Here we will solve simple 2D wave-equation and visualize it in realtime using polygons
Download original notebookInitialize mesh
50x50 equally distributed points on 2D plane
field0 = Table[0., {i,50}, {j,50}]; field1 = field0; field2 = field0; (* initial perturbation *) field2[[25,25]] = 0.01; lattice = Table[{i,j}, {i, 50}, {j,50}]; mesh = Flatten[lattice, 1]; makeVertices = Compile[{{f, _Real, 2}, {pairs, _Integer, 2}, {scale, _Real}}, Join[#, {scale f[[#[[1]], #[[2]]]]}] &/@ pairs ]; vertices = NumericArray[makeVertices[field0, mesh, 300]];
Triangulation
This is necessary for plotting the data using 3D polygons. That we keep indexes of all polygons the same, but will update vertices
Needs["ComputationalGeometry`"]; triangles2[points_] := Module[{tr, triples}, tr = DelaunayTriangulation[points]; triples = Flatten[Function[{v, list}, Switch[Length[list], (* account for nodes with connectivity 2 or less *) 1, {}, 2, {Flatten[{v, list}]}, _, {v, ##} & @@@ Partition[list, 2, 1, {1, 1}] ] ] @@@ tr, 1]; Cases[GatherBy[triples, Sort], a_ /; Length[a] == 3 :> a[[1]]]] triangles = triangles2[mesh];
Let us check the result
ListPlot[mesh, PlotStyle->{Red, PointSize[0.01]}, Prolog->{Line[mesh[[#]]] &/@ triangles}]
(*VB[*)(FrontEndRef["51f56b92-40e0-47fb-8b1a-9df9b700dd1d"])(*,*)(*"1:eJxTTMoPSmNkYGAoZgESHvk5KRCeEJBwK8rPK3HNS3GtSE0uLUlMykkNVgEKmxqmmZolWRrpmhikGuiamKcl6VokGSbqWqakWSaZGxikpBimAACAORXR"*)(*]VB*)
Define our equation
This is discretization of a wave-equation in a form
we follows the following tutorial and rewrite it in a form
update = Compile[{{i, _Integer}, {j, _Integer}, {f, _Real, 2}, {p, _Real, 2}}, 0.001 (If[i-1 > 0, f[[i-1, j]], 0] + If[i+1 < 51, f[[i+1, j]], 0] + If[j+1<51, f[[i, j+1]], 0] + If[j-1 > 0, f[[i, j-1]], 0] - 4 f[[i,j]]) + 2 f[[i,j]] - p[[i,j]] ];
Update function calculated a new value at vertex .
Realtime modelling
For displaying we use simple Polygon
and GraphicsComplex
Run the cell below after
evaluation of the initialization cells
Graphics3D[{Shadows[True], SpotLight[Blue, {2.4463, 60.1604, 9.3104}, 1.5, 3000], SpotLight[Red, {48.1378, 47.8679, 12.4545}, 1.5, 3000], Roughness[0], GraphicsComplex[vertices // Offload, Polygon[triangles]], AnimationFrameListener[vertices // Offload, "Event" -> "frame calc"] }, ImageSize->500]
(*VB[*)(FrontEndRef["93deb6c9-ae46-42de-b1aa-0dd3fdbb85eb"])(*,*)(*"1:eJxTTMoPSmNkYGAoZgESHvk5KRCeEJBwK8rPK3HNS3GtSE0uLUlMykkNVgEKWxqnpCaZJVvqJqaamOmaGKWk6iYZJibqGqSkGKelJCVZmKYmAQCW7Bbt"*)(*]VB*)
try to pan and zoom it ! 😉
Now we wrap our calculations into a animation cycle
Evaluate this if animation did not start
(*BB[*)(*reset again*)(*,*)(*"1:eJxTTMoPSmNhYGAo5gcSAUX5ZZkpqSn+BSWZ+XnFaYwgCS4g4Zyfm5uaV+KUXxEMUqxsbm6exgSSBPGCSnNSg9mAjOCSosy8dLBYSFFpKpoKkDkeqYkpEFXBILO1sCgJSczMQVYCAOFrJEU="*)(*]BB*) field0 = Table[0., {i,50}, {j,50}]; field1 = field0; field2 = field0; (* initial perturbation *) field2[[25,25]] = 0.01; EventHandler["frame calc", Function[Null, With[{i = #[[1]], j = #[[2]]}, field0[[i,j]] = update[i,j, field1, field2]; ]&/@ RandomSample[mesh]; field2 = field1; field1 = field0; vertices = NumericArray[makeVertices[field0, mesh, 300]]; ]]; (*BB[*)(*kickstarter*)(*,*)(*"1:eJxTTMoPSmNhYGAo5gcSAUX5ZZkpqSn+BSWZ+XnFaYwgCS4g4Zyfm5uaV+KUXxEMUqxsbm6exgSSBPGCSnNSg9mAjOCSosy8dLBYSFFpKpoKkDkeqYkpEFXBILO1sCgJSczMQVYCAOFrJEU="*)(*]BB*) EventFire["frame calc", True];
To stop it
EventRemove["frame calc"]