-using different element/cell types (beam, quadrilaterals, triangles, hexa- hedrons, tetrahedrons, . .... face of a 10-nodes tetra- hedron or quadratic tri- angle.
A FLUID-STRUCTURE INTERFACING TECHNIQUE FOR COMPUTATIONAL AEROELASTIC SIMULATIONS F. MOYROUD, N. COSME, M. JO CKER AND T.H. FRANSSON Royal Institute of Technology Chair of Heat and Power Technology Brinellvagen 60 S-100 44 Stockholm, Sweden D. LORNAGE AND G. JACQUET-RICHARDET Institut National des Sciences Appliquees de Lyon Laboratoire de Mecanique des Structures (UMR CNRS 5006) B^at. 113, 20 Avenue Albert Einstein F-69 621 Villeurbanne Cedex, France
Abstract. This paper presents an approach to transfer information be-
tween a set of independently designed structural and aerodynamic grids. The approach combines, a parent volume grid and child surface grid concept, a triangulation of the parent structural and aerodynamic surface grids at the uid-structure interface, and a robust three-dimensional search and interpolation/extrapolation scheme for 3D unstructured triangular grids. The robustness, accuracy and general character of the present CFD/CSD interfacing methodology are demonstrated on four 3D test cases covering the elds of turbomachinery blading and rotordynamics, a highly twisted transonic fan, a transonic turbine rotor, an hydrodynamic wheel-casing element and an hydrodynamic bearing. The results show an excellent behaviour of the proposed technique to prolongate mode shapes, in particular in regions of the uid-structure interface where CFD and CSD models have di erent requirements and sensibilities, e.g. blade leading and trailing edges.
1. Introduction The simulation of uid-structure engineering problems is increasingly becoming a subject of interest after considerable progress in computational
uid dynamics (CFD), computational structural dynamics (CSD), and in hardware and parallel processing technologies. Most computational aeroelastic (CAE) simulations rely on the so-called loose coupling method, [1{3]
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which allows to combine existing discretizations and solution strategies for the ow and structural domains with an exchange of coupling quantities at the uid-structure interface in the frequency or time domain. Three main reasons for the success of this approach are that, 1) the full functionalities of both existing uid and structural codes are preserved and can be combined leading to limited developments to treat the coupled- eld problem, 2) the years of experience of structural and uid dynamics engineers can be used in a joint solution procedure, 3) the problem is inherently and readily decomposed into two domains and this represents ideal conditions for the application of parallel processing technologies. However, as a result of the possibility of combining di erent discretizations, the uid and structural grids are in general incompatible at the uid-structure interface leading to the problem of transferring information between the two domains. The development of a suitable strategy for solving this interfacing problem is by no means trivial and this diculty strikes right at the heart of the aeroelastic analysis process. The problem of transferring mode shapes from an aerodynamic grid to a structural grid was rst addressed by Rauscher [4] in the late 1940s. This work has been followed by the development of a number of transformations developed for wing bodies based on one-dimensional splines, planar surface splines, least squares polynomial approximations, multiquadraticbiharmonics, inverse isoparametric mapping, Non-Uniform B-Splines (NUBS) and some more. Excellent reviews and evaluations of these classical methods are published by Hounjet and Meijer [5] and Smith at al. [6]. Since the 1990s, new methods have been proposed which are consistent with the maturation of CFD, CSD and CAE models, starting with the 3D interpolation and search algorithm for unstructured grids proposed by Lohner [7], latter supplemented with a Gaussian integration scheme by Cebral and Lohner [8]. Surface tracking and data projection algorithms have also been proposed by Maman and Farhat [9] for parallel processing platforms and by Brakkee et al. [10] for exchanging coupling quantities between industrial codes. More recently, a novel and promising structural boundary element method has been published by Chen and Jadic [11]. The objective of the present work is to develop an approach suitable for implementation in an aeroelastic master program supporting a library of CSD and CFD codes [12]. The requirements are therefore very similar to those studied by Brakkee et al. [10] in the framework of the CISPAR project. In particular, the approach must be suciently exible to support new mesh topologies and element/cell types without requiring any modi cation of the interpolation and search algorithm. The present approach combines, 1) a parent volume grid and child surface grid concept, 2) a triangulation of the parent structural and aerodynamic surface grids, and 3) a robust three-dimensional search and interpolation/extrapolation scheme for 3D unstructured triangular grids. The parent and child grid concept makes the surface tracking and data projection algorithm independent of the characteristics of the parent structural and aerodynamic volume grids,
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at the expenses of the creation of one additional surface grid. The approach is of general character and suitable for a number of aeroelastic problems. Its applicability, robustness and accuracy are demonstrated on four test cases covering the elds of turbomachinery blading and rotordynamics.
