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Unstructured Elliptic Smoothing Revisited

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

Chattanooga, Tennessee, 37403

A new approach for solving Winslow mesh smoothing equations on unstructured meshes is presented. The approach uses virtual control volumes to discretize the domain. The virtual control volumes can be defined with uniform edge length and equal angles, resulting in smooth meshes in the physical domain. Or the virtual control volumes can be manipulated to control mesh quality and produce clustering, such as required in viscous boundary layers. A detailed description of these virtual control volumes in included. Examples of mesh smoothing using this new approach include general smoothing, moving boundary problems and viscous clustering.

Nomenclature α, α1−3 = Winslow coefficient (2D, 3D) β, β1−3 = Winslow coefficient (2D, 3D) γ = Winslow coefficient (2D) AR = element aspect ratio J = element Jacobian x = computational coordinate y = computational coordinate z = computational coordinate u = physical coordinate v = physical coordinate w = physical coordinate

I. Introduction LLIPTIC mesh smoothing schemes are commonly used with structured meshes to improve mesh quality and enforce desired grid spacing and angularity. The governing partial differential 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 exert more control on the grid point placement near boundaries to enforce normal spacing requirements and grid line angularity These methods are typically employed for meshes created for use with viscous flow analyses. Elliptic smoothing can also be used to improve poor quality inviscid meshes resulting from mesh initialization procedures, such as Trans-Finite-Interpolation (TFI).

The authors demonstrated elliptic smoothing of unstructured meshes for cases where a computational mesh can be defined.5 Smoothing using Winslow equations without forcing functions was demonstrated in two dimensions and three dimensions in the reference. Fixed grid forcing functions were also demonstrated for a two dimensional case. In that approach an existing valid mesh was copied to serve as a computational mesh. The actual physical mesh could then be manipulated, such as moving boundaries or specifying viscous spacing. The interior physical mesh points were then recomputed by solving the Winslow equations on the computational mesh.

This paper continues that research and develops an alternate method for solving Winslow equations on unstructured mesh that does not require a computational mesh to pre-exist. In this new approach virtual control volumes are constructed for solving the Winslow equations. Manipulation of angles and edge lengths of these virtual control volumes enables control of the physical mesh angles and edge lengths. Details of their construction and manipulation will be provided. Results in two and three dimensions are included. * Research Professor, Graduate School of Computational Engineering, and AIAA Associate Fellow. † Research Associate and Graduate Student, Graduate School of Computational Engineering.

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47th AIAA Aerospace Sciences Meeting Including The New Horizons Forum and Aerospace Exposition5 - 8 January 2009, Orlando, Florida

AIAA 2009-1362

Copyright © 2009 by University of Tennessee at Chattanooga. Published by the American Institute of Aeronautics and Astronautics, Inc., with permission.

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II. Unstructured Mesh Smoothing This section will briefly describe Winslow elliptic smoothing applied to unstructured meshes. The staring point

is the same differential form of the equations solved by structured smoothing schemes. For unstructured smoothing these equations are recast in integral form to be solved using a finite volume discretization.

A. Elliptic (Winslow) Smoothing The three-dimensional Winslow equations are given in Equation 1.5 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 assumed equal to zero for this implementation.

�

α1uxx + α2uyy + α3uzz + 2(β1uxy + β2uyz + β3uxz) = 0α1vxx + α2vyy + α3vzz + 2(β1vxy + β2vyz + β3vxz) = 0

α1wxx + α2wyy + α3wzz + 2(β1wxy + β2wyz + β3wxz) = 0 (1)

�

α1 = ( r y • r y )( r z • r z) − (

r y • r z)( r y • r z)

α2 = ( r z • r z )( r x • r x ) − (

r z • r x )( r z • r x )

α3 = ( r x • r x )( r y • r y ) − (

r x • r y )( r x • r y )

β1 = ( r y • r z )( r z • r x ) − (

r x • r y )( r z • r z)

β2 = ( r z • r x )( r x • r y ) − (

r y • r z)( r x • r x )

β3 = ( 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.

