Research on the Internal Thermal Boundary Conditions of Concrete Closed Girder Cross-Sections under Historically Extreme Temperature Conditions

Featured Application: The temperature distributions on concrete closed girder cross-sections can be accurately simulated by using the ﬁnite element models to establish air elements as the internal thermal boundary condition. Using the ﬁnite element models to establish air elements as the internal thermal boundary condition has a wide application range because it does not require the ﬁeld measurement of the temperature inside the cavity, which is cost- and time-prohibitive. Moreover, this method can predict the temperature distributions on concrete closed girder cross-sections under historically extreme temperature conditions. Abstract: The accuracy of the ﬁnite element model (FEM) for concrete closed girder cross-sections is signiﬁcantly inﬂuenced by thermal boundary conditions. The internal thermal boundary conditions can be simulated by inputting the convection heat transfer coe ﬃ cient and the temperatures inside the cavities or by establishing air elements in the FEM. In order to analyze the inﬂuence of di ﬀ erent simulation methods for the internal thermal boundary conditions on temperature distributions for concrete closed girder cross-sections, the temperature distributions on the cross-sections of a box girder, small box girders, and adjacent box girders were monitored, and the corresponding FEMs were implemented. By comparing the temperature data obtained from the ﬁeld test and FEMs, the numerical hourly temperature curves calculated by using the measured temperatures inside the cavities provide the closest agreement with the measured results; however, the measurements of the temperatures on site are cost- and time-prohibitive. When there is a lack of measured temperatures inside the cavities, the numerical hourly temperature curves calculated by establishing air elements in the FEM provide the closest agreement. The inﬂuences of di ﬀ erent simulation methods for the internal thermal boundary conditions on the highest hourly average e ﬀ ective temperatures and the trends of the vertical temperature gradients for concrete closed girder cross-sections were small. The FEM with air elements can be adopted to analyze the temperature distributions on concrete closed girder cross-sections under historically extreme temperature conditions. It can be predicted that the longitudinal thermal movement of concrete closed girders would be underestimated by considering variations in the one-year measured average e ﬀ ective temperature of the cross-sections or the Chinese-code-speciﬁed design e ﬀ ective temperature for the highway bridge structures, which are thus unconservative for engineering applications. The Chinese-code-speciﬁed design vertical temperature gradients are conservative for the bridge deck surface and unconservative for the bottom ﬂange.

The temperature effects on structures vary with time during the service periods of structures. In order to ensure the safety and durability of bridges, it is necessary to analyze the temperature distributions on concrete girder cross-sections under historically extreme temperature conditions. The Measured Temperature Method, Ambient Temperature Method, and Mean Temperature Method can only be used to simulate the temperature inside the cavity in a specific period based on the measured data. Therefore, it is difficult for them to predict the temperature distributions on concrete girder cross-sections under historically extreme temperature conditions. However, the Air Element Method is expected to be used to simulate the temperature distributions on concrete girder cross-sections under historically extreme temperature conditions.
In this paper, three bridges with different kinds of concrete closed girder cross-sections, including a box girder, small box girders, and adjacent box girders, were chosen as case studies. The temperature distributions on three kinds of concrete closed girder cross-sections were monitored. The corresponding FEMs of the concrete closed girder cross-sections were implemented by using the midas FEA software. Four methods were used to simulate the internal thermal boundary conditions for the concrete closed girder cross-sections. The influences of different simulation methods on the temperature distributions on concrete closed girder cross-sections were analyzed. The Air Element Method was adopted to analyze the temperature distributions on concrete closed girder cross-sections under historically extreme temperature conditions.

Box Girder
One concrete bridge with a three-cell single box girder in Shenzhen, China was chosen as the case study. The dimensions of the box girder cross-section are illustrated in Figure 1. Twenty-one temperature sensors were installed in the box girder during the construction process, as shown in Figure 1. These include five sensors in the top flange (T-1 to T-5), three sensors in the bottom flange (B-1 to B-3), eight sensors in the exterior webs (W-1 to W-4 in the left exterior web and W-7 to W-10 in the right exterior web), two sensors in the interior webs (W-5 in the left interior web and W-6 in the right interior web), and three sensors (C-1 to C-3) in three cells of the box girder. It can be observed from the field test data that the differences between the temperatures inside the cavities and the ambient air temperature were large in the summer because of high solar radiation, while the temperature differences were small in the winter. The daily measured temperature responses obtained at different times of year have been used by researchers to analyze the temperature distributions on cross-sections [13,22,24,25,39]. Due to space limitations, the data from 13 August 2017 in summer, which is a sunny day with the highest ambient air temperature during the monitoring period, as well as high solar radiation and low wind speed, were selected for analysis. The meteorological data, including ambient air temperature, hourly global solar radiation, hourly diffuse solar radiation, and wind speed, were measured using a movable automatic meteorological station at the bridge site with a measurement time interval of one hour. Method can only be used to simulate the temperature inside the cavity in a specific period based on the measured data. Therefore, it is difficult for them to predict the temperature distributions on concrete girder cross-sections under historically extreme temperature conditions. However, the Air Element Method is expected to be used to simulate the temperature distributions on concrete girder cross-sections under historically extreme temperature conditions. In this paper, three bridges with different kinds of concrete closed girder cross-sections, including a box girder, small box girders, and adjacent box girders, were chosen as case studies. The temperature distributions on three kinds of concrete closed girder cross-sections were monitored. The corresponding FEMs of the concrete closed girder cross-sections were implemented by using the midas FEA software. Four methods were used to simulate the internal thermal boundary conditions for the concrete closed girder cross-sections. The influences of different simulation methods on the temperature distributions on concrete closed girder cross-sections were analyzed. The Air Element Method was adopted to analyze the temperature distributions on concrete closed girder crosssections under historically extreme temperature conditions.

Box Girder
One concrete bridge with a three-cell single box girder in Shenzhen, China was chosen as the case study. The dimensions of the box girder cross-section are illustrated in Figure 1. Twenty-one temperature sensors were installed in the box girder during the construction process, as shown in Figure 1. These include five sensors in the top flange (T-1 to T-5), three sensors in the bottom flange (B-1 to B-3), eight sensors in the exterior webs (W-1 to W-4 in the left exterior web and W-7 to W-10 in the right exterior web), two sensors in the interior webs (W-5 in the left interior web and W-6 in the right interior web), and three sensors (C-1 to C-3) in three cells of the box girder. It can be observed from the field test data that the differences between the temperatures inside the cavities and the ambient air temperature were large in the summer because of high solar radiation, while the temperature differences were small in the winter. The daily measured temperature responses obtained at different times of year have been used by researchers to analyze the temperature distributions on cross-sections [13,22,24,25,39]. Due to space limitations, the data from 13 August 2017 in summer, which is a sunny day with the highest ambient air temperature during the monitoring period, as well as high solar radiation and low wind speed, were selected for analysis. The meteorological data, including ambient air temperature, hourly global solar radiation, hourly diffuse solar radiation, and wind speed, were measured using a movable automatic meteorological station at the bridge site with a measurement time interval of one hour. The midas FEA software was chosen to establish the FEM for the box girder cross-section. Twodimensional plane strain elements, which can be used for steady-state or transient analyses, were used, as shown in Figure 2. Each node has a single temperature degree of freedom. The measurement time interval in the FEM was set with the same as that for the field test. For the thermal parameters of the concrete, density, heat conductivity, and specific heat values of 2500 kg/m 3 , 1.5 W/(m·°C), and 900 J/(kg·C), respectively, were used [14,15,17,47]. Based on the measured ambient air temperature, wind speed, and solar radiation obtained from the movable automatic meteorological station at the The midas FEA software was chosen to establish the FEM for the box girder cross-section. Two-dimensional plane strain elements, which can be used for steady-state or transient analyses, Appl. Sci. 2020, 10, 1274 4 of 28 were used, as shown in Figure 2. Each node has a single temperature degree of freedom. The measurement time interval in the FEM was set with the same as that for the field test. For the thermal parameters of the concrete, density, heat conductivity, and specific heat values of 2500 kg/m 3 , 1.5 W/(m· • C), and 900 J/(kg· • C), respectively, were used [14,15,17,47]. Based on the measured ambient air temperature, wind speed, and solar radiation obtained from the movable automatic meteorological station at the bridge site, the parameters associated with solar radiation, convection, irradiation, and the temperature of the surrounding fluid medium were used as the external thermal boundary conditions for the box girder in the FEM. The influence of solar radiation on different parts of the box girder was considered by using the following rules and assumptions: (a) the external surface of the top flange is influenced by both the beam and diffuse solar radiation; (b) the external surfaces of the two exterior webs in the shadow due to the top flange are influenced by the diffuse solar radiation and ground reflection; (c) the external surfaces of the two exterior webs not in the shadow are influenced by the beam solar radiation, diffuse solar radiation, and ground reflection; and (d) the undersides of the top and bottom flanges are influenced by the ground reflection alone. The environmental and climatic conditions existing on previous days must be considered to accurately determine the initial temperature distribution [13,14,17,25,39]. A 120-h period with the same environmental conditions imposed cyclically was considered in the analysis.
determine the initial temperature distribution [13,14,17,25,39]. A 120-h period with the same environmental conditions imposed cyclically was considered in the analysis.
The four simulation methods for the internal thermal boundary conditions summarized in Section 1 were considered in the FEMs of the box girder. The internal thermal boundary condition was not influenced by solar radiation. Therefore, the internal thermal boundary condition with a convection heat transfer coefficient of 3.5 W/(m 2 ·C) was set in the FEMs by using the Measured Temperature Method, Ambient Temperature Method, or Mean Temperature Method. For the Measured Temperature Method, the measured hourly temperature curves inside each cell of the box girder (13 August 2017) obtained from the temperature sensors C-1 to C-3 were chosen, as shown by the hollow points in Figure 3. For the Ambient Temperature Method, the measured hourly ambient air temperature curve (13 August 2017) was used, as shown by the solid line in Figure 3. For the Mean Temperature Method, the mean temperature (30.7 C) over the previous two days (12 and 13 August 2017) was calculated, as shown by the dashed line in Figure 3. The mesh size of the FEMs using the Measured Temperature Method, Ambient Temperature Method, or Mean Temperature Method was set as 20 mm. The total numbers of nodes and elements were 37,017 and 35,659, respectively, as illustrated in Figure 2a. For the Air Element Method, the heat conduction between the air and girder cross-section can be simulated by establishing the air elements inside the cavities without the input of the temperature curves and the convection heat transfer coefficient inside the cavities. The air elements were also two-dimensional plane strain elements with the thermal parameters of air under 20 C and one standard atmospheric pressure (density, heat conductivity, and specific heat of 1.205 kg/m 3 , 0.0259 W/(m·C), and 1005 J/(kg·C), respectively) [48]. The mesh size of the air element was set as 20 mm, which is the same as those of the girder cross-section. The total numbers of nodes and elements for the FEM using the Air Element Method were 52,458 and 51,669, respectively, as illustrated in Figure 2b.  The four simulation methods for the internal thermal boundary conditions summarized in Section 1 were considered in the FEMs of the box girder. The internal thermal boundary condition was not influenced by solar radiation. Therefore, the internal thermal boundary condition with a convection heat transfer coefficient of 3.5 W/(m 2 · • C) was set in the FEMs by using the Measured Temperature Method, Ambient Temperature Method, or Mean Temperature Method. For the Measured Temperature Method, the measured hourly temperature curves inside each cell of the box girder (13 August 2017) obtained from the temperature sensors C-1 to C-3 were chosen, as shown by the hollow points in Figure 3. For the Ambient Temperature Method, the measured hourly ambient air temperature curve (13 August 2017) was used, as shown by the solid line in Figure 3. For the Mean Temperature Method, the mean temperature (30.7 • C) over the previous two days (12 and 13 August 2017) was calculated, as shown by the dashed line in Figure 3. The mesh size of the FEMs using the Measured Temperature Method, Ambient Temperature Method, or Mean Temperature Method was set as 20 mm. The total numbers of nodes and elements were 37,017 and 35,659, respectively, as illustrated in Figure 2a. For the Air Element Method, the heat conduction between the air and girder cross-section can be simulated by establishing the air elements inside the cavities without the input of the temperature curves and the convection heat transfer coefficient inside the cavities. The air elements were also

