Online Sensing of Thermal Deformation in Complex Space Bulkheads Driven by Temperature Field Measurements

Abstract

In the assembly of spacecraft cabins, the presence of uncertain and time-varying temperature environments can induce thermal deformation in bulkheads, potentially affecting dimensional stability. Online sensing of thermal deformation is critical for mitigating such risks. However, conventional finite element methods (FEMs) rely on cascading thermal and structural analyses, which suffer from inefficiency. To address this issue, we propose a methodology that integrates a physical model with a data-driven temperature field measurement technique, demonstrated through case studies involving a spacecraft porthole bulkhead. First, leveraging the geometric invariance of the bulkhead during assembly, a purely static FE model is established offline. Second, multi-point temperature measurements combined with Kriging estimation are employed to directly reconstruct the temperature field, circumventing the computationally intensive FEM-based thermal analysis process. Finally, by utilizing the precomputed inverse stiffness matrix and performing an online conversion from temperature to equivalent forces, thermal deformation is rapidly resolved. The numerical results demonstrate that the root-mean-square errors of the predicted full-field deformation are maintained at the micron level, with an average computation time of less than 0.14 s. Furthermore, a meticulously designed experiment was conducted, where the predicted thermal displacements of several key points showed good agreement with measurements by means of a laser tracker. This research provides a promising tool to achieve digital twinning of thermal deformation states for aerospace components.

1. Introduction

Spacecraft cabins serve as the primary load-bearing structures of aerospace vehicles, assembled from numerous bulkheads. The dimensional stability of these structures during both manufacturing and service is critical to maintaining the safety of space stations operating in the extreme environment of outer space [1,2]. Existing engineering practices have demonstrated that thermal deformation is one of the main factors affecting the dimensional stability of cabin bulkheads [3,4,5]. These large-scale, thin-walled, and geometrically complex aluminum alloy structures are highly sensitive to temperature variations. Both the complex thermal conditions encountered during manufacturing [6] and solar radiation exposure in orbit can induce thermal deformations on the order of millimeters, thereby compromising manufacturing precision and potentially even structural integrity.

Online sensing of thermal deformation plays a critical role in mitigating such risks [7,8,9,10,11]. Extensive research efforts in this area can be broadly classified into data-driven and physics-based approaches. A major advantage of the former is their suitability for online deployment, which has led to widespread application in machine tool systems [12]. However, these methods heavily depend on large volumes of on-site data for model training, which poses a significant challenge for space cabins due to their typically small-batch production.

In contrast, physics-based approaches, such as finite element methods (FEMs), directly model thermal deformation behavior based on prior knowledge of heat transfer and constitutive relations, thereby obviating the need for extensive observational data [13,14]. For instance, Zhang et al. [15] developed a thermo-structural FEM model for thermal deformation analysis of a satellite camera. They first solved the temperature field based on periodic solar radiation conditions and then assessed the impact of structural thermal deformation on imaging quality. Wang et al. [16] employed the FEM to investigate the thermal deformation characteristics of a rotary air-preheater. The temperature distribution was numerically calculated and then utilized as a boundary condition to calculate the thermal deformation. These studies demonstrate the capability of FEM in thermo-structural analysis, and a common methodology can be summarized from these works: Firstly, conduct a FEM-based thermal analysis to solve the temperature field, where careful treatment of boundary conditions—such as conduction, convection, and radiation—is required due to their complexity and difficulty in quantification. Subsequently, perform a static structural analysis to solve the thermal deformation. However, this cascading process is not only time-consuming but also prone to errors, rendering FEM an inefficient approach [17].

Recently, new efforts have been made in this regard. Several scholars have recognized the importance of integrating in situ sensor data into the FEM to enhance accuracy. Yang et al. [18] proposed a FEM-based hybrid approach to evaluate the thermal deformation of a simplified aircraft structure. In their study, actual temperatures measured by thermocouples were incorporated into a transient thermal analysis to obtain temperature distribution. Similarly, Zhang et al. [14], in a study on the thermal deformation analysis of a magnetron injection gun, combined FEM with measured temperature boundary conditions obtained via an infrared thermal camera to generate accurate temperature fields. The introduction of temperature measurements indeed provides more reliable boundary conditions, thereby enhancing the calculation accuracy. However, these studies predominantly focus on determining heat transfer boundary conditions using limited temperature measurements rather than directly acquiring global temperature field. Consequently, it still relies heavily on FEM-based heat transfer analysis.

The motivation behind this work is to develop a general online sensing method for thermal deformation of complex aerospace components. A scaled-down spacecraft bulkhead employed as a demonstrator, as depicted in Figure 1. The main contributions of this study are summarized as follows:

• A restructuring strategy based on the geometric invariance of the bulkhead is proposed, which decouples the time-consuming stiffness matrix assembly and inversion process from the conventional FEM, thereby enabling FEM-based online computation.
• A data-driven temperature field reconstruction technique is introduced, which allows for fast and accurate capture of full-field temperature distributions. This not only ensures the real-time performance of thermal deformation prediction but also improves its accuracy.