�

αuxx − 2βuxy + γuyy = 0αvxx − 2βvxy + γvyy = 0

α = uy2 + vy

2

β = uxuy + vxvyγ = ux

2 + vx2

(2)

Equations 1 & 2 represent non-linear partial differential equations, with the coupling provided in the Winslow coefficients. The coefficients α, β , and γ are assumed frozen in order to linearize the system. An outer iteration loop is required to update these coefficient values. No forcing functions are used to exert control on the mesh. Instead, the control is achieved through modification of the computational stencil. 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 the reference. For this paper, Winslow smoothing is applied to the interior nodes only. The boundary nodes are either fixed or moved through a specified translation and/or rotation.

A. Virtual Control Volume (2D) The unstructured implementation of Winslow smoothing, such as Reference 5, requires a computational mesh

that must match the element topology of the physical mesh. But therein lies the problem. A mesh is needed to make a mesh! Structured mesh generation has an implied computational mesh. It is typically an equally spaced mesh aligned with the ξ, η, and ζ coordinate directions. For unstructured Winslow smoothing one also needs a defined

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computational coordinate system. But a unified global computational coordinate system is not possible and is not actually necessary. If one examines the discrete linear system that is created when solving a partial differential equation, like the Winslow equations, each row of the global matrix has non-zero entries in columns of the nearest neighbors.

�

A11 A12 A14 A15A21 A22 A23 A25

A32 A33 A36 A37A41 A44 A47A51 A52 A55 A56 A57

A63 A65 *A73 A74 A75 *

*

⎡

⎣

⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢

⎤

⎦

⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥

X1X2

X3

X4

X5

***

⎛

⎝

⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜

⎞

⎠

⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟

=

B1B2B3B4B5***

⎛

⎝

⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜

⎞

⎠

⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟

So it is only necessary to temporarily define a local coordinate system for each row of the matrix as the global

matrix is constructed. A different local coordinate system, i.e. a local control volume, can be used for each node. These “virtual control volumes” (VCV) can be constructed with uniform edge lengths and equal angles between the edges emanating from the node in question. Figure 1 shows the virtual control volumes (areas) for three interior nodes of a section of a triangular mesh. The angular orientation of the edges in the virtual control volume space does not matter, because the relationship between the entries in a row of the matrix is invariant to the orientation of the local coordinate system used. In other words, the direction of each edge does not matter, but the relationship between the edges through the angles is important. The edge lengths do matter and can be used to influence the spacing in the physical mesh, such as clustering in viscous regions. The physical mesh will attempt to mimic the computational mesh. Therefore, specifying equal angles in the virtual control volumes should produce similar angles in the physical mesh. Specifying uniform edge lengths in the virtual control volumes should result in near uniform spacing in the physical mesh.

Figure 1. Two-dimensional virtual control volumes for three interior nodes.

The Winslow equations are discretized using these virtual control volumes. Each node carries it’s own computational stencil. The topology of the stencil coincides with the topology of the physical mesh surrounding the

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point; i.e. the same number of edges and elements. The center point for each stencil is located at the origin and this corresponds to the computational coordinate (x, y) of the node in question. The computational node for the first neighboring point is typically set to x=1, y=0. The remaining surrounding computational nodes are placed on the unit circle distributed using equal angles. The element connectivity defines the order of the nodes around the circle. The “equal” angle is simply the 2π divided by the number of surrounding triangles. The computational elements are always triangles in 2D. For quadrilateral elements the control volume element for each node is the triangle creates using the corner node and it’s nearest neighbors; i.e. cut the corner of the quad.

III. Two-Dimensional Results Several examples are included to demonstrate the use of Winslow smoothing on unstructured meshes.6 The first

cases illustrate pure smoothing of an existing mesh without any edge spacing or element angle adjustments. Some examples show the use of the method where a boundary is moved and the Winslow smoothing returns the interior mesh to a valid and high quality state. Other examples show how layers can be inserted into the element connectivity and smoothed using smoothed Winslow equations.

A. Rotated and Translated Airfoil The first example shows how Winslow smoothing can be used to smooth a mesh where the airfoil boundary

points are rotated and translated significantly. Figure 2 shows the initial mesh for a NACA0012 airfoil. The boundary points are rotated 90 degrees and translated down and aft in Figure 3. The Winslow smoothed mesh is shown in Figure 4 and Figure 5. A completely valid and smooth mesh is produced. The mesh appears as though it was constructed in this orientation.

