48th AIAA Aerospace Sciences Meeting Including the New Horizons Forum and Aerospace Exposition 4 - 7 January 2010, Orlando, Florida

AIAA 2010-165

Explicit and Robust Inverse Distance Weighting Mesh Deformation for CFD Jeroen A.S. Witteveen∗ Center for Turbulence Research, Stanford University, Building 500, Stanford, CA 94305–3035, USA Mesh deformation algorithms usually require solving a system of equations which can be computationally intensive for complex three-dimensional applications. In this paper an explicit mesh deformation method is proposed based on Inverse Distance Weighting (IDW) interpolation of the boundary node displacements to the interior of the flow domain. The point-by-point mesh deformation algorithm results in a straightforward implementation and parallelization. The formulation is extended to a robust Extremum Conserving (ED) mesh deformation method. An IDW mesh optimization method is also introduced based on IDW mesh deformation, which reaches virtually perfect mesh orthogonality. It is also applied to mesh motion in the fluid-structure interaction simulation of the three-dimensional AGARD 445.6 aeroelastic wing, in which IDW mesh motion shows a reduction of the computational costs up to a factor 50 with respect to Radial Basis Function (RBF) mesh deformation for a comparable simulation accuracy.

I.

Introduction

an Arbitrary Lagrangian–Eulerian (ALE) formulation of a dynamic fluid–structure interaction problem Iinnthedeformation flow and the structure forces and displacements are coupled at the interface. The flow forces result and displacement of the structure, which in turn leads to moving boundaries for the flow domain. In order to interpolate the boundary displacements to the interior of the flow mesh an automatic mesh deformation method is required. The mesh deformation algorithm needs to be robust in case of large deformations and result in sufficiently high grid quality after deformations for arbitrary mesh topologies. In dynamic 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 coupling or 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 the fluid–structure interaction simulation. Mesh deformation is also used in static aeroelastic computations, aerodynamic shape optimization, and treating geometrical uncertainties.32 In these applications an easy implementation and parallelization of the mesh deformation algorithm is of interest. One of the available mesh deformation techniques for continuous surface deformations is the Transfinite Interpolation (TFI) method.12, 30 This is a fast method that interpolates the boundary displacement to interior points along grid lines, which makes it only applicable to single–block structured meshes. For unstructured meshes often methods are used that model the motion of the flow mesh by equations governing structural deformation.22, 27 For example, equilibrium equations for linear body elasticity have been used with 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 which the 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 the edges.13, 14 Other methods are based on solving Laplacian and biharmonic elliptic operator equations.16, 20 ∗ Postdoctoral

Fellow, Member AIAA, Phone: +1 650 723 9601, Fax: +1 650 723 9617, [email protected].

1 of 10 American Institute of Aeronautics and Astronautics Copyright © 2010 by J.A.S. Witteveen. Published by the American Institute of Aeronautics and Astronautics, Inc., with permission.

These methods have in common that they use mesh connectivity information to construct a system of equations 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 practical relevance is computationally intensive. In contrast to mesh connectivity approaches, point–to–point methods determine the displacement of the internal flow points based on their relative position with respect to the flow domain boundary only. One such point–to–point method is the recently developed mesh deformation technique5, 17, 24, 34 based on radial basis function (RBF) interpolation.8, 31 This flexible method robustly handles large deformations and hanging nodes 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 of boundary points nb to determine the mesh deformation. Although this nb × nb matrix problem is smaller than for mesh connectivity methods, since the topology of the boundary is a dimension lower than that of the volume mesh, the matrix is often dense and ill-conditioned. Solving this matrix equation using a direct Gauss elimination results in the computational costs of O(n3b ) operations, which can still be a significant part of the computational time of the fluid–structure interaction simulation. Other more efficient solution strategies and pre–conditioners are available, however, they add to the implementation complexity of the method. After solving the system of equations, the second step is the evaluation of the interpolation function in all mesh points, which is equivalent to O(ni nb ) operations. It can be estimated that in large–scale three–dimensional applications the computational costs of the mesh motion are dominated by the matrix solution step.6 In order to significantly reduce the computational costs of mesh deformation it is, therefore, necessary to focus on reducing the computational costs for solving the system of equations. One approach that has been used is to reduce the number of boundary nodes used by the mesh deformation method by combining radial basis function mesh deformation with a greedy data reduction algorithm,25 however, this introduces an error tolerance in the boundary displacements. RBF mesh deformation has earlier also been used for displacing the block vertices of multi–block meshes, in which the structured blocks are updated using TFI mesh deformation.23, 29 A mesh deformation method based on Delaunay 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 and rotations. The point–to–point interpolation technique has the flexibility to handle arbitrary mesh topologies with hanging nodes and is robust in case of large deformations. In contrast to RBF interpolation it results in 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 and parallelization 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 analysis and the generation of contour maps in geography,3 meteorology,11 and hydrology.7 The interpolated value is an average of the known values at the data points weighted by the inverse of the distance to the unsampled point. For mesh deformation that means that the influence of a boundary node displacements on the displacement of an internal mesh point is inversely proportional to the distance between the two points. The distance–decay effect can be influenced by a power parameter c, which is usually set to a value of 2, which gives satisfactory results and the simplest formulation.26 A number of more advanced IDW formulations have also been developed to improve on the limitations of pure IDW interpolation. For example, spatially non–uniform power parameter distributions based on data density have been used in adaptive IDW21 and through a Fourier integral representation.2 Other approaches use variate finite radius of compact support without the singularity at zero distance, which requires an iterative procedure to solve the interpolation problem.11 Due to the specific topology of the mesh deformation interpolation problem, pure IDW interpolation with a power parameter is suitable in this case. Other algebraic mesh deformation approaches are often not generally applicable to unstructured meshes.1 For example, methods based on nearest neighbor interpolation in combination with decay functions can 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 formulation is developed in Section III. In Section IV IDW mesh deformation is used as a mesh optimization tool 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 simulation of the AGARD 445.6 wing with an unstructured hexahedral mesh in Section V. In section VI the main conclusions are summarized.

