Function Repository Resource:

BoundaryElementMeshJoin

Source Notebook

Join multiple boundaries to form a single boundary ElementMesh

Contributed by: Oliver Rübenkönig

ResourceFunction["BoundaryElementMeshJoin"][bmesh₁,bmesh₂,…]

returns a boundary ElementMesh consisting of joined boundary MeshRegion or ElementMesh objects bmesh_i.

Details and Options

ResourceFunction["BoundaryElementMeshJoin"] accepts the options for ToBoundaryMesh.

Inputs can be either MeshRegion or ElementMesh objects.

Examples

Basic Examples (4)

Load the Finite Element Method package:

In[1]:=

Define two boundary meshes:

In[2]:=

$mr1 = \!\(\* GraphicsBox[ TagBox[ DynamicModuleBox[{ Typeset`mesh = {MeshRegion, {Properties -> {{1, {1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 52, 53, 54, 55, 56, 57, 58, 59, 60, 61, 62, 63, 64, 65, 66, 67, 68, 69, 70, 71, 72, 73, 74, 75, 76, 77, 78, 79, 80}} -> ( MeshCellMarker -> {1, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 3, 2, 1, 3, 1, 3, 1, 3, 1, 3, 1, 3, 1, 3, 1, 3, 1, 3, 1, 3, 1, 3, 1, 3, 1, 3, 1, 3, 1, 3, 1, 3, 1, 3, 1, 3, 1, 3, 1, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 3}), {1, Default} -> ( MeshCellMarker -> 0)}}}}, TagBox[GraphicsComplexBox[CompressedData[" 1:eJx1ktEJwjAYhIObOIMDZAVxBEFfnUc3cIQ++1RQKBSEgg8tLSWlIWQEbfGv 5MP/h1KOJJfL3a33p91hZYzZfr7pr83lPM3Vpjhb8GaeG9Zzm7IU2F8u+DjP E+erBT/u07zAV2N/A/4W5zvc14PPgW8A3wg+D74Avmj/+yZY1jPF12jpa4rl fK74Hr//QslB+Esll2iZS4rl/krJLVrmlmLRVyu5Cn+j5Cz6W+hvcX+H+zu8 r8f7euhz0Oegb4C+AfpG6Buhz0Ofh74AfQH6IvSxd+wZe8UesTfsCXvBHjB3 5sxcmSNzY07MhTnQd/pMX38+vgGZwkZb "], {Hue[0.6, 0.3, 0.75], TagBox[LineBox[{{1, 22}, {2, 1}, {3, 2}, {4, 3}, {5, 4}, {6, 5}, {7, 6}, {8, 7}, {9, 8}, {10, 9}, {11, 10}, {12, 11}, {13, 12}, {14, 13}, {15, 14}, {16, 15}, {17, 16}, {18, 17}, { 19, 18}, {20, 19}, {23, 21}, {21, 20}, {22, 24}, {25, 23}, {24, 26}, {27, 25}, {26, 28}, {29, 27}, {28, 30}, { 31, 29}, {30, 32}, {33, 31}, {32, 34}, {35, 33}, {34, 36}, {37, 35}, {36, 38}, {39, 37}, {38, 40}, {41, 39}, { 40, 42}, {43, 41}, {42, 44}, {45, 43}, {44, 46}, {47, 45}, {46, 48}, {49, 47}, {48, 50}, {51, 49}, {50, 52}, { 53, 51}, {52, 54}, {55, 53}, {54, 56}, {57, 55}, {56, 58}, {59, 57}, {58, 60}, {60, 61}, {61, 62}, {62, 63}, { 63, 64}, {64, 65}, {65, 66}, {66, 67}, {67, 68}, {68, 69}, {69, 70}, {70, 71}, {71, 72}, {72, 73}, {73, 74}, { 74, 75}, {75, 76}, {76, 77}, {77, 78}, {78, 79}, {79, 80}, {80, 