Small Box Girder
One prestressed concrete bridge with a cross-section consisting of seven small box girders in Handan, China was chosen as the case study. A Portland cement concrete overlay with a thickness of 12 cm and an asphalt concrete overlay with a thickness of 10 cm were used. The dimensions of the small box girder cross-sections are illustrated in Figure 4. Six temperature sensors were installed in the top flanges (1-T and 7-T), webs (1-W and 7-W), and bottom flanges (1-B and 7-B) of the small box girders #1 and #7, as shown in Figure 4. Due to space limitations, the data from 17 July 2015 in summer, which is a sunny day with the highest ambient air temperature during the monitoring period, as well as high solar radiation and low wind speed, were selected for analysis. The field test was carried out after bridge opening. Therefore, there was no temperature gauge placed inside the cells of the small box girders. The Measured Temperature Method was not considered in the analyses of the temperature distributions on concrete small box girder cross-sections. The meteorological data, including ambient air temperature, hourly global solar radiation, hourly diffuse solar radiation, and wind speed, were measured using a movable automatic meteorological station at the bridge site with a measurement time interval of one hour. The midas FEA software was chosen to establish a FEM of the small box girder cross-sections by using two-dimensional plane strain elements, as shown in Figure 5. The three simulation methods for the internal thermal boundary conditions summarized in Section 1, except the Measured Temperature Method, were considered in the FEMs of small box girders. For the Ambient Temperature Method, the measured hourly ambient air temperature curve (17 July 2015) was used, as shown by the solid line in Figure 6. For the Mean Temperature Method, the mean temperature (27.1 C) over the previous two days (16 and 17 July 2015) was calculated, as shown by the dashed line in Figure 6. The mesh size of the FEMs using the Ambient Temperature Method or Mean Temperature Method was set as 20 mm. The total numbers of nodes and elements were 34,020 and 32,462, respectively, as illustrated in Figure 5a. For the Air Element Method, the total numbers of nodes and elements for the FEM were 45,632 and 44,786, respectively, as illustrated in Figure 5b.

Small Box Girder
One prestressed concrete bridge with a cross-section consisting of seven small box girders in Handan, China was chosen as the case study. A Portland cement concrete overlay with a thickness of 12 cm and an asphalt concrete overlay with a thickness of 10 cm were used. The dimensions of the small box girder cross-sections are illustrated in Figure 4. Six temperature sensors were installed in the top flanges (1-T and 7-T), webs (1-W and 7-W), and bottom flanges (1-B and 7-B) of the small box girders #1 and #7, as shown in Figure 4. Due to space limitations, the data from 17 July 2015 in summer, which is a sunny day with the highest ambient air temperature during the monitoring period, as well as high solar radiation and low wind speed, were selected for analysis. The field test was carried out after bridge opening. Therefore, there was no temperature gauge placed inside the cells of the small box girders. The Measured Temperature Method was not considered in the analyses of the temperature distributions on concrete small box girder cross-sections. The meteorological data, including ambient air temperature, hourly global solar radiation, hourly diffuse solar radiation, and wind speed, were measured using a movable automatic meteorological station at the bridge site with a measurement time interval of one hour.

Small Box Girder
One prestressed concrete bridge with a cross-section consisting of seven small box girders in Handan, China was chosen as the case study. A Portland cement concrete overlay with a thickness of 12 cm and an asphalt concrete overlay with a thickness of 10 cm were used. The dimensions of the small box girder cross-sections are illustrated in Figure 4. Six temperature sensors were installed in the top flanges (1-T and 7-T), webs (1-W and 7-W), and bottom flanges (1-B and 7-B) of the small box girders #1 and #7, as shown in Figure 4. Due to space limitations, the data from 17 July 2015 in summer, which is a sunny day with the highest ambient air temperature during the monitoring period, as well as high solar radiation and low wind speed, were selected for analysis. The field test was carried out after bridge opening. Therefore, there was no temperature gauge placed inside the cells of the small box girders. The Measured Temperature Method was not considered in the analyses of the temperature distributions on concrete small box girder cross-sections. The meteorological data, including ambient air temperature, hourly global solar radiation, hourly diffuse solar radiation, and wind speed, were measured using a movable automatic meteorological station at the bridge site with a measurement time interval of one hour. The midas FEA software was chosen to establish a FEM of the small box girder cross-sections by using two-dimensional plane strain elements, as shown in Figure 5. The three simulation methods for the internal thermal boundary conditions summarized in Section 1, except the Measured Temperature Method, were considered in the FEMs of small box girders. For the Ambient Temperature Method, the measured hourly ambient air temperature curve (17 July 2015) was used, as shown by the solid line in Figure 6. For the Mean Temperature Method, the mean temperature (27.1 C) over the previous two days (16 and 17 July 2015) was calculated, as shown by the dashed line in Figure 6. The mesh size of the FEMs using the Ambient Temperature Method or Mean Temperature Method was set as 20 mm. The total numbers of nodes and elements were 34,020 and 32,462, respectively, as illustrated in Figure 5a. For the Air Element Method, the total numbers of nodes and elements for the FEM were 45,632 and 44,786, respectively, as illustrated in Figure 5b. The midas FEA software was chosen to establish a FEM of the small box girder cross-sections by using two-dimensional plane strain elements, as shown in Figure 5. The three simulation methods for the internal thermal boundary conditions summarized in Section 1, except the Measured Temperature Method, were considered in the FEMs of small box girders. For the Ambient Temperature Method, the measured hourly ambient air temperature curve (17 July 2015) was used, as shown by the solid line in Figure 6. For the Mean Temperature Method, the mean temperature (27.1 • C) over the previous two days (16 and 17 July 2015) was calculated, as shown by the dashed line in Figure 6. The mesh size of the FEMs using the Ambient Temperature Method or Mean Temperature Method was set as 20 mm. The total numbers of nodes and elements were 34,020 and 32,462, respectively, as illustrated in Figure 5a. For the Air Element Method, the total numbers of nodes and elements for the FEM were 45,632 and 44,786, respectively, as illustrated in Figure 5b. Other detailed information was the same as the FEM of the box girder, as reported in Section 2.1.

Adjacent Box Girder
One concrete girder with a cross-section consisting of three adjacent box girders in Fuzhou, China was chosen as the case study. The thickness of the Portland cement concrete overlay was 10 cm. The dimensions of the adjacent box girder cross-sections are illustrated in Figure 7. Eight temperature sensors were installed in the top flanges (1-T and 3-T), webs (1-W and 3-W), bottom flanges (1-B and 3-B), and cells (1-C and 3-C) of the adjacent box girders #1 and #3 during the construction process, as shown in Figure 7. Due to space limitations, the data from 24 July 2017 in summer, which is a sunny day with the highest ambient air temperature during the monitoring period, as well as high solar radiation and low wind speed, were selected for analysis. The meteorological data, including ambient air temperature, hourly global solar radiation, hourly diffuse solar radiation, and wind speed, were measured using a movable automatic meteorological station at the site with a measurement time interval of one hour.

