APPLICATION OF VARIABLE-FIDELITY MODELS TO AERODYNAMIC OPTIMIZATION

Applied Mathematics and Mechanics (English Edition), 2006, 27(8):1089 1095 c Editorial Committee of Appl. Math. Mech., ISSN 0253-4827 APPLICATION OF VARIABLE-FIDELITY MODELS TO AERODYNAMIC OPTIMIZATION XIA Lu (g ), GAO Zheng-hong (p»ù) (School of Aeronautics, Northwestern Polytechnical University, Xi an 710072, P. R. China) (Communicated by DENG Xue-ying) Abstract: For aerodynamic shape optimization, the approximation management framework (AMF) method is used to organize and manage the variable-fidelity models. The method can take full advantage of the low-fidelity, cheaper models to concentrate the main workload on the low-fidelity models in optimization iterative procedure. Furthermore, it can take high-fidelity, more expensive models to monitor the procedure to make the method globally convergent to a solution of high-fidelity problem. Finally, zero order variable-fidelity aerodynamic optimization management framework and search algorithm are demonstrated on an airfoil optimization of UAV with a flying wing. Compared to the original shape, the aerodynamic performance of the optimal shape is improved. The results show the method has good feasibility and applicability. Key words: aerodynamic optimization; variable-fidelity; approximation management framework Chinese Library Classification: V221.3; O221 2000 Mathematics Subject Classification: 76B07; 76B47; 76B75 Digital Object Identifier(DOI): 10.1007/s 10483-006-0809-z Introduction There are many kinds of aerodynamic analysis methods applied to aerodynamic numerical optimization, such as engineering methods, potential flow equations, Euler equations and N-S equations, etc. The Euler equations could well simulate the shock wave in the flow field and flow properties related to nonlinear effects. The N-S equations could provide information about viscous effects and flow separation. By using these two methods, accurate numerical solutions can be attained, but the compute expense is huge. Using engineering methods, potential flow equations, although the compute expense is relatively less, the credibility of the solutions is relatively lower. In this paper, those models, which can describe the physical phenomena more accurately, such as Euler equations and N-S equations, are called the high-fidelity models and those which describe the physical phenomena relatively less accurately, such as potential flow equations, are called the low-fidelity models. For the optimization problem, since there is an expensive iterative procedure, the analysis methods for different objectives should be efficient, fast, stable and accurate. Since the use of the high-fidelity models is usually expensive, they are hard to apply to optimization design especially 3D complex configuration design, while the optimal solutions by using the low-fidelity models are not so credible. If the high-fidelity models and the low-fidelity models could be well organized and managed during the optimization procedure, the expense relied exclusively on the high-fidelity models can be alleviated and the fidelity of optimal solution relied exclusively on the low-fidelity models can be increased. Received Jul.6, 2004; Revised Dec.26, 2005 Project supported by the National Natural Science Foundation of China (No.10502043) Corresponding author XIA Lu, Doctor, E-mail: xialu@nwpu.edu.cn

