Prospects of Geometry Parameterization based on Freeform Deformation in MDO

A. Ronzheimer

Institute of Aerodynamics and Flow Technology German Aerospace Center (DLR), Member of the Helmholtz-Association.

Lilienthalplatz 7, D-38108 Braunschweig, Germany. E-mail: Arno.Ronzheimer@dlr.de - Web page: http://www.dlr.de/as

Key Words: Geometry Parameterization, Freeform Deformation, CFD, Shape Optimization

ABSTRACT

In MDO, related to aircraft design, geometry parameterization has been identified to play a key role. In the present approach the freeform deformation technique has been adopted, which requires a preparation of adequate lattices to achieve a desired variation of a given geometry. For the generation of those lattices the DLR grid generation system MegaCads is utilized. Practicability and prospects of the present method are demonstrated for typical optimization problems in aerodynamic shape design.

Introduction In aircraft design related MDO applications which include disciplines as computational fluid dynamics and computational structural mechanics, geometry parameterization plays a fundamental role. In MDO loops the geometry generation module, which relies on a geometry parameterization, represents the important link between the optimization module and the CFD/CSM simulation module. During optimization the optimization module persistently sends varying design parameters, which have to be converted by the geometry generation module into an explicit mathematical description of the shape and the geometry. Further on, the geometrical output of the geometry generation module has to be in a format, which must be immediately usable by the simulation. In case of CFD simulation the geometry acquired by a grid generator has to be of high accuracy and has to be watertight.

In the present approach geometry parameterization is based on the freeform deformation method, which has been initially developed to animate objects in computer graphics [1]. A first successful application of the freeform deformation method in MDO of a supersonic transport aircraft has been reported in [2]. In this application freeform deformation has been tied to specific parts of the geometry, namely to the wing and to the fuselage. However, to cover a huge variety of applications the implementation of freeform deformation, described in the following, is at first detached from any specific geometry and may also be used for object animation. However the main focus of this paper is to demonstrate in several examples the prospects of freeform deformation in aerodynamic shape design.

Present Freeform Deformation Approach Freeform deformation can be outlined as a mapping and re-mapping of object points into tri-variate B-spline volumes, which are defined by their control point lattices, as shown in Figure 1. During the mapping step parametric coordinates are calculated from given Cartesian coordinates via a Newton iteration method.

Figure 1: Principle of freeform deformation

A variation of the object is now achieved indirectly by the movement of certain control points of the corresponding lattice before the re-mapping is carried out. In the simplest case coordinate values of the control points may represent the design variables.

For the use of the method in aircraft design optimization adequate control point lattices have to be prepared to deform certain parts of the geometry, while other parts and also the connection to other parts should remain unchanged. Therefore, the previously described freeform deformation algorithm has been implemented into the DLR grid generation system MegaCads, which provides a huge palette of functionalities for structured grid generation, which have been found to be well suited for the construction of proper control point lattices [3].

The general procedure is now to split the geometry into parts and to construct adequate control point lattices around those parts, which are designated for optimization [4]. All single construction steps are recorded in a replay file, which is later replayed in any optimization step. Consequently the extended MegaCads system represents the geometry generation module, which varies an initial geometry via freeform deformation, and which in turn is controlled via the movement of control points representing the design variables.

Applications The challenge is now to explore ways and means to tackle a large variety of design problems. Since the geometry is varied indirectly via control points the compliance with geometric constraints may be cumbersome for specific problems. However, prior to this, the quantity of control points, which are related to the design variables, becomes a major aspect in design optimization. Therefore, present work is mainly focused on efficient application of the freeform deformation method to real world problems, while in earlier applications, reported in [4], the focus was on verification of the method.

Transonic Airfoil Shape Optimization Aerodynamic airfoil design is of fundamental importance, since the wing of an aircraft has to provide the necessary amount of lift with a minimum of drag. Additionally, geometric constraints concerning the compliance with a prescribed thickness had to be maintained. Since earlier applications of the present freeform deformation method had shown, that it was delicate to preserve the exact thickness of an airfoil section of a wing [5], in the present study additionally steps have been added to overcome this problem. Therefore, in a first step the airfoil coordinates are mapped from their initial bounding box to a new bounding box with vertical coordinates +1 and -1 for practical reasons, as shown in Figure 2. This step has been done with freeform deformation, since simple geometrical transformations as translation, rotation and scaling are included.

ObjectPv

kjiQ ,,

v

( )OOO wvu ,,

Mapping

NewObjectPv

kjiR ,,

v

Re-Mapping

Figure 2: Principle steps of airfoil parameterization using free form deformation (FFD)

At next freeform deformation is applied to the airfoil contour, where now the vertices of the control point lattice may be moved. Finally, the vertical coordinate values represent the design parameters. This simple parameterization allows the variation of typical properties of an airfoil as cambering, nose radius, and also a change of the angle of attack is enabled. At next, a bounding box was calculated which encloses the deformed airfoil, and which was finally used for the back transformation into the former bounding box of the initial airfoil contour. The result was now, that the airfoil thickness was exactly preserved. For this step again freeform deformation was utilized.