Adjacent Box Girder
One concrete girder with a cross-section consisting of three adjacent box girders in Fuzhou, China was chosen as the case study. The thickness of the Portland cement concrete overlay was 10 cm. The dimensions of the adjacent box girder cross-sections are illustrated in Figure 7. Eight temperature sensors were installed in the top flanges (1-T and 3-T), webs (1-W and 3-W), bottom flanges (1-B and 3-B), and cells (1-C and 3-C) of the adjacent box girders #1 and #3 during the construction process, as shown in Figure 7. Due to space limitations, the data from 24 July 2017 in summer, which is a sunny day with the highest ambient air temperature during the monitoring period, as well as high solar radiation and low wind speed, were selected for analysis. The meteorological data, including ambient air temperature, hourly global solar radiation, hourly diffuse solar radiation, and wind speed, were measured using a movable automatic meteorological station at the site with a measurement time interval of one hour.

Adjacent Box Girder
One concrete girder with a cross-section consisting of three adjacent box girders in Fuzhou, China was chosen as the case study. The thickness of the Portland cement concrete overlay was 10 cm. The dimensions of the adjacent box girder cross-sections are illustrated in Figure 7. Eight temperature sensors were installed in the top flanges (1-T and 3-T), webs (1-W and 3-W), bottom flanges (1-B and 3-B), and cells (1-C and 3-C) of the adjacent box girders #1 and #3 during the construction process, as shown in Figure 7. Due to space limitations, the data from 24 July 2017 in summer, which is a sunny day with the highest ambient air temperature during the monitoring period, as well as high solar radiation and low wind speed, were selected for analysis. The meteorological data, including ambient air temperature, hourly global solar radiation, hourly diffuse solar radiation, The midas FEA software was chosen to establish a FEM of the adjacent box girder cross-sections by using two-dimensional plane strain elements, as shown in Figure 8. The four simulation methods for the internal thermal boundary conditions summarized in Section 1 were considered in the FEM of the adjacent box girders. For the Measured Temperature Method, the measured hourly temperature curves inside the cell of the adjacent box girder (24 July 2017) obtained from temperature sensors 1-C and 3-C were chosen, as shown by the hollow points in Figure 9. For the Ambient Temperature Method, the measured hourly ambient air temperature curve (24 July 2017) was used, as shown by the solid line in Figure 9. For the Mean Temperature Method, the mean temperature (35.7 C) over the previous two days (23 and 24 July 2017) was calculated, as shown by the dashed line in Figure 9. The mesh size of the FEMs using the Measured Temperature Method, Ambient Temperature Method, or Mean Temperature Method was set as 20 mm. The total numbers of nodes and elements were 4,459 and 4,211, respectively, as illustrated in Figure 8a. For the Air Element Method, the total numbers of nodes and elements for the FEM were 4,954 and 4,868, respectively, as illustrated in Figure 8b. Other detailed information was the same as the FEM of the box girder reported in Section 2.1.   The midas FEA software was chosen to establish a FEM of the adjacent box girder cross-sections by using two-dimensional plane strain elements, as shown in Figure 8. The four simulation methods for the internal thermal boundary conditions summarized in Section 1 were considered in the FEM of the adjacent box girders. For the Measured Temperature Method, the measured hourly temperature curves inside the cell of the adjacent box girder (24 July 2017) obtained from temperature sensors 1-C and 3-C were chosen, as shown by the hollow points in Figure 9. For the Ambient Temperature Method, the measured hourly ambient air temperature curve (24 July 2017) was used, as shown by the solid line in Figure 9. For the Mean Temperature Method, the mean temperature (35.7 • C) over the previous two days (23 and 24 July 2017) was calculated, as shown by the dashed line in Figure 9. The mesh size of the FEMs using the Measured Temperature Method, Ambient Temperature Method, or Mean Temperature Method was set as 20 mm. The total numbers of nodes and elements were 4,459 and 4,211, respectively, as illustrated in Figure 8a. For the Air Element Method, the total numbers of nodes and elements for the FEM were 4954 and 4868, respectively, as illustrated in Figure 8b. Other detailed information was the same as the FEM of the box girder reported in Section 2.1. The midas FEA software was chosen to establish a FEM of the adjacent box girder cross-sections by using two-dimensional plane strain elements, as shown in Figure 8. The four simulation methods for the internal thermal boundary conditions summarized in Section 1 were considered in the FEM of the adjacent box girders. For the Measured Temperature Method, the measured hourly temperature curves inside the cell of the adjacent box girder (24 July 2017) obtained from temperature sensors 1-C and 3-C were chosen, as shown by the hollow points in Figure 9. For the Ambient Temperature Method, the measured hourly ambient air temperature curve (24 July 2017) was used, as shown by the solid line in Figure 9. For the Mean Temperature Method, the mean temperature (35.7 C) over the previous two days (23 and 24 July 2017) was calculated, as shown by the dashed line in Figure 9. The mesh size of the FEMs using the Measured Temperature Method, Ambient Temperature Method, or Mean Temperature Method was set as 20 mm. The total numbers of nodes and elements were 4,459 and 4,211, respectively, as illustrated in Figure 8a. For the Air Element Method, the total numbers of nodes and elements for the FEM were 4,954 and 4,868, respectively, as illustrated in Figure 8b. Other detailed information was the same as the FEM of the box girder reported in Section 2.1.

Influence on the Hourly Temperature Curves
Due to space limitations, the measured hourly temperature curves obtained from the temperature sensors installed in the right part of the box girder cross-section (T-5, T-4, W-7, W-9, and B-3 as illustrated in Figure 1) were selected for comparison with the numerical hourly temperature curves calculated by the FEMs considering different simulation methods for the internal thermal boundary conditions, as shown in Figure 10. The solid points denote the measured data, and the solid lines with hollow points denote the calculated curves. The root-mean-square error (RMSE) and mean absolute percentage error (MAPE) were used to check the agreement between the measured and calculated temperature curves [49]. The RMSE and MAPE of the numerical hourly temperature curves of the box girder cross-sections obtained from the FEMs with different simulation methods for the internal thermal boundary conditions are illustrated in Figure 11.

Influence on the Hourly Temperature Curves
Due to space limitations, the measured hourly temperature curves obtained from the temperature sensors installed in the right part of the box girder cross-section (T-5, T-4, W-7, W-9, and B-3 as illustrated in Figure 1) were selected for comparison with the numerical hourly temperature curves calculated by the FEMs considering different simulation methods for the internal thermal boundary conditions, as shown in Figure 10. The solid points denote the measured data, and the solid lines with hollow points denote the calculated curves. The root-mean-square error (RMSE) and mean absolute percentage error (MAPE) were used to check the agreement between the measured and calculated temperature curves [49]. The RMSE and MAPE of the numerical hourly temperature curves of the box girder cross-sections obtained from the FEMs with different simulation methods for the internal thermal boundary conditions are illustrated in Figure 11.

