[American Institute of Aeronautics and Astronautics 43rd AIAA Aerospace Sciences Meeting and Exhibit...

American Institute of Aeronautics and Astronautics1

Mesh Generation Using Unstructured ComputationalMeshes and Elliptic Partial Differential Equation Smoothing

Steve L. Karman Jr.*, W. Kyle Anderson† and Mandar Sahasrabudhe‡

University of Tennessee at ChattanoogaChattanooga, Tennessee, 37403

A novel approach for generating unstructured meshes using elliptic smoothing ispresented. Like structured mesh generation methods, the approach begins with theconstruction of a computational mesh. The computational mesh is used to solve ellipticpartial differential equations that control grid point distributions and improve mesh quality.Two types of elliptic partial differential equations are employed; modified linear-elastictheory and Winslow equations, with or without forcing functions. Results are included toillustrate the use of these methods for unstructured mesh generation, mesh boundary shapemodification, mesh untangling and mesh movement.

Nomenclatureα = Winslow or linear-elasticity coefficientβ = Winslow or linear-elasticity coefficientγ = Winslow or linear-elasticity coefficientθ = linear-elasticity coefficientν = Poisson’s ratioΦ = physical u coordinate forcing functionψ = physical v coordinate forcing functionΩ = Element area or physical w coordinate forcing functionΓ = Element boundary edgeV, dV = Element volumeS, dS = Element surface areaξ = computational coordinateη = computational coordinateζ = computational coordinatex = physical coordinate or computational coordinatey = physical coordinate or computational coordinatez = physical coordinate or computational coordinateu = physical coordinate or physical perturbation componentv = physical coordinate or physical perturbation componentw = physical coordinate or physical perturbation component

I. IntroductionHE mesh generation process followed by most unstructured mesh generation methods is very similar to theprocess followed by structured mesh generation methods. The process for generating structured meshes in two

dimensions involves distributing grid points along edges of the domain, followed by distributing points in theinterior of the domain. The process in three dimensions adds a step for distributing grid points on faces of thedomain before distributing points in the interior of the domain. The latter steps in these processes may require meshsmoothing to improve the quality of the final mesh. This may be necessary because the methods for initializing the

* Research Professor, Graduate School of Computational Engineering, and AIAA Associate Fellow.† Professor, Graduate School of Computational Engineering, and AIAA Associate Fellow.‡ Research Associate and Graduate Student, Graduate School of Computational Engineering.

T

43rd AIAA Aerospace Sciences Meeting and Exhibit10 - 13 January 2005, Reno, Nevada

AIAA 2005-923

Copyright © 2005 by University of Tennessee at Chattanooga. Published by the American Institute of Aeronautics and Astronautics, Inc., with permission.


interior grid point locations may result in grid skewness or inverted elements. Unstructured mesh generationmethods typically follow a similar process and the final mesh may also require smoothing or optimizing to improvethe quality. Smoothing is also required when boundary motion is involved. How the smoothing is accomplished isusually quite different between structured and unstructured methods.

Structured grid generation methods often employ elliptic partial differential equation smoothing techniques toimprove mesh quality. This type of smoothing offers users a high degree of control over grid point spacing and gridline angularity. On the other hand, unstructured grid generation methods tend to use techniques that involve sometype of node averaging or node perturbation/optimization to improve mesh quality.1,2 Unfortunately, averaging andoptimizing methods are not always robust and can actually degrade the quality of the mesh. This is especially truefor meshes that contain sharp edges and viscous clustering.

Elliptic partial differential equation smoothing methods solving the Laplace or Poisson equations have not beenapplied to the generation of unstructured style meshes for one basic reason; the nonexistence of a mapping from thephysical domain to computational domain where the smoothing equations are solved. As such, the benefits of usingelliptic smoothing methods have not been realized in the unstructured grid environment. This paper describes anovel approach for generating meshes in the computational domain that can be used to solve the elliptic smoothingequations to produce smooth, high quality meshes in the physical domain.

A brief review of structured elliptic smoothing will be provided to establish a foundation for the unstructuredelliptic smoothing. Then two types of elliptic partial differential equation techniques for use with unstructuredmeshes will be described. The first technique uses a modified linear-elastic theory to control grid point distributionfor moving boundary problems. A variation of the method can be used to generate meshes for non-movingboundaries and viscous layers. The second technique solves the same elliptic partial differential equations, Laplaceand Poisson, often used by structured mesh smoothing methods. Several examples will be used to illustrate varioususes of these elliptic smoothing techniques on unstructured meshes.

II. Structured Mesh SmoothingElliptic smoothing has been used for a long time to generate and improve the quality of structured meshes. The

origin of the method can be traced back to Winslow3 and made popular by Thompson, Thames and Mastin4.Structured elliptic smoothing is incorporated in many commercial mesh generation packages, including Gridgen5.The success of the method is due to the mathematical basis of the derivation and the relationships inherent in thetransformation between computational and physical domains. Examining the details of this transformation isinstructive and is included for reference.

The two-dimensional computational domain is represented by (ξ, η) space and the two-dimensional physicaldomain is represented by (x, y) space. The mapping between these two domains is defined in the followingtransformations.

