Mesh quality improvement

See also: improvement, quality, Improvement

High quality meshes are essential for accuracy in simulations (e.g. finite element, finite volume methods) and other mesh-based computation. There are different approaches to improving mesh quality, including smoothing and modifications to the mesh connectivity (see also: Improvement). Often, the best approach is to use a combination of these methods [KS08].

The mymesh.improvement.Improve() function combines edge contraction for coarsening (Contract()), edge splitting for refinement (Split()), edge flipping (Flip()), and smoothing to improve mesh quality while also achieving a target edge length for triangular or tetrahedral meshes.

Improve() repeats a schedule of operations, which is by default splitting (‘s’), contraction (‘c’), flipping (‘f’), and smoothing (‘S’) defined by a string ('scfS'), however the schedule can be customized to remove or rearrange the order of operations (e.g. 'Sfc').

2D Surface Improvement

Improve() can be used on a two dimensional triangular surface while preserving the original boundaries of the mesh

from mymesh import implicit, improvement, primitives, quality, visualize
import numpy as np

C = primitives.Circle([0,0,0], 1, ElemType='tri',
                        theta_resolution=40, radial_resolution=20)
C.ElemData['skew'] = quality.Skewness(*C)

target_edge_length = 0.1
C2 = improvement.Improve(C, target_edge_length, schedule = 'scfS',
                         repeat=10, verbose=False)
C2.ElemData['skew'] = quality.Skewness(*C2)

means = [np.mean(C.ElemData['skew']), np.mean(C2.ElemData['skew'])]

C.plot(scalars='skew', clim=(0,1), show_edges=True, view='xy',
       title=f'Original\nMean Skewness={means[0]:.2f}')
C2.plot(scalars='skew', clim=(0,1), show_edges=True, view='xy',
        title=f'Improved\nMean Skewness={means[1]:.2f}')

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RFBOutputContext()
RFBOutputContext()

3D Surface Improvement

Similarly, three dimensional triangular surfaces can be improved. Implicit or image-based surfaces often contain some highly skewed elements as a result of the contouring process, which can be improved by Improve().

S = implicit.SurfaceMesh(implicit.sphere([0,0,0], 1), [-1,1,-1,1,-1,1], .1)
S.ElemData['skew'] = quality.Skewness(*S)

target_edge_length = 0.2
S2 = improvement.Improve(S, target_edge_length, schedule='scfS',
                         repeat=10, verbose=False)
S2.ElemData['skew'] = quality.Skewness(*S2)

means = [np.mean(S.ElemData['skew']), np.mean(S2.ElemData['skew'])]
S.plot(scalars='skew', clim=(0,1), show_edges=True, view='trimetric',
       title=f'Original\nMean Skewness={means[0]:.2f}')
S2.plot(scalars='skew', clim=(0,1), show_edges=True, view='trimetric',
        title=f'Improved\nMean Skewness={means[1]:.2f}')

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RFBOutputContext()
RFBOutputContext()

3D Volume Improvement

The tetrahedral meshes generated by MarchingTetrahedra() (mymesh.implicit.TetMesh(), mymesh.image.TetMesh()) generally structured interiors with some low quality elements on the surface, however, even the structured interiors have elements of only ‘average’ quality.

S = implicit.TetMesh(implicit.sphere([0,0,0], 1), [-1,1,-1,1,-1,1], .1)
S.ElemData['skew'] = quality.Skewness(*S)

target_edge_length = 0.25
S2 = improvement.Improve(S, target_edge_length, schedule='scfS',
                         repeat=20, verbose=False)
S2.ElemData['skew'] = quality.Skewness(*S2)

means = [np.mean(S.ElemData['skew']), np.mean(S2.ElemData['skew'])]
S.Clip().plot(scalars='skew', clim=(0,1), show_edges=True, view='trimetric',
       title=f'Original\nMean Skewness={means[0]:.2f}')
S2.Clip().plot(scalars='skew', clim=(0,1), show_edges=True, view='trimetric',
        title=f'Improved\nMean Skewness={means[1]:.2f}')

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RFBOutputContext()
RFBOutputContext()