Mesh Quality Effects on Accuracy of Node-Centered FiniteVolume Discretization Schemes Boris Diskin
Presented @ NIA CFD Seminar October 18, 2011
What does mesh quality mean? •
Traditional mesh quality indicators are purely geometric. Optimization of shapes, sizes, angles, aspect ratio, etc. of elements. Encourage “pretty” meshes. No dependence on solution, equations, desired computational output.
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In direct contradiction with modern unstructured-grid practice Accurate solutions on “ugly” meshes.
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While there is no doubt that certain mesh characteristics critically affect accuracy of CFD solutions and gradients, the precise nature of this influence (what affects what) is far from clear.
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Common sense definition: High quality grid facilitates computation of the output of interest with the desired accuracy.
efficient
Can finite volumes use accuracy measures employed by other discretization methods? The ultimate accuracy measures are discretization error and accuracy of the output. May be difficult to estimate
Finite differences: the main accuracy measure is truncation error Truncation errors of finite-volume discretization (FVD) schemes is reliable only on regular grids. Discretization errors of unstructured FVD solutions can supra-converge (H.-O. Kreiss et al., Math. of Comp. 47 (176),1986), i.e., converge with the design order even when truncation errors exhibit a lower-order convergence or, in some cases, do not converge at all.
Finite elements: the accuracy is typically estimated in Sobolev norms. The estimates simultaneously bound solution and gradient errors Recent FVD computations indicate that accurate solutions can be obtained in spite of formally poor accuracy of gradients
What is this study about?
Identify grid qualities that significantly affect specific accuracy measures for viscous and inviscid FVD schemes Identify FVD schemes that are least sensitive to grid qualities Truncation, discretization, and gradient errors are considered on grids ranging from regular (“good quality”) to highly irregular (“bad quality”) deliberately constructed through random perturbation of regular grids.
Avoid other factors
Node-centered FVD schemes provide the same degrees of freedom on tets, quads, prisms, mixed, etc. No variations because of different degrees of freedom. Consistent grid refinement. No errors because some cells are not refined Smooth solutions. No errors because of lack of resolution Manufactured solutions. Exact solutions are known Over-specified boundary conditions. No boundary condition errors
Simple model problems for inviscid and viscous fluxes • Model problem for inviscid fluxes is 2D convection
• Model problem for viscous fluxes is 2D diffusion
Outline
– – – – –
Grid specifications Finite-volume discretizations (FVD) Previous studies Class A: isotropic irregular grids Class B: anisotropic irregular grids (e.g., adaptive grids) – Class C: high-aspect ratio grids around curved bodies (e.g., boundary-layer grids) – Summary and recommendations
Grid Specifications
Three classes of representative grids Class A: Isotropic grids in rectangular geometry
Class B: Anisotropic grids in rectangular geometry (adaptive grids)
Class C: Anisotropic grids around a curved body (advancing-layer high-Reynolds RANS grids)
Regular Quadrilateral and Triangular Grids
Type I
Type II
Random Triangular and Mixed-Element Grids Type III
Type IV
Irregular Perturbed Quadrilateral, Triangular, and Mixed-Element Grids
Type Ip
• •
Type IIIp
Typical node perturbations are random shifts in the range of ¼ of the local mesh size in the corresponding direction Triangles can approach zero volumes
Type IVp
Consistent Grid Refinement • •
•
All grids are derived from the underlined (mapped) Cartesian grids Irregularities (triangulation and node perturbation) are introduced randomly and independently Perturbed triangular and mixed-element grids are grids with discontinuous metrics – Can approach topological degeneration (zero-volume cells) – No improved regularity in the limit of grid refinement
•
Consistent refinement –
Effective mesh sizes (3-D):
hN (1 / N )1/ 3 ; N degrees of freedom h V || (V ) || 1/ 3
; V control volume
– Effective mesh size in 2D computations is
hV h N norm
Node-Centered Finite Volume Discretizations
Finite-volume discretizations Median-dual partition
5
1 2
6
0 3
7
12
4 11
8
9
10
Edge-based FVD scheme for inviscid fluxes
1 2
•Flux is reconstructed at the edge median •Gradient is reconstructed at the node by a least-squares (LSQ) minimization •1st order accurate with a linear fit •2nd order accurate with a quadratic fit •Weighted (WLSQ) and unweighted (ULSQ) methods use edge-connected neighbors
0
3
4
Edge-based control-volume integration •3rd order for homogeneous solutions on regular simplicial grids •2nd order on general simplicial grids •1st order on general mixed elements
3rd order discretization errors on general simplicial grids with 2 nd-order gradients (Katz & Sankaran, AIAA 2011-0652)
Green-Gauss FVD scheme for viscous fluxes
1 2
0
A 3
Face-tangent augmentation
B
P 4
Edge-based on simplicial grids Element-based on other grid types 2nd order accurate on all grids
Previous studies
Previous studies •
Vast literature on supra-convergence publications since 1965
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Assessment of local grid irregularity effects on solution accuracy. Shortcomings of truncation error analysis. Downscaling tests. Diskin and Thomas, NIA Report 2007-08; Thomas, et. al, AIAAJ 2008
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Assessment of gradient accuracy on irregular grids. Diskin and Thomas, NIA Report 2008-12
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Comparative study of common viscous fluxes on unstructured grids. Diskin et. al, AIAAJ 2010
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Comparative study of common inviscid fluxes on unstructured grids. Diskin and Thomas, AIAAJ 2011
Current results Search for schemes which are least sensitive to mesh quality. Results are shown only for current frontrunners: – 3rd order ULSQ scheme with a quadratic fit for inviscid fluxes – Green-Gauss scheme for viscous fluxes
Preview of some conclusions • Gradient errors: – Largely insensitive to mesh qualities on isotropic grids. – Error magnitude proportional to aspect ratio on irregular grids .
