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

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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 Reconstruction on Unstructured Meshes

  • 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 Reconstruction on Unstructured Meshes

  • 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 Reconstruction on Unstructured Meshes

  • 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 Reconstruction on Unstructured Meshes

  • 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 Reconstruction on Unstructured Meshes

  • 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 Reconstruction on Unstructured Meshes

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 Reconstruction on Unstructured Meshes

  • 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 Reconstruction on Unstructured Meshes

    • 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 Reconstruction on Unstructured Meshes

    • 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 Reconstruction on Unstructured Meshes

    • 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 Grad. Rec.)

    • 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 Grad. Rec.)

    • 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) Grad. Rec.)

    • 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 Grad. Rec.)

    • Non-linear function required for adequate test

      • a = 200 (reduces roundoff error for small D)

    • Exact Gradient :

      Since


    Gradient error study
    Gradient Error Study Grad. Rec.)

    • 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 Grad. Rec.)

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


    Simpler flat plate geometry
    Simpler Flat Plate Geometry Grad. Rec.)

    • Rounded/Tapered Leading Edge


    Flat plate geometry
    Flat Plate Geometry Grad. Rec.)

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


    Accuracy failure mechanism
    Accuracy Failure Mechanism Grad. Rec.)

    • 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 Grad. Rec.)

    • 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 Grad. Rec.)

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


    Vertex discretization on triangles
    Vertex Discretization on Triangles Grad. Rec.)

    • Similar behavior to vertex discretization on quadrilaterals


    Cell centered discretizations
    Cell Centered Discretizations Grad. Rec.)

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

    • Cell-centered on triangles: No close neighbors


    Cell centered on triangles
    Cell-Centered on Triangles Grad. Rec.)

    • Unweighted and Weighted Least Squares Inadequate

    • Green-Gauss varies by 10% depending on diagonal edge orientation


    Effect on solution accuracy
    Effect on Solution Accuracy Grad. Rec.)

    • 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 Grad. Rec.)

    • 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 Grad. Rec.)

    • Normal dissipation << streamwise dissipation

    • 1st order normal dissip. < 2nd order streamwise dissp.


    Flow alignment2
    Flow Alignment Grad. Rec.)

    • 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 Grad. Rec.)

    • Weighted LS gradients for vertex discretizations

      • Accurate gradients

      • Reduced sensitivity to entropy fix


    Implications1
    Implications Grad. Rec.)

    • 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 Grad. Rec.)

    • 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 Grad. Rec.)

    • 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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