Holes in ElementMesh with ToElementMesh of ImplicitRegion
$begingroup$
I am trying to plot a function in a region below a level curve of the function and within a cell. I have been doing this by calculating an ElementMesh
using ImplicitRegion
and ToElementMesh
, but the result has holes.
Here is the cell (it's just a square),
cell = Parallelogram[{-0.5`, -0.5`}, {{1.`, 0.`}, {0.`, 1.`}}];
Graphics[{Transparent, EdgeForm[Thick], cell}]
and the function,
f[kx_, ky_, n_] :=
Sort[Eigenvalues[{{(-1. + kx)^2 + (-1. + ky)^2, -0.23, 0., -0.23,
0.12, 0., 0., 0.,
0.}, {-0.23, (-1. + kx)^2 + (0. + ky)^2, -0.23, 0.12, -0.23,
0.12, 0., 0., 0.}, {0., -0.23, (-1. + kx)^2 + (1. + ky)^2, 0.,
0.12, -0.23, 0., 0., 0.}, {-0.23, 0.12,
0., (0. + kx)^2 + (-1. + ky)^2, -0.23, 0., -0.23, 0.12,
0.}, {0.12, -0.23,
0.12, -0.23, (0. + kx)^2 + (0. + ky)^2, -0.23, 0.12, -0.23,
0.12}, {0., 0.12, -0.23, 0., -0.23, (0. + kx)^2 + (1. + ky)^2,
0., 0.12, -0.23}, {0., 0., 0., -0.23, 0.12,
0., (1. + kx)^2 + (-1. + ky)^2, -0.23, 0.}, {0., 0., 0.,
0.12, -0.23,
0.12, -0.23, (1. + kx)^2 + (0. + ky)^2, -0.23}, {0., 0., 0.,
0., 0.12, -0.23, 0., -0.23, (1. + kx)^2 + (1. + ky)^2}}]][[
n]];
Plot3D[f[x, y, 4], {x, y} [Element] cell, PlotPoints -> 50]
and what the region should look like,
isovalue = 1.29897233417072;
ContourPlot[f[x, y, 4], {x, y} [Element] cell,
Contours -> {isovalue}, ColorFunction -> GrayLevel,
PlotPoints -> 100]
This is what I have tried
reg = ToElementMesh[
ImplicitRegion[
f[x, y, 4] < isovalue && {x, y} [Element] cell, {x, y}],
"MaxBoundaryCellMeasure" -> 0.01, MeshQualityGoal -> 1,
PerformanceGoal -> "Quality", MaxCellMeasure -> 0.01,
"BoundaryMeshGenerator" -> "Continuation"];
RegionPlot[reg]
The region is no more accurate when I decrease MaxCellMeasure
or MaxBoundaryCellMeasure
. I also tried the solution suggested here.
plotting finite-element-method mesh implicit
$endgroup$
add a comment |
$begingroup$
I am trying to plot a function in a region below a level curve of the function and within a cell. I have been doing this by calculating an ElementMesh
using ImplicitRegion
and ToElementMesh
, but the result has holes.