In the present case 6 design parameters represent the parameterization, but only D1 to D4 will have a reasonable effect, while the influence of D5 and D6 might be compensated by the final re-fitting step. Therefore, for the following airfoil shape optimization runs, using an optimizer based on the SUBPLEX method, these parameters had been fixed. Results of the optimization run are shown in Figure 3. In each optimization cycle an unstructured hybrid 2-dimensional grid was generated around the deformed contour using the CENTAURSoft® grid generation system [6]. The objective, the CL/CD-ratio, is then calculated from a RANS-flow computation for which the DLR-TAU-code is used [7]. Finally a portion of lift induced drag from a fictitious wing with an aspect ratio of 20 was added to the calculated airfoil drag.

Figure 3: History of transonic airfoil shape optimization with the SUBPLEX method

Transformation Back-Transformation

FFD D2 D6

D4

D1 D5

D3

Initial Geometry Final Geometry

As can be seen from Figure 3, an improvement of the lift to drag ratio of about 12% has been achieved, with only 4 design variables used for the parameterization. However, the question arose, if more design variables may result in a more notable improvement. Therefore the prescribed study was repeated with a parameterization which was based on 8 design variables, as this is shown in Figure 4.

Figure 4: Airfoil shape parameterization with 8 design parameters

Again, two parameters D9 and D10 had been fixed and any other parameters concerning grid generation and flow computation had been kept constant from the previous case. While now the number of design variables had been doubled, the necessary number of optimization cycles to reach a converged maximum of lift to drag ratio had to be increased to 100. Finally an improvement of about 16% had been achieved as shown in Figure 5.

Figure 5: History of airfoil shape optimization with 8 design parameters

The final airfoil shapes together with the resulting pressure distributions are compared in Figure 6 for both optimization runs. As had been seen from the previous figures, notable changes of the design variables, which initially started with +1 and -1, could be observed for the optimized airfoil shapes. In case of 8 design parameters the curvature of the upper airfoil contour was reduced while on the lower contour curvature has been increased.

Figure 6: Comparison of final airfoil shapes and calculated pressure distributions

FFD

D2 D6

D5 D1 D3

D4 D8

D7

D10

D9

Flying Wing Shape Optimization As the results from the previous study were very promising, the strategy of airfoil parameterization had been adopted to define 4 sections of a rectangular wing as illustrated in Figure 7. After application of freeform deformation each section is moved to its corresponding span wise station and is then fitted into a predefined rectangle. Based on these sections lofts were constructed to form a rectangular wing shape. By this procedure it is also allowed to have different numbers of parameters in different sections. As it was observed from the previous example, that beneath changes of the airfoil cambering also a notable change of the angle of attack happened, no additional effort was necessary to incorporate twisting.

Figure 7: Rectangular wing loft from airfoil sections parameterized with FFD

As until now freeform deformation was used to deform a given geometry the present example has been chosen to demonstrate how freeform deformation may be utilized to model a non trivial shape of a flying wing configuration, as shown in Figure 8. For this purpose a wire frame has been defined which was based on the plan form of the VELA-configuration [8]. At next all previously defined rectangles and also the wire frame were used to build control point lattices. Segment by segment has been carried to the wire frame via freeform deformation.

Figure 8: Freeform deformation of a rectangular wing into a flying wing configuration

The final lattices have been defined in a way to create smooth contour changes between the inner body and the outer wing segment. Finally the most outer part of the wing has bended to form a wing tip. All geometric processing is carried out with MegaCads, which also handles the export of the geometry in a clean format, which is immediately usable for 3-dimensonial unstructured grid generation with CENTAURSoft.

To keep the computational effort as small as possible in the following optimization run inviscid drag for a constant lift at typical cruise conditions has been minimized. Furthermore, design parameters of the first 2 inner sections had been fixed. The history of the optimization, which has been stopped after 45 cycles, is shown in Figure 9 for the remaining 8 design parameters. These have been equally distributed to the inner and outer wing section and therefore affect mainly wing shape and partially the fairing to the body.

Figure 9: History of flying wing shape optimization with 8 design parameters

For the flying wing case finally a total reduction of inviscid drag of about 30 drag counts has been achieved. Resulting surface Mach number distributions for initial and optimized geometry are shown in Figure 9. It can be recognized that the surface Mach number in the outer wing region has been reduced significantly and in conjunction with this the wave drag has been decreased.

Figure 10: Comparison of surface Mach number distributions of initial and optimized geometry

Optimization of a Belly-Fairing Shape To exploit the use and to point out certain advantages of the present freeform deformation method in detailed aerodynamic shape optimization with a geometrical description relying on CAD-data, exemplarily a belly-fairing of a transport aircraft has been parameterized.

The underlying geometry is the DLR F6 wing/body/nacelle/pylon configuration, without any belly-fairing. Therefore, the initial geometry had to be prepared by splitting the fuselage along predefined curves to yield a typical looking contour of a belly-fairing. The geometry would now be intrinsically ready for grid generation and CFD analysis, but instead the freeform deformation process has been setup interactively with MegaCads.

