分布式超声波探测2维气体温度场方法研究

Study on Distributed Ultrasonic Detection Method of Two-dimensional Gas Temperature Field

摘要

在现有的热压罐中，设备主要使用若干温度传感器来实现复材构件成型温度控制，但这种点测的方式难以实现空间温度场的测量，同时，这种接触式测温方式会直接影响模具和构件接触面温度分布。这些局限性大大影响了构件成型的品质，准确测量大型空间构件成型气体温度场成为航空复合材料部件成型固化亟需解决的难题。为此，本文开展分布式超声波探测2维气体温度场方法研究。首先，通过数值仿真进行热压罐中的超声传播特性分析研究，并在此基础上依据时移特性建立分布式超声气体测温模型；其次，从热压罐实际工况出发，发展基于全局径向基函数（LQ）和奇异值分解（SVD）的2维气体温度场重建算法（LQ‒SVD），并通过和其他几个常见温度场重建算法作对比及带噪重建的方式进行该算法的考核与精度分析。数值模拟和热压罐现场实验表明：当温度场发生变化时，超声在热压罐中的传播会出现明显的时移特性；以该时移特性建立的测温模型可有效描述温度分布变化与超声传播时间的关系；LQ‒SVD重建算法在热压罐工况下具有较高精度和抗噪性，在数值模拟中该算法抗噪性较强且和其他算法相比具有更低的重建均方根误差，在实验验证中重建值和真值吻合较好；本文建立的分布式超声波探测2维气体温度场方法可较为准确地测量出大型空间2维气体温度场。