The remainder of this paper is organized as follows. Section 2 presents a detailed illustration of the proposed methodology. Section 3 introduces the numerical and experimental investigation for the studied bulkhead. Section 4 discusses the results and provides a comprehensive analysis. Finally, Section 5 concludes the research work.

2. Methodology

The conventional FEM-based approach typically involves performing a thermal analysis to solve the temperature field, coupled or inherited with static analysis. However, as mentioned earlier, this cascading process is both error-prone and inefficient. To this end, we propose a methodology that allows FEM to be deployed online by introducing a data-driven temperature field measurement technique, as shown in Figure 2. The proposed approach involves restructuring the conventional FEM framework, and decomposing it into an offline phase and an online phase, as outlined below.

• Offline phase: (a) Construct a finite element model for purely static analysis. (b) Assemble the inverse stiffness matrix incorporating the mesh, material properties, and constrained degrees of freedom (DOFs).
• Online phase: (a) Reconstruct the temperature field based on multi-sensor measurements. (b) Convert the temperature field into equivalent nodal forces, which are then applied as boundary conditions. (c) Solve the linear system of equations using the inverse stiffness matrix to obtain the thermal deformation.

2.1. Finite Element Modeling for Static Analysis

For the spacecraft bulkheads in the open workshop, temperature variations inside the component are far below the melting point of materials. Therefore, the induced thermal deformation follows the linear elastic constitutive law. In terms of the Cartesian coordinate system, this law is expressed as follows:

$$\begin{array}{c}{ϵ}_{xx}={\displaystyle \frac{1}{E}}\left({\sigma }_{xx}-\nu \left({\sigma }_{yy}+{\sigma }_{zz}\right)\right)\hfill \\ {ϵ}_{yy}={\displaystyle \frac{1}{E}}\left({\sigma }_{yy}-\nu \left({\sigma }_{zz}+{\sigma }_{xx}\right)\right)\hfill \\ {ϵ}_{zz}={\displaystyle \frac{1}{E}}\left({\sigma }_{zz}-\nu \left({\sigma }_{xx}+{\sigma }_{yy}\right)\right)\hfill \\ {ϵ}_{xy}={\displaystyle \frac{1}{2G}}{\sigma }_{xy};{ϵ}_{yz}={\displaystyle \frac{1}{2G}}{\sigma }_{yz};{ϵ}_{zx}={\displaystyle \frac{1}{2G}}{\sigma }_{zx}\hfill \end{array}$$

(1)

where $\sigma$ represents stress, $ϵ$ represents strain. E, $\nu$, and G are well-known material properties, namely Young’s modulus, Poisson’s ratio, and shear modulus, respectively.

The stress can also be expressed in terms of strain, which is given in matrix form as

$$\mathit{\sigma }=\mathit{D}\mathit{ϵ}$$

(2)

where $\mathit{\sigma }$ and $\mathit{ϵ}$ are the vector forms of stress and strain, respectively. $\mathit{D}$ is the stress–strain matrix. The thermal deformation is essentially the displacement of points within the continuum. According to the strain–displacement relations [19], strain can be expressed in terms of displacement and written in matrix form as

$$ϵ=\mathit{L}\mathit{u}$$

(3)

where $\mathit{u}$ is the point displacement in three directions, and $\mathit{L}$ is a linear differential operator as follows:

$$\mathit{L}={\left[\begin{array}{cccccc}{\displaystyle \frac{\partial }{\partial x}}& 0& 0& {\displaystyle \frac{\partial }{\partial y}}& 0& {\displaystyle \frac{\partial }{\partial z}}\\ 0& {\displaystyle \frac{\partial }{\partial y}}& 0& {\displaystyle \frac{\partial }{\partial x}}& {\displaystyle \frac{\partial }{\partial z}}& 0\\ 0& 0& {\displaystyle \frac{\partial }{\partial z}}& 0& {\displaystyle \frac{\partial }{\partial y}}& {\displaystyle \frac{\partial }{\partial x}}\end{array}\right]}^{\mathrm{T}}$$

(4)

To address the difficulty of obtaining analytical solutions for continuum displacements, the finite element approximation is employed. The continuous bulkhead is discretized into multiple finite elements, and each node of the element corresponds to a shape function $\mathit{N}$. Accordingly, the displacement at any point within the element can be expressed by means of shape functions as

$$\mathit{u}=\mathit{N}\mathit{U}$$

(5)

By substituting Equation (5) into Equation (3), the relationship between strain and nodal displacement can be expressed as

$$ϵ=\mathit{L}\mathit{N}\mathit{U}=\mathit{B}\mathit{U}$$

(6)

where $\mathit{B}$ is called the strain matrix, and $\mathit{B}=\mathit{L}\mathit{N}$.

