45th Aerospace Sciences Meeting and Exhibit, January 8th – January 11th, 2007, Reno, NV

1 American Institute of Aeronautics and Astronautics

Unstructured Adaptive Elliptic Smoothing

Steve L. Karman Jr.* and Mandar Sahasrabudhe† University of Tennessee at Chattanooga

Chattanooga, Tennessee, 37403

Forcing functions for controlling grid spacing have been investigated for an unstructured elliptic smoothing scheme. The forcing functions can provide control of grid spacing for feature-based solution adaptation and normal grid spacing for viscous clustering. An alternate approach, using Riemannian metric tensors to define a spacing field, is also described. This approach solves the same Winslow equations, but without forcing functions. Instead, the Riemannian metrics are used to alter the spacing of the computational mesh, which directly affects the spacing of the physical mesh. Two- and three-dimensional results are included to illustrate the use of these two different techniques.

Nomenclature α = Winslow coefficient β = Winslow coefficient γ = Winslow coefficient λ = eigenvalues, inverse of principal length scales squared in Riemannian tensor Φ = physical u coordinate forcing function Ψ = physical v coordinate forcing function Ω = physical w coordinate forcing function A, B = forcing function adaptation parameters C = equidistribution adaptation parameter a, b = forcing function adaptation parameters d = desired distance based on equidistribution scheme e1,2,3 = principal direction vectors in Riemannian tensor f, ∇f = scalar field and gradient vector of scalar field h1,2,3 = Riemannian tensor specified distances x = computational coordinate y = computational coordinate z = computational coordinate u = physical coordinate v = physical coordinate w = physical coordinate M = Riemannian metric tensor R, R-1 = rotation matrix and eigenvector system

I. Introduction LLIPTIC mesh smoothing schemes are commonly used with structured meshes. The governing equations are typically the Winslow equations, with or without forcing functions.1 Forcing functions have been developed for

structured meshes by many researchers.2, 3, 4 With these forcing functions users can control the grid point placement near boundaries to enforce normal spacing requirements and grid line angularity for use with viscous flow analyses and improve overall mesh quality. Other researchers have developed equidistribution schemes or forcing functions for Winslow schemes to adapt a given mesh to flowfield features contained in the solution computed on the mesh.5,6 The premise for the * Research Professor, Graduate School of Computational Engineering, and AIAA Associate Fellow. † Research Associate and Graduate Student, Graduate School of Computational Engineering.

E

45th AIAA Aerospace Sciences Meeting and Exhibit8 - 11 January 2007, Reno, Nevada

AIAA 2007-559

Copyright © 2007 by University of Tennessee at Chattanooga. Published by the American Institute of Aeronautics and Astronautics, Inc., with permission.

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equidistribution schemes is to equally distribute a weight function over the mesh. The weight function is typically based on first and second derivatives of features, such as shock, contained in the existing flowfield solution on the mesh. The resulting mesh will exhibit tighter grid spacing in region of high gradients of the flowfield feature. Anderson showed the relationship between the equidistribution equations and the forcing function for Winslow equations.5 His conclusions were that Winslow schemes with these adaptive forcing functions were superior to equidistribution schemes because the Winslow equations contains a smoothness property inherent in the governing equations. Elliptic smoothing of unstructured meshes is possible if a computational mesh can be defined.7 Elliptic smoothing using Winslow equations without forcing functions was demonstrated in two dimensions and three dimensions in the reference. Fixed grid forcing functions were demonstrated for a two dimensional case. This paper continues that research and examines forcing functions for unstructured elliptic smoothing. The adaptive technique described by Anderson is extended to unstructured meshes. Additional forcing functions are investigated to control grid spacing in the normal direction of viscous boundaries. An alternate method for controlling the physical mesh was suggested in Reference 7. Instead of specifying an equally spaced computational mesh one could incorporate clustering in the computational mesh and the solution to the Winslow equations without forcing functions would incorporate the clustering in the physical mesh. But, instead of actually clustering the computational mesh as demonstrated in Reference 7, Riemannian metric tensors can be used to specify the desired clustering. The use of these Riemannian metric tensors will be investigated for controlling mesh smoothness, mesh spacing in boundary layer regions mesh spacing in the high gradient regions for solution-based adaptation.

II. Unstructured Mesh Smoothing This section will briefly describe Winslow elliptic smoothing applied to unstructured meshes. Then forcing

functions will be defined for feature-based adaptation based on an existing flowfield solution on the given mesh. This will be followed by a discussion of forcing functions for the control of normal grid spacing of viscous meshes. Finally, the use of Riemannian metric tensors applied to the computational mesh will be discussed with emphasis on controlling viscous spacing at the boundaries and interior mesh spacing in solution-based adaptation.

A. Elliptic (Winslow) Smoothing An approach to solving the Winslow equations on unstructured meshes was described in Reference 7. The three-

dimensional Winslow equations are given in Equation 1. The computational coordinates are x, y, and z. The physical coordinates are u, v and w. Winslow coefficients are given in α1-3 and β1-3. The forcing functions are Φ, Ψ, and Ω.

�

!1(uxx + "ux ) + !

