Category Management and Minimisation of Uncertainties and Errors in Numerical Aerodynamics

The parameter variations presented for the GSB method were repeated with the MLS method. In the top left image of Fig. 13 the RMS deviations are shown in dependence of the number of support midpoints NM. As before the reference de­formed wetted surface is provided by the FIE method with default parameter set­tings. In the MLS method, during selection of support points an edge midpoint is always considered jointly with its surrounding additional support points. Hence the total number of support points of a given surface point is always a fixed multiple of the supporting edge midpoints Ng = (nFB + 1) x NM, at least outside of intersec­tion regions. For two edge midpoints the MLS projection with quadratic interpola­tion functions fails; the resulting deformed wetted surface is completely distorted. Even with the numerically more robust formulation (15) the MLS method requires a minimum number of support points which should be at least twice the number of monomials Q. With four or more supporting edge midpoints the MLS method results in only small deviations relative to the FIE method. Neither with linear nor with quadratic interpolation functions does the deviation exceed 10~4. In the inset of the top left diagram, a typical distribution of the normalised deflection differences АиУ/иУгtip| is given. These are present all over the wing. The finite support radius of the MLS method slightly smears the deformation during projection as compared to the FIE method’s. This represents a systematic discrepancy largely independent of the choice of MLS parameters. There are minor differences between the res­ulting wetted surface, though, as can be seen from the top right diagram. Here, the deformed wetted surface obtained with MLS for NM = 8and rFB = 0.15 m is the ref­erence. The different approximation orders of the interpolation functions are weakly reflected in the RMS values, as is the number of support points.

In the bottom images of Fig. 13 the influence of the radii of the additional sup­port points is examined. The bottom left diagram underscores the systematic dis­crepancy between FIE and MLS which goes completely unaffected by the choice of rFB. Again the smeared projection of deformations with MLS is apparent in the inset. The largest deviations occur in the region of the beam kink, where the shape of the surface deformed with FIE also depends on the choice of projection para­meters (see cases 0 to ® in Fig. 11). In the bottom right diagram of Fig. 13, the deformed wetted surface obtained with MLS and NM = 8 and rFB = 0.15 mis again the reference. For all contemplated parameter settings the deviations are very small. The systematic discrepancy between the results for linear and quadratic interpola­tion functions is due to the single peak at the leading edge wing root visible in the

inset of the bottom right image. This peak can be traced to the lack of additional support points in its vicinity, as can be identified in Fig. 5.

In the top left image of Fig. 12, the RMS deviations in the flap-wise bending deflec­tion are plotted over the minimum number of supporting edge midpoints NM, min. The radius of the additional support points was fixed at rFB = 0.15 m. In the GSB method the actual number of support points differs all over the wetted surface, as was explained on page 193. The support points of a given surface point are all the edge midpoints and additional support points within the support radius 5. The de­formed wetted surface used here as a reference was obtained with the FIE method and its default parameter settings. The deviations do not seem to be influenced by the number of support points. The RMS values hover at 1.7% for linear global inter­polation functions and at 0.12% for quadratic ones. Yet the wetted surfaces obtained with MLS are not completely identical, as is documented in the bottom left diagram. It shows the RMS deviation relative to the deformed surface resulting from the GSB method with NM, min = 8. These deviations, however, are at least one order of mag­nitude smaller than those relative to the FIE reference case.

The top right image of Fig. 12 shows the influence of the radius of the additional support points. Eight fixed radii are investigated as well as the alignment of the additional support points with the wetted surface. The number of supporting edge midpoints is NM, min = 8; the reference wetted surface as before resulted from the FIE method. The better the spatial arrangement of the additional support points ap­proximates the wetted surface, the lower the average deviations come to be. For both linear and quadratic global contributions to the interpolation function the smallest RMS values are achieved with an alignment of the additional support points. Next best is rFB = 0.15 m, which is approximately half the mean chord length. Between rFB = 0.276 m and rFB = 0.474 m a marked increase in deviations occurs. This is be­cause rFB becomes larger than the fixed support radius 5, and the additional support points cease to come into play. Case 0 in the bottom right image shows the distri­bution of the normalised deviations. These increase over the length of the span and values of Auy/uytip| > 2 are reached at the tip. Cases 0 and 0 highlight a general

