Weierstraß-Institut für Angewandte Analysis und Stochastik

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Accuracy Tests for COMSOL -and Delaunay Meshes

Ekkehard HolzbecherHang Si

European COMSOLConference 2008

Hannover

Presented at the COMSOL Conference 2008 Hannover

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

TestCase 1, Definition

2D single subdomain, potential equation:

Dirichlet

Dirichlet

Neumann

2 0u∇ =

TestCase 1, Analytical Solution

• Analytical solution by Schwarz-Christoffel Transformation

• using MATLAB SC*-toolbox by Driscoll & Trefethen

* Schwarz-Christoffel

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

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

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

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

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.)

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∇ ∇ =

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

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)

TestCase 3 (3D), Set-up

The 3D domain is produced by performing a shift and a rotation on a trianglesimultaneously. The angle is 165°.

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

=+ + + + +

TestCase 3 (3D), Solution

• Laplace equation • Dirichlet conditions at the two ‚end‘-positions of

the triangle• Neumann conditions at all other boundaries

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

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

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

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My affiliation has changed:

Ekkehard HolzbecherGeorg-August University GöttingenGZG, Applied Geology

ekkehard.holzbecher@geo.uni-goettingen.de