Influence on the Hourly Temperature Curves
Due to space limitations, the measured hourly temperature curves obtained from the temperature sensors installed in the right part of the box girder cross-section (T-5, T-4, W-7, W-9, and B-3 as illustrated in Figure 1) were selected for comparison with the numerical hourly temperature curves calculated by the FEMs considering different simulation methods for the internal thermal boundary conditions, as shown in Figure 10. The solid points denote the measured data, and the solid lines with hollow points denote the calculated curves. The root-mean-square error (RMSE) and mean absolute percentage error (MAPE) were used to check the agreement between the measured and calculated temperature curves [49]. The RMSE and MAPE of the numerical hourly temperature curves of the box girder cross-sections obtained from the FEMs with different simulation methods for the internal thermal boundary conditions are illustrated in Figure 11. In Figures 10a and 11, it can be observed that the influence of the different simulation methods for the internal thermal boundary conditions on the numerical hourly temperature curves of T-5 is negligible. This is because T-5 was in the flange overhang and far from the cavities. Therefore, the temperature distributions on the flange overhangs were not influenced by the internal thermal boundary conditions for the box girder. The RMSE and MAPE in Figure 11 indicate that the differences between the measured and numerical hourly temperature curves were very small (the RMSEs were all 0.5 C, and the MAPEs were all 1.2%).
In Figures 10b-d, and 11, it can be observed that the trends and the times at which the maximum temperatures occurred are similar for the measured and numerical hourly temperature curves of T-4 and W-7 (which was installed in the web near the top flange). For B-3 and W-9 (which was installed in the web near the bottom flange), it can be observed that the trends for the numerical hourly temperature curves calculated by the Measured Temperature Method, Air Element Method, and Mean Temperature Method are similar to the measured curves. When the Ambient Temperature Method was used, the trends for the numerical curves of B-3 and W-9 are similar to the measured hourly ambient air temperature curve. This is because the temperature change in the hourly ambient air temperature curve was much larger than those in the measured curves inside the cavities. Compared with the measured hourly temperature curves, the numerical curves calculated by the Measured Temperature Method provided the closest agreement (the maximum RMSE was 0.6 C, and the maximum MAPE was 1.3%). When there is a lack of measured temperatures inside the cavities, the numerical hourly temperature curves calculated by the Air Element Method (the In Figures 10a and 11, it can be observed that the influence of the different simulation methods for the internal thermal boundary conditions on the numerical hourly temperature curves of T-5 is negligible. This is because T-5 was in the flange overhang and far from the cavities. Therefore, the temperature distributions on the flange overhangs were not influenced by the internal thermal boundary conditions for the box girder. The RMSE and MAPE in Figure 11 indicate that the differences between the measured and numerical hourly temperature curves were very small (the RMSEs were all 0.5 C, and the MAPEs were all 1.2%).
In Figures 10b-d, and 11, it can be observed that the trends and the times at which the maximum temperatures occurred are similar for the measured and numerical hourly temperature curves of T-4 and W-7 (which was installed in the web near the top flange). For B-3 and W-9 (which was installed in the web near the bottom flange), it can be observed that the trends for the numerical hourly temperature curves calculated by the Measured Temperature Method, Air Element Method, and Mean Temperature Method are similar to the measured curves. When the Ambient Temperature Method was used, the trends for the numerical curves of B-3 and W-9 are similar to the measured hourly ambient air temperature curve. This is because the temperature change in the hourly ambient air temperature curve was much larger than those in the measured curves inside the cavities. Compared with the measured hourly temperature curves, the numerical curves calculated by the Measured Temperature Method provided the closest agreement (the maximum RMSE was 0.6 C, and the maximum MAPE was 1.3%). When there is a lack of measured temperatures inside the cavities, the numerical hourly temperature curves calculated by the Air Element Method (the In Figures 10a and 11, it can be observed that the influence of the different simulation methods for the internal thermal boundary conditions on the numerical hourly temperature curves of T-5 is negligible. This is because T-5 was in the flange overhang and far from the cavities. Therefore, the temperature distributions on the flange overhangs were not influenced by the internal thermal boundary conditions for the box girder. The RMSE and MAPE in Figure 11 indicate that the differences between the measured and numerical hourly temperature curves were very small (the RMSEs were all 0.5 • C, and the MAPEs were all 1.2%).
In Figure 10b-d, and Figure 11, it can be observed that the trends and the times at which the maximum temperatures occurred are similar for the measured and numerical hourly temperature curves of T-4 and W-7 (which was installed in the web near the top flange). For B-3 and W-9 (which was installed in the web near the bottom flange), it can be observed that the trends for the numerical hourly temperature curves calculated by the Measured Temperature Method, Air Element Method, and Mean Temperature Method are similar to the measured curves. When the Ambient Temperature Method was used, the trends for the numerical curves of B-3 and W-9 are similar to the measured hourly ambient air temperature curve. This is because the temperature change in the hourly ambient air temperature curve was much larger than those in the measured curves inside the cavities. Compared with the measured hourly temperature curves, the numerical curves calculated by the Measured Temperature Method provided the closest agreement (the maximum RMSE was 0.6 • C, and the maximum MAPE was 1.3%). When there is a lack of measured temperatures inside the cavities, the numerical hourly temperature curves calculated by the Air Element Method (the maximum RMSE was 1.1 • C, and the maximum MAPE was 2.8%) provided a closer agreement with the measured curves than the curves calculated by the Ambient Temperature Method (the maximum RMSE was 2.0 • C, and the maximum MAPE was 5.9%) and the Mean Temperature Method (the maximum RMSE was 1.4 • C, and the maximum MAPE was 4.1%).
The measured hourly temperature curves of all temperature sensors installed in the box girder cross-section were compared with the numerical hourly temperature curves calculated by the FEM using the Air Element Method as the simulation method for the internal thermal boundary condition, as shown in Figure S1. It can be observed that the trends and the times at which the maximum temperatures occurred are similar for the measured hourly temperature curves and numerical curves calculated by the Ambient Temperature Method for all temperature sensors installed in the box girder cross-section (the maximum RMSE was 1.3 • C, and the maximum MAPE was 3.8%).

Influence on the Temperature Contour Plots
The temperature contour plots (15:00, 13 August 2017) obtained from the FEMs considering different simulation methods for the internal thermal boundary conditions are compared in Figure 12. It can be observed that the highest temperature (39.

Influence on the Temperature Contour Plots
The temperature contour plots (15:00, 13 August 2017) obtained from the FEMs considering different simulation methods for the internal thermal boundary conditions are compared in Figure  12. It can be observed that the highest temperature (39.

Influence on the Average Effective Temperatures and Vertical Temperature Gradients
The hourly average effective temperature (TAVG) of the box girder cross-sections can be estimated using the following rules. The temperature of each element in the FEM is multiplied by its respective element area, and the results for all elements in the cross-section are added together. Then, TAVG can be obtained by dividing the sum by the total area of the cross-section [13,22]. The highest TAVG values of the box girder cross-sections for 13 August 2017, obtained from the field test and the FEMs with different simulation methods for the internal thermal boundary conditions, are listed in Table 1. It can be observed that the influence of different simulation methods for the internal thermal boundary conditions on the highest TAVG was small, with the maximum difference of 0.7 °C. The numerical maximum vertical temperature gradients of the box girder cross-section on 13 August 2017, which were obtained from the FEMs with different simulation methods for the internal thermal boundary conditions, are compared in Figure 13. It can be observed that the influence of different simulation methods for the internal thermal boundary conditions on the trends of the vertical temperature gradients for the box girder cross-section was small, with a maximum difference of 0.4 °C. The maximum vertical temperature differences on the top flanges and bottom flanges calculated by the Ambient Temperature Method and the Mean Temperature Method were slightly larger than those calculated by the Measured Temperature Method and the Air Element Method. The maximum vertical temperature differences on the webs calculated by the Ambient Temperature Method and the Mean Temperature Method were slightly smaller than those calculated by the Measured Temperature Method and the Air Element Method. The numerical maximum vertical temperature gradient of the box girder cross-section calculated by the Air Element Method was close to that calculated by the Measured Temperature Method.

Influence on the Average Effective Temperatures and Vertical Temperature Gradients
The hourly average effective temperature (T AVG ) of the box girder cross-sections can be estimated using the following rules. The temperature of each element in the FEM is multiplied by its respective element area, and the results for all elements in the cross-section are added together. Then, T AVG can be obtained by dividing the sum by the total area of the cross-section [13,22]. The highest T AVG values of the box girder cross-sections for 13 August 2017, obtained from the field test and the FEMs with different simulation methods for the internal thermal boundary conditions, are listed in Table 1. It can be observed that the influence of different simulation methods for the internal thermal boundary conditions on the highest T AVG was small, with the maximum difference of 0.7 • C. The numerical maximum vertical temperature gradients of the box girder cross-section on 13 August 2017, which were obtained from the FEMs with different simulation methods for the internal thermal boundary conditions, are compared in Figure 13. It can be observed that the influence of different simulation methods for the internal thermal boundary conditions on the trends of the vertical temperature gradients for the box girder cross-section was small, with a maximum difference of 0.

Influence on the Hourly Temperature Curves
Due to space limitations, the measured hourly temperature curves obtained from the temperature sensors installed on the small box girder #7 (7-T, 7-W, and 7-B, as illustrated in Figure 4) were selected for comparison with the numerical hourly temperature curves calculated by the FEMs considering different simulation methods for the internal thermal boundary conditions, as shown in Figure 14. The solid points denote the measured data, and the solid lines with hollow points denote the calculated curves. The RMSE and MAPE of the numerical hourly temperature curves of the small box girder cross-sections obtained from the FEMs with different simulation methods for the internal thermal boundary conditions are illustrated in Figure 15.

Influence on the Hourly Temperature Curves
Due to space limitations, the measured hourly temperature curves obtained from the temperature sensors installed on the small box girder #7 (7-T, 7-W, and 7-B, as illustrated in Figure 4) were selected for comparison with the numerical hourly temperature curves calculated by the FEMs considering different simulation methods for the internal thermal boundary conditions, as shown in Figure 14. The solid points denote the measured data, and the solid lines with hollow points denote the calculated curves. The RMSE and MAPE of the numerical hourly temperature curves of the small box girder cross-sections obtained from the FEMs with different simulation methods for the internal thermal boundary conditions are illustrated in Figure 15.

Influence on the Hourly Temperature Curves
Due to space limitations, the measured hourly temperature curves obtained from the temperature sensors installed on the small box girder #7 (7-T, 7-W, and 7-B, as illustrated in Figure 4) were selected for comparison with the numerical hourly temperature curves calculated by the FEMs considering different simulation methods for the internal thermal boundary conditions, as shown in Figure 14. The solid points denote the measured data, and the solid lines with hollow points denote the calculated curves. The RMSE and MAPE of the numerical hourly temperature curves of the small box girder cross-sections obtained from the FEMs with different simulation methods for the internal thermal boundary conditions are illustrated in Figure 15. In Figures 14 and 15, it can be observed that the trends and the times at which the maximum temperatures occurred are similar for the measured and numerical hourly temperature curves of 7-T. For 7-W and 7-B, it can be observed that the trends of the numerical hourly temperature curves calculated by the Air Element Method and Mean Temperature Method are similar to the measured curves. When the Ambient Temperature Method was used, the trends of the numerical curves of 7-W and 7-B are similar to the measured hourly ambient air temperature curve, which is the same as the results for the box girder cross-section. Compared with the measured hourly temperature curves, the numerical hourly temperature curves calculated by the Air Element Method (the maximum RMSE was 0.7 °C, and the maximum MAPE was 1.8%) provided a closer agreement than the curves calculated by the Ambient Temperature Method (the maximum RMSE was 1.5 °C, and the maximum MAPE was 4.9%) and the Mean Temperature Method (the maximum RMSE was 1.5 °C, and the maximum MAPE was 4.5%).

