A temperature difference-deformation decoupling method for in-situ monitoring of thermal expansion coefficient

Abstract

The application discloses a temperature difference-deformation decoupling method for in-situ monitoring of thermal expansion coefficient, and belongs to the technical field of material physical performance testing. The method comprises the following steps: acquiring high spatial resolution temperature distribution data and full-field deformation data; performing three-dimensional reconstruction on the high spatial resolution temperature distribution data to generate dynamic temperature field data; calculating theoretical thermal stress based on the dynamic temperature field data and material mechanics parameters to generate interference deformation data; introducing a dynamic correction coefficient to decouple the full-field deformation data and output a real thermal expansion deformation field; calculating the thermal expansion coefficient based on the real thermal expansion deformation field and temperature change trend data; monitoring the uniformity index of the real thermal expansion deformation field, and adjusting the dynamic correction coefficient and the material mechanics parameters online. The application can realize high-precision dynamic monitoring of the thermal expansion coefficient of a material under non-uniform temperature conditions by synchronously collecting a temperature field and a deformation field, theoretically calculating and removing thermal stress interference deformation caused by temperature difference, and combining a closed-loop feedback optimization mode.

Description

Technical Field

[0001] This invention relates to the field of material physical property testing technology, and in particular to a temperature difference-deformation decoupling method for in-situ monitoring of the coefficient of thermal expansion. Background Technology

[0002] The coefficient of thermal expansion (CTE) is an inherent physical property that characterizes the degree to which a material changes size with temperature. In aerospace, precision instruments, and microelectronic packaging, materials and structures need to operate over a wide temperature range; therefore, accurate determination of the CTE is crucial for structural design, reliability assessment, and failure analysis. Characterization of this property falls under the category of thermophysical property analysis of materials.

[0003] In existing technologies, common methods for measuring the coefficient of thermal expansion of materials include pushrod dilatometers, optical interferometry, and strain gauge methods. Pushrod dilatometers calculate the average linear expansion coefficient by measuring the elongation of a small sample heated in a uniformly heated furnace. Some advanced methods are also beginning to employ optical measurement techniques such as digital image correlation to obtain the strain field during heating by tracking the speckle displacement on the material surface, and then calculate the coefficient of thermal expansion accordingly.

[0004] However, the aforementioned existing technical solutions are usually based on an idealized premise that the sample being tested is in a completely uniform temperature field. In practical applications and many experimental scenarios, especially for large components or during rapid heating and cooling, achieving an absolutely uniform temperature distribution is extremely difficult, and temperature gradients are inevitable. This uneven temperature distribution will induce an uncoordinated expansion tendency within the material, thereby generating internal stress. The mechanical deformation caused by this thermal stress will be superimposed on the deformation caused by pure thermal expansion, making the directly measured total deformation unable to accurately reflect the true dimensional change, thus leading to deviations in the calculated coefficient of thermal expansion. Summary of the Invention

[0005] To address the aforementioned issues, this invention provides a temperature difference-deformation decoupling method for in-situ monitoring of the coefficient of thermal expansion. This method employs simultaneous acquisition of the temperature field and deformation field, theoretical calculation and elimination of thermal stress interference deformation caused by temperature difference, and closed-loop feedback optimization to achieve high-precision dynamic monitoring of the coefficient of thermal expansion of materials under non-uniform temperature conditions.

[0006] The above objectives can be achieved through the following approach:

[0007] A temperature difference-deformation decoupling method for in-situ monitoring of the coefficient of thermal expansion includes: acquiring high spatial resolution temperature distribution data of the surface or key area of ​​the material under test, and simultaneously acquiring full-field deformation data of the surface of the material under test to generate a temperature-deformation multi-source dataset; performing three-dimensional reconstruction on the high spatial resolution temperature distribution data to generate dynamic temperature field data; calculating the theoretical thermal stress caused by uneven temperature distribution based on the dynamic temperature field data and material mechanical parameters to generate interference deformation data; performing real-time decoupling calculation on the full-field deformation data using the interference deformation data, introducing a dynamic correction coefficient to obtain the true thermal expansion deformation field; calculating the coefficient of thermal expansion based on the true thermal expansion deformation field and temperature change trend data extracted from the dynamic temperature field data; monitoring the uniformity index of the true thermal expansion deformation field, determining whether the uniformity index exceeds a preset threshold, and adjusting the dynamic correction coefficient and the material mechanical parameters online.

[0008] Optionally, the generation of the temperature-deformation multi-source dataset includes: measuring the material under test using a fast scanning infrared thermal imager or a distributed multi-point micro temperature sensor array to generate high spatial resolution temperature distribution data; measuring the surface of the material under test using digital image correlation or laser speckle interferometry to generate full-field deformation data; and synchronously integrating the high spatial resolution temperature distribution data with the full-field deformation data to generate a temperature-deformation multi-source dataset.

[0009] Optionally, generating dynamic temperature field data includes: establishing three-dimensional geometric information of the material under test, and using the high spatial resolution temperature distribution data as boundary conditions to generate three-dimensional boundary temperature data; and based on the three-dimensional boundary temperature data, solving the unsteady heat conduction equation inside the material to generate dynamic temperature field data.

[0010] Optionally, generating the disturbance deformation data includes: calculating the theoretical thermal stress distribution based on the dynamic temperature field data and the material mechanical parameters to generate theoretical thermal stress data; and calculating the mechanical strain based on the theoretical thermal stress data and performing integration processing to generate disturbance deformation data.

[0011] Optionally, the calculation and integration of mechanical strain includes: based on the theoretical thermal stress data, using the material's elastic modulus and Poisson's ratio, and according to linear elasticity theory, calculating the mechanical strain tensor to generate mechanical strain data; and performing numerical integration of the mechanical strain data in the three-dimensional geometric domain of the material to generate disturbance deformation data.

[0012] Optionally, the real-time decoupling operation includes: calculating the disturbance deformation data and the dynamic correction coefficient to generate corrected disturbance deformation data; subtracting the corrected disturbance deformation data from the full-field deformation data to obtain the true thermal expansion deformation field.

[0013] Optionally, the calculation of the coefficient of thermal expansion includes: extracting the deformation of the real thermal expansion deformation field in the target area or key point to generate thermal expansion deformation data; extracting the overall average temperature change of the dynamic temperature field data in the corresponding area or key point to generate temperature change trend data; and calculating the coefficient of thermal expansion based on the thermal expansion deformation data and the temperature change trend data, according to the definition of the coefficient of thermal expansion.

