# A Hybrid Algorithm Based Static Model Updating Method and Its Application to Spacecraft

Authors: Jutao Wang; Zhenzhong Liu; Xiaofei Wang
Journal: Advances in Materials Science and Engineering
 (2022)

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## Abstract

In some cases, the static model updating requires more precise results to satisfy the requirement of simulating the structural response. The corresponding optimization algorithm not only should have a stronger capability to find the global optimum, but also should achieve an optimal result with high accuracy. The intelligent algorithm and direct searching method have the advantages of global and local optimization, respectively. In the present study, a hybrid strategy based on the intelligent algorithm and direct method are proposed, and a typical test function is initially utilized to test its accuracy. The re-entry module is taken as the case study, whose thermal deformation is used to construct the objective function for optimization, and precise results are achieved. Moreover, the high-accuracy results in different objective functions are obtained. In order to investigate the prediction capability of the proposed method, the displacements in another structural direction are extracted and show high accuracy, which demonstrates the feasibility and efficiency of the proposed approach. The results show that the hybrid algorithm can achieve a better result than the single algorithm in the static model updating.


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## Body

## 1. Introduction

Model updating of the finite element model (FEM) is categorized into static model updating and dynamic model updating based on the loading types of structural experiments. [1, 2] Different dynamic response data are used, including frequencies, mode, velocity, and acceleration. Therefore, the dynamic model updating has achieved many theoretical results and engineering results, which field from space to the ocean. [3, 4] The data type of the static response is single that stress, strain, displacement, and constraints [5] are the main responses to be used. Few investigations have been performed on the static model updating. However, the response data of the structural displacement has advantages, including good stability, high accuracy, and easy measuring. [6, 7] Since the results of the static model updating have high accuracy, the applications of the deflection measurement of the bridge, structural damage, and displacement controlling [8–12] have received more attention in the past few years.A novel probabilistic procedure, defined in the Bayesian framework, is proposed to calibrate the parameters of finite element models of dams. [13] The static and dynamic data are used for the structural health monitoring via the model updating method, [14, 15] where friction and temperature are considered. The Kriging model and response surface model are taken as a metamodel and applied to the model updating of bridges FEM that is based on the static or dynamic data, respectively, where the results demonstrate adequate updating accuracy and more accurate prediction. [16, 17] A practical FEM of the pyramid grid structure is obtained via the model updating, which is based on static measured data [18] for safety assessment. Similarly, based on uncertain static data of deformation, new model updating methods are proposed to update the frame structure, plate structure, and beam structures, respectively, and the accuracy is good. [19, 20].As mentioned above, static model updating is commonly used in civil structures with huge dimensions. However, the microstructure, the sensor structures, and even spacecraft structures are usually sensitive to tiny deformation, which is supposed to be controlled. Since the static model updating is of great importance, the results of the model updating require a much smaller error between simulation results and experimental results. This challenges the model updating based on the optimization algorithm for searching for the ideal solution. In recent years, meta-heuristics are applied to solve optimization problems more and more. Such as the monarch butterfly optimization (MBO) is inspired by the behavior of monarch butterflies,[21] and it can solve the dynamic problem. [22] The slime mold algorithm (SMA) is based on the oscillation mode of slime mold in nature, [23] the colony predation algorithm (CPA) is based on the corporate predation of animals, [24] and Harris hawks optimization (HHO) inspired by the hunting strategy, [25] they have applied to solve the design problem of the beam structures. Additionally, hunger games search (HGS) is a population-based algorithm that emulates the hunger-driven activity and social behavior of animals and is applied in parameter identification, [26] and the moth search algorithm (MSA) can be used for global optimization problems. [27] The present study demonstrates a method of hybrid optimization to satisfy the requirement of the high accuracy of the model updating.


## 2. The Main Procedure of the Static Model Updating

Model updating is defined as minimizing errors between the FEM response and experiment response via modifying the design parameter of FEM, which can be described in mathematical optimization as finding a group of design parameters to make the simulation values as consistent as possible with the experiment values, and the value ranges of design parameters are given. The mathematical model for the static model updating is described as follows:(1)minFx,s.t.Xl≤x≤Xu,where x, Xu, and X1 denote the parameter set of the updated structural FEM, upper bound, and lower bound of x for the optimization constraint, respectively. Moreover, F(x) denotes the objective function that reflects the response discrepancy between the simulation and experiment.It should be indicated that the construction of the objective function can affect the success and the accuracy of the model updating. [28] The structural responses of static experiments mainly include displacement, strain, rotation angle, and curvature. The objective function F(x) can be constructed in two forms based on absolute errors and relative error, [29] which are mathematically expressed as the following:(2)Obj1=∑i=1nWiDa,j−Dt,j2,Obj2=∑i=1nWiDa,j−Dt,jDt,j2,j=1,2,⋯,k,where Wi, Da,j, and Dt,j denote the weight coefficient of different types of responses; the jth response of the simulated model and the jth response of the experiment, respectively.Obj1 and Obj2 are the objective function of absolute error and the objective function of relative error, respectively. The weight of Obj1 should balance the order of magnitude for the response in different types.During the static model updating based on the static experiment, the FEM is initially established as the simulated model and the parameters for updating are selected. The objective function is constructed based on the response error between the FEM and the experiment, where the user is concerned. Then, the optimization process is done to find the optimal parameter value to minimize the error of the updated FEM.