Here is the cell (it's just a square),
cell = Parallelogram[{-0.5`, -0.5`}, {{1.`, 0.`}, {0.`, 1.`}}];
Graphics[{Transparent, EdgeForm[Thick], cell}]
and the function,
f[kx_, ky_, n_] :=
Sort[Eigenvalues[{{(-1. + kx)^2 + (-1. + ky)^2, -0.23, 0., -0.23,
0.12, 0., 0., 0.,
0.}, {-0.23, (-1. + kx)^2 + (0. + ky)^2, -0.23, 0.12, -0.23,
0.12, 0., 0., 0.}, {0., -0.23, (-1. + kx)^2 + (1. + ky)^2, 0.,
0.12, -0.23, 0., 0., 0.}, {-0.23, 0.12,
0., (0. + kx)^2 + (-1. + ky)^2, -0.23, 0., -0.23, 0.12,
0.}, {0.12, -0.23,
0.12, -0.23, (0. + kx)^2 + (0. + ky)^2, -0.23, 0.12, -0.23,
0.12}, {0., 0.12, -0.23, 0., -0.23, (0. + kx)^2 + (1. + ky)^2,
0., 0.12, -0.23}, {0., 0., 0., -0.23, 0.12,
0., (1. + kx)^2 + (-1. + ky)^2, -0.23, 0.}, {0., 0., 0.,
0.12, -0.23,
0.12, -0.23, (1. + kx)^2 + (0. + ky)^2, -0.23}, {0., 0., 0.,
0., 0.12, -0.23, 0., -0.23, (1. + kx)^2 + (1. + ky)^2}}]][[
n]];
Plot3D[f[x, y, 4], {x, y} [Element] cell, PlotPoints -> 50]
and what the region should look like,
isovalue = 1.29897233417072;
ContourPlot[f[x, y, 4], {x, y} [Element] cell,
Contours -> {isovalue}, ColorFunction -> GrayLevel,
PlotPoints -> 100]
This is what I have tried
reg = ToElementMesh[
ImplicitRegion[
f[x, y, 4] < isovalue && {x, y} [Element] cell, {x, y}],
"MaxBoundaryCellMeasure" -> 0.01, MeshQualityGoal -> 1,
PerformanceGoal -> "Quality", MaxCellMeasure -> 0.01,
"BoundaryMeshGenerator" -> "Continuation"];
RegionPlot[reg]
The region is no more accurate when I decrease MaxCellMeasure
or MaxBoundaryCellMeasure
. I also tried the solution suggested here.
plotting finite-element-method mesh implicit
$endgroup$
add a comment |
$begingroup$
I am trying to plot a function in a region below a level curve of the function and within a cell. I have been doing this by calculating an ElementMesh
using ImplicitRegion
and ToElementMesh
, but the result has holes.
Here is the cell (it's just a square),
cell = Parallelogram[{-0.5`, -0.5`}, {{1.`, 0.`}, {0.`, 1.`}}];
Graphics[{Transparent, EdgeForm[Thick], cell}]
and the function,
f[kx_, ky_, n_] :=
Sort[Eigenvalues[{{(-1. + kx)^2 + (-1. + ky)^2, -0.23, 0., -0.23,
0.12, 0., 0., 0.,
0.}, {-0.23, (-1. + kx)^2 + (0. + ky)^2, -0.23, 0.12, -0.23,
0.12, 0., 0., 0.}, {0., -0.23, (-1. + kx)^2 + (1. + ky)^2, 0.,
0.12, -0.23, 0., 0., 0.}, {-0.23, 0.12,
0., (0. + kx)^2 + (-1. + ky)^2, -0.23, 0., -0.23, 0.12,
0.}, {0.12, -0.23,
0.12, -0.23, (0. + kx)^2 + (0. + ky)^2, -0.23, 0.12, -0.23,
0.12}, {0., 0.12, -0.23, 0., -0.23, (0. + kx)^2 + (1. + ky)^2,
0., 0.12, -0.23}, {0., 0., 0., -0.23, 0.12,
0., (1. + kx)^2 + (-1. + ky)^2, -0.23, 0.}, {0., 0., 0.,
0.12, -0.23,
0.12, -0.23, (1. + kx)^2 + (0. + ky)^2, -0.23}, {0., 0., 0.,
0., 0.12, -0.23, 0., -0.23, (1. + kx)^2 + (1. + ky)^2}}]][[
n]];
Plot3D[f[x, y, 4], {x, y} [Element] cell, PlotPoints -> 50]
and what the region should look like,
isovalue = 1.29897233417072;
ContourPlot[f[x, y, 4], {x, y} [Element] cell,
Contours -> {isovalue}, ColorFunction -> GrayLevel,
PlotPoints -> 100]
This is what I have tried
reg = ToElementMesh[
ImplicitRegion[
f[x, y, 4] < isovalue && {x, y} [Element] cell, {x, y}],
"MaxBoundaryCellMeasure" -> 0.01, MeshQualityGoal -> 1,
PerformanceGoal -> "Quality", MaxCellMeasure -> 0.01,
"BoundaryMeshGenerator" -> "Continuation"];
RegionPlot[reg]
The region is no more accurate when I decrease MaxCellMeasure
or MaxBoundaryCellMeasure
. I also tried the solution suggested here.
plotting finite-element-method mesh implicit
$endgroup$
I am trying to plot a function in a region below a level curve of the function and within a cell. I have been doing this by calculating an ElementMesh
using ImplicitRegion
and ToElementMesh
, but the result has holes.