59}}], Annotation[#, "Geometry"]& ], PointBox[{1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 52, 53, 54, 55, 56, 57, 58, 59, 60, 61, 62, 63, 64, 65, 66, 67, 68, 69, 70, 71, 72, 73, 74, 75, 76, 77, 78, 79, 80}]}], MouseAppearanceTag["LinkHand"]], AllowKernelInitialization->False], "MeshGraphics", AutoDelete->True, Editable->False, Selectable->False], DefaultBaseStyle->{"MeshGraphics", FrontEnd`GraphicsHighlightColor -> Hue[0.1, 1, 0.7]}]\); mr2 = \!\(\* GraphicsBox[ TagBox[ DynamicModuleBox[{ Typeset`mesh = {MeshRegion, {Properties -> {{1, {1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48}} -> ( MeshCellMarker -> {1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1}), {1, Default} -> (MeshCellMarker -> 0)}}}}, TagBox[GraphicsComplexBox[CompressedData[" 1:eJw90m1Ik1EUwPFpW5SFUkYfbIhb0UyzxFARggPFlkXNUoxgvc1ZbBZNU2Ji y5fESP2QNm2W2JQ0RWNhA8Oak4xlTelx6abN1V50j9oSV/Zm0cp7rztwOV/P j/vnZMnTzwUzGIz4/29545kBvB1wJEmo7ZunoaqiUXlorRMe9TeWaXpoGFLW 32/e4oKjTTvsbSIaJo0Hkq1b3bDPpBq6M+mBs95vvgX2FARV3vP69nrgeHZG bVLwNMzXPxn/UjwNOl5oouDtNCwKouVdHVPQopiRxZR4YBvHoE8xuKFV2j+v 2EBDgmrPjYRXLsiWsbjVJTRsypQN1L1xQkthUbLfRAO3XRR22+IAX8Gqi72L NGgiYpnLd/vR0OD1GpkXPtthNhE75D/N/c8HbRBiwI58UW7NsacTkNeAHe1/ wtmRRisU92GHa/aErfb3GMzlYMeaaHddvngUhoXYEcfbtbPTb4bKVh5yVPcq uaH2EYh8oUeOk6yvz+i/FIR1xiNHCH91TZCUgv2pEuRgX/3Oq9pMweG0NOTA f0DBL/5G5JhTSZrLuBQcpLHDW2pc6rtFwSXiiBru4nenjkAucTCvW85TAjO0 EcfumNfb2RXvwEEc0qwGYQRjDFjEsaB7zHnYZYFY4nhZVF5eXToecLg6mnx3 894HHFfkKYWjOZOwnjgkYrFzIv1DwLGuwPbD7/8YcJy6ibvhEoeJdNN2DTts pJszA9hxmnSjkGFHBulm6dMgcnSTbqLMauTQkG7CL2uR4wHpRqcvRQ4J6UZt VSPHSjfmzDjkWOmmR5uKHP8A2tqKSg== "], {Hue[0.6, 0.3, 0.75], TagBox[LineBox[{{1, 2}, {2, 3}, {3, 4}, {4, 5}, {5, 6}, {6, 7}, {7, 8}, {8, 9}, {9, 10}, {10, 11}, {11, 12}, {12, 13}, {13, 14}, {14, 15}, {15, 16}, {16, 17}, {17, 18}, {18, 19}, { 19, 20}, {20, 21}, {21, 22}, {22, 23}, {23, 24}, {24, 25}, {25, 26}, {26, 27}, {27, 28}, {28, 29}, {29, 30}, { 30, 31}, {31, 32}, {32, 33}, {33, 34}, {34, 35}, {35, 36}, {36, 37}, {37, 38}, {38, 39}, {39, 40}, {40, 41}, { 41, 42}, {42, 43}, {43, 44}, {44, 45}, {45, 46}, {46, 47}, {47, 48}, {48, 1}}], Annotation[#, "Geometry"]& ], PointBox[{1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48}]}], MouseAppearanceTag["LinkHand"]], AllowKernelInitialization->False], "MeshGraphics", AutoDelete->True, Editable->False, Selectable->False], DefaultBaseStyle->{"MeshGraphics", FrontEnd`GraphicsHighlightColor -> Hue[0.1, 1, 0.7]}]\);$

Join the two:

In[3]:=

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Visualize the resulting boundary ElementMesh:

In[4]:=

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Scope (7)

Create boundary element meshes directly:

In[5]:=

Join the boundary meshes:

In[6]:=

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Visualize the resulting boundary ElementMesh:

In[7]:=

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Generate a full ElementMesh with the internal circle boundary:

In[8]:=

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Create three boundary meshes:

In[9]:=

bmesh1 = ToBoundaryMesh[Rectangle[], "MaxBoundaryCellMeasure" -> Infinity];
bmesh2 = ToBoundaryMesh["Coordinates" -> {{1/4, 1/2}, {2/3, 1/2}}, "BoundaryElements" -> {LineElement[{{1, 2}}]}];
bmesh3 = ToBoundaryMesh[Rectangle[{2/3, 1/3}, {4/5, 2/3}], "MaxBoundaryCellMeasure" -> Infinity];

Join the boundary meshes:

In[10]:=

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Visualize the resulting boundary ElementMesh:

In[11]:=

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Generate a full ElementMesh with the internal line boundary and a region hole in the box:

In[12]:=

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Create two boundary meshes:

In[13]:=

Join the boundary meshes:

In[14]:=

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Visualize the resulting boundary ElementMesh:

In[15]:=

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Generate a full ElementMesh with the internal boundaries:

In[16]:=

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Create two boundary meshes:

In[17]:=

Join the boundary meshes:

In[18]:=

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Visualize the resulting boundary ElementMesh:

In[19]:=

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Generate a full ElementMesh with an internal boundary at the connection:

In[20]:=

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Create two boundary meshes:

In[21]:=

Join the boundary meshes:

In[22]:=

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Visualize the resulting boundary ElementMesh:

In[23]:=

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Generate a full ElementMesh with the internal boundaries:

In[24]:=

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Generate a full ElementMesh with the internal boundaries and a region hole:

In[25]:=

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Create two boundary meshes in 3D:

In[26]:=

Join the boundary meshes:

In[27]:=

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Visualize the resulting boundary ElementMesh:

In[28]:=

Out[28]=

Generate a full ElementMesh with the internal ball boundary:

In[29]:=

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Create two boundary meshes in 3D:

In[30]:=

Join the boundary meshes:

In[31]:=

Out[31]=

Visualize the resulting boundary ElementMesh:

In[32]:=

Out[32]=

Generate a full ElementMesh:

In[33]:=

Out[33]=

Applications (2)

Use an image to generate a mesh that has material regions. Set up the image:

In[34]:=

$\!\(\* GraphicsBox[ TagBox[RasterBox[CompressedData[" 1:eJztnWlwE+cZx02OSYCZTtp8KFO+NITppEOadDjCWQikgZCkJIGQDk3bBBpI kzYkgaSUEKaZ6ZeSA3zr8oEtW7bkC9vYsmVbt2Vbh20ZHxhCTIqNAzHhtAGD oY/02otqC+0arfZ9sZ7//K2RVrvvvtrn+e17SLt+ZNP7a7fcExMTs/1BeFi7 ccfybds27lz3ELxYv3X7O29v3fzW6q0fbX5787YFm+6FhVvg7+FJMTG+5zdR KBQqpLp7TqLRY9135gzt3GRazd5DaPQoNza1HPumi3ZuMi1vSysaPcrAzjdd x2nnJtOiHiM0g0ZweEU9RmgGjeDwinqM0AwaweEV9RihGTSCwyvqMUIzaASH V9RjhGbQCA6vqMcIzaARHF5RjxGaQSM4vKIeIzSDRnB4RT1GaAaN4PCKeozQ DBrB4RX1GKEZNILDK+oxQjNoBIdX1GOEZtAIDq+oxwjNoBEcXlGPEZpBIzi8 oh4jNINGcHhFPUZoBo3g8Ip6jFhwy6E2geYt6lBre2tbBzxKadijkLohOCKK etJKQASXXT77n3Np1tTc0tjk9TQ2uz1NLnej0+VpADvdt+zywEJ4C1aA1WBl 2IQr3FfsSMmkWIvVXmM0G00WKQ17hEqKyA6CwyvquS0uIFwOw0JIcqer0VHX AMlcVWMqrzAUl5YVFBbn6gqyNdoMtSY9Q52anqFKTVeo0uTKVJkiRaZQJctV STJlstxn/xMVLIS3YAVYTZW6HzaBDTPVGihEqyuAAotLy6FwyF7wq6++tnTp 08uXPyOlFy/+jVKV2nH4CIIjmajn/B2TwnWK4CW0BXX1TrPFpq+sKjpQmpOb tz8zG5IcEj4xWRGfKItLSI6NT/I7GZ4Tw/KxTkiSj3LQ1bhCoEBSMizcF5e4 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eWDXUIFRVWKZFARnvKIeo8iZo4kYEpjkMHe2h55SU7MXkhzaApe7ERIeDD2o +gDDS7Lc5W6C1WBl2ATyyjvS6o0q+a5jJKgRHF5RjxF1kzwXbuoVlsAIDq+o xwjNoBEcXlGPEZpBIzi8oh4jNINGcHhFPUZoBo3g8Ip6jNAMGsHhFfUYoRk0 gsMr6jFCM2gEh1fUY4Rm0AgOr6jHCM2gERxeUY8RmkEjOLyiHiM0g0ZweEU9 RmgGjeDwinqM0AwaweEV9RihGTSCwyvqMUIzaASHV9RjhGbQCA6vqMcIzaAR HF5RjxGaQSM4vKIeIzSDRnB4RT1GaAYN4HQdR3BCqePwETR6lNs7Ok9099DO TRQqKvQ/7RRk6g== "], {{0, 102}, {275, 0}}, {0, 255}, ColorFunction->RGBColor], BoxForm`ImageTag["Byte", ColorSpace -> "RGB", Interleaving -> True], Selectable->False], DefaultBaseStyle->"ImageGraphics", ImageSize->{137.5, Automatic}, ImageSizeRaw->{275, 102}, PlotRange->{{0, 275}, {0, 102}}]\)$

Find the image components:

In[35]:=

Generate mesh regions:

In[36]:=

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Generate boundary ElementMesh objects and join them:

In[37]:=

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Visualize the boundary ElementMesh:

In[38]:=

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Find the centroids of the meshes:

In[39]:=

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Generate the full ElementMesh with region markers:

In[40]:=

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Solve a PDE with different coefficient values in the different material regions:

In[41]:=

$uif = NDSolveValue[{Inactive[Div][ Piecewise[{{-10, ElementMarker == 1}, {-25, ElementMarker == 2}}, -1] Inactive[Grad][ u[x, y], {x, y}], {x, y}] == 0, DirichletCondition[u[x, y] == 1, x == 0], DirichletCondition[u[x, y] == 0, x == 1264]}, u, {x, y} \[Element] mesh];$

Visualize the solution:

In[42]:=

$Show[ContourPlot[uif[x, y], {x, y} \[Element] mesh], bm["Wireframe"], AspectRatio -> Automatic]$

Out[42]=

Create two boundary meshes in 3D:

In[43]:=

Join the meshes:

In[44]:=

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Visualize the boundary ElementMesh:

In[45]:=

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Numerically integrate over the boundary ElementMesh:

In[46]:=

$NIntegrate[1, {x, y, z} \[Element] bmesh]$

Out[46]=

Possible Issues (6)

Sometimes the boundary mesh quality is not good enough to generate a full ElementMesh. Create two boundary ElementMesh objects:

In[47]:=

Join the boundary meshes:

In[48]:=

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Visualize the resulting boundary ElementMesh:

In[49]:=

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Generate a full ElementMesh with the internal circle boundary:

In[50]:=

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In that case, generate a small offset at the intersecting boundary:

In[51]:=

$offset = 10^-3; bmesh1 = ToBoundaryMesh[Rectangle[]]; bmesh2 = ToBoundaryMesh[Disk[{1, 1/2}, 1/2 - offset]]; bmesh = ResourceFunction["BoundaryElementMeshJoin"][bmesh1, bmesh2];$