Inverse design of low boom conf igurations using proper orthogonal decomposition and augmented Burgers equation

Yidin ZHANG , Jingto HUANG , Zhenghong GAO ,*, Cho WANG ,Bowen SHU

a School of Aeronautics, Northwestern Polytechnical University, Xi'an 710072, China

b Computational Aerodynamics Institute, China Aerodynamics Research and Development Center, Mianyang 621000, China

KEYWORDS Aeroacoustics;Augmented Burgers equation;Low boom conf iguration;Optimization;Supersonic aerodynamics

Abstract Mitigation of sonic boom to an acceptable stage is a key point for the next generation of supersonic transports.Meanwhile,designing a supersonic aircraft with an ideal ground signature is always the focus of research on sonic boom reduction. This paper presents an inverse design approach to optimize the near-f ield signature of an aircraft, making it close to the shaped ideal ground signature after the propagation in the atmosphere.Using the Proper Orthogonal Decomposition(POD)method,a guessed input of augmented Burgers equation is inversely achieved.By multiple POD iterations, the guessed ground signatures successively approach the target ground signature until the convergence criteria is reached. Finally, the corresponding equivalent area distribution is calculated from the optimal near-f ield signature through the classical Whitham F-function theory. To validate this method, an optimization example of Lockheed Martin 1021 is demonstrated. The modif ied conf iguration has a fully shaped ground signature and achieves a drop of perceived loudness by 7.94 PLd B. This improvement is achieved via shaping the original near-f ield signature into wiggles and damping it by atmospheric attenuation. At last, a nonphysical ground signature is set as the target to test the robustness of this inverse design method and shows that this method is robust enough for various inputs.

1. Introduction

The physical phenomenon of sonic boom attracted attention in 1947 when the trail aircraft Bell X-1 crossed sonic barrier for the f irst time in human history.However,over almost ten years after this milestone moment, the physical formation of sonic boom was understood as the natural nonlinear evolution of the near-f ield pressure as it propagated away from the aircraft1(see Fig. 12).

The sad failure of the f irst generation of supersonic aircraft,such as Concorde and Tupolev TU-144, is largely due to the intensive noise resulted from sonic boom. It has long been a constant pursuit for aircraft designers to achieve transports that f ly at supersonic speed quietly and economically.To meet the harsh requirement of noise level,a large number of studies have been conducted which could be roughly divided into two categories: sonic boom simulation or assessment3,4and sonic boom optimization.5,6

The modern sonic simulation approach is normally solved by two steps.First,the near-f ield overpressure is obtained.Second, the near-f ield overpressure is extrapolated along the ray path to a concerned altitude (normally the ground). The Whitham F-function theory7from linearized supersonic aerodynamics is widely used in the f irst step to get the near-f ield signature while Computational Fluid Dynamics (CFD) is gradually adopted in recent years8accounting for the nonlinear effects around aircrafts of complex conf igurations. For the second step, wave parameter method9and augmented Burgers equation10are the mainstream of today's approach.The main improvement of the latter is that it takes into consideration the attenuation of propagation in real atmosphere.

The prototype f light test in the Shaped Sonic Boom Demonstrator (SSBD)11verif ied the effectiveness of aircraft shaping to modify the boom ground signature,which provides a powerful factual basis for sonic boom optimization. Compared with looking for an optimal signature in countless waveforms by forward search, it seems to be more eff icient to specify an ideal target signature and achieve it. For that reason, inverse design approaches are mainly used in low-boom designs.Basically,most of the methods for sonic boom inverse design can be categorized into two groups:(A)near-f ield target matching12,13; (B) ground target matching.14,15Besides, Rallabhandi16proposed a novel way to match the equivalent area.The near-f ield matching method designates a target near-f ield overpressure distribution and achieves it by shaping the aircraft while the far-f ield matching method specif ies a target ground signature. Compared with the second method, the near-f ield target matching method is easier to operate because the relationship between the near-f ield signature and aircraft conf iguration, such as the distribution of equivalent area, is straightforward which is helpful for the shaping of aircraft.However,a well-shaped near-f ield signature does not necessarily ensure an optimal ground signature after a propagation in the atmosphere. In this work, a desired ground signature is given to match.