2. Aeroelastic con gurations
We consider an aeroelastic con guration with two contiguous domains, Fig. 1, a) a structural domain S and a corresponding CSD discretization GS , b) a uid domain F and a corresponding CFD discretization GF . The physical (as opposed to numerical) uid-structure interface is denoted ;F=S .
Figure 1. Fluid domain, structural domain and uid-structure interface.
As mentionned earlier, the GF and GS meshes have in general two independent representations of the physical uid-structure interface ;F=S . When these representations are identical, that is when every uid point on that surface is also a structural node and vice-versa, the evaluation of the pressure forces and the transfer of the structural motions to the uid mesh become trivial operations. However, CFD and CSD analysts are usually keen on ; using di erent CAD representations of the geometry (blade geometry at rest for CSD models but running geometry for CFD models, CFD models are sensitive to the blade leading/trailing edge and tip clearance descriptions, some components may be present in one model but not in the other, ...), ; using di erent coordinate systems (Cartesian/cylindrical, global/local), ; using di erent element/cell types (beam, quadrilaterals, triangles, hexahedrons, tetrahedrons, ...) and mesh topologies (single/multi-block structured, unstructured, hybrid, ...), ; re ning each mesh independently (for usual applications, the CFD mesh is much ner than the CSD mesh at the uid-structure interface), ; and more globally combining CFD and CSD analyses that have been independently designed and validated.
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A major diculty in the CFD/CSD interfacing problem is clearly the diversity of CFD and CSD models and analyses to be combined. The CFD and CSD analyses of particular interest in this work are brie y listed in Tab. 1. For example, it may be required to combine, 1) a beam CSD model with a three-dimensional CFD model, 2) a single blade CSD model with a multiple blade passages CFD model and vice versa (with the underlying problem of matching blade positions and blade surfaces), 3) a shrouded CSD model with an unshrouded CFD model. CFD analysis 2D/Q3D/3D, inviscid/viscous, types time linearized/nonlinear, single/multi blade passages, single/multi blade rows mesh single/multi-block structured, untopologies structured, hybrid, ... cell/element tri., quad., tetra., hexa., ... types
CSD 3D, single blade modal analysis, cyclic symmetric modal analysis, full assembly modal analysis unstructured linear/quadratic beam, tri., quad., hexa., tetra., ...
TABLE 1. CFD and CSD analyses according to turbomachinery engineering practices.
3. Parent and child, volume and surface meshes
3.1. PARENT VOLUME AND SURFACE GRIDS The parent volume grids, denoted GF and GS , are the uid and structural grids independently designed and validated by the CFD and CSD analysts. The parent aerodynamic volume grid is 1) a fully three-dimensional volume grid, or 2) a series of S blade-to-blade grids stacked along the blade span and denoted [Ss=1 G(Fs) . The parent structural volume grid is assumed threedimensional based on usual CSD engineering practices. The parent surface grids, denoted ;F ;p and ;S ;p, are the representations of the respective parent volume meshes GF and GS at the uid-structure interface ;F=S . 3.2. CHILD (SURFACE) GRID The child grid denoted ;X ;c , X = F or S , refers to an unstructured triangular surface mesh, based on the linear triangular element shown in Fig. 2, which is equivalent to the parent surface grid, generally made of mixed surface element types. The child element supports an arbitrary number of degrees of freedom per node (DOFs). For CSD applications, there are typically 3 translational DOFs and 2 or 3 rotational DOFs. For CFD applications, there is typically 1 DOF which is pressure.
