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Explore Stack Internal then ToElementMesh does have access to the symbolic region. In fact internally, it generates a NumericalRegion just as in this post and proceeds like shown here.
You can also set a (boundary) mesh to a NumericalRegion:
SetNumericalRegionElementMesh[nr, bmesh] mesh2 = ToElementMesh[nr] Pi - Total[mesh2["MeshElementMeasure"], 2] Hope that helps.
then ToElementMesh does have access to the symbolic region. In fact internally, it generates a NumericalRegion just as in this post and proceeds like shown here. Hope that helps.
then ToElementMesh does have access to the symbolic region. In fact internally, it generates a NumericalRegion just as in this post and proceeds like shown here.
You can also set a (boundary) mesh to a NumericalRegion:
SetNumericalRegionElementMesh[nr, bmesh] mesh2 = ToElementMesh[nr] Pi - Total[mesh2["MeshElementMeasure"], 2] Hope that helps.
When you call
ToElementMesh[Disk[]] then ToElementMesh does have access to the symbolic region. In fact internally, it generates a NumericalRegion just as in this post and proceeds like shown here. Hope that helps.
When you call
ToElementMesh[Disk[]] then ToElementMesh does have access to the symbolic region. In fact internally, it generates a NumericalRegion just as in this post and proceeds like shown here. Hope that helps.
I'll try do give a different explanation than given in the tutorial next. Let's consider for a minute that we have a boundary element mesh in input form. Likelike the following:
Needs["NDSolve`FEM`"] bmesh = ToBoundaryMesh["Coordinates" -> { {1., 0.}, {0.9125378206934781, 0.4089923297618155},{0.6654505497212123, 0.7464419373774067}, {0.32914518683708227, 0.9442793262493796}, {2.8415758474179748*^-8, 0.9999999999999996}, {-0.40899232976181543, 0.9125378206934781}, {-0.7464419373774067, 0.6654505497212122}, {-0.9442793262493796, 0.3291451868370823}, {-0.9999999999999996, 2.8415758313367482*^-8}, {-0.9125378206934783, -0.40899232976181493}, {-0.6654505497212126, -0.7464419373774064}, {-0.3291451868370832, -0.9442793262493793}, {-2.841576059504587*^-8, -0.9999999999999996}, {0.40899232976181327, -0.9125378206934791}, {0.7464419373774028, -0.6654505497212166}, {0.9442793262493757, -0.3291451868370933}}, "BoundaryElements" -> {LineElement[{{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, 1}}]}]; {0.6654505497212123, 0.7464419373774067}, {0.32914518683708227, 0.9442793262493796}, When we pass that to {2.8415758474179748*^-8, 0ToElementMesh it will generate a full mesh that approximates the boundary given.9999999999999996}, {-0.40899232976181543, 0.9125378206934781},ToElementMesh has no way of knowing what the original input to {-0.7464419373774067, 0ToBoundaryMesh was when it is given a boundary mesh representation.6654505497212122},
mesh = ToElementMesh[bmesh]; So how well does {-0.9442793262493796, 0.3291451868370823},ToElementMesh approximate {-0.9999999999999996, 2.8415758313367482*^-8},bmesh? We can not really tell because we do not know what {-0bmesh is.9125378206934783
Now, I am telling you that -0.40899232976181493},bmesh is supposed to represent a {-0Disk[].6654505497212126, -0 Then and only then we can check:
Pi - Total[mesh["MeshElementMeasure"], 2] 0.0819661 And it is a poor presentation.7464419373774064}, {-0 If you do have a symbolic representation of your region it's a good idea to pass that along.3291451868370832, That is what -0ToNumericalRegion is for.9442793262493793}, {-2 Let's look at an example.841576059504587*^-8, This generates a numerical region of a -0Disk[]:
nr = ToNumericalRegion[Disk[]]; We can now 'fill in' a boundary mesh like so:
bem2 = ToBoundaryMesh[nr, "MaxBoundaryCellMeasure" -> .5, AccuracyGoal -> 1] These options are in fact the ones I used to generate the above example boundary mesh.9999999999999996}, Note that now the {0.40899232976181327,NumericalRegion has a boundary mesh -0.9125378206934791}, {0.7464419373774028, the same as -0.6654505497212166},bem2