## 3. The Hybrid Strategy with High Accuracy

Besides the selection of design parameters and the construction of objective functions, the optimization algorithm is important to obtain satisfactory results in the optimization problem of the model updating. The optimization problems in engineering are usually complicated and the design parameters and the constraint function are diverse. Moreover, the constructed objective functions are multimodal, nonlinear, discontinuous, and non-differential in some cases. The conventional gradient algorithm can achieve local optimal solutions even if the derivative and gradients can be obtained. In recent year, soft computer strategies have gotten much attention, it has advantages in solving scheduling problems in the fields of electric vehicles and flow-shop based on optimization method [30–34]. The intelligent optimization algorithm has the advantages in solving global optimal solutions and is the main method to solve the complex optimization problems in model updating.The direct search methods, including the univariate search method, simplex algorithm, and Hooke-Jeeves method do not require the objective function to be differentiable. Therefore, the method can achieve an accurate search around the design point with robust calculation. The static problem always requires high accuracy results by the model updating. Then, the robust direct search method can be used to find the accurate optimal value. It is worth noting that both the capabilities of the global optimization (intelligent algorithm) and local optimization (direct search algorithm) can be mixed, and the advantages of the two types of algorithms can then be applied to solve the model updating with high accuracy. Thereby, the hybrid strategy that combines the multi-island genetic algorithm and simplex algorithm is proposed to solve the high accuracy problem in the static model updating.


### 3.1. The Multi-Island Genetic Algorithm

The multi-island genetic algorithm (MIGA) is an improved form of the parallel distribution genetic algorithm. [35] The MIGA divides all the initial population into sub-populations (islands). The conventional genetic algorithm operates on each island for optimization, therefore, the MIGA can enhance the capability of multi-peak problems, and can improve convergence speed. Moreover, when the predetermined generation is evolved, the units are migrated in a certain proportion between the islands, and the genetic optimization continues to operate on each island and waits for the next migration until the convergence condition is satisfied. The migration between islands can cause diversity in the population, which can reduce the probability of falling into the local optimum on each island. Therefore, the MIGA has a stronger capability of searching for the global optimum than the conventional genetic algorithm in the design domain. Figure 1 illustrates the evolution process between two successive generations.Figure 1
The evolution process of two successive generations of MIGA.The population of MIGA is exchanged proportionally between each island which increases the fitness value continuously until convergence. The computing process can be summarized as follows: First, the population is initialized and divided into several islands, and the fitness value of each unit is calculated. The unit is selected for the next generation in a certain principle based on the obtained fitness values. Then, a new population is created, where the crossover and mutation of the genetic operation are executed in proportion. Then, the fitness values are re-calculated. When the end condition is satisfied, the unit with the highest fitness value in all of the islands is taken as the optimal result. However, if the end condition is not satisfied, it is continued to perform genetically and exchange operations until meeting the end condition.


### 3.2. The Simplex Algorithm

The simplex algorithm is a direct search method to find the extremum of the function. “Simplex” means creating an n-dimension design space based on the design variables, and then constructing a (n + 1) dimension triangle for searching the optimal value. The triangle is assembled by the (n + 1) vertices and the worst vertex is replaced in each iteration. Therefore, a new triangle is re-constructed for searching the optimal value until convergence is achieved. The triangle is determined by the variable number in the simplex method, and the triangle moves to the optimal position by constantly eliminating the worst vertex, until shrinking to a sufficiently small position. The movement is a process of continuously reconstructing the triangle, which is calculated by four searching operations: reflection, expansion, shrinkage, and edge contraction. The simplex method is always obtained for searching the minimum value at any given initial vertex via the four operations. Figure 2(a) illustrates the main search process. The star in the center is the goal, XhXlXg is the initial searching vertex and the worst point Xh will always be replaced during the optimization. Moreover, Figure 2(b) illustrates the edge contraction.Figure 2
(a) Schematic of the simplex vertex; (b) the schematic of the edge contraction.
(a)(b)


