Center for Turbulence Research, Stanford University,
Building 500, Stanford, CA 94305–3035, USA
Mesh deformation algorithms usually require solving a system of equations which canbe computationally intensive for complex three-dimensional applications. In this paper anexplicit mesh deformation method is proposed based on Inverse Distance Weighting (IDW)interpolation of the boundary node displacements to the interior of the flow domain. Thepoint-by-point mesh deformation algorithm results in a straightforward implementation andparallelization. The formulation is extended to a robust Extremum Conserving (ED) meshdeformation method. An IDW mesh optimization method is also introduced based on IDWmesh deformation, which reaches virtually perfect mesh orthogonality. It is also applied tomesh motion in the fluid-structure interaction simulation of the three-dimensional AGARD445.6 aeroelastic wing, in which IDW mesh motion shows a reduction of the computationalcosts up to a factor 50 with respect to Radial Basis Function (RBF) mesh deformation fora comparable simulation accuracy.
I. Introduction
In an Arbitrary Lagrangian–Eulerian (ALE) formulation of a dynamic fluid–structure interaction problemthe flow and the structure forces and displacements are coupled at the interface. The flow forces result
in deformation and displacement of the structure, which in turn leads to moving boundaries for the flowdomain. In order to interpolate the boundary displacements to the interior of the flow mesh an automaticmesh deformation method is required. The mesh deformation algorithm needs to be robust in case of largedeformations and result in sufficiently high grid quality after deformations for arbitrary mesh topologies. Indynamic fluid–structure interaction simulations a low computational cost for the mesh motion is important,since the flow mesh has to be updated every time step or even multiple times per time step for strong couplingor higher–order multi–stage time integration schemes. Solving the mesh deformation problem can in large–scale cases with complex geometries consume a significant portion of the total computational time for thefluid–structure interaction simulation. Mesh deformation is also used in static aeroelastic computations,aerodynamic shape optimization, and treating geometrical uncertainties.32 In these applications an easyimplementation and parallelization of the mesh deformation algorithm is of interest.
One of the available mesh deformation techniques for continuous surface deformations is the TransfiniteInterpolation (TFI) method.12, 30 This is a fast method that interpolates the boundary displacement tointerior points along grid lines, which makes it only applicable to single–block structured meshes. Forunstructured meshes often methods are used that model the motion of the flow mesh by equations governingstructural deformation.22, 27 For example, equilibrium equations for linear body elasticity have been usedwith the cell modulus of elasticity equal to the reciprocal of the cell volume or the distance to the boundary.9
A discrete version of this approach is often used in terms of the tension spring analogy method, in whichthe grid connectivity edges are modeled as springs with a stiffness inversely proportional to their length.4, 28
In order to improve the mesh quality of high aspect ratio cells in case of large deformations and rotations,a torsional spring method has been introduced, in which the stiffness is a function of the angle between theedges.13, 14 Other methods are based on solving Laplacian and biharmonic elliptic operator equations.16, 20
These methods have in common that they use mesh connectivity information to construct a system ofequations of the size of the number of internal flow points ni to determine the deformed state of the mesh.Solving this ni×ni matrix problem repeatedly in dynamic fluid–structure interaction simulations of practicalrelevance is computationally intensive.
In contrast to mesh connectivity approaches, point–to–point methods determine the displacement of theinternal flow points based on their relative position with respect to the flow domain boundary only. One suchpoint–to–point method is the recently developed mesh deformation technique5, 17, 24, 34 based on radial basisfunction (RBF) interpolation.8, 31 This flexible method robustly handles large deformations and hangingnodes in arbitrary mesh topologies and it is easily implemented in parallel.