[0014] Optionally, the online adjustment of the dynamic correction coefficient and the material mechanical parameters includes: calculating the displacement gradient standard deviation of the real thermal expansion deformation field to generate a uniformity index; when the uniformity index exceeds a preset threshold, adjusting the dynamic correction coefficient to generate an updated dynamic correction coefficient; and when a region of displacement discontinuity is detected in the real thermal expansion deformation field, optimizing the elastic modulus parameter in the material mechanical parameters.

[0015] Optionally, optimizing the elastic modulus parameter in the material's mechanical parameters includes: calculating the elastic modulus parameter in reverse based on the objective of minimizing the standard deviation of the displacement gradient, generating optimized elastic modulus parameters; and recalculating the theoretical thermal stress distribution using the optimized elastic modulus parameters to generate new disturbance deformation data.

[0016] Based on the same inventive concept, this invention also provides a temperature difference-deformation decoupling system for in-situ monitoring of the coefficient of thermal expansion. The system includes: a synchronous acquisition module for acquiring high spatial resolution temperature distribution data of the surface or key area of ​​the tested material, and simultaneously acquiring full-field deformation data of the surface of the tested material to generate a temperature-deformation multi-source dataset; a temperature field reconstruction module for performing three-dimensional reconstruction of the high spatial resolution temperature distribution data to generate dynamic temperature field data; an interference deformation module for calculating the theoretical thermal stress caused by uneven temperature distribution based on the dynamic temperature field data and material mechanical parameters, generating interference deformation data; a real-time decoupling module for performing real-time decoupling calculations on the full-field deformation data using the interference deformation data, introducing a dynamic correction coefficient to obtain the true thermal expansion deformation field; a coefficient of thermal expansion calculation module for calculating the coefficient of thermal expansion based on the true thermal expansion deformation field and temperature change trend data extracted from the dynamic temperature field data; and a collaborative optimization module for monitoring the uniformity index of the true thermal expansion deformation field, determining whether the uniformity index exceeds a preset threshold, and adjusting the dynamic correction coefficient and the material mechanical parameters online.

[0017] Compared with the prior art, the present invention has the following advantages:

[0018] This invention actively identifies, calculates, and decouples the thermal stress interference deformation caused by non-uniform temperature from the total measured deformation, thereby separating the pure, true thermal expansion deformation. This method fundamentally eliminates the main source of error faced by traditional techniques under non-uniform thermal conditions, significantly improving the accuracy and reliability of thermal expansion coefficient measurement.

[0019] This invention utilizes full-field synchronous measurement technology to acquire temperature and deformation data, enabling continuous monitoring of the coefficient of thermal expansion throughout the heating or cooling process. This provides a dynamic relationship between the coefficient of thermal expansion and temperature. Compared to traditional methods that only provide average values ​​over a wide temperature range, this method achieves more comprehensive material property characterization and possesses the capability for real-time in-situ evaluation of materials.

[0020] This invention integrates a closed-loop feedback and optimization mechanism, which can adjust correction coefficients and even optimize core material mechanical parameters online by monitoring the rationality of the decoupled physical field. This self-calibration capability makes the system highly adaptable and robust to uncertainties in the initial model parameters, ensuring stable and accurate measurement results even under varying test conditions.

[0021] Other features and advantages of the invention will be set forth in the description which follows, and will be apparent in part from the description, or may be learned by practicing the invention. The objects and other advantages of the invention may be realized and obtained by means of the structures pointed out in the description, claims and drawings. Attached Figure Description

[0022] To more clearly illustrate the technical solutions in the embodiments of the present invention or the prior art, the drawings used in the description of the embodiments or the prior art will be briefly introduced below. Obviously, the drawings described below are some embodiments of the present invention. For those skilled in the art, other drawings can be obtained based on these drawings without creative effort.

[0023] Figure 1 This is a schematic flowchart of a temperature difference-deformation decoupling method for in-situ monitoring of the coefficient of thermal expansion according to an embodiment of the present invention.

[0024] Figure 2 This is a three-dimensional distribution diagram of the actual thermal expansion deformation field in an embodiment of the present invention.

[0025] Figure 3 This is a graph showing the relationship between the coefficient of thermal expansion and temperature in an embodiment of the present invention.

[0026] Figure 4 This is a schematic diagram illustrating the adaptive adjustment of the dynamic correction coefficient in an embodiment of the present invention.

[0027] Figure 5This is a schematic diagram of a temperature difference-deformation decoupling system for in-situ monitoring of thermal expansion coefficient according to an embodiment of the present invention. Detailed Implementation

[0028] To make the objectives, technical solutions, and advantages of the embodiments of the present invention clearer, the technical solutions of the embodiments of the present invention will be clearly and completely described below with reference to the accompanying drawings. Obviously, the described embodiments are only some embodiments of the present invention, not all embodiments. Based on the embodiments of the present invention, all other embodiments obtained by those skilled in the art without creative effort are within the scope of protection of the present invention.

[0029] Reference Figure 1 One embodiment of the present invention proposes a temperature difference-deformation decoupling method for in-situ monitoring of the coefficient of thermal expansion. By simultaneously acquiring the temperature field and deformation field, theoretically calculating and eliminating the thermal stress interference deformation caused by the temperature difference, and combining it with closed-loop feedback optimization, it is possible to achieve high-precision dynamic monitoring of the coefficient of thermal expansion of materials under non-uniform temperature conditions.

[0030] The method described in this embodiment specifically includes:

[0031] Acquire high spatial resolution temperature distribution data of the surface or key area of ​​the material under test, and simultaneously collect full-field deformation data of the surface of the material under test to generate a temperature-deformation multi-source dataset.

[0032] The high spatial resolution temperature distribution data is reconstructed in three dimensions to generate dynamic temperature field data.

[0033] Based on the dynamic temperature field data and material mechanical parameters, the theoretical thermal stress caused by uneven temperature distribution is calculated, and interference deformation data is generated.

[0034] The interference deformation data is used to perform real-time decoupling calculation on the full-field deformation data, and a dynamic correction coefficient is introduced to obtain the real thermal expansion deformation field;

[0035] The coefficient of thermal expansion is calculated based on the actual thermal expansion deformation field and the temperature change trend data extracted from the dynamic temperature field data.

[0036] Monitor the uniformity index of the real thermal expansion deformation field, determine whether the uniformity index exceeds a preset threshold, and adjust the dynamic correction coefficient and the material mechanical parameters online.

[0037] Specifically, by simultaneously acquiring high-precision temperature and full-field deformation fields from the surface of the tested material, a multi-source dataset is constructed that precisely correlates thermal input and mechanical response in space and time. Its core is a decoupling concept based on a physical model, assuming that the experimentally measured total deformation is a linear superposition of the material's intrinsic thermal expansion deformation and the internal thermal stress deformation caused by non-uniform temperature distribution. The method reconstructs a three-dimensional dynamic temperature field from the measured surface temperature data and uses this as a thermal load. Combined with material mechanical parameters (mainly including the material's elastic modulus, Poisson's ratio, and an initially estimated coefficient of thermal expansion), a thermo-mechanical coupling simulation is performed to theoretically calculate the thermal stress deformation caused by temperature difference, i.e., the interference deformation data. Subsequently, this interference deformation field, calibrated with a dynamic correction coefficient, is subtracted from the experimentally measured total deformation field, thus separating the pure, true thermal expansion deformation field. Finally, a closed-loop feedback mechanism is established. By monitoring the physical property of the uniformity of the actual thermal expansion deformation field, the dynamic correction coefficient and material mechanical parameters are adjusted online in reverse, enabling the decoupled model to adaptively approximate physical reality and thus achieve accurate calculation of the thermal expansion coefficient. This invention enables in-situ, dynamic, and high-precision monitoring of the thermal expansion coefficient of materials. By actively calculating and removing the core interference term—thermal stress deformation caused by temperature gradients—this method fundamentally overcomes the systematic errors inherent in traditional measurement methods under non-uniform temperature field conditions, greatly improving the accuracy and reliability of the measurement results. Its full-field measurement characteristic captures the overall behavior of the material rather than its local response, while the closed-loop design of dynamic correction and online optimization endows the system with strong adaptive capabilities and robustness, enabling it to effectively cope with uncertainties such as inaccurate model parameters.

[0038] Optionally, the generation of the temperature-deformation multi-source dataset includes:

[0039] The material under test is measured using a fast scanning infrared thermal imager or a distributed multi-point miniature temperature sensor array to generate high spatial resolution temperature distribution data.

[0040] The surface of the material under test is measured using digital image correlation or laser speckle interferometry to generate full-field deformation data;

[0041] The high spatial resolution temperature distribution data and the full-field deformation data are synchronously integrated to generate a temperature-deformation multi-source dataset.

[0042] Specifically, the surface temperature and deformation of the material under test are measured simultaneously. For temperature data acquisition, a rapid scanning infrared thermal imager can be used directly on the surface of the material to capture the thermal radiation intensity in real time through non-contact measurement and convert it into a two-dimensional temperature distribution map. To ensure data accuracy, the rapid scanning infrared thermal imager needs to have a high refresh rate and high pixel resolution to generate a series of temporally continuous and spatially detailed high spatial resolution temperature distribution data. Alternatively, a distributed multi-point micro-temperature sensor array, such as a micro-thermocouple or fiber optic grating sensor array, can be arranged in key areas of the material surface to directly measure and collect the real-time temperature values ​​of multiple points, also constituting high spatial resolution temperature distribution data. For deformation data acquisition, a digital image correlation method is used simultaneously. This method requires pre-preparing a random speckle pattern on the surface of the material and using a high-resolution camera to continuously capture a series of digital images during the material's thermal deformation process. By comparing and analyzing the displacement and changes of the same sub-region in the images at different times, the displacement vector field of all points within the entire field of view is calculated using a digital image correlation algorithm, thereby generating full-field deformation data. This full-field deformation data comprehensively describes the displacement of every point on the material surface during the heating process. To achieve effective data integration, the temperature acquisition system and the deformation acquisition system must be controlled by a synchronous triggering device to ensure that the acquisition time of each frame of temperature distribution map and each frame of deformation field is strictly aligned. Finally, these high spatial resolution temperature distribution data with the same timestamp are paired and integrated with the full-field deformation data to form a time-series dataset. Each data point contains surface temperature field and deformation field information at a specific moment, ultimately constituting the temperature-deformation multi-source dataset necessary for subsequent decoupling analysis. This technical solution, by synchronously acquiring high spatial resolution temperature field and full-field deformation data, constructs a comprehensive dataset that accurately correlates thermal input and mechanical response in time and space. This provides essential high-fidelity input for subsequent accurate separation of thermal expansion deformation and thermal stress deformation, thereby significantly improving the accuracy and reliability of in-situ monitoring of the coefficient of thermal expansion.

[0043] Optionally, the generation of dynamic temperature field data includes:

[0044] Establish the three-dimensional geometric information of the material under test, and use the high spatial resolution temperature distribution data as boundary conditions to generate three-dimensional boundary temperature data;

[0045] Based on the three-dimensional boundary temperature data, the unsteady heat conduction equation inside the material is solved to generate dynamic temperature field data.

[0046] Specifically, the first step is to establish accurate three-dimensional geometric information of the material under test. This can be done directly using computer-aided design software or by reverse engineering the actual material using technologies such as 3D laser scanning to obtain its complete three-dimensional digital model. Then, the high spatial resolution temperature distribution data obtained in the previous steps is used as the boundary conditions for thermal analysis. Specifically, the synchronously acquired two-dimensional surface temperature field map is precisely assigned to the corresponding outer surface of the three-dimensional geometric model using a coordinate mapping algorithm, forming three-dimensional boundary temperature data that varies over time. For surfaces not directly measured, boundary conditions can be set as either adiabatic or subject to convective heat transfer with the environment, depending on the actual situation. Based on this three-dimensional boundary temperature data, the internal temperature field is reconstructed by solving the unsteady-state heat conduction equation within the material. This equation describes the fundamental physical law describing the propagation of heat in a medium. Its mathematical expression is as follows:

[0047] ,

[0048] In this equation, Represents spatial location With time The changing temperature field, i.e., the dynamic temperature field data to be obtained; The density of the material being tested. The specific heat capacity of the material, These three parameters—thermal conductivity, thermal conductivity, and thermal conductivity—are inherent physical parameters of the material and can be obtained by consulting material handbooks or experimental measurements. Represents the divergence operator. This represents the gradient operator. The partial differential equation is typically solved using numerical methods such as the finite element method. First, the three-dimensional geometric information is meshed. Then, the three-dimensional boundary temperature data is applied to the surface nodes of the model. The solution is iteratively solved at each time step, calculating the temperature values ​​of all nodes within the model at the corresponding time. By repeating this calculation process over the entire time series of the temperature-deformation multi-source dataset, a series of three-dimensional data describing the complete temperature distribution within the material over time can be obtained, constituting the final dynamic temperature field data. This technical solution, by establishing a three-dimensional geometric model and solving the unsteady-state heat conduction equation, successfully extends the two-dimensional surface temperature information measured non-contactly to the three-dimensional temperature field within the entire material. This overcomes the limitation of surface measurements not being able to obtain internal temperature gradient information, thus enabling precise understanding of the temperature change history and spatial distribution at any point within the material during heating.

[0049] Optionally, the generation of interference deformation data includes:

[0050] Based on the dynamic temperature field data and the material mechanical parameters, the theoretical thermal stress distribution is calculated, and theoretical thermal stress data is generated.

[0051] Based on the theoretical thermal stress data, mechanical strain is calculated and integrated to generate disturbance deformation data.

[0052] Specifically, to generate interference deformation data, this method first performs a thermo-mechanical coupling analysis based on the generated dynamic temperature field data and pre-determined or set material mechanical parameters to calculate the theoretical thermal stress distribution. The material mechanical parameters mainly include the material's elastic modulus. Poisson's ratio And an initial estimate of the coefficient of thermal expansion. This calculation process is typically performed using finite element analysis software. Dynamic temperature field data is applied to the three-dimensional geometric model of the material under test at specific time points as a thermal load. Due to the non-uniform temperature distribution within the material, the free thermal expansion trends of different parts differ. This incompatibility causes mutual constraints within the material, resulting in internal stress, i.e., theoretical thermal stress. Based on thermo-elastic-plastic mechanics theory, the software solves the equilibrium equations under given temperature field and mechanical boundary conditions, outputting the stress tensor distribution throughout the entire three-dimensional model; this distribution data is the theoretical thermal stress data. After obtaining the theoretical thermal stress data, mechanical strain is calculated and integrated. Mechanical strain is the elastic or plastic strain caused by theoretical thermal stress, distinct from the purely thermal expansion strain caused directly by temperature changes. According to the generalized Hooke's law, the corresponding mechanical strain tensor can be calculated from the theoretical thermal stress data. Finally, spatial integration of this mechanical strain field over the entire three-dimensional geometric domain of the material yields the displacement field determined by this strain field. This displacement field fully describes the structural shape change caused by internal stress due to temperature inhomogeneity. It is not the intrinsic thermal expansion behavior of the material, but rather a deformation component that disturbs it; therefore, it is defined as interference deformation data. By introducing finite element simulation based on a physical model, this method quantifies the internal thermal stress and its resulting deformation, which cannot be directly measured. It separates the interference component induced by temperature inhomogeneity from the total deformation of the measured material, generating interference deformation data that can be used for subsequent decoupling calculations. This lays a crucial physical foundation for accurately extracting true, pure thermal expansion information from complex mixed deformation signals, greatly improving the targeting and effectiveness of subsequent decoupling calculations, thereby ensuring that the final calculated coefficient of thermal expansion more accurately reflects the intrinsic physical properties of the material.

[0053] Optionally, the calculation of mechanical strain and integration includes:

[0054] Based on the theoretical thermal stress data, the mechanical strain tensor is calculated using the material's elastic modulus and Poisson's ratio, according to the linear elasticity theory, to generate mechanical strain data.

[0055] The mechanical strain data is numerically integrated within the three-dimensional geometric domain of the material to generate disturbance deformation data.

[0056] Specifically, to convert theoretical thermal stress data into disturbance deformation data, it is first necessary to calculate the mechanical strain tensor caused by theoretical thermal stress based on linear elasticity theory. This process utilizes the material's elastic modulus. Compared to Poisson These two parameters are inherent mechanical properties of the material, obtained through experimental measurement or by consulting standard material databases. For three-dimensional isotropic materials, the relationship between their mechanical strain components and theoretical thermal stress components follows the generalized Hooke's law, and its specific calculation expression is:

[0057] ,

[0058] ,

[0059] ,

[0060] ,

[0061] ,

[0062] ,

[0063] in, , , The positive mechanical strain is along the coordinate axis direction; , , To cut mechanical strain; , , The normal stress component is extracted from theoretical thermal stress data; , , For the corresponding shear stress components. The shear modulus of a material is related to its elastic modulus. Compared to Poisson The relationship is Substituting the stress components of each point or element in the 3D model based on the theoretical thermal stress data into the above formula allows for the calculation of the mechanical strain distribution within the entire model, thus generating mechanical strain data. Then, this mechanical strain data needs to be numerically integrated within the 3D geometric domain of the material to solve for the resulting displacement field. In continuum mechanics, displacement and strain are linked through geometric equations; solving the displacement field is equivalent to solving this system of differential equations. This integration process is typically performed in finite element analysis software. The software, based on the calculated mechanical strain data and applying appropriate displacement constraints to prevent rigid body displacement, solves for the nodal displacements throughout the entire 3D geometric domain, ultimately obtaining a displacement field with the same format and dimension as the full-field deformation data. This displacement field is the disturbance deformation data. This technical solution, through the explicit application of linear elasticity theory and numerical integration, provides a rigorous and operable computational path for the conversion from theoretical thermal stress to disturbance deformation. It accurately transforms an abstract internal force field, i.e., theoretical thermal stress, into a physical quantity that can be directly compared with experimental measurements—the deformation field.

[0064] Optionally, the real-time decoupling operation includes:

[0065] The interference deformation data and dynamic correction coefficients are calculated to generate corrected interference deformation data;

[0066] The true thermal expansion deformation field is obtained by subtracting the corrected interference deformation data from the full-field deformation data.

[0067] Specifically, the experimentally measured full-field deformation data is decomposed into two parts: the actual thermal expansion deformation and the disturbance deformation caused by temperature difference. This calculation is based on the assumption that the total deformation is a linear superposition of these two components. First, since there may be discrepancies between the theoretical model and the actual physical process—for example, the material mechanical parameters used may not be entirely accurate—directly subtracting the theoretically calculated disturbance deformation data from the full-field deformation data would introduce errors. Therefore, a dynamic correction coefficient is introduced to calibrate the theoretical calculation results. This dynamic correction coefficient is mainly used to comprehensively compensate for the discrepancies between theoretical calculations and physical reality caused by factors such as the temperature dependence of material mechanical parameters, model simplification, and measurement system errors, thereby improving the adaptability and accuracy of the decoupled model. The calculation process is as follows: First, the theoretically calculated disturbance deformation data is multiplied by an initially set or real-time updated dynamic correction coefficient to generate corrected disturbance deformation data. This correction process bridges the gap between the theoretical model and physical reality. Subsequently, the full-field deformation data and the corrected disturbance deformation data are subtracted point-by-point in space. This operation is mathematically expressed as:

[0068] ,

[0069] In this expression, At any point on the surface of the material being tested and time The true thermal expansion deformation field, which is the final output of this calculation; It is the total displacement vector at that point, which is obtained synchronously through experimental methods such as digital image correlation. It is based on dynamic temperature field data and material mechanical parameters, and is calculated to represent the displacement vector caused by thermal stress. This is a time-varying dynamic correction coefficient, a dimensionless scalar, whose initial value can be set to 1 and adjusted online in subsequent steps based on feedback from the decoupling results. This subtraction operation is performed synchronously at all data points within the entire measured field of view, thereby decomposing the experimentally measured mixed deformation field in real time, ultimately obtaining a pure, true thermal expansion deformation field caused solely by the intrinsic thermal expansion effect of the material. For example... Figure 2 As shown, the three-dimensional map is... coordinates and Coordinates serve as a spatial reference. The axes represent displacement, and the actual thermal expansion deformation field obtained after decoupling calculations is intuitively displayed through grayscale color mapping. The spatial distribution characteristics. This adaptive decoupling strategy can effectively compensate for systematic errors caused by inaccurate material parameters or model simplification, making the real thermal expansion deformation field separated from complex measured deformation more accurate and reliable, and greatly improving the accuracy and robustness of the final in-situ monitoring of the thermal expansion coefficient.

[0070] Optionally, the calculated coefficient of thermal expansion includes:

[0071] Extract the deformation of the real thermal expansion deformation field in the target area or key points to generate thermal expansion deformation data;

[0072] Extract the overall average temperature change of the dynamic temperature field data in the corresponding region or key point to generate temperature change trend data;

[0073] Based on the thermal expansion deformation data and the temperature change trend data, the thermal expansion coefficient is calculated according to the definition of the thermal expansion coefficient.

[0074] Specifically, to calculate the coefficient of thermal expansion, this method utilizes the real thermal expansion deformation field and the reconstructed dynamic temperature field data obtained in the previous steps. First, a quantitative amount of thermal expansion deformation data needs to be extracted from the real thermal expansion deformation field. This operation can be performed on a predefined target area on the surface of the material being measured or between two key points along a specific direction. For example, selecting an initial length of... Using a line segment as the measurement benchmark, the change in length of the line segment in a certain direction is calculated by querying the displacement vectors of the two endpoints of the line segment at different times in the actual thermal expansion deformation field. This is the thermal expansion deformation data. The next step is to extract the corresponding temperature changes over the exact same time and space dimensions. Specifically, this involves extracting the temperature values ​​of the target area or key points from the dynamic temperature field data, calculating their overall average temperature change over the corresponding time period, and generating temperature change trend data. Finally, based on these two interconnected datasets—thermal expansion deformation data and temperature change trend data—the coefficient of thermal expansion is calculated according to the physical definition of the linear thermal expansion coefficient. The formula is as follows:

[0075] ,

[0076] In this formula, It is the coefficient of thermal expansion to be determined; It is extracted from the real thermal expansion deformation field, and is the length change that occurs within a specific time period, i.e. thermal expansion deformation data; At the initial reference temperature, the initial length between the selected target area or key points is a known geometric quantity; It is extracted from dynamic temperature field data, representing the average temperature change occurring at the same spatial location within the same time period, i.e., temperature change trend data. By repeating this calculation within different temperature ranges, the relationship curve between the coefficient of thermal expansion and temperature can be obtained. Figure 3 As shown, the graph uses temperature changes The x-axis represents the coefficient of thermal expansion. The vertical axis represents the calculated and actual values, respectively, using a scatter plot and a dashed line to reflect the dynamic variation of the material's coefficient of thermal expansion across different temperature ranges. This technical solution fundamentally eliminates the interference of deformation caused by thermal stress on the calculation results by basing the calculations on a precisely decoupled real thermal expansion deformation field, ensuring that the deformation used for calculation is purely due to thermal expansion. Simultaneously, this method emphasizes the strict spatial correspondence between the deformation measurement region and the temperature measurement region, guaranteeing the physical authenticity of the causal chain.

[0077] Optionally, the online adjustment of the dynamic correction coefficient and the material mechanical parameters includes:

[0078] Calculate the standard deviation of the displacement gradient of the real thermal expansion deformation field to generate a uniformity index;

[0079] When the uniformity index exceeds a preset threshold, the dynamic correction coefficient is adjusted to generate an updated dynamic correction coefficient.

[0080] When a region of displacement discontinuity is detected in the actual thermal expansion deformation field, the elastic modulus parameter in the material mechanical parameters is optimized.

[0081] Specifically, to achieve online adjustment of the dynamic correction coefficient, this method establishes a closed-loop feedback control mechanism based on the quality of the decoupling results. The core idea is that ideal pure thermal expansion should result in a uniform deformation field. Therefore, the uniformity of the actual thermal expansion deformation field obtained after decoupling can be used as an indicator to evaluate the decoupling effect. The first step is to calculate the standard deviation of the displacement gradient of the actual thermal expansion deformation field, which serves as the uniformity indicator. Specifically, the spatial derivative of the actual thermal expansion deformation field obtained in the previous step is taken over the entire measurement area to obtain the displacement gradient field, which physically corresponds to the pure thermal expansion strain field. Then, the standard deviation of all data points in this gradient field is calculated. An ideal uniform deformation field should have a constant displacement gradient and a standard deviation close to zero. Therefore, the smaller the standard deviation, the better the uniformity of the actual thermal expansion deformation field, and the more ideal the decoupling effect. This standard deviation is defined as the uniformity indicator. The second step is to compare this uniformity indicator with a preset threshold. The preset threshold... The settings are based on the signal-to-noise ratio of the measurement system. Specifically, threshold Typically set to the displacement resolution of the measurement system. Three times the order of magnitude, that is This ensures that when the uniformity index exceeds a threshold, the indicated deformation nonuniformity has statistical significance, rather than being measurement noise. Displacement resolution. It can be related to digital images ( The calibration experiment of the system or laser speckle interferometry system determines the dynamic correction coefficient. If the index is below the threshold, it indicates that the current dynamic correction coefficient is appropriate and the decoupling operation is successful. If the uniformity index exceeds the preset threshold, it indicates that the decoupling is insufficient, and there are still non-uniform deformation components caused by thermal stress in the real thermal expansion deformation field. This usually means that the calculated interference deformation data is underestimated. At this time, the system will automatically adjust the dynamic correction coefficient and generate an updated dynamic correction coefficient. The adjustment adopts a proportional adjustment strategy, that is, the adjustment amount of the dynamic correction coefficient. Compared with current uniformity index Exceeding the preset threshold The degree is directly proportional. The specific adjustment rule can be expressed as:

[0082] ,

[0083] in, The proportionality coefficient ( The value of this factor affects the speed and stability of the adjustment. A baseline value (0.1-0.5) was determined through preliminary experiments. The upper limit was used for high thermal conductivity materials to accelerate convergence, while the lower limit was used for high-noise systems to ensure stability. Updated dynamic correction coefficients. for:

[0084] ,

[0085] in, These are the original dynamic correction coefficients. These updated coefficients will be used in the real-time decoupling operation at the next time step. For example... Figure 4 As shown in the figure, this diagram reflects the closed-loop adaptive adjustment law of the dynamic correction coefficient of this invention. The horizontal axis represents the number of iterations, and the left vertical axis represents the uniformity index. The right ordinate represents the dynamic correction coefficient. The curves show that the uniformity index gradually decreases and stabilizes under feedback adjustment, while the dynamic correction coefficient is adjusted proportionally according to the degree to which the uniformity index exceeds the threshold, until the system converges to the optimal decoupling state. Furthermore, the system also monitors whether there are discontinuous displacement regions in the actual thermal expansion deformation field, i.e., local regions where the displacement gradient abruptly changes.

[0086] This discontinuity represents a physically unrealistic pure thermal expansion behavior, strongly indicating that the material mechanical parameters used in the theoretical model, especially the elastic modulus, may deviate significantly from reality. When such a displacement discontinuity is detected, the system will pause the adjustment of the dynamic correction coefficients and instead trigger the optimization procedure for the elastic modulus parameter. The detection of this displacement discontinuity is achieved through the following algorithm: calculating the displacement gradient values ​​between adjacent pixels in the actual thermal expansion deformation field. ,in Indicates the first The displacement of each pixel It represents the displacement of its adjacent pixels. The absolute value of the displacement gradient between the two points; followed by... The pixel region is a sliding window; all displacement gradient values ​​within each window are calculated. Standard deviation This value characterizes the fluctuation intensity of the local deformation field; With the standard deviation of the global displacement gradient (Obtained through calculations of the entire deformation field) For comparison, when The time is marked as an abnormal window ( (The threshold is set based on the statistical principle of normal distribution). Continuous abnormal windows are detected in the spatial domain. If three or more abnormal windows appear consecutively along the normal or tangential direction of the material surface, a displacement discontinuity region is determined, triggering elastic modulus optimization. This technical solution, by introducing a uniformity index and a closed-loop feedback mechanism, transforms the entire decoupling process from an open-loop, unidirectional calculation flow into an intelligent system capable of self-evaluation and self-correction. It can identify and trigger corrections to the core physical model parameters, fundamentally improving the robustness of the entire in-situ monitoring system and the reliability of the final measurement results.

[0087] Optionally, the optimization of the elastic modulus parameter in the material's mechanical parameters includes:

[0088] Based on the objective of minimizing the standard deviation of the displacement gradient, the elastic modulus parameters are calculated in reverse to generate optimized elastic modulus parameters;

[0089] The theoretical thermal stress distribution is recalculated using the optimized elastic modulus parameters to generate new disturbance deformation data.

[0090] Specifically, to optimize the elastic modulus parameter, this method transforms the detected inhomogeneity of the real thermal expansion deformation field into an inverse optimization problem aimed at minimizing the standard deviation of the displacement gradient. This process is triggered when a region of displacement discontinuity is detected in the real thermal expansion deformation field. Its core is to iteratively adjust the elastic modulus parameter until the theoretical calculation and experimental observation achieve a physically optimal match. First, an objective function is established with the elastic modulus as the independent variable, where the value of the function is the standard deviation of the displacement gradient of the real thermal expansion deformation field. The optimization process can be described as seeking an optimal elastic modulus. , so that the objective function The minimum, the formula is:

[0091] ,

[0092] In this optimization problem, The elastic modulus parameter to be optimized; The objective function is... This indicates taking the parameter that makes the function reach its minimum value; For standard deviation operator; For spatial gradient operators; These are the full-field deformation data measured in the experiment; This is the current dynamic correction factor; This represents the disturbance deformation data, which is the elastic modulus. The function, because the calculation of theoretical thermal stress and mechanical strain both depend on The optimization process is implemented through an iterative loop. In each iteration, the system selects an experimental elastic modulus parameter and re-executes the complete forward calculation from calculating theoretical thermal stress to generating disturbance deformation data. Then, the disturbance deformation data is used for decoupling to obtain a temporary real thermal expansion deformation field, and its displacement gradient standard deviation is calculated. Based on this standard deviation, optimization algorithms, such as gradient descent or particle swarm optimization, will provide the next better experimental elastic modulus parameter. This loop is repeated until the displacement gradient standard deviation converges to a minimum or is below a set tolerance; the resulting elastic modulus parameter is the optimized elastic modulus parameter. After obtaining this optimized parameter, the system will use this parameter value to recalculate the theoretical thermal stress distribution and ultimately generate a new set of more accurate disturbance deformation data for subsequent real-time decoupling monitoring. This technical solution, by introducing a reverse calculation and optimization mechanism, endows the entire monitoring system with self-calibration and model correction capabilities. When simple dynamic coefficient adjustments are insufficient to eliminate decoupling errors, this method can delve into the root of the physical model to identify and optimize core material mechanics parameters. This not only solves the systematic error caused by inaccurate elastic modulus, but also makes the whole method adaptable to situations where material properties are unknown or change with temperature.

[0093] Based on the same inventive concept, such as Figure 5 As shown, the present invention also provides a temperature difference-deformation decoupling system for in-situ monitoring of the coefficient of thermal expansion, the system comprising:

[0094] The synchronous acquisition module is used to acquire high spatial resolution temperature distribution data of the surface or key area of ​​the material under test, and simultaneously acquire full-field deformation data of the surface of the material under test to generate a temperature-deformation multi-source dataset.

[0095] The temperature field reconstruction module is used to perform three-dimensional reconstruction of the high spatial resolution temperature distribution data to generate dynamic temperature field data.

[0096] The interference deformation module is used to calculate the theoretical thermal stress caused by uneven temperature distribution based on the dynamic temperature field data and material mechanical parameters, and generate interference deformation data.

[0097] The real-time decoupling module is used to perform real-time decoupling calculations on the full-field deformation data using the interference deformation data, introduce dynamic correction coefficients, and obtain the real thermal expansion deformation field.

[0098] The thermal expansion coefficient calculation module is used to calculate the thermal expansion coefficient based on the real thermal expansion deformation field and the temperature change trend data extracted from the dynamic temperature field data.

[0099] The collaborative optimization module monitors the uniformity index of the real thermal expansion deformation field, determines whether the uniformity index exceeds a preset threshold, and adjusts the dynamic correction coefficient and material mechanical parameters online.

[0100] To verify the effectiveness and superiority of the temperature difference-deformation decoupling method for in-situ monitoring of the coefficient of thermal expansion proposed in this invention, this embodiment uses an aerospace-grade high-temperature alloy plate specimen as the test object and conducts an in-situ monitoring experiment in a controlled-temperature thermodynamic chamber. The specimen dimensions are... In the experiment, one end of the specimen was locally heated to create a typical non-uniform temperature distribution and complex thermal stress field inside the specimen.

[0101] In this embodiment, the synchronous acquisition module uses a high-resolution fast-scanning infrared thermal imager and a set of binocular digital image correlation (DIR) devices. The system uses an infrared thermal imager to continuously capture images of the specimen surface at a frequency of 50 Hz, generating high spatial resolution temperature distribution data. The system simultaneously captures pre-prepared random speckle images on the specimen surface, generating full-field deformation data. The two systems utilize an external synchronization trigger to ensure strict temporal alignment of data acquisition, forming a temperature-deformation multi-source dataset.

[0102] The temperature field reconstruction module first establishes three-dimensional geometric information that is completely consistent with the specimen, and uses the collected surface temperature data as boundary conditions. By solving the unsteady heat conduction equation, it generates dynamic temperature field data of the entire specimen's interior as it evolves over time.

[0103] The interference deformation module is based on this dynamic temperature field data and uses the initial mechanical parameters (elastic modulus) provided in the alloy material handbook. Poisson's ratio The theoretical thermal stress caused by the non-uniform temperature distribution was calculated using finite element analysis software, and the initial disturbance deformation data were generated through integration.

[0104] In the initial stage of the experiment, the real-time decoupling module used a dynamic correction coefficient with an initial value of 1.0. By subtracting the interference deformation data from the full-field deformation data, a preliminary true thermal expansion deformation field was obtained. However, the collaborative optimization module detected that the uniformity index (displacement gradient standard deviation) of this deformation field was... Significantly higher than The preset threshold was exceeded, and local discontinuities were found in the deformation field. This indicates that the initial mechanical parameters deviate from the actual conditions of the material at the current high temperature.

[0105] Accordingly, the collaborative optimization module triggered the optimization procedure for the elastic modulus parameters. With the objective of minimizing the standard deviation of the displacement gradient in the real thermal expansion deformation field, the system iteratively adjusted the elastic modulus parameters through inverse calculation. After several iterations, the system determined the optimal elastic modulus to be 195 GPa and simultaneously optimized the dynamic correction coefficient to 1.29. The optimized parameters were then used to regenerate and decouple the disturbance deformation data, resulting in a final real thermal expansion deformation field with a homogeneity index reduced to [value missing]. It is far below the threshold, and the discontinuous areas within the field have disappeared.

[0106] Finally, based on this highly uniform real thermal expansion deformation field and the corresponding dynamic temperature field data, the thermal expansion coefficient calculation module calculated the thermal expansion coefficient values ​​in different temperature ranges. To verify the effectiveness of the invention, the measurement results of the invention were compared with those of traditional measurement methods (i.e., calculations directly using total deformation and average temperature without decoupling) and the standard reference values ​​for the material.

[0107] Table 1. Comparison of Coefficients of Thermal Expansion (CTE) Measured by Different Methods

[0108]

[0109] Table 2. Data table of online adjustment process of collaborative optimization module

[0110]

[0111] Table 3 Error Analysis of Measurement Results of the Invention and Standard Reference Values

[0112]

[0113] As can be seen from the data in Tables 1, 2 and 3 above, the method of the present invention has significant advantages in dealing with the problem of monitoring the coefficient of thermal expansion under non-uniform temperature fields.

[0114] The data comparison in Table 1 clearly shows that, due to the inability to isolate the interfering deformation caused by thermal stress, the traditional method results in increasingly larger deviations from the standard reference values ​​as the temperature rises, leading to serious errors. In contrast, the present invention, through precise decoupling and optimization, achieves measurement results that highly match the standard reference values, proving that this method can effectively eliminate systematic errors caused by temperature differences.

[0115] Table 2 details the online adjustment process of the collaborative optimization module. The data shows that by optimizing the elastic modulus and dynamic correction coefficients in reverse, with the uniformity of the real thermal expansion deformation field as the objective, the system's uniformity index converged to an ideal state far below the threshold after only a few iterations. This fully demonstrates the powerful adaptive and model correction capabilities of the closed-loop feedback mechanism of this invention.

[0116] The error analysis in Table 3 further quantitatively demonstrates the accuracy of the present invention. Throughout the entire test temperature range, the relative error of the measurement results obtained by the present invention remains within 1%, achieving the high-precision measurement requirements. This indicates that the true thermal expansion deformation field obtained by the present invention can highly and realistically reflect the intrinsic physical behavior of the material, thereby ensuring the accuracy and reliability of the finally calculated coefficient of thermal expansion.

[0117] It should be noted that the electrical connections between the various units described above do not necessarily represent direct or indirect connections. Any indirect connection method can be applied to the embodiments of the present invention as long as it achieves the purpose of the present invention. The above descriptions are merely exemplary embodiments of the present invention and should not be construed as limiting the scope of the present invention.

[0118] All equivalent changes and modifications made in accordance with the teachings of this invention are still within the scope of this invention. Those skilled in the art will readily conceive of other embodiments of this invention upon considering the specification and the disclosure of practical truth. This application is intended to cover any variations, uses, or adaptations of this invention that follow the general principles of this invention and include common knowledge or conventional techniques in the art not described herein.

