[American Institute of Aeronautics and Astronautics 42nd AIAA Aerospace Sciences Meeting and Exhibit...

American Institute of Aeronautics and Astronautics

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A MESH MOVEMENT ALGORITHM FOR HIGH QUALITY GENERALISED MESHES

- D G Martineau*, J M Georgala† - Aircraft Research Association Limited,

Manton Lane, Bedford, United Kingdom, MK41 7PF

ABSTRACT

A new dynamic mesh movement algorithm designed for the robust and efficient deformation of high quality generalised meshes for viscous flow calculations, in response to perturbations of the underlying geometry, is presented. The algorithm is based on a technique developed for the deformation of inviscid meshes, which uses a modified form of the popular elastic spring analogy approach involving a novel predictor-corrector scheme to improve the robustness of the method significantly. The new algorithm extends the existing approach to generalised highly anisotropic meshes, and introduces rotational terms in the deformation process, resulting in a capability which combines the efficiency of the spring-analogy approach with the robustness of more sophisticated methods which model the computational domain as an elastic solid. Navier-Stokes computations for a 2D airfoil show the improvement in the accuracy of viscous flow solutions obtained on meshes deformed using the new mesh movement algorithm. The ability of the new algorithm to handle deformation of volume meshes for complex 3D configurations is demonstrated through application of the algorithm both to the simulation of a missile release from a cavity, and to the aeroelastic analysis of a generic civil aircraft.

NOMENCLATURE

x,y,z = Cartesian co-ordinate directions s,t = parametric co-ordinate directions x = position vector of point (x,y,z) x′ = position vector in initialised grid u = mesh velocity vector w = angular velocity vector n = unit normal vector d = wall normal distance e = finite-volume element α(x) = distance metric function β(x) = modified distance metric function γ = variable blending factor used in β(x) V = element volume A = element face area

* Project Scientist † Principal Project Scientist

f,g = mappings from parametric to Cartesian co-ordinates

T(x) = parametric re-definition function R(x) = rotational operator Ω = computational domain Γ = boundary of domain Ω ε = smoothing relaxation parameter τ = diffusion coefficient σ = face-based weight used in modified

Laplacian smoothing ω = weight used in interpolation of mesh

velocity to nodes λ = vector of Cartesian coefficients L(φ) = pseudo-Laplacian operator η = sliver metric θ = angle of attack M∞ = Mach number Re = Reynolds number

1. INTRODUCTION

The ability to modify a pre-existing computational grid, subject to a perturbation of the domain boundaries, is a requirement common to many areas of computational fluid dynamics (CFD). In some cases, the surfaces of the configuration remain fixed in shape, but undergo relative motion, for example in the simulation of a store-release trajectory, and in propeller modelling. Examples in which the underlying geometry of the configuration is perturbed include areas such as aeroelastic distortion, flutter prediction, and aerodynamic shape optimisation studies. There are several means by which the deformation of the computational domain boundaries can be accommodated, ranging from complete or partial regeneration to deformation of the existing mesh. At the Aircraft Research Association (ARA), techniques are being developed to produce a generic re-meshing capability that employs the most appropriate approach for each of the possible applications required in aerodynamic simulations. However, this paper focuses solely on the mesh deformation component of the re-meshing capability, and uses examples from 2D high-lift modelling, store-release simulation and aeroelasticity to illustrate the techniques developed.

42nd AIAA Fluid Dynamics Conference and Exhibit 5-8 January 2004, Reno, Nevada

42nd AIAA Aerospace Sciences Meeting and Exhibit5 - 8 January 2004, Reno, Nevada

AIAA 2004-614

Copyright © 2004 by the American Institute of Aeronautics and Astronautics, Inc. All rights reserved.