However, RBF mesh deformation still requires solving a system of equations of the size of the number ofboundary points nb to determine the mesh deformation. Although this nb × nb matrix problem is smallerthan for mesh connectivity methods, since the topology of the boundary is a dimension lower than that ofthe volume mesh, the matrix is often dense and ill-conditioned. Solving this matrix equation using a directGauss elimination results in the computational costs of O(n3
b) operations, which can still be a significantpart of the computational time of the fluid–structure interaction simulation. Other more efficient solutionstrategies and pre–conditioners are available, however, they add to the implementation complexity of themethod. After solving the system of equations, the second step is the evaluation of the interpolation functionin all mesh points, which is equivalent to O(ninb) operations.
It can be estimated that in large–scale three–dimensional applications the computational costs of themesh motion are dominated by the matrix solution step.6 In order to significantly reduce the computationalcosts of mesh deformation it is, therefore, necessary to focus on reducing the computational costs for solvingthe system of equations. One approach that has been used is to reduce the number of boundary nodes usedby the mesh deformation method by combining radial basis function mesh deformation with a greedy datareduction algorithm,25 however, this introduces an error tolerance in the boundary displacements. RBFmesh deformation has earlier also been used for displacing the block vertices of multi–block meshes, in whichthe structured blocks are updated using TFI mesh deformation.23, 29 A mesh deformation method based onDelaunay mapping has recently also been developed.19
In this paper an explicit mesh deformation method is developed based on Inverse Distance Weighting(IDW) interpolation, which does not require solving a system of equations for deforming the volume mesh.It results for general geometries automatically in a global parameterization that can handle translations androtations. The point–to–point interpolation technique has the flexibility to handle arbitrary mesh topologieswith hanging nodes and is robust in case of large deformations. In contrast to RBF interpolation it resultsin an algebraic expression for the internal flow point displacements as function of the boundary deformation.This explicit evaluation reduces the computational costs significantly and simplifies the implementation andparallelization of the mesh deformation routines.
IDW is a weighted average interpolation technique for multivariate interpolation of scattered data points.It is widely used for fitting a continuous surface through irregularly-spaced data in spatial objective analysisand the generation of contour maps in geography,3 meteorology,11 and hydrology.7 The interpolated value isan average of the known values at the data points weighted by the inverse of the distance to the unsampledpoint. For mesh deformation that means that the influence of a boundary node displacements on thedisplacement of an internal mesh point is inversely proportional to the distance between the two points. Thedistance–decay effect can be influenced by a power parameter c, which is usually set to a value of 2, whichgives satisfactory results and the simplest formulation.26
A number of more advanced IDW formulations have also been developed to improve on the limitationsof pure IDW interpolation. For example, spatially non–uniform power parameter distributions based ondata density have been used in adaptive IDW21 and through a Fourier integral representation.2 Otherapproaches use variate finite radius of compact support without the singularity at zero distance, whichrequires an iterative procedure to solve the interpolation problem.11 Due to the specific topology of themesh deformation interpolation problem, pure IDW interpolation with a power parameter is suitable inthis case. Other algebraic mesh deformation approaches are often not generally applicable to unstructuredmeshes.1 For example, methods based on nearest neighbor interpolation in combination with decay functionscan for not sufficiently small deformations result in irregular meshes.15, 23
In this paper IDW mesh deformation is introduced in Section II. A robust extremum conserving for-mulation is developed in Section III. In Section IV IDW mesh deformation is used as a mesh optimizationtool for a two–dimensional structured hexahedral mesh around the RAE2822 airfoil. Mesh motion based
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on IDW interpolation is applied to an aeroelastic three–dimensional fluid–structure interaction simulationof the AGARD 445.6 wing with an unstructured hexahedral mesh in Section V. In section VI the mainconclusions are summarized.
II. Inverse Distance Weighting mesh deformation
Inverse distance weighting interpolation26 is an explicit method for multivariate interpolation of scattereddata points. The interpolation surface w(x) through n data samples v = {v1, .., vn} of the exact functionu(x) with vi ≡ u(xi) is given in inverse distance weighting by
w(x) =
∑n
i=1 viφ(ri)∑n
i=1 φ(ri), (1)
with weighting functionφ(r) = r−c, (2)
where ri = ‖x−xi‖ ≥ 0 is the Euclidean distance between x and data point xi, and c is a power parameter.In the sampling points a the interpolation surface w(a) is not differentiable for low values of the powerparameter 0 ≤ c ≤ 1 and smooth for c > 1. In the mesh deformation algorithm separate shape parametersare introduced for dynamic and static boundary nodes, and for translations and rotations: cd,T, cd,R, cs,T,and cs,R.