problem of the GSB method: The deformation distribution is generally not approx­imated well by the global polynomial contribution to the interpolation function, not even by a quadratic one. This is to be compensated by the local RBF contributions of the edge midpoints and of the additional support points. The deviations become large in regions of the wetted surface far-off the support points, for instance at the leading and trailing edges. Consequently, case © with alignment of the additional support points with the wetted surface reveals significant improvement over cases with fixed values of rFB.

Ahrem et al. [1] propose breaking down the configuration into sections and ap­plying the GSB method on each one by itself. The projection results are smoothly interpolated by a partition of unity algorithm. In each section a different fit for the global polynomial contribution is obtained which results in a better approximation of the wetted surface. Simultaneously the memory requirements and the numerical effort are reduced. The downside is the introduction of yet another interpolation scheme, and this approach has not been included in the ACM.

The newly-implemented projection methods MLS and GSB have more control para­meters than the FIE method. Only the (minimum) number of supporting edge mid­points NM, the polynomial order of the interpolation functions and the radius of the additional support points rFB are examined here. The results for the remaining parameters are briefly summarised beforehand. As reference for the deviations the deformed wetted surface obtained by application of the FIE projection with default parameter settings is used.

The number of additional support points to be generated around each edge mid­point nFB depends on the type of structural model. In previous extensive tests, nFB = 5 was determined as the recommended number for beam models. With lower numbers reliable solutions could neither be obtained with the MLS method nor with the GSB method. Then again, greater numbers do not yield noticeable improvements in robustness or accuracy. With structural models consisting of higher-dimensional elements, the number of additional support points generally can be reduced. For shell models nFB = 2 is often sufficient. For volume models, additional support points are likely to be omitted. The choice of the RBF has only very small influ­ence on the solution both in the MLS and the GSB method. Wendland RBFs with different orders of smoothness have been tried out as well as other RBF with com­pact support, like Euclid’s hat functions or the Thin Plate Spline, with hardly any effect on the wetted surface. Because the mechanism to perform the interpolation in intersection regions between assemblies in the MLS method is very similar to the mechanism in the FIE method, the same effects on the wetted surface can be expected from variations of alimit.

The configuration used here is similar to the one presented in the previous section in Fig. 11, but without the dummy fuselage. It comprises only one component and thus allows direct comparison between GSB and MLS. The surface mesh has 31245 points. The structural model with 654 nodes and the applied load distribution are the same as before.

The interpolation parameters of the FIE method are the angular limit of the interpol­ation area in the vicinity of beam kinks внтіь the limit ratio of projection distances in the vicinity of beam kinks dlimit, and the width of the intersection region between assemblies alimit. They are varied and their respective influence on the shape of the deformed wetted surface is assessed. The test case is the HIRENASD configura­tion with a dummy fuselage, which constitutes a second assembly next to the wing. The CFD surface mesh sketched in the top left of Fig. 11 has 46919 points and the structural beam model has 654 nodes. These components are subjected to forces and moments which resulted from a previous aeroelastic simulation. These are kept constant during the parameter study. The deformed wetted surface acting as a ref­erence for the deviations was obtained with the default settings of the ACM, which are elimit = 0.08rad, dljmit = 1.2 and alimit = 0.1 m. This last value amounts to ap­proximately 8% of the model half-span of 1.29 m. As the parameters dlimit and eljmit both relate to the interpolation in the vicinity of kinks, these parameters are studied together. The RMS deviation of the surface point coordinates eS is determined by analogy to eu as defined in Eq. (19). Also in this case only the differences in the flap-wise deflection are considered.