bem2 === nr["BoundaryMesh"] True When you pass the numerical region to {0.9442793262493757,ToElementMesh things are very different as now -0ToElementMesh has access to the boundary representation and the symbolic representation of the region and can thus generate a better mesh.3291451868370933}}, "BoundaryElements" -> {LineElement[{{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, 1}}]}];
mesh2 = ToElementMesh[nr]; Pi - Total[mesh2["MeshElementMeasure"], 2] 3.893310501545955`*^-6 I'll try do give a different explanation than given in the tutorial next. Let's consider for a minute that we have a boundary element mesh in input form. Like the following:
Needs["NDSolve`FEM`"] bmesh = ToBoundaryMesh["Coordinates" -> { {1., 0.}, {0.9125378206934781, 0.4089923297618155}, {0.6654505497212123, 0.7464419373774067}, {0.32914518683708227, 0.9442793262493796}, {2.8415758474179748*^-8, 0.9999999999999996}, {-0.40899232976181543, 0.9125378206934781}, {-0.7464419373774067, 0.6654505497212122}, {-0.9442793262493796, 0.3291451868370823}, {-0.9999999999999996, 2.8415758313367482*^-8}, {-0.9125378206934783, -0.40899232976181493}, {-0.6654505497212126, -0.7464419373774064}, {-0.3291451868370832, -0.9442793262493793}, {-2.841576059504587*^-8, -0.9999999999999996}, {0.40899232976181327, -0.9125378206934791}, {0.7464419373774028, -0.6654505497212166}, {0.9442793262493757, -0.3291451868370933}}, "BoundaryElements" -> {LineElement[{{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, 1}}]}];
I'll try do give a different explanation than given in the tutorial next. Let's consider for a minute that we have a boundary element mesh like the following:
Needs["NDSolve`FEM`"] bmesh = ToBoundaryMesh["Coordinates" -> { {1., 0.}, {0.9125378206934781, 0.4089923297618155},{0.6654505497212123, 0.7464419373774067}, {0.32914518683708227, 0.9442793262493796}, {2.8415758474179748*^-8, 0.9999999999999996}, {-0.40899232976181543, 0.9125378206934781}, {-0.7464419373774067, 0.6654505497212122}, {-0.9442793262493796, 0.3291451868370823}, {-0.9999999999999996, 2.8415758313367482*^-8}, {-0.9125378206934783, -0.40899232976181493}, {-0.6654505497212126, -0.7464419373774064}, {-0.3291451868370832, -0.9442793262493793}, {-2.841576059504587*^-8, -0.9999999999999996}, {0.40899232976181327, -0.9125378206934791}, {0.7464419373774028, -0.6654505497212166}, {0.9442793262493757, -0.3291451868370933}}, "BoundaryElements" -> {LineElement[{{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, 1}}]}]; When we pass that to ToElementMesh it will generate a full mesh that approximates the boundary given. ToElementMesh has no way of knowing what the original input to ToBoundaryMesh was when it is given a boundary mesh representation.
mesh = ToElementMesh[bmesh]; So how well does ToElementMesh approximate bmesh? We can not really tell because we do not know what bmesh is.
Now, I am telling you that bmesh is supposed to represent a Disk[]. Then and only then we can check:
Pi - Total[mesh["MeshElementMeasure"], 2] 0.0819661 And it is a poor presentation. If you do have a symbolic representation of your region it's a good idea to pass that along. That is what ToNumericalRegion is for. Let's look at an example. This generates a numerical region of a Disk[]:
nr = ToNumericalRegion[Disk[]]; We can now 'fill in' a boundary mesh like so:
bem2 = ToBoundaryMesh[nr, "MaxBoundaryCellMeasure" -> .5, AccuracyGoal -> 1] These options are in fact the ones I used to generate the above example boundary mesh. Note that now the NumericalRegion has a boundary mesh - the same as bem2
bem2 === nr["BoundaryMesh"] True When you pass the numerical region to ToElementMesh things are very different as now ToElementMesh has access to the boundary representation and the symbolic representation of the region and can thus generate a better mesh.
mesh2 = ToElementMesh[nr]; Pi - Total[mesh2["MeshElementMeasure"], 2] 3.893310501545955`*^-6