II.

Inverse Distance Weighting mesh deformation

Inverse distance weighting interpolation26 is an explicit method for multivariate interpolation of scattered data points. The interpolation surface w(x) through n data samples v = {v1 , .., vn } of the exact function u(x) with vi ≡ u(xi ) is given in inverse distance weighting by Pn vi φ(ri ) w(x) = Pi=1 , (1) n i=1 φ(ri ) with weighting function φ(r) = r−c ,

(2)

where ri = kx − xi k ≥ 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 power parameter 0 ≤ c ≤ 1 and smooth for c > 1. In the mesh deformation algorithm separate shape parameters are 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 deformation method can be constructed by applying IDW interpolation directly to the location of the boundary nodes and in the internal mesh points. This is, however, only applicable to translations and not to rotations. A more 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 external flow problem, or a wall and inflow and outflow boundaries in an internal flow configuration. One can then easily check whether each of the n contributions in (1) of the n boundary nodes separately to the displacement of an internal node, displaces the internal node outside the rectangular bounding box around the flow mesh. If that is the case, then the corresponding term vi φ(ri ) in (1) is limited such that its separate effect is a displacement of the internal node exactly onto the outer mesh boundary. In this way the internal mesh points can never cross the outer mesh boundary, which leads to an EC formulation of IDW mesh deformation. Modifying the terms vi φ(ri ) is straightforward owing to the explicit nature of IDW interpolation. 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 displacements and boundary rotations.

IV.

Mesh optimization

In this section the property that IDW mesh deformation treats boundary rotations separately is used as a mesh optimization tool to improve the orthogonality of the cells adjacent to the surface. The orthogonality of boundary layer cells close to solid surfaces is often of principal importance for the accurate resolution of flow boundary layers. As a first step a simple single–block structured C–type inviscid Euler mesh with 12k nodes is generated around the RAE2822 airfoil using Gambit, see Figure 1. The second step is to compute the angle between the airfoil surface and the mesh edges emanating from the surface. These angles β should be 90o for perfect orthogonality. The boundary nodes are then rotated over an angle α = 90o − β to rotate the first cell layer to an orthogonal location with respect to the airfoil surface. These boundary rotations are finally extrapolated into the flow mesh using the ability of IDW mesh deformation to separately treat boundary rotations 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 the curvilinear abscissa s/c along the airfoil surface, where c stands for the airfoil chord length. This should not be confused with the IDW power parameter c, for which the values c = {1, 2, 3, 4} are considered. The dashed 3 of 10 American Institute of Aeronautics and Astronautics

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For power parameter c = 1 the β–angles further reduce, however, for increasing c the mesh quality improves spectacularly. The mesh quality measure shows a perfect value of β = 90o over more than 95% of the airfoil surface for c = 3. The minimal local mesh quality of β = 85o can be found at the leading and trailing edge. The local minimum at the trailing edge is caused by the application of the Kutta condition for the mesh rotation at the trailing edge. The β = 85o mesh quality at the leading edge is caused by the finite and relatively large height of the first cell layer of the Euler mesh. Further increasing c does not change the mesh quality significantly. Results for the global mesh quality metric of the average percentage β/90o over all surface nodes are given in Table 1. This metric gives for the generated original mesh a quality of 88.98%. Using IDW mesh optimization 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 shows visually a perfect orthogonality of the first cell layer. The mesh lines downstream of the airfoil also follows the trailing edge camberline angle, which can improve the resolution of the flow wake important for drag prediction in a viscous computation. These results demonstrate that IDW mesh optimization enables the use of relatively simple mesh generation tools, since the significant improvement of the mesh quality in the optimization step results in sufficient mesh quality for accurate predictive computations. The easy implementation and parallelization of the fast explicit IDW interpolation forms also an advantage over other available mesh optimization techniques. The effect on the global mesh quality of the whole internal mesh will 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.