The principle of virtual work is introduced to solve the nodal displacement, which reveals that for a system in equilibrium, the total virtual work of forces acting on the system is zero for any virtual displacement; hence, we have

$${\int }_{\Omega }\delta {\mathit{ϵ}}^{\mathrm{T}}\mathit{\sigma }d\Omega ={\int }_{\Omega }\delta {\mathit{u}}^{\mathrm{T}}\mathit{b}d\Omega +{\int }_{\Gamma }\delta {\mathit{u}}^{\mathrm{T}}\mathit{t}d\Gamma$$

(7)

where $\mathit{b}$ denotes the body force vector, and $\mathit{t}$ is the surface traction vector. $\Omega$ and $\Gamma$ represent the volume and surface domains over which $\mathit{b}$ and $\mathit{t}$ act, respectively.

The solution of Equation (7) can be obtained by substituting Equations (2), (5), and (6), yielding

$$\left({\int }_{\Omega }{\mathit{B}}^{\mathrm{T}}\mathit{D}\mathit{B}d\Omega \right)\mathit{U}={\int }_{\Omega }{\mathit{N}}^{\mathrm{T}}\mathit{b}d\Omega +{\int }_{\Gamma }{\mathit{N}}^{\mathrm{T}}\mathit{t}d\Gamma$$

(8)

or in a more compact form as

$$\mathit{K}\mathit{U}=\mathit{F}$$

(9)

where $\mathit{K}$ is the stiffness matrix, and $\mathit{F}$ is the force vector.

Additionally, the constraints should also be added in $\mathit{K}$ to account for the Dirichlet boundary conditions. After that, the inverse of the stiffness matrix can be computed, enabling the nodal displacements in Equation (9) to be obtained through simple matrix multiplication.

Since the bulkhead geometric shapes and materials do not undergo significant changes after being created, the stiffness matrix remains unchanged, and the same holds for its inverse. Conversely, the suffered temperatures are variable and uncertain. This geometric invariance in assembly inspires us that no matter how complex the components are, the tedious process of solving the inverse stiffness matrix can be completed in the offline phase, before identifying the force vectors online. Based on this premise, the subsequent challenges are twofold: (1) the real-time reconstruction of the full-field temperature and (2) the feasibility of relating this physical field to equivalent nodal forces.

2.2. Temperature Field Reconstruction Based on Multi-Sensors

To accurately obtain time-varying temperature fields in real time and account for thermal gradients, a temperature field reconstruction method based on multi-point measurements and Kriging estimation [20] is introduced to replace conventional thermal analysis. Suppose there are m sensors distributed within the volume region $\Omega$, with their coordinates forming the set $\mathit{S}={[s_1,\dots ,s_m]}^{\mathrm{T}}$ with $s_i\in {\mathbb{R}}^3$. Their corresponding temperature measurements are represented as $\mathit{T}={[T\left(s_1\right),\dots ,T\left(s_m\right)]}^{\mathrm{T}}$ with $T\left(s_i\right)\in \mathbb{R}$. The field reconstruction problem involves identifying a deterministic model $T\left(s\right)$ that relates the set of sample points $\mathit{S}$ and their responses $\mathit{T}$. In this manner, it becomes possible to reconstruct the temperatures at non-sample points and thereby the entire field. According to the Kriging method, the model $T\left(s\right)$ can be approximated as a weighted linear estimator based on the known samples and their responses:

$$\widehat{T}\left(s\right)=\sum _{i=1}^m{\omega }_i\left(s\right)·T\left(s_i\right),s\in \Omega \subseteq {\mathbb{R}}^3.$$

(10)

To determine the weight vector $\mathit{\omega }={[{\omega }_1\dots {\omega }_m]}^{\mathrm{T}}$, a statistical assumption is introduced that treats the deterministic model $T\left(s\right)$ as a realization of a random function $\mathcal{T}\left(s\right)$:

$$\mathcal{T}\left(s\right)=\sum _{j=1}^m{\beta }_{j}h\left(s_j\right)+Z\left(s\right)$$

(11)

The first term represents a regression model, for which a zero-order polynomial function is adopted in this study [21]. The second term $Z(·)$ is a stationary stochastic process with zero mean and covariance given by $E[Z\left(s_i\right),Z\left(s_j\right)]={\sigma }^2\mathcal{R}(s_i,s_j)$. The $\mathcal{R}(s_i,s_j)$ is the correlation function that depends solely on the distance between $s_i$ and $s_j$. By substituting $T(·)$ in Equation (10) with the random vector ${\mathcal{T}}_s={[\mathcal{T}\left(s_1\right),\dots ,\mathcal{T}\left(s_m\right)]}^{\mathrm{T}}$, the objective becomes minimizing the mean squared error of the Kriging estimation expressed by Equation (10) as follows:

$$min\mathrm{MSE}\left[\widehat{T}\left(s\right)\right]=E\left[{\left({\mathit{\omega }}^{\mathrm{T}}{\mathcal{T}}_s-\mathcal{T}\left(s\right)\right)}^2\right]$$