Here is the cell (it's just a square),
cell = Parallelogram[{-0.5`, -0.5`}, {{1.`, 0.`}, {0.`, 1.`}}];
Graphics[{Transparent, EdgeForm[Thick], cell}]
and the function,
f[kx_, ky_, n_] :=
Sort[Eigenvalues[{{(-1. + kx)^2 + (-1. + ky)^2, -0.23, 0., -0.23,
0.12, 0., 0., 0.,
0.}, {-0.23, (-1. + kx)^2 + (0. + ky)^2, -0.23, 0.12, -0.23,
0.12, 0., 0., 0.}, {0., -0.23, (-1. + kx)^2 + (1. + ky)^2, 0.,
0.12, -0.23, 0., 0., 0.}, {-0.23, 0.12,
0., (0. + kx)^2 + (-1. + ky)^2, -0.23, 0., -0.23, 0.12,
0.}, {0.12, -0.23,
0.12, -0.23, (0. + kx)^2 + (0. + ky)^2, -0.23, 0.12, -0.23,
0.12}, {0., 0.12, -0.23, 0., -0.23, (0. + kx)^2 + (1. + ky)^2,
0., 0.12, -0.23}, {0., 0., 0., -0.23, 0.12,
0., (1. + kx)^2 + (-1. + ky)^2, -0.23, 0.}, {0., 0., 0.,
0.12, -0.23,
0.12, -0.23, (1. + kx)^2 + (0. + ky)^2, -0.23}, {0., 0., 0.,
0., 0.12, -0.23, 0., -0.23, (1. + kx)^2 + (1. + ky)^2}}]][[
n]];
Plot3D[f[x, y, 4], {x, y} [Element] cell, PlotPoints -> 50]
and what the region should look like,
isovalue = 1.29897233417072;
ContourPlot[f[x, y, 4], {x, y} [Element] cell,
Contours -> {isovalue}, ColorFunction -> GrayLevel,
PlotPoints -> 100]
This is what I have tried
reg = ToElementMesh[
ImplicitRegion[
f[x, y, 4] < isovalue && {x, y} [Element] cell, {x, y}],
"MaxBoundaryCellMeasure" -> 0.01, MeshQualityGoal -> 1,
PerformanceGoal -> "Quality", MaxCellMeasure -> 0.01,
"BoundaryMeshGenerator" -> "Continuation"];
RegionPlot[reg]
The region is no more accurate when I decrease MaxCellMeasure
or MaxBoundaryCellMeasure
. I also tried the solution suggested here.
plotting finite-element-method mesh implicit
plotting finite-element-method mesh implicit
edited 47 mins ago
user21
21.1k55999
21.1k55999
asked 8 hours ago
jerjorgjerjorg
874
874
add a comment |
add a comment |
2 Answers
2
active
oldest
votes
$begingroup$
I hope I interpreted your question correctly that you want a more accurate ElementMesh
representation of the region.
First we create a high quality Graphics
of the region of interest.
isovalue = 1.29897233417072;
(* Add some margins to plot range to get connected region. *)
tolerance = 0.05;
plot = ContourPlot[
f[x, y, 4],
{x, y} ∈ Cuboid[{-0.5, -0.5} - tolerance, {0.5, 0.5} + tolerance],
Contours -> {isovalue},
ColorFunction -> GrayLevel,
(* We need high quality plot for ImageMesh later. *)
PlotPoints -> 200,
Frame -> None
]
Create MeshRegion
from Graphics
object.
mreg = ImageMesh[ColorNegate[plot]]
And convert it to ElementMesh
.