),(

),(

yx

yx

ηηξξ

==

and),(

),(

ηξηξ

yy

xx

==

(1)

The elliptic smoothing equations are expressed as a Laplacian operator on the computational coordinates setequal to zero or set equal to forcing functions in the form of P and Q or Φ and Ψ, as follows:


( )

( )Ψ∇•∇=∂∂+

∂∂=∇

Φ∇•∇=∂∂+

∂∂=∇

ΨΦ

=∂∂+

∂∂=∇

=∂∂+

∂∂=∇

ηηηηη

ξξξξξ

ηηη

ξξξ

2

2

2

22

2

2

2

22

2

2

2

22

2

2

2

22

:Form-

:FormQ-P

yx

yx

Qyx

Pyx

(2)

These Poisson equations are currently cast in physical space and represent the smooth variation of computationalcoordinates in physical space. In other words, the gradients of ξ and η will be constant and the second derivatives ofξ and η will be zero. When the forcing functions are set equal to zero the equations convert to the simple Laplaceequations. Solutions to Laplace equations satisfy the maximum/minimum principle, which means the dependentvariables on the interior of the domain are bounded by the values on the boundary of the domain. This helps assurethat grid lines do not cross if the boundaries of the domain are chosen to be constant ξ and constant η grid lines.

These equations are difficult to solve in the current reference frame because the computational coordinates areknown and the physical coordinates are desired. So the equations are transformed to computational space in order tosolve for the desired physical coordinates. The final transformed equations are sometimes referred to as theWinslow equations.3 The final form of the equations using the Φ−Ψ formulation is:

22

22

0)(2)(

0)(2)(

ξξ

ηξηξ

ηη

ηηηξηξξξ

ηηηξηξξξ

γ

β

α

γβα

γβα

yx

yyxx

yx

yyyyy

xxxxx

+=

+=

+=

=Ψ++−Φ+

=Ψ++−Φ+

(3)

Smoothing methods that solve these equations aresometimes referred to as Winslow smoothing methods.Solutions to these equations will produce a smooth meshin the physical domain. These solutions are typicallyobtained using an equally spaced mesh in computationalspace as shown on the left in Figure 0. The physicalmesh is shown on the right side of the figure. Eventhough the side boundaries in the physical domain areclustered toward the bottom boundary, the interior gridlines do not protrude through the spike on the bottomwall. The mesh is smooth and does not contain any gridline crossing.

It is possible to use a computational domain that is not equally spaced. The difference formulae for first andsecond derivatives of a function, f, in the x and y directions, derived from Taylor Series expansion are as follows.

Figure 0. Computational and physical domainsfor structured elliptic smoothing.


( )

( )

( )( )

( )( )

( )( )( ) ( )( )

( )( )( )( )2222

1,12

,122

1,122

1,2

,22

1,222

1,12

,122

1,122

22

,1,1,

22

,,1,1

22

1,2

,22

1,2

22

,12

,22

,12

1,,

,1,

,1,

,,1

2

2

jjjjiiii

jijjijjjiji

jijjijjjijii

jijjijjjiji

xy

jjjj

jijjjijjijyy

iiii

jiiijiijiixx

jjjj

jijjijjjijy

iiii

jiijiiijiix

jijij

jijij

jijii

jijii

fff

fff

fff

f

ffff

ffff

ffff

ffff

yy

yy

xx

xx

∇∆+∆∇∇∆+∆∇

∆−∇−∆+∇∆−∆−∇−∆+∇∇−∆+

∆−∇−∆+∇∇

=

∇∆+∆∇

∆+∇−∇+∆=

∇∆+∆∇

∆+∇−∇+∆=

∇∆+∆∇

∆−∇−∆+∇=

∇∆+∆∇

∆−∇−∆+∇=

−=∇

−=∆

−=∇

−=∆

−−−+−

−+

−++++

+−

+−

−+

−+

−

+

−

+

(4)

If these formulae are used with an unequally spaced, but still Cartesian computational mesh that mimics thedesired clustering of the boundaries of the physical mesh the result is the mesh shown in Figure 0. The sideboundaries of the computational domain are now clustered to match the spacing of the side boundaries in thephysical domain. The interior of the physical mesh is also clustered to the bottom boundary and the grid lines do notcross.

This result is similar to traditional Winslow methodsthat solve Poisson’s equations with control functions.Poisson control functions must be defined to producethe desired grid clustering and/or the desired grid lineangularity near the boundaries. Many differenttechniques for constructing these control functions havebeen developed over the years.6,7 The results in Figure 0indicate that it is possible to manipulate thecomputational mesh and control the spacing in thephysical mesh without using forcing functions.

III. Unstructured Mesh SmoothingUntil recently, smoothing methods for unstructured meshes have been limited to simple averaging techniques,

sometimes called spring analogy methods, or optimization techniques using node perturbations that attempt tomaximize or minimize a function that is a measure of smoothness or quality.1,2 These methods have been used withsome success to untangle meshes and improve element shape quality, but they do not offer the robustness affordedby elliptic methods, especially when sharp geometry shapes are involved. A linear-elastic theory method ofsmoothing was described in Ref. 8 that showed increased robustness and improved results when applied tounstructured dynamic mesh problems. This paper revisits the linear-elastic approach and introduces an approach forsolving the Winslow equations on unstructured meshes. This section will describe the details of the existing linearelastic theory approach first, followed by a detailed description of the Winslow approach.

Figure 0. Clustered computational mesh andphysical mesh with elliptic-Laplacian smoothing.


A. Linear-Elastic Theory SmoothingLinear-elastic theory smoothing is generally used for mesh movement where boundaries of the mesh are

deforming and interior mesh points need to be adjusted to produce a valid mesh8. It can also be used to generate theinitial mesh under certain circumstances, as will be shown later.

In this approach it is assumed the mesh obeys the isotropic linear elastic relations shown below.

021

1

021

1

2

2

=•∇∂∂

−+∇

=•∇∂∂

−+∇

Vy

v

Vx

u

r

r

ν

ν(5)

where ν in the denominator is Poisson’s ratio and the nodal displacement vector is given by jviuV ˆˆ +=r

. The

parameter, ν, is typically manipulated so the coefficient 1/(1-2ν) is equal to the aspect ratio of the local cell. Thisproduces stiffness in regions with high aspect ratio cells and ensures that boundary layer elements track closely withthe local boundary as it moves. The solution to these equations is a vector field defining the displacement of eachnode.