Since the shape of fuselage and wing should remain unaffected, in a first process step freeform deformation was utilized to transform the shape of the belly-fairing into a cuboid, shown in Figure 11, with the result that all boundaries of the transformed shape lay either on faces or on edges of the cuboid.

Figure 11: Freeform Deformation Sequence of Belly-Fairing Shape Parameterization

At next freeform deformation is applied to the transformed shape in a way that all boundaries of the transformed shape remain unaffected and finally, the deformed shape is transformed back using the reverse transformation from the first step and the deformed belly-fairing will fit exactly to the previous cut out. However, the initial intersection of the belly fairing with the wing has become invalid and has to be recalculated to re-establish a watertight geometry, which also includes a pylon and a nacelle, to be used for grid generation. Further on, to complete the optimization loop the hybrid unstructured grid generation system from CENTAURSoft was used again, and to calculate viscous turbulent flow at constant lift the DLR TAU-code has been applied. As can be seen in Figure 11, a total number of 4 control-points of the lattice would be variable for freeform deformation. However, for simplification, design parameters have been linked pair wise together, which result in at least 2 design parameters. In Figure 12 the convergence history is shown after 70 optimization cycles with the SUBPLEX method.

Transformation to Cuboid

FFD

Back-Transformamation

Figure 12: Optimization history of belly fairing with SUBPLEX method

It turns out, that drag can be reduced by about 5 drag counts implicated by a significant swelling of the belly-fairing in the rear part. However, due to the fact that the changes of the shape were restricted to the belly fairing, while the drag of the whole configuration was evaluated the objective function appeared to be more noisy than in the cases discussed before. The initial and a final geometry are compared in Figure 13.

Figure 13: Initial and final geometry with computed viscous flow solution at cruise conditions

Extension of the Freeform Deformation Method to CSM As freeform deformation affects the whole space enclosed by the corresponding B-Spline volume the extension to CSM-models is straight forward. Therefore similar strategies as shown in the previously cases may be applied simultaneously. In the following example this is demonstrated for a wing of supersonic transport aircraft. Since the plan form of the given wing was curved in the kink area, a similar strategy of sequential applications of the freeform deformation method as demonstrated in the belly-fairing case was applied. At first both, the aerodynamic surfaces and the CSM-model were transformed to a cuboid using freeform deformation, as shown in Figure 14.

In a next step freeform deformation was applied to vary aerodynamic surface and CSM-model simultaneously, with design parameters associated to the vertical coordinates of the control point lattice. Finally the inverse of the initial transformation step was applied and last the compliance between CFD- and CSM-model was achieved.

Figure 14: Sequence of FFD steps applied to CFD-Shape and CSM-model

Conclusion The present studies have demonstrated the applicability of the described freeform deformation method in shape optimization, and since the whole space is affected, it would be also possible to apply the same deformation to the geometry used for CFD and to a structural model to be used for CSM. The practicability of the method is mainly enhanced by the combination with the MegaCads grid generation system. For the present cases the time required to setup the freeform deformation procedure was in the order of hours.

Furthermore, it turns out, that multiple sequential applications of FFD to transform a non-trivial geometry in a first step to a simpler geometry offer certain advantages. First of all, the number of design parameters may be kept very small, even though the shape may be complex, and at last the possibility is given to vary the number of design parameters very easily by refining the lattices used for deformation.

Back-Transformamation

FFD

Transformation to Cuboid

Initial Geometry Final Geometry

REFERENCES

[1] T. W. Sederberg, S. R. Parry: „Freeform Deformation of Solid Geometric Models“, Proceedings of SIGGRAPH’ 86, Dallas, Texas, Aug. 18-22, 1986.

[2] J. A. Samareh: "A Novel Shape Parameterization Approach ", NASA TM-1999-209116, NASA Langley Research Center, Hampton, VA 23681.

[3] O. Brodersen, A. Ronzheimer, R. Ziegler, T. Kunert, J. Wild, M. Hepperle: "Aerodynamic Applications using MegaCads", 6th Intern. Conf. on Numerical Grid Generation, Ed.: M.Cross et.al., London, 1998, pp. 793-802.

[4] A. Ronzheimer: “Shape Parameterization Based On Freeform Deformation In Aerodynamic Design Optimization”, ERCOFTAC Design Optimization International Conference, March 31.-April 2. 2004 Athenas, Greece.

[5] A. Ronzheimer: “Post-Parameterization of CAD-Geometries Using Freeform Deformation and Grid Generation Techniques”, Contributions to the 13th STAB/DGLR Symposium, pp. 382-389, ISBN 3-540-20258-7, Munich Germany, 2002.

[6] www.centaursoft.com

[7] T. Gerhold, J. Evans: “Efficient computation of 3D-flows for complex configurations with the DLR-TAU code using automatic adaption”, New Results in Numerical and Experimental Fluid Mechanics II, Vieweg Notes on Numerical Fluid Mech anics, Vol 72, 1998, pp 178-185.

[8] B. Mialon, M. Hepperle: “Flying Wing Aerodynamics Studies at ONERA and DLR”, CEAS(KATnet Conference on Key Aerodynamic Technologies, 20-22 June 2005, Bremen.