In the top right diagram of Fig. 11 the RMS deviations of the normalised de­flection are plotted over the investigated parameter combinations. Whereas the minimum valid values of ^limit and dljmit are defined inside the ACM, the upper values were chosen arbitrarily. The choice of dljmit has a more profound effect on the shape of the wetted surface than ftimit. There is a weak interdependence visible between the two parameters. The indication of a global RMS value is somewhat misleading here, as both parameters lead to highly localised deviations. This can be seen in the bottom right image. The distributed values are displayed as bars over the configuration’s planform for combinations of the maximum and the default set­tings of ftimit and dljmit. Distinct peaks close to the leading edge are apparent for the maximum value of dljmit, whereas for the maximum value of elimit there are dif­ferences visible in the wedge-shaped areas of non-unique projection mentioned in Chap. 3.2. The choice of the width of the intersection region between fuselage and wing not only has a more widespread influence. It also produces deviations an order of magnitude higher and it thus gives far higher RMS errors. To put the given norm­alised distributed values into perspective: Assuming a bending deflection of 5% of the model half-span of 1.29 m, the peak deviation for parameter combination 0 is less than 0.5 mm, but for setting 0 with alimit = 1.0 m it exceeds 3 mm.

The spatial coupling methods available in the ACM all base the transfer of loads and deformations on geometric neighbourhood relations between the wetted surface and the structure. For this, the methods require different additional interpolation para­meters. As explained in Chap. 3.2, the FIE method for beam elements has three additional interpolation parameters: The weighting parameters ^ljmit and dlimit ap­ply for projections of surface points in the vicinity of kinks of the beam model. The width of the intersection region alimit has to be defined if assemblies border on each other. For shell and volume models, the FIE method currently does not call on interpolation parameters. Once it has been extended to configurations with mul­tiple assemblies, aljmit also will come into play. The GSB and the MLS methods are derived from general interpolation algorithms. They require the definition of the (minimum) number of supports Ng or NM, the number of additional support points per edge midpoint nFB, their radius rFB, the polynomial degree of the interpolation functions (7) or (11) and the type of RBF function ф. The MLS method is also suitable for configurations with multiple assemblies and thus again alimit has to be provided. None of these parameters is directly based on physical considerations, so the optimal values are not obvious.

All interpolation parameters of the FIE method required for beam models shall be examined here, whereas for the MLS and GSB methods only a selection is presented in detail. As became clear from the results presented in Chap. 4.1.2, distinct load dis­tributions can result in identical deformation distributions. Therefore, the influence of the projection parameters is evaluated via the deformation projection. In order to have identical input data for all cases, no aerodynamic loads are imposed on the wet­ted surface. Instead, a force distribution is applied directly to the structural nodes. The structural deformations then become independent of the projection method. The differences in the shape of the wetted surface can consequently be attributed solely to the deformation projection. Also in this investigation exact reference solution is not available, and as in the previous chapter the deviations should not be interpreted as absolute errors.

With the beam model test configuration, the different interpolation schemes and sur­face mesh resolutions do not produce profound local load incidence effects or major differences in the global deformation. A third test setup was investigated which bears more resemblance to a real-world configuration. The wetted surface is the HIRENASD wing scaled to a half-span of 29 m. The structural model is a shell model kindly supplied by the Institute of Aircraft Design and Lightweight Structures (IFL) of the Technical University of Braunschweig. It is akin to a modern transport aircraft wing box and has been dimensioned to real-world design loads. As such it has a realistic ratio of local sheet flexibility to total cantilever flexibility. The model is depicted in the left image of Fig. 9; a detailed description can be found in the contribution of Reich et al. in this volume. Also in this test case on wetted surfaces with varying resolutions a fictitious pressure distribution was applied. However, it cannot be exactly integrated here and thus no reference load or deformation distri­bution is available. In the right image of Fig. 9, the loads projected on the structure with the MLS method are depicted for the coarsest and the finest surface mesh. With decreasing number of points on the wetted surface, the absolute values of the aerodynamic surface loads increase due to the larger area of each individual sur­face cell. At least for the structured CFD meshes used here, simultaneously a con­centration of the surface loads occurs. Potentially both can cause local load in­cidence effects, i. e. local “bumps” on the shell model which are then projected back to the wetted surface and might locally alter the flow field. The bumps are a

result of the mismatch between the structural and fluid meshes; if forces rather than surface strains are projected, any such appearance must be examined: does it repres­ent a valid structural deformation or is it merely an artifact of the spatial coupling method.