Abstract

Objective This study applies several temperature sensors in current autoclave systems to control the forming temperature of composite components. This point-measurement method presents challenges in capturing the spatial temperature field. In addition, this contact-based temperature measurement approach directly influences the temperature distribution at the interface between the mold and the components. These limitations significantly affect the quality of component formation. Accurately capturing the gas temperature field during the forming of large-scale spatial components remains a critical challenge in the curing process of aerospace composite material parts. This study investigates a method for distributed ultrasonic detection of the two-dimensional gas temperature field. Methods This study first employed numerical simulation software to analyze the ultrasonic propagation characteristics in an autoclave under steady-state temperature field conditions from the perspective of thermo-acoustic coupling. The analysis was then extended to ultrasonic propagation characteristics under varying temperature fields. Based on these analyses, a distributed ultrasonic gas temperature measurement model was established using time-of-flight characteristics. In addressing issues associated with inverse problems, such as the limitations of least squares and algebraic iterative methods, where the number of discrete points cannot exceed the number of propagation paths and the temperature resolution is low, a two-dimensional temperature field reconstruction algorithm based on logarithmic-quadratic (LQ) functions and singular value decomposition (SVD) was developed for autoclave conditions. The core idea of this algorithm was to fit the distribution of the reciprocal of sound speed using a linear combination of LQ functions to establish an inversion model. SVD was then employed to address the ill-posedness in the inversion process, which enabled the model to be solved. The accuracy of the LQ‒SVD algorithm was analyzed by evaluating the maximum absolute error, mean error, and root mean square error between the reconstructed temperature field and the original temperature field, and the results were compared to those of three other common algorithms. In accounting for practical errors, white noise was added to the theoretical actual values to simulate system errors, and the noise resistance of the algorithm was tested based on this. A distributed ultrasonic temperature measurement system was finally set up in the autoclave for experimental validation. Several thermocouples were utilized to measure the actual temperature field, and the reconstruction error between the experimental system's reconstructed values and the actual values was analyzed to verify the effectiveness of the proposed method. Results and Discussions In addition to the primary longitudinal wave pulses in the propagation of ultrasound within the gas space of an autoclave, multiple reflection waves and wave interferences were present, resulting in complex propagation behaviors. Due to varying temperature distributions, the propagation time of ultrasound along its path differed. This study examined the “time-of-flight characteristics” at the microsecond level, which were influenced by factors such as medium properties, temperature fields, and the propagation distance of ultrasound. The LQ‒SVD algorithm demonstrated the following reconstruction metrics for different types of temperature fields: for a single-peak symmetric temperature field, the maximum absolute error, mean error, and root mean square error were 0.812 3 K, 0.161 1 K, and 0.057 4%, respectively; for a single-peak biased temperature field, these metrics were 0.043 1 K, 0.008 7 K, and 0.002 6%; for a double-peak biased temperature field, they were 9.982 9 K, 0.828 3 K, and 0.317 5%; for a triple-peak biased temperature field, they were 16.301 7 K, 2.125 0 K, and 0.651 6%; and for a quadruple-peak biased temperature field, they were 95.163 8 K, 11.184 3 K, and 3.878 4%. Based on root mean square error, the temperature field reconstruction errors for the Multi-Quadratic (MQ), Markov radial basis function (MK), and Gaussian function-based (GS) algorithms for single-peak symmetric temperature fields were 0.058 3%, 1.249 0%, and 0.529 1%, respectively; for single-peak biased temperature fields, these errors were 0.203 0%, 0.501 7%, and 0.049 4%; for double-peak biased temperature fields, they were 0.588 7%, 1.220 8%, and 1.800 2%; for triple-peak biased temperature fields, they were 1.238 7%, 1.801 7%, and 3.696 5%; and for the quadruple-peak biased temperature fields, they were 3.884 3%, 3.964 4%, and 10.735 7%. These results indicated that regardless of the temperature field model, the LQ‒SVD algorithm consistently achieved the best reconstruction performance. In the noise resistance test of the LQ‒SVD algorithm, with a noise standard deviation of 0.5 us, the three reconstruction metrics for single-peak symmetric temperature fields were 22.409 0 K, 3.965 9 K, and 1.338 6%; for single-peak biased temperature fields, they were 21.174 7 K, 3.507 9 K, and 1.090 3%; for double-peak biased temperature fields, they were 20.757 7 K, 3.417 6 K, and 1.051 0%; for triple-peak biased temperature fields, they were 25.588 9 K, 4.663 4 K, and 1.383 5%; and for quadruple-peak biased temperature fields, they were 85.480 4 K, 11.704 6 K, and 3.9158%. With a noise standard deviation of 1 us, these metrics for single-peak symmetric temperature fields were 36.578 1 K, 5.779 2 K, and 1.956 3%; for single-peak biased temperature fields, they were 46.651 1 K, 7.260 9 K, and 2.581 6%; for double-peak biased temperature fields, they were 67.944 5 K, 8.371 8 K, and 2.647 0%; for triple-peak biased temperature fields, they were 59.341 8 K, 7.221 1 K, and 2.333 9%; and for quadruple-peak biased temperature fields, they were 90.738 4 K, 14.397 9 K, and 4.435 0%. In the on-site experimental validation within the autoclave, the average relative error between reconstructed temperatures and thermocouple measurements was 4.48%, which indicated the effective performance of the distributed ultrasound temperature measurement system in temperature field reconstruction. Conclusions The results demonstrate that the method established in this study for distributed ultrasound detection of two-dimensional gas temperature fields precisely describes the propagation characteristics of ultrasound under thermal/acoustic coupling effects within the autoclave. This method accurately reconstructs the temperature field distribution in the autoclave space. Its effectiveness significantly alleviates the current limitation of relying solely on a few sensor points for temperature measurement in autoclaves. It preliminarily fulfills the requirement for accurate measurement of the temperature field of forming gas in significant spatial components during the molding and curing of aerospace composite material parts, providing robust support for enhancing the quality of component formation in the future.

Graphical abstract

关键词

超声气体测温 / 分布式 / 2维气体温度场 / 温度场重建 / 径向基函数

本刊网刊

热压罐工艺是目前先进复合材料构件成型固化的主要制备工艺之一^[1‒3]。为满足大型构件的高品质制备需求^[4]，热压罐的直径和长度也须增大。同时，为提高制备效率，还面临多个构件同时成型需求^[5]。这些新的需求使得热压罐内部空间对流、导热、辐射、材料固化散热之间耦合加剧，空间环境、模具、构件组成的成型固化温度场测量与控制面临巨大压力，构件成型品质受到严重制约^[6‒8]，如何准确测量大型空间构件成型气体温度场是航空复合材料部件成型固化亟需解决的难题。

