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Revisiting the Least-Squares Procedure for Gradient Reconstruction on Unstructured Meshes

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### Revisiting the Least-Squares Procedure for Gradient Reconstruction on Unstructured Meshes

Dimitri J. Mavriplis

National Institute of Aerospace

Hampton, VA 23666

Motivation

- Originated from study of matrix dissipation versus upwind schemes for unstructured mesh RANS solver
- Least Squares Gradient now standard technique for higher order accuracy with upwind schemes
- Unexpected behavior observed (with entropy fix)

- 1 week project 3 month investigation

Summary of Findings

- Least squares gradient construction may under-predict gradients by orders of magnitude (~100% error)
- Vertex, cell centered, simplicial, mixed elements
- Subtle mechanism
- Apparently has gone unnoticed in literature
- May not show up in standard test cases
- Similar results: N.B. Petrovskaya: ``The impact of grid cell geometry on the least squares gradient reconstruction’’, Keldysh Institute of Applied Math., Russian Academy of Sciences, April 2003

Spatial Discretization

- Mixed Element Meshes
- Tetrahedra, Prisms, Pyramids, Hexahedra
- Control Volume Based on Median Duals
- Fluxes based on edges
- Single edge-based data-structure represents all element types

Fik = F(uL) + F(uR) + T |L| T-1 (uL –uR)

- Upwind discretization

- Matrix artificial dissipation

Upwind Discretization

- First order scheme

- Second order scheme

- Gradients evaluated at vertices by Least-Squares
- Limit Gradients for Strong Shock Capturing

Matrix Artificial Dissipation

- First order scheme

- Second order scheme

- By analogy with upwind scheme:

- Blending of 1st and 2nd order schemes for strong shock capturing

Entropy Fix

L matrix: diagonal with eigenvalues:

u, u, u, u+c, u-c

- Robustness issues related to vanishing eigenvalues
- Limit smallest eigenvalues as fraction of largest eigenvalue: |u| + c
- u = sign(u) * max(|u|, d(|u|+c))
- u+c = sign(u+c) * max(|u+c|, d(|u|+c))
- u – c = sign(u -c) * max(|u-c|, d(|u|+c))

Entropy Fix

- u = sign(u) * max(|u|, d(|u|+c))
- u+c = sign(u+c) * max(|u+c|, d(|u|+c))
- u – c = sign(u -c) * max(|u-c|, d(|u|+c))

d = 0.1 : typical value for enhanced robustness

d = 1.0 : Scalar dissipation

- L becomes scaled identity matrix

- T |L| T-1 becomes scalar quantity
- Simplified (lower cost) dissipation operator
- Applicable to upwind and art. dissipation schemes

Green-Gauss Gradient Construction

- Contour integral around control volume
- Generally NOT Exact for linear functions
- Only for vertex discretizations on triangles/tetrahedra
- Accuracy dependant on cell shapes
- Poor solver robustness reported for RANS cases

Least Squares Gradient Construction

- Formally unrelated to grid topology
- Natural to base point sample on grid stencil
- Exact for linear functions on all grid/discretization types
- More accurate gradients on distorted meshes
- Reported to be more robust for viscous flows

Drag Prediction Workshop I

- DLR-F4: Mach=0.75, CL=0.6, Re=3M
- Baseline grid: 1.65 million vertices, mixed elements

Comparison of Discretization Formulation (Art. Dissip vs. Grad. Rec.)

- Least squares approach slightly more diffusive
- Extremely sensitive to entropy fix value

Reduce to Simpler 2D Case

- RAE 2822 Airfoil, Mach=0.73, alpha=2.31, Re= 6.5M
- Least-square gradient upwind scheme with entropy fix overly diffusive

Gradient Accuracy Study

- Least-Squares, Green-Gauss, Finite Difference
- Discretization type (cell-vertex), element type
- Exact analytic function (non-linear)
- Compute exact error
- Function similar to flow gradients
- Boundary layer regions

Distance Function: D(x,y)

- Similar to boundary layer velocity gradients
- Available (required by turbulence model)
- Approximately linear:
- Good accuracy of estimated gradient with all methods

Non-Linear Function

- Non-linear function required for adequate test
- a = 200 (reduces roundoff error for small D)
- Exact Gradient :

Since

Gradient Error Study

- Compare calculated and exact Gradient of function F at vertices of mixed element unstructured mesh (quadrilateral elements near airfoil surfaces)

Vertex Discretization on Quadrilaterals

- Unweighted Least Squares Gradients under-predicted by order of magnitude in inner BL

Simpler Flat Plate Geometry

- Rounded/Tapered Leading Edge

Flat Plate Geometry

- Unweighted Least Squares gradients underpredicted up to point of vanishing curvature

Accuracy Failure Mechanism

- All stencil points contribute equally (unweighted)
- Upstream/Downstream Points contribute to
- H > h (due to surface curvature)

Grid Requirements for Unweighted LS

- h > H for accurate grads
- eg: Unit circle, 100 surface points: h > 10-4
- Inv.Distance weighting OK
- S >> h

Vertex Discretization on Quadrilaterals

- Unweighted Least Squares Gradients under-predicted by order of magnitude in inner BL

Vertex Discretization on Triangles

- Similar behavior to vertex discretization on quadrilaterals

Cell Centered Discretizations

- Cell-centered on quads: similar to vertex-based stencil

- Cell-centered on triangles: No close neighbors

Cell-Centered on Triangles

- Unweighted and Weighted Least Squares Inadequate
- Green-Gauss varies by 10% depending on diagonal edge orientation

Effect on Solution Accuracy

- How can good solution accuracy be obtained in the presence of poor gradient estimates ?
- Why is accuracy so sensitive to small values of entropy fix ?

- Flow alignment phenomena
- Occurs in exact same regions as inaccurate gradients
- Inner BL region

Flow Alignment

- Flow solution on RAE Airfoil Grid at x=0.3
- Normal velocity << Streamwise velocity
- Normal convective eigenvalues (u.ds) can be largest (stiff)

Flow Alignment

- Normal dissipation << streamwise dissipation
- 1st order normal dissip. < 2nd order streamwise dissp.

Flow Alignment

- Entropy fix: ufix = sign(u) . min (|u|, d(|u|+c))
- For aligned flow
- Large increase in ufix for small values of d
- Explains solution sensitivity to entropy fix
- Flow alignment irrelevant for acoustic modes
- Good overall accuracy retained in spite of poor resolution of acoustic modes in BL (?)

Implications

- Weighted LS gradients for vertex discretizations
- Accurate gradients
- Reduced sensitivity to entropy fix

Implications

- Unweighted LS more accurate on isotropic grids
- Unweighted LS inaccurate on stretched meshes
- Effect mitigated by flow alignment
- Inaccurate grads only in presence of curvature
- Problem not seen for flat plate BL test case

Implications

- Weighted LS or Green-Gauss gradients more accurate overall
- Robustness issues reported
- Unweighted LS Grads more robust
- Not because of superior gradient estimates
- Because solution is 1st order (limited) in BL
- Viscous (NS) terms based on LS grads could pass flat plate test, but be disastrous

Conclusions

- Unweighted LS grads acceptable
- Must be used only for reconstruction in convective terms
- No entropy fix
- Weighted LS grads offer superior accuracy
- Result in well conditioned system of equations for gradient calculation
- Stencils require close normal neighbor points
- Semi-structured BL meshes
- Robustness issues remain (further investigation)
- Alternate construction techniques (further investigation)
- Dimensional splitting
- Gradient projection (Desideri), SLIP (Jameson)
- Other approaches (Frink, Rausch, Batina and Yang) etc.

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