(12)

subject to the unbiased constraint:

$$E\left[{\mathit{\omega }}^{\mathrm{T}}{\mathcal{T}}_s\right]=E\left[\mathcal{T}\left(s\right)\right]$$

(13)

The Lagrange multiplier method is utilized to solve the above optimization problem, and the detailed procedure can be found in [22]. Herein, the solution is provided as

$$\mathit{\omega }={\mathit{R}}^{-1}\left[\mathit{r}\left(s\right)-\mathit{H}{\left({\mathit{H}}^T{\mathit{R}}^{-1}\mathit{H}\right)}^{-1}\left({\mathit{H}}^{\mathrm{T}}{\mathit{R}}^{-1}\mathit{r}\left(s\right)-h\left(s\right)\right)\right]$$

(14)

where

$$\begin{array}{cc}\hfill \mathit{H}& ={\left[h\left(s_1\right)\dots h\left(s_{\mathrm{m}}\right)\right]}^{\mathrm{T}}\hfill \\ \hfill \mathit{R}& =\left[\begin{array}{ccc}\mathcal{R}(s_1,s_1)& \dots & \mathcal{R}(s_1,s_m)\\ ⋮& \ddots & ⋮\\ \mathcal{R}(s_m,s_1)& \dots & \mathcal{R}(s_m,s_m)\end{array}\right]\hfill \\ \hfill \mathit{r}\left(s\right)& ={\left[\mathcal{R}(s_1,s)\dots \mathcal{R}(s_m,s)\right]}^{\mathrm{T}}\hfill \end{array}$$

(15)

Substitute Equations (14) and (15) into Equation (10) and rearrange them, yielding

$$\begin{array}{cc}\hfill \widehat{T}\left(s\right)& =h{\left(s\right)}^{\mathrm{T}}{\beta }^{*}+\mathit{r}{\left(s\right)}^{\mathrm{T}}{\mu }^{*}\hfill \\ \hfill {\beta }^{*}& ={\left({\mathit{H}}^{\mathrm{T}}{\mathit{R}}^{-1}\mathit{H}\right)}^{-1}{\mathit{H}}^{\mathrm{T}}{\mathit{R}}^{-1}\mathit{T}\hfill \\ \hfill {\mu }^{*}& ={\mathit{R}}^{-1}\left(\mathit{T}-\mathit{H}{\beta }^{*}\right)\hfill \end{array}$$

(16)

Here, ${\beta }^{*}$ is the least squares estimation of the parameter ${\beta }_i$ in Equation (11), which is related to the selected regression model $h(·)$. Since multiple sensors measure the temperature of the bulkhead, the sample point set $\mathit{S}$ and their corresponding temperature values $\mathit{T}$ are known, allowing $\beta$ and ${\mu }^{*}$ to be pre-calculated. For any non-sample point s, the temperature will be obtained by computing the correlation function value $\mathit{r}\left(s\right)$ and the selected regression model response $h\left(s\right)$, and then summing the two resulting terms. This algebraic computation is very fast, making real-time reconstruction of the temperature field of the bulkhead possible. This computationally efficient algebraic operation enables real-time reconstruction of the bulkhead’s temperature field.

2.3. Online Solution of Thermal Displacements

Based on the obtained full-field temperature information, the generated thermal strain can be determined according to the theory of thermoelasticity [23], as given by

$$\begin{array}{c}{ϵ}_{xx,T}={ϵ}_{yy,T}={ϵ}_{zz,T}=\alpha \Delta T\hfill \\ {ϵ}_{xy,T}={ϵ}_{yz,T}={ϵ}_{zx,T}=0\hfill \end{array}$$

(17)

where $\Delta T$ denotes the temperature change, and $\alpha$ is the coefficient of thermal expansion (CTE). Since thermal strains are additive to elastic strains, the constitutive law described by Equation (1) is accordingly modified to

$$\begin{array}{c}{ϵ}_{xx}={\displaystyle \frac{1}{E}}\left({\sigma }_{xx}-\nu \left({\sigma }_{yy}+{\sigma }_{zz}\right)\right)+\alpha \Delta T\hfill \\ {ϵ}_{yy}={\displaystyle \frac{1}{E}}\left({\sigma }_{yy}-\nu \left({\sigma }_{zz}+{\sigma }_{xx}\right)\right)+\alpha \Delta T\hfill \\ {ϵ}_{zz}={\displaystyle \frac{1}{E}}\left({\sigma }_{zz}-\nu \left({\sigma }_{xx}+{\sigma }_{yy}\right)\right)+\alpha \Delta T\hfill \\ {ϵ}_{xy}={\displaystyle \frac{1}{2G}}{\sigma }_{xy};{ϵ}_{yz}={\displaystyle \frac{1}{2G}}{\sigma }_{yz};{ϵ}_{zx}={\displaystyle \frac{1}{2G}}{\sigma }_{zx}\hfill \end{array}$$