Influence on the Temperature Contour Plots
The temperature contour plots (15:00, 17 July 2015) obtained from the FEMs considering different simulation methods for the internal thermal boundary conditions are compared in Figure  16. It can be observed that the highest temperature (49.  In Figures 14 and 15, it can be observed that the trends and the times at which the maximum temperatures occurred are similar for the measured and numerical hourly temperature curves of 7-T. For 7-W and 7-B, it can be observed that the trends of the numerical hourly temperature curves calculated by the Air Element Method and Mean Temperature Method are similar to the measured curves. When the Ambient Temperature Method was used, the trends of the numerical curves of 7-W and 7-B are similar to the measured hourly ambient air temperature curve, which is the same as the results for the box girder cross-section. Compared with the measured hourly temperature curves, the numerical hourly temperature curves calculated by the Air Element Method (the maximum RMSE was 0.7 °C, and the maximum MAPE was 1.8%) provided a closer agreement than the curves calculated by the Ambient Temperature Method (the maximum RMSE was 1.5 °C, and the maximum MAPE was 4.9%) and the Mean Temperature Method (the maximum RMSE was 1.5 °C, and the maximum MAPE was 4.5%).

Influence on the Temperature Contour Plots
The temperature contour plots (15:00, 17 July 2015) obtained from the FEMs considering different simulation methods for the internal thermal boundary conditions are compared in Figure  16. It can be observed that the highest temperature (49.0 °C) was located on the external surfaces of the overlays in all FEMs. The influence of different simulation methods for the internal thermal boundary conditions on the highest temperature of the small box girder cross-sections is negligible because it is far from the cavities. The temperatures of the top flanges (30.8 °C) and bottom flanges In Figures 14 and 15, it can be observed that the trends and the times at which the maximum temperatures occurred are similar for the measured and numerical hourly temperature curves of 7-T. For 7-W and 7-B, it can be observed that the trends of the numerical hourly temperature curves calculated by the Air Element Method and Mean Temperature Method are similar to the measured curves. When the Ambient Temperature Method was used, the trends of the numerical curves of 7-W and 7-B are similar to the measured hourly ambient air temperature curve, which is the same as the results for the box girder cross-section. Compared with the measured hourly temperature curves, the numerical hourly temperature curves calculated by the Air Element Method (the maximum RMSE was 0.7 • C, and the maximum MAPE was 1.8%) provided a closer agreement than the curves calculated by the Ambient Temperature Method (the maximum RMSE was 1.5 • C, and the maximum MAPE was 4.9%) and the Mean Temperature Method (the maximum RMSE was 1.5 • C, and the maximum MAPE was 4.5%). In Figure S2, it can be observed that the trends and the times at which the maximum temperatures occurred are similar for the measured hourly temperature curves and numerical curves calculated by the Ambient Temperature Method (the maximum RMSE was 0.8 • C, and the maximum MAPE was 2.6%).
The measured hourly temperature curves of all temperature sensors installed in the small box girder cross-sections were compared with the numerical hourly temperature curves calculated by the FEM using the Air Element Method as the simulation method for the internal thermal boundary condition, as shown in Figure S2. It can be observed that the trends and the times at which the maximum temperatures occurred are similar for the measured hourly temperature curves and numerical curves calculated by the Ambient Temperature Method for all temperature sensors installed in the small box girder cross-sections (the maximum RMSE was 0.8 • C, and the maximum MAPE was 2.6%).

Influence on the Temperature Contour Plots
The temperature contour plots (15:00, 17 July 2015) obtained from the FEMs considering different simulation methods for the internal thermal boundary conditions are compared in Figure 16. It can be observed that the highest temperature (49.

Influence on the Average Effective Temperatures and Vertical Temperature Gradients
The highest TAVG values of the small box girder cross-sections for 17 July, 2015 obtained from the field test and the FEMs with different simulation methods for the internal thermal boundary conditions, are listed in Table 2. It can be observed that the influence of different simulation methods for the internal thermal boundary conditions on the highest TAVG was small, with the maximum difference of 1.1 °C.

Influence on the Average Effective Temperatures and Vertical Temperature Gradients
The highest T AVG values of the small box girder cross-sections for 17 July, 2015 obtained from the field test and the FEMs with different simulation methods for the internal thermal boundary conditions, are listed in Table 2. It can be observed that the influence of different simulation methods for the internal thermal boundary conditions on the highest T AVG was small, with the maximum difference of 1.1 • C. The numerical maximum vertical temperature gradients of the small box girder cross-sections on 17 July 2015 that were obtained from the FEMs with different simulation methods for the internal thermal boundary conditions are compared in Figure 17. It can be observed that the maximum vertical temperature differences on the top flanges and bottom flanges calculated by the Air Element Method

Influence on the Hourly Temperature Curves
Due to space limitations, the measured hourly temperature curves obtained from the temperature sensors installed on the adjacent box girder #3 (3-T, 3-W, and 3-B as illustrated in Figure  7) were selected for comparison with the numerical hourly temperature curves calculated by the FEMs considering different simulation methods for the internal thermal boundary conditions, as shown in Figure 18. The solid points denote the measured data, and the solid lines with hollow points denote the calculated curves. The RMSE and MAPE of the numerical hourly temperature curves of the adjacent box girder cross-sections obtained from the FEMs with different simulation methods for the internal thermal boundary conditions are illustrated in Figure 19.

Influence on the Hourly Temperature Curves
Due to space limitations, the measured hourly temperature curves obtained from the temperature sensors installed on the adjacent box girder #3 (3-T, 3-W, and 3-B as illustrated in Figure 7) were selected for comparison with the numerical hourly temperature curves calculated by the FEMs considering different simulation methods for the internal thermal boundary conditions, as shown in Figure 18. The solid points denote the measured data, and the solid lines with hollow points denote the calculated curves. The RMSE and MAPE of the numerical hourly temperature curves of the adjacent box girder cross-sections obtained from the FEMs with different simulation methods for the internal thermal boundary conditions are illustrated in Figure 19. 7) were selected for comparison with the numerical hourly temperature curves calculated by the FEMs considering different simulation methods for the internal thermal boundary conditions, as shown in Figure 18. The solid points denote the measured data, and the solid lines with hollow points denote the calculated curves. The RMSE and MAPE of the numerical hourly temperature curves of the adjacent box girder cross-sections obtained from the FEMs with different simulation methods for the internal thermal boundary conditions are illustrated in Figure 19. In Figures 18 and 19, it can be observed that the trends and the times at which the maximum temperatures occurred are similar for the measured and numerical hourly temperature curves of 3-T, 3-W, and 3-B. The influence of the Ambient Temperature Method on the trends of the numerical hourly temperature curves of the bottom flanges on the adjacent box girders is much smaller than that on the box girder or small box girders. This is because the area of the cavities in the adjacent box girders illustrated in Figure 7 is quite small. Compared with the measured hourly temperature curves, the numerical curves calculated by the Measured Temperature Method (the maximum RMSE was 1.1 °C, and the maximum MAPE was 2.6%) and the Air Element Method (the maximum RMSE In Figures 18 and 19, it can be observed that the trends and the times at which the maximum temperatures occurred are similar for the measured and numerical hourly temperature curves of 3-T, 3-W, and 3-B. The influence of the Ambient Temperature Method on the trends of the numerical hourly temperature curves of the bottom flanges on the adjacent box girders is much smaller than that on the box girder or small box girders. This is because the area of the cavities in the adjacent box girders illustrated in Figure 7 is quite small. Compared with the measured hourly temperature curves, the numerical curves calculated by the Measured Temperature Method (the maximum RMSE In Figures 18 and 19, it can be observed that the trends and the times at which the maximum temperatures occurred are similar for the measured and numerical hourly temperature curves of 3-T, 3-W, and 3-B. The influence of the Ambient Temperature Method on the trends of the numerical hourly temperature curves of the bottom flanges on the adjacent box girders is much smaller than that on the box girder or small box girders. This is because the area of the cavities in the adjacent box girders illustrated in Figure 7 is quite small. Compared with the measured hourly temperature curves, the numerical curves calculated by the Measured Temperature Method (the maximum RMSE was 1.1 • C, and the maximum MAPE was 2.6%) and the Air Element Method (the maximum RMSE was 1.2 • C, and the maximum MAPE was 3.0%) provided the closest agreement. When there is a lack of measured temperatures inside the cavities, the numerical hourly temperature curves calculated by the Air Element Method provided a closer agreement with the measured curves than the curves calculated by the Ambient Temperature Method (the maximum RMSE was 2.0 • C, and the maximum MAPE was 4.8%) and the Mean Temperature Method (the maximum RMSE was 1.7 • C, and the maximum MAPE was 4.2%).
The measured hourly temperature curves of all temperature sensors installed in the adjacent box girder cross-sections were compared with the numerical hourly temperature curves calculated by the FEM using the Air Element Method as the simulation method for the internal thermal boundary condition, as shown in Figure S3. It can be observed that the trends and the times at which the maximum temperatures occurred are similar for the measured hourly temperature curves and numerical curves calculated by the Ambient Temperature Method for all temperature sensors installed in the adjacent box girder cross-sections (the maximum RMSE was 1.2 • C, and the maximum MAPE was 3.0%).

