accuracy tests for comsol - and delaunay meshes
TRANSCRIPT
![Page 1: Accuracy Tests for COMSOL - and Delaunay Meshes](https://reader034.vdocuments.us/reader034/viewer/2022042707/586accc21a28aba3348b89b3/html5/thumbnails/1.jpg)
Weierstraß-Institut für Angewandte Analysis und Stochastik
Mohrenstr. 39 10117 Berlin +49 30 | 2 03 72 0 www.wias-berlin.de 1/n
Accuracy Tests for COMSOL -and Delaunay Meshes
Ekkehard HolzbecherHang Si
European COMSOLConference 2008
Hannover
Presented at the COMSOL Conference 2008 Hannover
![Page 2: Accuracy Tests for COMSOL - and Delaunay Meshes](https://reader034.vdocuments.us/reader034/viewer/2022042707/586accc21a28aba3348b89b3/html5/thumbnails/2.jpg)
Mesh Test - General
• We test the influence of the mesh on the accuracy of a COMSOL Finite Element solution
• We choose 2D and 3D testcases – with classical differential equation– and a complex geometry
• We compare linear and quadratic elements • We study regular mesh refinement and adaptive mesh
refinement• We study meshes with and without Delaunay property
![Page 3: Accuracy Tests for COMSOL - and Delaunay Meshes](https://reader034.vdocuments.us/reader034/viewer/2022042707/586accc21a28aba3348b89b3/html5/thumbnails/3.jpg)
TestCase 1, Definition
2D single subdomain, potential equation:
Dirichlet
Dirichlet
Neumann
2 0u∇ =
![Page 4: Accuracy Tests for COMSOL - and Delaunay Meshes](https://reader034.vdocuments.us/reader034/viewer/2022042707/586accc21a28aba3348b89b3/html5/thumbnails/4.jpg)
TestCase 1, Analytical Solution
• Analytical solution by Schwarz-Christoffel Transformation
• using MATLAB SC*-toolbox by Driscoll & Trefethen
* Schwarz-Christoffel
![Page 5: Accuracy Tests for COMSOL - and Delaunay Meshes](https://reader034.vdocuments.us/reader034/viewer/2022042707/586accc21a28aba3348b89b3/html5/thumbnails/5.jpg)
TestCase 1, Mesh Quality (2D)
2 2 21 2 3
4 3Aqh h h
=+ +
Element quality:
Mesh quality is defined as the minimum element quality
with: area A and sidelengths h1, h2 and h3
Element qualitydistribution
![Page 6: Accuracy Tests for COMSOL - and Delaunay Meshes](https://reader034.vdocuments.us/reader034/viewer/2022042707/586accc21a28aba3348b89b3/html5/thumbnails/6.jpg)
TestCase 1, Results for Quadratic Elements
2e
131295362602254 2.973632384653453 0.94698096164812 0.91129202441931 1.2530150610850
ϑ104No. elements
DOFRefine-ments
( ) ( )( ) ( )
1 2
1 2
ln ln2
ln lne e
DOF DOFϑ
−= −
−with convergence order defined by
Quadratic elements; refinements are regular.
Degrees of freedom
Convergence order
![Page 7: Accuracy Tests for COMSOL - and Delaunay Meshes](https://reader034.vdocuments.us/reader034/viewer/2022042707/586accc21a28aba3348b89b3/html5/thumbnails/7.jpg)
Delaunay Meshes
The Delaunay triangulation is defined by the property that there are no further nodes within the circumspheres of the triangles(Delaunay 1934, russ.)
p
q
r
s
p
q
r
s
![Page 8: Accuracy Tests for COMSOL - and Delaunay Meshes](https://reader034.vdocuments.us/reader034/viewer/2022042707/586accc21a28aba3348b89b3/html5/thumbnails/8.jpg)
Voronoi Diagrams
In the Voronoi diagram each cell consists of points closest to one node
The Voronoi diagram is the dual representation of the Delaunay triangulation;
-- Delaunay triangulation
-- Voronoi diagram
![Page 9: Accuracy Tests for COMSOL - and Delaunay Meshes](https://reader034.vdocuments.us/reader034/viewer/2022042707/586accc21a28aba3348b89b3/html5/thumbnails/9.jpg)
TestCase 1, Delaunay Meshes (Quadratic Elements)
e
7867831384010-3/81013375695010-3/41261693352610-3/2
26897833176410-3
2 104# elem.DOFMean
elem. size
e
7569541425710-3/8*
7367281380510-3/81023352697710-3/41331678353510-3/2177854184910-3
2 104#
elem.DOFMean
elem. size
Delaunay meshes, produced with ‚triangle‘ (Shewchuk)
Default option D option (improved quality)* q option (30° angle restr.)
