How to maximize the number of defect placements on a sphere with exclusion zones

Question

I am working on a problem where I need to place defects (dimples) on a hemispherical shell while ensuring that the angular distance between any two defects is greater than or equal to a fixed value. This creates an exclusion zone around each defect where no other defect can be placed.

So far, I’ve tried two approaches:

Manual Placement:

I manually place defects and define an exclusion zone around them. I try to place the next defect on the edge of the exclusion zone. This method is very tedious and hard to optimize.

Random Placement with Constraints:

I randomly generate points on the hemisphere and ensure that each new point satisfies the minimum angular distance constraint. While this avoids overlaps, it does not maximize the number of defects I can place on the hemisphere.

My Questions

Can Delaunay triangulation (using convex hulls) help me find the maximum number of defects I can place? I know it can create a triangular mesh, but I’m unsure how to use it for optimizing placement. Are there better algorithms or approaches (e.g., optimization methods or packing algorithms) to maximize defect placement while maintaining exclusion zones?

The defects are represented as points on the sphere, and the exclusion zones are defined by a minimum angular distance between these points. The triangulation method I used partitions the hemisphere into triangles, but I’m unsure how to apply this information to optimize defect placement.

Any insights, suggestions, or references to tools/algorithms that could help me solve this would be greatly appreciated!

Here is the figure of how I manually placed the centers of defects on the hemisphere.