Influence on the Temperature Contour Plots
The temperature contour plots (15:00, 24 July 2017) obtained from the FEMs considering different simulation methods for the internal thermal boundary conditions are compared in Figure 20. It can be observed that the influence of different simulation methods for the internal thermal boundary conditions on the adjacent girder #1 is negligible because girder #1 was covered by a shadow. The highest temperature (57.8 • C) was located on the external surface of the top flange in girder #3, which was exposed to sunlight in all the FEMs. The influence of different simulation methods for the internal thermal boundary conditions on the highest temperature of the adjacent box girder cross-section is negligible because it is far from the cavities. The temperature distributions on the adjacent box girder cross-sections calculated by the Measured Temperature Method and Air Element Method were similar and smaller than those calculated by the Ambient Temperature Method and larger than those calculated by the Mean Temperature Method. methods for the internal thermal boundary conditions on the highest temperature of the adjacent box girder cross-section is negligible because it is far from the cavities. The temperature distributions on the adjacent box girder cross-sections calculated by the Measured Temperature Method and Air Element Method were similar and smaller than those calculated by the Ambient Temperature Method and larger than those calculated by the Mean Temperature Method.

Influence on the Average Effective Temperatures and Vertical Temperature Gradients
The highest TAVG values of the adjacent box girder cross-sections for 24 July 2017 obtained from the field test and the FEMs with different simulation methods for the internal thermal boundary conditions are listed in Table 3. It can be observed that the influence of different simulation methods for the internal thermal boundary conditions on the highest TAVG was small, with a maximum difference of 0.8 °C.

Influence on the Average Effective Temperatures and Vertical Temperature Gradients
The highest T AVG values of the adjacent box girder cross-sections for 24 July 2017 obtained from the field test and the FEMs with different simulation methods for the internal thermal boundary conditions are listed in Table 3. It can be observed that the influence of different simulation methods for the internal thermal boundary conditions on the highest T AVG was small, with a maximum difference of 0.8 • C. The numerical maximum vertical temperature gradients of the adjacent box girder cross-sections on 24 July 2017, obtained from the FEMs with different simulation methods for the internal thermal boundary conditions are compared in Figure 21. It can be observed that the influence of different simulation methods for the internal thermal boundary conditions on the trends of the vertical temperature gradients for the adjacent box girder cross-sections was small, with a maximum difference of 0.5 • C. The maximum vertical temperature differences on the top flanges and bottom flanges calculated by the Ambient Temperature Method were slightly smaller than those calculated by the other three methods. The maximum vertical temperature differences on the webs calculated by different simulation methods for the internal thermal boundary conditions were almost the same.  The numerical maximum vertical temperature gradients of the adjacent box girder cross-sections on 24 July 2017, obtained from the FEMs with different simulation methods for the internal thermal boundary conditions are compared in Figure 21. It can be observed that the influence of different simulation methods for the internal thermal boundary conditions on the trends of the vertical temperature gradients for the adjacent box girder cross-sections was small, with a maximum difference of 0.5 °C. The maximum vertical temperature differences on the top flanges and bottom flanges calculated by the Ambient Temperature Method were slightly smaller than those calculated by the other three methods. The maximum vertical temperature differences on the webs calculated by different simulation methods for the internal thermal boundary conditions were almost the same.

Temperature Distributions on Concrete Closed Girder Cross-Sections under Historically Extreme Temperature Conditions
The temperature curves inside the cavities under historically extreme temperature conditions are difficult to measure. However, the Air Element Method can be used as a simulation method for the internal thermal boundary conditions in the FEM to predict the temperature distributions on concrete closed girder cross-sections under historically extreme temperature conditions.

Meteorological Data
In order to simulate the external thermal boundary conditions in the FEMs, the hourly ambient air temperature curves, hourly solar radiation curves, and wind speed under historically extreme temperature conditions should be assumed. The sinusoidal function was used to simulate the hourly ambient air temperature curves by determining the maximum and minimum ambient air temperatures (Tmax and Tmin) using the following Equation [15]:

Temperature Distributions on Concrete Closed Girder Cross-Sections under Historically Extreme Temperature Conditions
The temperature curves inside the cavities under historically extreme temperature conditions are difficult to measure. However, the Air Element Method can be used as a simulation method for the internal thermal boundary conditions in the FEM to predict the temperature distributions on concrete closed girder cross-sections under historically extreme temperature conditions.

Meteorological Data
In order to simulate the external thermal boundary conditions in the FEMs, the hourly ambient air temperature curves, hourly solar radiation curves, and wind speed under historically extreme temperature conditions should be assumed. The sinusoidal function was used to simulate the hourly Appl. Sci. 2020, 10, 1274 20 of 28 ambient air temperature curves by determining the maximum and minimum ambient air temperatures (T max and T min ) using the following Equation [15]: where T i is the ambient air temperature at the ith hour, a and β are the model coefficients calculated by a = (T max − T min )/2 and β = (T max + T min )/2, and ω i is the hour angle at the ith hour.
The designed life of a highway bridge is set at 100 years, according to the Chinese code (JTG D60-2015) [50]. Based on the historical extreme temperatures obtained from the China Meteorological Administration [51] and an assumption of a normal distribution with a confidence of 95% [52], the T max and T min for the 100-year return period can be predicted, as listed in Table 4. The monthly average daily temperature range can be obtained from the China Meteorological Administration [51]. The idealized hourly temperature curve on the hottest day was based on T max , and the corresponding lowest temperature of the idealized curve was selected by subtracting the monthly average daily temperature range in July. The idealized hourly temperature curve on the coldest day was based on T min , and the corresponding highest temperature of the idealized curve was selected by adding the monthly average daily temperature range in January [13,53]. The times at which T max and T min occurred in the idealized hourly temperature curves under historically extreme temperature conditions were assumed to be 14:00 for the idealized hourly temperature curve on the hottest day and 06:00 for the idealized hourly temperature curve on the coldest day. According to the bridge locations introduced in Chapter 2, the hourly ambient air temperature curves (continued for five days) under historically extreme temperature conditions in Shenzhen, Handan, and Fuzhou are illustrated in Figure 22. Table 4. Meteorological parameters of the hourly ambient air temperature curves under historically extreme temperature conditions (Unit: • C). where Ti is the ambient air temperature at the ith hour, a and β are the model coefficients calculated by a = (Tmax-Tmin)/2 and β = (Tmax+Tmin)/2, and ωi is the hour angle at the ith hour.

Cross-Section
The designed life of a highway bridge is set at 100 years, according to the Chinese code (JTG D60-2015) [50]. Based on the historical extreme temperatures obtained from the China Meteorological Administration [51] and an assumption of a normal distribution with a confidence of 95% [52], the Tmax and Tmin for the 100-year return period can be predicted, as listed in Table 4. The monthly average daily temperature range can be obtained from the China Meteorological Administration [51]. The idealized hourly temperature curve on the hottest day was based on Tmax, and the corresponding lowest temperature of the idealized curve was selected by subtracting the monthly average daily temperature range in July. The idealized hourly temperature curve on the coldest day was based on Tmin, and the corresponding highest temperature of the idealized curve was selected by adding the monthly average daily temperature range in January [13,53]. The times at which Tmax and Tmin occurred in the idealized hourly temperature curves under historically extreme temperature conditions were assumed to be 14:00 for the idealized hourly temperature curve on the hottest day and 06:00 for the idealized hourly temperature curve on the coldest day. According to the bridge locations introduced in Chapter 2, the hourly ambient air temperature curves (continued for five days) under historically extreme temperature conditions in Shenzhen, Handan, and Fuzhou are illustrated in Figure 22. Table 4. Meteorological parameters of the hourly ambient air temperature curves under historically extreme temperature conditions (Unit: °C).  Under the highest historical temperature condition, the yearly maximum hourly solar radiation occurs on 21 June, which has the maximum sunshine duration and extraterrestrial radiation [54]. The hourly beam solar radiation and diffuse solar radiation on 21 June can be simulated by using numerical models [25] based on the solar radiation data obtain from the National Aeronautics and Space Administration (NASA) Langley Research Center [55], as illustrated in Figure 23. Under the lowest historical temperature condition, the influence of solar radiation was not considered. Under the highest historical temperature condition, the yearly maximum hourly solar radiation occurs on 21 June, which has the maximum sunshine duration and extraterrestrial radiation [54]. The hourly beam solar radiation and diffuse solar radiation on 21 June can be simulated by using numerical models [25] based on the solar radiation data obtain from the National Aeronautics and Space Administration (NASA) Langley Research Center [55], as illustrated in Figure 23. Under the lowest historical temperature condition, the influence of solar radiation was not considered. The smaller the wind speed, the higher the temperature of the girder cross-section [17]. However, the probability that zero wind speed and the maximum temperature will occur together is extremely small. A wind speed of 1.0 m/s [14,56] was used to analyze the temperature of the concrete bridge structure under the highest historical temperature conditions. The maximum basic wind speeds for the 100-year return period, which is 37.5 m/s for Shenzhen, 27.9 m/s for Handan, and 34.0 m/s for Fuzhou [57], were chosen as the wind speeds under the lowest historical temperature conditions.

Cross-Section Location
In order to reduce the influence of the initial thermal boundary condition, the same idealized hourly ambient air temperature curves, hourly solar radiation curves, and wind speed were imposed cyclically for five days.

