*4.1. Baseline Noise of the Experimental Setup*

The baseline noise of the experimental setup in the test conditions of the laboratory without the introduction of a heat source was measured first. The laboratory was in a closed environment with no airflow disturbance. The room temperature was approximately 25 ◦C, and the hot plate was not activated. When the laboratory environment was stable, the images of the specimen were recorded at 1 Hz over 5 min, and a total of 300 images were obtained. The middle 100 images were selected. The first image of the 100 images was taken as the reference image, and the remaining 99 images were correlated to the reference image to complete the DIC measurement. The displacement and strain fields of the 99 images were obtained. The Region of Interest (ROI) in DIC measurement was set to 452 × 452 pixels, as shown in Figure 7. The pixel equivalent was 0.109 mm/pixel. The subset radius was 14 pixels and the subset spacing was 1 pixel. An example of the calculation result is shown in Figure 8.

**Figure 7.** Region of Interest (ROI) in DIC measurement.

**Figure 8.** Representative contour plots of the displacements and strains computed in the DIC measurement without the introduction of a heat source: (**a**) Displacement in U direction; (**b**) displacement in V direction; (**c**) strain εxx; (**d**) strain εxy; (**e**) strain εyy.

In order to quantify the errors of the displacement field and strain field, two standard deviations, the spatial standard deviation and the temporal standard deviation, were calculated according to Reference [30]. The spatial standard deviation is calculated by calculating the standard deviation of each displacement field and strain field data to quantify the spatial variation of the displacement and strain. Then, the spatial standard deviation is averaged over 99 images. The temporal standard deviation is calculated by calculating the standard deviation of the 99 displacement fields or the strain field corresponding subsets to quantify the temporal variation. Then, in the ROI region, the standard deviations of all the subsets are spatially averaged. Table 1 shows the calculated spatial standard deviation and temporal standard deviation. According to References [31–34], for a DIC system with extremely well-controlled experimental noise sources, the error does not exceed 0.001 pixels, and the DIC system in a general laboratory environment does not exceed 0.01 pixels. From Table 1, it can be seen that the displacement standard deviations, both spatial and temporal, did not exceed 0.005 pixels, which is generally accepted by the DIC community as being a reasonable noise floor for typical experiments.


**Table 1.** Baseline noise floor for the experimental setup, without the purposeful introduction of a heat source, quantified by spatial and temporal standard deviations (STD) of the data.

### *4.2. Characteristics of Distortions due to Heat Waves*

Next, an experiment was conducted to investigate how heat waves affect DIC measurements. First, an image of the specimen was taken without the influence of a heat source as a reference image. The hot plate was turned on and the temperature set to 300 ◦C by the controller. After the temperature of the hot plate was stabilized at 300 ◦C, 200 images of the specimen were recorded at 1 Hz. The 200 images taken through heat waves and the reference image were used for DIC measurement. The result of the DIC measurement of the image taken at 100 s is shown in Figure 9.

**Figure 9.** Representative contour plots of the displacements and strains computed in DIC measurement affected by heat waves: (**a**) Displacement in U direction; (**b**) displacement in V direction; (**c**) strain εxx; (**d**) strain εxy; (**e**) strain εyy.

Comparing Figures 8 and 9, the effect of heat waves on the DIC measurement results is evident. The spatial and temporal standard deviations of the displacement fields and strain fields under the influence of heat waves are shown in Table 2. It can be seen from Table 2 that the spatial standard deviation and temporal standard deviation are significantly increased due to the effect of heat waves. The spatial standard deviation was increased by nearly ten times, and the standard deviation of the displacement fields was more than 0.05 pixels, which is unacceptable.


**Table 2.** Mean error of displacements and strains caused by imaging through heat waves, quantified by spatial and temporal standard deviations (STD) of the data.

Figure 10 shows the variation of the displacement of the ROI central subset over time. It can be seen that the swing of the curves of the displacement in both directions of U and V are not surrounding zero, but there is an offset, the Mu and Mv, as shown in Figure 10. The offsets are the main distortion mentioned above, and the swing around the main distortion is random distortion.

**Figure 10.** Displacement of the center subset as a function of time: (**a**) U direction; (**b**) V direction.

Next, the situation in which the temperature of the heat source changes drastically was analyzed. The heat source temperature was adjusted to 100, 150, 200, 250, 300 degrees Celsius, respectively. At each temperature, 200 unloaded sample images were taken; the calculated main distortion is shown in Figure 11. As the temperature increases, the main distortion tends to increase, but at different temperatures, this offset is different. Therefore, in the case where the temperature of the heat source changes drastically, the method of correcting the main distortion based on the BOS technique is no longer applicable. But the time-average method can also be used to improve the accuracy of the measurement.

**Figure 11.** The average main distortion at each temperature.

#### *4.3. Influence of Heat Waves on DIC Measurement Results*

Then, the screw bar behind the specimen was rotated and the in-plane displacement to the disc was loaded. DIC was used to measure the in-plane displacement of the disc. The measurement results with and without the heat waves were compared. First, in the absence of the heat source, the micrometer screw bar behind the specimen was rotated to make a slight rotation of the disc. The angle of the rotation was 0.2 degrees. The image before the rotation is the reference image, and the rotated specimen image is correlated with the reference image. The measured displacement field is shown in Figure 12.

**Figure 12.** Displacement field measured without the influence of heat waves.

The components of the displacement field in the U and V directions are shown in Figure 13. The displacement fields are expressed in pixel values to facilitate the precision analysis.

Next, the image taken without the heat source and rotation was used as the reference image. The specimen was kept still, the heat source turned on, and the temperature of the hot plate adjusted to 300 ◦C. After the environment was stable, the micrometer screw bar was rotated to rotate the disc, and the angle of the rotation was still 0.2 degrees. Then, 100 images at a frequency of 1 Hz were taken, the result of the DIC measurement of the images taken at the 50th second is shown in Figure 14. The components of the displacement field in the U and V directions are shown in Figure 15. It can be seen that under the influence of heat waves, the measurement results of the DIC were severely distorted.

**Figure 13.** Displacement fields obtained from the DIC measurement without the influence of heat waves: (**a**) Component in U direction; (**b**) component in V direction.

**Figure 14.** Displacement field measured with the influence of heat waves.

**Figure 15.** Displacement fields obtained from DIC measurement with the influence of heat waves: (**a**) Component in U direction; (**b**) component in V direction.

The 100 displacement fields were time-averaged to remove random disturbances, and the result is shown in Figure 16. It can be seen that the result improved, but it is still not ideal due to the main disturbance not having been removed.

**Figure 16.** Displacement field after time average: (**a**) Component in U direction; (**b**) Component in V direction.

#### *4.4. Verification of the Correction Algorithm*

In order to verify the proposed method, the following experiment was performed. First, in the case where the hot plate was not turned on, and the specimen was not loaded in displacement, an image was taken as the reference image of the DIC and BOS technique. Next, the hot plate was turned on, the temperature of the hot plate adjusted to 300 ◦C, and 200 images taken after the environment was stable. Using the particle image velocimetry (PIV) algorithm in the BOS technique, the distortion vector field of each image was obtained by calculating the amount of distortion of every particle in the 200 images compared with the corresponding particle in the reference image. By time-averaging 200 distortion vector fields, the vector field of the main distortion was obtained, as shown in Figure 17.

**Figure 17.** Main distortion measured by the BOS technique: (**a**) Displacement vector distribution (the box represents local amplification results); (**b**) resultant displacement vector distribution.

The micrometer screw was gently turned to rotate the disc by 0.2 degrees. The in-plane displacement to the disc was loaded, and then 100 images of the specimen were taken through the heat waves. The main distortion displacement field was utilized to remap the 100 images to remove the main distortion on them. Then, using the time-average method, 100 images with the main distortion removed were averaged to remove the random distortion. The image with the heat wave disturbance removed was finally obtained, as shown in Figure 18. The corrected image was correlated to the reference image to complete the DIC measurement. The measurement result is shown in Figure 19, and the components of the displacement field in the U and V directions are shown in Figure 20. It can be seen that the influence of the heat waves on the DIC measurement result was corrected, as compared with Figures 14 and 15.

**Figure 18.** Corrected image obtained by the proposed method (The blue box is the region where the DIC calculation is performed).

**Figure 19.** Displacement field after correction.

**Figure 20.** Displacement fields obtained from DIC measurement using the corrected image: (**a**) Component in U direction; (**b**) component in V direction.

In order to further prove that the proposed method can improve the measurement accuracy of DIC, a plot of displacement in U direction vs. Y (X = 226) and a plot of displacement in V direction vs. X (Y = 226) are shown in Figure 21.

**Figure 21.** Plots of displacement in U vs. Y and in V vs. X: (**a**) Displacement component in U direction vs. Y (X = 226); (**b**) displacement component in V direction vs. X (Y = 226).

The root means square error (RMSE) is used to evaluate the displacement measurement results before and after the correction. The RMSE reflects the degree to which the measured value deviates from the real value, and can reflect the accuracy of the measurement. The smaller the root mean square error, the higher the measurement accuracy. The calculation formula of the RMSE is shown in Equation (5) [35].

$$RMSE = \sqrt{\frac{\sum\_{i=1}^{n} (X\_i - X\_{turc,i})^2}{n}} \tag{5}$$

where *Xi* is the measured value, the *Xture*,*<sup>i</sup>* is the real value and the n is the number of measurements. The *RMSE* of the measurement of the displacement component in the U direction before correction was 0.0732 pixels, while it was 0.0126 pixels after correction. The *RMSE* of the measurement of the displacement component in V direction before correction was 0.0711 pixels, while it was 0.0102 pixels after correction.

#### **5. Conclusions**

This paper proposes a correction method based on the background-oriented schlieren technique. The method can correct the distortion caused by heat waves to digital image correlation measurements. Through theoretical analysis and experiments, the characteristics of the distortion due to heat waves on the images were researched. The effectiveness of the proposed method was verified by experiments. Through the research of this paper, the following conclusions are drawn:


Although the proposed method can effectively improve the measurement accuracy of DIC, it also has a certain limitation. This method is especially suitable for cases where the temperature of the heat source is stable. When the temperature changes drastically, the correction is less than ideal.

**Author Contributions:** Conceptualization, C.M.; Data curation, Z.Z.; Formal analysis, X.R.; Funding acquisition, H.Z.; Investigation, C.M.; Methodology, C.M.; Project administration, Z.Z.; Resources, Z.Z.; Software, X.R.; Supervision, Z.Z.; Validation, H.Z.; Visualization, C.M.; Writing—original draft, C.M.; Writing—review & editing, H.Z. and X.R.; All authors reviewed the manuscript.

**Funding:** This research was funded by the National Key R&D Program of China (No. 2016YFF0101802) and Natural Science Foundation of Tianjin (No. 17JCYBJC19300).

**Conflicts of Interest:** The authors declare no conflict of interest.