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Figure 2. 3-nodes triangular element, (1 ; 2 ; 3 ): area coordinates of a point q within element plane, (xl ; yl ; zl ): local element coordinate system, (x; y; z): global coordinate system.
3.3. GENERATION OF ;X ;C FROM GX The generation of a child grid is in practice straight forward and can be generated from the following information, 1) the list of node numbers and nodal coordinates of GX , 2) the list of element types and table of element connectivities of GX , 3) a list of GX node numbers on ;X ;p . From a code development point of view, a GX -to-;X ;c transfer procedure must be available for all relevant pairs of parent volume grid topology and cell/element type. The generation of ;X ;c from GX can be decomposed into the following steps, Fig. 3, 1. identify the GX element faces at the uid-structure interface, 2. generate a parent surface grid denoted ;X ;p , out of the identi ed parent element faces, with mixed n-linear triangular/quadrilateral surface elements, see rst column of Tab. 2, 3. generate the child grid ;X ;c by triangulating the parent surface elements following the conventions established in Tab. 2, i.e. each n-nodes surface element is converted into an equivalent set of linear triangular elements of the type shown in Fig. 2. The main concern in the parent-to-child element conversion process is the treatment of possibly all kinematic nodes/DOFs of the parent element and to create additional nodes to support the child 3-nodes elements whenever this enhances the child grid quality.
4. 3D interpolation/extrapolation scheme
We consider now the problem of transferring a nodal solution from an unstructured triangular surface mesh ;1;c to a target surface mesh or eld of points denoted ;2 . ;1;c is typically a child surface grid of a CFD or
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Figure 3. Child grid generation from a parent surface mesh.
CSD parent volume grid and ;2 can be a parent or child surface grid or a three-dimensional aerodynamic blade section. The data transfer procedure is totally independent of the topology of the parent volume grid G1 , the characteristics of the parent element, and the topology of the target grid (set of points, surface mesh or volume mesh). The basic idea of the proposed 3D interpolation scheme is, for a given point N2(n) of ;2 , to identify the host triangular element of ;1;c according to Fig. 4, and then to interpolate from the host element nodes to the point using the linear shape functions of the 3-nodes nite element of Fig. 2. This node-cell matching or pairing process works well in the interior domain of the uid-structure interface. However, special treatments are necessary at the boundaries where ;1;c and ;2 usually do not perfectly match for various reasons. For example, Fig. 5 shows an aeroelastic con guration of two grids, i.e. CSD surface mesh and CFD volume mesh, with di erent mesh resolutions in a region of high curvature of the uid-structure interface. For turbomachinery applications, this would correspond to the blade leading/trailing edge and hub/tip regions. In these areas, there are points of ;2 that do not have a projection on any element of ;1;c (Fig. 5) or that have a projection on an element which is not the most appropriate host element.
A F-S INTERFACING TECHNIQUE FOR CAE SIMULATIONS parent cell/element
7
equivalent set of child cells/elements
face of a 4-nodes tetrahedron or linear triangle
face of a 10-nodes tetrahedron or quadratic triangle face of a 8-nodes hexahedron or linear quadrilateral face of a 20-nodes hexahedron or quadratic quadrilateral (1) TABLE 2. Library of supported 3D parent and child surface elements. (1): DOFs at the new node can be obtained by averaging DOFs at surrounding nodes or by using the shape functions of the quadratic quadrilateral.