Figure 2. Original triangular mesh for NACA0012

airfoil.

Figure 3. Airfoil boundary points translated and

rotated 90 degrees.

Figure 4. Interior points smoothed using Winslow

equations.

Figure 5. Magnified view of rotated mesh near

airfoil.

The resulting mesh quality metrics are shown in Table 1. The normalized Jacobian value for an equilateral triangle is 0.866. A 90-degree angle produces a value of 1.0. Aspect ratio is defined as the circumscribing radius divided by

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twice the inscribed radius, which can range from 1 to ∞. The condition number is weighted for an ideal triangle and can range from 1 to ∞. These are very good values for an inviscid type mesh.

Table 1. Mesh quality metrics for smoothed rotated and translated NACA-0012 airfoil.

Minimum Average Maximum

Normalized Jacobian 0.29 0.83 1.0

Aspect Ratio 1.0 1.08 2.98

Condition Number 1.0 1.06 2.23

B. Extreme Rotation of Airfoil The next case uses the same airfoil but simply rotates it 178 degrees. The rotated boundary points with the

original interior points are shown in Figure 6. The smoothed mesh is shown in Figure 7 through Figure 11. The outer boundary points are held fixed. As a result, a twisting of the interior mesh is evident in Figure 10 and Figure 11. The choice of a rotation angle less than 180 degrees is necessary to define a unique path for the smoothing scheme. If the rotation angle was 180 degrees there exist two paths for the interior nodes; one rotated clockwise and one rotated counter-clockwise. One could conceivably perform additional rotations to increase the winding further.

The resulting mesh quality metrics are shown in Table 2. The minimum Jacobian is slightly lower and the maximum aspect ratio and condition number are slightly higher, but these are still in a very good range.

Table 2. Mesh quality metrics for extremely rotated airfoil.

Minimum Average Maximum

Normalized Jacobian 0.18 0.82 1.0

Aspect Ratio 1.0 1.16 6.71

Condition Number 1.0 1.12 3.55

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Figure 6. Same initial triangular mesh for

NACA0012 airfoil, but with the boundary points rotated 178 degrees.

Figure 7. Zoomed view of Winslow smoothed mesh

near airfoil.

Figure 8. Magnified view of Winslow smoothed mesh

near leading edge.

Figure 9. Magnified view of Winslow smoothed mesh

near trailing edge.

Figure 10. Winslow smoothed mesh view in a larger

viewing region.

Figure 11. View of total domain. Outer boundary

points were held fixed.

C. General Smoothing A triangular mesh for a multi-element airfoil is deliberately scrambled by randomizing the interior points, Figure

12. The connectivity is already defined, so the virtual control volumes are constructed with uniform edge lengths and equal angles. Winslow smoothing produces a very smooth mesh, shown in Figure 13.

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Figure 12. Interior points of multi-element airfoil

scrambled.

Figure 13. Winslow smoothing unscrambled the

mesh, producing a high quality mesh.

The corresponding mesh quality metrics, shown in Table 3, are very good. The minimum Jacobian corresponds to an angle of 26 degrees, which is certainly acceptable. The maximum aspect ratio and condition number are considered very low.

Table 3. Quality metrics for smoothed multi-element airfoil.

Minimum Average Maximum

Normalized Jacobian 0.44 0.87 1.0

Aspect Ratio 1.0 1.04 2.75

Condition Number 1.0 1.03 1.97

D. Flap Deflection The same multi-element mesh is manipulated by rotating the flap 40 degrees downward. The rotated geometry

and original mesh is shown in Figure 14 and Figure 15. The Winslow smoothed mesh is shown in Figure 16 and Figure 17. This is achieved using uniform edge lengths and uniform angles in the virtual control volumes. The mesh quality metrics are shown in Table 4. These values are more extreme than those reported in the previous cases. The extremely low Jacobian and high aspect ratio and condition number is due some severe skewing in the gap between the main airfoil element and the flap. However, this valid mesh was achieved with constant equal angles for the VCV with no edge length manipulation.

Table 4. Mesh quality metrics for multi-element airfoil with flap deflection.