(18)

Similar to Equation (2), the modified matrix form of the stress expressed in terms of strains can be deduced as

$$\mathit{\sigma }=\mathit{D}\left(\mathit{ϵ}-{\mathit{ϵ}}_T\right)$$

(19)

where ${\mathit{ϵ}}_{\mathit{T}}={\left[\alpha \Delta T\alpha \Delta T\alpha \Delta T000\right]}^{\mathrm{T}}$.

Substituting this new constitutive law into Equation (7) and using Equation (9), yields

$$\mathit{K}\mathit{U}-{\int }_{\Omega }{\mathit{B}}^{\mathrm{T}}\mathit{D}{\mathit{ϵ}}_{T}d\Omega =\mathit{F}$$

(20)

which can be rewritten as

$$\mathit{K}\mathit{U}={\mathit{F}}_T+\mathit{F}$$

(21)

Equation (21) indicates that the spontaneous thermal deformation can also be equivalently generated by an equivalent nodal force vector ${\mathit{F}}_T$. Furthermore, since the inverse of the stiffness matrix $\mathit{K}$ has been pre-calculated and full temperature field has been reconstructed in real time, the nodal thermal displacement vector $\mathit{U}$ can be solved online via simple matrix multiplication, yielding the following solution:

$$\mathit{U}=\left[{\mathit{K}}^{-1}\right]\left({\mathit{F}}_T+\mathit{F}\right)$$

(22)

3. Numerical and Experimental Investigation

Following the proposed methodology, both numerical and experimental studies were conducted on the porthole bulkhead. The main process is illustrated in Figure 3, which consists of two phases. First, in the offline phase, the bulkhead is modeled to yield the inverse stiffness matrix, and the positions of temperature measurement points are defined. This information is shared across numerical and experimental cases. Second, in the online phase, considering the practical difficulty of verifying the full-field temperature and thermal deformation in practice, a numerical investigation is conducted to assess accuracy and efficiency on a full-field scale. As a supplement, an experiment is designed to validate the performance at several key points (KPs). In this study, the offline phase is performed using ABAQUS 2022, while the online phase is implemented through custom MATLAB scripts. The procedures for numerical investigation and the experiment are detailed in this section, and the results and discussion are presented in the following section.

3.1. Offline Phase

During the offline phase, a FE model of the bulkhead was established using ABAQUS. The material is aluminum–magnesium alloy, with its properties provided by the manufacturer as follows: $E=70\mathrm{GPa}$, $\nu =0.33$, and $\alpha =2.4\times {10}^{-5} °{\mathrm{C}}^{-1}$. Additionally, the density is $2.7\times {10}^3\mathrm{kg}/{\mathrm{m}}^3$, specific heat capacity is 1300 J/(kg·°C), and thermal conductivity is 170 W/(m·°C). Before meshing, some local fine features including fillets and holes are simplified to allow for the creation of a more regular mesh. Subsequently, the C3D8R hexahedral element was employed to mesh the component. Various mesh sizes are numerically tested to examine the grid dependency of maximum deformations, as shown in Figure 4a. Accordingly, a global mesh size of 10 mm was chosen, resulting in a total of 2095 elements and 4348 nodes. Furthermore, a Dirichlet boundary condition was applied where a short edge of the bulkhead is fixed, as illustrated in Figure 4b. Finally, the stiffness matrix of the bulkhead is assembled by combining the discretized nodes, material properties, and the constrained DOFs. It is further inverted and stored in a compact format.

3.2. Online Phase

3.2.1. Numerical Validation

As shown on the left side of Figure 3, two types of ideal temperature fields (linear and nonlinear) are first defined, with the reference temperature set to 0 °C. These temperature fields are analytically expressed as follows:

$$\begin{array}{c}{T}_1=0.1x+0.15(y-500)\hfill \\ T_2=0.02\left[zsin\left({\displaystyle \frac{\pi }{150}}x\right)-xcos\left({\displaystyle \frac{\pi }{150}}z\right)\right]+50\hfill \end{array}$$

(23)

In light of (23), the values at the 62 temperature measurement points in Figure 4b,c are used to reconstruct the full temperature fields, which are then compared with their corresponding “prototypes”, as indicated by the label “Comparison A” in Figure 3. Subsequently, the reconstructed temperature field, combined with the inverse stiffness matrix obtained offline, is employed to solve for the bulkhead’s full-field thermal deformation. These results are subsequently compared with thermo-structural simulation results from ABAQUS, as indicated by the label “Comparison B”.