The terms in the equations can be expanded and cast in a form similar to the Winslow equations.

021

11

21

1

021

1

21

11

=

−++

−+

=+

−+

−+

yyxyxx

yyxyxx

vuv

uvu

νν

νν(6)

Linear-elastic theory smoothing can be used togenerate the spike mesh shown earlier by deforming thebottom boundary from the initial horizontal line in thecomputational mesh to the final geometry through aseries of smaller incremental steps. The resultingtriangular and quadrilateral meshes are shown in Figure0. The clustering of the original meshes is maintained asthe bottom boundary is moved from the originallocation to the final position. Incremental steps areemployed to increase the robustness of the technique.The size of the steps is limited by the height of the firstlayer of points. Too large of a step size will sometimespush the boundary points too fast and may causeelements to become inverted.

B. Elliptic (Winslow) SmoothingThe use of Winslow smoothing with unstructured

meshes is difficult because a mapping from anunstructured physical domain to a computationaldomain is not available. A local (ξ, η) coordinatesystem can be constructed at the cell or node level, but aglobal mapping of the entire domain is generally notdefined.9 Ref. 9 used trigonometric functions to approximate the computational coordinates and Taylor-seriesexpansions of these coordinates to compute first and second derivatives for the Winslow equations. The resultsindicated slightly improved mesh quality with comparable computational performance. The possibility of extendingthe method to three dimensions was questioned in the conclusions.

However, if a “valid” mesh with the same element topology is available then the Winslow equations can besolved on that mesh. This was the basic technique used in Ref. 8 to solve the linear-elastic equations. The starting

Figure 0. Linear-elastic smoothing applied to thespike geometry.

Final triangular mesh

Final quadrilateral mesh

Initial triangular mesh

Initial quadrilateral mesh


mesh was a valid mesh for the current boundary position. The deformation of the boundary nodes was defined usingthe displacement vector, V

r. The solution to the linear elastic relations was obtained on the original mesh. That

solution is the displacement vector for each node of the entire mesh. The new grid point positions were computed byadding this displacement vector to the old positions.

Winslow equations can also be solved using an existing computational mesh that matches the element topologyof the physical mesh. The equations must first be discretized for an unstructured mesh. For convenience, anexchange of variables is performed from (x, y) to (u, v) and from (ξ, η) to (x, y). Then the Winslow equations arereformulated, consistent with the nomenclature used in the linear-elastic equations, as follows.

22

22

0)(2)(

0)(2)(

xx

yxyx

yy

yyyxyxxx

yyyxyxxx

vu

vvuu

vu

vvvvv

uuuuu

+=

+=

+=

=Ψ++−Φ+

=Ψ++−Φ+

γ

β

α

γβα

γβα

(7)

In this context the x and y coordinates are the computational coordinates and the u and v coordinates are the physicalcoordinates.

One approach to solving these equations is to consider the coefficients, α, β, γ, Φ and Ψ as constant and integratethe first equation over x-y space.

0)(2)( =ΩΨ++Ω−ΩΦ+ ∫∫∫∫∫∫ duududuu yyyxyxxx γβα (8)

Ω represents the area in two dimensions. Examine the first integral and define a vector, vr

, such that

xx

x

uv

jiuv

=•∇+=

r

r ˆ0ˆ(9)

The divergence theorem can be used to convert the double integral to a line integral.

Γ•=Ω•∇ ∫∫∫ dnFdF ˆ (10)

Γ represents the boundary in two dimensions. Therefore, the first term in the first integral becomes

∫∫∫ Γ=Ω dnudu xxxx ˆ (11)

A similar operation is used to convert the other double integrals with second derivative terms. The forcing functionterms are converted by defining the vector using u and v instead of components of the gradients of u and v. Thensimilar operations are performed on each term in both equations. The resulting set of integral equations become

0)ˆˆ(ˆ2)ˆˆ(

0)ˆˆ(ˆ2)ˆˆ(

=ΓΨ+Γ+Γ−ΓΦ+Γ

=ΓΨ+Γ+Γ−ΓΦ+Γ

∫∫∫∫∫∫∫∫∫∫

dnvdnvdnvdnvdnv

dnudnudnudnudnu

yyyxyxxx

yyyxyxxx

γβα

γβα(12)

These integral equations are discretized at the nodes using the control volume (area) for the node. Severalchoices for the control volume are shown in Figure 0. For the first option, the faces of the control volume extendfrom one cell center to the mid-edge to the neighboring cell center. This is a dual of the mesh and results in no


overlapping regions. The second option for the control volume would include the complete volumes (areas) of thesurrounding elements. This would result in overlapping control volumes, but is still a valid option. The final optionis a modification of the second whereby only a portion of a quadrilateral element is included in the node controlvolume. For quadrilateral elements only the triangle consisting of the node in question and its adjacent neighbors inincluded in the control volume. The advantage of this choice is the ability to formulate the system of equations usingonly triangular shaped elements. All of these control volume options extend to three dimensions, where the thirdoption would require only tetrahedral shaped elements to formulate the system of equations.

Most of the integrals in equation 12 contain a first derivative of u or v with respect to the x or y computationaldirections. These first derivative terms are simply components of the gradient of a scalar functions. In this case thescalar functions are the physical coordinates, u and v. If the third option for the control volume is chosen, Green-Gauss theorem can be used to compute the gradient of a scalar, φ, over a triangle as follows.

[ ][ ])()()(

2

1

)()()(21

213132

213132

321

321

yyyyyy

xxxxxx

nnnnnnAy

nnnnnnAx

+++++=∂∂

+++++=∂∂

φφφφ

φφφφ

(13)

The area of the triangle is represented by A. Components of the edge normal vectors are shown in Figure 0.

