Thermal protection system reliability optimization design method based on double-approximation quantization model

Abstract

The application provides a thermal protection system reliability optimization design method based on a double-approximation quantization model, which comprises the following steps: determining a thermal protection system uncertain parameter argument domain, using test points to establish a double-approximation quantization model for the uncertain parameter set in the argument domain; using a Latin hypercube sampling method to obtain sample points in the argument domain and design domain, and calculating response values through finite elements; establishing a polynomial model of the response and the uncertain parameter, and using a Kriging model to determine the mapping relationship between the design variable and the polynomial coefficients in the polynomial model, so as to obtain a nested mixed proxy model; solving the response interval of the double-approximation quantization model based on the mixed proxy model, and determining conservative and aggressive reliability indexes according to the response safety threshold; and performing thermal protection system lightweight design with the conservative and aggressive reliability indexes as constraints. On the basis of ensuring design safety, the structural efficiency of the thermal protection system and the calculation efficiency of reliability optimization are improved.

Description

Technical Field

[0001] This invention belongs to the field of mechanical engineering, specifically relating to a reliability optimization design method for thermal protection systems based on a dual approximation quantization model. Background Technology

[0002] Reliability refers to the ability to effectively perform required tasks under given conditions and within a given timeframe. System reliability is closely linked to uncertainty. In recent years, aircraft development has faced strategic needs such as high speed and high maneuverability. As a structural system that protects the internal structure of an aircraft from external aerodynamic loads, the safety of thermal protection systems has received increasing attention. For thermal protection systems, uncertainties such as load and material property variations, processing and measurement errors further exacerbate the nonlinearity of thermo-mechanical coupling, thus severely impacting stability and reliability.

[0003] Reliability optimization design is divided into reliability analysis and optimization design. Currently, commonly used reliability analysis models mainly include probabilistic reliability models, fuzzy reliability models, and non-probabilistic reliability models. The probabilistic reliability theory upon which probabilistic reliability models are based is quite mature, while the fuzzy reliability theory upon which fuzzy reliability models are based is still being refined. These theories have played an important role in addressing the numerous random and fuzzy phenomena present in engineering practice. However, both probabilistic and fuzzy reliability models require substantial data to describe the probability distribution or membership function of uncertain parameters, and obtaining this data often relies on random and fuzzy experiments. For thermal protection systems, the objective constraints of actual engineering projects make it difficult to meet the basic requirements of these two reliability theories, thus hindering the accuracy of reliability index calculations. Under conditions of few samples, non-probabilistic reliability models, based on convex set theory, utilize bounded regions (such as intervals, ellipsoids, parallelepipeds, etc.) to characterize the variation region of uncertain parameters. Based on this, they evaluate the reliability of thermal protection systems using metrics such as volume ratio. However, the complex correlations among the uncertain parameters of thermal protection systems result in irregular boundaries and other characteristics in the actual uncertain regions. Therefore, when using non-probabilistic reliability models based on convex set models for reliability assessment, the results are poor and the accuracy is low.

[0004] Based on reliability analysis models, reliability optimization design is an important way to achieve lightweighting and other requirements while ensuring the safety of thermal protection systems. However, with the increasing complexity of engineering problems, existing reliability optimization design methods, such as approximate analytical methods and numerical sampling methods, face an irreconcilable contradiction between computational accuracy and efficiency.

[0005] Therefore, there is an urgent need for a reliability optimization design method for thermal protection systems that offers accurate reliability assessment and high optimization efficiency. Summary of the Invention

[0006] In view of the above problems, embodiments of the present invention provide a reliability optimization design method for thermal protection systems based on a dual approximation quantization model, which is used to improve the reliability assessment accuracy and reliability optimization efficiency of thermal protection systems.

[0007] This invention provides a reliability optimization design method for a thermal protection system based on a dual approximation quantization model, comprising:

[0008] Step 1: Determine the universe of discourse for the uncertain parameters of the thermal protection system, and based on the obtained experimental data, establish a double approximation quantization model for the set of uncertain parameters with irregular boundaries within the universe of discourse;

[0009] Step 2: Determine the design domain based on the maximum and minimum values ​​of the design variables. Use the Latin hypercube sampling method to sample uncertain parameter points and design variable points in the domain of discourse and the design domain respectively to form a sample point matrix. Then, obtain the response value of the thermal protection system at each element in the sample point matrix through finite element calculation to form a response value matrix.