Fig. 1 Sonic boom generation and propagation.2

The critical point of this inverse design framework is to f ind the corresponding near-f ield signature of the given ground signature, or a near-f ield signature adjacent to it in case it does not exist physically. Based on classical wave parameter method, the inverse design approach has been proposed by George and Seebass.17However, there are few counterparts for high-f idelity augmented Burgers equation method. Li and Rallabhandi18presented a novel reversed augmented Burgers to reversely propagate the signature from ground to a nearf ield position. Despite the straightforwardness of this method,the reverse diffusion equation is inherently ill-posed and hence needs regularization to stabilize the numerical solution. Like the artif icial viscosity in CFD, the regularization process contaminates the real solution.Moreover,if the given ground signature is non-physical,some other numerical problems may be caused as it does happen in other inverse problems of partial differential equations.19,20

The Proper Orthogonal Decomposition (POD) is widely used in a number of f ields, including dimensionality reduction,21image inpainting,22pattern recognition23and signal processing.24In addition, it also yields satisfactory results in aerodynamics, comprising reduced-order model for unsteady aerodynamics,25,26inverse design for airfoils27and analysis of turbulent f lows.28This technique extracts a set of empirical modes from samples, which illustrate the dominant behavior of the concerned system.Impressively,the POD based method,which will be showed in this paper,exerts no restriction on the objective of inverse design, even if it is non-physical.

This paper introduces an extended POD method, so-called Gappy (incomplete) POD (GPOD) method,29,30to the inverse design of sonic boom.In short,the method is stated as follows:given a target ground signature, the optimal near-f ield signature can be determined by appropriate interpolation of known samples. First, the mathematical models used in this work is introduced in Section 2. Then, the optimization framework is constructed in Section 3. Section 4 exhibits an optimization example of Lockheed Martin 1021. Section 5 concludes this paper.

2. Mathematical models

2.1. Proper orthogonal decomposition

The fundamental POD method is introduced here in brief.Detailed deduction and discussion are provided by Berkooz et al.28.

Despite the big success in many nonlinear problems, POD is essentially a linear method. The core thought of it is to f ind a set of optimal POD modes(Φ)to maximize the projection of the raw data on it. Mathematically, it can be described as a maximum problem with a constraint:

where ·〈〉is an averaging operation,which may be time,space,or ensemble average. (·) means the inner product of two vectors and ·｜｜represents the norm induced by inner product:

From Eq. (1), it can be shown that, Φ is the eigenfunction of the correlation tensor:

where U*is the hermitian of U.

The method of snapshots, firstly introduced by Sorovich,31is a great improvement for POD. This method transforms the original analytical problem to a sample-based problem which is easy to solve numerically. Most of today's applications of POD are based on the method of snapshots.

In the linear space spanned by a set of linearly independent snapshots (samples), Φ is simply a linear combination of the M centralized snapshots:

For f inite samples,Eq.(3)can be converted to the following eigenproblem:

where R is the correlation matrix:

The correlation matrix has M eigenvectors:, which are corresponding to M eigenvalues: λ= [λ1,λ2,...,λM]. Based on Eq. (4), the M optimal POD basis vectors (also called empirical basis) are determined. The degree of dominance of each basis is given by the corresponding eigenvalue. Provided the abovementioned eigenvalues are in descending order, the ‘‘energy” captured by the f irst N modes is determined by:

To approximate the desired space, a linear combination of L selected modes is used:

where the selection of modes is determined by Eq. (7), typically, an energy percent of 99% is selected.

POD method also offers a good way to f ill the incomplete data, which is regarded as gappy POD method. The incomplete data can be given in the following form:

where K represents the known elements while X represents the unknowns.In this paper,K stands for the target ground signature and X means the unknown near-f ield signatures.