Claims

1. A method for decoupling temperature difference and deformation for in-situ monitoring of coefficient of thermal expansion, characterized in that, The method comprises: acquiring high spatial resolution temperature distribution data of a surface or a key area of a measured material and synchronously acquiring full-field deformation data of the surface of the measured material to generate a temperature-deformation multi-source data set; reconstructing the high spatial resolution temperature distribution data in three dimensions to generate dynamic temperature field data; calculating theoretical thermal stress caused by uneven temperature distribution based on the dynamic temperature field data and material mechanical parameters to generate interference deformation data; real-time correcting the interference deformation data by a dynamic correction coefficient to obtain corrected interference deformation data, subtracting the full-field deformation data from the corrected interference deformation data in a spatial domain point by point to obtain a real thermal expansion deformation field; calculating a thermal expansion coefficient based on the real thermal expansion deformation field and temperature change trend data extracted from the dynamic temperature field data; The displacement gradient standard deviation of the real thermal expansion deformation field is calculated to obtain a uniformity index, and when the uniformity index exceeds a preset threshold, the dynamic correction coefficient is adjusted, and the adjustment amount of the dynamic correction coefficient wherein is a proportional coefficient, the basic value is determined by pre-experiment, the upper limit is taken for high thermal conductivity material to accelerate convergence, and the lower limit is taken for high noise system to ensure stability, is a current uniformity index, is a preset threshold, and the updated dynamic correction coefficient wherein is the original dynamic correction coefficient, when it is detected that the real thermal expansion deformation field has a displacement discontinuous region, the optimization elastic modulus parameter is calculated through an iterative loop according to a displacement gradient standard deviation minimization target wherein is an elastic modulus parameter to be optimized, is an objective function, represents a parameter that makes the function reach a minimum value, is a standard deviation operator, is a spatial gradient operator, is full-field deformation data measured by experiment, is a current dynamic correction coefficient, represents interference deformation data, which is a function of the elastic modulus and the theoretical thermal stress distribution is recalculated based on the optimized elastic modulus parameter to generate new interference deformation data.

2. The method of claim 1, wherein the method further comprises, the generation of the temperature-deformation multi-source data set comprises: measuring the measured material by using a fast scanning infrared thermal imager or a distributed multi-point micro temperature sensor array to generate high spatial resolution temperature distribution data; measuring the surface of the measured material by using a digital image correlation method or a laser speckle interference method to generate full-field deformation data; synchronously integrating the high spatial resolution temperature distribution data and the full-field deformation data to generate a temperature-deformation multi-source data set.

3. The method of claim 2, wherein the method further comprises, the generation of the dynamic temperature field data comprises: establishing three-dimensional geometric information of the measured material and taking the high spatial resolution temperature distribution data as a boundary condition to generate three-dimensional boundary temperature data; solving a non-steady-state heat conduction equation inside the material based on the three-dimensional boundary temperature data to generate dynamic temperature field data.

4. The method of claim 3, wherein, the generation of the interference deformation data comprises: calculating a theoretical thermal stress distribution based on the dynamic temperature field data and the material mechanical parameters to generate theoretical thermal stress data; calculating mechanical strain and performing integral processing based on the theoretical thermal stress data to generate interference deformation data.

5. The method of claim 4, wherein the method further comprises, the calculation of the mechanical strain and the integral processing comprises: calculating a mechanical strain tensor based on the theoretical thermal stress data by using the elastic modulus and the Poisson's ratio of the material according to the linear elasticity theory to generate mechanical strain data; performing numerical integration of the mechanical strain data in a three-dimensional geometric domain of the material to generate interference deformation data.

6. The method of claim 1, wherein, the calculation of the thermal expansion coefficient comprises: extracting deformation values of the real thermal expansion deformation field at target areas or key points to generate thermal expansion deformation value data; extracting overall average temperature changes of the dynamic temperature field data at corresponding areas or key points to generate temperature change trend data; calculating a thermal expansion coefficient based on the thermal expansion deformation value data and the temperature change trend data according to a definition relationship of the thermal expansion coefficient.

7. A temperature difference-deformation decoupling system for in-situ monitoring of coefficient of thermal expansion, characterized in that, the system comprises: a synchronous acquisition module configured to acquire high spatial resolution temperature distribution data of a surface or a key area of a measured material and synchronously acquire full-field deformation data of the surface of the measured material to generate a temperature-deformation multi-source data set; a temperature field reconstruction module configured to reconstruct the high spatial resolution temperature distribution data in three dimensions to generate dynamic temperature field data; and a thermal expansion coefficient calculation module configured to calculate a thermal expansion coefficient based on a real thermal expansion deformation field and temperature change trend data extracted from the dynamic temperature field data. an interference deformation module, configured to calculate theoretical thermal stress caused by uneven temperature distribution based on the dynamic temperature field data and material mechanics parameters, and generate interference deformation data; a real-time decoupling module, configured to correct the interference deformation data in real time by a dynamic correction coefficient, obtain corrected interference deformation data, and subtract the full-field deformation data and the corrected interference deformation data in a spatial domain point by point to obtain a real thermal expansion deformation field; a thermal expansion coefficient calculation module, configured to calculate a thermal expansion coefficient based on the real thermal expansion deformation field and temperature change trend data extracted from the dynamic temperature field data. A cooperative optimization module is configured to calculate a uniformity index of a displacement gradient standard deviation of the real thermal expansion deformation field, and adjust the dynamic correction coefficient when the uniformity index exceeds a preset threshold value, and the adjustment amount of the dynamic correction coefficient wherein is a proportional coefficient, a basic value is determined by a pre-experiment, an upper limit is taken for the high thermal conductivity material to accelerate convergence, and a lower limit is taken for the high noise system to ensure stability, is a current uniformity index, is a preset threshold value, and the updated dynamic correction coefficient wherein is the original dynamic correction coefficient, and when it is detected that the real thermal expansion deformation field has a displacement discontinuity region, an optimization elastic modulus parameter is calculated according to a displacement gradient standard deviation minimization target iterative cycle wherein is an elastic modulus parameter to be optimized, is an objective function, represents a parameter that makes the function reach a minimum value, is a standard deviation operator, is a spatial gradient operator, is full-field deformation data measured by an experiment, is a current dynamic correction coefficient, represents interference deformation data, is a function of the elastic modulus , and the theoretical thermal stress distribution is recalculated based on the optimized elastic modulus parameter, and new interference deformation data is generated.

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