Explicit Mesh Deformation Using Inverse Distance Weighting Interpolation

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1 19th AIAA Computational Fluid Dynamics June 2009, San Antonio, Texas AIAA Explicit Mesh Deformation Using Inverse Distance Weighting Interpolation Jeroen A.S. Witteveen, Hester Bijl Faculty of Aerospace Engineering, Delft University of Technology, Kluyverweg 1, 2629HS Delft, The Netherlands Mesh deformation algorithms usually require solving a system of equations which can be computationally intensive for complex three-dimensional applications. Here 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. Fluid-structure interaction applications to a two-dimensional airfoil and the three-dimensional AGARD wing benchmark show a reduction of the computational costs up to a factor 100 with respect to Radial Basis Function (RBF) mesh deformation for a comparable simulation accuracy. I. Introduction In fluid-structure interaction simulations the dynamics of the structure and the flow are coupled by forces and displacements on their interface. Flow forces result through deformations of the structure in moving boundaries for the flow domain. There is, therefore, a need for accurate and efficient mesh motion algorithms to propagate these boundary displacements to the mesh in the interior of the flow domain. For structured meshes the efficient Transfinite Interpolation method is available. Methods for the deformation of unstructured meshes often use the connectivity of the flow mesh in algorithms based on spring analogy, body elasticity, or Laplacian and Biharmonic operators. These methods can be computationally intensive, since they result in solving a system of equations of the size of the number of flow points. Point-by-point methods do not rely on connectivity information as they determine the displacement of each point in the flow mesh based on its relative position with respect to the domain boundary. Recently, a point-by-point mesh deformation method was developed based on Radial Basis Function (RBF) interpolation. 1,6 The resulting flexible and robust mesh motion algorithm can deal with large deformations, hanging nodes, and is easily implemented in parallel. However, radial basis function mesh motion requires the solution of a system of equations of the size of the number of boundary nodes. Solving this system can still be expensive as it accounts for a significant part of the computational time in large-scale three-dimensional simulations. In this paper, we present an explicit mesh motion algorithm based on Inverse Distance Weighting (IDW) interpolation, 7 which does not lead to solving a system of equations. The proposed point-by-point method maintains the robustness and flexibility for dealing with large deformations, hanging nodes, and parallelization of the previous method. In addition, the explicit formulation results in a faster mesh motion algorithm and an easier implementation. The mesh deformation method is studied for unstructured triangular and hexahedral meshes in a NACA0012 airfoil fluid-structure interaction to optimize the shape parameter value c. Subsequently IDW mesh deformation is applied to fluid-structure interaction simulation for a twodimensional airfoil and the three-dimensional AGARD wing benchmark. The results show an decrease of computational costs up to a factor 100 with respect to the RBF method, while maintaining comparable simulation result accuracy. Postdoctoral researcher, Member AIAA, +31(0) , j.a.s.witteveen@tudelft.nl, Full Professor, Member AIAA, +31(0) of 10 Copyright 2009 by J.A.S. Witteveen, H. Bijl. Published by the American American Institute Institute of of Aeronautics Aeronautics and Astronautics, and Astronautics Inc., with permission.

II. Inverse distance weighting interpolation Inverse distance weighting interpolation 7 is an explicit method for multivariate interpolation of scattered data points.

2 II. Inverse distance weighting interpolation Inverse distance weighting interpolation 7 is an explicit method for multivariate interpolation of scattered data points. The interpolation surface w(x) through n data samples v = {v 1,.., v n } of the exact function u(x) with v i u(x i ) is given in inverse distance weighting by n i=1 w(x) = v iφ(r i ) n i=1 φ(r i), (1) with weighting function φ(r) = r c, (2) where r i = x x i 0 is the Euclidian distance between x and data point x i, and c is a power parameter. In the mesh deformation algorithm separate shape parameters are introduced for dynamic and static boundary nodes, and for translations and rotations: c d,t, c d,r, c s,t, and c s,r. III. Triangular grid around a NACA0012 airfoil First the mesh deformation of an unstructured triangular mesh around a NACA0012 airfoil in a square domain of size 10c 10c is considered with c the airfoil chord, as shown in Figure 1. The mesh consists of 1524 cells with 112 nodes on the airfoil, 24 nodes on the outer boundary, and 694 internal nodes. The mesh is subject to a given displacement of the airfoil consisting of a translation in both directions of x = y = 2.5c and a rotation over α = 60deg in one step from the equilibrium position of the initial mesh. The mesh quality of the resulting mesh is shown in Figure 2. A combination of a size and skew measure between 0 and 1 is used as measure of the mesh quality. 4 The proposed explicit mesh deformation method results in a mesh of good quality without degenerate cells even for this extreme displacement of the airfoil in one step. Especially near the airfoil the grid is practically undistorted with a mesh quality close to 1, see Figure 2b. This is important for a robust deformation of the boundary layer cells and an accurate resolution of the aerodynamic forces on the airfoil. The overall quality of the mesh is in this case This mesh quality is obtained by optimizing the shape parameter values, which results in this case in c d,t = 2.2, c d,r = 2.2, c s,t = 2.0, and c s,r = 0.1. Similar optimizations for several other displacements point to general optimal integer values of the shape parameters of c d,t = c d,r = 2, c s,t = c s,r = 1. These values are used in the fluid-structure applications below. (a) Initial mesh (b) Initial mesh (zoom) Figure 1. Unstructured triangular mesh around a NACA0012 airfoil with 1524 cells. A. Two-dimensional airfoil fluid-structure interaction example The two-dimensional example is the post-flutter simulation of an elastically mounted rigid airfoil with nonlinear structural stiffness in an inviscid flow. The nonlinear structural stiffness is modeled by a cubic spring stiffness term in the following two-degree-of-freedom model for coupled pitch and plunge motion of the airfoil: 3 ( ω ) 2 ξ + x α α + (ξ + βξ U ξ 3 ) = 1 πµ C l(τ), (3) 2 of 10