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mesh quality 88.98% 37.67% 91.80% 98.82% 99.40% 99.48% 99.51%

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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 considered here with a NACA 65A004 symmetric airfoil, taper ratio of 0.66, 45o quarter-chord sweep angle, and a 2.5foot semi-span subject to an inviscid flow. The structure is described by a nodal discretization using an undamped linear finite element model in the Matlab finite element toolbox OpenFEM. The discretization contains 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 the density ρ∞ = 0.099468kg/m3 and the pressure p∞ = 7704.05Pa.33 Time integration is performed using a third-order implicit multi-stage Runge-Kutta scheme with step size ∆t = 2.5 · 10−3 s until t = 1.25s to determine the stochastic solution until t = 1s. The first bending mode with a vertical tip displacement of ytip = 0.01m is used as initial condition for the structure. The coupled fluid-structure interaction system is solved using a partitioned IMEX scheme with explicit treatment of the coupling terms without sub-iterations. An Arbitrary Lagrangian-Eulerian formulation is employed to couple the fluid mesh with the movement of the structure. The flow forces and the structural displacements are imposed on the structure and the flow using nearest neighbor and radial basis function interpolation,34 respectively. The results for IDW and RBF mesh deformation are compared from three different meshes with increasing number of cells shown in Figure 4. In addition a formulation IDWnorot is considered which does not take into account the effect of rotations of the mesh boundary normal vector rotation, since rotations are expected to be small for this problem. This is a possible approach to reduce the computational costs of IDW mesh deformation further, since IDW treats boundary node displacements and rotations separately. The RBF algorithm handles them combined by implicitly interpreting relative translations as rotations.

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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. The maximum 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.

RBF IDWnorot

15122 cells 5.90 · 10−3 2.95 · 10−2

31274 cells 2.08 · 10−3 1.21 · 10−2

75656 cells 9.57 · 10−5 1.24 · 10−2

The computational costs per time step are in this case the sum of the CPU times for the three stages per time step of the multi-stage Runge-Kutta time integration scheme employed here. The mesh deformation algorithms result in three dimensions in higher computational costs due to the additional translation and

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rotation dimensions as can be concluded from Figure 6 and Table 3. The RBF method results again in a fast increase of computational costs with increasing mesh size up to 2544 seconds per time step for the finest mesh with 75656 cells, which constitutes an impractical contribution to the total CPU time for the fluid-structure interaction simulation of approximately 90%. The reduction of the CPU time by IDW with a factor 20 demonstrates that the efficiency gain of the explicit IDW algorithm increases with dimension compared to RBF mesh deformation. Neglecting the contributions of rotations reduces the computational costs for IDWnorot further to 39.7 seconds which corresponds to a factor 50 decrease with respect to the RBF method. 3000

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However, not taking into account rotations results in a mean mesh quality18 fmean of 95% to 96% as shown in Figure 7. The mesh quality metric increases with time for this case due to the decaying oscillation amplitude illustrated by the lift force L(t) in Figure 5. RBF mesh deformation gives here a mean mesh quality of more than 99%, which is 2% higher than the quality for IDW of 97%. This difference in mesh quality would in general not justify the significant additional computational costs for the RBF algorithm presented in Table 3. Moreover the mesh qualities of the three methods are not notably reflected in the aeroelastic simulation results in terms of the lift force amplitude AL as illustrated in Table 2. An implementation of the IDWeig formulation based on the structural eigenmodes of the AGARD 445.6 wing is a direction to further reduced the computational costs possibly in combination with neglecting the effect of boundary normal vector rotations in IDWnorot. 8 of 10 American Institute of Aeronautics and Astronautics

Table 3. Average computational time per time step in seconds as function of the number of cells for the three-dimensional AGARD 445.6 wing aeroelastic benchmark.

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VI.

Conclusions

An explicit mesh deformation method is presented based on Inverse Distance Weighting (IDW) interpolation of the boundary node displacements. The point-by-point approach results in a significant reduction of computational costs, since the proposed mesh deformation algorithm does not involve the solution of a matrix system of equations. This enables an easy implementation and parallelization of the IDW mesh deformation 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 optimization to improve the orthogonality of the cells adjacent to the surface. IDW mesh optimization with c = 6 improves the mesh quality a simple single–block structured C–type inviscid Euler mesh with 12k nodes around the RAE2822 airfoil from 88.98% to a virtually perfect value of 99.51%. These results demonstrate that IDW mesh optimization enables the use of relatively simple mesh generation tools, while still resulting in 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 AGARD 445.6 aeroelastic wing with an unstructured hexahedral mesh demonstrate a reduction of computational costs for IDW mesh deformation with respect to the RBF method of a factor 20. To additional formulation of neglecting boundary normal vector rotations in IDWnorot further reduce the CPU time to a factor 50 with respect to RBF mesh deformation. The 4% higher mesh quality of the RBF algorithm is not reflected in the simulation results for the amplitude of the lift force L, for which the mesh deformation methods agree up to a relative error of 3 · 10−2 .

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