[0010] Step 3: Based on the sample point matrix and the response value matrix, establish a polynomial model of the response with respect to the uncertain parameters;

[0011] Step 4: Based on the polynomial model and the sample point matrix, use the Kriging model to determine the mapping relationship between the design variables and the polynomial coefficients in the polynomial model, and obtain the hybrid surrogate model of the response, the uncertain parameters, and the design variables;

[0012] Step 5: Using the hybrid proxy model, solve the response range of the thermal protection system in the double approximation quantization model, and determine the conservative and aggressive reliability indices based on the set response safety threshold.

[0013] Step Six: Using the conservative and aggressive reliability indices as constraints and the quality of the thermal protection system as the optimization objective, establish corresponding conservative and aggressive optimization models, and optimize the conservative and aggressive optimization models respectively to obtain conservative and aggressive lightweight design schemes.

[0014] In some possible embodiments, a polynomial model of the response with respect to uncertain parameters is established based on the sample point matrix and the response value matrix, including:

[0015] Based on the sample point matrix and the corresponding response value matrix, a complete quadratic polynomial of the response with respect to the uncertain parameter is established for each of the design variable sample points.

[0016] The significance of the fitted terms in the complete quadratic polynomial is verified by the t-test method. The fitted terms that meet the significance requirements are retained. The fitted terms include constant terms, linear terms, cross terms, and quadratic terms.

[0017] Based on the retained fitting terms, the least squares method is used to re-establish the polynomial model of the response with respect to the uncertain parameters, and the polynomial coefficients corresponding to the retained fitting terms are obtained.

[0018] In some possible embodiments, based on the polynomial model and the sample point matrix, a kriging model is used to determine the mapping relationship between the design variables and the polynomial coefficients in the polynomial model, thereby obtaining a hybrid surrogate model of the response, the uncertain parameters, and the design variables, including:

[0019] Using the polynomial coefficients as output and the design variable sample points as input, a Kriging model of the polynomial coefficients with respect to the design variables is established.

[0020] By nesting the Kriging model with the polynomial model, a hybrid surrogate model of the response with respect to the design variables and the uncertain parameters is obtained.

[0021] In some possible embodiments, the response range of the thermal protection system is solved in the dual approximation quantization model using the hybrid proxy model, and conservative and aggressive reliability indices are determined based on a set response safety threshold, including:

[0022] According to the hybrid proxy model, the response range of the thermal protection system is defined within the upper approximation set and the lower approximation set of the dual approximation quantization model, respectively.

[0023] Based on the response safety threshold and the response interval, the response-threshold interference interval of the upper approximation set is obtained. and the response-threshold interference interval of the lower approximation set Where the subscript i represents the i-th response;

[0024] Based on the response-threshold interference intervals of the upper approximation set and the lower approximation set, an aggressive reliability index and a conservative reliability index for the response are obtained, wherein the aggressive reliability index η 1,i and the conservative reliability index η 2,i They are respectively:

[0025]

[0026]

[0027] The reliability optimization design method for thermal protection systems based on a dual approximation quantization model provided by this invention has at least the following advantages:

[0028] (1) Compared with existing reliability analysis models, the upper and lower approximation sets based on rough set theory in this invention are used as a double approximation quantization model, which is applicable to uncertain parameter sets of any complex shape and improves the accuracy of parameter uncertainty characterization.

[0029] (2) When performing system reliability optimization design, this invention defines conservative and aggressive reliability indices based on the boundary of the response interval of the double approximation quantization model, providing two optimization design schemes for lightweight design of thermal protection systems.

[0030] (3) This invention separates design variables from uncertain parameters, establishes a polynomial model of the system response for the uncertain parameters, and determines the significant terms of the polynomial through significance analysis; it also establishes a kriging model of the polynomial coefficients for the design variables, forming a nested hybrid surrogate model. Using this hybrid surrogate model as a black-box operation for reliability optimization can reduce the number of calculations and improve computational efficiency while ensuring computational accuracy. Attached Figure Description

[0031] The accompanying drawings, which are incorporated in and form part of this specification, illustrate embodiments consistent with the invention and, together with the description, serve to explain the principles of the invention.