• Truncation errors: – Inadequate accuracy metric on irregular grids.
• Inviscid discretization errors: – Insensitive to mesh quality on triangular grids.
• Viscous discretization errors: – Insensitive to mesh quality on grids of Classes (A) and (C)
Class A: Isotropic Irregular Grids
Class A: gradient errors (L1 norm) ULSQ at node
GG at element
ULSQ is 2nd order on all grids GG is 2nd order on Cartesian grids, 1st order on all other grids Insensitive to mesh quality
Class A: truncation error (L1 norm), inviscid fluxes Triangular meshes
Mixed and quad. meshes
Extremely sensitive to any mesh perturbation •3rd order on regular tets, •2nd order on regular quads and irregular tets, •0th order on mixed and perturbed quads
Class A: truncation error (L1 norm), viscous fluxes Triangular meshes
Mixed and quad. meshes
Extremely sensitive to any mesh perturbation •2nd order on regular tets and quads, •0th order on other grids
Class A: discretization errors (L1-norm), inviscid fluxes Triangular meshes
Mixed and quad. meshes
No sensitivity to mesh quality on tets, there is sensitivity on quads •3rd order on tets •2nd order on regular quads, •1st order on mixed and irregular quads
Class A: discretization errors (L1-norm), viscous fluxes
No sensitivity to mesh quality 2nd order on all grids
Conclusions for isotropic grids of Class (A)
• Truncation errors are very sensitive to grid irregularities Do not adequately represent accuracy of the solution • Accuracy of gradients is insensitive to grid irregularities • Discretization errors of viscous fluxes converge with 2nd order and are insensitive to grid irregularities
• Discretization errors of inviscid fluxes converge with 3rd order and are insensitive to grid quality on triangular grids. Lose 3rd order convergence on quadrilateral and mixed grids
Class B: Anisotropic Irregular Grids
Class B: gradient errors (L∞ norm) ULSQ at node
GG at element
Class B: gradient reconstruction accuracy
Grids
(I)
(II)
(III)
(IV)
(Ip)-(IVp)
ULSQ GG
Gradients are sensitive to irregularities on high-aspect ratio grids • • • • •
ULSQ gradients converge with 2nd order GG gradients converge with at least 1st order All methods may generate large relative errors on perturbed grids ULSQ gradients are accurate on regular grids GG gradients are accurate for all unperturbed grids
Class B: discretization errors, inviscid fluxes A = 1000 Convergence of
norm is shown
Class B: discretization errors, viscous fluxes
A = 1000 Convergence of
norm
Conclusions for anisotropic grids of Class (B)
• All ULSQ gradients converge with 2nd order The GG gradients converge with 1st order All gradients produced large errors on perturbed grids • Discretization errors of viscous fluxes converge with 2 nd order on all grids. Error magnitude is sensitive to node perturbations • Discretization errors of inviscid fluxes converge with 3rd order and are insensitive to grid quality on triangular grids. Lose 3rd order convergence on quadrilateral and mixed grids
Class C: High-Aspect Ratio Grids Around Curved Bodies
Random node perturbations are not applied to high-aspect-ratio grids with curvature; even small circumferential perturbations may lead to non-physical control volumes
Γ - grid deformation parameter
In refinement with fixed A, Γ asymptotes to 0 Range of parameters:
High-Γ grids • inherit all difficulties of Class B grids for solutions varying in the circumferential direction of larger mesh spacing • have additional difficulties for solutions varying in the direction of small mesh spacing
Class C: gradient reconstruction accuracy on high-Γ grids Radial solution
Grids
(I)
(II)
(III)
(IV)
GG ULSQ, linear fit
ULSQ, quadratic fit
Accurate GG and ULSQ (quadratic fit) gradients on all grids
Class C: gradient errors (L∞ norm) ULSQ at node
GG at element
Discretization Errors on High-Γ Grids inviscid
viscous
Solution:
Maximum aspect ratio A = 1000
Stretching factor: β≈1.25, 1.11, 1.06, 1.03, 1.01
Γ ≈ 24, 12, 6, 3, 1.5
Conclusions for grids of Class (C)
• ULSQ gradients with a quadratic fit converge with 2nd order and do not suffer from accuracy degradation on curved highaspect-ratio grids The GG gradients converge with 1st order • Discretization errors of viscous fluxes converge with 2 nd order on all grids and insensitive to quality of advanced-layer grids
• Discretization errors of inviscid fluxes converge with 3rd order and are insensitive to grid quality on triangular grids.
Summary and recommendations
Summary •
Gradient errors: – Largely insensitive to mesh qualities on isotropic grids of Class (A). – Converge with the design order (1st and 2nd for Green-Gauss and ULSQ methods, respectively) on all grids (order is not sensitive to grid quality) – Error magnitude proportional to aspect ratio on grids with node perturbation. – ULSQ with a quadratic fit does not suffer from accuracy loss on curved highaspect-ratio grids of Class (C).
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Truncation errors: – Very sensitive to mesh qualities – Converge with a lower-than-design order on all irregular meshes. – Viscous errors do not converge at all on perturbed grids.
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Inviscid discretization errors: – Converge with 3rd order on all triangular grids – Practically insensitive to deterioration of mesh quality on triangular grids. – More sensitive to mesh quality on quadrilateral grids.
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Viscous discretization errors: – Converge with 2nd order on all grids – Insensitive to grid irregularities on grids of Classes (A) and (C)
Recommendations
• The ULSQ with a quadratic fit is highly recommended as a robust way to compute accurate gradients on all grids. • The edge-based scheme based on 2nd-order ULSQ gradients is recommended for inviscid fluxes.
• The Green-Gauss scheme is recommended for viscous fluxes.
Thank You