2(uyy + #uy ) + !

3(uzz + $uz) + 2(%

1uxy + %

2uyz + %

3uxz) = 0

!1(vxx + "vx ) + !

2(vyy + #vy ) + !

3(vzz + $vz) + 2(%

1vxy + %

2vyz + %

3vxz) = 0

!1(wxx + "wx ) + !

2(wyy + #wy ) + !

3(wzz + $wz) + 2(%

1wxy + %

2wyz + %

3wxz) = 0

(1)

�

!1

= (! r y •! r y )(! r z •! r z) " (

! r y •! r z)(! r y •! r z)

!2

= (! r z •! r z )(! r x •! r x ) " (

! r z •! r x )(! r z •! r x )

!3

= (! r x •! r x )(! r y •! r y ) " (

! r x •! r y )(! r x •! r y )

#1

= (! r y •! r z )(! r z •! r x ) " (

! r x •! r y )(! r z •! r z)

#2

= (! r z •! r x )(! r x •! r y ) " (

! r y •! r z)(! r x •! r x )

#3

= (! r x •! r y )(! r y •! r z) " (

! r z •! r x )(! r y •! r y )

! r = (u,v,w)

The two-dimensional form of the Winslow equations are given in Equation 2. Winslow coefficients are given as

α, β and γ. The forcing functions are Φ and Ψ.

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22

22

0)(2)(

0)(2)(

xx

yxyx

yy

yyyxyxxx

yyyxyxxx

vu

vvuu

vu

vvvvv

uuuuu

+=

+=

+=

=!++"#+

=!++"#+

$

%

&

$%&

$%&

(2)

Equations 1 & 2 represent non-linear partial differential equations, with the coupling provided in the Winslow coefficients. The coefficients α, β, γ, Φ, Ψ and Ω are assumed frozen in order to linearize the system. An outer iteration loop is required to update these coefficient values. The linear system of equations can be solved using any number of techniques, such as a simple point-implicit method or more complicated Generalized Minimum Residual method. A description of the solution method is given in Reference 7. The unstructured implementation of Winslow smoothing requires a computational mesh that must match the element topology of the physical mesh. The computational mesh can also match the initial physical mesh exactly. In other words, for this investigation, the computational mesh can be a copy of the initial physical mesh. The computational mesh typically remains constant while the physical mesh is change based on the solution to the Winslow equations. If the boundaries of the computational mesh exactly match the boundaries of the physical mesh and no forcing functions are used then Winslow equations will produce a physical mesh that exactly matches the computational mesh at every grid point.

B. Forcing Functions Forcing functions for use with an existing flowfield solution will be described first. Two forms of the weight

function will be described; one uses only first derivatives of the solution adaptation parameter and the second uses first and second derivatives of the solution adaptation parameter. These techniques can be extended to define forcing functions to control the normal spacing of the physical mesh near viscous boundaries.

1. Feature-based solution adaptation Anderson equated the two-dimensional equidistribution scheme with the Winslow forcing functions using

equations 3.5

�

! =1

w1

"w1

"x

# =1

w2

"w2

"y

(3)

The weight functions, w1 and w2, defined for the equidistribution scheme are given by equations 4.

�

w1

=1+ a!f!x"

# $

%

& '

2

w2

=1+ b!f!y

"

# $

%

& '

2 (4)

Given an adaptation function f, such as pressure, velocity magnitude or Mach number, the appropriate weight functions can be computed on the computational mesh. Parameters a and b are user defined variables that control the amount of adaptation that takes place. These weight functions can then be used to construct the Winslow forcing functions, Φ and Ψ.

Another formulation was given by Eiseman that accounts for the transverse variation of the weights in each direction.6 The two-dimensional forcing functions are given in equations 5.

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�

! =1

""

w1( )x

w1

#$w1( )y

w1

%

& ' '

(

) * *

+ =1

,#$

w2( )

x

w2

+ ,w2( )

y

w2

%

& ' '

(

) * *

(5)

The two-dimensional Winslow coefficients, α, β & γ, are given by equations 2. An alternative form for the weight function includes first and second derivatives of the adaptation parameter.

�

w1

=1+ A!f!x"

# $

%

& '

2

+ B! 2 f!x 2

"

# $

%

& '

2

w2

=1+ A!f!y

"

# $

%

& '

2

+ B! 2 f!y 2

"

# $

%

& '

2 (6)

Parameters A and B are also user defined variables that control the amount of adaptation that takes place. These weight functions can also be used to construct the Winslow forcing functions.

2. Viscous normal spacing One approach for controlling normal spacing is an extension of the feature based adaptation functions. The

control functions can be based on a contrived adaptation field in the form of a ramp function defined in the viscous region of the mesh. The portion of the mesh in the viscous region is generally created with a specified distribution of points in the normal direction. The elements are typically quadrilaterals in two dimensions and prisms or hexahedra in three dimensions. Therefore, it is reasonable to define a ramp function that is large at the surface and ramps to zero over a specified number of elements in the normal direction. The depth of the ramp function affects both the first and second order derivatives and hence directly affects the clustering. The steepness of the ramp functions also has an effect on the clustering, but that same influence is also afforded by the user defined adaptation parameters, A and B. The defined viscous region ramp function can then be used to compute the weight functions and the Winslow control functions in the same manner as used for feature-based adaptation.