A set of snapshots are sampled and the f irst L modesare extracted via POD analysis. The selected modes can be reorganized into the following form:

The subscripts K and X represent the components of the basis corresponding to knowns and unknowns respectively.

Then, the known components in the incomplete data is reconstructed by the corresponding POD basis using the least square method:

where Γ= [γ1,γ2,...,γL]Tis the new coordinates of the known data in the coordinates system def ined by the L eigenvectors.Finally, the repaired vectorof GPOD is given by:

It should be noted that,the POD makes no special assumption about the analyzed data. In addition, no restriction is exerted on the incomplete data which ensure the robustness of this method.

2.2. Augmented Burgers equation

The augmented Burgers equation is an accurate physical model to simulate the propagation of sonic boom in a stratif ied real atmosphere with low computational cost compared with direct CFD simulation. Therefore, this method is widely adopted by sonic boom researchers. In the second sonic boom prediction workshop (2nd SBPW),32ten of the eleven participants selected Burgers equation based method as the tool for the propagation cases in that workshop.The form of it is stated here brief ly. A comprehensive introduction is provided by Cleveland.10Eq. (13) is the nondimensional form of it:

where the f ives terms represent nonlinear distortion, classical attenuation, atmospheric stratif ication, geometry spreading and molecular relaxation respectively.The dimensionless pressure is P=p′/p0, where p0is the reference pressure. S is the area of the ray tube.σ is the non-dimensional distance normalized by shock formation distance xsf=/(βω0p0).ρ0,c0are the ambient density and sound speed.β is a constant def ine by specif ic heat ratio γ=1.4, where β=1+(γ-1)/2. The dimensionless time is τ=ω0t′where 1/ω0is the reference time and t′is the reduced time. The classical dimensionless attenuation parameter is def ined by Γ=bω0/(2βp0), where b is the viscosity coeff icient. The dimensionless relaxation time is θv=ω0τυ. The dimensionless time is given by Cυ=/(2c0), where the dimensionless sound speed disturbance is mυ=2(Δc)υ/(c0).

Eq.(13)can be solved by splitting method.The initial equation is split into f ive equations:

Eqs.(14)-(18)are solved sequentially and periodically from the near-f ield to a desired altitude.Lee and Hamilton33showed that the error of splitting is small if the time step is small enough.

3. Inverse design framework

This paper focuses on the development of an eff icient methodology to determine an optimal near-f ield signature for a given ground signature using GPOD. In essence, GPOD of snapshots is a sample-based method. To enhance the optimizer, the sample library is dynamically adjusted during the design process by means of the following two ways: (A) The guessed near-f ield signature that fails to approach the given ground signature is added to the sample library as a calibration sample; (B) The sampling range shrinks as the optimizer advances for f ine search at later stages.In each GPOD process,using the dominant modes extracted from the samples,the target ground signature is f itted. Then, the same f itting coeff icients are applied to the samples to get the predicted nearf ield signature as is introduced in Section 2.1. A f low chart of the whole framework is depicted in Fig. 2. The operations of each step are listed as follows:

Step 1. Initialization: Def ine the initial near-f ield signature to optimize; generate an ideal ground signature as the target for the following steps.

Step 2. Sampling: A set of samples are generated based on Latin hypercube sampling design, the sampling range is:R=(rlb,rub).

Fig. 2 Sonic boom inverse design framework (the objective function will be given in Section 4.2).

Step 3. Gappy POD reconstruction: The guessed near-f ield signature is got by GPOD method. The true ground signature for the guessed near-f ield signature is calculated.

Step 4. Comparison and selection: Select the best near-f ield signature from the samples and the guessed near-f ield signature as the design result for current design loop.

Step 5. Enhancement judgement: Judge whether the cost drops. If it drops, update the current optimum and go to Step 6. If not, then add the current guessed near-f ield signature to sample library as the calibration sample and go to Step 3. If it fails to drop more than 10 times, then shrink the sampling range: R=SR and go to Step 2. S is the constriction factor(0 ＜S ＜1).