Minimum Average Maximum

Normalized Jacobian 0.04 0.86 1.0

Aspect Ratio 1.0 1.07 1,110

Condition Number 1.0 1.05 41.5

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Figure 14. Multi-element mesh with main flap

geometry rotated 40 degrees.

Figure 15. Magnified view of rotated flap geometry.

Figure 16. Winslow smoothed mesh near rotated

flap.

Figure 17. Magnified view of region near main

element and flap.

E. Layer insertion A uniform collection of quadrilateral elements in shown on the left side of Figure 18. Layers of additional

quadrilaterals are added to the connectivity and smoothed using the Winslow equations. The resulting mesh is shown on the right side of the figure. No attempt has been made to control the normal spacing of the newly added quadrilaterals. This shows pure smoothing of the resulting unstructured quadrilateral mesh.

Figure 18. Initial unstructured quadrilateral mesh on left and layers inserted at boundaries on right.

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F. Layers Added to Circle Mesh The same strategy was used to insert layers at the circle boundary shown on the left in Figure 19. Viscous layers

were inserted by adding element connectivity for 15 layers of quadrilaterals. The initial normal mesh spacing at the circle boundary was specified as 0.001. Normal grid spacing was defined following a geometric progression distribution using an initial progression factor of 1.15. The progression factor was multiplied by a growth rate of 1.01 for each subsequent layer. The edge lengths of the corresponding computational stencils were modified to reflect the new distribution. The smoothed mesh is shown in the right side of the figure.

Figure 19. Viscous layers inserted at circle boundary.

G. Shape Deformation The multi-element airfoil case is used to demonstrate how Winslow smoothing may be used to perform shape

deformation. A sinusoidal function is used to deform the shape of the main element, shown in Figure 20. This is a severe deformation, but the Winslow smoothing generate a valid mesh with reasonable mesh quality.

Figure 20. Main element boundary points are

deformed.

Figure 21. Smoothed mesh produced high quality

mesh.

Magnified views of the deformations and smoothed meshes for the trailing edge and leading edge are shown in Figure 22 and Figure 23, respectively. The deformation is applied to the main element only.

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Figure 22. Shape change near trailing edge of main element and resulting smoothed mesh.

Figure 23. Shape change near leading edge of main element and resulting smoothed mesh.

The mesh quality metrics are shown in Table 5. The extremes are higher than desired, but the quality is certainly

acceptable. The overall mesh quality is quite good. This case demonstrates the potential of Winslow smoothing for shape optimization.

Table 5. Mesh quality metrics for shape deformation case.

Minimum Average Maximum

Normalized Jacobian 0.06 0.86 1.0

Aspect Ratio 1.0 1.06 141

Condition Number 1.0 1.05 14.5

IV. Virtual Control Volume Construction (3D) The 3D computational stencil consists of a collection of tetrahedra surrounding each node. For an all-tetrahedral

mesh this is simply the collection of the elements associated with each node. For other element types the stencil is constructed from tetrahedra formed using the nearest nodes from each surrounding element. In essence, cut-the-corner, use the corner node and the nearest neighboring nodes of the surrounding elements to form the tetrahedron for the computational stencil. This information can be extracted from the mesh connectivity.

The goal is to construct the computational stencil without knowledge of the physical coordinates. The mesh connectivity provides the elements, but the computational nodes must be created. Each element of the computational stencil shares the central node, which is placed at the origin of the local computational space. The triangular faces

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opposite the central node in each computational tetrahedron are placed on a unit sphere that is centered on the central node. The analogy of the equal angle approach used in 2D is to specify equal solid angles in 3D. The solid angle of these triangular faces is defined as the surface area projected onto the unit sphere subtended by the edges emanating from the central node to the nodes of the triangles. Solid angles are measured in steradians, ranging from 0 to 4π, the total surface area of a unit sphere. Figure 24 shows examples of virtual control volumes for two different computational stencils. The central nodes are not shown. The simple VCV on the left contains 10 surrounding tetrahedra resulting in 10 triangles on the surface of the unit sphere. The more complex VCV on the right contains 22 triangles. In two dimensions the number of triangles surrounding any given node is generally on the order of 5 or 6. The number of surrounding computational tetrahedra for 3D meshes can vary greatly, from 6 to 40 or more.