C. Riemannian Metric Tensor Scaling Mesh generation researchers are starting to define target element sizes through the use of a Riemannian metric

tensor field.8,9,10 The desired spacing and orientation can be defined by a symmetric, positive-definite matrix M as a product of a rotation matrix R and a scaling matrix λ.11 The columns of R are the eigenvectors of M and correspond to the principal directions. The scaling matrix λ is a diagonal matrix of eigenvalues of M, which are specified as the inverse square of the desired distances along principal directions.

M = R[ ] ![ ] R[ ]"1

2-D # M =!e

1

!e

2[ ]h

1

"20

0 h2

"2

$

%&'

()

!e

1

T

!e

2

T

*

+,

-

./

3-D # M =!e

1

!e

2

!e

3[ ]

h1

"20 0

0 h2

"20

0 0 h3

"2

$

%

&&&

'

(

)))

!e

1

T

!e

2

T

!e

3

T

*

+

,,,

-

.

///

(7)

Tchon et. al. provided a geometrical interpretation of the defined Riemannian metric, shown in Figure 1.11 The two-dimensional metric defines a transformation of an ellipse in physical space to a unit circle in control space. A

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similar construction in three dimensions results in a Riemannian tensor that can be interpreted as a transformation of an ellipsoid in physical space to a unit sphere in control space. To use this tensor field, mesh refinement algorithms perform local refinement to enforce the prescribed distances and orientations, which requires the calculation of the metric length between two points. The formula for the metric length is given in Equation 8. If the result from Equation 8 equals unity the mesh edge A-B matches the desired length defined by the tensor.

Figure 1. Geometric interpretation of prescribed Riemannian metric.

dAB

= ABT

! "!!!

M[ ] AB

! "!! (8)

These Riemannian metric tensors can be used to control the grid spacing in the context of Winslow elliptic smoothing as well. But instead of defining forcing functions to control the physical spacing, the Riemannian tensors are used to define the desired lengths in the computational mesh. The spacing in the physical mesh will mimic the spacing in the computational mesh when no forcing functions are used with the Winslow equations. Therefore, the physical mesh will mimic the Riemannian-metric prescribed spacing of the computational mesh. The scaling is applied during the discretization of the Winslow equations. The computational mesh is not actually altered. Hansen et. al. used Riemannian metric tensors to control grid smoothness in a finite element implementation of a Winslow smoothing scheme.12,13 Their approach was to alter the shape function by computing the metric tensors on the dual mesh. They demonstrated the ability to smooth two- and three-dimensional meshes to improve mesh quality. The present approach uses a finite volume method to solve the Winslow equations and applies the Riemannian metric tensors to smoothing, viscous layer control and solution adaptation.

The effect of the Riemannian metric tensor is shown in Figure 2 where a collection of triangles is flattened in the vertical direction by a horizontally oriented tensor, shown as an ellipse in the figure. In this example the more vertically aligned edges are reduced in length to a larger extent than the more horizontally aligned edges. Also note that all edges appear to be reduced to some extent, even the horizontally aligned edges. This would not be true if the prescribed tensor field defined a larger spacing in the horizontal direction. In that case the elements would be elongated and compressed at the same time.

r = 1

physical space control space

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Figure 2. Example of Riemannian metric tensor applied to a collection of triangles.

The Riemannian tensors can be prescribed on a background field or at elements or edges of the current mesh. The tensors are used to scale the edges of the computational mesh in the discretization of the Winslow equations. If a background tensor field were prescribed the tensor for a given edge would be interpolated from the background field. If element values of the metric are prescribed the edges are scaled using an average of the Riemannian tensors for the two elements on each side of the edge. If edge values of the metric are prescribed then no averaging is necessary; the edge tensor is applied directly.

When formulating the Winslow system of equations the edges of the computational mesh must be scaled consistently from one element to another. If an edge is scaled differently when processing one triangle versus the neighboring triangle then the discretization does not employ a closed control volume for the nodes of that edge. Whereas, if the tensors are prescribed at the edges or at the elements and averaged to scale the edge then a consistent scaling and control volume is achieved.

In order to compute this tensor field for an existing mesh one needs to be able to compute a tensor for an existing triangle in 2D or tetrahedra in 3D.8 The existing tensor for a single simplex element is the tensor that returns a unit metric length for each edge of the simplex. Writing an expression of unit metric length for each edge of the simplex, as shown below, creates the system of equations for computing the elements of the Riemannian metric tensor. Since this tensor is symmetric 3 unknowns are required in two dimensions (d = 3) and 6 unknowns are required in three dimensions (d = 6).

Pj ! Pi( )T

M[ ] Pj ! Pi( ) = 1 for 1 " i < j " d (9)

If the three vertices of a triangle were numbered 1-3 then the resulting system of equations for computing the elements of the 2-D Riemannian metric tensor are shown in equations 10.

