If you discretize, your system will not work against the analytical description of the lens, but rather against a bunch of hyper polygons (in 3D), and any computation you do, will have to consider them all, unless you do some programming yourself. This is why I recommended to postpone discretization (if at all) as much as possible.

Good luck with your project

yehuda

Hi Tim,

This is extremely useful. Maybe, more like a game-changer. I got it working myself about an hour ago. Next week, I will hook it all up with NDSolve properly.

Thank you for telling me OpenCascadeLink. It really will help.

Antony

Hi GianLuca,

I cannot help with either of these problems. I am still too new to Mathematica.

Regards

Antony

Hi Tim,

What you have posted is amazing. The OpenCascade Link will work for me because I am using Mathematica 12.1. However, I am not really in a position to dive in with OpenCascade yet because I have to get to grips with the core Mathematica. In particular, I must get to grips with NDSolve and investigate the heat performance of this lens.

I have been reading the documentation for OpenCascade and it looks like it is able to Import Step files - something Mathematica can't do. This is interesting. I may be able to get the .Step files for the optics I am interested in - including the one I have recreated in this post. This is something I am going to look into.

Thank you for your help.

Antony

If you have version 12.1, then you can use the OpenCascade Link to give access to the open source CAD program. Like most CAD packages, it can do a good job with boolean operations.

I am by no means an expert, but I was able to throw this workflow together based on the tutorial.

(* Load Required Packages *)
Needs["OpenCascadeLink`"]
Needs["NDSolve`FEM`"]
(* parameters *)
z0 = 3.1981; z1 = 0.9; z2 = -0.036; z3 = 0.001;
lowz =  0.0001;
highz = 9.05;
mainR = (z0 + z*(z1 + z*(z2 + z3*z))) /. z -> highz;
highPinz = 4.88;
(* create curve for LED *)
pp = ParametricPlot[{(z0 + z*(z1 + z*(z2 + z3*z))), z}, {z, lowz, 
   highz}, AspectRatio -> Automatic, PlotRange -> {{0, All}, All}, 
  MaxRecursion -> 1, PlotPoints -> 20]
(* Extract points from plot *)
coordinates = First[Cases[Normal[pp], Line[l_] :> l, \[Infinity]]];
(* Convert coordinates to 3D *)
pts = ArrayPad[coordinates, {{0, 0}, {0, 1}}];
pts[[All, {1, 2, 3}]] = pts[[All, {1, 3, 2}]];
(* Create surface of revolution in OpenCascade *)
ll = Line[pts];
wire = OpenCascadeShape[ll];
axis = {{0, 0, 0}, {0, 0, 1}};
sweep = OpenCascadeShapeRotationalSweep[wire, axis];
bmesh = OpenCascadeShapeSurfaceMeshToBoundaryMesh[sweep];
Show[Graphics3D[{{Red, Thickness[0.02], ll}, {Blue, Thick, 
    Arrow[10 axis]}}], bmesh["Wireframe"], Boxed -> False]
(* Cap ends of swept shell *)
crd = bmesh["Coordinates"];
lowcap = Polygon@Select[crd, (#[[3]] == lowz) &];
highcap = Polygon@Select[crd, (#[[3]] == highz) &];
p1 = OpenCascadeShape[lowcap];
p2 = OpenCascadeShape[highcap];
shape = OpenCascadeShapeSewing[{sweep, p1, p2}];
(* Create main body *)
cylshape = 
  OpenCascadeShape[
   c1 = Cylinder[{{0, 0, lowz}, {0, 0, highPinz}}, mainR]];
(* Difference LED from main in OpenCascade *)
diff = OpenCascadeShapeDifference[cylshape, shape];
(* Visualize *)
bmeshshape = OpenCascadeShapeSurfaceMeshToBoundaryMesh[shape];
bmeshcyl = OpenCascadeShapeSurfaceMeshToBoundaryMesh[cylshape];
bmeshdiff = OpenCascadeShapeSurfaceMeshToBoundaryMesh[diff];
groups = bmeshdiff["BoundaryElementMarkerUnion"];
temp = Most[Range[0, 1, 1/(Length[groups])]];
colors = ColorData["BrightBands"][#] & /@ temp;
Show[bmeshshape["Wireframe"], bmeshcyl["Wireframe"]]
bmeshdiff["Wireframe"["MeshElementStyle" -> FaceForm /@ colors]]

It appears that OpenCascade does a good job maintaining sharp edges.

Hi Yehuda,

The command

DiscretizeRegion[df, MaxCellMeasure -> 0.005]

works really well. Thank you, so much. I was really stuck.

My intention is to investigate the thermal performance of the lens with different LEDs. The prospect is that I will need to make the lens just once and once I have it, I can loop over many thermal simulations. The speed of execution for generating the lens is only expected to be a minor issue.

Again thank you.

Antony

I tried again now, with version 12.0 on MacBook Pro, Mojave 10.14.6. Crash again with DiscretizeRegion[mainBody].