III. Robust Extremum Conserving formulation
Since IDW is an Extremum Conserving (EC) interpolation method, an EC–IDW mesh deformationmethod can be constructed by applying IDW interpolation directly to the location of the boundary nodesand in the internal mesh points. This is, however, only applicable to translations and not to rotations. Amore practical EC–IDW formulation that can handle rotations can be derived as follows.
Consider a rectangular other boundary of the flow mesh. This can be a far field boundary in an externalflow problem, or a wall and inflow and outflow boundaries in an internal flow configuration. One canthen easily check whether each of the n contributions in (1) of the n boundary nodes separately to thedisplacement of an internal node, displaces the internal node outside the rectangular bounding box aroundthe flow mesh. If that is the case, then the corresponding term viφ(ri) in (1) is limited such that itsseparate effect is a displacement of the internal node exactly onto the outer mesh boundary. In this waythe internal mesh points can never cross the outer mesh boundary, which leads to an EC formulation ofIDW mesh deformation. Modifying the terms viφ(ri) is straightforward owing to the explicit nature of IDWinterpolation. The modification does also not affect meeting the known boundary displacement conditions.The limiting of the mesh point displacements can be used for contributions of both boundary displacementsand boundary rotations.
IV. Mesh optimization
In this section the property that IDW mesh deformation treats boundary rotations separately is used asa mesh optimization tool to improve the orthogonality of the cells adjacent to the surface. The orthogonalityof boundary layer cells close to solid surfaces is often of principal importance for the accurate resolution offlow boundary layers.
As a first step a simple single–block structured C–type inviscid Euler mesh with 12k nodes is generatedaround the RAE2822 airfoil using Gambit, see Figure 1. The second step is to compute the angle betweenthe airfoil surface and the mesh edges emanating from the surface. These angles β should be 90o for perfectorthogonality. The boundary nodes are then rotated over an angle α = 90o − β to rotate the first celllayer to an orthogonal location with respect to the airfoil surface. These boundary rotations are finallyextrapolated into the flow mesh using the ability of IDW mesh deformation to separately treat boundaryrotations independent of boundary translations, which are absent in this case.
The results for IDW mesh optimization are shown in Figure 2 in terms of angle β as function of thecurvilinear abscissa s/c along the airfoil surface, where c stands for the airfoil chord length. This should notbe confused with the IDW power parameter c, for which the values c = {1, 2, 3, 4} are considered. The dashed
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Figure 1. Original single–block structured C–type around the RAE2822 airfoil.
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line shows that the mesh quality of the originally generated mesh leaves to be desired with a minimum ofβ = 65o.
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Figure 2. Local mesh orthogonality β for the optimized single–block structured C–type around the RAE2822
airfoil.
For power parameter c = 1 the β–angles further reduce, however, for increasing c the mesh qualityimproves spectacularly. The mesh quality measure shows a perfect value of β = 90o over more than 95% ofthe airfoil surface for c = 3. The minimal local mesh quality of β = 85o can be found at the leading andtrailing edge. The local minimum at the trailing edge is caused by the application of the Kutta condition forthe mesh rotation at the trailing edge. The β = 85o mesh quality at the leading edge is caused by the finiteand relatively large height of the first cell layer of the Euler mesh. Further increasing c does not change themesh quality significantly.
Results for the global mesh quality metric of the average percentage β/90o over all surface nodes aregiven in Table 1. This metric gives for the generated original mesh a quality of 88.98%. Using IDW meshoptimization improves the mesh quality to a virtually perfect 99.51% for c = 6.