(x2! x

1)2

2(x2! x

1)(y

2! y

1) (y

2! y

1)2

(x3! x

1)2

2(x3! x

1)(y

3! y

1) (y

3! y

1)2

(x3! x

2)22(x

3! x

2)(y

3! y

2) (y

3! y

2)2

"

#

$$$

%

&

'''

M11

M12

M22

(

)*

+*

,

-*

.*=

1

1

1

(

)*

+*

,

-*

.*

(10)

These systems of equations are typically ill conditioned so methods such as Gauss elimination or Gauss-Seidel do not work. Therefore, Singular Value Decomposition was employed to solve for the unknowns, Mij. The simplex element metric tensors can then be averaged to edges to define consistent edge-based metric tensors. This is possible due to the associative properties of these tensors. The element tensors that share an edge will return a unit metric

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length for the common edge, so the average of the element tensors will also return a unit metric length for the common edge. This makes it possible to construct edge-based tensors for an existing mesh that, if used in the Winslow smoothing, will not alter the lengths of an existing computational mesh and Winslow smoothing will behave exactly as it does without Riemannian metric scaling. The benefit of this edge-based construction is that one can manipulate tensors locally, such as in viscous layers or in solution-based adaptation, and not adversely change the behavior of Winslow smoothing globally.

To achieve the desired local clustering of the physical mesh in one or more directions simply replace the existing Riemannian metric tensor with one that conforms to the desired spacing applied in the appropriate directions. For instance, the Riemannian metric tensor for an edge in the viscous layer region aligned in the normal direction can be constructed using the local normal vector as one of the eigenvectors and the inverse square of the desired normal spacing as the eigenvalues in the construction of the tensor. If the desired tangential spacing in unknown the existing spacing in the tangential direction can be obtained by taking the inverse of the metric length for a unit vector in the tangential direction. In other words, the existing tangential spacing is determined by taking the inverse of the metric length for a unit tangential vector using the existing Riemannian metric. This form of tensor modification is applicable to viscous layer spacing control.

An alternate approach is to modify the eigenvalues of the existing Riemannian tensor, keeping the original eigenvectors. The eigenvalues can be scaled based on the desired distance in the direction of the existing eigenvectors, or principal directions. In the case of solution-based adaptation, the gradient vector of the adaptation function can be used to determine a desired spacing in the direction of the gradient. The equidistribution law can be employed to compute the desired spacing. In equation 11 the gradient vector for adaptation function is dotted with an edge normal vector and multiplied by distance in the direction of the edge raised to the power p. The minimum, average and maximum values of C are computed over the entire mesh. The average value of C, or a user specified value, is then used to determine the new local desired spacing. Edges with a value of this function greater than Cavg will be reduced in length. Edges with a value of this function less than Cavg will be increased in length. The equidistribution law is a statement of error in the existing solution and the goal is to drive C towards zero. The power of the distance was chosen to provide more influence in adapting the mesh in regions where the is larger grid spacing as opposed to regions of small spacing in the presence of discontinuities, such as shocks.

!f • e "dp= C

"dnew =Cavg

!f • ep

(11)

This computed distance, Δd, replaces the existing distance in the calculation of the Riemannian tensor. But, some limiting is imposed to prevent drastic changes from the existing spacing and limit the minimum and maximum size of the distance specified. One control limits the percent change in the computed distance compared to the existing distance. Typically the specified size is prevented from decreasing or increasing by more than 25 percent. Another control prevents the specified size from exceeding minimum and maximum global spacing parameters.

The discretization of the Winslow equations will use modified control volumes in the computational mesh. If the control volume for a node is the collection of elements surrounding the node then for the center node on the left in Figure 2 the discretization will use the collection elements shown on the right. By using Riemannian metric tensors the use of forcing functions in Winslow equations are not necessary. Besides, prescribing forcing functions does not guarantee a valid mesh will result. One can always compute forcing functions for a given physical mesh, but the reverse is not always true. Arbitrary forcing functions do not always result in a valid physical mesh. Since the Riemannian metric is a positive definite matrix it would seem impossible to prescribe a metric that will result in an inverted element in computational space. The scaling term is always positive, so no edge flips will result, thus no element inversion of the computational mesh. The discretization should remain valid and the robustness and smoothing qualities of the Winslow equations without forcing functions would not be sacrificed.

In practice, however, this is not the case. Scaling the control volume edges does not invert the computational elements but it can induce a rotation of the element if edges are not scaled equally. This will be reflected in the physical mesh. It is also possible to prescribe Riemannian tensor that attempts to elongate the edges of the mesh in a way that will no longer fit within the confines of the boundaries of the physical mesh. Experience has shown that valid meshes can be obtained if the tensor field, for the most part, attempts to reduce the spacing in the mesh. If the tensor field attempts to increase the spacing in parts of the mesh the solution to Winslow equations may produce

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inverted physical elements. This can be explained using Figure 3. Winslow smoothing, using the Riemannian metric scaling of 10 layers of points near the surface produced the adapted mesh on the right side of the figure. The local Riemannian metrics for these 10 layers of edges was redefined using the desired normal spacing and direction while retaining the existing tangential spacing of the original mesh. That tangential spacing was valid for the elements in their original location. As the Winslow smoothing pulls the tetrahedra toward the sphere the original tangential spacing is too large for the new location closer to the sphere. As a result the specified tangential spacing cannot be satisfied by the Winslow equations. However, even with the excessive tangential spacing, the Winslow equations did converge to produce a valid mesh for this case. A tighter initial spacing could result in a buckling of the elements away from the sphere where the tangential spacing cannot be satisfied.

