Abstract
Increased energy costs make optimal aerodynamic design even more critical today as even small improvements in aerodynamic performance can result in significant savings in fuel costs. Energy conscious industries like transportation (aviation and ground based) are particularly affected.
There have been a number of different optimization methods, some of which require geometrically parameterized models. For non-parameterized models (as it is the casa often in reality where models and shapes are very complex). Shape optimization and adjoint solvers are some of the latest approaches.
In our study we are focusing on generating best practices and investigating different strategies of employing the commercially available shape optimizer tool from ANSYS’CFD solver Fluent. The shape optimizer is based on a polynomial mesh- morphing algorithm. The simple case of a low speed, airfoil/flap combination is used as a case study with the objective being the lift to drag ratio.
A number of different built-in optimization algorithms and the way to best employ them are investigated.
- I. Introduction
Computationally expensive simulation codes based on mathematical models of a relevant system are in wide-spread use throughout the engineering industry. For instance, in the field of computational fluid dynamics (CFD) a single evaluation of the model may take several hours of computer run time. With the help of modern computer facilities, numerous difficult and mathematically intensive problems are now readily solved.
Towards significantly reducing user expertise requirements (Zingg et al, 2002), the solutions of the Reynolds-averaged Navier-Stokes (RANS) equations in the application of computational fluid dynamics (CFD) to aircraft design, involve challenges in the following three areas.
Computational Efficiency: There has to be a reduction in the computing time required to achieve appropriately resolved solutions. This is an urgent need in aerodynamic and multi-disciplinary design optimization, as an outcome of the trend towards an integrated product and process development environment. For complete integration into the design process, the time required for solution of the RANS equations should be restricted to a few minutes, over three-dimensional configurations, which is two orders of magnitude faster than capability, a new development at the start of the new millennium. Although there is a definite advantage of parallel architectures and increased computer speeds, there is a need for improvements in algorithms. In design optimization also, algorithm reliability has greater significance. “Modern design optimization algorithms, such as adjoint methods cannot be effective if the flow solver does not converge in relevant areas of the design space” (Zingg et al, 2002, p.347).
Human Efficiency: There is a pressing need for a reduction in the “human effort and expertise required for computing flows over complex configurations since humans are not governed by Moore’s law” (Zingg et al, 2002, p.348) which states that computer power doubles every eighteen months, and the rapid increase in power on this scale is a unique and tremendous advancement in the history of technology (Kaku, 1999). Though expertise in CFD and aerodynamics is vital for carrying out the computations, there should be a minimization in the expertise required in the selection of solver and grid parameters. Another issue to be resolved is the estimation of global numerical error
Accuracy of Physical Modeling: Through methods such as appropriate grid resolution, numerical errors can be carefully controlled. However, errors resulting from “physical models, including turbulence models and prediction of laminar turbulent transition are more difficult to estimate and control” (Zingg et al, 2002, p.348). The eddy-viscosity turbulence models used at present for computing aerodynamic flows are unable to predict with accuracy, subtle phenomena such as Reynolds number or flap gap effects on high-lift configurations.
The most optimal approach appears to be Reynolds stress models or second moment closures. It is essential to considerably increase the speed with which such models are incorporated into aerodynamic flow solvers, for production use at this stage, and to assess and guide the development of such models. Further development of turbulence models requires reduction of time needed to compute three dimensional flows; also there is a need for more high quality experimental datasets which include all the boundary condition data needed for computations. To increase the computational efficiency of RANS solvers, two new developments are considered: an inexact Newton-Krylo algorithm which decreases the time needed for computing towards achieving a steady state solution; and “a higher order spatial discretization which decreases the grid resolution requirements for a given level of numerical accuracy” (Zingg et al, 2002, p.348), consequently also reducing computing expense. Moreover, objective measures of algorithm performance are required.
This paper is organized as follows.