In Fig. 10 the deviations in load and deformation distributions are compared for the FIE and MLS projection and the four investigated surface meshes. No results were obtained with the GSB method owing to the high numerical effort brought about by the inversion of the RBF weights matrix C in Eq. (10). The data plotted in Fig. 10 are not be understood as absolute projection errors. The reference values do not result from exact solution, which is not available. Here, the results obtained for the surface mesh with 107703 points and the FIE method were chosen, this however does not implicitly make them the "correct” values.

In the top left diagram there is an inherent deviation visible between the load distributions obtained with the projection methods which does not decrease signi­ficantly with increasing mesh resolution. The deviations in the deformation distri­bution, though, are strongly dependant on the mesh resolution rather than on the projection method, as can be seen from the top right diagram. This effect can once again be attributed to the offset moments which partially compensate the differences between the load distributions. A distributed deformational deviation was extracted along a line in span-wise direction and is shown in the bottom left image of Fig. 10. The graphs for the coarsest surface mesh and projection with FIE and MLS are virtually identical. They exhibit the mentioned bumps, but also a global bending deflection higher than in the reference case. The bumps are in the order of tenths of a per cent of the total bending deflection, which for this model equates to local

differences in the contour of several millimetres. The difference in global deflection is in the order of one per cent. The deviations between the projection methods for the finest mesh level are also plotted. They are close to zero all along the span. For a rough assessment of the influence of the differing structural deformations on the shape of the wetted surface, in the bottom right image two sections through the wet­ted surface at the spanwise position n = 0.84 are superimposed, which is where the distinct peak in the deviations is visible in the bottom left image. This comparison is slightly marred by the fact that the respective projection methods are applied twice, first for the loads and again for the deflections. Nonetheless, both the difference in global deflection and a bump on the suction side are apparent for the coarsest wetted surface. It can be concluded that with thin-walled structural shell models and coarse

CFD meshes load incidence effects can indeed have an influence on results, but it is seen to diminish rapidly with the finer CFD meshes. Even though such coarse meshes are not regularly used for standard steady simulations, they still play a role in unsteady simulations and in design, where accuracy is sacrificed for the sake of solution speed. For instance during the preceding project MEGADESIGN [20], a design case was investigated using a volume CFD mesh with approximately 170000 points; 4425 points thereof made up the surface mesh.

While these studies revealed first clues of the influence of the structural and CFD surface mesh spacings, the test configuration is very abstract and the conclusions

Fig. 7 Influence of the CFD mesh resolution on the structural load distribution. top left: Definition of the fictitious pressure field imposed on the wing. top right: Nodal forces in flap – wise direction resulting from the pressure distributions on the coarsest and the finest surface mesh and the projection with FIE. These loads are juxtaposed with those obtained by exact integration of the pressure distribution. bottom left: RMS error of the nodal forces in flap-wise direction for the three projection methods. bottom right: Nodal forces in flap-wise direction resulting from the pressure distributions on the finest surface mesh and projection with MLS and with GSB.