[0032] Figure 1 This is a flowchart of the reliability optimization design method for a thermal protection system based on a dual approximation quantization model in an embodiment of the present invention;

[0033] Figure 2 This is a simplified flowchart of the reliability optimization design method for a thermal protection system based on a dual approximation quantization model in an embodiment of the present invention.

[0034] Figure 3 This is a schematic diagram of the thermal protection system in an embodiment of the present invention. Detailed Implementation

[0035] To make the above-mentioned objectives, features, and advantages of the embodiments of the present invention more apparent and understandable, the technical solutions of the embodiments of the present invention will be clearly and completely described below with reference to the accompanying drawings. Obviously, the described embodiments are only a part of the embodiments of the present invention, and not all of them. Based on the embodiments of the present invention, all other embodiments obtained by those skilled in the art without creative effort are within the scope of protection of the present invention.

[0036] refer to Figure 1This invention provides a reliability optimization design method for a thermal protection system based on a dual approximation quantization model, comprising the following steps:

[0037] Step 1: Determine the universe of discourse for the uncertain parameters of the thermal protection system, and based on the obtained experimental data, establish a double approximation quantization model for the set of uncertain parameters with irregular boundaries within the universe of discourse.

[0038] Thermal protection systems typically have multiple uncertainties, each of which is a i The range of variation can be represented using intervals, specifically:

[0039]

[0040] Among them, a i This represents the i-th uncertain parameter. This represents the range of variation for the i-th uncertain parameter. and These are the lower and upper bounds of the variation interval, respectively, and m represents the number of uncertain parameters. All uncertain parameters are represented as a = (a1, a2, ..., a...). m ) T Therefore, the universe of discourse ψ of the uncertain parameters of the thermal protection system can be expressed as the variation range of multiple uncertain parameters. The Cartesian product, i.e., the domain ψ, can be specifically expressed as:

[0041]

[0042] The aforementioned uncertain parameters were measured experimentally, resulting in d experimental measurement points, which constitute the experimental measurement point set W = {w1, w2, ..., w...}. d}. Among them, w i =(w i1 ,w i2 ,...,w im ) T Let w represent the i-th experimental measurement point. ij This represents the value of the j-th uncertain parameter at the i-th experimental measurement point.

[0043] Sort the values ​​of the j-th uncertain parameter for each experimental measurement point in the experimental measurement point set W, remove duplicate values, and add the uncertain parameter a. j The lower bound of the range of change and the Upper Realm Obtain the uncertain parameter a j This corresponds to a set of breakpoints arranged in ascending order. Using the method described above, we can obtain m sets of breakpoints corresponding to all uncertain parameters (m uncertain parameters). These m sets of breakpoints can be represented as:

[0044]

[0045] Among them, w i,j Indicates the uncertain parameter a j The corresponding breakpoint in ascending order, index k j Indicates the uncertain parameter a j The corresponding number of breakpoints.

[0046] Based on the above set of m breakpoints, the variation interval of each uncertain parameter can be discretized into the union of multiple sub-intervals, specifically represented as:

[0047]

[0048] For each uncertain parameter, arbitrarily select one subinterval from the multiple subintervals. The Cartesian product of the selected subintervals for each uncertain parameter is taken as the equivalence class of the uncertain parameter, which can be expressed as:

[0049]

[0050] in, Represents the set of breakpoints W j The rth j One breakpoint, Indicates the uncertain parameter a j Corresponding to Let be the discrete interval with the left boundary. When When the first breakpoint is reached, the interval is... Transform into a closed interval For any elements b, c, and d in the above equivalence class, we consider them to satisfy the equivalence relation R, i.e. and

[0051] The strong correlation among the uncertain parameters in a thermal protection system causes the region characterizing these uncertain parameters to exhibit a bounded but irregular feature, i.e., irregular boundaries. This region of variation is called the set of uncertain parameters A, which is a subset of the universe of discourse. Using the aforementioned equivalence classes, upper and lower approximate sets of the set of uncertain parameters A are constructed as follows:

[0052]

[0053]

[0054] in, Let U be the upper approximate set of the uncertain parameter set A, which is composed of elements in the universe of discourse U that may belong to set A. R(A) is the lower approximation set of the uncertain parameter set A, which consists of elements whose universe of discourse U definitely belongs to set A. The upper and lower approximation sets approximate the uncertain parameter set A from both internal and external perspectives. Therefore, the upper and lower approximation sets are the double approximation quantization model of the uncertain parameter set A.