Needs["NDSolve`FEM`"]
mesh = ToElementMesh[mreg,"MeshOrder"->1]
(* ElementMesh[{{7., 353.}, {7., 353.}}, {TriangleElement["<" 1057 ">"]}] *)
mesh["Wireframe"]
$endgroup$
add a comment |
$begingroup$
Another approach is:
reg = ToElementMesh[
ImplicitRegion[
f[x, y, 4] < isovalue && {x, y} [Element] cell, {x, y}],
"MaxBoundaryCellMeasure" -> 0.01, MeshQualityGoal -> 1,
PerformanceGoal -> "Quality", MaxCellMeasure -> 0.01,
"BoundaryMeshGenerator" -> {"RegionPlot", "SamplePoints" -> 41}];
reg["Wireframe"]
One thing to be a bit careful about is the question if the holes intersect the boundary. From the mesh it does not look like it but the math might say it.
$endgroup$
add a comment |
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2 Answers
2
active
oldest
votes
2 Answers
2
active
oldest
votes
active
oldest
votes
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oldest
votes
$begingroup$
I hope I interpreted your question correctly that you want a more accurate ElementMesh
representation of the region.
First we create a high quality Graphics
of the region of interest.
isovalue = 1.29897233417072;
(* Add some margins to plot range to get connected region. *)
tolerance = 0.05;
plot = ContourPlot[
f[x, y, 4],
{x, y} ∈ Cuboid[{-0.5, -0.5} - tolerance, {0.5, 0.5} + tolerance],
Contours -> {isovalue},
ColorFunction -> GrayLevel,
(* We need high quality plot for ImageMesh later. *)
PlotPoints -> 200,
Frame -> None
]
Create MeshRegion
from Graphics
object.
mreg = ImageMesh[ColorNegate[plot]]
And convert it to ElementMesh
.
Needs["NDSolve`FEM`"]
mesh = ToElementMesh[mreg,"MeshOrder"->1]
(* ElementMesh[{{7., 353.}, {7., 353.}}, {TriangleElement["<" 1057 ">"]}] *)
mesh["Wireframe"]
$endgroup$
add a comment |
$begingroup$
I hope I interpreted your question correctly that you want a more accurate ElementMesh
representation of the region.
First we create a high quality Graphics
of the region of interest.
isovalue = 1.29897233417072;
(* Add some margins to plot range to get connected region. *)
tolerance = 0.05;
plot = ContourPlot[
f[x, y, 4],
{x, y} ∈ Cuboid[{-0.5, -0.5} - tolerance, {0.5, 0.5} + tolerance],
Contours -> {isovalue},
ColorFunction -> GrayLevel,
(* We need high quality plot for ImageMesh later. *)
PlotPoints -> 200,
Frame -> None
]
Create MeshRegion
from Graphics
object.
mreg = ImageMesh[ColorNegate[plot]]
And convert it to ElementMesh
.
Needs["NDSolve`FEM`"]
mesh = ToElementMesh[mreg,"MeshOrder"->1]
(* ElementMesh[{{7., 353.}, {7., 353.}}, {TriangleElement["<" 1057 ">"]}] *)
mesh["Wireframe"]
$endgroup$
add a comment |
$begingroup$
I hope I interpreted your question correctly that you want a more accurate ElementMesh
representation of the region.
First we create a high quality Graphics
of the region of interest.
isovalue = 1.29897233417072;
(* Add some margins to plot range to get connected region. *)
tolerance = 0.05;
plot = ContourPlot[
f[x, y, 4],
{x, y} ∈ Cuboid[{-0.5, -0.5} - tolerance, {0.5, 0.5} + tolerance],
Contours -> {isovalue},
ColorFunction -> GrayLevel,
(* We need high quality plot for ImageMesh later. *)
PlotPoints -> 200,
Frame -> None
]
Create MeshRegion
from Graphics
object.
mreg = ImageMesh[ColorNegate[plot]]
And convert it to ElementMesh
.