1 2

3Normal vector components

nx1

= y3 – y2 ny1

= -(x3 – x2)

nx2

= y1 – y3 ny2

= -(x1 – x3)

nx3

= y2 – y1 ny3

= -(x2 – x1)

n1

n3

n2

Figure 0. Normal vectors defined for a triangle.

Figure 0. Several options for control volume surrounding a node.


Expressions for ux, uy, vx, and vy can now be substituted in equations 12, resulting in the following equations fora triangular element. These are the discretized Winslow equations for a triangle and can be used to construct theglobal system of equations in order to solve for the coordinates u and v.

[ ][ ][ ]

[ ][ ][ ]

0

)()()(2

)()()(

)()()(2

0

)()()(2

)()()(

)()()(2

213132

213132

213132

213132

213132

213132

321

321

321

321

321

321

=Ψ+Φ

++++++

++++++

−+++++

=Ψ+Φ

++++++

++++++

−+++++

yfxf

yyyyyyy

xyyyyyy

xxxxxxx

yfxf

yyyyyyy

xyyyyyy

xxxxxxx

tvtv

tnnvnnvnnvA

tnnvnnvnnvA

tnnvnnvnnvA

tutu

tnnunnunnuA

tnnunnunnuA

tnnunnunnuA

γα

γ

β

α

γα

γ

β

α

(14)

The coefficients tx and ty are the appropriate normal components. When updating node 1, the components of thenormal vector for side 1 would be used, etc. The uf and vf terms are the averaged values for the face control volume.When updating node 1, uf would be the average of u2 and u3. The terms for each node can be regrouped based on thenode values of u and v resulting in the following form.

[ ][ ][ ]

[ ][ ][ ]

0][

)()(2)(2

)()(2)(2

)()(2)(2

0][

)()(2)(2

)()(2)(2

)()(2)(2

213132

213132

213132

213132

213132

213132

3

2

1

3

2

1

=Ψ+Φ

++++−+

++++−+

++++−+

=Ψ+Φ

++++−+

++++−+

++++−+

yxf

yyyxyyxyy

yyyxyyxyy

yxxxxxxxx

yxf

yyyxyyxyy

yyyxyyxyy

yxxxxxxxx

ttv

tnntnntnnA

v

tnntnntnnA

v

tnntnntnnA

v

ttu

tnntnntnnA

u

tnntnntnnA

u

tnntnntnnA

u

γα

γβα

γβα

γβα

γα

γβα

γβα

γβα

(15)


Equations 15 represent a linear system of equations for a triangle with unknown variables u1, u2, u3, v1, v2, andv3. Similar equations can be constructed for all triangles in the mesh, as well as sub-triangles for the corners of thequadrilateral elements. Combining the equations from all elements in the mesh will result in a sparse matrix linearsystem of equations that can be solved for new physical coordinates using techniques such as point-implicit withunder-relaxation. Since the coefficients α, β, γ, Φ and Ψ were assumed frozen, but are actually functions of thephysical locations, an outer iteration loop is required to update these coefficient values. When a point-implicitmethod is used an inner iteration loop is required to converge the linear system.

IV. Computational MeshesWinslow smoothing requires a computational mesh in order to solve the partial differential equations. Structured

mesh smoothing typically employs a rectangular Cartesian mesh for the computational mesh. This computationalmesh is naturally implied in the meshing process, in that the domains are generated assuming a rectangular topologywith dimensions specified by the user. The structured computational mesh is assumed to be uniform and equallyspaced. A computational mesh is not implied for unstructured meshes, therefore one must be provided with the sameelement topology as the resulting physical mesh.

The most common forms of unstructured meshes are triangular meshes in two dimensions and tetrahedralmeshes in three dimensions. Hybrid meshes are also possible, where quadrilateral elements are introduced in twodimensions and triangular prisms, pyramid and hexahedral elements are introduced in three dimensions. The processof generating these meshes can involve several steps, such as extruding meshes from surfaces and tessellatingvolume regions with an advancing front technique or a Delaunay-based insertion method. If the process is successfulthe resulting mesh can be used as a computational mesh. But at this point, unless boundary motion is involved,smoothing may not be required. If the process is not successful then neither a physical mesh nor a computationalmesh is obtained, and the process must be modified to produce a usable mesh.

A different method for generating hybrid unstructured meshes was presented by Karman that producesquadrilateral-dominant meshes in two dimensions and hexahedral-dominant meshes in three dimensions10. Thismesh generation process is reversed from traditional approaches. The volume mesh is constructed first, followed bythe creation of body-conforming elements at the boundaries. Figure 0 shows part of the process applied to aconfiguration containing a circle in a square box. A root cell is initially generated that encompasses the entiredomain. Then recursive subdivision is used to produce a mesh that will adequately resolve the pertinent geometricfeatures, guided by some spacing parameters that are automatically generated for each boundary part. The caseshown requires seven refinements to produce the desired resolution in the vicinity of the circle. The next stepinvolves removing the elements that are cut by the geometry or that exist exterior to the domain of interest.Quadrilaterals that contain mid-edge nodes are replaced with triangles to eliminate hanging nodes and produce ahybrid mesh with point-to-point connectivity. The edges formed by the exposed nodes comprise the voxel surface ofthe mesh. This surface is stair-stepped and resides just above the true geometry surface.