Construction of these virtual control volumes is critical to solving the 3D Winslow equations on unstructured meshes. The quality of the computational tetrahedra will impact the quality of the elements in the physical mesh. Therefore, it is desirable to have high quality tetrahedra that form the computational control volume. This is accomplished by distributing the computational nodes of the triangular faces opposite the central node so as to maximize the quality of the computational tetrahedra. A smoothing scheme was developed based on perturbations of these

surface nodes that minimizes a cost function.7 The cost function is a combination of the aspect ratio for each tetrahedron and the desired solid angle for each triangle on the surface of the unit sphere.

There are two modes for controlling physical mesh using the computational stencil. One mode manipulates the solid angles and controls the quality of the computational elements, which is reflected in the physical elements. Therefore, the cost function takes into account the desired solid angle for each surface triangle. The other mode of manipulation of the control volume is to alter the lengths of the edges emanating from the central node of the computational stencil. That can influence the clustering that required for viscous layers and solution adaptation.

The VCV smoothing scheme affects the solid angle aspect of control and attempts to minimize the cost function for all computational nodes on the surface of the unit sphere. The smoothing scheme perturbs the computational surfaces nodes and accepts perturbations that improve (reduces) the cost function. Perturbations are made in multiple directions on the surface of the unit sphere. The cost function for each surface node is a weighted average of the cost of the surrounding computational tetrahedra. The cost for each computational tetrahedron is computed, using computational coordinate, as

�

If J < 0c = 1− J

elsec = 1−1/AR*

AR* = AR(1 + α −θ )2

(3)

where α is the current value of the solid angle. If the tetrahedron is inverted the Jacobian, J, will be negative and the cost function will be greater than one. Otherwise, The value of c will range from 0 for a perfectly formed element to 1 for an element that has collapsed to zero volume. The multiplier on AR further enforces the desired solid angle. The weighted average of the cost for each surface node is given by

Figure 24. Examples of 3D Virtual Control Volumes

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�

C =c 2

1

ntet

∑ntet

C* = rCmax + Max(0,1− r)Cr = Cmax

3 , Cmax is the maximum value (4)

The cost function for each surface node, C*, is a blending of the maximum cost and the average cost. As the maximum cost approaches one the multiplier, r, becomes large and the blended cost is dominated by the maximum value. Conversely, with low maximum values the blended cost favors the average cost. Equal angles were used to construct virtual control volumes for a tetrahedral mesh about a sphere, original mesh shown in Figure 25. The mesh contained 60,333 nodes and 354,571 tetrahedra. The smoothed mesh is shown in Figure 26. A significant movement of points toward the sphere is evident. The magnified views, shown in Figure 27 and Figure 28, reveal the collapse of the mesh through the surface of the sphere. The minimum solid angle in the computational mesh was 0.16048. The maximum solid angle in the computational mesh was 1.50027. The minimum solid angle in the physical mesh was -3.14155 and the maximum solid angle was 3.14135. The minimum and maximum cost functions for all VCV creation was 0.00391342 and 0.370437, respectively. The equal angle approach obviously does not work in three dimensions as well as it did in two dimensions. The construction of the VCVs is significantly different between the two-dimensional algorithm and the three-dimensional algorithm. The 2D control volumes were explicitly defined. The number of triangles for each node was used to determine the value of the angles and the computational nodes were explicitly places on the unit circle surrounding each node. In 3D the number of computational tetrahedra is known, but the distribution of the computational nodes on the unit sphere is controlled by the VCV smoothing method, which is based on a perturbation/optimization scheme. One may specify equal solid angles, but the cost function is also attempting to create quality element shapes, which may prevent the method from achieving the desired solid angles. And the number of computational triangles surrounding a 2D node is relatively constant compared to the number of computational tetrahedra surrounding a 3D node. A close examination of the original mesh reveals that solid angles are not actually equal. The solid angles on the sphere side of interior nodes tend to be smaller than the solid angles on the opposite side of the nodes. The original mesh was created using a Delaunay scheme and the point distribution is based on the boundary distribution. The Delaunay scheme does not produce a mesh with equal solid angles. The Winslow smoothing, as applied, is already specifying uniform edge lengths and defining equal solid angles forces Winslow smoothing to move the mesh points toward the sphere to achieve equal solid angles in the physical mesh.