A solution to circumvent this problem consists in supplementing the interpolation scheme for the inner domain with an extrapolation scheme for the boundaries. The basic idea is to identify if a given point of ;2 has a projection contained within an opening angle of an element of ;1;c as shown in Fig. 6. The best associate is the element for which the projection q of N2(n) is closest to one of the element edges, i.e. edge Ni3 Ni1 in Fig. 6.
INNER DOMAIN (1ST ORDER INTERPOLATION) For a prescribed node N2(n) of ;2 to be matched or paired with an element of ;1;c , out of E elements, the interpolation algorithm can be written as follows in pseudo-code form. (ID-0) Initialization: set e = 1 and E h = 0, where e is the index of an element of ;1;c and E h is the number of potential host elements for the current
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Figure 4. Pairing an interior node of ;2 with an element/cell of ;1;c (orthogonal projection).
Figure 5. A uid node without an associate in a 2D problem- node in a region of high curvature of the uid-structure interface -.
Figure 6. Pairing a boundary node of ;2 with an element/cell of ;1;c , i2 < 0.
node N2(n) . (ID-1) Project N2(n) on ;(1e;)c according to Fig. 4, and calculate
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a) the coordinates of q in the element local coordinate system, ;;;! b) the projection distance d(n) = jjN2(n) q jj, Fig. 4, c) the area coordinates (1 ; 2 ; 3 ) of q on ;(1e;)c . (ID-2) Is the node-cell pairing successful ? If 0:0 1 ; 2; 3 1:0 then mark element ;(1e;)c as a potential host element and increment E h . (ID-3) If e < E then increment e and goto (ID-1), i.e. check next target element. If e = E , i.e. all elements of ;1;c have been tested, and a) E h = 0 then switch to the extrapolation scheme, i.e. goto (B-0), b) E h = 1 then the search is completed and the host element is identi ed, h i.e. node N2(n) and ;(1e;1c) are paired, c) E h > 1 then the best candidate for the node-cell pairing is element No. ehi (1 i E h ) such that d(n) is minimum.
BOUNDARIES (0TH ORDER EXTRAPOLATION) Under the condition that the node-cell pairing using the interpolation algorithm fails, the algorithm below is used.
(B-0) Initialization: set e = 1 and E h = 0. (B-1) Project N n on ; e;c according to Fig. 6. (B-2) Determine the area coordinates ( ; ; ) of q. (B-3) Check the following two criterion with (i ; i ; i ) = (1; 2; 3) and (2; 3; 1) ( ) 2
( ) 1
1
2
3
1
2
3
and (3; 1; 2): criteria 1: q is within region A1 (note that i2 < 0), criteria 2: q is within region A2 (note that i1 < 0 and i3 < 0). If criteria 1 (resp. 2) is ful lled then mark element ;(1e;)c as a potential host element, increment E h and evaluate the parameter = i2 (resp. = Min(i1 ; i3 )). (B-4) If e < E then increment e and goto (B-1), i.e. check next target element. If e = E and h a) E h = 1 then node N2(n) and ;(1e;1c) are paired, b) E h > 1 then the best candidate for the node-cell pairing is element No. ehi (1 i E h ) such that j j is minimum. For each pair of node N2(n) and identi ed host element ;(1e;)c, the nodal values are calculated as follows, X (n) = X (q ) = X1 1 + X2 2 + X3 3 (1) (e) where (X1 ; X2 ; X3 ) are the nodal values of ;1;c and (1 ; 2 ; 3 ) are the area coordinates of the orthogonal projection q of N2(n) on ;(1e;)c .
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The proposed node-cell pairing process requires a loop over all pairs of elements of ;1;c and points of ;2 . In fact, the CPU cost of the overall data transfer increases as the number of ;1;c elements and number of ;2 points increase. However, the child grid concept allows to work with surface grid rather than with the parent volume grid thereby reducing considerably the number of elements to sweep through. To enhance the computational performances, for example in the framework of time domain CAES where a search may be needed at each time step, the proposed so-called linear search algorithm can be vectorized or parallelized like in [9]. Less straightforward but faster and more ecient search algorithms discussed in [7, 10] can also be used.