3.2.2. Experimental Validation

The primary idea of the experiment is to actively heat the bulkhead to induce thermal deformation and then compare the predicted deformation derived from multiple temperature sensors with the actual measurements obtained from a laser tracker. Based on this concept, an experimental apparatus was developed, as illustrated in Figure 5a. It consists of five main components: a self-developed active temperature control system (ATCS), the porthole bulkhead, a temperature measurement system, a laser tracker, and a PC.

Referring to Figure 5b, the configuration of the experimental apparatus is described as follows: First, the bulkhead was fixed to the experimental platform using hot-melt glue along its lower edge, in order to apply fixed constraints consistent with the constrained DOFs defined in the offline phase. Second, an ATCS is developed to generate a temperature field in the bulkhead. The system is powered by a 6-channel temperature controller, Omega CN616A (DwyerOmega, Michigan City, IN, USA), and communicates with a PC (Lenovo, Beijing, China) via RS485 bus. Moreover, as shown in Figure 5c, six 100 mm × 100 mm polyimide heating films (24 V, 100 W) were attached to the backside of the bulkhead using 3M adhesive, forming six independently controllable heating zones.

Third, the temperature measurement system is also multi-channel, consisting of a Toprie TP1000 data logger and eight TP1608 data collectors (Toprie, Shenzhen, China). Each collector was connected to 8 K-type thermocouples, with a measurement accuracy of ±(0.5 °C + 0.005% rdg), as specified by the manufacturer. The TP1000 unit synchronized the acquisition of temperature signals from all the data collectors through the RS485 bus. Out of the total 64 thermocouples, 62 were taped close to the predefined temperature measurement points (as seen in Figure 4b,c). Fourth, a Leica AT960MR laser tracker (Hexagon AB, Stockholm, Sweden) was employed for measuring thermal displacements at the KPs. The specified maximum permissible error is 15 µm + 6 µm/m, with a measurement range of up to 20 m. Three 0.5-inch spherically mounted retro-reflectors (SMRs) and their adapters (SBNs) were used as cooperative measurement targets, as depicted in Figure 5b. The positions of SMRs had been mapped to the mesh coordinate system, and their nodal number and coordinates are listed in

Table 1. The three KPs, their corresponding nodes, and nodal coordinates in the mesh coordinate system, in millimeters.

Key Point	Corresponding Node	Nodal Coordinates
1	259	(0, 760.000, −112.500)
2	3123	(−102.127, 743.014, −145.000)
3	2257	(174.839, 729.336, −227.500)

.

The experimental process, as outlined in Figure 3, is detailed as follows:

Active thermal control: The ATCS was activated to heat the porthole bulkhead. The feedback temperature signals from the six heating films are shown in Figure 6. As illustrated in the figure, the temperature rise in Zone 4 was intentionally made distinct from the others to create a deliberately non-uniform temperature field, thereby simulating a representative thermal condition.

Measurement: The temperature measurement system was configured to synchronously collect data from thermocouples at 1 Hz. Simultaneously, the laser tracker operated in Fast Measurement Mode to record the Cartesian coordinates of the three KPs. Each KP was measured 150 times, with the initial values serving as the reference. The KP displacements were subsequently calculated using the following equation:

$$U_i^j=∥{\mathit{s}}_i^j-{\mathit{s}}_i^0∥$$

(24)

where ${\mathit{s}}_i^j$ represents the measured coordinates of i-th KP at the j-moment.

Simulation: A thermo-structural simulation was carried out in ABAQUS to compute the thermal deformation of the bulkhead throughout the entire experimental process. In this simulation, the feedback temperature signals from the six heating zones, as shown in Figure 6, were applied as thermal boundary conditions, as indicated in the subplot. Meanwhile, the convective heat transfer coefficient between the bulkhead and the ambient environment was empirically set to 10 $\mathrm{W}/{\mathrm{m}}^2·$°C. The resulting thermal displacements at the three KPs were then extracted and used as the simulated values.

Prediction: Based on the measured data from the 62 thermocouples and the proposed method, the thermal displacements at the three KPs were predicted. It is worth noting that the initial temperature readings from the 62 thermocouples were used as reference values, consistent with the initial coordinate measurements of the KPs.

Comparison: The predicted and simulated results were subsequently compared with the laser tracker measurements, as indicated by the label “Comparison C” in Figure 3.

4. Results and Discussion

The results of the three comparisons are presented in this section. Data processing and visualization were performed using the MATLAB 2022b on a computer equipped with an Intel i5-2.9 GHz processor (Intel, Santa Clara, CA, USA) and 16 GB of RAM.