Step 6. Convergence judgement: Judge whether the convergence criteria is reached,if it converges then end the optimization, if not go to Step 2.

4. Validation and an optimization example

In this section,the code for the simulation of sonic boom propagation is validated f irstly.Then an optimization example is set and demonstrated. Finally, the results of the optimizer are discussed.

4.1. Validation of code for sonic boom propagation

The accurate physical model is the basis for optimization. To validate the code used in this paper, two cases in 2nd SBPW32are utilized. The calculated ground signature are compared against existing validated code sBOOM34developed by NASA.

The two cases of Lockheed Martin 1021 (LM 1021) and axis-symmetric body in 2nd SBPW are selected due to the reason that most of the participants have close results.The atmospheric conditions for the selected cases are standard atmosphere prof ile and constant relative humidity of 70%across the prof ile. Fig. 3 shows the near-f ield signatures which are extracted from several body lengths away from the aircraft.The axial location is def ined by the distance away from a reference point at the selected altitude (normally a point outside the Mach cone).Often,the perturbation pressure is normalized by the ambient pressure p. Fig. 4 gives the comparison of the ground signatures of both two cases against sBOOM.The signatures in both cases conform very well and validate that the results obtained from the present code are reliable for the following optimization.

4.2. Setting of the optimization example

For the sake of f idelity and convenience, the same case of LM 1021 is selected as the initial conf iguration for the optimizer.The conf iguration of it is illustrated in Fig. 5.32

Before starting the optimization process in Section 3, an interesting nature of the signature of sonic boom is introduced f irst.

Fig. 3 Near-f ield signatures of two cases in 2nd SBPW (rolling angle: 0°).

As is illustrated in Fig.6,if we impose perturbation to two isolated regions in the near-f ield signature, the corresponding ground signature is also disturbed in two isolated regions. In Fig.6,two pairs of red solid boxes and blue dashed boxes represent the corresponding signature sections before and after propagation. It shows that the local disturbance only exert inf luence on a f inite region rather than the whole waveform.From the macroscopic aspect,these two pairs of regions in signature can be regarded as two independent physical based input-output systems. The mechanism behind this phenomenon is that the inf luence of the nonlinear effect and attenuation is limited and only affects the local characteristics of the whole waveform.This nature can be exploited in the following optimization.The inverse design framework only considers the disturbed regions in Fig. 6. Besides, these two pair of regions are treated as two independent subsystems. For instance, the guessed near-f ield signature of the f irst disturbed region only depends on the signatures in the two red solid boxes in Fig.6.

Based on the low-boom principles summarized by Plotkin et al.,35the original ground signature is shaped into a sinelike boom which would remove the audible high frequency energy. The target ground signature is shown in Fig. 7.

The perceived noise level of the ground boom is evaluated by the method proposed by Stevens36and adapted by Shepherd and Sullivan.37This metric is widely used in today's sonic boom research which correlates with human perception of sonic booms very well.The perceived noise level of the shaped target gains a reduction of 8.7 PLdB from 91.93 PLdB to 83.23 PLd B.

Fig. 4 Comparisons of ground signatures in 2nd SBPW against sBOOM (atmospheric conditions: standard atmosphere prof ile,constant relative humidity of 70%, rolling angle: 0°).

Fig. 5 Conf iguration of LM 1021.32

To describe the degree of proximity of the intermediate design ground signature to the target, the Euclidean metric is naturally selected as the objective function:

where D is the design variables, PDesignis the ground signature of the design and PTargetis the target ground signature. The objective function has the dimension of Pascal. N is the number of discrete ground signature points. Although this objective function is ubiquitous in sonic boom design, some questions are still pending and a discussion on it will be given in Section 4.4.