II. Research strategy plan
The strategy of this research is to develop an airfoil design and optimization system that can modify an airfoil shape (section geometry) yielding improved aerodynamic performance in terms of maximum lift to drag ratio under landing and takeoff flight condition. Lift-to-drag ratio is simply defined as the ratio of the magnitude of the lift to the amount of turbulence, or drag, of an aerodynamic structure (Encyclopedia Britannica, 2011).
Computerizing the optimization progression can considerably shorten development product lifecycle and bring forth improved designs compared to conventional tactical design alteration approaches (Andreoli, 2003). An efficient and effective optimization technique is developed by a combination of high-fidelity commercial CFD tools with mathematical optimization techniques. This system not only offers a consistent, automated optimization tool for blueprint engineers, but also considerably diminishes the cost and the manufacturing time for a design procedure.
The development will be to combine the high fidelity commercial CFD tools (Fluent) with numerical optimization techniques to morph high lift system. In research strategy as shown in the figure below, we will perform morphing (grid deformation) directly inside the fluent code without rebuilding geometry and the mesh with an external tool. Direct search method algorithms such as the Simplex, Compass, and Torczon are investigated. The user can choose any of those to utilize in optimization of aerodynamic shapes characteristics.
Fig. 1 Research Strategy Plan
III. Formulation
- A. Flow Solver
FLUENT is a computational fluid dynamics (CFD) solver. It is a software package to simulate fluid flow problems. It is based on a finite-volume method to solve the governing equations for a fluid. It provides the capability to use different physical models such as incompressible or compressible, inviscid or viscous, laminar or turbulent, steady- state or transient analyses.
In the CFD formulation as well, the conservative forms of continuity and Navier-Stoke’s equations in integral form for incompressible flow of constant viscosity were solved by the built-in functions of the Fluent 13 CFD software. The current work used two equation turbulent models: the realizable k-e model that solves one transport equation to allow the turbulent kinetic energy, and its dissipation rate to be independently determined. The realizable k-e model is particularly suitable for our model, as the model uses enhanced wall treatment based on the law of the wall.
In the present work, steady-state, incompressible two dimensional flow was assumed. The numerical simulations were carried out by solving the conservation equations for mass and momentum by using structured and an unstructured grid finite volume methodology.
The sequential algorithm, semi-implicit method for pressure linked equation (SIMPLE), was used in solving all the scalar variables. For the convective terms of the continuity and momentum equations, and also for the turbulence equations, the first order upwind interpolating scheme was applied in order to achieve more accurate results compared to experimental results. The computational conditions are given in the next section.
- B. Design Variables
As shown from Figure 2.1, the overlap and the gap between the flap and main airfoil are used as the design variable. Each design variable is limited as shown in Figure 2.2.
Fig. 2.1 Horizontal (overlap) and vertical (gap) translation design variables in multi-element configurations.
The control points will be situated around the areas of the gap and the overlap sections. Thus, the shape optimization, i.e., expansion, retraction, etc., will be imposed in these areas in order to achieve optimal values of L/D ratio.
Fig. 2.2 Design Variables
The figure above illustrates the possible locations of the control points (shown here, for instance, are points p1 and p2).
- Objective functions
The problem that is considered in current work is single-objective programming problem. A single-objective programming problem can be stated as:
Find X which
Minimize f(X) = [f1(X)/ f2(X)]
Where f1, f2 are drag and lift functions, respectively, X is called the design vectors.
The vector of design variables, X, primarily contains parameters that control the shape of the airfoil. Depending on the problem of interest, additional design variables may include the angle of attack, the horizontal and vertical translation design variables that control the position of flaps in multi-element configurations as shown in Figure 2.2.
During this work the objective function will be formulated by C++ language programming and built on the User Defined Function (UDF) library which takes control points and angle of attack as input parameters, and values of lift to drag and drag to lift ratio are output parameters.
