An improved global-direction stencil based on the face-area-weighted centroid for the gradient reconstruction of unstructured finite volume methods
Ling-Fa Kong(孔令发)1, Yi-Dao Dong(董义道)1,†, Wei Liu(刘伟)1, and Huai-Bao Zhang(张怀宝)2
1College of Aerospace Science and Engineering, National University of Defense Technology, Changsha 410073, China 2School of Physics, Sun Yat-sen University, Guangzhou 510275, China
The accuracy of unstructured finite volume methods is greatly influenced by the gradient reconstruction, for which the stencil selection plays a critical role. Compared with the commonly used face-neighbor and vertex-neighbor stencils, the global-direction stencil is independent of the mesh topology, and characteristics of the flow field can be well reflected by this novel stencil. However, for a high-aspect-ratio triangular grid, the grid skewness is evident, which is one of the most important grid-quality measures known to affect the accuracy and stability of finite volume solvers. On this basis and inspired by an approach of using face-area-weighted centroid to reduce the grid skewness, we explore a method by combining the global-direction stencil and face-area-weighted centroid on high-aspect-ratio triangular grids, so as to improve the computational accuracy. Four representative numerical cases are simulated on high-aspect-ratio triangular grids to examine the validity of the improved global-direction stencil. Results illustrate that errors of this improved methods are the lowest among all methods we tested, and in high-mach-number flow, with the increase of cell aspect ratio, the improved global-direction stencil always has a better stability than commonly used face-neighbor and vertex-neighbor stencils. Therefore, the computational accuracy as well as stability is greatly improved, and superiorities of this novel method are verified.
* Project supported by the National Key Project, China (Grant No. GJXM92579).
Cite this article:
Ling-Fa Kong(孔令发), Yi-Dao Dong(董义道)†, Wei Liu(刘伟), and Huai-Bao Zhang(张怀宝) An improved global-direction stencil based on the face-area-weighted centroid for the gradient reconstruction of unstructured finite volume methods 2020 Chin. Phys. B 29 100203
Fig. 1.
The process of cell-centered finite volume discretization on triangular cells.
Fig. 2.
Face-neighbor and vertex-neighbor stencil cells in different layers. The different numbers in these two figures represent the diverse stencil layers (e.g., for vertex-neighbor stencil, the first layer stencil is composed of all cells that share vertices with the central cell, and the second layer stencil consists of cells that share vertices with the first layer stencil). (a) Face-neighbor stencil. (b) Vertex-neighbor stencil.
Fig. 3.
Local direction stencils on triangular grids with minor aspect ratio. (a) Grid with straight boundary. (b) Grid with curved boundary.
Fig. 4.
Local direction stencils on triangular grids with high aspect ratio. (a) Grid with straight boundary. (b) Grid with curved boundary.
Fig. 5.
The global-direction stencils on triangular girds with high aspect ratio. (a) Grid with straight boundary. (b) Grid with curved boundary.
Fig. 6.
Global-direction stencils at internal field and the curved boundary on high-aspect-ratio triangular grids. (a) Internal field. (b) Curved boundary.
Fig. 7.
The optimized global-direction stencil cells on grids with the curved boundary and high aspect ratio. (a) Grid 1. (b) Grid 2.
Fig. 8.
The geometric centroid and face-area-weighted centroid of triangular cells. (a) Geometric centroid. (b) Face-area-weighted centroid.
Fig. 9.
The grid skewness measure of cells in different aspect ratios.
Fig. 10.
A typical high-aspect-ratio triangular grid in Cartesian-coordinate system.
Fig. 11.
The geometric centroid and face-area-weighted centroid with different parameter p on grids with straight and curved boundary respectively. (a) Grid with straight boundary. (b) Grid with curved boundary.
Fig. 12.
The global-direction stencils combined with geometric centroid and face-area-weighted centroid on regular triangular grid used for the boundary layer type flow. (a) Geometric centroid. (b) Face-area-weighted centroid.
Fig. 13.
The global-direction stencils combined with geometric centroid and face-area-weighted centroid on triangular grid with the curved boundary. (a) Geometric centroid. (b) Face-area-weighted centroid.