(a) Mesh quality (b) Mesh quality (zoom) Figure 2. Mesh quality of the unstructured triangular mesh around a NACA0012 airfoil for a deflection of x = y = 2.5c and α = 60deg.

3 (a) Mesh quality (b) Mesh quality (zoom) Figure 2. Mesh quality of the unstructured triangular mesh around a NACA0012 airfoil for a deflection of x = y = 2.5c and α = 60deg. x α rα 2 ξ + α + 1 U 2 (α + β αα 3 ) = 2 πµrα 2 C m (τ), (4) where β ξ = 0m 2 and β α = 3rad 2 are nonlinear spring constants, ξ(τ) = h/b is the nondimensional plunge displacement of the elastic axis, α(τ) is the pitch angle, and ( ) denotes differentiation with respect to nondimensional time τ = Ut/b, with half-chord length b = c/2 = 0.5m and free stream velocity U = 103.6m/s, which corresponds to a Mach number of M = 0.3 for free stream density ρ = 0.12kg/m 3 and pressure p = Pa. The radius of gyration around the elastic axis is r α b = 0.25m, bifurcation parameter U is defined as U = U/(bω α ), and the airfoil-air mass ratio is µ = m/πρ b 2 = 100, with m the airfoil mass. The elastic axis is located at a distance a h b = 0.25m from the mid-chord position and the mass center is located at a distance x α b = 0.125m from the elastic axis. The ratio of natural frequencies ω(ω) = ω ξ /ω α = 0.2, with ω ξ and ω α the natural frequencies of the airfoil in pitch and plunge, respectively. 5 The nondimensional aerodynamic lift and moment coefficients, C l (τ) and C m (τ), for the NACA0012 airfoil are determined by solving the nonlinear Euler equations for inviscid flow 2 using a second-order finite volume scheme on an unstructured hexahedral mesh in spatial domain D with dimensions 30c 20c. An Arbitrary Lagrangian-Eulerian formulation is employed to couple the fluid mesh with the movement of the structure. Time integration is performed using the BDF-2 method with stepsize τ = 0.4 until τ max = Initially the airfoil is at rest at a deflection of α(0) = 0.1deg and ξ(0) = 0 from its equilibrium position. The initial condition of the flow field is given by the steady state solution for the initial deflection. In the numerical results we compare the results of Inverse Distance Weighting (IDW) and Radial Basis Function (RBF) mesh deformation on three different meshes with increasing number of cells. The meshes with 4198, 11112, and cells are shown in Figure 3. Euler flow simulations are used here in order to be able to use this large range of mesh sizes including the coarse mesh of 4198 cells. For the Inverse Distance Weighting mesh deformation two different implementations are used. The first formulation, IDW, is the straightforward implementation based on the boundary node displacements with respect to the previous time step, which is in general robust and accurate. In the second approach the boundary node displacements are written as function of the generalized structural eigenmode displacements. For this eigenmode formulation, IDWeig, the displacements are computed with respect to the equilibrium position by storing the significantly smaller interpolation matrix of size n dof n in, with n dof = 2 the number of eigenmodes and n in the number of internal mesh points. The results are compared with a Radial Basis Function mesh deformation reference based on the Thin Plate Splines (TPS) function deformation with respect to the previous time step, where the system of equations is solved directly. Other matrix solution methods are possible here, such that the CPU results should be interpret as a first comparison. The simulation results are compared in terms of the pitch angle α(τ), the computational time, and the mesh quality f opt (τ). The pitch angle α(τ) as function of nondimensional time is given in Figure 4 for the IDW, IDWeig, and RBF mesh deformation algorithms on the finest mesh with cells. The three time histories show a practically identical oscillatory increase of α(τ) until the system reaches a periodic limit cycle oscillation with constant amplitude. The quantitative differences in the prediction of the limit cycle oscillation amplitude with respect to the IDW results are given in Table 1 for the three mesh sizes. These 3 of 10