- Different Strategies in Optimization
There will be five (5) strategies of optimization to be studied in this paper. The strategies are the following:
1.) Changing the position of the control points
2.) Changing the number of control points
3.) Applying the Compass Algorithm
4.) Applying the Simplex Algorithm
5.) Applying the Torczon Algorithm
Simulations will be executed using each of these strategies. The resulting L/D ratio will be closely monitored. Comparison between the before and after optimization scenarios will be noted. Each strategy will be evaluated of its effectiveness in improving the performance of the system. Also, analogous strategies will be compared to each other.
IV. Results and Discussion
In this section, we show the influence of algorithms on the value of L/D ratio at varying angle of attack . As shown from Figure 4.20, all data sets indicate a nearly identical lift to drag ratio value at (-4 to 0) degrees angle of attack and the data continues to match quite well at (6 to 20) degrees. There are variance values at angle of attack from (0 to 6) degrees. The figure below shows that the Torczon and simplex algorithms produced the maximum lift-to-drag ratio, while the compass algorithm produced the worst value.
Fig. 4.20 Effect of algorithms on lift to drag ratio.
Those variances arise from the specific behavior of each algorithm; the differences between the various algorithms of the direct search method are largely in the choice of the step length and search direction. For example, in compass algorithm, first choose an initial point (x0,y0) for 2D or ( x0,y0,z0) for 3D, an initial step length Δ, compute the objective function at initial point then at east, west, north, south respectively. If one of these steps yields to a smaller f(x ,y) for 2D,or f(x ,y ,z)for 3D there is new iteration (x1,y1) or (x1,y1.z1) as shown in Figure 4.21 . The calculation steps continue by the same initial step length and search direction until converge optimal shape, while the step length and search direction are variables depend on objective function value at Torczon and simplex algorithms.
Fig. 4.21 New Iteration (x1, y1)
As we mentioned from previous description, the compass algorithm has few steps as compared to other algorithms such as the Simplex and Torczon algorithms. These latter two need more steps to approach a minimum objective as explained by (Torczon, 2009), (Dantzig, 1953). This makes the compass algorithm easy to describe, easy to implement. Also, it may quickly approach a minimum, but may be slow to detect it, if the step size is large.
This seems clear from present work, that the number of design approaches in the compass algorithm are less than the other algorithms; for example at angle of attack (0, 4) degrees the maximum number of designs were 31 and 32 respectively in compass algorithm, while in Torczon, there were 40 and 41 at 0 and 4 degrees respectively, and in simplex algorithm, there were 40 and 44 for 0 and 4 degree respectively. This means that the convergence of the solution for compass algorithm is achieved faster.
As we have mentioned before, the objective of current work is the improved the value of L/D ratio at landing and take-off flight conditions by aerodynamic shape optimization using grid deformation .
As shown in Figures 4.22-4.24, it is clear that the optimization system is successful in achieving the goal. In the comparison between L/D ratio before optimization and after optimization, we see clearly how values of L/D ratio increases with the angles of the attack varying from -4 to 20 degrees. The highest percentage of improvement was 33.919% at angle of attack 4 degrees in simplex algorithm, and the lowest percentage of improvement was 6.925% at 0 degree angle of attack in compass algorithm, as presented in appendix A.
Figures 4.25-4.27 show the final shape for airfoil after optimization processes. These figures show the influence of different algorithms on final shape optimization. As compared to the initial configuration, there are miniscule variations in the shapes of the airfoil and the flap.
Fig. 4.22 Comparion L/D ratio before and after optimization for Compass algorithm
Fig. 4.23 Comparion L/D ratio before and after optimization for Simplex algorithm
Fig. 4.24 Comparion L/D ratio before and after optimization for Torczon algorithm
As we mentioned before every algorithm has specific behavior and search method. The compass algorithm has bounded search direction and length size, while the simplex and Torczon algorithms have unbounded length size and search direction that are shown clearly in Figures 4.26-4.27.
Both the algorithms, the Simplex and the Torczon, produced approximately the same optimal shape. This is a very useful observation because it probably translates to the intrinsic similarities of the two algorithms in terms of computational methods. Nevertheless, their L/D ratio plots are also essentially identical.