To begin with, the evaluation metrics are defined. The mean absolute percentage error (MAPE) is utilized to assess the accuracy of the reconstructed temperature fields, whereas the maximum absolute error (MaAE), mean absolute error (MAE), and root-mean-square error (RMSE) are introduced to assess the predicted thermal deformation, as defined by

$$\begin{array}{c}\mathrm{MAPE}={\displaystyle \frac{1}{n}}\sum _{i=1}^n\left|{\displaystyle \frac{{\widehat{T}}_i-T_i}{T_i}}\right|\hfill \\ \mathrm{MaAE}=\max \left(\left|{\widehat{u}}_i-u_i\right|\right)\hfill \\ \mathrm{MAE}={\displaystyle \frac{1}{n}}\sum _{i=1}^n\left|{\widehat{u}}_i-u_i\right|\hfill \\ \mathrm{RMSE}=\sqrt{{\displaystyle \frac{1}{n}}{\sum }_{i=1}^n{\left({\widehat{u}}_i-u_i\right)}^2}\hfill \end{array}$$

(25)

Here, ${\widehat{T}}_i$ and $T_i$ represent the estimated and reference temperatures at i-th node, respectively. Similarly, ${\widehat{u}}_i$ and $u_i$ denote the predicted displacement and the reference value at i-th node, respectively. The reference thermal displacement are obtained from ABAQUS simulation (in the numerical study) or laser tracker measurements (in the experiment).

4.1. Comparison A

The results of “Comparison A” are presented in Figure 7, where both the reference and reconstructed temperature fields, defined by Equation (23), are illustrated. Subsequently, an error analysis was conducted by calculating the point-wise differences. As shown in Figure 7, both reconstructed temperature fields exhibit high accuracy, with no notable deviations observed. From Figure 7c,f, the MAPE in the linear temperature field is 0.26%, while for the nonlinear case, it is 0.38%. Moreover, the distribution of errors resembles white noise, indicating that the reconstruction is unbiased. This underscores the efficacy of the constraints introduced in Equation (13). Additionally, these results confirm the robustness of the temperature measurement point layout.

Meanwhile, the time consumption was also assessed. To ensure reliability of the results, each reconstruction was performed ten times, and the average runtime were taken. The results listed in

Table 2. Reconstruction time for the two fields in second, where the “tx” denotes the time consumed of x-th trial.

Field	t1	t2	t3	t4	t5	t6	t7	t8	t9	t10	Avg.
${\widehat{T}}_1$	0.035	0.031	0.031	0.048	0.036	0.035	0.032	0.031	0.042	0.029	0.035
${\widehat{T}}_2$	0.044	0.035	0.041	0.033	0.033	0.031	0.032	0.031	0.032	0.031	0.035

indicate that there is no significant difference in the reconstruction time between the linear and nonlinear temperature field, with both cases requiring approximately 0.035 s. This demonstrates that, by means of multi-sensor measurements, the entire temperature field can be reconstructed both accurately and efficiently.

4.2. Comparison B

The reconstructed temperature fields were then used to compute the full-field thermal deformations, which were subsequently compared with those obtained from ABAQUS simulation using the ideal temperature fields as input. The results and error analysis are presented in Figure 8 and Figure 9, where $U_x$, $U_y$, and $U_z$ denote the displacement components along the x, y, and z directions, respectively; $U_d$ represents the total displacement value.

Based on the analysis of the results presented in Figure 8 and Figure 9, the following observations can be made:

First, Figure 8 shows a strong consistency between the predicted and simulated outcomes in various directions across the bulkhead, which can be attributed to the well-reconstructed temperature fields that serve as boundary conditions. Referring to the prediction error depicted in Figure 9, for the thermal deformation induced by the linear field $T_1$, the MaAE, MAE, and RMSE account for only 1.67%, 0.19%, and 0.28% of the maximum deformation magnitude, respectively. These metrics increase slightly for the nonlinear case, reaching 1.15%, 0.51%, and 0.60%, respectively.

Second, by comparing the contour map of thermal deformation in Figure 8 with the corresponding temperature field in Figure 7, it is found that the deformation at the location of the highest temperature is not necessarily the largest, and vice versa. This suggests that even for predicting the thermal displacement of individual points, relying solely on the local temperature is insufficient; instead, global temperature field information is necessary.

The third point to note is that the prediction error in the y direction is relatively large. The reduced accuracy may be partly attributed to the insufficient number of integration points in this direction, which corresponds to the thickness of the bulkhead. Achieving higher accuracy may indeed require the use of shell elements. However, overall, the micron-level accuracy produced by the C3D8R element is still sufficient in these cases.

Moreover, the computational efficiency of the online phase is evaluated by repeating the calculations ten times. The results are presented in

Table 3. The time consumption of the online phase in two studied cases in seconds.