目前，主要通过若干温度传感器联网，测量空间环境和模具若干点温度来实现复材构件温度的控制^[9‒10]。由于该方式测点有限，对于大型空间较难实现瞬态温度场的测量。另外，由于该方式是接触式，较难在不影响模具和构件接触的情况下实现模具接触面温度分布的无损测量^[11]。随着构件尺寸增加，这些局限对于成型品质的影响越来越突出。超声测温是一种基于声信号无损测量内部温度分布的新方法，具有非接触、实时和场测量等特点^[12‒14]。但目前尚缺少热压罐中的分布式超声探测2维气体温度场方法。

为此，本文开展了分布式超声波探测2维气体温度场方法研究。首先，研究热压罐中的超声传播特性，在此基础上建立分布式超声气体测温模型；其次，从热声耦合反问题角度入手，发展基于全局径向基函数（LQ）和奇异值分解（SVD）2维气体温度场重构算法（LQ‒SVD），并进行算法的考核与精度分析；最后，通过数值模拟和热压罐现场实验进行方法有效性验证。

1 成型空间气体温度场中超声传播特性

基于超声波建立复材构件成型空间气体温度场感知模型的前提是分析超声波在热压罐中具有怎样的传播特性，以及当温度场变化时声波的变化情况。通过数值模拟，首先分析研究稳态温度场条件下热压罐中的超声传播特性，接着进一步分析不同温度场下超声传播特性。

算例中气体区域为直径为0.1 m的圆形；换能器压电材料为PZT‒5H；激励信号是频率为100 kHz的3周期正弦调制高斯脉冲；壁面边界条件为硬声场边界。算例模型如图1所示，计算流程如图2所示。被测温度场（K）为单峰对称温度场，记为T_{\mathrm{1}}(x,y)，其中，x\mathrm{和}y分别为点的横、纵坐标，该温度场函数表达式如下：

T_{\mathrm{1}}(x,y)=\mathrm{500}{\mathrm{e}}^{-\frac{x^{\mathrm{2}}+y^{\mathrm{2}}}{\mathrm{0.5}}}（1）

不同时刻气体空间的声压分布如图3所示，接收端随时间的声压变化情况计算结果如图4所示。

压电换能器受脉冲电压激发后，因逆压电效应，产生主脉冲纵波，呈辐射状向被测气体介质内部传播，沿程衰减（图3（a）、（b））；受固壁硬声场边界影响，超声波在壁面发生反射，产生壁面反射纵波（图3（c））；由于壁面是圆形，随着超声传播的深入，反射角逐渐增大，壁面反射纵波与壁面发生2次反射（图3（d）），产生壁面2次反射纵波。当主脉冲纵波传播到接收探头时，由于界面处声阻抗的变化，一部分波透射，一部分波则被反射回来，形成主脉冲反射波。随后，壁面反射纵波相继传播到接收探头，并发生相互干扰，再反射，形成多次壁面反射波（图3（d）、（e））。反射的主脉冲纵波和壁面波又继续在气体内部传播、衰减、再反射，形成多种2次反射波（图3（f））。波的传播、反射、衰减符合物理规律，表明算法有效。

接着，改变温度场条件，模型仍为单峰对称，分别记为T_{\mathrm{2}}(x,y)和T_{\mathrm{3}}(x,y)，函数表达式如下：

T_{\mathrm{2}}(x,y)=\mathrm{450}{\mathrm{e}}^{-\frac{x^{\mathrm{2}}+y^{\mathrm{2}}}{\mathrm{0.5}}}（2）

T_{\mathrm{3}}(x,y)=\mathrm{400}{\mathrm{e}}^{-\frac{x^{\mathrm{2}}+y^{\mathrm{2}}}{\mathrm{0.5}}}（3）

图5为不同温度场接收端声压计算结果。由图5可知，3种不同温度场条件下到达接收端的主脉冲纵波存在微秒级的时间差。当热压罐空间温度分布变化时，波速分布发生变化，导致路径上的超声传播时间发生变化，因此，接收端信号会出现“时移特性”。

超声波在热压罐内部气体空间的传播中，除主脉冲纵波外，还存在多种反射波及波的相互干扰，传播行为比较复杂。且随着温度分布不同，超声在路径上的传播时间不同，在本算例中“时移特性”为微秒量级，这与介质的物性、温度场以及超声波的传播距离等因素有关。