Figure 3. Riemannian metric scaling applied to tetrahedral mesh about a sphere. Normal spacing specified

using an initial spacing of 0.04 and a geometric progression factor of 1.0 through 10 layers of edges.

The color contours show the approximate specified distance. It is actually the inverse of the square root of the Frobenius norm of the Riemannian metric tensor. The Frobenius norm is the root mean square value of the elements of the matrix, which is driven by the scaling specified through the eigenvalues. So the Frobenius norm is a measure of the eigenvalues and the inverse of the square root of the Frobenius norm would be an approximate measure of the collective magnitude of the specified distances.

III. Results Several examples are included to demonstrate the use of forcing functions and Riemannian metric scaling for

viscous clustering and solution-based mesh adaptation. The first three cases illustrate the use of forcing functions to control the spacing of two-dimensional meshes. The remaining cases use Riemannian metric scaling to control the spacing of two- and three-dimensional meshes.

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A. Forcing Functions 1. Diagonal Shock Adaptation with Forcing Functions

The first case recreates one of the cases described in Anderson’s paper.5 An equally spaced quadrilateral mesh on a square domain is converted to a triangular unstructured mesh, shown in Figure 4. A function is programmed to stored scalar values of f at the nodes using the following formula.

�

if x <y + 10

2

!

" #

$

% & ' f = 1

if x (y + 10

2

!

" #

$

% & and x )

y + 14

2

!

" #

$

% & ' f =

7 * x2

+y

4

if x >y + 14

2

!

" #

$

% & ' f = 0

(12)

Equation 12 defines a ramp function from 1 to 0 that is skewed at an angle. The function ramps over 4 mesh points in the horizontal direction. The origin is located at the lower left corner of the mesh. The adapted mesh was computed using the expression in equations 5 and 6 to compute the forcing functions. Adaptation parameters A and B were set to 10.0. The final mesh is shown in Figure 5. The boundary points are held fixed in this operation, so the points along the top and bottom boundary do not cluster to the shock. The adaptation to the prescribed function is clear but extent of the clustering appears to be less than found in Anderson’s paper. This could be due to the unstructured discretization versus the structured discretization. The structured mesh solution would use a 3 X 3 stencil centered at each point. The unstructured solution would use a stencil comprised of all neighboring points connected by an edge. This could include as few as 5 neighbors and as many as 7 neighbors for this triangulation.

Figure 4. Original mesh and computational mesh

for diagonal shock case.

Figure 5. Adapted mesh using first and second

derivative formula and A=10, B=10.

2. Viscous Layer Control The second case illustrates the use of a ramp function to influence the normal spacing of layers of quadrilateral

elements surrounding an array of cylinders in a two-dimensional mesh. The original mesh is shown in Figure 6. The ramp function varied from 0.2 at the cylinder to 0.0 at the edge of the quadrilateral layers with a slope of 0.05. The constants A and B in equations 6 are calculated as the minimum of the local edge length surrounding the node. The control functions can be further scaled upon calculation to give additional control on the grid point distribution. The compression case, Figure 7, was achieved by multiplying the control function by +50. The expansion case, Figure 8, was obtained when the control functions were multiplied by -80. From preliminary experiments, scaling the control

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functions seems to give a better grid quality than scaling the depth or steepness of the viscous ramp function. Further investigation is required to examine the effect of each of these parameters in more detail, to accurately match the required grid point distribution.

Figure 6. Original (computational) mesh for array of cylinders with quadrilateral layers.

Original & computational mesh Ramp function inducing compression of layers

Figure 7. Forcing functions applied to compress the quadrilateral layers near cylinders.

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Original & computational mesh Ramp function inducing expansion of layers

Figure 8. Forcing functions applied to expand the quadrilateral layers near the cylinders.

3. Transonic Airfoil Solution Adaptation With Forcing Functions

A two-dimensional inviscid solution was computed for a NACA 0012 airfoil at a freestream Mach number of 0.8 and 1.5 degrees angle of attack. The initial unstructured triangular mesh is shown in Figure 9. Color contour of Mach number are also included in the image. Feature-based mesh adaptation was performed using Mach number as the adaptation function. Equations 6 were used to compute the weight functions. The adaptation parameters, A and B, were set tot 10 and 0.1, respectively. The adapted mesh is shown in Figure 10. The boundary points were held fixed. Clustering occurs in the region of the shock near the airfoil and diminishes further away from the surface, as the magnitude of the Mach gradient decreases. A solution on the adapted mesh resulted in the location of the shock shifting outside the clustered region. Adaptation based on the new solution clustered the mesh to the new shock location. This oscillatory behavior persisted and no greater clustering was achieved. Further research is required to increase the amount of adaptation in coarser regions of the mesh and to allow convergence of the adaptation-solution cycle.