[0055] For example, a corrugated sandwich thermal protection system mainly includes an upper panel, a lower panel, a web, and a sandwich insulation layer, made of different materials, namely Inconel 718, Al2O24 alloy, Ti alloy, and Saffil fiber material, respectively. During aircraft service, the upper panel is subjected to aerodynamic heating over time, while the lower panel, located inside the fuselage, is subject to natural convection with the air. Furthermore, the corrugated sandwich structure is subject to static constraints such as… Figure 3 As shown, boundaries 1 and 2 are subject to displacement constraints along the z-axis and cannot rotate in any direction. Boundaries 3, 4, 5, and 6 are symmetrical boundaries, and the upper panel is subjected to aerodynamic loads. Taking this corrugated sandwich thermal protection system as an example, the thermal conductivity λ, thermal expansion coefficient α, elastic modulus E, and convective heat transfer coefficient h of the web are uncertain parameters of this corrugated sandwich thermal protection system. λ∈[6.4,13.2]W / (m·K), α∈[6×10⁻⁶ ... -6 1.3×10 -5 ]K -1 , E∈[96,120]GPa, h=[1,10]W / (m 2 ·K) are their respective intervals of variation, and the universe of discourse ψ of the uncertain parameter is represented as the Cartesian product of the above four intervals of variation, specifically:

[0056] ψ=[6.4,13.2]×[6×10 -6 1.3×10 -5 [96,120]×[1,10];

[0057] The aforementioned uncertain parameters were measured experimentally, resulting in the experimental measurement point set W = {w1, w2, ..., w...}. 15 Sort all experimental measurement points with respect to the j-th uncertain parameter and remove duplicate values, then add the uncertain parameter a. j The lower bound of the range of change and the Upper Realm Obtain the uncertain parameter a j This corresponds to a set of breakpoints. Taking thermal conductivity as an example, the corresponding set of breakpoints is:

[0058] W1={6.4,6.5,7.0,7.4,8.1,8.4,9.0,9.4,9.9,10.3,10.7,11.4,11.6,12.2,12.6,13.1,13.2};

[0059] The breakpoint sets for the four uncertain parameters can be obtained through the above method. These breakpoint sets for the four uncertain parameters are represented as follows:

[0060]

[0061] Based on the set of breakpoints corresponding to each uncertain parameter, the variation range of that uncertain parameter can be discretized into the union of multiple sub-intervals. Taking thermal conductivity as an example, the variation range of thermal conductivity is discretized as follows:

[0062]

[0063] For each of the multiple subintervals corresponding to the aforementioned uncertain parameters, one subinterval can be selected. Multiple uncertain parameters can result in multiple subintervals. The equivalence class is the Cartesian product of the selected subintervals, and is represented as follows:

[0064]

[0065] in, Represents the set of breakpoints W j The rth j One breakpoint, Indicates the uncertain parameter a j Corresponding to Let be the discrete interval with the left boundary. When When the first breakpoint is reached, the interval is... Transform into a closed interval Taking the first subinterval corresponding to the thermal conductivity coefficient λ, thermal expansion coefficient α, elastic modulus E, and convective heat transfer coefficient h as an example, an equivalence class is established as follows:

[0066] [a] R = [6.4, 6.5] × [6 × 10 -6 6.4×10 -6 ]×[96,96.56]×[1,1.67];

[0067] Among these, the three uncertain parameters—thermal conductivity λ, thermal expansion coefficient α, and elastic modulus E—are correlated, meaning they are correlated parameters. The convective heat transfer coefficient h is uncorrelated with these three uncertain parameters, meaning it is an independent parameter. Therefore, Boolean operations on a three-dimensional ellipsoid are used as the uncertain set of correlated parameters. The equation for ellipsoid Ω1 is:

[0068]

[0069] The equation of the ellipsoid Ω2 is:

[0070]

[0071] According to Boolean operations, the set of uncertain parameters corresponding to the three uncertain parameters—thermal conductivity coefficient λ, thermal expansion coefficient α, and elastic modulus E—is A1 = Ω1 - Ω2. The set of uncertain parameters corresponding to the convective heat transfer coefficient h is represented by the interval A2 = [2, 9]. The set of uncertain parameters A for the corrugated sandwich thermal protection system is the Cartesian product of the set of uncertain parameters A1 (related parameters) and A2 (independent parameters), i.e., A = A1 × A2. It can be understood that the set of uncertain parameters A lies within the universe of discourse and has irregular boundaries.

