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 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

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|, (|u|+c))

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

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

Entropy Fix

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

= 0.1 : typical value for enhanced robustness = 1.0 : Scalar dissipation becomes scaled identity matrix– T || 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

= 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)

n

F

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|, (|u|+c))

• For aligned flow– Large increase in ufix for small values of – 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.