Average Effective Temperatures of Cross-Sections
Based on the aforementioned meteorological data under historically extreme temperature conditions and the Air Element Method, the FEMs of the concrete closed girder cross-sections were established to predict the peak average effective temperatures of the cross-sections under historically extreme temperature conditions (hereafter referred to as TE). The one-year measured average effective temperatures of the cross-sections can be obtained from the field test (hereafter referred to as TM). The maximum and minimum effective temperatures of highway concrete bridges were recommended by the Chinese code JTG D60-2015 [50] (hereafter referred to as TC). China is partitioned into four climatic regions that are categorized as severe cold, cold, warm, and "other" regions [50]. The smaller the wind speed, the higher the temperature of the girder cross-section [17]. However, the probability that zero wind speed and the maximum temperature will occur together is extremely small. A wind speed of 1.0 m/s [14,56] was used to analyze the temperature of the concrete bridge structure under the highest historical temperature conditions. The maximum basic wind speeds for the 100-year return period, which is 37.5 m/s for Shenzhen, 27.9 m/s for Handan, and 34.0 m/s for Fuzhou [57], were chosen as the wind speeds under the lowest historical temperature conditions.
In order to reduce the influence of the initial thermal boundary condition, the same idealized hourly ambient air temperature curves, hourly solar radiation curves, and wind speed were imposed cyclically for five days.

Average Effective Temperatures of Cross-Sections
Based on the aforementioned meteorological data under historically extreme temperature conditions and the Air Element Method, the FEMs of the concrete closed girder cross-sections were established to predict the peak average effective temperatures of the cross-sections under historically extreme temperature conditions (hereafter referred to as T E ). The one-year measured average effective temperatures of the cross-sections can be obtained from the field test (hereafter referred to as T M ). The maximum and minimum effective temperatures of highway concrete bridges were recommended by the Chinese code JTG D60-2015 [50] (hereafter referred to as T C ). China is partitioned into four climatic regions that are categorized as severe cold, cold, warm, and "other" regions [50]. Handan is in the cold region, with the highest T C of 34 • C and the lowest T C of −10 • C; Shenzhen is in the warm region, with the highest T C of 34 • C and the lowest T C of −3 • C; and Fuzhou is in the "other" region, with the highest T C of 34 • C and the lowest T C of 0 • C. Comparisons of T M , T C , and T E are illustrated in Figure 24. The smaller the wind speed, the higher the temperature of the girder cross-section [17]. However, the probability that zero wind speed and the maximum temperature will occur together is extremely small. A wind speed of 1.0 m/s [14,56] was used to analyze the temperature of the concrete bridge structure under the highest historical temperature conditions. The maximum basic wind speeds for the 100-year return period, which is 37.5 m/s for Shenzhen, 27.9 m/s for Handan, and 34.0 m/s for Fuzhou [57], were chosen as the wind speeds under the lowest historical temperature conditions.
In order to reduce the influence of the initial thermal boundary condition, the same idealized hourly ambient air temperature curves, hourly solar radiation curves, and wind speed were imposed cyclically for five days.

Average Effective Temperatures of Cross-Sections
Based on the aforementioned meteorological data under historically extreme temperature conditions and the Air Element Method, the FEMs of the concrete closed girder cross-sections were established to predict the peak average effective temperatures of the cross-sections under historically extreme temperature conditions (hereafter referred to as TE). The one-year measured average effective temperatures of the cross-sections can be obtained from the field test (hereafter referred to as TM). The maximum and minimum effective temperatures of highway concrete bridges were recommended by the Chinese code JTG D60-2015 [50] (hereafter referred to as TC). China is partitioned into four climatic regions that are categorized as severe cold, cold, warm, and "other" regions [50]. Handan is in the cold region, with the highest TC of 34 °C and the lowest TC of −10 °C; Shenzhen is in the warm region, with the highest TC of 34 °C and the lowest TC of −3 °C; and Fuzhou is in the "other" region, with the highest TC of 34 °C and the lowest TC of 0 °C. Comparisons of TM, TC, and TE are illustrated in Figure 24. The comparisons of TM, TC, and TE for the box girder cross-section are illustrated in Figure 24a. It can be observed that the highest TM was close to the highest TC (a difference of 0.9 °C), and these values were lower than the highest TE (differences of 4.9 °C for TM and 5.8 °C for TC). The lowest TM was much higher than the lowest TE (a difference of 12.2 °C) and the lowest TC (a difference of 17.8 °C). The variation of TC was close to that of TE (a difference of only 0.2 °C), and they were larger than that of TM (differences of 16.9 °C for TC and 17.1 °C for TE). It can be predicted that the longitudinal thermal movement of the concrete box girder under historically extreme temperature conditions would be underestimated by considering the variation of TM. The variations of TC (37.0 °C) and TE (37.2 °C) were almost the same. However, assuming that the datum temperatures (T0) were the same, the thermal expansion of the concrete box girder calculated by the difference between the highest TC and T0 would be smaller than that calculated by the difference between the highest TE and T0, while the thermal contraction of the concrete box girder calculated by the difference between T0 and the lowest TC would be larger than that calculated by the difference between T0 and the lowest TE.
The comparisons of TM, TC, and TE for the small box girder cross-sections are illustrated in Figure  24b. It can be observed that the highest TE (44.7 °C) was significantly higher than the highest TM (a difference of 8.8 °C) and the highest TC (a difference of 10.7 °C); meanwhile, the lowest TE (−18.2 °C) was significantly lower than the lowest TM (a difference of 20.2 °C) and the lowest TC (a difference of 8.2 °C). The variation of TE (62.9 °C) was significantly larger than that of TM (a difference of 29.0 °C) and TC (a difference of 18.9 °C), which was possibly because Handan is in Northern China, where the variation of ambient air temperature is large. It can be predicted that the longitudinal thermal movement of the concrete small box girders under historically extreme temperature conditions would be underestimated by considering variations of TM or TC.
The comparisons of TM, TC, and TE for the adjacent box girder cross-sections are illustrated in Figure 24c. It can be observed that the highest TE (39.8 °C) was higher than the highest TM (a difference of 4.9 °C) and the highest TC (a difference of 5.8 °C); meanwhile, the lowest TE (0.1 °C) was lower than the lowest TM (a difference of 8.4 °C) and similar to the lowest TC (a difference of only 0.1 °C). The variation of TE (39.8 °C) was larger than that of TM (a difference of 10.8 °C) and TC (a difference of 5.8 °C). It can be predicted that the longitudinal thermal movement of the concrete adjacent box girders under historically extreme temperature conditions would be underestimated by considering variations of TM or TC.
Therefore, it can be concluded that the longitudinal thermal movement of the concrete closed girders calculated by the one-year measured average effective temperature of cross-sections or by the Chinese-code-specified effective temperatures for highway bridge structures is smaller than that under historically extreme temperature conditions, which are thus unconservative for engineering applications. It is suggested that the average effective temperature for the concrete closed girder cross-sections under historically extreme temperature conditions should be calculated for each city using the FEM with the Air Element Method described in this paper. The comparisons of T M , T C , and T E for the box girder cross-section are illustrated in Figure 24a. It can be observed that the highest T M was close to the highest T C (a difference of 0.9 • C), and these values were lower than the highest T E (differences of 4.9 • C for T M and 5.8 • C for T C ). The lowest T M was much higher than the lowest T E (a difference of 12.2 • C) and the lowest T C (a difference of 17.8 • C). The variation of T C was close to that of T E (a difference of only 0.2 • C), and they were larger than that of T M (differences of 16.9 • C for T C and 17.1 • C for T E ). It can be predicted that the longitudinal thermal movement of the concrete box girder under historically extreme temperature conditions would be underestimated by considering the variation of T M . The variations of T C (37.0 • C) and T E (37.2 • C) were almost the same. However, assuming that the datum temperatures (T 0 ) were the same, the thermal expansion of the concrete box girder calculated by the difference between the highest T C and T 0 would be smaller than that calculated by the difference between the highest T E and T 0 , while the thermal contraction of the concrete box girder calculated by the difference between T 0 and the lowest T C would be larger than that calculated by the difference between T 0 and the lowest T E .
The comparisons of T M , T C , and T E for the small box girder cross-sections are illustrated in Figure 24b. It can be observed that the highest T E (44.7 • C) was significantly higher than the highest T M (a difference of 8.8 • C) and the highest T C (a difference of 10.7 • C); meanwhile, the lowest T E (−18.2 • C) was significantly lower than the lowest T M (a difference of 20.2 • C) and the lowest T C (a difference of 8.2 • C). The variation of T E (62.9 • C) was significantly larger than that of T M (a difference of 29.0 • C) and T C (a difference of 18.9 • C), which was possibly because Handan is in Northern China, where the variation of ambient air temperature is large. It can be predicted that the longitudinal thermal movement of the concrete small box girders under historically extreme temperature conditions would be underestimated by considering variations of T M or T C .
The comparisons of T M , T C , and T E for the adjacent box girder cross-sections are illustrated in Figure 24c. It can be observed that the highest T E (39.8 • C) was higher than the highest T M (a difference of 4.9 • C) and the highest T C (a difference of 5.8 • C); meanwhile, the lowest T E (0.1 • C) was lower than the lowest T M (a difference of 8.4 • C) and similar to the lowest T C (a difference of only 0.1 • C). The variation of T E (39.8 • C) was larger than that of T M (a difference of 10.8 • C) and T C (a difference of 5.8 • C). It can be predicted that the longitudinal thermal movement of the concrete adjacent box girders under historically extreme temperature conditions would be underestimated by considering variations of T M or T C .
Therefore, it can be concluded that the longitudinal thermal movement of the concrete closed girders calculated by the one-year measured average effective temperature of cross-sections or by the Chinese-code-specified effective temperatures for highway bridge structures is smaller than that under historically extreme temperature conditions, which are thus unconservative for engineering applications. It is suggested that the average effective temperature for the concrete closed girder cross-sections under historically extreme temperature conditions should be calculated for each city using the FEM with the Air Element Method described in this paper.