2 分布式超声气体测温模型

气体介质中的温度分布发生变化时，随着超声波的传播，声能发生衰减，同时声速发生变化声时出现“时移特性”。本质上，超声渡越时间反映了波传播路径上温度分布的累积量。因此，从波速与温度的相关性出发，建立分布式超声波感知气体介质中温度分布变化的分析模型。

对于空间温度场分布的超声探测，本文采用分布式超声探测方案，如图6所示。图6中，红色的A1、B1、C1、D1为4个激发换能器，蓝色的A2、B2、C2、D2为4个接收换能器。方案的核心思想是将被测区域划分为若干个视为温度均匀分布的子温区，通过布设多组声波换能器形成多条声波路径，每条路径都会穿过多个子温区，每个子温区也会被多条路径穿过。通过测量每条路径的超声传播时间，重构出被测区域的完整温度分布。

多路超声传播时间\{t_{\mathrm{t}\mathrm{o}\mathrm{f}}{\}}_M描述如下：

\{t_{\mathrm{t}\mathrm{o}\mathrm{f}}{\}}_M={\left\{\left.{\int }_{}^{}\frac{\mathrm{1}}{V\left(T(x,y,t)\right)}\mathrm{d}s\right\}\right.}_M（4）

式中：M为超声传播路径数；{\{·\}}_M表示公式适用于所有路径，共M条；s为当前传播路径；V\left(T(x,y,t)\right)为坐标为(x,y)的点在时刻t的绝对温度T（K）所决定的超声传播速度。

每个检测周期内，各声波换能器按顺序依次发声，当某个换能器发声时，其余所有换能器接收信号。经过计算机处理分析，得到各条路径的声波飞渡时间，结合温度场重构算法求解出声波的速度分布，最终通过式（5）^[15‒16]计算出被测区域的温度分布。

\left\{\begin{array}{l}V=\sqrt[]{\gamma RT}=Z\sqrt[]{T},\\ Z=\sqrt[]{\gamma R}\end{array}\right.（5）

式中：V为气体中声波的传播速度，m/s；\gamma为气体绝热系数，由气体的物理性质决定；Z为气体的声音传播常数；R为气体常数。

3 基于全局径向基函数的温度场重建算法

现有的大多温度场重建算法如代数迭代方法、最小二乘法等，存在离散点数不能大于传播路径数、温度分辨率不高等问题^[17‒18]，因此，本文使用基于LQ和SVD的2维温度场重建算法（LQ‒SVD）。该算法主要思想是先利用LQ函数的线性组合对声速的倒数分布进行拟合，建立反演模型；再利用SVD解决反演过程中的病态问题，实现模型求解，其重构流程如图7所示。

假设被测区域内的声速的倒数分布为连续函数，记为f(x，y)，则区域中第k条超声传播路径的声波飞渡时间理论值t_k 可表示为：

t_k=\int f(x,y)\mathrm{d}{l}_k（6）

式中，l_k 为第k条声波传播路径长度。

将被测区域划分为N个子温区，f(x，y)可表示为N个子温区几何中心点径向基函数的线性组合^[19]。

f(x,y)={\displaystyle \sum _{i=\mathrm{1}}^N}{\xi }_i{\phi }_i(x,y)（7）

式中：{\xi }_i为待定系数；{\phi }_i(x,y)为第i个子温区几何中心点的径向基函数，i=1,2,…,N。

2维LQ径向基函数的表达式为：

{\phi }_i(x,y)=\mathrm{l}\mathrm{n}\left(\frac{{\delta }^{\mathrm{2}}}{{\left(x-x_i\right)}^{\mathrm{2}}+{\left(y-y_i\right)}^{\mathrm{2}}+{\delta }^{\mathrm{2}}}\right)（8）

式中：(x_i, y_i )为LQ径向基函数的中心，即第i个子温区几何中心点的坐标；δ为基函数形状参数。

联立式（7）和（8）可得：

t_k={\displaystyle \sum _{i=\mathrm{1}}^N}{\xi }_i\int {\phi }_i(x,y)\mathrm{d}{l}_k={\displaystyle \sum _{i=\mathrm{1}}^N}{h}_{k,i}{\xi }_i（9）