5. Applications
Although the present data transfer scheme has been originally developed for turbomachinery applications, we found it to be applicable to a number of aeroelastic components, for example to wing bodies, helicopter rotor blades, hydrodynamic bearings and seals. The four test examples discussed below illustrates its applicability to a variety of geometries and aeroelastic con gurations. For all four combinations of uid and structural grids, the CSD grid (GS ) is much coarser than the CFD grid (GF ) at the uidstructure interface and this is not unusual. In the rst two examples, the problems of transferring blade mode shapes from a stand-alone blade/cyclic symmetric bladed-disk mesh sector to a 3D/Q3D aerodynamic mesh/blade section is considered. This is of interest in the framework of unsteady aerodynamic utter/ uid-structure coupled simulations towards an assessment of blade utter (in)stabilities. The last two examples relate to some studies of the uid-structure coupled behaviour of rotating shaft-disk assemblies and uid- lm supporting elements [13] towards an accurate prediction of rotordynamic response amplitudes and instabilities. 5.1. NASA ROTOR 67 Figure 7 shows typical uid and structural meshes used for a number of unsteady aerodynamic and aeroelastic analyses performed on the NASA Rotor 67 transonic fan rotor [14]. The mode shapes of a reference clamped blade are obtained with the FE mesh GS shown in Fig. 7a. Three-dimensional steady and unsteady aerodynamic calculations are performed with the 5blocks structured mesh GF of a reference blade passage shown in Fig. 7c. Of interest here are the two blocks surrounding the blade surface and forming an O-mesh. Figure 7d is the intersection of the O-mesh with the uidstructure interface ;F=S (blade suction and pressure sides). This is the parent CFD surface grid ;F ;p as de ned in section 3 and the corresponding child grid is shown in Fig. 7e. For the transfer of blade mode shapes from GS to GF , the 3D interpolation/extrapolation scheme of section 4 is applied with Fig. 7b as source grid and Fig. 7d as target grid. Figure 8 shows
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the blade mode shape on the structural and aerodynamic surface grids. A perfect agreement is obtained, which quanti es the level of accuracy of the proposed method on a highly twisted blade. 5.2. ADTURB TURBINE ROTOR
Figure 9 shows an FE surface mesh of the ADTurB turbine rotor with a mid-span aerodynamic blade section [15, 16]. The problem is here to transfer the mode shapes from the structural mesh to the blade section at which Q3D unsteady aerodynamic analyses are to be performed. The mid-span blade pro le is de ned by a series of points in 3D Cartesian coordinates. Figures 10a-b show the real and imaginary parts of the displacement eld on the FE mesh GS ([15, 16]) as well as the mapped movement of the target blade pro le for a rst torsion with 21 nodal diameters. Figures 11a-b show the corresponding vibrational displacements of the blade section in a plane parallel to the blade platform. The results show a perfect behaviour of the interpolation/extrapolation scheme in the leading and trailing edge regions where the blade curvature is the largest. In these regions, the interpolated displacement eld over the blade pro le is perfectly smooth. 5.3. HYDRODYNAMIC WHEEL-CASING ELEMENT
Figures 12a-b show a combination of structural and uid grids used to model the uid-structure coupled behaviour of a wheel-casing hydrodynamic element with a leakage oil- lm clustered between the wheel and the casing. The structural domain consists of the wheel (elements in grey) mounted on a shaft (elements in blue) modelled with the nite element grid shown in Fig. 12a. The uid domain consists of the oil- lm discretized with the two-dimensional axi-symmetric grid GF of Fig. 12b. The dynamic behaviour of the structure is governed by the structural equations for rotating shaft-disk assemblies and a modal reduction is applied to reduce the nite element system of equations. The steady state and time dependent pressure elds in the oil- lm are governed by the Reynolds equation [17] which links the pressure to the thickness of the lm at all radial and circumferential positions. In order to evaluate the modal forces from the unsteady pressure eld (in the oil- lm) acting on the wheel and the set of mode shapes included in the modal basis, the mode shapes are transfered from GS to GF . Figures 12c-d is a comparison of the axial component of the rst bending mode shape of the shaft on the source and target grids. The mode shapes are found to be in good agreement especially considering the large di erences in mesh resolution along the circumferential direction between the two grids.