The last step is a projection step that can be performed in a number of ways. The lower-right image in Figure 0shows one possible approach. In this case the projection is in the normal direction from the voxel surface andextends only a fraction of the local element size, typically one third. The resulting layer of boundary elementsexhibits a stair-stepped shape. The boundary points of the corresponding physical mesh will be positioned on thetrue surface during the smoothing process. Another approach uses a closest-point projection to position the projectednodes on the true surface. This works well in most regions but can produce undesirable results in concave andconvex corners. A third option combines the normal projection and the closest point projection schemes. The pointis projected in the voxel normal direction the fractional distance. If an intersection with the geometry is detected theintersection point becomes the boundary point. Otherwise, closest point projection is used from that location. Anyof these approaches results in valid meshes that can be used as computational meshes for solving Winslowequations. The first approach results in a computational mesh that is not body-conforming while the latterapproaches result in body conforming computational meshes.

Experience has shown that some additional enhancements can improve the quality of the mesh, especially atsharp corners. Sharp corners are easily detected in the geometry and introduced into the mesh as new boundarypoints. These points can be connected to the voxel nodes by locating the cut cell that contains the sharp point. Theconnection process will result in a number of triangles and/or quadrilaterals connected to the sharp point. Theprocess is shown on the left side of Figure 0. The resulting mesh can contain small, high-aspect ratio elements next tolarger elements. This can be improved using simple averaging with neighboring nodes, as shown on the right side ofFigure 0. Notice the averaged mesh does not conform to the true geometry. Again, this is not necessary. Theboundary points in the physical mesh will be forced to lie on the true geometry during the Winslow smoothingprocess.

Figure 0. Recursive subdivision used to produce fine resolution near circle. Cut cells and exterior cellsremoved to create voxel grid.

Refinement Refinement 6

Cut & ExteriorCells Removed

Refinement 2 Refinement 3

Layer ofboundaryelements

near circle

Refinement 4

Refinement

Refinement 1


The most important point is that the computational mesh is easily generated and is a valid mesh with all positivevolumes (areas). The various approaches just described can be used to produce a valid mesh every time. TheWinslow smoothing will be used to produce the physical mesh with all boundary points lying on the true geometry,similar to the process followed to generate structured meshes.

V. Robust Generation of High Quality Viscous Meshes

Two different approaches have beenexplored for producing high aspect ratioelements for viscous simulations. The firstapproach continues with the recursivesubdivision technique. An example of thecomputational mesh generated for thecircle is shown on the left in Figure 0. Inthe subdivision technique thecomputational mesh is further subdividednear the surface to define viscous spacing.These subdivisions will not generate anyinverted elements. The subdividedelements will resemble the parent cell.The refined computational mesh is shownin the both images on the left side ofFigure 0. After applying Winslowsmoothing the physical mesh willmaintain the clustering that exists in thecomputational mesh near the surface, aswas shown earlier with the spikeconfiguration. The smoothed physicalmesh for this refined computational meshis shown on the right of the figure. Noticethat the mesh away from the surfacemaintains the spacing and character of thecomputational mesh. If the spring analogymethod were used to smooth this mesh theviscous clustering at the boundary wouldbe sacrificed.

The difference between the computational and physical meshes becomes apparent in magnified views near thesurface, as shown in the bottom images of Figure 0. This particular computational mesh exhibits an undesirable

Figure 0. Computational mesh generated near sharp trailing edge.

Figure 0. Subdivision meshing process applied to viscouscylinder configuration.


stair-stepped character that is reflected in thephysical mesh. The grid line angularity at thesurface is also less than desirable.

An alternate approach combines Winslowsmoothing with linear-elastic smoothing toproduce higher quality viscous meshes.Winslow smoothing without forcingfunctions is used to generate the inviscidmesh and linear-elastic smoothing is used toinsert viscous layers at solid boundaries. Aviscous mesh for the circle example wasrecreated with this approach and is shown inFigure 0. The computational mesh for thiscase has been modified by inserting trianglesin the stair-stepped regions, resulting in acomputational mesh that better conforms tothe geometry. In addition, a few averagingpasses are applied to the boundary nodes andthe immediate interior nodes of thecomputational mesh. These changes reducedthe stair-stepping effect and do not detractfrom the ability to generate a validcomputational mesh, even as the geometrybecomes more complicated. Close inspectionof the upper left image in the figure revealsthe true physical boundary, in this case thecircle. The physical mesh was initializedwith the computational mesh coordinates.Then Winslow smoothing was used tosmooth the mesh, resulting in the inviscidmesh shown in the upper right.

Next multiple layers were inserted at thecircle boundary. As each layer is inserted the previous layer of boundary points is pushed a short distance into theinterior of the domain in the “normal direction” using linear-elastic smoothing. The “normal direction” can bedefined in a number of ways. Sharp corners similar to the trialing edge shown in Figure 0 can have multiple normaldirections. Normal vectors for other boundary nodes can be defined by averaging neighboring edge normals. Or thenormal direction can be defined as the direction along the edge emanating from the boundary point in the directionof the interior point. The latter method was chosen because it does not require computation of a normal direction andtreatment of sharp points is similar to all other points.

The short distance that each node is pushed depends on the current local normal spacing and the desired normalspacing for the current layer. No point is pushed more than 80% of the current edge length or less than the fractionconsistent with a specified geometric progression factor, typically around 50%. These are the bounds of theprojection distance. If possible, the node is pushed a distance corresponding to the current layer spacing, assuming adesired initial spacing and the same geometric progression factor. As each layer is pushed into the interior, theelement shapes are preserved by the Poisson’s ratio term in the linear-elastic smoothing equations. Following thisscenario, the normal direction is always the same for each layer and that direction corresponds to the edges of theinitial valid inviscid mesh.