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The preservation of the original grid connectivity, inherent in a mesh deformation approach, gives an intuitive guarantee that any mesh-induced errors in the flow solution (i.e. due to truncation error) will be consistent between the initial and deformed mesh, providing the element quality is preserved. This is an important feature for design optimisation applications, where any changes in the grid topology can adversely affect the computed sensitivity derivatives. A further benefit of deforming the computational mesh, as opposed to regeneration, is that the converged flow solution obtained on the original grid can be used to initialise the flowfield on the deformed grid, without having to resort to interpolation of flow variables between the two grids. The most common methods employed in mesh movement algorithms applicable to unstructured grids are variations of the elastic spring analogy1-6, where the edges of the mesh are treated as linear elastic springs, with the spring constants inversely proportional to the square of the edge length. Whilst the spring analogy is efficient and works well for inviscid grids on relatively simple configurations, it is not adequate when applied to the deformation of highly refined grids generated for viscous flow solutions, and can lead to crossing of grid lines and the formation of negative volumes5. The inadequacies of the basic spring analogy have led to modifications to the spring analogy based on the addition of torsional spring elements by Farhat et al7 for mesh movement in two dimensions, and extended to three-dimensional unstructured grids by Murayama et al8. This approach, whilst improving the robustness of the spring analogy method, is of limited use in structured or hybrid meshes, due to the possibility of distorted quadrilateral and hexahedral elements9. Recently, mesh movement algorithms have been developed that model the grid as an elastic solid10,11,12, and obtain the deformed grid by solving the equilibrium equations for the stress field. Although far more robust than the spring analogy approach, this method incurs a much higher computational cost. A more cost-effective approach is proposed by Gao et al13, based on a non-linear elastic boundary element method (NBEM). Although demonstrated to be a robust and effective capability, the NBEM approach requires the use of an incremental procedure for large-amplitude deformations. Motivated by the limitations of existing mesh movement techniques discussed above, a new algorithm is described that exhibits all of the following key attributes required of a generic mesh movement capability for CFD:

i) preservation of the connectivity of the original grid

ii) ability to generate a valid deformed grid for a wide range of perturbations of the boundary of the domain, including: – relative motion of surfaces in close

proximity. – large-scale geometry deformations

iii) applicability to a variety of mesh types, including structured, unstructured, and hybrid meshes

iv) preservation of grid quality (in terms of orthogonality and smoothness)

v) effective handling of multiple moving surfaces

vi) acceptable efficiency The approach to mesh movement that has been adopted in the current work is derived from a technique that was originally developed at ARA for inviscid, hybrid meshes produced by the SAUNA CFD system14. These meshes were composed of hexahedra, pyramids, and tetrahedra. The ability to deform more general meshes, composed of arbitrary polyhedra, became a necessity with the creation of the SOLAR CFD system, which is described in the following section. In section 3 of this paper, an overview of the initial algorithm developed for SAUNA meshes is given as a precursor to the description given in section 4 of the enhancements made to extend the technique to generalised meshes and improve the quality of the deformed meshes. Section 5 of the paper is given over to illustrating the power of the improved mesh deformation approach through demonstration of its use on single and multi-element airfoil sections, the release of a store from a cavity, and the aeroelastic analysis of a civil aircraft.

2. THE SOLAR CFD SYSTEM

Since the original work on deformation of SAUNA meshes, ARA has been involved in a collaborative programme developing the SOLAR15,16 CFD suite. The BAE SYSTEMS’ SOLAR suite was instigated in 2000, and since then this suite has been extended and developed by a collaboration involving BAE SYSTEMS, ARA, Airbus UK and QinetiQ to provide a rapid-response complex configuration RANS CFD capability. The SOLAR suite includes an automatic mesh generation method capable of producing high quality meshes for viscous flow simulations. The meshes are created using an Advancing-Layer17 approach to march a near-field mesh away from a surface mesh composed of quadrilaterals and triangles. Edge-collapsing and enrichment algorithms drive the topology of the layer to change automatically to take into account the underlying concavity or convexity of the region being meshed. The layer growth is terminated locally when cells reach approximately unit aspect ratio, or when further


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growth would lead to cells overlapping. A Cartesian mesh of variable density is then cut to conform to the outer shell of the near-field mesh. Thus, the complete mesh is composed largely of hexahedra, but will also contain simpler elements and general polyhedra, as well as hanging faces, as illustrated in Fig. 1. Fig. 1. a) A SOLAR volume mesh (surface mesh

not shown for clarity). Fig. 1. b) Detail of a SOLAR volume mesh

3. ORIGINAL MESH MOVEMENT ALGORITHM

The original mesh movement algorithm developed at ARA by Leatham et al18 involves a modification to the elastic spring approach based on the addition of a rigid-body initialisation (RBI) procedure. The RBI procedure generates an initial guess of the position of each grid node based on a distance-weighted average of the movement of the two nearest boundaries (lines in 2D, surfaces in 3D). A Laplacian-based smoothing step, similar to the elastic spring smoothing algorithm, is then applied to the position of the nodes to improve the mesh quality in the interior of the domain.