式中，h_{k,i}为第i个径向基函数在第k条路径上的线积分，h_{k,i}=\int {\phi }_i(x,y)\mathrm{d}{l}_k。

将式（9）改写为矩阵形式：

\mathit{t}=\mathit{H}\mathit{\xi }（10）

式中，\mathit{t}为M条声波路径的飞渡时间构成的矩阵，\mathit{\xi }为M个径向基函数权重所构成的系数矩阵，\mathit{H}为N个子温区的LQ径向基函数在M条声波路径上的积分所构成的路径矩阵。

式（10）的求解是一个不适定问题，多数情况下矩阵\mathit{H}为病态矩阵，不存在逆矩阵，但存在伪逆矩阵。对于矩阵\mathit{P}，如果存在一个矩阵\mathit{Q}使得二者满足关系\mathit{P}\mathit{Q}\mathit{P}=\mathit{P},\mathit{Q}\mathit{P}\mathit{Q}=\mathit{Q},{(\mathit{P}\mathit{Q})}^{\mathbf{T}}=\mathit{P}\mathit{Q},{(\mathit{Q}\mathit{P})}^{\mathbf{T}}=\mathit{Q}\mathit{P}，则矩阵\mathit{Q}称为矩阵\mathit{P}的伪逆矩阵。

SVD^[20‒21]是处理矩阵病态问题的一种有效手段，它通过提取矩阵的特征向量计算矩阵的伪逆，为矩阵方程提供最优解。大量仿真实验证明了SVD在声学测温技术中的有效性^[22‒23]。

因此，式（10）中待求解的权重矩阵\mathit{\xi }可表示为：

\mathit{\xi }={\mathit{H}}^{+}\mathit{t}（11）

式中，{\mathit{H}}^{+}为\mathit{H}的伪逆。

将\mathit{\xi }代入式（7）并结合式（5）可以得到温度场函数T(x,y)：

T(x,y)=\frac{\mathrm{1}}{Z^{\mathrm{2}}{f}^{\mathrm{2}}(x,y)}（12）

4 算法考核与精度分析

为了验证LQ‒SVD 算法的重建性能，构造单峰对称、单峰偏置、双峰偏置、三峰偏置以及四峰偏置的典型2维温度场，并分析算法对以上5种温度场模型的重建精度。另外，通过在飞渡时间模拟真值的基础上叠加不同标准差的高斯白噪声，进行算法抗噪性分析。

图8为2维温度场声学测量方案示意图。

图8中：验证算例的被测区域为直径为0.1 m的圆形；在测量区域周围均匀地布置了16 个声波换能器（编号为s1~r8）；红色实线表示所形成的有效声波传播路径，共52条（图8（a））；测量区域根据其外切正方形等分的共16×16个子温区划分了圆内的172个子温区（图8（b））。

以被测区域圆心坐标为原点建立2维直角坐标系，各换能器位置如表1所示。

在被测区域内分别构造单峰对称、单峰偏置、双峰偏置、三峰偏置以及四峰偏置的典型2维温度场作为仿真重建对象，其函数表达式分别如下。

2维单峰对称温度场函数为\mathrm{T}{\mathrm{M}}_{\mathrm{1}}(x,y)：

\mathrm{T}{\mathrm{M}}_{\mathrm{1}}(x,y)=\frac{\mathrm{5}\mathrm{000}}{\mathrm{2}\mathrm{600}(x^{\mathrm{2}}+y^{\mathrm{2}})+\mathrm{10}}（13）

2维单峰偏置温度场函数为\mathrm{T}{\mathrm{M}}_{\mathrm{2}}(x,y)：

\mathrm{T}{\mathrm{M}}_{\mathrm{2}}(x,y)=\frac{\mathrm{5}\mathrm{000}}{\mathrm{800}\left({(x-\mathrm{0.03})}^{\mathrm{2}}+{(y-\mathrm{0.03})}^{\mathrm{2}}\right)+\mathrm{10}}（14）