Trail	${\mathit{T}}_1$			${\mathit{T}}_2$
	Total	TFR	SFD	Total	TFR	SFD
1	0.137	0.030	0.052	0.154	0.034	0.059
2	0.137	0.029	0.052	0.139	0.037	0.052
3	0.135	0.029	0.049	0.136	0.036	0.051
4	0.141	0.029	0.052	0.137	0.032	0.050
5	0.134	0.031	0.048	0.135	0.036	0.049
6	0.136	0.033	0.051	0.134	0.033	0.049
7	0.136	0.029	0.050	0.136	0.032	0.051
8	0.140	0.031	0.051	0.145	0.031	0.048
9	0.138	0.046	0.049	0.134	0.049	0.049
10	0.138	0.029	0.053	0.134	0.033	0.049

, which reveal that the average time consumption for the online phase is about 0.138 s for both two cases. Among them, the average time for temperature field reconstruction (TFR) is about 0.034 s; solving the thermal deformation field (SFD) consumes 0.051 s. In comparison, solving each of these cases using ABAQUS takes 27 s. This comparison demonstrates the efficiency advantage of the proposed method.

4.3. Comparison C

In this section, the proposed method is experimentally validated using the thermal displacements at three KPs as representative cases, involving comparisons among the simulated values, predicted values, and experimentally measured values. The simulated values were directly extracted from ABAQUS. In contrast, the predicted values were obtained through a two-step process:

The first step is to reconstruct the temperature field, leveraging the 62 thermocouple measurements as the input. Due to the abundance of results throughout the experimental process, we solely selected four typical moments and illustrated the corresponding contour plots in Figure 10, where the blue curve represents the average values of real-time measurements from the 62 thermocouples. As different colors progressively light up, the expected process of increasing temperature on the bulkhead can be observed.

The second step involves the online computation of thermal displacements at the three KPs based on the real-time reconstructed temperature field. Finally, the simulated values, predicted values, and the laser tracker measurements of thermal displacements at the three key points were compared, as shown in Figure 11. In addition,

Table 4. Quantification summary of simulation and prediction errors for the three KPs, in millimeters.

Key Point	MaAE		MAE		RMSE
	Simulation	Prediction	Simulation	Prediction	Simulation	Prediction
1	0.228	0.121	0.065	0.026	0.095	0.042
2	0.098	0.078	0.027	0.018	0.037	0.026
3	0.122	0.081	0.026	0.017	0.037	0.025

presents the prediction errors and simulation errors relative to the experimental measurements.

A comprehensive analysis of Figure 11 and Table 4 leads to the following conclusions:

First, both the simulation based on measured temperature boundary conditions and the proposed method driven by temperature field measurements show good agreement with the experimental results in terms of both trend and magnitude, which is consistent with findings reported in existing studies [14,18].

However, when the prediction and simulation errors are quantitatively evaluated (see Table 4), subtle differences become evident. Compared with FEM simulations that rely on sparse input data, the proposed temperature field measurement-driven approach demonstrates higher accuracy. Specifically, for the three key points, the MaAE is reduced by 47.17%, 20.33%, and 33.67%, respectively; the MAE is reduced by 59.70%, 31.71%, and 35.04%; and the RMSE is reduced by 56.40%, 29.41%, and 31.42%, respectively. These improvements can be attributed to the introduction of a data-driven temperature field measurement technique, which enables direct acquisition of more accurate thermal loads, thereby enhancing the predictive performance of the conventional FEM approach.

Moreover, the proposed method exhibits a significant advantage in terms of computational efficiency. According to the program timer, the simulation consisting of 150 incremental steps required a total of 238 s to complete. In contrast, the proposed method took only 20.3 s, resulting in an efficiency improvement of approximately 11.7 times. This confirms the potential of the method for online deployment.

Finally, it is worth noting that in the later stages of Figure 11a,b, the measured values tend to diverge from the other two curves. We speculate that this may be due to the presence of complex structures near these two points, where residual stresses may exist. Part of the thermal stress may engage in the “smoothing out” of the residual stress, and the remaining part produces the macroscopic displacement. This also explains why KP3 performs better, as it is located at the bulkhead edge with fewer residual stresses.

5. Conclusions

This study introduces an online sensing method for monitoring thermal deformation in complex aerospace components, comprising an offline phase and an online phase. By integrating a data-driven global temperature field measurement technique, the computational burden of conventional FEM is significantly reduced, achieving improved efficiency without sacrificing accuracy. Numerical and experimental studies were conducted on a spacecraft porthole bulkhead. The numerical results demonstrate that both linear and nonlinear temperature fields can be reconstructed with high accuracy and efficiency. The prediction errors of thermal deformation fields are maintained at the micron level, with computational time of less than 0.14 s. Experimental validation further confirms the method’s effectiveness, with good agreement observed between predicted and measured displacements at KPs, and RMSE remaining below 0.042 mm.

Future work will explore the application of the proposed method to thermal deformation sensing in composite aerospace structures, where anisotropic material properties and more complex boundary conditions should be carefully incorporated into the finite element modeling and measurement strategies.