Figure 25. Original sphere mesh.

Figure 26. Smoothed mesh with equal solid angles.

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Figure 27. Original mesh, magnified view.

Figure 28. Smoothed mesh with equal solid angle

VCVs, magnified view.

Since the original mesh does not contain equal solid angles, can the existing angles in the physical mesh be used

to construct the virtual control volumes? The actual solid angles of the original physical mesh were specified as the desired computational solid angles for the mesh shown in Figure 29. A magnified view of the mesh near the sphere is shown in Figure 30. The minimum solid angle in the computational mesh was 0.108655. The maximum solid angle in the computational mesh was 2.40931. The minimum solid angle in the physical mesh was 0.00421259 and the maximum solid angle was 3.13896. The minimum and maximum cost functions for all VCV creation was 0.0140168 and 0.346293, respectively.

The use of the existing solid angles improved the result, but now the elements are elongated in the direction normal to the sphere. This is expected since uniform edge lengths are used for the computational elements. Winslow smoothing is attempting to make equal edge lengths in the physical mesh. And the mesh does not contain any inverted elements.

Figure 29. Smoothed mesh with existing solid angle

VCVs, global view.

Figure 30. Smoothed mesh with existing solid angle

VCVs, magnified view.

V. Adjusted Virtual Control Volume Angles It is obvious from the previous two examples that the solid angles of the virtual control volumes greatly impact

the resulting physical mesh. Therefore, dynamic adjustments to the VCV through modified solid angles should lead to improved results in the physical mesh. This is also necessary to overcome tangled meshes that can be created

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through a variety of means, such as the use of equal solid angle control volumes in the first example for the sphere mesh.

The simplest approach is to combine the equal solid angle strategy with the existing solid angle strategy. At each outer iteration the desired solid angles for each computational element is adjusted to be the average of the equal angle value and the current solid angle of the physical mesh. The VCV smoothing algorithm is employed to adjust the computational coordinates for nodes where the desired solid angles changed from the previous iteration. The sphere mesh computed with adjusted solid angle VCVs is shown in Figure 31 and Figure 32. The minimum solid angle in the computational mesh was 0.177182. The maximum solid angle in the computational mesh was 1.78772. The minimum solid angle in the physical mesh was 0.0511013 and the maximum solid angle was 3.04576.

Figure 31. Smoothed mesh with adjusted solid angle

VCVs, global view.

Figure 32. Smoothed mesh with adjusted solid angle

VCVs, magnified view.

This result is greatly improved over the constant equal angle case and the constant existing angle case. The

resulting mesh is very similar to the original mesh. No attempt was made to alter the edge lengths of the computational stencil, so this result is purely from manipulation of the solid angles.

VI. Three Dimensional Results Two additional three-dimensional cases are included. In each case solid angle adjustment is used to alter the

virtual control volumes to produce the physical meshes. The edge lengths of the computational stencils are fixed at unity.

A. Cube The sphere from the previous cases is replaced with a cube to demonstrate the method for geometries with sharp

edges. The original mesh, shown in Figure 33, contained 42,757 nodes and 253,009 tetrahedra. When constant equal solid angles are used the result is collapsing of elements at the cube boundary, similar to the sphere case, see Figure 34. The minimum solid angle in the computational mesh was 0.155406. The maximum solid angle in the computational mesh was 1.43975. The minimum solid angle in the physical mesh was -3.14154 and the maximum solid angle was 3.14115. The minimum and maximum cost functions for all VCV creation was 0.00422734 and 0.370668, respectively.

When the constant existing solid angles are used the result is an elongation of the elements away from the cube, shown in Figure 35. The minimum solid angle in the computational mesh was 0.0993382. The maximum solid angle in the computational mesh was 2.2.425. The minimum solid angle in the physical mesh was -0.00910635 and the maximum solid angle was 3.1363. The minimum and maximum cost functions for all VCV creation was 0.0108693 and 0.294552, respectively.

Finally, smoothing with dynamically adjusting the solid angles recovers a mesh similar to the original mesh, shown in Figure 36. The minimum solid angle in the computational mesh was 0.174268. The maximum solid angle in the computational mesh was 1.6496. The minimum solid angle in the physical mesh was 0.00974328 and the maximum solid angle was 2.55622.