In Figure 3 the corresponding mesh after IDW mesh optimization for c = 6 is shown. The mesh showsvisually a perfect orthogonality of the first cell layer. The mesh lines downstream of the airfoil also followsthe trailing edge camberline angle, which can improve the resolution of the flow wake important for dragprediction in a viscous computation. These results demonstrate that IDW mesh optimization enables theuse of relatively simple mesh generation tools, since the significant improvement of the mesh quality inthe optimization step results in sufficient mesh quality for accurate predictive computations. The easyimplementation and parallelization of the fast explicit IDW interpolation forms also an advantage over otheravailable mesh optimization techniques. The effect on the global mesh quality of the whole internal meshwill be considered in future work.
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Table 1. Global mesh quality metric for the optimized single–block structured C–type around the RAE2822
airfoil.
c mesh quality
original 88.98%
1 37.67%
2 91.80%
3 98.82%
4 99.40%
5 99.48%
6 99.51%
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Figure 3. Optimized single–block structured C–type around the RAE2822 airfoil for c = 6.
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V. Mesh motion for the three-dimensional AGARD 445.6 aeroelastic wing
The AGARD aeroelastic wing 445.6 configuration number 3 known as the weakened model33 is consideredhere with a NACA 65A004 symmetric airfoil, taper ratio of 0.66, 45o quarter-chord sweep angle, and a 2.5-foot semi-span subject to an inviscid flow. The structure is described by a nodal discretization using anundamped linear finite element model in the Matlab finite element toolbox OpenFEM. The discretizationcontains in the chordal and spanwise direction 6×6 brick-elements with 20 nodes and 60 degrees-of-freedom,and at the leading and trailing edge 2× 6 pentahedral elements with 15 nodes and 45 degrees-of-freedom.34
Orthotropic material properties are used and the fiber orientation is taken parallel to the quarter-chord line.The Euler equations for inviscid flow10 are solved using a second-order central finite volume discretization
on a 60 × 15 × 30m domain using an unstructured hexahedral mesh. The free stream conditions are for thedensity ρ∞ = 0.099468kg/m3 and the pressure p∞ = 7704.05Pa.33 Time integration is performed usinga third-order implicit multi-stage Runge-Kutta scheme with step size ∆t = 2.5 · 10−3s until t = 1.25s todetermine the stochastic solution until t = 1s. The first bending mode with a vertical tip displacement ofytip = 0.01m is used as initial condition for the structure.
The coupled fluid-structure interaction system is solved using a partitioned IMEX scheme with explicittreatment of the coupling terms without sub-iterations. An Arbitrary Lagrangian-Eulerian formulation isemployed to couple the fluid mesh with the movement of the structure. The flow forces and the structuraldisplacements are imposed on the structure and the flow using nearest neighbor and radial basis functioninterpolation,34 respectively.
The results for IDW and RBF mesh deformation are compared from three different meshes with increasingnumber of cells shown in Figure 4. In addition a formulation IDWnorot is considered which does not takeinto account the effect of rotations of the mesh boundary normal vector rotation, since rotations are expectedto be small for this problem. This is a possible approach to reduce the computational costs of IDW meshdeformation further, since IDW treats boundary node displacements and rotations separately. The RBFalgorithm handles them combined by implicitly interpreting relative translations as rotations.
(a) 15122 cells (b) 31274 cells (c) 75656 cells
Figure 4. Three surface meshes for the three-dimensional AGARD 445.6 wing aeroelastic benchmark.
The response output functional that is considered in this case is the lift force L(t) as function of time.The time series for L(t) of the three mesh deformation algorithms closely agree as shown in Figure 5. Themaximum relative error in the lift amplitude AL with respect to the IDW solution is smaller than 3 · 10−2,see Table 2.
Table 2. Relative error in lift force amplitude AL with respect to the IDW solution as function of the number
of cells for the three-dimensional AGARD 445.6 wing aeroelastic benchmark.
