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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

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revisiting the least squares procedure for gradient reconstruction on unstructured meshes

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

Dimitri J. Mavriplis

National Institute of Aerospace

Hampton, VA 23666

motivation
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
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
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
Upwind Discretization
  • First order scheme
  • Second order scheme
  • Gradients evaluated at vertices by Least-Squares
  • Limit Gradients for Strong Shock Capturing
matrix artificial dissipation
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
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 fix1
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
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
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
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
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
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
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
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
  • Non-linear function required for adequate test
    • a = 200 (reduces roundoff error for small D)
  • Exact Gradient :

Since

gradient error study
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
Vertex Discretization on Quadrilaterals
  • Unweighted Least Squares Gradients under-predicted by order of magnitude in inner BL
simpler flat plate geometry
Simpler Flat Plate Geometry
  • Rounded/Tapered Leading Edge
flat plate geometry
Flat Plate Geometry
  • Unweighted Least Squares gradients underpredicted up to point of vanishing curvature
accuracy failure mechanism
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
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 quadrilaterals1
Vertex Discretization on Quadrilaterals
  • Unweighted Least Squares Gradients under-predicted by order of magnitude in inner BL
vertex discretization on triangles
Vertex Discretization on Triangles
  • Similar behavior to vertex discretization on quadrilaterals
cell centered discretizations
Cell Centered Discretizations
  • Cell-centered on quads: similar to vertex-based stencil
  • Cell-centered on triangles: No close neighbors
cell centered on triangles
Cell-Centered on Triangles
  • Unweighted and Weighted Least Squares Inadequate
  • Green-Gauss varies by 10% depending on diagonal edge orientation
effect on solution accuracy
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 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 alignment1
Flow Alignment
  • Normal dissipation << streamwise dissipation
  • 1st order normal dissip. < 2nd order streamwise dissp.
flow alignment2
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
Implications
  • Weighted LS gradients for vertex discretizations
    • Accurate gradients
    • Reduced sensitivity to entropy fix
implications1
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
implications2
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
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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