### 3.3. The Procedure of the Hybrid Optimization

Based on the above two sections, the global optimization and the local optimization are combined as a strategy to improve the accuracy of the model updating. The result of the model updating shows that the error between the simulation results and experimental results is minimal. Therefore, the optimization algorithm can achieve the global optimum with high accuracy. The genetic algorithm considers the updating parameters as the population. Moreover, during population evolution, genetics operates in each unit based on fitness, and a new population is generated globally. The unit with greater fitness value is searched continuously by the iteration method, and the globally optimal result is achieved consequently, which satisfies the requirement of the global optimization. Moreover, the simplex algorithm can search the extremity of the objective function step by step based on the controllable search length. Therefore, the optimal result shrinks to any small region, and a more accurate optimal result is obtained.The searching capability of multiple peaks can be strengthened by MIGA, and the optimal result is obtained with a higher probability like the ordinary genetic algorithm. Moreover, the insufficiency of the weak local optimization and slow convergence speed [36, 37] can be improved. The simplex algorithm has the advantage of precise searching capability, while it easily falls into the local optimization and strongly depends on the initial parameter values.Based on the advantages of the above-mentioned two algorithms, a strategy that combines the two algorithms is proposed and is described as: The MIGA is initially applied to locate the globally optimal result “roughly.” Then, the location is searched precisely by the simplex method, and the precise value of the global optimum can be obtained. The specific flowchart is plotted in Figure3 and can be summarized as follows:Step 1: the design variables are encoded, and the population is generated.Step 2: the population is divided into different islands. Then, the MIGA operates with fitness judgment and unit migration.Step 3: the obtained optimization result is considered as the vertex of the simplex method.Step 4: the precise searching is achieved by the simplex algorithm with reflection, expansion, shrink and edge contraction operations.Step 5: when the error of the consecutive iteration is less than the given error, the final optimal result is achieved, or we go back to step 2.Step 6: the optimal results are used to calibrate the initial model, and the model updating is finished after checking the updated effect.Figure 3
The flowchart of the hybrid optimization.


### 3.4. Accuracy Test

In order to investigate the searching capability of the proposed optimization strategy, the Styblinski–Tang function [38] with a known optimal value is taken as the test function. Figure 4(a) shows its 3-dimensional shape and it is mathematically expressed as:(3)fx=12x14−16x12+5x1+x24−16x22+5x2.Figure 4
The Styblinski–Tang function: (a) The shape of the 3 dimensions; (b) The contour.
(a)(b)The two variables vary in the range of [−5, 5]. Moreover, the global minimum of (x1, x2) is at the position of (−2.903534, −2.903534), and the minimum is −78.33198. It should be indicated that the function has three different local minimums. Figure 4(b) shows the contour of the Styblinski–Tang function, where the global minimum has a larger area than the three local minimums. Accordingly, it is a challenge to find the precise value of the optimization algorithm. Table 1 presents the values of the parameters of the hybrid optimization. The optimization results after 175 iterations are listed in the 2nd line in Table 2. Moreover, optimization results obtained by the MIGA and simplex algorithm are listed in Table 2. It is observed that the hybrid optimization results in a higher accuracy result, while the simplex algorithm falls into the local optimization.Table 1
Parameter setting of the hybrid algorithm.
MIGASimplex algorithmSub population sizeIslands numberGenerations numberCrossover rateMutation rateMigration rateMigration intervalInitial size of triangle (%)Maximum iterations5550.80.020.251050Table 2
The results of the hybrid algorithm.
Updating parametersHybrid algorithmMIGASimplex algorithmResultsError (%)ResultsError (%)ResultsError (%)x1 = -2.903534−2.9035341≈0−3.14328228.262.7468018194.6x2 = -2.903534−2.9035338≈0−3.204318310.42.7468028194.6f(x) = -78.33198−78.332334.47 × 10−4−77.245381.39−50.0588936.1In order to test the influence of iteration times, the island number is increased to 10, evolved generation is increased to 10, and the max iteration times of the simplex algorithm are set to 100. Table3 lists the comparison of optimization results between hybrid optimization, MIGA, and simplex algorithm. It is observed that the results of the hybrid optimization and simplex algorithm are not improved, while the error of design variables and objective function increases significantly. The results in Tables 2 and 3 demonstrate that hybrid optimization is a high-accuracy algorithm.Table 3
The results of the hybrid algorithm with more iterations.
Updating parametersHybrid algorithm (600 iterations)MIGA (500 iterations)Simplex algorithm (500 iterations)ResultsError (%)ResultsError (%)ResultsError (%)x1 = -2.903534−2.9035341≈0−2.83169302.472.7468027194.6x2 = -2.903534−2.9035341≈0−2.86831461.212.7468028194.6f(x) = -78.33198−78.332334.47 × 10−4−78.224030.138−50.0588936.1