4 (a) 4198 cells (zoom) (b) cells (zoom) (c) cells (zoom) (d) cells Figure 3. Three meshes for the two-dimensional airfoil fluid-structure interaction example. 4 of 10

5 results confirm that the relative errors in the amplitude prediction are small with a maximum of RBF IDW IDWeig angle of attack α [deg] nondimensional time τ Figure 4. Pitch angle α(τ) of the mesh with cells for the two-dimensional airfoil fluid-structure interaction example. Table 1. Relative error in pitch angle amplitude A α with respect to the IDW solution as function of the number of cells for the two-dimensional airfoil fluid-structure interaction example cells cells cells RBF IDWeig In contrast, the CPU time results for the average computational costs for the mesh deformation per time step shows significant differences. The computational times for the three mesh deformation methods increase monotonically with the mesh size in Figure 5. The explicit IDW algorithm results, however, in a substantial slower increase of the computational costs with increasing number of mesh cells than RBF mesh deformation, since it does not involve solving a system of equations. Table 2 indicates that IDW reduces the computational work for mesh deformation up to a factor 10 compared to RBF for the mesh with cells. The IDWeig formulation based on the structural eigenmodes further reduces the costs to a factor 100 compared to RBF. A linear increase of the CPU time is actually achieved here for IDWeig, where the other algorithms show a quadratic growth rate. Table 2. Average computational time per time step in seconds as function of the number of cells for the two-dimensional airfoil fluid-structure interaction example cells cells cells RBF IDW IDWeig The differences in computational costs are reflected to some extend in the mesh qualities given in Figure 6. The global mesh quality metric f opt shows for the three methods an oscillatory behavior due to the oscillation of the airfoil around the equilibrium position. The mesh quality decreases up to τ = 2 due to the increasing amplitude in the transient part of the airfoil oscillation. For IDW and RBF the metric f opt further decreases for τ > 2 caused by the build up of errors due to the deformation of the mesh with respect to the previous time step. Since IDWeig is based on the deformation with respect to the equilibrium position, the mesh quality does not detoriate after the transient part with maxima close to f opt = 100%. Overall the RBF mesh deformation reaches a factor 2 higher mesh quality than IDW and IDWeig. This leads, however, not 5 of 10

6 25 20 RBF IDW IDWeig average CPU time [s] number of cells x 10 4 Figure 5. Average computational time per time step as function of the number of cells for the two-dimensional airfoil fluid-structure interaction example. in significantly more accurate fluid-structure interaction response predictions as discussed in relation with Figure 4 and Table 1. Also IDW and IDWeig mesh deformation result in high mesh quality of f opt > 96.5%. The IDW mesh quality can be improved by using optimized shape parameter values c for this specific problem instead of the used integer values to enable fast evaluation IDW global mesh quality f opt IDWeig global mesh quality f opt RBF global mesh quality f opt nondimensional time τ (a) IDW nondimensional time τ (b) IDWeig nondimensional time τ (c) RBF Figure 6. Mesh quality f opt(τ) of the mesh with cells for the two-dimensional airfoil fluid-structure interaction example. B. Three-dimensional AGARD wing aeroelastic benchmark The AGARD aeroelastic wing configuration number 3 known as the weakened model 8 is considered here with a NACA 65A004 symmetric airfoil, taper ratio of 0.66, 45 o 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 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. 9 Orthotropic material properties are used and the fiber orientation is taken parallel to the quarter-chord line. The Euler equations for inviscid flow 2 are solved using a second-order central finite volume discretization on a m domain using an unstructured hexahedral mesh. The free stream conditions are for the density ρ = kg/m 3 and the pressure p = Pa. 8 Time integration is performed using a third-order implicit multi-stage Runge-Kutta scheme with step size t = s until t = 1.25s to determine the stochastic solution until t = 1s. The first bending mode with a vertical tip displacement of y tip = 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 6 of 10

7 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, 9 respectively. This three-dimensional test problem is studied in analogy to the analysis for the two-dimensional airfoil fluid-structure interaction application. The results for IDW and RBF mesh deformation are compared from three different meshes with increasing number of cells shown in Figure 7. 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. (a) cells (b) cells (c) cells Figure 7. Three surface meshes for the three-dimensional AGARD 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 8. The maximum relative error in the lift amplitude A L with respect to the IDW solution is smaller than , see Table RBF IDW IDWnorot 100 lift force [N] time t Figure 8. Lift force L(t) of the mesh with cells for the three-dimensional AGARD wing aeroelastic benchmark. 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 rotation dimensions as can be concluded from Figure 9 and Table 4. 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 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 7 of 10