Figure 9. Initial mesh and Mach contours for

NACA0012 airfoil.

Figure 10. Adapted mesh based on gradients of

Mach number.

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B. Riemannian Metric Scaling 1. Mesh Smoothing

Mesh smoothing is required when the grid obtained from the grid generator is lacking in overall grid quality. Mesh smoothing typically refers to an improvement in grid quality by relocating mesh points. The simplest and most inexpensive way to do mesh smoothing on unstructured meshes is to use Laplacian based smoothing. Laplacian based smoothing is simply an averaging of neighboring nodes and can generate invalid meshes, especially at sharp corners. Spring analogy smoothing is based on treating the edge connections as linear and torsional springs. Smoothing is based on the equidistribution of energy in the loaded springs at equilibrium. A newer and more expensive mesh smoothing process involves the minimization of a cost function based on a particular grid metric such as Aspect Ratio, Condition Number. These methods typically use a perturbation technique to move the points and check for an improvement in the cost function.

The Riemannian tensors are also extendable for the purpose of mesh smoothing. In the Riemannian tensor process outlined above specifying a uniform tensor on the computational mesh implies a uniform edge spacing being imposed over the physical mesh. Specifying a unit tensor results in an unchanged metric length for a given edge. Since the metric lengths scale the computational mesh we get a uniformly spaced computational mesh. This is mirrored in the smoothed physical mesh.

The first mesh smoothing case is a square domain populated with points generated by a random number generator and the computational mesh is obtained by a Delaunay triangulation method. The Riemannian tensors are uniformly assigned to be a unit tensor. Figure 11 shows the initial mesh and Figure 12 shows the final mesh after 5 iterations. The improvement in mesh quality is obvious.

The second mesh smoothing case is a circle inside square domain. The grid was generated using the commercial code Gridgen.14 The smoothing objective for this case was to generate a more uniform mesh with a smooth transition from inner elements to the outer elements and without sacrificing element quality. A background Riemannian tensor field was specified in this case, which was proportional to the distance of the edge from the center of the circle. Limits were put on the minimum and the maximum specified edge lengths. The initial mesh and point distribution are shown in the top of Figure 13 and the smoothed mesh and point distribution are shown in the bottom of the figure.

Figure 11. Initial 2-D mesh prior to smoothing.

Figure 12. Final 2-D after Riemannian metric scaling

with Winslow equations.

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Original mesh

Original point distribution

Smoothed mesh

Smoothed point distribution

Figure 13. Riemannian metric applied to smooth point distribution of 2-D cylinder mesh.

Mesh smoothing was also attempted in the 3-D case shown in Figure 14. The original mesh was generated in Gridgen and is shown as the gray mesh on the left side of the top images in the figure. The Riemannian metric tensors were specified as a spherical tensor with the length determined from an inverse distance weighted average of the desired spacing for each boundary. The inner boundary, the sphere, had a specified spacing of 0.04 and the outer boundary had a specified spacing of 1.0. The color contours on the left side in the bottom of the figure show the specified spacing field. Some slight changes can be seen in the smoothed mesh on the right side of the top images in the figure.

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Figure 14. Smoothing tensor applied to a 3-D sphere mesh. Top two images show original mesh on left and smoothed mesh on right. Bottom image shows contours of specified spacing parameter.

2. 3-D Hollow Sphere Solution Adaptation Using Riemannian Tensors

Solution-based adaptation was performed on a three-dimensional hollow spherical shaped scalar field in a box. A nearly uniform tetrahedral mesh was created for a 20 X 20 X 20 box, shown on the left side in Figure 15 through Figure 18. This mesh served as the computational mesh. The initial physical mesh was an exact copy of the

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computational mesh. A scalar function was artificially imprinted on the physical mesh in the shape of a step function of unit height between radii of 2 and 3 from the origin followed by a ramp function back down to a value of zero between radii of 3 to 5. The color contours in the figures represent this scalar function. The gradient of this function was computed at each node of the physical mesh. The gradient vectors were normalized by the maximum magnitude of the gradient vectors over the entire mesh. Equation 11 was then used to modify the eigenvalues (h’s in equation 7) of the Riemannian metric tensor for each edge in the mesh. Averaging the nodal values created edge values of the gradient vector. The prescribed distance was prevented from exceeding less than 50% of the computational mesh spacing or exceeding a minimum spacing of 0.05. The prescribed distance was also prevented from exceeding more that 120% of the computational spacing or exceeding a maximum spacing of 0.6.

Six adaptation cycles were performed. The first mesh adaptation is shown in Figure 15. The left-side mesh is the computational mesh, which does not change. The right-side mesh is the physical mesh after the adaptation pass. The locations of the beginning and ending points of the function profile are inaccurate, as can be seen on the right side of Figure 15. Winslow with Riemannian metric scaling clustered the mesh based on the gradients of the scalar field, but does not impose the location where the clustering takes place. This behavior was also seen with the use of forcing functions for the airfoil case, section III.A.3.

Figure 15. Hollow sphere mesh after 1st adaptation.