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5.4. HYDRODYNAMIC BEARING The rotating assembly of a machine interacts with the static support structure through a variety of components. Figures 13a-b show the case of a rotating shaft-disk assembly clamped on one side and supported on the other side by an oil- lm bearing. The location of the oil- lm bearing with respect to the rotating assembly coincides with the mesh re nement (elements in grey) in Fig. 13a. The steady state and unsteady pressure elds in the oil- lm are solved on the mesh shown in Fig. 13b. The coupled behaviour of the shaft-disk assembly and the oil- lm is obtained using a time domain uid-structure coupling method. Before solving the uid-structure coupled problem, the mode shapes included in the modal basis are transfered from GS to GF with the proposed data transfer method. Figures 13c-d are contour lines of the y-component of the rst torsion mode shape of the shaft, on the child structural surface mesh and on the parent uid mesh. Here again, the mode shapes compare excellent well.
6. Conclusions
Fluid-structure engineering problems in turbomachines are presently tackled by combining existing discretizations and solution strategies for the two domains. However, as a result of the possibility of combining di erent discretizations, the uid and structural grids are in general incompatible at the uid-structure interface leading to the problem of transferring coupling quantities between the two domains. Furthermore, in an industrial or research environment where a library of CSD and CFD codes with di erent functionalities can be used, the data transfer procedure must be suciently
exible to cope with a large variety of aeroelastic con gurations and must be opened to new element/cell types. The proposed interfacing strategy has been designed to meet these requirements. New element/cell types can be introduced by simply updating the parent-to-child grid transformation algorithm but without modifying the data projection algorithm. The discussed aeroelastic test cases demonstrate the robustness, accuracy and general character of the present 3D CFD/CSD interfacing methodology. Future work should assess the performance and robustness of more advanced search algorithms as well as address the problem of conservative load projection.
7. Acknowledgments This research was funded by the Swedish `Nationella Flygtekniska Forsknings Programmet', project `Forced Response and High Cycle Fatigue Prediction of Bladed-Disk Assemblies' under contract No. 342, and by the European Community, Brite/Euram project "Aeromechanical Design of Turbine Blades" (ADTurB) under contract No. BRPR-CT95-0124.
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References
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1. F. Moyroud. A Review of Computational Methods for Fluid-Structure Coupled Systems. Technical Report KTH/HPT 95/06, Stockholm (Sweden) : Chair of Heat and Power Technology, Royal Institute of Technology, 1995. 2. J.G. Marshall and M. Imregun. A Survey of Aeroelasticity Methods with Emphasis on Turbomachinery Applications. Journal of Fluid & Structures, 10:237{267, 1996. 3. M. Imregun. Recent Developments in Turbomachinery Aeroelasticity. In Proceedings of the ECCOMAS 98 Conference, Athene, pages 524{533. New York (USA): John Wiley & Sons Ltd, 1998. 4. M. Rauscher. Station Functions and Air Density Variations in Flutter Analysis. Journal of The Aeronautical Sciences, 16(6):345{353, 1949. 5. M.H.L. Hounjet and J.J. Meijer. Evaluation of Elastomechanical and Aerodynamic Data Transfer for Non-Planar Con gurations in CAE Analysis. In Proceedings of the International Forum on Aeroelasticity and Structural Dynamics, Manchester, England, pages 10.1{10.25. London : Royal Aeronautical Society, 1995. 