## 3.1. The Multi-Island Genetic Algorithm

The multi-island genetic algorithm (MIGA) is an improved form of the parallel distribution genetic algorithm. [35] The MIGA divides all the initial population into sub-populations (islands). The conventional genetic algorithm operates on each island for optimization, therefore, the MIGA can enhance the capability of multi-peak problems, and can improve convergence speed. Moreover, when the predetermined generation is evolved, the units are migrated in a certain proportion between the islands, and the genetic optimization continues to operate on each island and waits for the next migration until the convergence condition is satisfied. The migration between islands can cause diversity in the population, which can reduce the probability of falling into the local optimum on each island. Therefore, the MIGA has a stronger capability of searching for the global optimum than the conventional genetic algorithm in the design domain. Figure 1 illustrates the evolution process between two successive generations.Figure 1
The evolution process of two successive generations of MIGA.The population of MIGA is exchanged proportionally between each island which increases the fitness value continuously until convergence. The computing process can be summarized as follows: First, the population is initialized and divided into several islands, and the fitness value of each unit is calculated. The unit is selected for the next generation in a certain principle based on the obtained fitness values. Then, a new population is created, where the crossover and mutation of the genetic operation are executed in proportion. Then, the fitness values are re-calculated. When the end condition is satisfied, the unit with the highest fitness value in all of the islands is taken as the optimal result. However, if the end condition is not satisfied, it is continued to perform genetically and exchange operations until meeting the end condition.


## 3.2. The Simplex Algorithm

The simplex algorithm is a direct search method to find the extremum of the function. “Simplex” means creating an n-dimension design space based on the design variables, and then constructing a (n + 1) dimension triangle for searching the optimal value. The triangle is assembled by the (n + 1) vertices and the worst vertex is replaced in each iteration. Therefore, a new triangle is re-constructed for searching the optimal value until convergence is achieved. The triangle is determined by the variable number in the simplex method, and the triangle moves to the optimal position by constantly eliminating the worst vertex, until shrinking to a sufficiently small position. The movement is a process of continuously reconstructing the triangle, which is calculated by four searching operations: reflection, expansion, shrinkage, and edge contraction. The simplex method is always obtained for searching the minimum value at any given initial vertex via the four operations. Figure 2(a) illustrates the main search process. The star in the center is the goal, XhXlXg is the initial searching vertex and the worst point Xh will always be replaced during the optimization. Moreover, Figure 2(b) illustrates the edge contraction.Figure 2
(a) Schematic of the simplex vertex; (b) the schematic of the edge contraction.
(a)(b)


## 3.3. The Procedure of the Hybrid Optimization

Based on the above two sections, the global optimization and the local optimization are combined as a strategy to improve the accuracy of the model updating. The result of the model updating shows that the error between the simulation results and experimental results is minimal. Therefore, the optimization algorithm can achieve the global optimum with high accuracy. The genetic algorithm considers the updating parameters as the population. Moreover, during population evolution, genetics operates in each unit based on fitness, and a new population is generated globally. The unit with greater fitness value is searched continuously by the iteration method, and the globally optimal result is achieved consequently, which satisfies the requirement of the global optimization. Moreover, the simplex algorithm can search the extremity of the objective function step by step based on the controllable search length. Therefore, the optimal result shrinks to any small region, and a more accurate optimal result is obtained.The searching capability of multiple peaks can be strengthened by MIGA, and the optimal result is obtained with a higher probability like the ordinary genetic algorithm. Moreover, the insufficiency of the weak local optimization and slow convergence speed [36, 37] can be improved. The simplex algorithm has the advantage of precise searching capability, while it easily falls into the local optimization and strongly depends on the initial parameter values.Based on the advantages of the above-mentioned two algorithms, a strategy that combines the two algorithms is proposed and is described as: The MIGA is initially applied to locate the globally optimal result “roughly.” Then, the location is searched precisely by the simplex method, and the precise value of the global optimum can be obtained. The specific flowchart is plotted in Figure3 and can be summarized as follows:Step 1: the design variables are encoded, and the population is generated.Step 2: the population is divided into different islands. Then, the MIGA operates with fitness judgment and unit migration.Step 3: the obtained optimization result is considered as the vertex of the simplex method.Step 4: the precise searching is achieved by the simplex algorithm with reflection, expansion, shrink and edge contraction operations.Step 5: when the error of the consecutive iteration is less than the given error, the final optimal result is achieved, or we go back to step 2.Step 6: the optimal results are used to calibrate the initial model, and the model updating is finished after checking the updated effect.Figure 3
The flowchart of the hybrid optimization.