![Page 10: Accuracy Tests for COMSOL - and Delaunay Meshes](https://reader034.vdocuments.us/reader034/viewer/2022042707/586accc21a28aba3348b89b3/html5/thumbnails/10.jpg)
TestCase 2, Definition
• 2D three subdomain set-up
• High permeability (diffusivity) in domain 1 (1)
• Low permeability (diffusivity) in domains 2 and 3 (10-4 and 10-5)
Dirichlet Dirichlet
Neumann
Neumann
( ) 0uσ∇ ∇ =
![Page 11: Accuracy Tests for COMSOL - and Delaunay Meshes](https://reader034.vdocuments.us/reader034/viewer/2022042707/586accc21a28aba3348b89b3/html5/thumbnails/11.jpg)
TestCase 2, Results 1
792401281205454
1.2518860032302573
1.214331500876252
1.23996375219371
1.2223029385000
ϑ104
# elem.DOFRefinements
e
142401284812174
1.6544600321205453
1.3311015008302572
1.23256375276251
1.2259393819370
ϑ104
# elem.DOFRefinements
Regular refinement e
Linear elements
Quadratic elements
![Page 12: Accuracy Tests for COMSOL - and Delaunay Meshes](https://reader034.vdocuments.us/reader034/viewer/2022042707/586accc21a28aba3348b89b3/html5/thumbnails/12.jpg)
TestCase 2, Results 2
e
101.0941884382913
101.1001722350512
101.0761566319311
121.0741456297310
201.078135627739
211.024125825778
241.050122825177
351.045117024016
551.055112023015
641.025106221854
1251.022103621333
1331.048101420892
2841.03296819971
Mesh increase
No. elements
DOFRefine-ments
Adaptive refinement
Quadratic elements, residual method: coefficient, refinement method: regular, element selection method: fraction of worst error (parameter 0.5)
![Page 13: Accuracy Tests for COMSOL - and Delaunay Meshes](https://reader034.vdocuments.us/reader034/viewer/2022042707/586accc21a28aba3348b89b3/html5/thumbnails/13.jpg)
TestCase 3 (3D), Set-up
The 3D domain is produced by performing a shift and a rotation on a trianglesimultaneously. The angle is 165°.
![Page 14: Accuracy Tests for COMSOL - and Delaunay Meshes](https://reader034.vdocuments.us/reader034/viewer/2022042707/586accc21a28aba3348b89b3/html5/thumbnails/14.jpg)
TestCase 3 (3D), Mesh
2465 elementsElement quality from 0 (blue) to 1 (red)Mesh quality: 0.1174
( )3 / 22 2 2 2 2 21 2 3 4 5 6
72 3Vqh h h h h h
=+ + + + +
![Page 15: Accuracy Tests for COMSOL - and Delaunay Meshes](https://reader034.vdocuments.us/reader034/viewer/2022042707/586accc21a28aba3348b89b3/html5/thumbnails/15.jpg)
TestCase 3 (3D), Solution
• Laplace equation • Dirichlet conditions at the two ‚end‘-positions of
the triangle• Neumann conditions at all other boundaries
![Page 16: Accuracy Tests for COMSOL - and Delaunay Meshes](https://reader034.vdocuments.us/reader034/viewer/2022042707/586accc21a28aba3348b89b3/html5/thumbnails/16.jpg)
TestCase 3 (3D), Results for Linear Elements
e
0.1090.219111063+2726extra fine
0.260.19543633+1028finer
0.380.20491407+464fine
0.600.1814676+256normal
1.470.0697342+141coarse
0.900.1934250+109coarser
2.580.2030161+75extra coarse
0.1270.019211801-2726extra fine
0.250.02703849-1028finer
0.420.02481492-464fine
0.550.0561714-255normal
1.460.0567352-140coarse
2.210.0508253-108coarser
2.210.0846162-74extra coarse
2 102Mesh quality
No. elementsQuality optim.
DOFMesh
![Page 17: Accuracy Tests for COMSOL - and Delaunay Meshes](https://reader034.vdocuments.us/reader034/viewer/2022042707/586accc21a28aba3348b89b3/html5/thumbnails/17.jpg)
TestCase 3 (3D), Results for Quadratic Elements
e
0.060.219111063+18158extra fine
0.110.19543633+6452finer
0.810.20491407+2736fine
1.690.1814676+1425normal
9.610.0697342+757coarse
12.20.1934250+572coarser
17.90.2030161+382extra coarse
0.090.019211801-18896extra fine
0.220.02703849-6668finer
0.790.02481492-2821fine
2.280.0561714-1461normal
9.720.0567352-765coarse
12.50.0508253-573coarser
16.80.0846162-381extra coarse
2 102Mesh quality
No. elementsQuality optim.
DOFRefine-ments
![Page 18: Accuracy Tests for COMSOL - and Delaunay Meshes](https://reader034.vdocuments.us/reader034/viewer/2022042707/586accc21a28aba3348b89b3/html5/thumbnails/18.jpg)
Summary
• For the same DOF quadratic elements deliver more accurate results then linear
elements
• The convergence rate for linear elements in 2D problems is ≈1.2
• For quadratic elements the convergence rate is only slightly increased in
comparison to linear elements, and lies significantly below the theoretical value of 2
• In comparison to globally refined meshes adaptive techniques deliver results with
same accuracy, but with significantly lower DOF
• Multiple application of adaptive mesh refinement shows reduced improvement with
each application
• For the chosen testcases Delaunay meshes do not offer advantages compared to
usual COMSOL meshing
• Quality and angle restriction of Delaunay triangulations do not lead to improved
results
• Mesh quality optimization is recommended
![Page 19: Accuracy Tests for COMSOL - and Delaunay Meshes](https://reader034.vdocuments.us/reader034/viewer/2022042707/586accc21a28aba3348b89b3/html5/thumbnails/19.jpg)
--------------------------------------
My affiliation has changed:
Ekkehard HolzbecherGeorg-August University GöttingenGZG, Applied Geology