This approach to viscous mesh generation is different from marching methods that are sometimes used togenerate viscous layers. Marching methods add the next layer on top of the previous layers. As new layers are addedthe normal direction must be recomputed, which can eventually result in poorly defined normal vectors. Marchingmethods also suffer from possible collisions in concave corners or with other fronts marching away from nearbysurfaces. The current approach always inserts new layers at the boundary where the normal direction is well definedand does not change. The Poisson’s ratio term in the smoothing scheme ensures that element quality is maintained.In addition, the largest layer displacement occurs during the first layer insertion. The distance specified is limited tonever be larger than the local element size. Each subsequent layer is pushed away from the boundary a smallerdistance, following a reverse geometric progression, until the final layer is inserted at the desired viscous spacing at

Figure 0. Viscous mesh for circle created usingcombination of Winslow and linear-elastic smoothing.

Inviscid computational mesh

Viscous mesh created usinglinear-elastic smoothing

Magnified view of viscousmesh near surface

Inviscid physical mesh


the wall. The process of inserting layers in this reversed mode using the linear-elastic smoothing is extremely robust.By combining the easily generated computational meshes with the two forms of elliptic smoothing, high qualityviscous meshes can generated in a nearly automatic fashion. An example of the method applied to a concave corneris shown in Figure 0. Element shape is maintained as each new layer is inserted.

VI. Discussion of Linear-Elastic Smoothing versus Winslow SmoothingThe equations for Winslow smoothing and linear-elasticity smoothing can be combined into a generalized set of

equations. When solving the linear-elastic equations u and v represent grid point velocities. When solving Winslowequations u and v represent the physical coordinates.

0,,2,

0,,2,

:wFor Winslo

021

1,

211

1,0,1

211

,1,0,21

11

:elasticitylinearFor

0)()(

0)()(

2222

1111

2222

1111

2222

1111

==−====−==

=Ψ=Φ−

=−

+===

−===

−+=

=+Ψ+++Φ+

=+Ψ+++Φ+

θγγββααθγγββαα

υθ

υγβα

υθγβ

υα

θγβα

θγβα

xyyyyxyxxx

xyyyyxyxxx

uvvvvv

vuuuuu

(16)

After 28 layers inserted

Figure 0. Multiple layers inserted in corner region.

Initial Mesh 1st layer inserted 2nd layer inserted

3rd layer inserted


Winslow smoothing and linear-elasticity smoothing can be used in a couple of different ways. When theboundaries are stationary Winslow smoothing can be used to improve the quality of the mesh. Forcing functions canprovide additional control over grid point distribution. The linear-elasticity smoothing method requires boundarymotion. If there is no boundary motion, linear-elasticity smoothing cannot be used to smooth the mesh. Whenboundary motion is involved either smoother can be used to control the mesh movement and quality.

For the stationary case, if the boundary distribution of the computational mesh matches the boundary distributionof the physical mesh and the forcing functions are equal to zero Winslow smoothing will result in the physical meshmatching the computational mesh, exactly. This means that the interior grid points can be scrambled and Winslowsmoothing can be used to recover the original mesh. This is obviously an academic exercise, as described. If theboundary distributions are identical, one could always obtain the original mesh by simply substituting thecomputational coordinates for the physical coordinates. The significance of this result occurs when boundary motionis involved. In a simulation of a moving boundary Winslow smoothing will return the original mesh if the boundarypoints return to their original positions, whereas the linear-elasticity smoothing method is not guaranteed to producethe original mesh with under similar circumstances.

Both smoothing methods can be used to reposition interior nodes when changes are made to the boundary nodes,such as in design optimization applications. In the case of Winslow smoothing, the character of the computationalmesh is maintained in the physical mesh. Recall that the clustering of the computational mesh in the structured gridwas reflected in the physical mesh for the spike problem. As the deformation from the computational mesh growsthe clustering of the mesh near the boundary will change to satisfy the governing grid equations. As the deviationfrom the computational mesh increases in Winslow smoothing, the clustering of the interior points will change in theusual manner associated with Winslow smoothing; namely pulling out of concave corners and clustered moretoward convex corners. This differs from the linear-elasticity method because the clustering of interior grid points iscontrolled by the Poisson’s ratio term. High aspect ratio cells are maintained relative to the local boundary motion.If more control over the point distribution is desired then Winslow smoothing with forcing functions can beemployed.

VII. Forcing FunctionsThe spacing and topology of the non-uniform computational meshes have a direct influence on the

characteristics of the physical mesh. This was demonstrated earlier on the spike example problem. However, it issometimes useful to add forcing functions to provide increased control over grid point placement and grid lineangularity. Research is underway to develop methods for utilizing forcing function for the general case. Currentlyonly a fixed grid method has been implemented and tested.

A. Fixed Grid MethodOne of the simplest uses of forcing functions is to recover an existing mesh or slightly modify an existing mesh

to improve element quality. Under these circumstances the required forcing functions can be computed from theexisting mesh. The Winslow equations must first be solved for the forcing functions, as shown below.

−−−−

=

ΨΦ

−−−−

=

ΨΦ

ΨΦ

=Ψ++−Φ+

=Ψ++−Φ+

−

yyxxxy

yyxxxy

yx

yx

yyxxxy

yyxxxy

yx

yx

yyyxyxxx

yyyxyxxx

vvv

uuu

vv

uu

vvv

uuu

vv

uu

vvvvv

uuuuu

γαβγαβ

γαγα

γαβγαβ

γαγα

γβα

γβα

2

2

2

2

andforsolve

0)(2)(

0)(2)(

1

(17)


The right-hand-side of the equation set is driven by the residual of the Winslow equations with no forcingfunctions, the terms contained in the brackets. So an existing mesh that satisfies the Winslow equations with noforcing functions will result in zero control functions. If one desires to slightly modify the existing physical mesh toimprove the smoothness and eliminate kinks in the mesh, the resulting forcing functions can be averaged withneighboring values. These averaged forcing functions can then be used with Winslow smoothing to improve anexisting mesh. If no averaging of the computed forcing functions is performed the original mesh can be fullyrecovered. To demonstrate, the original spike problem is initialized with a transfinite interpolation procedure. Thecomputational mesh is an equally spaced Cartesian mesh shown in the upper left corner of Figure 0. The forcingfunctions are computed using equations 17. The physical mesh is scrambled to produce the mesh in the lower leftcorner of Figure 0. The original forcing functions are applied to the Winslow equations to produce the physical meshon the lower right corner of the figure. In this case no averaging of the forcing functions was performed and theresulting physical mesh is the original mesh created with the transfinite interpolation procedure. If the forcingfunctions are not used the resulting physical mesh is shown in the upper right corner of the figure.