Fig. 2. Control flow in the original mesh movement algorithm

The original algorithm splits the mesh movement into two separate processes for the surface and volume mesh, as illustrated in Fig. 2, within which the same operations are applied. These two key operations, rigid-body initialisation and smoothing, can be viewed as forming predictive and corrective steps of the mesh movement algorithm and are reviewed in the following sections.

3.1. Rigid-Body Initialisation

In the rigid-body initialisation an initial estimate, x′, for the location of each interior mesh node is derived from a distance-weighted average of two position

vectors, n

x′ , n = 1,2. The vector n

x′ gives the

position that the node would have if it were translated with the movement of the nearest node nx on

boundary Γn.

nnxxx ∆+=′

(1)

The appropriate weighting of these two values is driven by the relative proximity of the node to the two nearest boundaries Γ1 and Γ2. These distances, d1 and d2, are determined using an iterative procedure, which defines the distance as the sum of edge lengths along the shortest path from the node to the boundary. The ‘distance metric’, α, is given by:

21

2)(dd

d

+=xα , 1)(0 ≤≤ xα (2)

The initial guess for the new location of the node is then evaluated as follows:

( )2 1

)(1)( xxxxx ′−+′=′ αα (3)

VOLUME DEFORMATION

SURFACE DEFORMATION

for each surface

Create initialised grid

Smooth node locations


Smooth node locations

far-field mesh

near-field mesh


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3.2. Laplacian Smoothing

The smoothing in the original algorithm is implemented through the iterative application of an explicit solver for a modified form of what is essentially a Laplacian filter. For the surface smoothing, a Laplacian filter applied edgewise, weighted by the reciprocal of edge length, is used. In the volume smoothing, a face based Laplacian operator is used instead, where the weighting term is based on the ratio of face area to element volume to ensure the desired behaviour for both hexahedral and tetrahedral elements. In order to avoid the grid folding that can occur as a result of the application of the Laplacian filter, the smoothing is heavily under-relaxed close to the boundaries (in what is subsequently referred to as the ‘near-field region’). The smoothing correction is relaxed by two factors; the first is the minimum edge length connected to a node, the second is a function of the distance from a boundary. The full relaxation parameter at node i, εi, is given by:

⎟⎟

⎠

⎞

⎜⎜

⎝

⎛−⋅= 0.1,min

min2

1jii

d

dxxε (4)

4. IMPROVED MESH MOVEMENT

ALGORITHM

The approach described above has been shown to extend the range of mesh movement permissible with the elastic spring analogy significantly. However, the algorithm suffers from a feature common to many mesh movement schemes in that it fails to preserve the orthogonality of the mesh close to boundaries undergoing rotational movement. This can be detrimental to the accuracy of the flow solution obtained on the mesh, particularly when modelling viscous flows where the ability of the grid to capture the strong flow gradients found in the boundary layer is important. The work described in the remainder of this paper focussed principally on the extension of the above algorithm to deformation of highly refined viscous meshes comprised of arbitrary polyhedra, within a wide range of application areas. As in the original algorithm, the mesh deformation procedure involves separate processes for the surface and volume mesh, but the new algorithm introduces a third stage, prior to the surface mesh deformation (see Fig. 3). The geometry manipulation stage covers the various means of perturbing the boundaries of the computational domain for various mesh movement applications. The following sections describe the developments made to the mesh movement algorithm in each of these three main stages.

Fig. 3. Control flow in the new algorithm. 4.1. Geometry Manipulation

The geometry manipulation stage essentially provides the interface between the mesh movement software and the various application areas, as illustrated in Fig. 4. To date, the mesh movement software has been coupled to the following applications:

• Flutter Prediction A rotation is applied to the relevant components of the geometry, based on a prescribed sinusoidal motion and the current and previous positions in the flutter cycle.

• Store Release Simulations The rigid-body motion of the store, calculated using a 6 degrees-of-freedom store trajectory prediction code, is applied directly to the geometric surfaces representing the store.

• Aero-structural Coupling The geometry of the configuration is deformed via the application of a structural mode shape, determined by a structural dynamics solver.

• Design Optimisation A new geometric definition of the configuration, perturbed from the original geometry by an external software system, is input from a file.

Fig. 4. Design of the geometry manipulation stage.

SURFACE DEFORMATION

VOLUME DEFORMATION

GEOMETRY MANIPULATION

Rotate geometry

Move geometry

Deform geometry

Re-calculate intersections

Input geometry

Store Release

Design Optimisation

Flutter Prediction

Aeroelastic Analysis


§4.1

SURFACE DEFORMATION

§4.2

VOLUME DEFORMATION

§4.3


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After creation/input of the new parametric surface definitions, any intersections affected by the geometry manipulation are automatically re-calculated. Beyond this stage, the procedure for the deformation of the surface and volume mesh is common for all the mesh movement applications. 4.2. Surface Mesh Deformation

The modifications made to the surface mesh movement stage relate primarily to the issues introduced by the necessity to perform the surface mesh movement in physical space, rather than parametric space. Realistic configurations from CAD models are often described by hundreds of sets of surface patches, and for this reason the SOLAR surface mesh generator, MERCURY, is designed to operate on a set of meshable entities referred to as ‘zones’. A zone consists of a number of surface patches, each with different parametric definitions, and is bounded by a closed loop of lines. An Advancing-Front method19 is used to create unstructured quadrilateral-dominant meshes on each zone. The method operates in physical space and projects the created nodes to the surface at various stages to ensure surface conformity. The surface mesh deformation is therefore also performed on a zonal basis, and in physical rather than parametric space. To accommodate the change from parametric space to physical space has necessitated the introduction of two new stages in the surface deformation process (as indicated by Fig. 5): • A parametric re-definition stage, prior to the

rigid-body initialisation stage. • A final re-projection stage, after application of

the mesh movement. Fig. 5. Control flow for the new surface mesh

deformation process

4.2.1. Parametric Re-definition

In addition to the stages described above, the surface mesh deformation includes a parametric re-definition stage before the rigid-body initialisation. In this stage, the surface mesh nodes are mapped onto the new geometry using the parametric co-ordinates derived from the underlying geometry of the original surfaces and the new physical definition of the geometry. Thus, if fi and gi are mappings from parametric to physical co-ordinates for each surface patch Si in the old and new geometry, respectively, then the operator Ti, which represents the parametric re-definition stage, is given by

( ) ( )( )xx 1−= iii fgT (5)

where

( ) ( )oldzyxtsf ,,, = (6)

( ) ( )newzyxtsg ,,, = (7)

The deformation of the surface mesh is thus driven by any changes in the intersection of two geometric components. If no intersections have changed due to the movement (e.g. in the case of a store released from an aircraft) then the surface mesh is simply mapped onto the new geometry and no further stages are required in the surface mesh deformation process.

4.2.2. Rigid-Body Initialisation

Since the rigid-body initialisation stage is performed in physical space, it is necessary to project the nodes in the surface mesh back onto the underlying geometry to ensure surface conformity, in the same manner as in the mesh generation process. If a surface exhibits a high degree of curvature this projection process can lead to some highly distorted surface mesh. To avoid this situation, the rigid body initialisation stage has been modified to ensure that any movement of the surface mesh nodes is in a direction tangential to the surface, thus minimising any subsequent displacement of the nodes as a result of the re-projection. Instead of applying a weighted average of the physical translation at each interior node, the contribution of the boundary movement ∆xc to the movement of an interior point xi is defined as follows:

( ) icci nxnx ×∆×=∆ (8)

where nc and ni are the unit vectors normal to the surface at the boundary curve and the interior mesh point, respectively. This modification essentially views the translation of the boundary as a movement tangential to the surface, which is then re-evaluated at each interior mesh point, as illustrated by Fig. 6:


VOLUME DEFORMATION

SURFACE DEFORMATION

Map nodes onto new geometry

for each zone


Smooth interior mesh points

Project points to surface

§4.2.1.

§4.2.2.


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Fig. 6. Constraint of the surface mesh movement to

the geometry. 4.3. Volume Mesh Deformation

The enhancements to the volume mesh deformation process were made primarily in order to enable the algorithm to function robustly on the more general types of meshes produced by the SOLAR volume mesh generator VENUS. The fundamental changes to the original algorithm for the deformation of the volume mesh are twofold:

• re-formulation of the mesh movement process using the ‘dual-mesh’

• incorporation of the boundary rotation in the mesh movement

DUAL-MESH FORMULATION In order to address some of the challenges posed by SOLAR meshes, the original mesh movement algorithm12 has been reformulated using a ‘dual-mesh’ approach in which cell centres take the place of the interior mesh nodes, and faces replace edges. As illustrated in Fig. 7, the dual-mesh has a simplified connectivity, compared to the original mesh, which has the advantage of effectively ‘removing’ the hanging nodes and creating a more even stencil for the smoothing algorithm. The dual-mesh formulation also reduces the memory storage required and permits the use of the face-based data structure used throughout the SOLAR CFD system. Fig. 7. Illustration of the dual-mesh