Fig. 4.25The shape after optimization ( compass algorithm at 4deg)
Fig. 4.26 The shape after optimization( simplex algorithm at 4deg)
Fig. 4.27 The shape after optimization (Torczon algorithm at 4deg)
V. Conclusions
Everything in the design process is implemented using the CFD-based software Fluent. Different approaches in attempting to optimize the aerodynamic design are suggested in this study. The design to be studied is basically a low speed, air-foil/flap combination.
In this study, the main objective is achieving the optimum lift-to-drag ratio. There are five (5) approaches, or strategies, that are included in this study for the optimization of the aerodynamic design. The five (5) approaches are the following:
1.) Changing the position of the control points
2.) Changing the number of control points
3.) Applying the Compass Algorithm
4.) Applying the Simplex Algorithm
5.) Applying the Torczon Algorithm
In each approach, the lift-to-drag ratio, or simply L/D ratio, is closely investigated. This is a very practical criterion since it greatly defines the performance of any aerodynamic structure. Simulations are run using the Fluent program. The L/D ratio is plotted against the angle of attack (in degrees).
- Changing the Position of the Control Points
The first strategy employed in the design is changing the position of the control points. The control points are critical in the design of an aerodynamic structure since they determine the factors in the system to be measured. Three (3) locations of the control points are examined in the study. The positions (A, B and C) are illustrated in Figure 4.28.
Fig 4.28 Location of the Control Points
Location A is directly in front of the flap and is very close to the air foil but not under the overlap. Location B is above the flap and is very close, directly below the overlap. Location C is above the flap and is quite far from the air foil. Location B is halfway between locations A and C. These locations are chosen strategically so as to achieve different measurements of L/D ratio. Also, these are the critical positions where in manipulations of the parameters in these positions could possibly translate to changes in the performance of the structure.
Figure 4.29 shows the L/D ratio graph against the angle of attack. Measurements are extended up until angle of attack = 16 degrees. The results of the three locations are all plotted in one graph.
This figure shows that at negative angles, the L/D ratio of locations A and C are slightly higher than location C, although the difference is not that significant. Across the majority of the angle measures, the values are virtually the same.
Fig. 4.29 L/D ratio at different locations
This test shows that changing the location of the control points does not affect the L/D ratio of the design structure.
- Changing the Number of Control Points
The second approach in this study is changing the number of control points. Intuitively, this criterion will affect the performance of the air foil structure since more system points are manipulated throughout the process (Gill, 1981).
There are three cases. The first case is using four (4) control points; the second case is using eight (8) control points; the third case is using twelve (12) control points. The L/D ratio is plotted against the angle of attack up to angle of attack = 20 degrees.
Fig. 4.30 L/D ratio for 4, 8, and 12 control points
Figure 4.30 shows the results for the different numbers of control points. At negative angles, the values of the L/D ratio of the three cases are virtually the same. At angles in the range [2,8] degrees, the third case is slightly higher than the other two, although this difference is not significant. Also, at this range, the third case achieved an L/D ratio slightly greater than 25. At angles above 10, the L/D ratios of the three cases are virtually the same.
The plot shows a pattern implies that the L/D ratio is slightly increased when the number of control points is increased. This pattern may be further verified by testing larger number of control points. The changes may be insignificant in value, but these miniscule changes would translate to higher energy efficiency. Thus, it can be concluded that the number of control points would affect the performance of the aerodynamic design.
- Applying the Compass Algorithm
Utilization of optimization algorithms is essential in establishing the foundation of this study. The first algorithm to be studied is the Compass algorithm.
The Compass algorithm is incorporated in the system using the Fluent software. The L/D ratios before and after the optimization is measured; measurements are done up until angle of attack = 20 degrees. Figure 4.31 shows the graphs before and after the application of the algorithm.
Fig. 4.31 L/D ratio of Compass Algorithm, before and after optimization
The L/D ratio is increased significantly all throughout the range of angle of attack. The maximum L/D ratio is 25, which is at angle of attack ≈ 5 degrees, wherein the increase in L/D ratio is approximately 2.5. At angles of attack between 8 to 12 degrees, the increase in L/D ratio is about 5.