The design variables in this problem is the discrete points in the near-f ield pressure distribution.The number of them is 200 in total. In this case, only the points inside the two disturbed regions are active. The settings of the optimization example are as follows. For the f irst disturbed region (region in red solid boxes in Fig. 6 (a)), 64 design variables are active. And for the second disturbed region (region in blue dashed boxes in Fig. 6 (a)), 42 design variables are active. For both of these two regions, the initial sampling range is:R=(-4×10-3,4×10-3) and the constriction factor in this case is S=0.8. In each sampling, 30 samples are generated and calculated. 99% of energy is chosen as the threshold for POD modes selection.In each optimization,a total of 50 times of sub-iterations are conducted.Since the main purpose of this research is to demonstrate the effectiveness of the method, a relatively sparse grid is used, which requires about 10 seconds for a call of sonic boom simulation routine on a 64-bit Inter Quad Core 3.8 GHz processor with 8 GB RAM.

4.3. Results of the optimization example

To eliminate the impact of the uncertainty of the samples, the optimizer is conducted 200 times in total. By checking the intermediate results of the optimization, it can be found that the designs of the lowest cost do not necessarily have the lowest noise level.Hence both of the design of the lowest cost and the design of the lowest noise level are stored during the optimization loops. The statistical results of them are listed in Table 1.

As reported in Table 1,the mean noise level of the designs is 87.01 PLd B which has an enhancement of 4.94 PLd B compared with the baseline by almost 600 calls of sonic boom propagation model. For the optimal design, a reduction of 7.34 PLd B is achieved, which is regarded as a huge improvement in sonic boom optimization because the perceived loudness of sonic boom is in logarithmic scale.

The ground signature of the design with the lowest noise level is plotted in Fig.8.As can be seen in Fig.8,the optimized ground signature is fully shaped.The sharp shocks are shaped into smooth wiggles across the target signature.

The corresponding designed near-f ield and the iteration history is presented in Fig.9.The former illustrates that the original smooth near-f ield signature is optimized into repeated oscillations. These sharp shocks will be damped by atmospheric attenuation and f inally become comparatively smooth waves near the ground. The latter of Fig. 9 reveals a typical phenomenon in the inverse design of sonic boom. Since the optimizer is driven by objective Eq.(19),undoubtedly,the cost of it declines strictly. However, the perceived noise level does not drop monotonously, although the downward trend of it is apparent. At the last step, a drop of the cost results in a small increase on the noise level.

Fig. 6 A set of disturbed near-f ield signatures and the corresponding ground signatures of LM 1021. (The signatures in red solid boxes and blue dashed boxed represent the corresponding signature sections before and after propagation.)

Fig. 7 Baseline and target ground signature for LM 1021 (a reduction of 8.7 PLdB after shaping).

Fig.8 Comparison of the designed ground signature against the target.

Fig. 9 Design result and iteration history of the optimizer.

Table 1 Statistical results of 200 times optimization.

Once the near-f ield signature is determined, the equivalent area could be obtained by means of the inverse Abel transform18based on the assumption for slender bodies38:

where, x is the coordinate in axial position, R is the near-f ield distance,Ma is the cruise Mach number,γ = 1.4 is the specif ic heat ratio, and d p/p is the non-dimension pressure. Based on Eq. (20), the distribution of the equivalent area is got by numerical integration and is plotted against the initial equivalent area distribution in Fig.10.The equivalent area consists of two parts accounting for the contributions of volume and lift respectively.

Fig.10 shows that,the equivalent area of the design is obviously tightened at the tail section, which means a volume loss for the aircraft if the lift distribution is assumed to be unchanged. In addition, the optimized equivalent area distribution is smoother than the baseline.

4.4. Discussions on the optimization example

In this section, some discussions about the intermediate and f inal results of the optimization is given. First the optimized ground signature is inspected in frequency domain. Then the‘energy distribution' of POD modes and the rationality of the current objective function is investigated. Finally, a nonphysical target ground signature is utilized to test the robustness of the presented method.

To check the result of the optimal design, the shaped ground signature is inspected in the frequency domain by one-third octave analysis.36The frequency spectra of Sound Pressure Level (SPL) and the loudness in the scale of sone is presented in Fig. 11 respectively, where one sone is referenced to a noise of 32 d B at 3150 Hz.As we can see clearly in Fig.11(b), the optimized ground signature reduces the loudness signif icantly at bands around 100 Hz, which are the dominant bands of the noise level.In the frequency domain,the strength of the ground boom is dominated by the bands of the highest SPL, the contributions from other bands are negligible. To attenuate the overall noise level, special attention should be given to noise components near 100 Hz.

Fig. 10 Equivalent area of the designed near-f ield signature.

Fig. 11 Frequency spectra of optimized ground signature against baseline and target. (Bands around 100 Hz are the dominant bands of the noise level.)

Fig. 12 Energy captured by each POD mode. (The f irst 16 modes capture more than 99% of the overall energy.)

Fig. 12 indicates the energy captured by each POD mode when 30 samples are selected.The f irst 16 modes capture more than 99% of the overall energy which are selected to reconstruct the target ground signature.

Fig.13 Cost versus the corresponding noise level when iteration ends. (The noise level here is the loudness corresponding to the waveform at the last iteration rather than the optimal noise level of the whole optimization.)

A question remained in Section 4.3 is that, a drop of the cost of the objective function cannot ensure a drop in perceived noise level.To illustrate this question more thoroughly,the cost and the noise level of all designs at the last step of iteration are plotted in pair in Fig. 13. As is presented in Fig. 13, although there are few f inal designs of high cost but low noise level and vice versa, the relationship between cost and noise level is not very clear. Despite this uncertainty, the presented target has achieved good results in previous optimization. From the author's point of view, the target guides the direction of the optimizer rather than a requirement must be met. For this reason, the storing of both the design closest to target and the design of the lowest noise level during the optimization process is strongly recommended. In addition,blindly pursuing the proximity to the target is meaningless.Furthermore, some advanced objective functions should be proposed to describe the closeness away from the target in term of the frequency domain rather than the stiff Euclidean metric.

In the last part of this section,the robustness of the present method is examined when non-physical solutions are set as the desired ground signature.A zigzag solution is made whose f irst order derivative is discontinuous at inf lection points.This kind of ground signature is physically non-existent because of the occurrence of the atmospheric attenuation. The parameters and convergence criteria are exactly the same as the optimization example in Section 4.2 expect for the desired ground signature. Fig. 14 gives the optimized ground signature against the baseline and target. As shown in Fig. 14, the optimized ground signature matches the target to a great extent,although the target ground signature is non-physical.

Fig. 14 Optimized ground signature matching a physically nonexistent target.

5. Conclusions

An inverse design framework using Gappy Proper Orthogonal Decomposition (GPOD) method combined with augmented Burgers equation was developed and investigated, aiming to shape the ground signature into a desired target. The developed framework was verif ied based on an optimization example to inversely design the distribution of the equivalent area of Lockheed Martin 1021. A non-physical target was generated to test the robustness of the developed method. Some conclusions can be drawn as follows:

(1) The proposed inverse design frame work provides a promising and eff icient avenue to f ind the corresponding near-f ield signature for a given target ground signature based on augmented Burgers equation of high-f idelity.The optimized ground signature achieves a huge enhancement of 7.34 PLd B in perceived noise level in spite of the big difference between the baseline and target signature.

(2) This method is robust enough for various inputs.Even if the target ground signature is non-physical, the present method could still offer a solution which has a ground signature very adjacent to the unreal target.This nature is designer-friendly which does not require rich experience in sonic boom engineering for aircraft designers.

(3) The Euclidean metric of the designed signature to the target is not always consistent to the proximity in frequency domain, although it does guide the direction of the optimization. Some advanced cost function should be proposed to describe to degree of closeness to the target in term of noise perceived level.

Until now, the proposed design methodology focuses on the f ield of purely deterministic approach. Since the atmospheric properties are of natural variability, the application of uncertainty quantif ication and robust design39-41is very demanding.Moreover,some further multidisciplinary research could be conducted on boom/aerodynamic integrated design using multi-objective algorithms.42,43

Acknowledgement

The author thanks Miss Siyi LI for her early work on POD and enlightening discussions with her.