June 23, 2008

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Variational tetrahedral meshing of mechanical models for FEA

Matthijs Sypkens SmitWillem F. Bronsvoort

CAD ’08 Conference, Orlando, Florida

Faculty of Electrical Engineering, Mathematics and Computer Science

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Outline

• Research motivation• Variational tetrahedral meshing (VTM)• The shortcomings of VTM for mechanical

models• Improvements / recommendations• Results• Conclusions

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Research motivation (1)

Analysis / product simulation:• Reduces need to construct

real world test models• Decreases length of product

development cycle• Increases quality/safety• Lowers total cost

Most popular method: finite element analysis• Requires a mesh / decomposition of geometry• Can work with many types of meshes

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Research motivation (2)

Meshes and mesh quality:• Zero bad quality elements• Near regular elements in computational space• Accurate representation of model boundary• Efficient variation in sizing w.r.t. to accuracy

Quality is context dependent. Generally:A higher quality mesh results in a quicker, more accurate, and more reliable analysis.

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Research motivation (3)

Variational tetrahedral meshing(VTM) offers:• Exceptional quality distribution:• Majority near regular elements• Effectively no bad quality

elements• Mesh sizing

VTM was not conceived for mechanical models. We have investigated this application We have made several improvements

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Variational tetrahedral meshing (1)

Central ideas:• Delaunay mesh• Optimisation through connectivity and node

locations• Boundary conformance as continuous process

simultaneously executed with optimisation

Delaunayproperty:

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Variational tetrahedral meshing (2)Delaunay optimisation:• Delaunay connectivity globally optimal• Node relocation improves local quality Combine these two into an optimisation

procedure1: 2: 3:

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Variational tetrahedral meshing (3)

Achieving boundary conformance:• “Shape” the mesh by pulling the outer nodes

towards the boundary:

• Use boundary samples to perform pulling• Separate treatment of corner, edge and face

samples

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Variational tetrahedral meshing (4)

The role of boundary samples:

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Variational tetrahedral meshing (5)

Interior mesh optimising Boundary shaping

The algorithm:• Initialise data structures• Spread out nodes• Optimisation loop• Extract mesh

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Variational tetrahedral meshing (6)

Mesh extraction:• Delaunay mesh covers the

convex hull• Remove elements that are

not part of the intended shape

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The weaker points of VTM (1)

Points of attentionfor mechanical models:

• Boundary conformance An accurate representation of the boundary should be present in the mesh

• Mesh extraction An accurate representation of the boundary should result from Delaunay mesh extraction

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The weaker points of VTM (2)

Boundary conformance:• There is a risk that no set of

tetrahedra exists in the mesh that is acceptable to represent the model boundary

3D: Tetrahedron “crossing” the boundary

2D: correct vs.

“crossing”

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The weaker points of VTM (3)

Mesh extraction:• The heuristics for mesh extraction frequently fail

to deliver an acceptable result

Excess andmissingtetrahedraafter extraction:

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Improving the results (1)

To get better results applying VTM on mechanical models,

we must first understand the cause(s) of the defects.

Boundary conformance:• Why do we expect boundary conformance in our

mesh?• How can it go wrong?• What can be done about it?

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Improving the results (2)Why do we expect boundary conformance?

The boundary samples always pull on their closest node.

Result: (near) Gabriel edges and triangles everywhere on the boundary.

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Improving the results (3)How can boundary conformance fail?

If the number of boundary samples is low, less of theboundary will be Gabriel.

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Improving the results (4)What can be done about it?

• The risk of failing boundary conformance can be reduced by increasing the number of boundary samples

• From experience: 10 samples per node on the boundary makes failure rare

Caveat:• Small dihedral angles

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Improving the results (5)Lack of resolution

A geometry needs a certain amount of nodes for an accurate representation by a conforming Delaunay triangulation

Even with ample boundary samples, boundary conformance can still fail due to lack of nodes

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Improving the results (6)Detect lack of resolution and locally fix it

Indication of lack of nodes: Samples from different geometrical features are

pulling on the same nodeFix: split the node (= add a new node right next to

it)

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Improving the results (7)Detect lack of resolution and locally fix it

Example of howeffective splitting is: Start with one interiornode and 54 corner nodes:continue splitting until nomore splits occur

Fix is a local solution; With many nodes missing better to

start with more nodes

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Improving the results (8)Mesh extraction

Improved mesh extraction logic:• Project boundary nodes to model boundary• Tetrahedra with an interior node are inside the model Only tetrahedra with 4 boundary nodes left to

consider

• If centroid of a tetrahedron falls outside, thentetrahedron outside the model

• Elsetetrahedron inside the model

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Improving the results (9)Mesh extraction

2D example:• Location of centroid is decisive• Works for both concave and convex regions

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Results (1)Gear mesh

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Results (2)Gear volume-length ratio

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Results (3)Gear min/max dihedral angle

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Conclusions

• Number of boundary samples and number of nodes are important for success

• Node splitting is effective at enforcing boundary conformance

• Improved mesh extraction recovers the intended boundary

• Improved VTM can be used for the construction of meshes of mechanical models

• Results in high quality meshes of mechanical models

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Credits

• Research supported by NWO:

• Computational Geometry Algorithms Library (CGAL) was used in the creation of some of the illustrations