The plot shows that the Compass algorithm is very effective in optimizing the value of the L/D ratio. Increase of 5 in the L/D ratio can be reached using this algorithm.
- Applying the Simplex Algorithm
The second algorithm studied is the Simplex algorithm. This algorithm is implemented in the design by changing the program options in Fluent.
Fig. 4.32 L/D ratio of Simplex Algorithm, before and after optimization
Figure 4.32 shows the results of the Simplex algorithm approach. The plots of with and without the algorithm are graphed in one plot. The measurements are done up to angles of attack = 20.
This plot shows that Simplex algorithm significantly increased the L/D ratio of the system. The maximum L/D ratio achieved is 25, which is at angle of attack = 5 degrees, with increase of ≈ 2.5 in the ratio. The minimum increase is ≈ 1. At angle of attack from 7 to 12, the increase in L/D ratio is ≈ 5.
The simplex algorithm is proven to be useful in optimizing the L/D ratio value. The range of angle of attack of maximum increase in L/D ratio is slightly higher than that of the Compass algorithm.
- Applying the Torczon Algorithm
The third and last algorithm tested is the Torczon algorithm. This algorithm is introduced in the system using the Fluent software. The L/D ratio is measured across angle of attack values up to 20 degrees.
Figure 4.33 shows the results of using the Torczon algorithm.
The maximum L/D ratio achieved is ≈ 25. This is at angle of attack = 5 degrees. The increase of L/D ratio at this point is ≈ 3. The maximum increase in L/D ratio is ≈ 5, which is achieved at angles between 9 and 13 degrees.
Fig 4.33 L/D ratio of Torczon Algorithm, before and after optimization
The Torczon algorithm is also an effective approach in increasing the L/D ratio of the aerodynamic structure. The value of the L/D ratio is increased in all the values of angle of attack.
- Comparison of the Compass, Simplex and Torczon Algorithms
Figure 4.34 shows the interleaving plots of the three algorithms.
At angle of attack = 5 degrees, the Torczon algorithm achieved a slightly higher L/D ratio. This slight difference could translate to magnified effects on the performance of the system. Nevertheless, the contours of the plots of the three algorithms are virtually the same.
Fig. 4.34 L/D ratio of Compass, Simplex, and Torczon Algorithms
This just shows that all the three algorithms are effective in optimizing the L/D ratio. The performances of the algorithms are similar. The effects of the three algorithms on the L/D ratio are virtually the same. The increase in L/D ratio, which ranges from 1 to 5, translates to decrease in energy costs of the entire system. Thus, it can be concluded that all these optimization techniques are meet the objective, which is attaining higher lift-to-drag ratio.
Further advances of this work could be the combination of any of these techniques to achieve better design optimization, i.e., Torczon algorithm using 14 control points, etc.
Conclusion
This study has tested and proven the effectiveness of five (5) different optimization strategies. Changing the location and the number of control points both affect the L/D ratio of the system. Compass, simplex, and Torczon algorithms are even more effective in optimizing the design. Combinations of these strategies could be a possible venture in the future of further optimization of aerodynamic systems.
REFERENCES
Andreoli, M., Janka, A., Desideri, J.A., Free-form-deformation parameterization for
multilevel 3D shape optimization in aerodynamics. INRIA, No.5019, November 2003.
Dantzig, G. B., Orden, A., Wolfe, P. The Generalized Simplex Method For Minimizing a Linear Form Under Linear Inequality Restraints. RAND Corporation, Copyright 1953.
Hong, L., Nelson, B., Discrete Optimization via Simulation Using Compass. Operations Research Vol. 54, No. 1, January-February 2006, pp. 115-129
Kaku, Michio (1999). Visions: How Science Will Revolutionize the 21st century and Beyond. New York: Oxford University Press
lift-to-drag ratio. (2011). In Encyclopædia Britannica. Retrieved from http://www.britannica.com/EBchecked/topic/340373/lift-to-drag-ratio
Gill, P. E., Murray, W., Wright, M. H., Practical Optimization, Academic Press, London, 1981.