1090 XIA Lu and GAO Zheng-hong Approximation management framework (AMF) [1] is based on the trust region methodology. It can provide a measure of the low-fidelity models. According to the measure the design system can provide a response, whether the low-fidelity model gives good or bad prediction of high-fidelity model s actual behavior. Furthermore, the low-fidelity model can be re-calibrate occasionally. So, the workload of the computation during iterations of search procedure concentrates on the calculations of the low-fidelity models and the convergence to a solution of the high-fidelity problem can be guaranteed. 1 Trust Region Methodology The idea of trust region methodology is based on using linear function fitting and approaching nonlinear function. Using the concept of local linearization, in a neighbor region of a point, the properties of an arbitrary nonlinear function can be described by a linear function. That is to say, in a certain region, the linear function can fit the original function to an acceptable degree of fidelity. So, the neighbor region of the point is called trust region in which a linear function can fit the nonlinear function. Applying the concept of trust region to optimization problem, the principal idea is to build an approximation of original optimization problem in a neighborhood of an original design point, and the computation cost of the approximation is lower. In the neighborhood, which is also the trust region, optimization is carried out to obtain the local optimal point by using the approximation to get the object value. And then, after updating the trust region and the approximation according to some principle, the local optimal point replaces the original one to carry out the next round. That is to say, for a single objective with unconstrained optimization problem: min f(x), the general strategy of trust region methodology is as follow: Step 1: Give original value X 0 R n,δ 0 > 0, i =0. Step 2: Search a step which length satisfies s i Δ i by solving the approximation f a (X), where is some norm of R n. Step 3: If f(x) satisfies some decrease condition with the step s i,thenx i+1 = X i + s i ; otherwise, X i+1 = X i. Step 4: Give Δ i+1, i = i +1,gotoStep2. By convergence analysis [2 4], the sufficient condition for the global convergence of the trust region methodology is that s i must satisfy as follows: (i) There exists δ>0, independent of i, forwhichthesteps i satisfies s i δδ i ; (1) (ii) FCD condition (Fraction of Cauchy Decrease): there exist β>0andc>0, independent of i, forwhichthesteps i satisfies ( f(x i ) f ai (X i + s i ) β f(x i ) min Δ i, f(x ) i). (2) C Since the approximation of original objective function is used to get object value, the computational cost is reduced, the optimization efficiency is increased and the optimal point of original problem could be obtained. Obviously, the key to determine the optimization quality and efficiency is how to build the approximation and define the trust region. Usually, the approximation f a is obtained by evaluating f in a neighborhood of an original design point at selecting one or several design points and using the Taylor series expansion or interpolated fitted function. Using such method, the number and value of the design sites immediately affect the approximation degree of the approximation function and the size of trust space, and then affect the optimal result and efficiency. For a complex multi-variable problem or a highly nonlinear problem, since the approximation degree may be low or the

Application of Variable-Fidelity Models 1091 size of trust space is small, using such algebraic smoothing methods may lost the inherence advantage. On the other hand, for practical physical problem, usually, different physical or mathematical models could be built according to different requirements. That is to say, there are high-fidelity and low-fidelity models for physical problem itself. So, those variable-fidelity models of physical problem itself could be used to build corresponding AMF(Approximation Management Framework) based on trust region methodology. And then combining searching algorithms, an efficient and accurate aerodynamic optimization method will be built. 2 AMF N. M. Alexandrov et al. first introduced AMF [4] to structure optimization, which based on trust region methodology to manage and organize variable fidelity models. The conceptual distinction between AMF and conventional optimization is showed in Fig.1 and Fig.2. In conventional optimization, the optimizer and the analysis software exchange information as follows: the analysis supplies the optimizer with objective and derivative information f, f while the optimizer produces new values of the design variables X for re-analysis. If evaluation of the problem function and derivatives involves a simulation of high-fidelity, repeated consultations with analysis required by the optimizer are expensive. Now suppose one also has a low-fidelity model of the same phenomenon, denoted by f a, f a. In a conceptual AMF scheme, the optimizer receives the function and sensitivity information f a, f a from the lowerfidelity model. Expensive, high-fidelity computations proceed outside the optimization loop and serve to re-calibrate the lower fidelity model occasionally, based on a set of systematic criteria [1]. Fig.1 Conventional optimization Fig.2 AMF According to the conformation rule of approximate models, there are two main kinds of AMF: zero-order and first-order consistency [4, 5]. Zero-order consistency condition is that at local point X c, the approximation f a agrees with actual objective f: f a (X c )=f(x c ). (3) First-order consistency condition is that at local point X c, the approximation f a and its first derivatives f a agree with those of the actual objective f and its derivatives: f a (X c )=f(x c ), f a (X c )= f(x c ). (4) The condition could be achieved by a correction technique introduced by Chang et al [6].

1092 XIA Lu and GAO Zheng-hong 3 Application of AMF to Aerodynamic Optimization AMF was applied in structural optimization firstly, and so the systematic responses to the accurate degree of the approximations, are determined according to variable-fidelity models of structural optimization. For the aerodynamic optimization, since the objective function is hard to be defined as an explicit form of the design variables, the value is obtained usually by solving corresponding flow equations. So, the low-fidelity models of physical problem could be used as the approximations of high-fidelity models and the AMF of aerodynamic optimization could be built. In Fig.3, the flowchart of zero-order AMF used in this paper is illustrated. Firstly, the start design point X 0 and the original trust region radius Δ 0 could be determined according to experiences. Secondly, at the original point X 0, the values of high-fidelity and low-fidelity models should be calculated to build the zero-order approximation model f a of the high-fidelity model f by using Eq.(3). Then, in the original trust region, the low-fidelity model is used to search an optimal point X + while high fidelity-model serves to re-calibrate the lower-fidelity Fig.3 Zero-order AMF