may not be transferable to real-world problems. Further test configurations were created for use together with the stand-alone version of the ACM which are based on the wetted surface of the HIRENASD wing [4]. The resolution of the wetted sur­face was varied again to quantify its effect on the structural load distribution. Four refinement levels with 2268, 8227, 31245 and 107703 nodes were realised by ex­tracting different multigrid levels from a structured FLOWer mesh. Certainly, with an actual flow solver any change of the mesh alters the flow solution. Because this problem is specific to the single-field solver and not to the spatial coupling method, a fictitious pressure distribution was again employed here. It is defined in terms of the rotated planform coordinates (x, z) shown on the top left of Fig. 7. The z-axis is chosen to connect the quarter-chord points at root and tip. The pressure distri­bution describes a quarter cosine wave along the z-axis and half a sine wave along

Fig. 8 Influence of the CFD mesh resolution on the structural deformation. left: RMS er­ror of the flapwise bending deflections resulting from the different load distributions. right: Comparison of the actual bending deflections and the distributed error for the case with the highest deflection error (FIE, nCFD = 103307) and the lowest (GSB, nCFD = 8227).

the perpendicular coordinate direction Jc. In the shaded areas the fictitious pressure distribution is zero. It is applied with opposite signs to both surfaces of the wing to produce a net positive bending moment in flap-wise direction. From the exact integ­ration of the pressure distribution a line load along the z-axis is obtained. If a beam stick model has parallel orientation, the line load can be divided consistently among the structural nodes to obtain a reference load distribution. This serves to define an error incurred by each projection scheme in dependence of the surface mesh resol­ution. Moreover, with the reference load distribution the structural deformation is computed and compared to those of the projected loads. As a measure for the error the root mean square (RMS) of the nodal forces in flap-wise direction, normalised by the total force in flap-wise direction, is used:

This is plotted in the bottom left image of Fig. 7 and different trends are apparent for the projection schemes. FIE exhibits a strong reduction in the error when the surface mesh resolution is increased, whereas the error with GSB remains almost constant. MLS ranges in between. In contrast to the FIE method the MLS and GSB methods are not able to capture the discontinuous onset of the pressure distribution near the root because they rely on finite support radii. Even with the finest surface mesh the load distribution at the root remains smeared out. Over the remaining span the MLS method approximates the exact loads well, whereas with the GSB method there are still large discrepancies, as can be seen from the bottom right diagrams of Fig. 7. In the left image of Fig. 8 the resulting bending deflections in flap-wise directions are compared by means of the RMS differences of the nodal deflections normalised by the tip deflection

It should be noted that because of their different normalisation the error values eu and eF cannot be directly compared.

For the investigated straight beam model, the differences in the load distribution do not translate in large differences in deformations. This is documented in the right diagram by a comparison of the bending deflections and their deviations from the reference distribution for the cases with the highest and the lowest total deflection error. The deformed structural models are almost identical because in all projection methods the redistribution of the bending forces along the beam axis is compensated by offset moments.

As the structural model is straight and the configuration comprises only a single assembly, no interpolation parameters come into play with the FIE method (cf. Chap. 3.2). For both the MLS and the GSB method, eight edge midpoints were set as supports, with nFB = 5 additional support points each. The radius of the sup­port points was chosen as rFB = 0.15m, which is approximately half the mean chord length. Quadratic polynomials were used for the global contribution to the interpol­ation function of the GSB method and for the local interpolation functions of the MLS method.

First investigations were carried out with one-dimensional test configurations loosely following an approach laid out by Jaiman et al. [17]: The one-dimensional fluid mesh and the structural mesh are colinear, but feature non-matching discretisations. An analytical pressure distribution is applied to the fluid mesh. The equivalent con­sistent nodal loads are then projected on the structural nodes. Furthermore, a refer­ence load distribution can be obtained by calculating the nodal loads consistent with the pressure distribution directly at the structural nodes. The reference loads can then be used to calculate a relative error eF of the projected loads. In the first three diagrams of Fig. 6, the analytical pressure distribution is plotted over the length of the one-dimensional domain together with the forces acting on the fluid nodes and on the structural nodes. The fluid mesh is discretised with an increasing number of elements nCFD, whereas the number of elements on the structural side is kept con­stant at nCSD = 20. In this example, linear shape functions are used for both the cal­culation of the consistent nodal forces and for their projection with the FIE method. With the coarser CFD meshes the distribution of the projected forces on the struc­tural nodes is highly irregular. Parameter studies indicate that the mesh spacings on