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Figure 33. Original cube mesh.

Figure 34. Smoothed mesh using equal solid angle

VCVs.

Figure 35. Smoothed mesh using existing solid angle

VCVs.

Figure 36. Smoothed mesh using adjusted solid angle

VCVs.

B. Multi-Element Wing The geometry and symmetry plane mesh for a multi-element wing is shown in Figure 37. A mid-span cut

through the original mesh is shown in Figure 38. The mesh contains 402,630 nodes and 2,226,199 tetrahedra. The same mid-span cut through the smoothed mesh, using dynamically adjusted VCVs, is shown Figure 39.

Figure 37. Multi-element wing geometry with symmetry plane mesh.

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Figure 38. Mid-span cutting plane through original

mesh.

Figure 39. Mid-span cutting plane through smoothed

mesh.

Magnified views of the leading edge region for the original and smoothed mesh are shown in Figure 40 and

Figure 41, respectively. Similar views of the trailing edge regions are shown in Figure 42 and Figure 43. The smoothed mesh appears to be very similar to the original mesh. Using the average of the existing physical solid angle and the equal solid angle appears to work on this more complicated case.

Figure 40. Leading edge region of mid-span cutting

plane through original mesh.

Figure 41. Leading edge region of mid-span cutting

plane through smoothed mesh.

Figure 42. Trailing edge region of mid-span cutting

plane through original mesh.

Figure 43. Trailing edge region of mid-span cutting

plane through smoothed mesh.

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VII. Conclusions A new approach for performing Winslow smoothing on unstructured meshes was presented. The approach uses

virtual control volumes to discretize the integral form of the Winslow equations. This approach enables general mesh smoothing, boundary movement and viscous layer insertion. Two-dimensional examples show the robustness of the method to perform large translations and rotations of boundaries in the mesh. Quality metrics indicate the overall mesh quality is very good. Some cases had isolated regions with lower quality elements. Layer insertion was also demonstrated with edge length control of the computational elements to enforce viscous spacing in the physical mesh. Shape deformation for an airfoil was also demonstrated and resulted in a high quality mesh.

The preliminary 3D examples illustrate the same type of control is possible. Three-dimensional virtual control volumes are constructed using an optimization-based smoothing scheme that distributes the computational nodes on the unit sphere. The VCV smoothing scheme attempts to enforce good element shape and desired solid angles. Results for cases using “equal” solid angles show the tendency to collapse elements into the interior boundaries. VCVs constructed using the existing physical solid angles as the desired solid angles improved the results. Element inversion was reduced or eliminated, but the elements near the inner boundary were elongated in the normal direction due to the use of unit length edges in the computational stencil. If the desired solid angles were specified as a simple average of the equal solid angle value and the current physical solid angle the smoothed meshes resembled the original meshes. However, some element inversion still existed at sharp corners for the multi-element wing case. Further research into the adjustments made to the VCV solid angles is necessary. And further use of the edge length manipulation could lead to additional control of mesh quality.

This new Winslow smoothing approach does not require the definition of a global computational mesh. A global connectivity is required, though. And this is no different than the situation found with structured mesh smoothing. The “connectivity” of the structured mesh is implied. For unstructured Winslow smoothing one must have a valid connectivity. So, in some sense, one must still have a mesh to define a mesh. However, it is now possible to manipulate that mesh to achieve different outcomes, such as general smoothing, boundary motion, or viscous layer insertion. Also, an inviscid mesh can be converted to a viscous mesh by simply adding the connectivity for boundary layer elements, specify the desired normal spacing distribution through the edges of the virtual control volume and solving the Winslow equations. Large deformations are also possible with this new approach, which could be significant in design optimization via shape change.

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 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. 6 Sahasrabudhe, M., “Unstructured Mesh Generation and Manipulation Based on Elliptic Smoothing and Optimization,” Ph.D. Dissertation, University of Tennessee at Chattanooga, August 2008. 7 Sahasrabudhe, M. S., Karman, S. L. Jr., and Anderson, W. K., “Grid Control of Viscous Unstructured Meshes Using Optimization,” AIAA-2006-0532, Reno, NV, January 2006.