VIII. ResultsSeveral examples are included to demonstrate the geometric complexity that can be modeled using these mesh

smoothing techniques and to illustrate the various uses of the two forms of elliptic smoothing described in thispaper.

Figure 0. Scrambled spike mesh recovered using Poisson smoothing.

Computational Mesh Physical Mesh: No forcing functions

Physical Mesh: scrambled Physical Mesh: With forcing functions


A. Three-element airfoilA mesh for a three-element airfoil is shown

in Figure 0.11,12 The corner region shown earlierin Figure 0 is the region located in the aft end ofthe main element. The mesh has 28 viscouslayers inserted using linear-elastic smoothingwith the spacing at the wall specified as 2.0e-6.The final mesh contains 101,787 points, 8,104triangles, 96,680 quadrilaterals, and 2,296boundary segments.

B. SubmarineTo further demonstrate the ability of

Winslow smoothing to recover thecomputational mesh, a physical mesh for a two-dimensional submarine shape was intentionallytangled and then smoothed using Winslowequations with zero forcing functions. Initiallythe computational mesh was copied into thephysical mesh, so the two meshes were exactlyidentical. Then the physical mesh was tangled,shown on the left in Figure 0. The recoveredmesh after smoothing is shown on the right sideof the figure. The Winslow equations were fullyconverged to machine zero and the resultingmesh coordinates compared exactly with theoriginal physical coordinates.

C. Oscillating NACA0012 AirfoilAn oscillating airfoil is used to demonstrate the use of the two smoothing techniques for moving boundary

problems. The first case uses Winslow smoothing with airfoil rotations of ±10 degrees about the quarter chordlocation. The mesh for the 10 degree rotation is shown on the left in Figure 0. The mesh for the original location, orzero degree rotation, is shown in the middle of the figure. And the mesh for the -10 degree rotation is shown on theright side of the figure.

Figure 0. Viscous mesh for 3-element airfoil.

Figure 0. Tangled mesh for submarine shape untangled using Winslowsmoothing without forcing functions.


The second case uses linear-elasticity smoothing to move the interior points to follow the prescribed rotation ofthe airfoil boundary points. The prescribe motion is the same ±10 degree rotation about the quarter chord location.The final mesh for the 10 degree rotation is shown on the left in Figure 0. The mesh for the original location isshown in the middle of the figure. And the mesh for the -10 degree rotation is shown on the right side of the figure.This particular mesh is the same mesh from Figure 0 converted to all triangles and smoothed using a separatecomputer program that did not handle quadrilateral elements.

This illustrates that both smoothing techniques are plausible methods to perform this type of mesh movement.Rotations of ±10 degrees are considered moderate rotations. Larger rotations are possible using these methods. Asthe rotation angle increases, the skewing of the elements near the leading edge and trailing edge increases andeventually both methods will fail.

D. 3D Wing/Body/Pylon/NacelleThe basic 3-D smoothing algorithms have been programmed and applied to several validation cases. Some cases

are generic 3-D shapes, such as spheres and cubes. One of the more interesting examples is thewing/body/pylon/nacelle configuration from the second AIAA Drag Prediction Workshop.13,14 This mesh is shownin Figure 0. The mesh does not contain viscous clustering, as the 3-D layer insertion routine is still underdevelopment. It does, however, contain complicated geometry and a mix of element types; namely tetrahedra,pyramid and hexahedra. The mesh is defined for the left half of the configuration as shown in the inset figures at thetop. The background image was generated by mirroring the mesh in the visualization package, Fieldview15. The finalmesh contained 584840 nodes, 394594 tetrahedra, 197889 pyramid, 10342 prisms, 398609 hexahedra, 10342triangle boundary faces and 106605 quadrilateral boundary faces.

Figure 0. NACA0012 mesh for 10 degree airfoil oscillation using Winslow smoothing.

Figure 0. NACA0012 mesh for 10 degree airfoil oscillation using linear-elasticity smoothing.


IX. ConclusionsA method for performing elliptic partial differential equation smoothing on unstructured meshes has been

described that can use two different partial differential equations. The first equation set is modified linear elasticitytheory. These equations can be used to generate initial meshes by moving boundary nodes of computational meshesto the geometry surface, and is also suitable for smoothing meshes that involve boundary motion. The secondequation set is Winslow’s equations. This equation set requires a computational mesh and can be used to smoothinitial meshes and meshes with boundary motion. With no forcing functions, the interior grid spacing of the physicalmesh mimics the spacing of the computational mesh. As a result, any clustering of the computational mesh will bereflected in the physical mesh. Forcing functions are possible and offer increased control over grid point placementand clustering.

A novel approach to generating computational meshes has been described. This method can producecomputational meshes consistently and robustly. The approach uses recursive cell subdivision to generate the initialinviscid mesh. Recursive subdivision can later be used to refine the mesh for viscous spacing at the boundaries andto refine the mesh for solution adaptation. The viscous and adaptive subdivision maintains the validity of thecomputational mesh and results in clustering that is reflected in the physical mesh after performing Winslowsmoothing.