## 3.4. Accuracy Test

In order to investigate the searching capability of the proposed optimization strategy, the Styblinski–Tang function [38] with a known optimal value is taken as the test function. Figure 4(a) shows its 3-dimensional shape and it is mathematically expressed as:(3)fx=12x14−16x12+5x1+x24−16x22+5x2.Figure 4
The Styblinski–Tang function: (a) The shape of the 3 dimensions; (b) The contour.
(a)(b)The two variables vary in the range of [−5, 5]. Moreover, the global minimum of (x1, x2) is at the position of (−2.903534, −2.903534), and the minimum is −78.33198. It should be indicated that the function has three different local minimums. Figure 4(b) shows the contour of the Styblinski–Tang function, where the global minimum has a larger area than the three local minimums. Accordingly, it is a challenge to find the precise value of the optimization algorithm. Table 1 presents the values of the parameters of the hybrid optimization. The optimization results after 175 iterations are listed in the 2nd line in Table 2. Moreover, optimization results obtained by the MIGA and simplex algorithm are listed in Table 2. It is observed that the hybrid optimization results in a higher accuracy result, while the simplex algorithm falls into the local optimization.Table 1
Parameter setting of the hybrid algorithm.
MIGASimplex algorithmSub population sizeIslands numberGenerations numberCrossover rateMutation rateMigration rateMigration intervalInitial size of triangle (%)Maximum iterations5550.80.020.251050Table 2
The results of the hybrid algorithm.
Updating parametersHybrid algorithmMIGASimplex algorithmResultsError (%)ResultsError (%)ResultsError (%)x1 = -2.903534−2.9035341≈0−3.14328228.262.7468018194.6x2 = -2.903534−2.9035338≈0−3.204318310.42.7468028194.6f(x) = -78.33198−78.332334.47 × 10−4−77.245381.39−50.0588936.1In order to test the influence of iteration times, the island number is increased to 10, evolved generation is increased to 10, and the max iteration times of the simplex algorithm are set to 100. Table3 lists the comparison of optimization results between hybrid optimization, MIGA, and simplex algorithm. It is observed that the results of the hybrid optimization and simplex algorithm are not improved, while the error of design variables and objective function increases significantly. The results in Tables 2 and 3 demonstrate that hybrid optimization is a high-accuracy algorithm.Table 3
The results of the hybrid algorithm with more iterations.
Updating parametersHybrid algorithm (600 iterations)MIGA (500 iterations)Simplex algorithm (500 iterations)ResultsError (%)ResultsError (%)ResultsError (%)x1 = -2.903534−2.9035341≈0−2.83169302.472.7468027194.6x2 = -2.903534−2.9035341≈0−2.86831461.212.7468028194.6f(x) = -78.33198−78.332334.47 × 10−4−78.224030.138−50.0588936.1


## 4. The Procedure of the Model Updating

The static model updating based on the hybrid optimization should initially construct FEM based on the experiment. Second, the design parameters are selected for creating the objective function, and the corresponding optimization problem is determined. Third, the hybrid optimization is applied to the optimization problem, and then the FEM is calibrated by the optimization results. Therefore, the deviation between FEM and the experiment can be minimized. Figure5 summarizes the flowchart of the proposed procedure for the model updating.Figure 5
Flowchart of the static model updating.


## 5. Case Study

As a typical spacecraft, the returnable spacecraft usually has different modules with different functions. Moreover, the modules are often separated or assembled in different missions. As an example, the re-entry module should separate from the service module after flying around the planet. The successful separation between the two modules is very important so that the re-entry module can re-enter into the Earth orbit. The reliability of the separable device determines whether the re-entry module separates successfully or not, which is illustrated in Figure6. It is observed that the re-entry module has separation dowels that connect to the service module, and the dowel is pulled out during the separation.Figure 6
The connection of the re-entry module and service module.However, spacecraft always flies in complex environments. [39] More specifically, the thermal load cannot be ignored, which may cause the deformation of the dowels. A large radial deformation may generate excessive normal pressure to the fixed rack, and the thrust for the dowels may be insufficient because of the friction of the pressure. Meanwhile, the dowels and their relative structure are always metallic and the metallic structure is sensitive to the deformation and very small deformation may generate a great resistance from friction. Therefore, it is significantly important to estimate the dowels deformation as accurately as possible. In this section, the proposed hybrid algorithm is applied to the FEM model updating the thermal deformation. Moreover, the dowels deformation is modified and a reliable basis is provided for the control of the structural deformation.