INTRODUCTION OF ROTATIONAL TERMS As mentioned earlier, the use of purely translational components of deformation in the original mesh movement algorithm allows the introduction of skewness in the deformed mesh. This is particularly evident, for instance, near the surface at the leading edge of an airfoil subjected to a rotation about a spanwise axis, as can be observed in Fig 8. Fig. 8. Mesh after application of original mesh

movement algorithm. In order to prevent this reduction in mesh orthogonality, and to enable the algorithm to function robustly on highly anisotropic meshes generated for Navier-Stokes flow simulations, rotational deformation of the mesh was incorporated in the new algorithm. Both the dual-mesh formulation and the addition of rotational terms have required significant modification and extension to all the stages involved in the volume mesh deformation process (see Fig. 9), which are described in the following sections. Fig. 9. Control flow of new volume deformation

process

mesh

dual-mesh

cell vertex

cell centre

DUAL-MESH DEFORMATION

VOLUME DEFORMATION


SURFACE DEFORMATION

Calculate distance metrics


Smooth mesh velocities

§4.3.1.

§4.3.2.

§4.3.3.

Interpolate movement to nodes

§4.3.4.

∆xi

xi

xc

∆xc

ni

nc


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4.3.1. Distance Metric Calculation

The distances to the nearest boundaries are still calculated using an iterative procedure, but looping over the faces in the grid, rather than edges. As it is necessary to store the index of the nearest boundary face during this process, it is possible to calculate the direct distance to this face. Although more expensive per iteration, the use of the exact distance to a boundary, as opposed to the sum of edge lengths, was found to improve the convergence of this procedure, as well as resulting in a smoother variation in the distance metric (see Fig. 10). a) detail of grid b) sum of edge lengths c) distance to face centre d) projection onto faces Fig. 10. Variation of the distance metric. Although the use of direct distances in the distance metric calculation showed benefits, the presence of hanging faces close to the boundary in the main mesh still caused a discontinuity in the variation of the distance metric in the dual mesh which led to a lack of robustness. A remedy was found by comparing the distance to the boundary face centre with the square root of the face area. If this test indicated that the point was close to the surface, then the distance to the boundary was re-evaluated using a projection of the interior mesh point onto the face. As Fig. 10 shows, this further improves the smoothness of the distance metric field. Despite the various improvements made to the calculation of the distance metrics, this part of the mesh movement process is still one of the most time-consuming stages. However, in many applications of

the mesh movement algorithm, such as flutter prediction and design optimisation, repeated deformations of the grid are required, but the overall shape of the configuration remains largely unchanged. In these instances, the indices of the two nearest boundary faces are stored at each cell and re-used in the calculation of the distance metrics in subsequent executions of the mesh movement code. This allows significant savings in overall execution time for the mesh movement process.

4.3.2. Rigid-Body Initialisation

In the original algorithm, the node locations in the initialised grid, 1x′ and 2x′ , were determined using a

simple translation of the original position vector using equation (1). In the new algorithm, the two possible locations are calculated for each cell centre by applying a combination of the translation and the rotation of the nearest boundary faces:

nnn R xxx ∆+=′ )( (9)

where Rn(x) gives the position of the point x as though it had rotated rigidly with the movement of the boundary face. The average rotation of each cell about its centre (i.e. the change in orientation of the cell) is also stored as a vector quantity. The calculation of the cell centre co-ordinates in the initialised grid has been modified slightly to improve the orthogonality of the mesh in the near-field region:

( ) 21 )(1)( xxxxx ′−+′=′ ββ (10)

where

[ ]γαβ ))((exp)( xx c= (11)

The function α(x) is still defined as in equation (2), but is bounded between 1.0 at the boundary of the domain, and 0.5 on the median surface of the grid, since d1 and d2 are chosen such that d1 ≤ d2. The factor γ determines how the blending function β(x) varies between the boundary and interior of the mesh, as illustrated in Fig. 11. Fig. 11. Variation of the blending function through

the domain.