[0072] Using equivalence classes, we establish upper and lower approximate sets of the uncertain parameter set A of the corrugated sandwich thermal protection system, which are expressed as:

[0073]

[0074]

[0075] The upper and lower approximate sets and the set of uncertain parameters A satisfy the inclusion relationship. Therefore, the upper and lower approximation sets can be used as a double approximation quantization model for an uncertain parameter set A to quantize the uncertain parameter set A.

[0076] Step 2: Determine the design domain based on the maximum and minimum values ​​of the design variables. Use the Latin hypercube sampling method to extract sample points of uncertain parameters and design variables in the universe of discourse and the design domain respectively, forming a sample point matrix. Then, obtain the response value of the thermal protection system at each element in the sample point matrix through finite element calculation, forming a response value matrix.

[0077] First, since the optimal design of thermal protection systems typically involves multiple design variables, based on the given maximum and minimum values ​​of these multiple design variables, the values ​​of each design variable x are determined using intervals. i The representation is as follows:

[0078]

[0079] Where, x i For the i-th design variable, Let i be the minimum value of the i-th design variable. Let x be the maximum value of the i-th design variable, and n be the number of design variables. All design variables are represented as x = (x1, x2, ..., xn). n ) T The design domain Γ of a thermal protection system can be expressed as a Cartesian product of intervals of multiple design variables:

[0080]

[0081] Within the design domain Γ, r design variable sample points are randomly selected using the Latin hypercube sampling method, forming a sample point set ξ = {ξ1, ξ2, ..., ξ} for the design variables. r}.in, This represents the i-th design variable sample point. Let represent the value of the j-th design variable at the i-th sample point. Within the universe of discourse ψ of the uncertain parameters, s sample points of the uncertain parameters are randomly selected using the Latin hypercube sampling method, forming the sample point set ζ = {ζ1, ζ2, ..., ζ...}. s}.in, This represents the i-th uncertain parameter sample point. This represents the value of the j-th uncertain parameter for the i-th sample point.

[0082] Establish a sample point matrix using sample points of design variables and uncertain parameters. It is represented as:

[0083]

[0084] in, It is the i-th design variable sample point ξ i With the j-th uncertain parameter sample point ζ j Composition. Calculation using finite element method. Response value at All calculated response values Form the corresponding response value matrix

[0085] Taking the aforementioned corrugated sandwich thermal protection system as an example, the height h of the sandwich insulation layer... s The inclination angle θ of the web, and the thickness t of the upper plate. u and the thickness t of the web s For design variables. Where h s ∈[8,12]mm,θ∈[70,85]°,t u ∈[1,3]mm,t s = [1,3]mm are their intervals in sequence, and the design domain Γ is the Cartesian product of the intervals of these four design variables. The design domain Γ is specifically expressed as:

[0086] Γ=[8,12]×[70,85]×[1,3]×[1,3];

[0087] Within the aforementioned design domain Γ, 30 design variable sample points are randomly selected using the Latin hypercube sampling method, forming the sample point set ξ = {ξ1, ξ2, ..., ξ} of the design variables. 30}.in, Let ξ represent the sample point of the i-th design variable.i j This represents the value of the j-th design variable for the i-th sample point.

[0088] Within the universe of discourse ψ of the uncertain parameter, 10 sample points of the uncertain parameter are randomly selected using the Latin hypercube sampling method, forming the sample point set ζ={ζ1,ζ2,...,ζ 10}.in, This represents the i-th uncertain parameter sample point. Let represent the value of the j-th uncertain parameter at the i-th sample point. A two-dimensional sample matrix is ​​formed using the set of sample points for the design variables and uncertain parameters. Specifically:

[0089]

[0090] in It is the i-th design variable sample point ξ i With the j-th uncertain parameter sample point ζ j Composition. Obtained through finite element analysis. Response value at All calculated response values Forming a response value matrix

[0091] Step 3: Based on the sample point matrix and the response value matrix, establish a polynomial model of the response with respect to the uncertain parameters.