A robust method for generating viscous meshes using layer insertion was described. This method used the linear-elastic form of smoothing to insert new layers at the boundary and push the interior points away from the surface in

Figure 0. Three-dimensional mesh for a wing/body/pylon/nacelle configuration.


a very robust manner. The resulting viscous meshes are generally of a higher quality than the meshes produced usingrecursive cell subdivision, as the “stair-stepping” effect is minimized due to this smoothing technique.

Various uses of these smoothing techniques were shown in several example cases, mostly in two dimensions.The methods have been extended to three dimensions and illustrated in a complex aircraft configuration case.Further development in three dimensions is underway to incorporate the viscous layer insertion technique, as well assolution adaptive mesh refinement. Additional research is also underway to take full advantage of the forcingfunctions in the Winslow equations to control grid points spacing and grid line angularity at the boundaries.

AppendixThe three-dimensional equations for Winslow smoothing and linear-elasticity smoothing can be combined into a

generalized set of equations and are included for reference.5,16 The numerical method used to discretize theseequations is an extension of the method described earlier for the two-dimensional form of the equations.

),,(

0

))(())((

))(())((

))(())((

))(())((

))(())((

))(())((

:wFor Winslo

021

1,0

211

1,1,1

1,21

11,1

1,1,21

11

:elasticitylinearFor

0)()(2

)()()(

0)()(2

)()()(

0)()(2

)()()(

3

2

1

332313

322212

312111

321

333231

232221

131211

321

333231

321

232221

321

131211

wvur

rrrrrrrr

rrrrrrrr

rrrrrrrr

rrrrrrrr

rrrrrrrr

rrrrrrrr

vuwww

wwwwww

wuvvv

vvvvvv

wvuuu

uuuuuu

yyxzzyyx

xxzyyxxz

zzyxxzzy

yxyxyyxx

xzxzxxzz

zyzyzzyy

yzxzxzyzxy

zzzyyyxxx

yzxyxzyzxy

zzzyyyxxx

xzxyxzyzxy

zzzyyyxxx

==

••−••=

••−••=

••−••=

••−••===••−••===

••−••===

=Ω=Ψ=Φ−

====

−+===

=−

+==

==−

+=

=++++

+Ω++Ψ++Φ+

=++++

+Ω++Ψ++Φ+

=++++

+Ω++Ψ++Φ+

r

rrrrrrrr

rrrrrrrr

rrrrrrrr

rrrrrrrr

rrrrrrrr

rrrrrrrr

θ

β

β

β

αααααα

ααα

υθβββ

υααα

αυ

αα

ααυ

α

θβββ

ααα

θβββ

ααα

θβββ

ααα

(18)


Acknowledgements

This work was sponsored by the University of Tennessee at Chattanooga through the Lupton Renaissance Fund.This support is greatly appreciated.

References

1 Freitag, L. A., Knupp, P. M., “Tetrahedral Element Shape Optimization via the Jacobian Determinant andCondition Number,” Proceedings of the 8th International Meshing Roundtable, South Lake Tahoe, Ca, October1999, pp. 247-258.2 Brewer, M., Diachin, L. F., Knupp, P., Leurent, T. and Melander, D., “The Mesquite Mesh Quality ImprovementToolkit,” Proceedings of the 12th International Meshing Roundtable, Sandia National Laboratories, September 2003,pp. 239-250.3 Winslow, A., “Numerical Solution of the Quasilinear Poisson Equation in a Nonuniform Triangle Mesh,” Journalof Computational Physics, Vol. 2, pp 149-172, 1967.4 Thompson, J. F., Thames, F. C., and Mastin, C. W., “Boundary-Fitted Curvilinear Coordinate Systems for Solutionof Partial Differential Equations on Fields Containing Any Number of Arbitrary Two-Dimensional bodies,” NASACR-2729, July 1977.5 Steinbrenner, J. P., Chawner, J. R., and Fouts, C. L., “The GRIDGEN 3D Multiple Block Grid GenerationSystem,” Final Report, WRDC-TR-90-3022, Volume 1, July 1990.6 Thomas, P. D., and Middlecoff, J. F., “Direct Control of the Grid Point Distribution in Meshes Generated byElliptic Equations,” AIAA Journal, Vol. 18, 1979, pp. 652-656.7 Sorenson, R. L., “A Computer Program to Generate Two-Dimensional Grids About Airfoils and Other Shapes byUse of Poisson’s Equations,” NASA TM-81198, 1980.8 Nielsen, Eric J., Anderson, W. Kyle, “Recent Improvements in Aerodynamic Design Optimization on UnstructuredMeshes,” AIAA-2001-0596, January 2001.9 Knupp, P., “Winslow Smoothing on Two-Dimensional Unstructured Meshes,” Engr. With Computers, 15:263-268,1999.10 Karman, Steve L. Jr., “Hierarchical Unstructured Mesh Generation,” AIAA-2004-0613, January 2004.11 Lin, John C. and Klausmeyer, Steven M., “Comparative Results From a CFD Challenge Over a 2D Three-Element High-Lift Airfoil,” NASA TM 112858, May 1997.12 Chin, V.D., Peters, D.W., Spaid, F.W. and McGhee, R.J., “Flowfield Measurements about Multi-Element Airfoilat High Reynolds Numbers,” AIAA-93-3137, July, 1993.13 Brodersen, O., Sturmer, A., “Drag Prediction of Engine-Airframe Interference Effects Using Unstructured Navier-Stokes Calculations,” AIAA-2001-2414, June 2001.14 AIAA Drag Prediction Workshop website, URL:http://aaac.larc.nasa.gov/tsab/cfdlarc/aiaa-dpw/Workshop2/DLR-F6-geom.html.15 Fieldview, Intelligent Light, Inc., URL:http://www.ilight.com.16 Pilkey, W.D., and Wunderlich, W., Mechanics of Structured: Variational and Computational Methods, CRCPress, Florida, 1994.

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