γ = 2 γ = 4 γ = 8

0.5

0.6

0.7

0.8

0.9

1

0.5 0.6 0.7 0.8 0.9 1

Ι ( x )

ϑ(x

)


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4.3.3. Laplacian Smoothing

The original mesh deformation algorithm featured a Laplacian-based filter for the smoothing of the volume mesh. Due to the transfer of the mesh movement algorithm to a dual-mesh formulation, a new discretisation of the Laplacian smoothing was required for the generalised meshes created by the SOLAR mesh generator. If the displacement of the mesh due to the deformation process is viewed as a mesh velocity, then a smooth displacement field can be obtained through the solution of the Laplace equation

0=∇⋅∇ u , where u ≡ ∆x is the mesh velocity. The displacement calculated in the rigid-body

initialisation stage, xxu −′=′ , is used as an initial guess, and the translation of the boundary faces are used to apply the Dirichlet boundary conditions. However, in order to preserve the quality of the mesh in the near-field region and restrict the smoothing to the larger volume cells in the interior of the domain, a modified form of the Laplacian operator is used:

0)( =∇⋅∇ uτ (12)

The factor τ is the variable diffusivity proposed by Baker20, based on the element volume V:

V

VV minmax1−+=τ (13)

The local diffusivity factor is evaluated only once, prior to the smoothing process, using the original undeformed mesh. The integral form of equation (12) is obtained by integrating the PDE form over some domain, Ω, and applying Gauss’ divergence theorem:

0d d )( =Γ⋅∇=Ω∇⋅∇ ∫∫Ω∂Ω

nuu ττ (14)

where n is an outward facing normal on the surface S bounding the volume Ω. For small volumes, this becomes:

0d )(1 =Γ⋅∇≈∇⋅∇ ∫

Ω∂Ωnuu ττ (15)

Discretisation of equation (15) on the arbitrary polyhedral element ei with cell centre xI, connected via the dual mesh to the cell centres xj, j = 1,N, yields the following formula:

01

1

=⋅∇∑=

ij

N

j

ijijij

i

AV

nuτ (16)

where A is the face area and the subscript ij indicates a quantity evaluated at the face between element ei and element ej. Equation (16) is simplified by making the following approximations:

2

ji

ij

τττ

+≈ (17)

ij

ijijij

xx

uunu

−−

≈⋅∇ (18)

Using a point iterative scheme, the updated mesh velocity at iteration t +1 is given by:

ij

N

j

j

ij

ti

ti σ

σεε ∑

=

+

∑+−=

1

1 )1( uuu (19)

The use of a variable diffusivity effectively increase the stiffness of the small cells in the near-field region of the mesh, and allows a global, rather than a local, relaxation parameter, 0 ≤ ε ≤ 1, to be used in the smoothing algorithm. The form of equation (19) is similar to the original edge-based formula, but instead of being a ratio of face area to element volume, the weighting σ is now defined as:

ij

ijijij

xx

A

−=τσ (20)

Although this approach clearly has links with the spring analogy approach, the weights given by equation (20) show more similarity with the smoothing operator adopted by Crumpton and Giles21 in their mesh movement algorithm. The update formula given by equation (19) is applied iteratively to the mesh velocities using a Gauss-Seidel scheme in which the updated mesh velocities are used as soon as they are available. The smoothing process is advanced until the r.m.s. value of the residual has converged to a specified tolerance. In practice, as a result of the predictor stage to generate an initialised grid, very few iterations of the smoothing algorithm are required to create a valid grid. As well as smoothing the translational velocity, the angular velocity, w, which is also stored as a vector quantity at each cell centre, is smoothed using an equation of the same form as equation (12):

0)( =∇⋅∇ wτ (21)

4.3.4. Interpolation to Nodes

After deformation of the dual-mesh, it is then necessary to interpolate the mesh movement to the


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cell vertices. The new node locations are given by the following formula:

∑ ′=∑ j

jj

j j

i xx ωω

1 (22)

where the vectors jx′ define the locus of possible

positions of the node if it were rotated and translated rigidly with each of the cells connected to the node:

jijj R xxx ∆+=′ )( (23)

After evaluating numerous interpolation schemes, the optimal values of ωj were found using an interpolation procedure similar to that presented by Coirier and Jorgenson22, where the weights are derived from the pseudo-Laplacian operator:

( )∑ −=j

ijjL ϕϕωϕ )( (24)

Linearity preservation is guaranteed by requiring that:

( ) ( ) ( ) 0=== zLyLxL (25)

Using linear basis functions to expand the weights about unity, the following expression is derived from equations (24) and (25):

)(1 Tijj xxλ −+=ω (26)