[0092] Specifically, based on the sample point matrix and the corresponding response value matrix, the sample points ξ of different design variables... k For (k = 1, 2, ..., r), construct a complete quadratic polynomial of the response with respect to all uncertain parameter inputs a, as expressed below:

[0093]

[0094] Among them, X i For the i-th fitting term, b i Let {b1, b2, ..., b} be the coefficients of the corresponding undetermined fitted terms, and {b1, b2, ..., b} be the coefficients of the undetermined fitted terms. o}={c0,c1,...,c mm}, where the subscript o represents the number of fitted terms, and ε represents the random error, which follows a normal distribution with a mean of 0. Substituting the s uncertain parameter sample points into the above complete quadratic polynomial and writing it in matrix form:

[0095]

[0096] The predicted value of the complete quadratic polynomial at the sample points of uncertain parameters Compared with the true response value f(ξ)k ,ζ i The error function for (i = 1, 2, ..., s) is defined as:

[0097]

[0098] With error function Q k To minimize the optimization objective, the least squares method is used to determine the coefficients b of the undetermined fitting term of the complete quadratic polynomial. i Furthermore, the variance estimator of the random error ε is:

[0099]

[0100] Undetermined fitting term coefficients b i It follows a t-distribution with (sq-1) degrees of freedom, i.e.:

[0101]

[0102] Among them, D ii For matrix (X) T X) -1 The main diagonal element, B i The coefficients are those of the true complete quadratic polynomial. Latin hypercube sampling is used to additionally extract multiple uncertain parameter sample points in the universe of discourse for significance analysis. These uncertain parameter sample points used for significance analysis are not identical (or not completely identical) to the uncertain parameter sample points used to form the sample point matrix, and the number of uncertain parameter sample points used for significance analysis is greater than the number of fitted terms of the complete quadratic polynomial. This is to verify the effect of the coefficients of the linear, intersection, and quadratic terms on the response. Based on the significance level, the following hypothesis is proposed:

[0103] Null hypothesis: B i =0; Alternative hypothesis: B i ≠0; Under the condition that the null hypothesis is true, the t-statistic is calculated as follows: Find the corresponding probability P from the t-distribution table. Given the significance level α. t If P < α t If the null hypothesis is rejected, meaning the true polynomial coefficients are not zero, then the corresponding fitted term is considered to have good fit to the response. It is significant. Therefore, based on the significance test results, we retain l highly significant fitted terms, re-establish the polynomial model using the least squares method, and construct the polynomial coefficient matrix C = (C1, C2, ..., C...). l ), where C i =(C i,1 C i,2 ,...,C i,r ) TThis is to design the i-th polynomial coefficient vector corresponding to the sample points of the variable.

[0104] Based on the example of 30 design variable sample points and 10 uncertain parameter sample points mentioned above, and according to the sample point matrix and its corresponding response value matrix, under different design variables ξ k Given (k = 1, 2, ..., 30), establish a response with respect to all uncertain parameters input a = (a1, a2, a3, a4). T The expression for a complete quadratic polynomial is as follows:

[0105]

[0106] Among them, X i For the i-th fitting term, b i Let {b1, b2, ..., b} be the coefficients of the corresponding undetermined fitted terms, and {b1, b2, ..., b} be the coefficients of the undetermined fitted terms. 15}={c0,c1,...,c 44}, where ε is the random error, which follows a normal distribution with a mean of 0.

[0107] The maximum displacement response F(x,a) on the upper surface of the corrugated sandwich thermal protection system is used as an example for illustration. To ensure the accuracy of the significance analysis, the design variable x = (9,82,2,2.6). T Under the given conditions, 25 additional sample points of uncertain parameters were drawn from the universe of discourse for significance analysis only. The corresponding coefficients of the undetermined fitted terms and the p-values ​​of the t-significance test are shown in the table below:

[0108] Table 1. Coefficients of the polynomial fit term of the displacement response with respect to the uncertain parameters and the p-value of the significance test.

[0109]

[0110] When the significance level α t When = 0.05, based on the verification results, retain {constant term, λ, h, λ×α, E×h, λ}. 2 ,α 2 ,h 2 A total of 8 terms are used as fitting terms for the subsequent polynomial model.