Full name
Abbreviation
Vertex-neighbor stencil
V-stencil
Face-neighbor stencil
F-stencil
Global-direction stencil with geometric centroid (i.e., p = 0)
G-stencil (p = 0)
Global-direction stencil with the face-area-weighted centroid and p = 1
G-stencil (p = 1)
Global-direction stencil with the face-area-weighted centroid and p = 2
G-stencil (p = 2)
Table 1.
The abbreviation of different stencils.
Fig. 14.
Flow fields of this numerical example. (a) Density. (b) Pressure.
Fig. 15.
Regular and randomly perturbed triangular grids with AR = 103: (a) regular, (b) randomly perturbed.
Grid name
Distribution in x and y directions (x,y ∈ [0, 1] × [0, 0.001]
AR = 102
AR = 5 × 102
AR = 103
vcoa
50 × 5
20 × 10
15 × 15
coa
100 × 10
40 × 20
30 × 30
med
150 × 15
60 × 30
45 × 45
fin
200 × 20
80 × 40
60 × 60
vfin
300 × 30
120 × 60
90 × 90
Table 2.
The distribution of five sets of background quadrilateral grid cells in different aspect ratios.
Fig. 16.
L2 and L∞ norms of pressure errors, when the aspect ratio AR = 102: (a) L2 norm, (b) L∞ norm.
Fig. 17.
The stencil size of different stencil selection methods.
Fig. 18.
The geometric centroid and face-area-weighted centroid of grid cells.
Different stencils
L2 norm of pressure errors
L∞ norm of pressure errors
V-stencil
0.000517445
0.00192892
F-stencil
0.000470333
0.00166419
G-stencil (p = 0)
0.000446073
0.00146549
G-stencil (p = 1)
0.000237936
0.000923927
G-stencil (p = 2)
0.000234345
0.000917014
Table 3.
Pressure errors of different stencils on the vfin grid.
Fig. 19.
L2 and L∞ norms of pressure errors, when the aspect ratio AR = 103: (a) L2 norm, (b) L∞ norm.
Fig. 20.
L2 and L∞ norms of pressure errors, when the aspect ratio AR = 103: (a) L2 norm, (b) L∞ norm.
Fig. 21.
The stencil size of different stencil selection methods.
Fig. 22.
The geometric centroid and face-area-weighted centroid of grid cells.
Fig. 23.
The flow field of manufactured boundary layer type flow: (a) μ = 10−6, (b) μ = 10−8.
Fig. 24.
Regular and randomly perturbed triangular grids with AR = 104: (a) regular, (b) randomly perturbed.
Grid name
Distribution in x and x directions (x,y ∈ [0, 1] × [0, 0.001]
AR = 102
AR = 103
AR = 104
AR = 105
vcoa
120 × 10
60 × 20
20 × 30
30 × 30
coa
180 × 15
90 × 30
30 × 45
45 × 45
med
240 × 20
120 × 40
40 × 60
60 × 60
fin
360 × 30
180 × 60
60 × 90
90 × 90
vfin
480 × 40
240 × 80
80 × 120
120 × 120
Table 4.
The distribution of five sets of background quadrilateral grid cells in different aspect ratios.
Fig. 25.
L2 and L∞ norms of solution errors, when the aspect ratio AR = 102: (a) L2 norm, (b) L∞ norm.
Fig. 26.
The stencil size of different stencil selection methods.
Fig. 27.
The geometric centroid and face-area-weighted centroid of grid cells.
Different stencils
L2 norm of solution errors
L∞ norm of solution errors
V-stencil
0.000170044
0.00202639
F-stencil
0.00015098
0.00205056
G-stencil (p = 0)
0.00014244
0.00195697
G-stencil (p = 1)
0.0000746072
0.00140981
G-stencil (p = 2)
0.0000734274
0.00143022
Table 5.
Solution errors of different stencils on the vfin grid.
Fig. 28.
L2 and L∞ norms of solution errors, when the aspect ratio AR = 105: (a) L2 norm, (b) L∞ norm.
Fig. 29.
L2 and L∞ norms of solution errors, with the aspect ratio AR = 104: (a) L2 norm, (b) L∞ norm.