8 Table 3. Relative error in lift force amplitude A L with respect to the IDW solution as function of the number of cells for the three-dimensional AGARD wing aeroelastic benchmark cells cells cells RBF IDWnorot a factor 20 demonstrates that the efficiency gain of the explicit IDW algorithm increases with dimension compared to RBF mesh deformation. Neglecting the constributions 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 RBF IDW IDWnorot average CPU time [s] number of cells x 10 4 Figure 9. Average computational time per time step as function of the number of cells for the three-dimensional AGARD wing aeroelastic benchmark. Table 4. Average computational time per time step in seconds as function of the number of cells for the three-dimensional AGARD wing aeroelastic benchmark cells cells cells RBF IDW IDWnorot However, not taking into account rotations results in a mean mesh quality f mean of 95% to 96% as shown in Figure 10. 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 8. RBF mesh deformation gives here a mean mesh quality of more than 99%, which is a factor 3 larger 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 4. Moreover the mesh qualities of the three methods are not notably reflected in the aeroelastic simulation results in terms of the lift force amplitude A L as illustrated in Table 3. An implementation of the IDWeig formulation based on the structural eigenmodes of the AGARD 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

9 1 mean global mesh quality f mean RBF IDW IDWnorot time t (a) IDW Figure 10. Mesh quality f mean(t) of the mesh with cells for the three-dimensional AGARD wing aeroelastic benchmark. IV. 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. For an unstructured triangular mesh around a NACA0012 airfoil the optimal integer values of the shape parameter values c d,t = c d,r = 2, c s,t = c s,r = 1 are established. These shape parameter values are subsequently implemented in a fluid-structure interaction solver applied to a two-dimensional airfoil and the three-dimensional AGARD wing benchmark. The results demonstrate a reduction of computational costs for IDW mesh deformation with respect to the RBF method of a factor 10 and 20, respectively. To addional formulations of IDWeig based on the structural eigenmodes and neglecting boundary normal vector rotations in IDWnorot further reduce the CPU time to a factor 100 and 50 with respect to RBF mesh deformation. The factor 2 5 higher mesh quality of the RBF algorithm are not reflected in the simulation results for the amplitudes of the pitch angle α and lift force L, for which the mesh deformation methods agree up to a relative error of In future work the mesh quality of IDW mesh deformation can be improved by optimizing the shape parameter values c for each case separately. An implementation of the IDWeig formulation for the AGARD wing can further reduce the computational costs in the three-dimensional problem. References 1 A. de Boer, M.S. van der Schoot, H. Bijl. Mesh deformation based on radial basis function interpolation, Comput. Struct. 85 (2007) A.J. Chorin, J.E. Marsden, A mathematical introduction to fluid mechanics, Springer-Verlag, New York (1979). 3 Y. Fung, An introduction to aeroelasticity, Dover Publications, New York (1969). 4 P.M. Knupp, Algebraic mesh quality metrics for unstructured initial meshes, Finite Elem. Anal. Des. 39 (2003) B.H.K. Lee, L.Y. Jiang, Y.S. Wong, Flutter of an airfoil with a cubic nonlinear restoring force, AIAA , 39th AIAA/ASME/ASCE/AHS/ASC Structures, Structural Dynamics, and Materials Conference, Long Beach, California (1998). 6 T.C.S. Rendall, C. Allen, Unified CFD-CSD interpolation and mesh motion using radial basis functions, Int. J. Numer. Meth. Eng. 74 (2008) D. Shepard. A two-dimensional interpolation function for irregularly-spaced data, Proceedings of the 1968 ACM National Conference (1968) E. Yates Jr., AGARD standard aeroelastic configurations for dynamic response. Candidate configuration I.-Wing 445.6, Technical Memorandum , NASA, of 10

10 9 A.H. van Zuijlen, A. de Boer, H. Bijl, Higher-order time integration through smooth mesh deformation for 3D fluidstructure interaction simulations, J. Comput. Phys. 224 (2007) of 10

39th AIAA Aerospace Sciences Meeting and Exhibit January 8 11, 2001/Reno, NV

39th AIAA Aerospace Sciences Meeting and Exhibit January 8 11, 2001/Reno, NV AIAA 1 717 Static Aero-elastic Computation with a Coupled CFD and CSD Method J. Cai, F. Liu Department of Mechanical and Aerospace Engineering University of California, Irvine, CA 92697-3975 H.M. Tsai,