The next adaptation cycle imprinted the same scalar field on the new physical mesh and then adapted the mesh using the modified Riemannian tensors based on equation 11. The second mesh is shown in Figure 16. This time the radius of the inner side of the profile is more accurately captured. The outer radius of the profile is also more accurately captured. The third mesh is shown in Figure 17. The inner and outer radii of the profile are accurately captured. The process was repeated another 3 adaptation cycles. Each time the scalar field was imprinted on the current physical mesh. The computational mesh was never changed. The Riemannian tensors of the computational mesh were modified based on equation 11 and the current gradient information. The expansion of the scalar function in the computational mesh can be seen on the left side of the figures. The final mesh is shown in Figure 18. The expansion of the profile into the computational mesh can be seen in the left-side mesh. The final profile, shown on the right side of the figure is crisply captured and accurately located. This adaptation process used an artificial scalar field, but represents the process that would be used to adapt a mesh to a solution obtained using a flow solver.

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Figure 16. Hollow sphere mesh after 2nd adaptation.

Figure 17. Hollow sphere mesh after 3rd adaptation.

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Figure 18. Hollow sphere mesh after final adaptation.

3. 3-D Sphere: Hybrid Prisms & Tetrahedra Viscous Layer Control Using Riemannian Tensors

Riemannian metric scaling was used to control viscous clustering for a hybrid tetrahedral/prism mesh about a sphere. The initial (and computational) mesh was created by inserting 11 layers of prism with a height of 0.01 into an existing all tetrahedral mesh about the same sphere shown in section II.C. Riemannian metric tensors for 15 layers of edges, including the 11 prism layers and 4 tetrahedral layers, were prescribed using an initial normal spacing of 0.001 and a geometric progression factor of 1.15. The initial mesh is shown on the left side of Figure 19. The re-clustered mesh is shown on the right side of the figure. The color contours represent the magnitude of the approximate specified distance.

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Figure 19. Prism layers and some tetrahedra are clustered toward sphere with initial spacing of 0.001.

4. 3-D Cube Viscous Layer Control Using Riemannian Tensors

Riemannian metric scaling was used to control the viscous spacing of a hybrid mesh about a cube with sharp edges. The initial mesh was created by inserting 11 layers of prismatic elements with a normal spacing of 0.01 in a tetrahedral mesh, shown on the left side of Figure 20 and Figure 21. The mesh was redistributed using Winslow smoothing to produce the mesh on the right side of the figures. The first case reduced the initial spacing by a factor of 10 and used a geometric progression factor of 1.15. The second case started with the physical mesh from Figure 20 and double the initial spacing to 0.002 with the same geometric progression factor.

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Figure 20. Prism layers clustered to surface with 1st layer spacing of 0.001.

Figure 21. Prism layers redistributed with 1st layer spacing of 0.002.

5. Onera M6 Solution Adaptation

The last example case is an adaptation of an all-tetrahedral mesh to a transonic solution of the Onera M6.15 The inviscid solutions were computed using the TENASI flow solver developed at the University of Tennessee

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SimCenter at Chattanooga. The freestream conditions were Mach 0.84 at an angle of attack of 3.06 degrees. This case produces a double shock structure on the upper surface of the wing, shown in Figure 22. The initial mesh was created using Gridgen is shown on the left side of Figure 23 through Figure 25. Since the Winslow smoother does not currently have the ability to allow boundary points movement the case was computed for a full span model. The adaptation parameter, C in equation 11, was set to a value of 0.0001. Since the desire was to show adaptation to the shock waves the value for the exponent was set to 1.0. Thus the adaptation will focus on the discontinuity resulting from the shock. Density served as the adaptation function. Due to time constraints, only one adaptation pass was performed. The edges were allowed to reduce to 50% of the computational mesh. Due to some grid crossing that occurred at the wing tip the adaptation was restricted to edges smaller than .007 in length. The maximum edge size was set to 1.0. The adapted mesh is shown on the right side of the figures. The solution shown is after another flow solver calculation, so the shock location is not oscillating as it did in the 2-D case. Clustering to the shock structure is evident at each span station. The contour plots for each mesh are shown in the bottom of each figure. The shocks are more crisply captured in the adapted mesh.

Figure 22. Surface density contours showing complex shock structure.

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Original

Adapted

Figure 23. Onera M6 original and adapted grid at Y=0 span station.

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Original

Adapted

Figure 24. Onera M6 original and adapted grid at Y=0.5 span station.

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Original

Adapted

Figure 25. Onera M6 original and adapted mesh at Y=1.0 span station.

IV. Conclusions Adaptive forcing functions, originally developed for structured grid smoothing, have been extended for use in

unstructured meshes. Weight functions were described that use first and second derivatives of the adaptation function. These weight functions are used to develop forcing functions for the Winlsow equations to influence the clustering of the mesh based on features contained in the solution stored on the mesh. Results on a diagonal shock problem with an unstructured mesh show agreement with structured mesh adaptation performed by Anderson.5 Initial efforts to adapt an inviscid solution for a transonic shock produced clustering near the strong shock on the upper surface, but oscillations in the location of the shock were observed with subsequent adaptations. Additional research is required to enable the adaptation process to converge and cluster the physical mesh and accurately capture the features in the solution.