both sides have to be fairly similar in order to achieve a sufficiently regular load dis­tribution. The graph on the bottom right of Fig. 6 underscores this result. It shows the relative error plotted over the mesh ratio у = i+^csD/wcro’ Values from -1 to 0 imply a fluid mesh coarser than the structural mesh, and values between 0 and 1 a finer one. For negative values of у the error is high, and only approaching у = 0 it decreases to an acceptable level. In this example, for у = 0 the structural nodes and the fluid nodes are placed at the same coordinates and the relative error becomes zero (not shown in the logarithmic diagram).

In this section, the investigation of error sources in the spatial coupling by means of model problems shall be detailed. Tracing the steps outlined in Section 2, sev­eral potential sources of error were identified and looked into during the project MUNA. Three of these investigations are presented here: The influence of the mesh spacing of the structural and surface mesh is determined as well as the effect of the deformation mapping on the shape of the deformed wetted surface. For all three available spatial coupling methods, parameter studies were carried out to ascertain the importance of the user-defined projection parameters.

4.1 CFD Mesh Spacing and Load Distribution

When discretising the solution domains of the flow problem and of the structural problem before a coupled aeroelastic simulation, one would prefer to choose the grid spacings only considering the requirements of the single-field solvers and the desired solution accuracy. Especially one would like to avoid having to match up the discretisations at the coupling surface, which might not be possible at all when reduced structural models are used. Therefore the question arises as to how the choice of mesh spacings of the structural model and of the subsets of the CFD volume mesh representing the wetted surface influence the projection results. The influence on the load projection is most conveniently analysed by regarding the calculation of the consistent nodal loads on the wetted surface together with their projection on the structure.

The solution accuracy and the robustness of the MLS projection algorithm im­proves with increasing number of support points, but only if these offer sufficient information density in all spatial directions. Moreover, with higher number of sup­port points necessarily also the support radius becomes larger; hence the desirable locality of the projection diminishes. These two problems can be alleviated by us­ing an idea also put forward by Quaranta et al. [23]: It is not the actual nodes of the

Fig. 5 left: Arrangement of the “fishbones” generated for a beam model. The structural nodes are represented by squares, the edge midpoints by diamonds and the additional support points by circles. Here the radius rFB is fixed to a value approximately half the average chord length. The number of additional support points per edge midpoint nFB is set to five, which is the recommended value for beam models. right: Arrangement of the additional support points after alignment with the wetted surface.

structural model that are taken as supports for the interpolation. Instead, the midpoints of the beam elements, or respectively the edge midpoints of higher­dimensional elements are used. Surrounding these points, additional supports are in­serted circumferentially, as shown in the left image of Fig. 5. Quaranta et al. coined the term “fishbones” for this arrangement. The additional support points not only as­sure adequate information distribution in all spatial directions, but also allow for the simple projection of rotational deformational components at the structural nodes: The rotations are interpolated to the edge midpoints and result in a translation of the additional support points according to their radius rFB. During the converse pro­jection of loads from the wetted surface to the structure the forces projected to the additional support points are combined at the edge midpoints and corresponding off­set moments are introduced. The loads at the edge midpoints are then split between the adjacent structural nodes.

The GSB and MLS projection methods were further enhanced by enabling an automatic alignment of the additional support points with the wetted surface, as shown on the right of Fig. 5. To this end, the information which surface points are in the vicinity of each edge midpoint is required. Therefore, in a preparatory step the mapping of supporting edge midpoints to surface points is inverted. The shape of the surface section normal to a given edge is approximated as an ellipse. The additional support points are then inserted along its circumference with equiangu­lar spacing. Special attention is needed when the wetted surface does not cover the whole circumference, like along the fuselage of a half-model suspended in the sym­metry plane.