  7. Press X > Faces to delete selected eyes
  8. Alt + LMB to select edge loops around the gaps and press F to Fill them
  9. Repeat the Cell Fracture operation and enjoy the result

enter image description here

Of course you can work on geometry more properly to get nicer eyes fo the monkey)

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