Application of Variable-Fidelity Models 1093 model. If the low-fidelity model f a does a very good job of predicting the actual behavior of high-fidelity model f, the radius of trust region Δ i can be increased or a much lower-fidelity model can be used to increase the optimization efficiency at the next optimization iteration. Furthermore, the optimal solution X + can be accepted. If the low-fidelity model does a bad job of predicting the actual behavior, including f actually increased or f did decreased but not as much as predicted by f a, the radius of trust region Δ i should be decreased and the optimal solution X + should not be accepted at the next optimization iteration. If the low-fidelity model f a does an acceptable job of predicting the actual behavior, the radius of trust region Δ i can remain unchanged and the optimal solution X + can be accepted at the next optimization iteration. Since the whole optimization procedure should be controlled quantificationally, the quantificational expression to define the approximation degree of the low-fidelity model to the highfidelity model should be provided: R = f(xi) f(x+) f(x i) f a(x +). The value of R reflects the comparison of results from the approximation to the high-fidelity model. If R 1, it means the low-fidelity model f a does a very good job of predicting the actual behavior of high-fidelity model while if R 0evenR<0, it means the low-fidelity model does a bad job of predicting the actual behavior. So, the approximation degree could be distinguished and then the size of trust region could be adjusted by selecting 0 <η 1 <η 2 < 1. In the whole optimization procedure, the low-fidelity model does not remain unchanged but updates to reflect current design space. Along with every optimal search the more accurate objective value is obtained by high-fidelity model. Since that, when using AMF, even if the low-fidelity model can not capture a particular feature of the physical phenomenon to the same degree as the higher-fidelity model, it is enough that the low-fidelity model have satisfactory predictive properties of finding a good direction of improvement. That is to say, a low-fidelity model may not capture all the important feature of the higher-fidelity model, but it is enough that the low-fidelity model can predict a satisfying step of improvement of the objective. Since the bulk of the computational expense involves calculations based on low-fidelity models in iterations of optimization procedure, the expense relying exclusively on high-fidelity models can be alleviated and as long as the above convergence conditions can be satisfied, the improvement for high-fidelity design can be guaranteed. In this paper, the high-fidelity model is Euler equations and the low-fidelity model is Green s function method based on potential flow equations [7]. Analyzing the properties of the two models and according to practical computational experiences, the calculation results at low angle of attack obtained by low-fidelity model which is Green s function method are much more accurate, especially the lift coefficient. Since in this paper, the objective is lift-to-drag ratio L/D at low angle of attack, the criterion of R which means the quantificationally distinction of the approximation degree for low-fidelity models follows the principle that the tolerance of these two models difference is widen, which means the size of the region reflects where the low-fidelity model f a does a very good or acceptable job of predicting the actual behavior of high-fidelity model could be defined much bigger. So, in this paper, η 1 and η 2 are selected relatively smaller as η 1 =0.10 and η 2 =0.60 while the size of the original trust region which is the original design space is defined relatively bigger [7]. 4 Results and Discussion To validate the feasibility of above method, computational analysis of an example for the design of UAV with a flying wing is carried out to obtain high aerodynamic efficiency. The design variables are parameters describing the airfoils at wing root and wing tip and the analytic function linear combination method [8] is used to describe the parameters of the airfoils. The objective functions are maximizing the lift-to-drag ratio L/D. According to structure, intensity, the feature of the objective functions and the engineering limits, the constrains are