6. M.J. Smith, C.E.S. Cesnik, D.H. Hodges, and K.J. Moran. An Evaluation of Computational Algorithms to Interface between CFD and CSD Methodologies. Reston (USA): AIAA, 1996. AIAA Paper 96-1400. 7. R. Lohner. Robust, Vectorized Search Algorithms for Interpolation on Unstructured Grids. Journal of Computational Physics, 118:380{387, 1995. 8. J.R. Cebral and R. Lohner. Conservative Load Projection and Tracking for FluidStructure Problems. AIAA Journal, 35(4):687{692, 1997. 9. N. Maman and C. Farhat. Matching Fluid and Structure Meshes for Aeroelastic Computations: A Parallel Approach. Computers & Structures, 54(4):779{785, 1995. 10. E. Brakke, K. Wolf, P. Post, and T. Schuller. Speci cation of the COupling COmmunications LIBrary. Deliverable 1.1, CISPAR Esprit project 20161, July 1998. Can be downloaded from . 11. P.C. Chen and I. Jadic. Interfacing of Fluid and Structural Models via Innovative Structural Boundary Element Method. AIAA Journal, 36(2):282{287, 1998. 12. F. Moyroud. Fluid-Structure Integrated Computational Methods for Turbomachinery Blade Flutter and Forced Response Predictions. Dual Ph.D. Thesis and These de Doctorat, Stockholm (Sweden) : Royal Institute of Technology, Villeurbanne (France) : Institut National des Sciences Appliquees, Dec. 1998. 300 p. 13. F.F. Ehrich. Handbook of Rotordynamics. New York (USA): McGraw-Hill, Inc., 1992. 14. J.R. Wood, T. Strazisar, and M. Hathaway. Test Case E/CO-2, Single Transonic Fan Rotor. In Test Cases for Computation of Internal Flows in Aero Engines Components, pages 165{213. Brussels (Belgium): AGARD, 1990. Manual AGARDAR-275. 15. M. Jocker and J. Jeanpierre. ADTurB database at KTH, Establishment and Access. Technical Report ADTB-KTH-1003, Stockholm (Sweden): Chair of Heat and Power Technology, Royal Institute of Technology, 1997. 16. J.S. Green and R. Elliott. DLR Rig Flexible Rotor Final Design Report. Technical Report ADTB-RR-4004, Rolls-Royce Aerospace Group, Derby (England), April 1997. 17. O. Reynolds. On the Theory of Lubri cation and its Application to Mr. Towers' experiments. Phil. Trans. Soc., London, 177:154{234, 1886.
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Z
a: GS grid (20-nodes hexahedral element).
b: ;S;c grid (suction surface).
c: GF grid. Z
d: ;F ;p grid (suction surface). e: ;F ;c grid (suction surface). Figure 7. Combination of volume and surface grids for the NASA Rotor 67.
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0.015
-1.000
a: on ;S;c (suction surface). b: on ;F ;p (suction surface). Figure 8. First bending blade mode shapes (component normal-to-page) on structural and aerodynamic grids.
Figure 9. Structural surface grid (faces of 10-nodes tetrahedral elements) with mid-span aerodynamic blade section.
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a: Real part. b: Imaginary part. Figure 10. Third blade mode shape (torsion) with 21D on CSD surface mesh with mid-span aerodynamic blade section, blue: undeformed structure, gray: deformed structure, red: undeformed & deformed blade sections (target).
a: Real part. b: Imaginary part. Figure 11. Blade section movement in (y,z) plane.
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a: GS grid.
b: GF grid. Y
Y
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X
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1.0
-1.0
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c: mode shape on GS . d: mode shape on GF . Figure 12. Combination of grids for the oil- lm wheel-casing case.
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a: GS grid.
b: GF grid.
Y
Y
0.2
0.2
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-0.2
X
-0.2
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c: mode shape on ;S;c. d: mode shape on GF . Figure 13. Combination of grids for the oil- lm bearing case.