### 5.1. The Construction of FEM

The main load-bearing structure of the re-entry module includes envelope, bottom, and truss structure. The bottom with dowels and truss is taken as the case study, which is the main control structure of the dowels deformation. Figure7 illustrates the FEM. It is observed that the bottom and dowels are shell elements, and the truss is the beam element. The dowels are made of titanium alloy, and the rests are made of aluminum alloy. In the known temperature field, the elastic modulus and expansion coefficient of the materials are the main parameters that influence the thermal deformation significantly. Therefore, the elastic modulus and expansion coefficient of the titanium alloy and aluminum alloy are considered as the updating parameter to be modified. In this case, the FEM with the theoretical values of the 2 parameters is taken as the “experiment” model. Moreover, the FEM with artificial errors of updating parameters is considered as the “FEM” to be updated, and the deviation of the FEM is given based on the theoretical values. Table 4 presents the parameter settings. Moreover, Figure 8 illustrates the temperature load of the structure.Figure 7
The structure of the bottom of the re-entry module.Table 4
The parameter setting of the re-entry modules.
Aluminum alloyTitanium alloyElastic modulusExpansion (°C)Elastic modulusExpansion (°C)Experiment model71 GPa2.3 × 10−5110 GPa0.9 × 10−5FEM92.3 GPa1.61 × 10−577 GPa1.17 × 10−5Deviation+30%−30%−30%+30%Value range[46.15, 184.6][0.805, 3.22]×10−5[38.5, 154][0.585, 2.34]×10−5Figure 8
Temperature field of the bottom of the re-entry module.


### 5.2. The Construction of the Objective Function

The temperature load is applied to the experiment model and FEM, and the fixed constraint is located at the upper edge of the bottom. Figure9 shows the nephogram of the composite displacement of the generated thermal deformation. It is observed that the maximum displacement is located at the bottom of the structure, while the location of the maximum displacement cannot affect the separation of the re-entry module. Moreover, it is found that all of the three separation dowels expand differently and the concern position should be updated. Therefore, the displacements of Point 1, Point 2, and Point 3 are considered as the observation of the thermal simulation of deformation.Figure 9
Displacement of the bottom of the re-entry module.TheX direction and Y direction of the displacement at each point can generate the normal pressure to the rack directly. Therefore, the displacement of direction X of the 3 Points is considered as the response for the model updating. Figure 9 shows that the magnitude differential between the displacement response is 101, i.e., 1 order of magnitude and the response is supposed to be normalized. the objective function of the optimization for the model updating is formulated as the following:(4)Fx=∑i=131−xidi,where xi and di denote the displacement of the point i from FEM, and the displacement of the point i from the experiment model, respectively.The construction of the objective function of the model updating may affect the accuracy of the results and even affect the success of the model updating. The results in different constructions of the objective function are compared in this study.


### 5.3. The Results and Discussion of the Model Updating

The hybrid algorithm is applied to the optimization problem, and Table1 lists the parameter values of the algorithm. The value range of each design variable (updating parameters) changes in the range of −50% to 200% of its initial deviated value, which is listed in the last line of Table 4. After 151 iterations, the optimal results are obtained and listed in the second line of data in Table 5. Moreover, a different objective function is formulated to investigate the influence of the optimization results of different functions. Then, the difference at each point is directly used to formulate the objective function as the following:(5)fx=∑i=13xi−di,where xi and di denote the displacement of Point i from FEM and the displacement of Point i from the experiment, respectively.Table 5
Results Comparison of different objective functions.
Aluminum alloyTitanium alloyMinimum value of objective functionElastic modulus (GPa)Expansion (°C)Elastic modulus (GPa)Expansion (°C)Experiment712.3 × 10−51109 × 10−5—Fx=∑i=131−xi/di87.4942.3040 × 10−5132.727.1180 × 10−51.03 × 10−3fx=∑i=13xi−di77.6692.3002 × 10−5121.689.0701 × 10−54.80 × 10−5F′x=∑i=131−xi/di269.9392.2907 × 10−5114.651.3905 × 10−53.46 × 10−6f′x=∑i=13xi−di268.9062.2982 × 10−5126.251.0962 × 10−51.21 × 10−7The optimization results are listed in the third line of data in Table5, which are based on (5). It is observed that the function achieves a required minimum value. Moreover, two different objective functions are formulated and their minimum values are listed in the last two lines of Table 5. It is found that the lowest value is achieved based on (5) which means the direct difference of the objective function has the advantage to solve the optimization problem of this case study.The optimal values of design variables in different objective functions are utilized to calibrate the FEM. The displacement responses of the 3 Points based on the second line and the last line of data in Table5 are listed in the 4th and 5th columns of Table 6. Moreover, the two columns are corresponding to the largest and smallest minimum value of Table 5, respectively. Based on the obtained two group optimal parameter values, the response errors between the updated FEM and the experiment are very small. Therefore, both optimizations have high accuracy.Table 6
Displacement ofX direction of the observed points.
PointExperimentInitial valueResults ofF (x)Results off′(x)Error ofF (x) (%)Error off′ (x) (%)10.5527330.3892440.5531470.5525220.0750.03820.02736630.02058400.02736640.02719243.7 × 10−40.643−1.03180−0.733618−1.03151−1.031830.0282.9 × 10−3Moreover, the accuracy of the updated FEM is investigated. The displacement of direction Y of 3 Points is extracted and listed in Table7. Table 7 shows that the updated results have a small error. The error comparison of the observation points before and after model updating is plotted in Figure 10, and the errors are reduced sharply can be seen. Figure 11 presents the convergence curve of design variables and objective functions, and each parameter can converge soon. Therefore, the proposed algorithm can improve the model updating, and it is found that the updated FEM has a precise prediction capability.Table 7
Displacement ofY direction of the observed points.
PointExperimentInitial valueResults ofF (x)Results off′ (x)Error ofF (x) (%)Error off′ (x) (%)1−0.342162−0.242313−0.342331−0.3418570.0490.08921.587441.120361.588641.586260.0760.0743−0.607994−0.433530−0.607688−0.6079390.0509.0 × 10−3Figure 10
Error comparison of observed points before and after model updating: (a) Error comparison atX direction of the observed points; (b) Error comparison at Y direction of the observed points.
(a)(b)Figure 11
Convergence curves of each parameter and objective function: (a) Convergence curve of the elastic modulus of the aluminum alloy; (b) Convergence curve of the expansion coefficient of the aluminum alloy; (c) Convergence curve of the elastic modulus of the titanium alloy; (d) Convergence curve of the expansion coefficient of the aluminum alloy; (e) Convergence curve of the objective function.
(a)(b)(c)(d)(e)