1094 XIA Lu and GAO Zheng-hong decided: the low and up bounds of the design variables; the area and the maximum thickness of the airfoil remain unchanged; pitching moment coefficient can not decrease; pitching moment derivative can not increase; the lift coefficient can not decrease. Search algorithm is the genetic algorithm. The population size is 80 and the number of maximum generation is 20 at every optimization iteration. The low-fidelity model is used to carry out 3 optimization iterations, the high-fidelity model is used to carry out 4 times of re-calibration calculation including original shape, the actual CPU times are approximately 82.6 hours on a Pentium IV PC, and then the convergence requirement is satisfied. Figure 4 shows the comparison of airfoil shape before and after optimization. Table 1 is the comparison of aerodynamic performance. Figure 4 and Table 1 show that the optimal shape has better aerodynamic performance and the optimization goal is achieved. If the conventional optimization exclusively with the high-fidelity to get objective values and the same genetic algorithm with the same control parameters are used during the optimization search, the CPU times will be approximately 1060 hours. It shows that the optimization efficiency is increased using AMF. Fig.4 Comparison of airfoil shape Table 1 Comparison of aerodynamic performance Aerodynamic Original performance shape Optimal result First iteration Second iteration Third iteration Low-fidelity High-fidelity Low-fidelity High-fidelity Low-fidelity High-fidelity Lift coefficient C L 0.2920 0.2950 0.2930 0.3500 0.3100 0.3100 0.2930 Drag coefficient C D 0.0207 0.0190 0.0200 0.0220 0.0205 0.0188 0.0179 Pitching moment coefficient C m Pitching moment 0.0400 0.0600 0.0440 0.0580 0.0450 0.0800 0.0700 derivative Cm α Lift-to-drag 0.0020 0.0002 0.0016 0.0005 0.0017 0.0004 0.0015 ratio L/D 14.10 15.50 14.65 15.63 15.10 16.50 16.40 5 Conclusions Many physical phenomena in engineering design can be described by high-fidelity models. Whereas, the use of high-fidelity models, such as the N-S equations or those based on fine

Application of Variable-Fidelity Models 1095 computational meshes, in iterative procedure can be considerably expensive. On the other hand, the use of corresponding lower-fidelity models alone does not guarantee improvement for higher-fidelity design. AMF based on the trust region methodology by constructing framework of variable fidelity models, can alleviate the expense relying exclusively on high-fidelity models. In aerodynamic optimization, variable-fidelity models can be managed by using AMF and be taken full advantage. In this paper, how to build aerodynamic optimization method by using AMF was studied elementarily and the conclusion could be drawn: (i) The value of R reflects the approximation degree for low-fidelity models to high-fidelity models. The selection of parameters η 1 and η 2 which are distinguish the different range of R mainly follows the tolerance of variable-fidelity models difference in design and practical computational experiences. (ii) For different aerodynamic optimization problem, since the bulk of the computational expense involves calculations based on low-fidelity models in iterations of optimization procedure, the low-fidelity model should be chosen according to the complexity of the problem and the cost that can be afforded while the high-fidelity model is chosen usually according to the precision of the design requirement. Furthermore, the change of R may be used to test whether the variable-fidelity models are properly chosen. If the value of R is always on the low side, the fidelity of the low-fidelity model should be increased while if the value of R is always on the high side, a much lower-fidelity model could be used as approximation. Generally, the high-fidelity model keeps unchanged to guarantee the accuracy of the optimal solution. (iii) There are still many problems to be solved to improve the optimization design methods, such as how to build AMF of multi-level variable-fidelity models, how to improve the searching iterations with the low-fidelity model and adjust the re-calibration with the high-fidelity model to increase the optimization efficiency, etc. References [1] Alexandrov N M, Lewis R M, Gumbert C R, Green L L, Newman P A. Optimization with variable-fidelity models applied to wing design[c]. AIAA Paper 2000 0841. 38th AerospaceSciences Meeting and Exhibit, Reno, NV, 2000. [2] Shi Guangyan, Dong Jiali. Optimization Method[M]. Higher Education Press, Beijing, 2000 (in Chinese). [3] Yuan Yaxiang, Sun Wenyu. Optimization Theory and Method[M]. Science Press, Beijing, 1999 (in Chinese). [4] Alexandrov N M, Dennis J E Jr, Lewis R M, et al. A Trust-region framework for managing the use of approximation models in optimization[j]. Structural Optimization, 1998, 15(1):16 23. [5] Booker A J, Dennis J E Jr, Frank P D, et al. A rigorous framework for optimization of expensive functions by surrogates[j]. Structural Optimization, 1999, 17(1):1 13. [6] Chang K J, Haftka R T, Giles G L, et al. Sensitivity-based scaling for approximating structural response[j]. Journal of Aircraft, 1993, 30(2):283 288. [7] Xia Lu. Aerodynamic and Stealth Synthesis Optimization Design for Aircraft Configuration[D]. Ph D Dissertation, Northwest Polytechnical University, Xi an, 2004 (in Chinese). [8] Hicks R, Henne P. Wing design by numerical optimization[j]. Journal of Aircraft, 1978, 15(7):407 413.