Needs["NDSolve`FEM`"]
mesh = ToElementMesh[mreg,"MeshOrder"->1]
(* ElementMesh[{{7., 353.}, {7., 353.}}, {TriangleElement["<" 1057 ">"]}] *)
mesh["Wireframe"]
$endgroup$
I hope I interpreted your question correctly that you want a more accurate ElementMesh
representation of the region.
First we create a high quality Graphics
of the region of interest.
isovalue = 1.29897233417072;
(* Add some margins to plot range to get connected region. *)
tolerance = 0.05;
plot = ContourPlot[
f[x, y, 4],
{x, y} ∈ Cuboid[{-0.5, -0.5} - tolerance, {0.5, 0.5} + tolerance],
Contours -> {isovalue},
ColorFunction -> GrayLevel,
(* We need high quality plot for ImageMesh later. *)
PlotPoints -> 200,
Frame -> None
]
Create MeshRegion
from Graphics
object.
mreg = ImageMesh[ColorNegate[plot]]
And convert it to ElementMesh
.
Needs["NDSolve`FEM`"]
mesh = ToElementMesh[mreg,"MeshOrder"->1]
(* ElementMesh[{{7., 353.}, {7., 353.}}, {TriangleElement["<" 1057 ">"]}] *)
mesh["Wireframe"]
answered 1 hour ago
PintiPinti
3,95211037
3,95211037
add a comment |
add a comment |
$begingroup$
Another approach is:
reg = ToElementMesh[
ImplicitRegion[
f[x, y, 4] < isovalue && {x, y} [Element] cell, {x, y}],
"MaxBoundaryCellMeasure" -> 0.01, MeshQualityGoal -> 1,
PerformanceGoal -> "Quality", MaxCellMeasure -> 0.01,
"BoundaryMeshGenerator" -> {"RegionPlot", "SamplePoints" -> 41}];
reg["Wireframe"]
One thing to be a bit careful about is the question if the holes intersect the boundary. From the mesh it does not look like it but the math might say it.
$endgroup$
add a comment |
$begingroup$
Another approach is:
reg = ToElementMesh[
ImplicitRegion[
f[x, y, 4] < isovalue && {x, y} [Element] cell, {x, y}],
"MaxBoundaryCellMeasure" -> 0.01, MeshQualityGoal -> 1,
PerformanceGoal -> "Quality", MaxCellMeasure -> 0.01,
"BoundaryMeshGenerator" -> {"RegionPlot", "SamplePoints" -> 41}];
reg["Wireframe"]
One thing to be a bit careful about is the question if the holes intersect the boundary. From the mesh it does not look like it but the math might say it.
$endgroup$
add a comment |
$begingroup$
Another approach is:
reg = ToElementMesh[
ImplicitRegion[
f[x, y, 4] < isovalue && {x, y} [Element] cell, {x, y}],
"MaxBoundaryCellMeasure" -> 0.01, MeshQualityGoal -> 1,
PerformanceGoal -> "Quality", MaxCellMeasure -> 0.01,
"BoundaryMeshGenerator" -> {"RegionPlot", "SamplePoints" -> 41}];
reg["Wireframe"]
One thing to be a bit careful about is the question if the holes intersect the boundary. From the mesh it does not look like it but the math might say it.
$endgroup$
Another approach is:
reg = ToElementMesh[
ImplicitRegion[
f[x, y, 4] < isovalue && {x, y} [Element] cell, {x, y}],
"MaxBoundaryCellMeasure" -> 0.01, MeshQualityGoal -> 1,
PerformanceGoal -> "Quality", MaxCellMeasure -> 0.01,
"BoundaryMeshGenerator" -> {"RegionPlot", "SamplePoints" -> 41}];
reg["Wireframe"]
One thing to be a bit careful about is the question if the holes intersect the boundary. From the mesh it does not look like it but the math might say it.
answered 32 mins ago
user21user21
21.1k55999
21.1k55999
add a comment |
add a comment |
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