15122 cells 31274 cells 75656 cells
RBF 5.90 · 10−3 2.08 · 10−3 9.57 · 10−5
IDWnorot 2.95 · 10−2 1.21 · 10−2 1.24 · 10−2
The computational costs per time step are in this case the sum of the CPU times for the three stages pertime step of the multi-stage Runge-Kutta time integration scheme employed here. The mesh deformationalgorithms result in three dimensions in higher computational costs due to the additional translation and
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Figure 5. Lift force L(t) of the mesh with 75656 cells for the three-dimensional AGARD 445.6 wing aeroelastic
benchmark.
rotation dimensions as can be concluded from Figure 6 and Table 3. The RBF method results again ina fast increase of computational costs with increasing mesh size up to 2544 seconds per time step for thefinest mesh with 75656 cells, which constitutes an impractical contribution to the total CPU time for thefluid-structure interaction simulation of approximately 90%. The reduction of the CPU time by IDW witha factor 20 demonstrates that the efficiency gain of the explicit IDW algorithm increases with dimensioncompared to RBF mesh deformation. Neglecting the contributions of rotations reduces the computationalcosts for IDWnorot further to 39.7 seconds which corresponds to a factor 50 decrease with respect to theRBF method.
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Figure 6. Average computational time per time step as function of the number of cells for the three-dimensional
AGARD 445.6 wing aeroelastic benchmark.
However, not taking into account rotations results in a mean mesh quality18 fmean of 95% to 96% asshown in Figure 7. The mesh quality metric increases with time for this case due to the decaying oscillationamplitude illustrated by the lift force L(t) in Figure 5. RBF mesh deformation gives here a mean mesh qualityof more than 99%, which is 2% higher than the quality for IDW of 97%. This difference in mesh qualitywould in general not justify the significant additional computational costs for the RBF algorithm presentedin Table 3. Moreover the mesh qualities of the three methods are not notably reflected in the aeroelasticsimulation results in terms of the lift force amplitude AL as illustrated in Table 2. An implementationof the IDWeig formulation based on the structural eigenmodes of the AGARD 445.6 wing is a directionto further reduced the computational costs possibly in combination with neglecting the effect of boundarynormal vector rotations in IDWnorot.
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Table 3. Average computational time per time step in seconds as function of the number of cells for the
Figure 7. Mesh quality fmean(t) of the mesh with 75656 cells for the three-dimensional AGARD 445.6 wing
aeroelastic benchmark.
VI. Conclusions
An explicit mesh deformation method is presented based on Inverse Distance Weighting (IDW) interpo-lation of the boundary node displacements. The point-by-point approach results in a significant reductionof computational costs, since the proposed mesh deformation algorithm does not involve the solution ofa matrix system of equations. This enables an easy implementation and parallelization of the IDW meshdeformation routine. The method is extended to a robust Extremum Conserving (EC) formulation.
The property that IDW mesh deformation treats boundary rotations separately is used for mesh opti-mization to improve the orthogonality of the cells adjacent to the surface. IDW mesh optimization withc = 6 improves the mesh quality a simple single–block structured C–type inviscid Euler mesh with 12k nodesaround the RAE2822 airfoil from 88.98% to a virtually perfect value of 99.51%. These results demonstratethat IDW mesh optimization enables the use of relatively simple mesh generation tools, while still resultingin sufficient mesh quality for accurate predictive computations after the optimization step.
The mesh motion results for the three–dimensional fluid–structure interaction simulation of the AGARD445.6 aeroelastic wing with an unstructured hexahedral mesh demonstrate a reduction of computationalcosts for IDW mesh deformation with respect to the RBF method of a factor 20. To additional formulationof neglecting boundary normal vector rotations in IDWnorot further reduce the CPU time to a factor 50with respect to RBF mesh deformation. The 4% higher mesh quality of the RBF algorithm is not reflectedin the simulation results for the amplitude of the lift force L, for which the mesh deformation methods agreeup to a relative error of 3 · 10−2.
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