2维双峰偏置温度场函数为\mathrm{T}{\mathrm{M}}_{\mathrm{3}}(x,y)：

\begin{array}{l}\mathrm{T}{\mathrm{M}}_{\mathrm{3}}(x,y)=\frac{\mathrm{1}\mathrm{900}}{\mathrm{1}\mathrm{900}\left({(x-\mathrm{0.03})}^{\mathrm{2}}+{(y-\mathrm{0.03})}^{\mathrm{2}}\right)+\mathrm{5}}+\\ \frac{\mathrm{2}\mathrm{000}}{\mathrm{1}\mathrm{800}\left({(x+\mathrm{0.03})}^{\mathrm{2}}+{(y+\mathrm{0.02})}^{\mathrm{2}}\right)+\mathrm{5}}\end{array}（15）

2维三峰偏置温度场函数为\mathrm{T}{\mathrm{M}}_{\mathrm{4}}(x,y)：

\begin{array}{l}\mathrm{T}{\mathrm{M}}_{\mathrm{4}}(x,y)=\frac{\mathrm{1}\mathrm{700}}{\mathrm{3}\mathrm{500}\left({(x+\mathrm{0.03})}^{\mathrm{2}}+{(y+\mathrm{0.03})}^{\mathrm{2}}\right)+\mathrm{5}}+\\ \frac{\mathrm{1}\mathrm{600}}{\mathrm{4}\mathrm{000}\left({(x+\mathrm{0.03})}^{\mathrm{2}}+{(y-\mathrm{0.03})}^{\mathrm{2}}\right)+\mathrm{5}}+\\ \frac{\mathrm{1}\mathrm{850}}{\mathrm{3}\mathrm{300}\left({(x-\mathrm{0.035})}^{\mathrm{2}}+{(y-\mathrm{0.02})}^{\mathrm{2}}\right)+\mathrm{5}}\end{array}（16）

2维四峰偏置温度场函数为\mathrm{T}{\mathrm{M}}_{\mathrm{5}}(x,y)：

\begin{array}{l}\mathrm{T}{\mathrm{M}}_{\mathrm{5}}(x,y)=\frac{\mathrm{700}}{\mathrm{3}\mathrm{000}\left({(x+\mathrm{0.03})}^{\mathrm{2}}+{(y+\mathrm{0.03})}^{\mathrm{2}}\right)+\mathrm{2}}+\\ \frac{\mathrm{750}}{\mathrm{3}\mathrm{000}\left({(x+\mathrm{0.03})}^{\mathrm{2}}+{(y-\mathrm{0.03})}^{\mathrm{2}}\right)+\mathrm{2}}+\\ \frac{\mathrm{750}}{\mathrm{3}\mathrm{000}\left({(x-\mathrm{0.03})}^{\mathrm{2}}+{(y-\mathrm{0.03})}^{\mathrm{2}}\right)+\mathrm{2}}+\\ \frac{\mathrm{700}}{\mathrm{3}\mathrm{000}\left({(x-\mathrm{0.03})}^{\mathrm{2}}+{(y+\mathrm{0.03})}^{\mathrm{2}}\right)+\mathrm{2}}\end{array}（17）

基于上述已知条件，可得到超声传播时间的理论值。然后通过前述LQ‒SVD算法进行温度场的重建，重建前后对比结果如图9所示。由图9可以看出：本文所使用的LQ‒SVD算法在5个温度场中都具有很好的表现，重建结果与原始温度分布具有较高的一致性。

为进一步评估不同算法的重建性能，对重建结果进行定量的误差分析。重建误差的评价指标包括最大绝对误差E_max、平均绝对误差E_mean及均方根误差E_rms，对应的计算公式如下：

E_{\mathrm{m}\mathrm{a}\mathrm{x}}=\mathrm{m}\mathrm{a}\mathrm{x}\left|T_j^{\mathrm{R}}-T_j^{\mathrm{M}}\right|（18）

E_{\mathrm{m}\mathrm{e}\mathrm{a}\mathrm{n}}=\frac{\mathrm{1}}{n}{\displaystyle \sum _{i=j}^n}\left|T_j^{\mathrm{R}}-T_j^{\mathrm{M}}\right|（19）

E_{\mathrm{r}\mathrm{m}\mathrm{s}}=\frac{\sqrt[]{\frac{\mathrm{1}}{n}{\displaystyle \sum _{i=\mathrm{1}}^n}{\left(T_j^{\mathrm{R}}-T_j^{\mathrm{M}}\right)}^{\mathrm{2}}}}{T_{\mathrm{m}\mathrm{e}\mathrm{a}\mathrm{n}}^{\mathrm{M}}}\times \mathrm{100}\mathrm{\%}（20）