Forcing functions were also used to control grid spacing in the normal direction for a hybrid 2-D grid comprised of multiple cylinders with quadrilateral elements near the surfaces of the cylinders. A scalar ramp function was generated and the Winslow forcing functions based on the gradients of this function were used to alter the normal grid spacing of the viscous layers of the mesh. The distribution could be expanded or contracted based on the scaling applied to the forcing functions.

A new approach was presented that uses Riemannian metric tensors to alter the computational mesh to control the spacing in the physical mesh. No forcing functions are required. Instead the user prescribes a Riemannian metric tensor field that is used to scale the edges of the computational mesh during the discretization of the Winslow equations. This field can be manipulated to control physical mesh spacing to smooth the mesh, control normal spacing in viscous layers and adapt to gradients in a flowfield solution. Two examples prescribed a tensor field that smoothly varied the desired edge spacing from the inner boundary of a cylinder in 2-D and a sphere in 3-D to the outer boundary. The resulting physical meshes showed a much smoother point distribution. Two other cases used Riemannian metric scaling to control the viscous layers near a sphere and a cube. An artificial scalar field was

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adapted to using modified Riemannian tensors based on an equidistribution technique. The adaptation process accurately converged to produce a physical mesh that clustered points in the gradient region of the scalar field. The final example showed the adaptation of a tetrahedral mesh to the transonic solution about an Onera M6 wing. The adapted mesh properly clustered to the double shock on the upper surface without exhibiting the oscillations observed using the forcing functions on the 2-D airfoil case.

The use of the Riemannian metric tensor to scale the computational edges opens up new possibilities for mesh manipulation. This paper describes only a few examples of constructing Riemannian tensor fields on simple configurations to accomplish mesh smoothing, viscous layer spacing control and mesh adaptation. Additional research is required to determine ways of constructing Riemannian tensor fields to accomplish the desired mesh control and to guarantee the construct tensor field produces valid physical meshes.

References 1 Winslow, A., “Numerical Solutions of the Quasilinear Poisson Equation in a Nonuniform Triangle Mesh,” Journal of Computational Physics, Vol. 1, No. 2, 1967, pp. 149-172. 2 Thompson, J. F., Thames, F. C., and Mastin, C. W., “Boundary-Fitted Curvilinear Coordinate Systems for Solution of Partial Differential Equations on Fields Containing Any Number of Arbitrary Two-Dimensional bodies,” NASA CR-2729, July 1977. 3 Thomas, P. D., and Middlecoff, J. F., “Direct Control of the Grid Point Distribution in Meshes Generated by Elliptic Equations,” AIAA Journal, Vol. 18, 1979, pp. 652-656. 4 Sorenson, R. L., “A Computer Program to Generate Two-Dimensional Grids About Airfoils and Other Shapes by Use of Poisson’s Equations,” NASA TM-81198, 1980. 5 Anderson, D. A., “Equidistribution Schemes, Poisson Generators, and Adaptive Grids,” Applied Mathematics and Computation, Vol. 24, 1987, pp. 211-227, 6 Eiseman, P. R., “Adaptive Grid Generation,” Computer Methods in Applied Mechanics and Engineering, Vol. 64, 1987, pp. 321-376. 7 Karman, S. L. Jr., Anderson, W. K., and Sahasrabudhe, M., “Mesh Generation Using Unstructured Computational Meshes and Elliptic Partial Differential Equation Smoothing,” AIAA Journal, Vol. 44, Number 6, 2006, pp. 1277-1286. 8 Labbe, P., Dompierre, J., Vallet, M-G., Guibault, F., and Trepanier, H-Y., “A Universal Measure of the Conformity of a Mesh with Respect to an Anisotropic Metric Field,” International Journal for Numerical Methods in Engineering 2003; 00:1-20. 9 Tchon, K-F and Camarero, R., “Quad-Dominant Mesh Adaptation Using Specialized Simplicial Optimization,” Proceedings of the 15th International Meshing Roundtable, pp. 21-38, Published by Springer 2006, Library of Congress Control Number 2006930257. 10 Venditti, David Anthony, “Grid Adaptation for Functional Outputs of Compressible Flow Simulations,” Ph.D. Dissertation, Massachusetts Institute of Technology, June 2002. 11 Tchon, K-F, Guibault, F., Dompierre, J., Camarero, R., “Adaptive Hybrid Meshing Using Metric Tensor Line Networks,” AIAA-2005-5332, June 2005. 12 Hansen, G., Zardecki, A., Greening, D., and Bos, R., “A Finite Element Method for Unstructured Grid Smoothing,” Journal of Computational Physics, 2004, Vol. 194, pp. 611-631. 13 Hansen, G., Zardecki, A., Greening, D., and Bos, R., “A Finite Element Method for Three-Dimensional Unstructured Grid Smoothing,” Journal of Computational Physics, 2005, Vol. 202, pp. 281-297. 14 Pointwise, Inc., Gridgen User Manual, version 15, http://www.pointwise.com/. 15 Onera M6 validation case, URL:http://www.grc.nasa.gov/WWW/wind/valid/m6wing/m6wing.html