Fig. 30.
The stencil size of different stencil selection methods.
Fig. 31.
The geometric centroid and face-area-weighted centroid of grid cells.
Fig. 32.
Flow fields of the supersonic vortex flow: (a) density, (b) mach number.
Fig. 33.
Regular and randomly perturbed triangular grids with AR ≈ 4: (a) regular, (b) randomly perturbed.
Grid name
The distribution in radial and circumferential directions
AR ≈ 2.5
AR ≈ 4
AR ≈ 8
vcoa
20 × 30
10 × 10
20 × 10
coa
30 × 45
20 × 20
40 × 20
med
40 × 60
40 × 40
80 × 40
fin
60 × 90
60 × 60
120 × 60
vfin
80 × 120
80 × 80
160 × 80
Table 6.
The distribution of background quadrilateral grid cells in radial and circumferential directions on three different grid categories.
Fig. 34.
L2 and L∞ norms of global pressure errors, and the aspect ratio AR ≈ 2.5: (a) L2 norm, (b) L∞ norm.
Fig. 35.
The geometric centroid and face-area-weighted centroid of grid cells.
Fig. 36.
L2 norm of wall pressure errors and the aspect ratio AR ≈ 2.5.
Different stencils
L2 norm of global pressure errors
L∞ norm of global pressure errors
Average stencil size
V-stencil
0.000109909
0.000631396
11.8339
F-stencil
0.000066257
0.000612726
8.896
G-stencil (p = 0)
0.0000541208
0.000457848
6.9375
G-stencil (p = 1)
0.0000454133
0.000366076
6.9375
G-stencil (p = 2)
0.0000427758
0.000331916
6.9375
Table 7.
Global pressure errors and average stencil size of different stencils on the vfin grid.
Fig. 37.
L2 and L∞ norms of global pressure errors, and the aspect ratio AR ≈ 8: (a) L2 norm, (b) L∞ norm.
Fig. 38.
The geometric centroid and face-area-weighted centroid of grid cells and the aspect ratio AR 7≈ 8.
Fig. 39.
L2 norm of wall pressure errors and the aspect ratio AR ≈ 8.
Different stencils
L2 norm of global pressure errors
L∞ norm of global pressure errors
Average stencil size
V-stencil
0.000261469
0.00144312
11.8008
F-stencil
0.00017859
0.00141231
8.87531
G-stencil (p = 0)
0.000151624
0.00112674
6.925
G-stencil (p = 1)
0.000128455
0.00085544
6.925
G-stencil (p = 2)
0.00012346
0.000787497
6.925
Table 8.
Global pressure errors of different stencils on the randomly perturbed vfin grid.
Fig. 40.
L2 and L∞ norms of global pressure errors on randomly perturbed grids, and the aspect ratio AR ≈ 4: (a) L2 norm, (b) L∞ norm.
Fig. 41.
The geometric centroid and face-area-weighted centroid of grid cells, and the aspect ratio AR ≈ 4.
Fig. 42.
L2 norm of wall pressure errors on randomly perturbed triangular grids, and the aspect ratio AR ≈ 4.
Different grids
Distribution in x and y directions
Aspect ratio (AR)
Grid 1
160 × 100
2.5
Grid 2
180 × 240
5.3
Grid 3
90 × 190
8.4
Grid 4
100 × 280
11.2
Table 9.
Cell aspect ratio and distribution of background quadrilateral grids in x and y directions.
Fig. 43.
Flow fields of four different stencils with AR = 2.5 and t = 0.2: (a) V-stencil, (b) F-stencil, (c) G-stencil (p = 0), (d) G-stencil (p = 2).
Fig. 44.
Flow fields of four different stencils with AR = 5.3 and t = 0.2, where “Diverged” in red represents the divergent computing. (a) V-stencil. (b) F-stencil. (c) G-stencil (p = 0). (d) G-stencil (p = 2).
Fig. 45.
Flow fields of four different stencils with AR = 11.2 and t = 0.2, where “Diverged” in red represents the divergent computing. (a) V-stencil. (b) F-stencil. (c) G-stencil (p = 0). (d) G-stencil (p = 2).
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