AIRFOIL STALL FLUTTER PREDICTION BASED ON DEEPONET
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摘要: 失速颤振是弹性结构大幅俯仰振动与动态失速气动力耦合所发生的一种单自由度失稳现象, 需有效预测其失稳分岔速度与失稳后的极限环振荡幅值. 针对NACA0012翼型大幅俯仰运动气动力预测问题, 发展了由嵌入门限循环单元或长短时记忆神经网络单元的分支网络(branch net)和主干网络(trunk net)组成的深度算子神经网络(deep operator network, DeepONet)结构. 通过给定大幅俯仰运动下的动态失速CFD气动力数据对深度算子神经网络参数进行训练, 建立了高精度动态失速气动力的数据驱动模型, 并有效预测其他俯仰运动下的非定常气动力. 更进一步, 将基于深度算子神经网络的非定常气动力数据驱动模型与结构动力学方程耦合, 采用数值积分方法预测失速颤振的失稳分岔速度和不同速度下的极限环振荡特性. 结果表明, 在动态失速气动力预测精度方面, 与普通循环神经网络相比, 深度算子神经网络通过引入主干网络结构, 可考虑运动与气动力间的迟滞特性, 气动力预测平均绝对误差降低2%, 误差分散性更低; 在失速颤振预测方面, 极限环振荡幅值误差在2%以内, 增加来流速度输入的深度算子神经网络模型预测误差显著小于固定速度输入的算子模型.Abstract: Stall flutter is a single degree-of-freedom instability phenomenon that occurs due to the coupling of large pitch motion of elastic structures and dynamic stall aerodynamic forces. It is necessary to effectively predict its bifurcation speed and the limit cycle amplitude. To address the problem of predicting the aerodynamic forces during large pitch oscillations of the NACA0012 airfoil, a deep operator network (DeepONet) structure was developed, consisting of a branch net with embedded gated recurrent units or long-short-term memory neural network units and a trunk net. The DeepONet structure was trained using dynamic stall CFD aerodynamic force data for large pitch oscillations, and a high-fidelity data-driven model for dynamic stall aerodynamic forces was established, which effectively predicted unsteady aerodynamic forces for other pitch oscillations. Furthermore, the data-driven model for unsteady aerodynamic forces based on the DeepONet was coupled with the structural dynamics equation, and numerical integration was used to predict the bifurcation speed of stall flutter and the limit cycle oscillation characteristics at different speeds. The results showed that, compared with ordinary recurrent neural networks, the DeepONet could consider the hysteresis characteristics between motion and aerodynamic forces by introducing a trunk net structure, resulting in a 2% reduction in the mean absolute error in predicting aerodynamic forces during dynamic stall. Regarding the prediction of stall flutter, the error in the limit cycle oscillation amplitude was within 2%, and the DeepONet model with inflow velocity input had significantly smaller prediction errors than the operator model without velocity input.
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Keywords:
- stall flutter /
- DeepONet /
- dynamic stall /
- unsteady aerodynamic force /
- neural network
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引 言
动气动弹性问题是指非定常气动力和柔性结构之间相互耦合作用导致的稳定性/动响应问题, 具有重要的研究价值: 一方面气动弹性动响应可以用于振动俘能, 张野等[1]利用翅片超表面将涡激振动转变为驰振, 显著提升了俘能性能; 另一方面动气动弹性会带来结构的大幅振动, 导致飞行器结构破坏, 比如失速颤振现象[2]. 失速颤振是弹性结构在初始大迎角下产生的一种单自由度自激振荡现象. 失速颤振主要出现在直升机旋翼与涡轮叶片中, 大迎角飞行、强阵风等情况下亦有可能诱发机翼失速颤振. 与经典弯扭耦合颤振不同的是, 失速颤振过程伴随着大尺度的流动分离与再附[3], 气动力非线性程度高, 因此对于失速颤振失稳速度和振荡幅值的高精度高效分析十分重要. Dimitriadis等[4]开展了二自由度翼型风洞试验研究, 测量了失速颤振的分岔速度. Bhat等[5]采用能量理论对失速颤振边界进行了辨识. Poirel等[6]在转捩雷诺数下进行了失速颤振风洞试验, 研究了雷诺数对失速颤振特性的影响.
随着计算流体力学(computational fluid dynamics, CFD)仿真技术的发展, CFD被逐渐运用于动态失速与失速颤振研究中. Spentzos等[7]基于CFD方法对三维机翼动态失速特性进行了仿真研究; Yabili等[8]基于OpenFOAM开发了一流固耦合求解器, 仿真求解二维翼型失速颤振问题. 目前针对二维翼型在不同雷诺数下的失速颤振速度和极限环幅值多采用CFD仿真进行研究[9-11]. 然而, 对于失速颤振等流固耦合研究而言, CFD方法存在计算时间长、迭代繁琐等问题. 为克服这些缺点, 提出了非定常气动力降阶模型[12-15], 旨在尽量保证精度的前提下提高计算效率.
降阶模型可分为线性模型与非线性模型两种. 线性模型包括自回归滑动平均模型[16-17]、Volterra级数模型[18]等, 对于线性或弱非线性气动力建模有着良好的效果. 然而对于动态失速这类强非线性数据, 则需要采用以神经网络为代表的非线性数据驱动降阶模型[19-21]进行建模. Wang等[22]建立了基于多保真度数据的机器学习框架, 对动态失速气动力进行机器学习建模, 展现了较好的泛化能力. 对于时域非定常气动力而言, 为时序数据所设计的循环神经网络(recurrent neural network, RNN)也具有良好的效果. Mohamed等[23]开展了二维翼型动态失速试验, 并采用双向长短时记忆(long short-term memory, LSTM)神经网络对实验数据进行学习与预测, 结果证明双向LSTM能够捕捉极端迎角下翼型非定常气动力的物理特性. Dai等[24]使用门限循环单元(gated recurrent unit, GRU)的循环神经网络对翼型动态失速气动力进行学习, 并基于所建立的降阶模型进行了失速颤振幅值与分岔速度预测, 获得良好效果. Li等[25]采用LSTM神经网络对二维翼型非定常气动力进行学习, 并进行气动弹性响应预测. Dou等[26]同样基于LSTM建立了二维翼型非定常气动力降阶模型, 与结构动力学方程进行耦合, 预测跨音速抖振现象.
Lu等[27]基于算子学习的理念, 提出了深度算子神经网络. 深度算子神经网络基于数据对线性或非线性算子进行学习, 具有收敛速度快、泛化能力强的优势, 被用于求解微分方程[28]、边界层不稳定波[29]和裂纹扩展[30]等问题. 同时, 深度算子神经网络具有极强的可扩展性, 其包含的两个子网络可根据需要选择合适的类型. Lu等[31]就提出了深度算子神经网络的多种扩展形式, 在Burgers方程、对流问题、可压缩Euler方程和小幅俯仰振荡翼型表面涡强度等算例上对这些扩展形式进行验证, 表明采用适当扩展形式的深度算子神经网络能够在多种问题中表现出良好的精度. Garg等[32]将动态系统的输入变量与时间变量分别输入深度算子神经网络的两个子网络, 对动态系统的时序响应进行学习与预测.
基于以上背景, 本文基于研究[24]数据, 将GRU和LSTM两种单元与深度算子神经网络相结合, 进行动态失速非定常气动力建模与失速颤振预测. 本文对研究[24]中神经网络的结构进行更改, 以获得更高的预测精度, 并对其中的机理进行了初步的分析.
1. 基于深度算子神经网络的非定常气动力数据驱动模型理论
传统神经网络学习的是数据$I$到数据$O$之间的函数关系$K$
$$ O = K(I) $$ (1) 与之不同的是, 深度算子神经网络学习的是从函数$u(x)$到函数$v(y)$的映射关系${\mathrm{G}}$[27]
$$ v(y) = {\mathrm{G}}(u(x))(y) $$ (2) 式中${\mathrm{G}}$即为算子. 深度算子神经网络中包含了branch net与trunk net两个子网络, 分别以$u$与$y$作为输入. 若$u$为连续函数, 则需要将其在定义域中取点集$[{x_1},{x_2}, \cdots ,{x_n}]$, 采样为离散值$[u({x_1}),u({x_2}), \cdots ,u({x_n})]$作为branch net的输入向量. 采样点的分布可根据需要进行选择, 但需在所有的输入函数上保持一致. 两个子网络的输出向量分别定义为${\boldsymbol{D}} = {[{d_1},{d_2}, \cdots ,{d_l}]^{\mathrm{T}}}$与${\boldsymbol{F}} = {[{f_1},{f_2}, \cdots ,{f_l}]^{\mathrm{T}}}$, ${\boldsymbol{D}}$与${\boldsymbol{F}}$做内积, 并加上一偏置量${b_0}$, 便得到深度算子神经网络的输出
$$ {\mathrm{G}}(u)(y) = \mathop \sum \limits_{i = 1}^l {d_i}{f_i} + {b_0} $$ (3) 为便于理解, 在此以积分算子为例, 对深度算子神经网络的输入输出结构进行说明. 定义一积分算子${\mathrm{G}}$, 采用深度算子神经网络对其进行学习. 满足
$$ {\mathrm{G}}(u(x))(y) = \int_0^y u (x){\mathrm{d}}x $$ (4) 其中$u(x)$为输入函数(被积函数), $y$为原函数的自变量. 在此取$u(x) = 2 x$, 则原函数为$G(u(x))(y) = {y^2}$. 设$u(x)$定义域为$[0,1]$, 在定义域上以平均间隔取20个点$[0.05,0.1, \cdots ,1]$对$u(x)$进行采样, 得到采样集$[0.1,0.2, \cdots ,2]$. 原函数的定义域同样为$[0,1]$, 在该域上随机取100个点$[{y_1},{y_2}, \cdots ,{y_{100}}]$, 计算原函数的值$[y_1^2,y_2^2, \cdots ,y_{100}^2]$, 与采样集一同构成训练集, 见下式
$$\begin{split} &\qquad\qquad\;\;\; {\mathrm{input}} \qquad\qquad\quad {\mathrm{output}}\\ &\left[\begin{array}{cccc:c} 0.1 & 0.2 & \cdots & 2 & y_1 \\ 0.1 & 0.2 & \cdots & 2 & y_2 \\ \vdots & \vdots & \vdots & \vdots & \vdots \\ 0.1 & 0.2 & \cdots & 2 & y_{100} \end{array}\right]\left[\begin{array}{c} y_1^2 \\ y_2^2 \\ \vdots \\ y_{100}^2 \end{array}\right]\\ &\qquad\quad\;\;\; {\mathrm{branch}} \qquad {\mathrm{trunk}}\end{split}$$ (5) 其中branch net的输入维度为20, 与采样集相匹配; trunk net的输入维度为1, 与原函数的自变量维度相匹配. 在训练集上对深度算子神经网络进行训练, 便可得到如图1所示的结果.
深度算子神经网络的branch net与trunk net多为多层感知器(multilayer perceptron, MLP)神经网络. 而在本研究中, 考虑到非定常气动力的长时滞与强非线性特性, 选择GRU与LSTM两种神经网络单元的RNN作为branch net, 而trunk net则为MLP神经网络, 形成了GRU-DeepONet (后称为G-DeepONet)与LSTM-DeepONet (后称为L-DeepONet).
本研究中, 深度算子神经网络的输入为长度$l$的迎角序列${\boldsymbol{\alpha }}$ = $ [{\alpha _{n - l + 1}},{\alpha _{n - l + 2}}, \cdots ,{\alpha _n}] $与相对时刻指示标量$t$, 输出为在给定迎角运动下相对时刻$t$处的俯仰力矩系数${C_m}$
$$ {C_m}(t) = G({\boldsymbol{\alpha }})(t) $$ (6) 其中, ${\boldsymbol{\alpha }}$直接与时滞效应相联系, 因而作为branch net的输入. $t$则作为trunk net的输入, 取值范围为$[0,1]$, 是时间步的归一化表示. 由于$t$的存在, 深度算子神经网络所学习和预测的时间步范围$ [{t_1},{t_2}] $构成一组超参数, 本研究中该范围设定为后1/2时间步, 即${t_1} = n/2$, ${t_2} = n$, $l = 50$.
branch net中的RNN网络均为两个隐含层, 每层神经元个数为100. trunk net均为3个隐含层, 每层神经元数量为100, 激活函数$\sigma $为tanh. 所构建的深度算子神经网络结构如图2所示. 同时, 加入了只有GRU与LSTM的两组传统RNN作为对照, 以${\boldsymbol{\alpha }}$为输入, 最后一个时间步的${C_m}$为输出, 网络结构与branch net保持一致.
2. 基于深度算子神经网络的翼型俯仰振荡气动力预测结果与分析
研究对象为NACA0012翼型, 在训练信号驱动下进行俯仰运动, 进行CFD计算得到相对应的气动力系数, 来流速度固定为8 m/s. 使用的训练数据与文献[24]相同, 训练输入信号为一正弦叠加形式的信号
$$ {{\alpha }} = \sum\limits_{i = 0}^5 {{A_i}} \sin [2\text{π} (2.5 + 0.2i)T] $$ (7) 其中, ${A_i}$为随机选择的大幅值, 且最大值大于失速颤振极限环振荡幅值, $T$为绝对时间. 通过CFD计算得到在如式(7)俯仰运动下NACA0012翼型的俯仰力矩系数${{{C}}_{{m}}}$, 如图3所示.
同时, 所有的数据均进行了z-score标准化处理[33]以提高训练速度, 见下式
$$ {\boldsymbol{y}}* = \frac{{{\boldsymbol{y}} - {\boldsymbol{\bar y}}}}{s} $$ (8) 其中${\boldsymbol{y}}$为原始数据, $ {\boldsymbol{\bar y}} $与$s$分别为${\boldsymbol{y}}$的均值与标准差. $ {\boldsymbol{y}}* $为标准化处理后的数据, 其均值为0, 标准差为1.
采用Adam算法对几种神经网络进行训练, 初始学习率设置为0.001, 并使用余弦退火方法[34]进行动态调节, 以进一步提高模型的收敛性与泛化能力. 损失函数为均方误差MSE (mean squared loss). 为更直观地监控训练过程, 采用误差$e$作为判定收敛的标准, 其定义见下式
$$\qquad\quad e = \frac{{||{{\boldsymbol{C}}_{{\boldsymbol{m}},{\mathrm{NN}}}} - {{\boldsymbol{C}}_{{\boldsymbol{m}},{\mathrm{CFD}}}}|{|_F}}}{{||{{\boldsymbol{C}}_{{\boldsymbol{m}},{\mathrm{CFD}}}}|{|_F}}} \times 100\text{%} $$ (9) 其中$ {{\boldsymbol{C}}_{{\boldsymbol{m}},{\mathrm{NN}}}} $为神经网络预测值, ${{\boldsymbol{C}}_{{\boldsymbol{m}},{\mathrm{CFD}}}}$为CFD计算值, 二者均为多个不同时间点的Cm值所拼接的矢量; 式中范数为F范数. 采用RTX3090 GPU进行训练, 过程中的误差曲线见图4, 待误差小于10%并稳定后完成训练. 在相同误差水平上, 深度算子神经网络训练所需迭代次数为200次左右, 小于普通循环神经网络的400次. 但在训练总时长方面, 深度算子神经网络进行一次训练所需时长为${\text{2580 s}}$, 高于RNN的${\text{721 s}}$.
对训练后的几种气动力数据驱动模型进行泛化测试. 选取了不同幅值与不同减缩频率的正弦俯仰信号进行CFD仿真计算, 作为泛化测试的测试数据, 测试结果见表1、图5和图6. 表1中减缩频率$k$的定义为$k = \omega c/(2 U)$, 式中$\omega $为正弦信号圆频率, $c$为弦长, $U$为来流速度. 图5是对表1所列相对误差的统计, 可见对于同一类型的RNN, trunk net的加入使得泛化误差平均值与中位数均降低了3%左右. 同时, 虽然最小误差有所提高, 但误差的分散性大幅降低. 可以认为深度算子神经网络具有更好的泛化能力. 而结构更复杂的LSTM则比GRU更好地捕捉到了气动力中的非线性特征, 体现在LSTM与L-DeepONet平均泛化误差均低于GRU与G-DeepONet. 深度算子神经网络与RNN最大的区别在于对时间进行编码的trunk net, 因而取训练完成后的G-DeepONet, 进一步对trunk net的作用进行研究.
表 1 泛化误差Table 1. Generalization error$k$ AoA amplitude/
radGRU/
%LSTM/
%G-DeepONet/
%L-DeepONet/
%0.165 0.524 21.1 25.6 16.9 20.5 0.171 0.524 16.8 18 12.7 13.8 0.184 0.349 18.4 15.5 12.6 10.3 0.184 0.524 15.4 9.8 11.8 9.5 0.184 0.698 7.5 7.2 10.6 10.4 0.196 0.524 18.7 16.6 15.7 14.4 0.202 0.524 21.1 21.1 18.2 17.1 0.202 0.698 15.3 12 14.6 10.8 0.214 0.524 26.1 29.6 22.6 21.8 average 17.8 17.3 15.1 14.3 根据式(3), trunk net的输出向量${\boldsymbol{F}}$与branch net的输出向量${\boldsymbol{D}}$进行了内积, 这一过程可以认为是对${\boldsymbol{D}}$的各分量进行了加权求和, ${f_i}$为${d_i}$的权重. 图7展示了${f_i}$的绝对值随$t$的变化情况, ${f_i}$绝对值越大, 则权重越高. 由第1节所述, 当$t = 0$时, 深度算子神经网络的输出值为绝对时间第$n/2$步时的气动力系数, 对应$i = 25$; 当$t = 1$时, 则对应$i = 50$. 能够发现, 上述两点之间存在一条明显的高权重带, 说明与$t$所对应的${f_i}$值具有最高的权重; 此外, 图中也存在其他规律性的结构, 说明深度算子神经网络所输出的气动力以$t$所对应的$ {d_i} $值为主, 并采用其他时间处的${d_i}$值进行微调, 在深度算子神经网络中体现了气动力的时滞特性, 因此比普通RNN提升了精度.
进一步对branch net的输出向量${\boldsymbol{D}}$进行拼接, 并与CFD计算的气动力系数${{\boldsymbol{C}}_{\boldsymbol{m}}}$进行对比, 二者均进行了z-score标准化处理, 结果见图8, 能够发现${\boldsymbol{D}}$与CFD结果具有一定的相似性. 可以认为, ${\boldsymbol{D}}$与${{\boldsymbol{C}}_{\boldsymbol{m}}}$具有相似的物理含义, 而${\boldsymbol{D}}$通过与${\boldsymbol{F}}$内积进行权重调整后得到了高精度结果.
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图 8 ${\boldsymbol{D}}$与CFD计算值的对比Figure 8. Comparison between ${\boldsymbol{D}}$ and CFD results3. 基于气动力数据驱动模型的失速颤振极限环预测与结果分析
单自由度二元翼型气动弹性模型如图9所示. 其气动弹性方程如下
$$\left.\begin{split} &{I_\alpha }\ddot \alpha + {C_\alpha }\dot \alpha + {K_\alpha }\alpha = M\\ &{C_\alpha } = 2\zeta \sqrt {{K_\alpha }/{I_\alpha }} \\ &M = \frac{1}{2}\rho {U^2}{c^2}b{C_m}\end{split} \right\}$$ (10) 其中, ${I_\alpha }$为转动惯量, ${C_\alpha }$, $\zeta $与${K_\alpha }$分别为阻尼系数、阻尼比与刚度系数, $M$为俯仰力矩. $c$与$b$分别为翼型的弦长与展长, $U$为来流速度, 这些参数取值与文献[24]保持一致. 将式(10)写为状态空间的形式, 即下式, 即可通过龙格-库塔方法进行求解
$$\left.\begin{split} &{\boldsymbol{\dot u}} = {\boldsymbol{Au}} + {\boldsymbol{B}}\\ &{\boldsymbol{u}} = \left[ {\begin{array}{*{20}{c}} \alpha \\ {\dot \alpha } \end{array}} \right]\\ &{\boldsymbol{A}} = \left[ {\begin{array}{*{20}{c}} 0&1 \\ { - {K_\alpha }/{I_\alpha }}&{ - {C_\alpha }/{I_\alpha }} \end{array}} \right]\\ &{\boldsymbol{B}} = \left[ {\begin{array}{*{20}{c}} 0 \\ {M/{I_\alpha }} \end{array}} \right]\end{split} \right\}$$ (11) 给定初始条件$\alpha = 0,\dot \alpha = 3{\text{rad/s}}$, 采用基于CFD的流固耦合方法求解失速颤振特性, 作为对比基准. 分别采用4种神经网络气动力数据驱动模型进行8 m/s下失速颤振极限环振荡幅值预测, 结果见表2与图10. G-DeepONet与L-DeepONet极限环振荡幅值预测误差分别为0.43%与−1.99%, 均低于2%, 而G-DeepONet预测精度更高.
表 2 失速颤振预测极限环振荡幅值Table 2. Predicted amplitude of stall flutterNeural network type GRU LSTM G-DeepONet L-DeepONet amplitude/rad 1.2197 1.2217 1.2732 1.2425 relative error −3.79% −3.63% 0.43% −1.99% 4. 基于气动力数据驱动模型的失速颤振分岔速度预测与结果分析
来流速度是影响失速颤振特性的重要因素之一, 翼型动态失速气动力特性随来流速度变化显著, 因而需要对不同来流速度下的气动力系数进行学习, 从而准确预测失速颤振分岔速度. 在此对第2节所建立的气动力数据驱动模型进行扩展, 将来流速度$U$扩展为迎角序列矢量${\boldsymbol{\alpha }}$的第2分量, 如下式所示, 分别输入RNN与branch net, trunk net则与上一节中相同
$$ {\boldsymbol{\alpha }} = \left[ {\begin{array}{*{20}{c}} {{\alpha _{n - l + 1}}}&{{U_{n - l + 1}}} \\ {{\alpha _{n - l + 2}}}&{{U_{n - l + 2}}} \\ \cdots & \cdots \\ {{\alpha _n}}&{{U_n}} \end{array}} \right] $$ (12) 同样地, 采用文献[24]中包含不同来流速度的训练信号进行训练. 其中的来流速度为$U = (6,7,8,9){\text{ m/s}}$, 呈阶梯状递增. 除输入向量维度外, 神经网络的其他结构保持不变. 训练过程如图11所示, 待误差稳定并小于10%后完成训练.
同样地, 采用正弦信号对训练后的神经网络进行泛化测试. 测试信号幅值为0.698 rad, 减缩频率为0.202, 在不同来流速度下进行泛化测试, 其中包括了$9.5{\text{ m/s}}$的外插工况, 结果见表3与图12. G-DeepONet与L-DeepONet平均泛化误差均低于GRU与LSTM. 在$9.5{\text{ m/s}}$的外插工况下, 预测误差均显著上升, 但深度算子神经网络组的预测误差仍低于RNN组, 体现出了更好的泛化能力.
表 3 不同速度输入下的极限环振荡幅值泛化误差Table 3. Generalization error with different flow velocitiesVelocity/(m·s−1) GRU/% LSTM/% G-DeepONet/% L-DeepONet/% 6 18.1 16.2 13.7 12.9 6.5 14.1 11.2 11.1 10.1 7 15.8 13.5 10.3 10.4 7.5 16.5 14.1 9.5 8.5 8 17.0 13.3 9.8 8.5 8.5 19.8 14.7 9.5 11.6 9 22.7 15.7 13.3 15.1 9.5 29.3 24.7 21.3 20.4 average 19.16 15.43 12.31 12.19 选取L-DeepONet进行失速颤振分岔速度预测, 并与流固耦合结果进行对比, 结果见图13. 有速度输入时, 所建立的数据驱动模型能够较好地预测失速颤振分岔现象, 分岔速度为7.5 m/s, 而流固耦合分岔速度在7 ~ 7.5 m/s之间, 取中值7.25 m/s, 相对误差为3.45%. 而无速度输入组则没有明显的分岔现象.
最后对流固耦合方法与数据驱动建模方法的预测时间进行对比. 基于CFD的流固耦合方法对图13中7个来流速度进行计算的总耗时为8404 s, 而两种L-DeepONet预测时长均为7.5 s, 具有更高的预测效率.
5. 结论
本文提出了一种基于深度算子神经网络的翼型非定常气动力数据驱动建模方法, 实现了基于深度算子神经网络气动力数据驱动模型的二维翼型失速颤振分析.
(1)在气动力预测方面, 深度算子神经网络数据驱动模型与传统RNN相比引入了主干网络, 通过主干网络对RNN的输出进行了不同时间步气动力的权重优化, 更能体现非定常气动力的时滞特性, 也具有更低的泛化误差, 同时误差分散性大幅降低.
(2)在失速颤振极限环振荡幅值预测方面, 在相同网络结构参数下, 基于深度算子神经网络模型的幅值预测误差在2%以内, 优于基于RNN模型的预测结果.
(3)在失速颤振分岔速度预测方面, 考虑速度输入的数据驱动模型预测精度显著高于没有速度输入的数据驱动模型.
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图 8 ${\boldsymbol{D}}$与CFD计算值的对比
Figure 8. Comparison between ${\boldsymbol{D}}$ and CFD results
表 1 泛化误差
Table 1 Generalization error
$k$ AoA amplitude/
radGRU/
%LSTM/
%G-DeepONet/
%L-DeepONet/
%0.165 0.524 21.1 25.6 16.9 20.5 0.171 0.524 16.8 18 12.7 13.8 0.184 0.349 18.4 15.5 12.6 10.3 0.184 0.524 15.4 9.8 11.8 9.5 0.184 0.698 7.5 7.2 10.6 10.4 0.196 0.524 18.7 16.6 15.7 14.4 0.202 0.524 21.1 21.1 18.2 17.1 0.202 0.698 15.3 12 14.6 10.8 0.214 0.524 26.1 29.6 22.6 21.8 average 17.8 17.3 15.1 14.3 
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表 2 失速颤振预测极限环振荡幅值
Table 2 Predicted amplitude of stall flutter
Neural network type GRU LSTM G-DeepONet L-DeepONet amplitude/rad 1.2197 1.2217 1.2732 1.2425 relative error −3.79% −3.63% 0.43% −1.99%
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表 3 不同速度输入下的极限环振荡幅值泛化误差
Table 3 Generalization error with different flow velocities
Velocity/(m·s−1) GRU/% LSTM/% G-DeepONet/% L-DeepONet/% 6 18.1 16.2 13.7 12.9 6.5 14.1 11.2 11.1 10.1 7 15.8 13.5 10.3 10.4 7.5 16.5 14.1 9.5 8.5 8 17.0 13.3 9.8 8.5 8.5 19.8 14.7 9.5 11.6 9 22.7 15.7 13.3 15.1 9.5 29.3 24.7 21.3 20.4 average 19.16 15.43 12.31 12.19
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