## 5.1. The Construction of FEM

The main load-bearing structure of the re-entry module includes envelope, bottom, and truss structure. The bottom with dowels and truss is taken as the case study, which is the main control structure of the dowels deformation. Figure7 illustrates the FEM. It is observed that the bottom and dowels are shell elements, and the truss is the beam element. The dowels are made of titanium alloy, and the rests are made of aluminum alloy. In the known temperature field, the elastic modulus and expansion coefficient of the materials are the main parameters that influence the thermal deformation significantly. Therefore, the elastic modulus and expansion coefficient of the titanium alloy and aluminum alloy are considered as the updating parameter to be modified. In this case, the FEM with the theoretical values of the 2 parameters is taken as the “experiment” model. Moreover, the FEM with artificial errors of updating parameters is considered as the “FEM” to be updated, and the deviation of the FEM is given based on the theoretical values. Table 4 presents the parameter settings. Moreover, Figure 8 illustrates the temperature load of the structure.Figure 7
The structure of the bottom of the re-entry module.Table 4
The parameter setting of the re-entry modules.
Aluminum alloyTitanium alloyElastic modulusExpansion (°C)Elastic modulusExpansion (°C)Experiment model71 GPa2.3 × 10−5110 GPa0.9 × 10−5FEM92.3 GPa1.61 × 10−577 GPa1.17 × 10−5Deviation+30%−30%−30%+30%Value range[46.15, 184.6][0.805, 3.22]×10−5[38.5, 154][0.585, 2.34]×10−5Figure 8
Temperature field of the bottom of the re-entry module.


## 5.2. The Construction of the Objective Function

The temperature load is applied to the experiment model and FEM, and the fixed constraint is located at the upper edge of the bottom. Figure9 shows the nephogram of the composite displacement of the generated thermal deformation. It is observed that the maximum displacement is located at the bottom of the structure, while the location of the maximum displacement cannot affect the separation of the re-entry module. Moreover, it is found that all of the three separation dowels expand differently and the concern position should be updated. Therefore, the displacements of Point 1, Point 2, and Point 3 are considered as the observation of the thermal simulation of deformation.Figure 9
Displacement of the bottom of the re-entry module.TheX direction and Y direction of the displacement at each point can generate the normal pressure to the rack directly. Therefore, the displacement of direction X of the 3 Points is considered as the response for the model updating. Figure 9 shows that the magnitude differential between the displacement response is 101, i.e., 1 order of magnitude and the response is supposed to be normalized. the objective function of the optimization for the model updating is formulated as the following:(4)Fx=∑i=131−xidi,where xi and di denote the displacement of the point i from FEM, and the displacement of the point i from the experiment model, respectively.The construction of the objective function of the model updating may affect the accuracy of the results and even affect the success of the model updating. The results in different constructions of the objective function are compared in this study.


## 5.3. The Results and Discussion of the Model Updating

The hybrid algorithm is applied to the optimization problem, and Table1 lists the parameter values of the algorithm. The value range of each design variable (updating parameters) changes in the range of −50% to 200% of its initial deviated value, which is listed in the last line of Table 4. After 151 iterations, the optimal results are obtained and listed in the second line of data in Table 5. Moreover, a different objective function is formulated to investigate the influence of the optimization results of different functions. Then, the difference at each point is directly used to formulate the objective function as the following:(5)fx=∑i=13xi−di,where xi and di denote the displacement of Point i from FEM and the displacement of Point i from the experiment, respectively.Table 5
Results Comparison of different objective functions.
Aluminum alloyTitanium alloyMinimum value of objective functionElastic modulus (GPa)Expansion (°C)Elastic modulus (GPa)Expansion (°C)Experiment712.3 × 10−51109 × 10−5—Fx=∑i=131−xi/di87.4942.3040 × 10−5132.727.1180 × 10−51.03 × 10−3fx=∑i=13xi−di77.6692.3002 × 10−5121.689.0701 × 10−54.80 × 10−5F′x=∑i=131−xi/di269.9392.2907 × 10−5114.651.3905 × 10−53.46 × 10−6f′x=∑i=13xi−di268.9062.2982 × 10−5126.251.0962 × 10−51.21 × 10−7The optimization results are listed in the third line of data in Table5, which are based on (5). It is observed that the function achieves a required minimum value. Moreover, two different objective functions are formulated and their minimum values are listed in the last two lines of Table 5. It is found that the lowest value is achieved based on (5) which means the direct difference of the objective function has the advantage to solve the optimization problem of this case study.The optimal values of design variables in different objective functions are utilized to calibrate the FEM. The displacement responses of the 3 Points based on the second line and the last line of data in Table5 are listed in the 4th and 5th columns of Table 6. Moreover, the two columns are corresponding to the largest and smallest minimum value of Table 5, respectively. Based on the obtained two group optimal parameter values, the response errors between the updated FEM and the experiment are very small. Therefore, both optimizations have high accuracy.Table 6
Displacement ofX direction of the observed points.
PointExperimentInitial valueResults ofF (x)Results off′(x)Error ofF (x) (%)Error off′ (x) (%)10.5527330.3892440.5531470.5525220.0750.03820.02736630.02058400.02736640.02719243.7 × 10−40.643−1.03180−0.733618−1.03151−1.031830.0282.9 × 10−3Moreover, the accuracy of the updated FEM is investigated. The displacement of direction Y of 3 Points is extracted and listed in Table7. Table 7 shows that the updated results have a small error. The error comparison of the observation points before and after model updating is plotted in Figure 10, and the errors are reduced sharply can be seen. Figure 11 presents the convergence curve of design variables and objective functions, and each parameter can converge soon. Therefore, the proposed algorithm can improve the model updating, and it is found that the updated FEM has a precise prediction capability.Table 7
Displacement ofY direction of the observed points.
PointExperimentInitial valueResults ofF (x)Results off′ (x)Error ofF (x) (%)Error off′ (x) (%)1−0.342162−0.242313−0.342331−0.3418570.0490.08921.587441.120361.588641.586260.0760.0743−0.607994−0.433530−0.607688−0.6079390.0509.0 × 10−3Figure 10
Error comparison of observed points before and after model updating: (a) Error comparison atX direction of the observed points; (b) Error comparison at Y direction of the observed points.
(a)(b)Figure 11
Convergence curves of each parameter and objective function: (a) Convergence curve of the elastic modulus of the aluminum alloy; (b) Convergence curve of the expansion coefficient of the aluminum alloy; (c) Convergence curve of the elastic modulus of the titanium alloy; (d) Convergence curve of the expansion coefficient of the aluminum alloy; (e) Convergence curve of the objective function.
(a)(b)(c)(d)(e)


## 6. Conclusions

In the present study, the static model updating algorithm with high accuracy is investigated. During the optimization solution, a certain optimization algorithm may reasonably search the global optimum, while the accuracy often cannot satisfy the requirement of the static model updating. The direct optimization algorithm is good at searching the optimum with high accuracy, while it may fall into the local optimum easily. Therefore, the roughly global optimum is initially located, and then the required optimum with high accuracy is investigated precisely by the direct searching algorithm. Moreover, a hybrid optimization algorithm that combines the MIGA and simplex algorithm is used for the static model updating, and a procedure for the static model updating is proposed based on the hybrid algorithm.The separation device of the re-entry module is considered as the case study and the model updating of the thermal deformation is studied. A little deformation of the important part may cause a harmful effect. Therefore, a high accuracy model updating is required to achieve precise results. In this study, the proposed model updating procedure is applied and a precise result is achieved. Moreover, different objective functions are formulated and the obtained results are considered precise results. Moreover, the displacements that are not constructed in the objective functions are investigated after the model updating, and the obtained precise results demonstrate the efficiency and reasonable prediction capability of the proposed procedure. The proposed procedure can be applied to a very small structure in future work.



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*Source: 7448549-2022-08-18.xml*