A 3x3 linear system is found for λ which is inverted to give values for the interpolation weights that are based on the products and moments of inertia. Although computationally more expensive than many other interpolation schemes, this method is more robust and gives better results in highly anisotropic regions of the grid. A further novel feature introduced in the new mesh movement algorithm is a method for avoiding the formation of negative volumes, without recourse to mesh topology modification resulting from local mesh regeneration or edge/face swapping techniques. After interpolation of the mesh movement to the cell vertices, the new volumes of the cells are calculated. If any negative volumes are calculated, the mesh movement is re-interpolated to any nodes connected to these elements using weightings modified by the following formula:

ave1VVm

jj

je+=′

ωω (27)

The cell volumes are re-calculated again and the procedure repeated, up to a maximum of 10 times, until there are no longer any negative volumes. As the iteration count m increases, the modification to

the interpolation weights given by equation (27) gives increasingly greater importance to small or negative volumes in the interpolation. This technique provides a powerful capability for dealing with small areas of poor quality in the initial grid.

5. RESULTS AND DISCUSSION 5.1. Multi-Element Airfoil

To illustrate the advantages of the new mesh movement strategy over the original algorithm, the flap on the multi-element airfoil shown in Fig. 12 has been deflected by 15°, and both the mesh deformation strategies applied. Fig. 12. Geometry of the multi-element airfoil Fig. 13 shows a close-up view of the baseline mesh in the region between the trailing edge of the main element and the leading edge of the flap. Figures 14 and 15 show the same region after deformation via the original and modified version of the mesh movement algorithm, respectively.

Fig. 13. Near-field view of baseline mesh.

Fig. 14. Near-field view of mesh after applying

original mesh movement algorithm

Fig. 15. Near-field view of mesh after applying

improved mesh movement algorithm


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In Fig. 15 it can be clearly seen how the addition of the rotational terms in the new algorithm reduces the skewness introduced in the near-field region of the mesh as a result of the deformation. 5.2. NACA0012 Airfoil

In order to quantify the differences between the two mesh movement schemes, each algorithm was used to deform the mesh on a NACA0012 airfoil section rotated by θ. Flow solutions were then obtained on the meshes at an angle of attack of –θ, which should be identical to the flow solution obtained at zero angle of attack on the undeformed mesh.

Fig. 16 shows the baseline mesh (mesh A) and the two deformed meshes at a pitch-up angle of 20 degrees: • Mesh B:

The deformed mesh obtained by applying the original mesh movement algorithm.

• Mesh C: The deformed mesh obtained by applying the new mesh movement algorithm.

Details of the meshes in the near-field region and at the leading edge of the airfoil are shown in Figures 17 and 18, respectively.

Mesh A Mesh B Mesh C Fig. 16. Meshes generated on NACA0012 airfoil. Fig. 17. Near-field view of meshes. Fig. 18. Detail of meshes at the leading edge of the NACA0012 airfoil


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Ideally, the results obtained on all three grids would be identical. In practice, however, the accuracy of the flow solution will be dependent on the quality of the grid. This dependence is illustrated in the results presented in Fig. 19, which shows the lift and drag coefficients obtained from flow solutions computed on the different meshes. The viscous flow solutions were performed using the SOLAR Navier-Stokes flow solver, JUPITER, with the k-g turbulence model, at M∞ = 0.7, Re = 2.5 million, and at θ = 0° to 20°. Fig. 19. a) Variation in lift coefficient Fig.19 shows that the results obtained using the original mesh deformation show a divergence from the baseline solution (obtained on the rotated mesh), which becomes more noticeable at high pitch angles. In contrast, the close agreement of the results on mesh C with the baseline results demonstrates the capability of the new mesh movement algorithm for preserving the high quality grid necessary for accurate computation of viscous flow solutions. 5.3. Missile / Cavity

The modelling of a store release is a particularly demanding application for a mesh movement algorithm, characterised by a computational domain in which different boundaries remain fixed in shape while undergoing significant relative motion. The computational domain changes substantially in shape during the course of a store release simulation, and

will typically involve relative movement of surfaces in close proximity. The objective of a mesh movement algorithm in this context is to permit as much deformation as possible before the quality of the grid is compromised to the extent that mesh regeneration is required. Application of the new mesh movement algorithm to store trajectory modelling is illustrated by the deformation of a SOLAR mesh in the simulation of a generic missile released from a cavity. A highly refined grid suitable for viscous flow calculations using wall function turbulence models was generated consisting of 1,008,630 nodes and 871,670 cells. The geometry of the missile used in the store release simulation is shown in Fig. 20. For clarity, two sides of the cavity are cut away.

Fig. 20. Geometry of the store and cavity Figures 21 and 22 show sectional cuts through the store/cavity volume mesh a) in the carriage position, and b) after 3 store trajectory steps. The mesh in parts b represents 12 successive applications of the mesh deformation algorithm, since the mesh is deformed after each step of the four-stage Runge Kutta integration scheme used in the 6-DOF calculation of the store trajectory. Figures 21 (b) and 22 (b) clearly show how the new mesh movement algorithm preserves the orthogonality of the grid in the immediate vicinity of the missile, and on the walls of the cavity. Also evident is the smooth transition in the mesh deformation between the moving and stationary boundaries of the domain.

a)

-0.005

0.000

0.005

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0.025

0 5 10 15 20

Angle of Mesh Rotation

Sec

tio

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oef

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Fig. 21. Cut at constant y through the store/cavity volume mesh a) before, and b) after mesh deformation. a) b) Fig. 22. Cut at constant x through the a) original,

and b) deformed store/cavity mesh

The CPU requirement for the mesh movement process, based on an SGI Octane/SE, 300MHz, R12000 processor, is summarised in the table below.

Execution Time (seconds)

Mesh generation 2435.402 Initial mesh deformation 3664.000 Repeat mesh deformation 943.652

Table 1. Comparison of CPU requirements 5.4. Civil Aircraft

A realistic civil aircraft configuration with nacelle and pylon demonstrates the use of the new mesh movement algorithm in a coupled aerodynamic / structural dynamics analysis, where the mesh deformation is driven by a perturbation to the underlying geometry via application of a structural mode shape. A structural solver provides displacement and rotation information at each finite element node on the beam-stick model of the aircraft wing. The structural displacement data is then interpolated to the corresponding geometry surfaces in the CFD representation of the configuration. Fig. 23 shows the geometry of the aircraft used for the aeroelastic analysis with the deformed geometry superimposed. An unstructured mesh containing 0.6 million nodes and 3.6 million tetrahedral elements was generated for the analysis.

Fig. 23. Baseline and deflected geometry

b)


13

In Fig. 24, detail of the surface mesh around one of the engine nacelles is illustrated. Fig. 24. Detail of the surface mesh on nacelle. The cells intersected by a spanwise sectional cut through the volume mesh, before and after application of the structural mode shape, are visualised in Fig. 25 in the region close to the wing tip, demonstrating the preservation in grid quality. This case illustrates the ability of the mesh movement algorithm to handle significant deformation of the computational domain in an acceptable manner. Fig. 25. Original and deformed volume mesh in

the region of the wing tip This test case also demonstrates the effectiveness of the technique developed for handling regions of poor quality grid, which in this case corresponds to the ‘sliver’ tetrahedra commonly found in unstructured meshes. Sliver tetrahedra are extremely flat tetrahedra with a low value of the sliver metric, η, defined as follows:

∑=

iA

V

41

3

η , 745.00 ≤≤η (28)

The theoretical upper bound on η of 0.745 occurs for equilateral tetrahedra. Fig. 26 shows a histogram of the sliver metric computed for all the elements in the original aircraft mesh, highlighting the extent of these sliver elements in unstructured meshes. Fig. 26. Sliver tetrahedra in the volume mesh

CONCLUSIONS

A dynamic mesh movement algorithm based on a modified form of the elastic spring analogy has been re-formulated using a dual-mesh approach to enable deformation of generalised meshes in response to perturbations of the underlying geometry. A number of improvements have been made to the existing approach, including the incorporation of the rotation of the domain boundaries in the deformation process, in order for the algorithm to function robustly on high quality meshes generated for viscous flow simulations. Comparison of lift and drag coefficients calculated for a 2D airfoil show the improvement in the accuracy of viscous flow solutions obtained on meshes deformed using the new mesh movement algorithm. Applications of the mesh movement algorithm to a store release simulation and the aeroelastic analysis of a generic civil aircraft demonstrate the ability of the new algorithm to handle deformation of volume meshes for complex 3D configurations.

ACKNOWLEDGEMENTS

Initial development of the mesh movement capability was funded by QinetiQ as part of the SOLAR store release software, which also received funding from BAE SYSTEMS. Further development of the capability has been carried out within the CAST23 project, which is jointly funded by Airbus UK and the CARAD programme from the UK’s Department of Trade and Industry (DTI). The authors would like to acknowledge the technical assistance provided by the members of the SOLAR development team from BAE SYSTEMS, QinetiQ and ARA.

Original Deformed

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Sliver Metric

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