Vertical Temperature Gradients
Based on the aforementioned meteorological data under historically extreme temperature conditions and the Air Element Method, the FEMs of the concrete closed girder cross-sections were established to predict the vertical temperature gradients under historically extreme temperature conditions. The highest temperatures for the bridge deck surface proposed by Chinese code JTG D60-2015 [50] are 25 • C for Portland cement concrete overlays and 14 • C for 100 mm asphalt concrete overlays (for positive vertical temperature differences), and the lowest temperatures of the bridge deck surface are −12.5 • C for Portland cement concrete overlays and −7 • C for 100 mm asphalt concrete overlays (for negative vertical temperature differences). The numerical vertical temperature gradients of cross-sections under historically extreme temperature conditions and the designed vertical temperature gradients for concrete superstructures proposed by Chinese code JTG D60-2015 [50]

Vertical Temperature Gradients
Based on the aforementioned meteorological data under historically extreme temperature conditions and the Air Element Method, the FEMs of the concrete closed girder cross-sections were established to predict the vertical temperature gradients under historically extreme temperature conditions. The highest temperatures for the bridge deck surface proposed by Chinese code JTG D60-2015 [50] are 25 °C for Portland cement concrete overlays and 14 °C for 100 mm asphalt concrete overlays (for positive vertical temperature differences), and the lowest temperatures of the bridge deck surface are −12.5 °C for Portland cement concrete overlays and −7 °C for 100 mm asphalt concrete overlays (for negative vertical temperature differences). The numerical vertical temperature gradients of cross-sections under historically extreme temperature conditions and the designed vertical temperature gradients for concrete superstructures proposed by Chinese code JTG D60-2015 [50] are compared in Figures 25-27.

Vertical Temperature Gradients
Based on the aforementioned meteorological data under historically extreme temperature conditions and the Air Element Method, the FEMs of the concrete closed girder cross-sections were established to predict the vertical temperature gradients under historically extreme temperature conditions. The highest temperatures for the bridge deck surface proposed by Chinese code JTG D60-2015 [50] are 25 °C for Portland cement concrete overlays and 14 °C for 100 mm asphalt concrete overlays (for positive vertical temperature differences), and the lowest temperatures of the bridge deck surface are −12.5 °C for Portland cement concrete overlays and −7 °C for 100 mm asphalt concrete overlays (for negative vertical temperature differences). The numerical vertical temperature gradients of cross-sections under historically extreme temperature conditions and the designed vertical temperature gradients for concrete superstructures proposed by Chinese code JTG D60-2015 [50] are compared in Figures 25-27.  For positive vertical temperature differences, the designed highest temperatures of the bridge deck surface were higher than those obtained from the FEMs with differences of 7.3 °C for the box girder cross-section, 9.1 °C for the small box girder cross-sections, and 18.2 °C for the adjacent box girder cross-sections, respectively. For negative vertical temperature differences, the designed lowest temperatures of the bridge deck surface were lower than those obtained from the FEMs with differences of 9.9 °C for the box girder cross-section, 6.8 °C for the small box girder cross-sections, and 12.1 °C for the adjacent box girder cross-sections, respectively. Therefore, it can be concluded that the Chinese-code-specified design temperatures of the bridge deck surfaces in vertical temperature gradients are conservative.
The temperature differences on the bottom flanges were not considered in Chinese code JTG D60-2015 [50]. However, the extreme temperature differences on the bottom flanges under historically extreme temperature conditions obtained from the FEMs were 3.3 °C and −2.7 °C for the box girder cross-section, and 1.2 °C and −2.0 °C for the small box girder cross-sections, respectively. The temperature differences on the bottom flanges of the adjacent box girder cross-sections are negligible because the bottom flange height of the adjacent box girders is small (0.44 m). Therefore, it can be concluded that the Chinese-code-specified design temperatures on the bottom flanges in the vertical temperature gradients are unconservative.

Summary
The following conclusions can be drawn based on the results, and within the limitations, of the research presented in this paper: (1) The Measured Temperature Method can reflect the actual temperatures inside the cavities, but measurements on site are cost-and time-prohibitive. Therefore, the application range of the Measured Temperature Method is limited. When there is no measurement on site to obtain the temperature inside the cavity, the Ambient Temperature Method, Mean Temperature Method, or Air Element Method can be used as alternative methods. (2) The influences of different simulation methods for the internal thermal boundary conditions on the numerical hourly temperature curves of the parts of cross-sections far from the cavities are negligible. Compared with the measured hourly temperature curves, the numerical hourly temperature curves of the parts of the cross-sections near the cavities calculated by the Measured Temperature Method provide the closest agreement. When there is a lack of measured temperature inside the cavity, the numerical hourly temperature curves calculated by the Air Element Method provide a closer agreement with the measured curves than the curves calculated by the Ambient Temperature Method and Mean Temperature Method. When the Ambient Temperature Method is used, the trends of the numerical curves of the bottom flanges and the webs near the bottom flanges are similar to the measured hourly ambient air For positive vertical temperature differences, the designed highest temperatures of the bridge deck surface were higher than those obtained from the FEMs with differences of 7.3 • C for the box girder cross-section, 9.1 • C for the small box girder cross-sections, and 18.2 • C for the adjacent box girder cross-sections, respectively. For negative vertical temperature differences, the designed lowest temperatures of the bridge deck surface were lower than those obtained from the FEMs with differences of 9.9 • C for the box girder cross-section, 6.8 • C for the small box girder cross-sections, and 12.1 • C for the adjacent box girder cross-sections, respectively. Therefore, it can be concluded that the Chinese-code-specified design temperatures of the bridge deck surfaces in vertical temperature gradients are conservative.
The temperature differences on the bottom flanges were not considered in Chinese code JTG D60-2015 [50]. However, the extreme temperature differences on the bottom flanges under historically extreme temperature conditions obtained from the FEMs were 3.3 • C and −2.7 • C for the box girder cross-section, and 1.2 • C and −2.0 • C for the small box girder cross-sections, respectively. The temperature differences on the bottom flanges of the adjacent box girder cross-sections are negligible because the bottom flange height of the adjacent box girders is small (0.44 m). Therefore, it can be concluded that the Chinese-code-specified design temperatures on the bottom flanges in the vertical temperature gradients are unconservative.

Summary
The following conclusions can be drawn based on the results, and within the limitations, of the research presented in this paper: (1) The Measured Temperature Method can reflect the actual temperatures inside the cavities, but measurements on site are cost-and time-prohibitive. Therefore, the application range of the Measured Temperature Method is limited. When there is no measurement on site to obtain the temperature inside the cavity, the Ambient Temperature Method, Mean Temperature Method, or Air Element Method can be used as alternative methods. (2) The influences of different simulation methods for the internal thermal boundary conditions on the numerical hourly temperature curves of the parts of cross-sections far from the cavities are negligible. Compared with the measured hourly temperature curves, the numerical hourly temperature curves of the parts of the cross-sections near the cavities calculated by the Measured Temperature Method provide the closest agreement. When there is a lack of measured temperature inside the cavity, the numerical hourly temperature curves calculated by the Air Element Method provide a closer agreement with the measured curves than the curves calculated by the Ambient Temperature Method and Mean Temperature Method. When the Ambient Temperature Method is used, the trends of the numerical curves of the bottom flanges and the webs near the bottom flanges are similar to the measured hourly ambient air temperature curve, because the temperature change in the hourly ambient air temperature curve was much larger than the temperature changes in the measured curves inside the cavities. (3) The comparisons of the temperature contour plots obtained from the FEMs considering different simulation methods for the internal thermal boundary conditions indicated that the temperature distributions on the parts near the cavities calculated by the Measured Temperature Method and Air Element Method were close to the measured values. (4) The influences of different simulation methods for the internal thermal boundary conditions on the highest hourly average effective temperature of concrete closed girder cross-sections and the trends of the vertical temperature gradients for the box girder and adjacent box girder cross-sections were small. The maximum vertical temperature gradients calculated by the Air Element Method on the top and bottom flanges were larger than those calculated by the Ambient Temperature Method and Mean Temperature Method for the small box girder cross-sections. (5) The Air Element Method can be used as a simulation method for the internal thermal boundary conditions in the FEM to predict the temperature distributions on concrete closed girder cross-sections under historically extreme temperature conditions. The longitudinal thermal movement of concrete closed girders calculated by the one-year measured average effective temperature of the cross-sections or by the Chinese-code-specified effective temperatures for the highway bridge structures would be smaller than those under historically extreme temperature conditions, which are thus unconservative for engineering applications. It is suggested that the average effective temperature of concrete closed girder cross-sections under historically extreme temperature conditions should be calculated for each city using the FEM with the Air Element Method, as described in this paper. The comparisons of vertical temperature gradients under historically extreme temperature conditions indicate that the Chinese-code-specified vertical temperature gradients are conservative for the bridge deck surfaces and unconservative for the bottom flanges. (6) Further research has been carried out to analyze the average effective temperatures and vertical temperature gradients of concrete closed girder cross-sections under historically extreme temperature conditions for each city in China using the FEM with the Air Element Method described in this paper.
Supplementary Materials: The following are available online at http://www.mdpi.com/2076-3417/10/4/1274/s1, Figure S1: Comparisons of the hourly temperature curves of the box girder cross-sections obtained by the field test and the FEM using the Air Element Method as the simulation method for the internal thermal boundary conditions: