Thermal field simulation and material parameter optimization for spaceborne annular truss antennas

The operational efficacy of large spaceborne annular truss antennas in orbit is significantly influenced by solar irradiation and alternating Earth shadow. This antenna system periodically encounters diverse extreme environments that impact the precision of the antenna surface performance. Consequently, this study presents an optimized thermal control design and conducts temperature field simulation calculations for such antennas. Initial efforts are directed toward analyzing the variables influencing the antenna structure’s temperature, with specific attention paid to the distinctive compositional characteristics of high-precision antennas. As a subsequent step, orthogonal tests are implemented, facilitating the development of an antenna thermal analysis model. This model assists in the identification of principal variables influencing the antenna’s temperature field. Finally, the antenna’s optimal thermal design is drawn upon the biogeography-based optimization (BBO) algorithm, enabling the derivation of ideal material parameters for the thermal design of the antenna. This methodology offers theoretical guidance for future thermal control design of large spaceborne annular truss antennas.


Introduction
In light of the sustained advancement in space technology, the large spaceborne annular truss antenna has become a prevalent component across various spacecraft, serving as an essential conduit for spacecraft information transmission.The antenna undergoes transient external heat flow as the spacecraft maneuvers through its orbit, transitioning intermittently into and out of Earth's shadow region.This transition incites substantial temperature fluctuations in the antenna, leading to an elevated thermogenic response.Such thermal phenomena can diminish the antenna's operational precision and disrupt the attitude stability of the spacecraft.As the frequency band of antennas has seen significant enhancements in recent years, thermal deformation has correspondingly become a more critical factor affecting antenna performance.This evolution necessitates heightened accuracy in the antenna-type surface.
Addressing the issue of thermogenic response control has been a pivotal aspect of antenna development.Depending on the chosen vibration control strategy, this control issue can be categorized into passive response control and active response control.Passive response control has seen extensive investigation by both domestic and international researchers.
Hedgepeth [1] compared the impacts of different structural forms, thermogenic deformation, and manufacturing errors on the precision of the antenna's shape surface.Hedgepeth [1] posited that the shape surface precision of space antennas could be enhanced through passive control strategies such as the optimization of structural design form, material parameters, and precision machining.The Hubble Telescope's solar wing support beam was also restored through passive control strategies, such as adding thermal protection layers and utilizing bellows [2].Song et al. [3] proposed the application of composite materials to mitigate the thermally induced deformation of the structure.Fan et al. [4] employed deterministic and robust optimization methods to perfect the design of geometric and material parameters of space structures, with the primary aim being to suppress the thermotropic deformation of large space structures.These methods effectively controlled the thermogenic deformation of large space structures like space parabolic antennas and masts.Wang et al. [5] executed a thermal deformation optimization design of a large-aperture high-precision satellite-based antenna to understand how component deformation influences antenna profile and pointing.Rong et al. [6] conducted a thermal deformation optimization analysis of the parabolic antenna reflecting surface, and a rational lay-up method was preferred.
Building on the preceding research, this study employs the comprehensive model of a satellite equipped with a large spaceborne annular truss antenna as the subject of investigation.The model is simulated using Thermal Desktop (TD) and Sinda/Fluint (S/F) to determine the average temperature field fluctuation of the antenna nodes during transitions into and out of Earth shadow.This study also incorporates the design of material parameter sensitivity tests to identify the principal material parameters influencing the temperature field of the antenna.Subsequently, the antenna's material thermal properties are optimized using the BBO algorithm to minimize temperature field variance during transitions in and out of Earth shadow, thereby reducing the antenna's thermotropic response.This method contributes a foundation for designing and selecting materials for large spaceborne annular truss antennas.

Spacecraft thermal analysis model
The spacecraft thermal-mathematical model is solved in the following sequence: First, the radiation calculations are performed based on the geometric-mathematical model, in which the radiative heat transfer calculations are performed using the Monte Carlo ray-tracing algorithm to calculate Q S-i , Q a-i , Q IRp-I , and G ji , and these outputs are input into the thermal-mathematical model.The second step is to solve the thermal mathematical model and obtain the temperature field of each surface element of the spacecraft.The thermal mathematical model is solved as follows: where the subscripts i and j denote surface element numbers.T is temperature.m is mass.c is specific heat.t is time.Q S-i is the external heat flow of solar radiation absorbed by surface element i. Q a-i is the external heat flow of Earth's albedo radiation absorbed by surface element i. Q IRp-i is the external heat flow of Earth's infrared radiation absorbed by surface element i. Q is the internal heat source power.D ji and G ji denote the linear thermal conductivity (heat transfer coefficient) and the radiative thermal conductivity between face elements j and i, respectively [7].

Orthogonal experimental design
The orthogonal experimental design method, a significant branch of statistical mathematics, leverages probabilistic mathematical statistics, professional expertise, and practical experience.An orthogonal table, compiled by applying mathematical and statistical principles, serves as an effective tool to address complexities.Utilizing orthogonal tables to design experimental protocols enables the selection of a subset of representative combinations from a substantial array of comprehensive tests.
With minimal computation, we can identify superior process conditions or optimal formulations.The subsequent test results analysis facilitates pursuing a potentially optimal testing solution.The standardized orthogonal table is employed to arrange the testing scheme and to compute and analyze the test results.Ultimately, it serves as a scientific calculation method aimed at minimizing the number of tests, curtailing the testing cycle, and promptly identifying the optimal solution.

Biogeography-based optimization algorithm
The fundamental essence of the BBO algorithm lies in migration and mutation, which serve as its mechanisms for information exchange.Migration and mutation enable the circulation of species information across habitats, enhancing the Habitat Suitability Index (HSI) values.The algorithm simulates natural hazard events and species mutations within the natural environment via a predetermined probability of mutation [8].

Migration.
The migration operation intends to facilitate the sharing of information amongst varied solutions.Superior solutions tend to disseminate their information to other solutions, whereas inferior solutions are more likely to assimilate information from other solutions.The migration model employed in this research is linear, wherein the migration rate primarily depends on the number of biological populations.With increased habitat species, the ascendant and descendant trajectories of migration and emigration rates mimic a primary function.The relationship between the emigration and immigration rates and the number of biological populations is articulated by the subsequent Equation (2) [9].
where λ is the migration rate; μ is the emigration rate; S and S max are the current number of species and the maximum number of species in the habitat, respectively; I and E are the maximum values of the migration and emigration rates, respectively.
In its specific implementation, each iteration of the BBO algorithm scrutinizes every solution, H i , within the population.Each component of H i has a likelihood, λ i , of undergoing modification (i.e., executing a migration).If migration is to occur, a solution, H j , is selected from the population for migration, with selection probability equal to the emigration rate, μ j .Subsequently, the current component of H i is supplanted by the analogous component of H j .Upon executing this operation on all components of H i , a new solution, H i 1 , is produced.The algorithm then compares the fitness of H i and H i 1 , retaining the solution with the superior fitness within the population.

Mutation.
Numerous critical emergencies can significantly alter certain properties of natural habitat, subsequently modifying the habitat suitability index.This scenario is modeled as a mutation in the context of the BBO algorithm.Within the specific implementation, every iteration of the BBO algorithm scrutinizes each solution, H i , within the population, with each solution component having a π i probability of mutation [10].In recognition of the thermogenic vibration phenomenon, which predominantly manifests around the transition points when a spacecraft enters and departs from Earth's penumbra, the focus of this paper is specifically targeted at calculating the spacecraft's thermal field within the orbital time frame of 8200 s to 10600 s.The thermal-physical properties of the materials constituting the various components of the satellite are presented in Tables 2 and 3.   Variables such as the inherent heat source within the onboard antenna, the angular coefficient pertinent to each surface about celestial bodies like the Sun and planets, and the radiant energy these bodies emit are contingent upon the specific mission and thus offer little room for modification.Hence, the primary focus of this paper lies in analyzing the impact of diverse physical parameters associated with the satellite antenna structure on the in-orbit thermal field of the same.
To ascertain the design variables essential for optimization, it is imperative to scrutinize key elements influencing the temperature fluctuation of the antenna as it transitions into and out of the terrestrial shadow.This scrutiny entails altering the material attributes and surface state of a predefined antenna structure of specific dimensions while keeping the orbit and attitude constant.
Primarily, the material thermal parameters that significantly impact the antenna temperature encompass the ratio of absorption to emission, thermal conductivity, and specific heat capacity of the material surface.This research concentrates on these three elements, rendering them pivotal for orthogonal tests.Each of these factors will be evaluated at three levels, and the factor level chart is depicted in Table 4.The metric under examination is the mean alteration in the temperature of the antenna's surface during its transition into and out of the Earth's shadowed region.They are drawing upon the three elements detailed in Table 4, and their respective three tiers, an orthogonal testing scheme was conceptualized in this study.The specifics of this scheme are depicted in Table 5.The final figures within Table 5 correspond to the level values of these influential factors.

BBO algorithm design
The parameters of the BBO algorithm chosen in this paper are as follows.The initial population size is 40.The maximum possible migration rate to the habitat is 1.The maximum possible migration rate from the habitat is 1.The variation rate is 0.02.The maximum number of function evaluations (termination condition) of the algorithm is 500.The orbits of the spacecraft and their attitudes chosen for the optimization analysis are those of the spacecraft and their attitudes for the orthogonal tests described above.To reflect the full orbit situation, the global adaptability of the optimization results is considered.The objective function is chosen as the rate of change of the average temperature of the antenna in and out of the shadow region of the Earth during one cycle of the orbit.
1 1 1 0 0 0 (4) (5) where ε 0 is the strain due to temperature change; F T is the external force due to temperature change; D is the elastic instanton matrix; B is the unit strain matrix; α is the coefficient of thermal expansion of the antenna surface element material.
The force due to temperature change is proportional to the thermal strain, which in turn is proportional to the temperature change.With a constant coefficient of thermal expansion of the material, the force due to temperature is proportional to the change in temperature.To minimize the thermotropic response of the antenna, it is necessary to ensure minimal temperature variation.So, this optimization selects the average change of ground shadow temperature in and out of the antenna-type surface as the objective function.

Results and discussion
As outlined in Table 5, the testing protocol was subjected to computational analysis using TD and S/F.The resulting values from each test were duly recorded, and these entries are represented in the farright column of Table 6.The cumulative data obtained from the initial set of tests were combined.In this vein, the summative test data for the 1st, 2nd, and 3rd trials, correlating with level 1 of the first column, was aggregated, and this total was designated as 'A'.The extreme difference signifies that the absorption-to-emission ratio holds the most significant influence on the temperature index.A comparative analysis between its optimal and sub-optimal levels reveals a substantial impact value of 59.87℃ on the mean temperature fluctuation within the ingress and egress of Earth's shadow regions.The magnitude of its influence considerably supersedes that of other parameters.The impacts of thermal conductivity and specific heat essentially coincide, and their alterations induce a minimal effect on the antenna-type surface temperature field distribution.Consequently, it can be inferred that the absorption-to-emission ratio is the predominant factor influencing the antenna surface temperature.
Implementing the BBO algorithm to optimize the absorption-to-emission ratio of the material allows for the selection of an improved thermal control coating for the antenna.This process is centered around utilizing this material to mitigate the influence of the thermal environment on the antenna's operational precision.Optimization results indicate that the antenna coating's absorbance and emissivity are 0.0971 and 0.3164, respectively.The average temperature discrepancy between the antenna structure's ingress and egress of Earth's shadow region is minimal, peaking at 2.2316℃.The spacecraft applies these optimized parameters to the coating's thermal properties.Figure 5 to Figure 8 depict the temperature distribution of the entire spacecraft at orbital moments of 8220 s, 9360 s, 10000 s, and 10600 s.

Conclusion
This paper presents an in-depth analysis of the numerous factors that influence the temperature regulation of a satellite-mounted antenna, with a primary focus on the thermal physical attributes of the antenna material.Using orthogonal experiments, simulations were conducted utilizing TD and S/F to generate data for analysis.Results reveal that the absorption-to-emission ratio is the primary influencer of the antenna surface temperature, given the fixed orbit and attitude of the spacecraft.Leveraging the findings from the orthogonal experiments, the solar absorption rate and emissivity of the thermally controlled coating on the space-borne antenna reflector's surface were designated design variables.The temperature fluctuation of the antenna within the Earth's shadow region was used as the objective function.The BBO algorithm was then employed to optimize the design of the material's thermal parameters for the satellite-based antenna.The optimization process yielded an antenna coating absorbance of 0.0971 and emissivity of 0.3164.The average temperature deviation between the antenna structure's entry and exit from the Earth's shadow region was negligible, reaching a maximum of 2.2316℃.This methodology offers a theoretical foundation for the thermal control design of large spaceborne annular truss antennas.

Figure 2 .
Figure 2. Spacecraft thermal analysis model.Figure3presents a schematic diagram of the satellite's orbit.The moment of entry into the Earth's shadow along the orbit is 8370.65 s, while the moment of exit is 10, 571.7 s.The orbital angles corresponding to these shadow ingress and egress events are 277.685°and 350.702°, respectively.

Figure 3 .
Figure 3. Satellite orbit diagram.In recognition of the thermogenic vibration phenomenon, which predominantly manifests around the transition points when a spacecraft enters and departs from Earth's penumbra, the focus of this paper is specifically targeted at calculating the spacecraft's thermal field within the orbital time frame of 8200 s to 10600 s.The thermal-physical properties of the materials constituting the various components of the satellite are presented in Tables2 and 3.

Figure 4 .
Figure 4. Parameters affecting the temperature of the antenna structure.

Figure 4
Figure 4 delineates the factors influencing the temperature dynamics of the antenna structure.Variables such as the inherent heat source within the onboard antenna, the angular coefficient pertinent to each surface about celestial bodies like the Sun and planets, and the radiant energy these bodies emit are contingent upon the specific mission and thus offer little room for modification.Hence, the primary focus of this paper lies in analyzing the impact of diverse physical parameters associated with the satellite antenna structure on the in-orbit thermal field of the same.To ascertain the design variables essential for optimization, it is imperative to scrutinize key elements influencing the temperature fluctuation of the antenna as it transitions into and out of the terrestrial shadow.This scrutiny entails altering the material attributes and surface state of a predefined antenna structure of specific dimensions while keeping the orbit and attitude constant.Primarily, the material thermal parameters that significantly impact the antenna temperature encompass the ratio of absorption to emission, thermal conductivity, and specific heat capacity of the material surface.This research concentrates on these three elements, rendering them pivotal for orthogonal tests.Each of these factors will be evaluated at three levels, and the factor level chart is depicted in Table4.The metric under examination is the mean alteration in the temperature of the antenna's surface during its transition into and out of the Earth's shadowed region.

Figure 6 .
Figure 6.Temperature field cloud of antenna after optimization parameters (Time=9360 s).

Figure 7 .
Figure 7. Temperature field cloud of antenna after optimization parameters (Time=10000 s).

Figure 8 .
Figure 8. Temperature field cloud of antenna after optimization parameters (Time=10600 s).

Table 2 .
Thermal property parameters of satellite component materials.

Table 3 .
Thermal property parameters of satellite component coatings.

Table 4 .
Factor table of orthogonal test.

Table 5 .
Orthogonal test protocol table.