式（18）~（20）中，n为被测区域内温度计算点的总个数，T_j^{\mathrm{M}}为第j个计算点的仿真温度值，T_j^{\mathrm{R}}为第j个计算点的重建温度值，T_{\mathrm{m}\mathrm{e}\mathrm{a}\mathrm{n}}^{\mathrm{M}}为仿真温度场平均温度值。

表2为5个典型2维温度场3种重建结果误差。表3为LQ‒SVD算法和马尔可夫径向基法（MK）^[24‒26]、基于高斯函数的温度场重建算法（GS）^[27‒28]、基于多2次函数的温度场重建算法（MQ）^[29‒30]3种常见算法对不同温度场的重建均方根误差对比。由表2、3可以看出，LQ‒SVD算法的各项误差指标均较优，有着高度的重建精度。

进一步对算法进行抗噪性分析。考虑到仿真的工况下温度引起的声时变化最小在20 us左右，取20 us的2.5%和5.0%作为高斯白噪声标准差，在模拟真值的基础上分别叠加均值为0，标准差σ为0.5 us及1.0 us的高斯白噪声，基于带噪声的声时矩阵重建温度场，重建结果和误差分别如图10和表4所示。从图10和表4对比可知，在叠加不同程度的噪声之后，重建值与真值仍然吻合较好。

综上可知，本文使用的LQ‒SVD算法在该工况下具有较好的抗噪性和适用性。

5 热压罐超声测温现场验证实验

从实验验证的角度，开展针对热压罐复材构件成型空间气体温度场的分布式超声气体测温实验工作。

实验用热压罐罐体结构及尺寸如图11所示。该罐体内径为60.0 cm，壁厚为18.7 cm，容积为0.43 m³，罐内底部布置了高为15.0 cm的架车用于放置构件。

超声测温系统框图和现场布置分别见图12和 13。受超声换能器指向性影响，在实测之后采用图12中的换能器布局和朝向，以确保每个发射信号（A~J）都能被10个接受端捕捉。实验中信号源使用普源信号发生器DG1000Z产生频率40 kHz的多周期正弦信号；超声换能器选择模块使用的是16路串口继电器；换能器型号为FUS‒40E，其谐振频率为40 kHz，超声信号采集模块使用NI USB数据采集卡，热风机作为热源将罐内升温，10个超声发射换能器和10个超声接收换能器组成的阵列共形成了100条超声传播路径。分布式超声测温系统现场测量100条路径得到的超声飞渡时间经过校准，可以得到如图14所示的100路超声飞渡时间。

将100路超声飞渡时间输入给LQ‒SVD重建算法，便可反演出热压罐此时的2维温度场分布，如图15所示。

图15中标注了16个热电偶（1~16）在温度场中的位置，表5给出了这16个温度场真值、重建值及重建的绝对和相对误差。从表5可以看出，分布式超声气体测温系统测量出的温度场数据与热电偶测量所得温度趋势基本一致。经计算，重建温度与热电偶测点实测温度的平均相对误差为4.48%，分布式超声测温系统温度场重建效果良好。

6 结 论

本文针对热压罐成型空间气体温度场测量难题，主要研究了构件成型空间热/声耦合作用下超声波传播特性分析、全局大型温度场重构和分布式多源超声探测等多项关键技术。通过数值模拟和热压罐现场实验验证，得到如下结论。

1）超声波在热压罐中的传播过程中除了会出现主脉冲回波，还会产生较为明显的多次壁面反射；随着空间温度分布不同，接收端信号产生明显的“时移特性”，基于此可建立分布式超声气体测温模型。

2）基于LQ‒SVD的2维温度场重建算法重建的空间温度分布，从云图及具体误差数值看，重建的温度场具有较高的分辨率，精度较高。从时间真值带噪测试结果看，算法抗噪性较好，算法有效。

3）本文建立的分布式超声探测2维气体温度场方法可以较为细致地描述热压罐中热声耦合作用下的超声波的传播特性，可以较精确地反演探测出热压罐空间温度场分布，计算的结果符合物理规律，方法有效。

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