Torczon, V., Lewis, R.M. Active set identification for linearly constrained minimization without explicit derivatives, SIAM Journal on Optimization, Volume 20, Issue 3, 2009, pages 1378-1405.
Appendix A. Algortihm Fluent Results
COMPASS ALGORITHM FLUENT RESULTS
| alpha |
L |
CL |
D |
CD |
(L/D)after |
(L/D)before |
improvement% |
|
-4
|
-155.328
|
-0.19831
|
20.385
|
0.02602
|
-7.6197204
|
-9.36422
|
18.629
|
|
0
|
193.67516
|
0.247414
|
15.52708
|
0.0198238
|
12.4733794
|
10.08724
|
23.655
|
|
4
|
572.0383
|
0.730338
|
23.95354
|
0.03058219
|
23.8811591
|
22.33445
|
6.925
|
|
8
|
922.85634
|
1.178237
|
41.2369
|
0.052648
|
22.3793821
|
18.7843
|
19.139
|
|
12
|
1206.4263
|
1.54027
|
71.4266
|
0.09119245
|
16.8904344
|
13.231
|
27.658
|
|
16
|
1362.6852
|
1.73978
|
117.0807
|
0.1494804
|
11.6388542
|
10.0595
|
15.700
|
|
20
|
1412.675
|
1.7990488
|
181.91876
|
0.20324918
|
7.76541683
|
6.2311
|
24.624
|
SIMPLEX ALGORITHM FLUENT RESULTS
| alpha |
L |
CL |
D |
CD |
L/D)after |
(L/D)B |
improvement% |
| -4 |
-141.052 |
-0.18 |
19.883 |
0.0253 |
-7.0941005 |
-9.36422 |
24.242 |
| 0 |
209.3865 |
0.267329 |
15.5001 |
0.019789 |
13.5087193 |
10.08724 |
33.919 |
| 4 |
593.6022 |
0.75786 |
24.3635 |
0.031105 |
24.3644058 |
22.33445 |
9.089 |
| 8 |
942.13833 |
1.202855 |
42.5575 |
0.0543344 |
22.1380093 |
18.7843 |
17.854 |
| 12 |
1219.0726 |
1.556425 |
70.95902 |
0.090595 |
17.1799526 |
13.231 |
29.846 |
| 16 |
1399.2947 |
1.7865207 |
118.31805 |
0.15106013 |
11.8265531 |
10.0044 |
18.213 |
| 20 |
1413.675 |
1.80488 |
182.91876 |
0.233537 |
7.72843092 |
6.2311 |
24.030 |
TORCZON ALGORITHM FLUENT RESULTS
| alpha |
L |
CL |
D |
CD |
L/D)after |
(L/D)B |
improvement% |
|
-4
|
-150.766
|
-0.192488
|
19.35581
|
0.024712
|
-7.7891858
|
-9.36422
|
16.820
|
|
0
|
202.4986
|
0.258536
|
15.90276
|
0.0203035
|
12.7335507
|
10.08724
|
26.234
|
|
4
|
578.8115
|
0.73898
|
23.36652
|
0.02983
|
24.7709757
|
22.33445
|
10.909
|
|
8
|
945.22
|
1.2067903
|
42.8034
|
0.054648
|
22.0828252
|
18.7843
|
17.560
|
|
12
|
1226.154
|
1.56546
|
73.4578
|
0.093785
|
16.691951
|
13.231
|
26.158
|
|
16
|
1409.3438
|
1.7993508
|
120.0749
|
0.153303
|
11.7372057
|
10.2303
|
14.730
|
|
20
|
1409.014
|
1.7989
|
184.38187
|
0.235405
|
7.64182509
|
6.2311
|
22.640
|
