Discretization

Meshes.discretizeFunction
discretize(geometry, [method])

Discretize geometry with discretization method.

If the method is ommitted, a default algorithm is used with a specific number of elements.

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Meshes.discretizewithinFunction
discretizewithin(boundary, method)

Discretize geometry within boundary with boundary discretization method.

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Meshes.simplexifyFunction
simplexify(object)

Discretize object into simplices using an appropriate discretization method.

Notes

This function is sometimes called "triangulate" when the object has parametric dimension 2.

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FanTriangulation

hexagon = Hexagon((0.,0.), (1.,0.), (1.,1.),
                  (0.75,1.5), (0.25,1.5), (0.,1.))

mesh = discretize(hexagon, FanTriangulation())

fig = Mke.Figure(size = (800, 400))
viz(fig[1,1], hexagon)
viz(fig[1,2], mesh, showsegments = true)
fig
Example block output

DehnTriangulation

# polygonal area
polyarea = PolyArea([(0.22926679, 0.47329807), (0.23094065, 0.44913536), (0.2569517, 0.38217533),
                     (0.3072999, 0.272418), (0.34814754, 0.18421611), (0.37949452, 0.11756973),
                     (0.4013409, 0.07247882), (0.41368666, 0.048943404), (0.42597583, 0.031655528),
                     (0.4382084, 0.0206152), (0.45038435, 0.015822414), (0.4625037, 0.017277176),
                     (0.47175184, 0.02439156), (0.47812873, 0.03716557), (0.4816344, 0.055599205),
                     (0.48226887, 0.07969247), (0.48172843, 0.10446181), (0.4800131, 0.12990724),
                     (0.47712287, 0.15602873), (0.47305775, 0.18282633), (0.47093934, 0.20558843),
                     (0.47076762, 0.22431506), (0.47254258, 0.23900622), (0.47626427, 0.24966191),
                     (0.47768936, 0.25845313), (0.47681788, 0.26537988), (0.4736498, 0.27044216),
                     (0.46818516, 0.27363995), (0.4613889, 0.27232954), (0.45326096, 0.2665109),
                     (0.44380143, 0.256184), (0.43301025, 0.24134888), (0.4246466, 0.22978415),
                     (0.41871038, 0.22148979), (0.4152017, 0.21646582), (0.4141205, 0.21471222),
                     (0.41227448, 0.21589448), (0.40966362, 0.22001258), (0.40628797, 0.22706655),
                     (0.40214747, 0.23705636), (0.40200475, 0.24653101), (0.40585983, 0.25549048),
                     (0.41371268, 0.2639348), (0.4255633, 0.2718639), (0.4378565, 0.28495985),
                     (0.4505922, 0.30322257), (0.46377045, 0.32665208), (0.47739124, 0.35524836),
                     (0.5046394, 0.36442512), (0.5455148, 0.35418236), (0.60001767, 0.32452005),
                     (0.66814786, 0.27543822), (0.7186763, 0.24664374), (0.75160307, 0.23813659),
                     (0.76692814, 0.2499168), (0.7646515, 0.28198436), (0.7769703, 0.29925033),
                     (0.8038847, 0.3017147), (0.84539455, 0.28937748), (0.9015, 0.26223865),
                     (0.94408435, 0.24899776), (0.9731477, 0.24965483), (0.98869, 0.26420987),
                     (0.9907113, 0.29266283), (0.9849871, 0.31338844), (0.97151726, 0.32638666),
                     (0.950302, 0.3316575), (0.9213412, 0.32920095), (0.8798396, 0.34078467),
                     (0.8257972, 0.36640862), (0.7592141, 0.40607283), (0.6800901, 0.4597773),
                     (0.6450007, 0.49104902), (0.6539457, 0.49988794), (0.7069251, 0.48629412),
                     (0.803939, 0.45026752), (0.877913, 0.4226481), (0.9288472, 0.40343583),
                     (0.9567415, 0.39263073), (0.961596, 0.39023277), (0.9419039, 0.40523484),
                     (0.89766514, 0.43763688), (0.8288798, 0.48743892), (0.7355478, 0.55464095),
                     (0.6655121, 0.60063523), (0.6187727, 0.6254217), (0.5953296, 0.62900037),
                     (0.5951828, 0.6113712), (0.57516366, 0.60261106), (0.53527224, 0.6027198),
                     (0.4755085, 0.6116975), (0.3958725, 0.6295441), (0.33913234, 0.6398651),
                     (0.30528808, 0.6426605), (0.2943397, 0.6379303), (0.30628717, 0.6256744),
                     (0.32149008, 0.6093727), (0.33994842, 0.5890249), (0.36166218, 0.5646312),
                     (0.38663134, 0.5361916), (0.3919681, 0.520893), (0.3776725, 0.5187355),
                     (0.34374446, 0.52971905), (0.29018405, 0.5538437), (0.25439468, 0.5678829),
                     (0.2363764, 0.5718367), (0.23612918, 0.56570506), (0.25365302, 0.549488),
                     (0.2733971, 0.5246488), (0.29536137, 0.49118724), (0.3195459, 0.4491035),
                     (0.34595063, 0.39839754), (0.3647463, 0.3590396), (0.37593287, 0.33102974),
                     (0.37951034, 0.31436795), (0.37547874, 0.30905423), (0.36070493, 0.3204269),
                     (0.33518887, 0.348486), (0.29893062, 0.3932315), (0.25193012, 0.45466346)])

mesh = discretize(polyarea, DehnTriangulation())

fig = Mke.Figure(size = (800, 400))
viz(fig[1,1], polyarea)
viz(fig[1,2], mesh, showsegments = true)
fig
Example block output

HeldTriangulation

Meshes.HeldTriangulationType
HeldTriangulation([rng]; shuffle=true)

Fast Industrial-Strength Triangulation (FIST) of polygons.

This triangulation method is the method behind the famous Mapbox's Earcut library. It is based on a ear clipping algorithm adapted for complex n-gons with holes. It has O(n²) time complexity where n is the number of vertices. In practice it is very efficient due to heuristics implemented in the algorithm.

The option shuffle is used to shuffle the order in which ears are clipped. It improves the quality of the triangles, which can be very sliver otherwise. Optionally, specify the random number generator rng.

References

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mesh = discretize(polyarea, HeldTriangulation())

fig = Mke.Figure(size = (800, 400))
viz(fig[1,1], polyarea)
viz(fig[1,2], mesh, showsegments = true)
fig
Example block output

DelaunayTriangulation

mesh = discretize(polyarea, DelaunayTriangulation())

fig = Mke.Figure(size = (800, 400))
viz(fig[1,1], polyarea)
viz(fig[1,2], mesh, showsegments = true)
fig
Example block output

As can be seen in the following example, all discretization methods for Polygon automatically work in the presence of holes:

outer = [(0.18142937, 0.54681134), (0.38282228, 0.107781954), (0.43220532, 0.013640274),
         (0.48068276, 0.019459315), (0.48322055, 0.11583236), (0.46696007, 0.2230227),
         (0.48184678, 0.2656454), (0.45998818, 0.2784367), (0.4168235, 0.2190962),
         (0.4124987, 0.21208182), (0.39593673, 0.2520411), (0.44333926, 0.28375763),
         (0.4978224, 0.3981428), (0.7703431, 0.20181546), (0.7612364, 0.33008572),
         (0.9856581, 0.2215304), (0.99374324, 0.3353423), (0.9688778, 0.38663587),
         (0.59554976, 0.655444), (0.59496254, 0.58492756), (0.27641845, 0.656314),
         (0.3242084, 0.6072907), (0.42408508, 0.49353212), (0.20984341, 0.59003067)]

inners = [[(0.87789994, 0.32551613), (0.5614043, 0.540334), (0.9494598, 0.39622766)],
          [(0.2799388, 0.52516246), (0.38555774, 0.32233855), (0.36943135, 0.30108362)]]

polyarea = PolyArea([outer, inners...])

mesh = discretize(polyarea, DelaunayTriangulation())

fig = Mke.Figure(size = (800, 400))
viz(fig[1,1], polyarea)
viz(fig[1,2], mesh, showsegments = true)
fig
Example block output

RegularDiscretization

Meshes.RegularDiscretizationType
RegularDiscretization(n1, n2, ..., np)

A method to discretize primitive geometries with n1×n2×...×np elements sampled regularly along each parametric dimensions. The adequate number of points is calculated for each type of geometry and passed to RegularSampling.

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sphere = Sphere((0.,0.,0.), 1.)

mesh = discretize(sphere, RegularDiscretization(10,10))

viz(mesh, showsegments = true)
Example block output

ManualSimplexification

box = Box((0., 0., 0.), (1., 1., 1.))

mesh = discretize(box, ManualSimplexification())

viz(mesh, colors = 1:nelements(mesh))
Example block output