[0111] Based on the sample point matrix and its corresponding response value matrix, the sample points ξ under different design variables... k Under the condition (k = 1, 2, ..., 30), the least squares method is used again to fit the polynomial model composed of the above 8 fitting terms, and the polynomial coefficients are obtained. The 30 design variable sample points correspond to 30 sets of polynomial coefficients, and a polynomial coefficient matrix C = (C1, C2, ..., C8) is constructed using these coefficients. Where C... i =(C i,1 Ci,2 ,...,C i,30 ) T This is the i-th coefficient vector corresponding to the sample points of the design variables. Furthermore, the same steps are used to obtain the corresponding coefficient matrices for the temperature response and stress response.

[0112] Step 4: Based on the polynomial model and the sample point matrix, use the Kriging model to determine the mapping relationship between the design variables and the polynomial coefficients in the polynomial model, and obtain a hybrid surrogate model of the response, uncertain parameters and design variables.

[0113] Specifically, based on different design variable sample points ξ k The coefficient matrix C = (C1, C2, ..., Cn) of the polynomial model under (k = 1, 2, ..., r) l Input any design variable x = (x1, x2, ..., x...) n ) T The corresponding polynomial coefficients C j (x) can be represented in the following Kriging model form:

[0114]

[0115] Where g(x)=(g1(x),g2(x),...,g p (x)) T Let p be a vector formed by p regression functions of x, such as a linear regression function where p = n + 1, and g1(x), g2(x), ..., g2(x). p (x))=(1,x1,...x n ), where n is the number of design variables, x j Let β represent the j-th design variable value of x, where β = (β1, β2, ..., β). p ) T Let z(x) represent the parameters of the regression function, where the mean is 0 and the variance is σ. 2 The stochastic process. Given any two design variables x within the design domain Γ. i and x j Define z(x) i ) and z(x j The covariance between them is:

[0116]

[0117] in, Represents a vector with correlation coefficients The relevant function is often based on the Gaussian model, and its expression is:

[0118]

[0119] in, and x i and x j The value of the k-th design variable, Correlation coefficient The component of the k-th design variable.

[0120] Polynomial coefficients caused by arbitrary design variable input x It can be a set of sample points ξ containing known design variables. k C caused by (k=1,2,...,r) j,k The linear weighting is thus established as follows:

[0121]

[0122] Where η=(η1,η2,...,η r ) T G represents the weight coefficient vector, and is an r×p matrix. ij =g j (ξ i ), Z=(z(ξ1),z(ξ2),...,z(ξ r )) T Let x be the error vector at the sample points of the design variable. Therefore, the prediction error at any design variable input x is calculated as follows:

[0123]

[0124] Among them, G T η is the basis function at the input x of the design variable, obtained by weighted summation of the sample points of the design variable. To ensure that the predicted value satisfies the unbiased estimation, the following equation relationship is obtained:

[0125] G T η = g(x);

[0126] Therefore, the mean square error at the design variable input x is calculated as follows:

[0127]

[0128] Represents the correlation coefficient matrix of a stochastic process among sample points of design variables. The design variable sample point set ξ = {ξ1, ξ2, ..., ξ3} represents the set of sample points. r The correlation coefficient matrix between η and any design variable input x within the design domain Γ is the stochastic process. The model prediction accuracy, i.e., minimizing the mean square error at the design variable input x, can be transformed into solving for the minimum mean square error with η as the independent variable. The optimization problem ultimately yields a polynomial coefficient mapping model for any design variable input x:

[0129]

[0130] Similarly, we can obtain the coefficients of all polynomials {C1(x), C2(x), ... C...} l The Kriging model of design variable x is used. Finally, an embedded hybrid proxy model is established using a multinomial model and a Kriging model, with the following expression:

[0131]

[0132] Among them, the complete quadratic polynomial contains an invalid term, namely c. i or c ij It could be 0, X (i) This is the i-th retained term after the significance analysis, which is only related to the uncertain parameter a. At this point, the embedded hybrid surrogate model of the thermal protection system response with respect to the design variables and the uncertain parameter is complete. This hybrid surrogate model can predict the response for any set of inputs (x, a) within both the uncertain domain and the design domain.

[0133] For example, based on 30 design variable sample points ξ k The coefficient matrix C = (C1, C2, ..., C8) of the polynomial model under (k = 1, 2, ..., 30) is given by the design variable input x = (h s ,θ,t u ,t s ) T The coefficients C1(x) of the first retained constant term are explained, and their relationship is expressed in the following Kriging model form:

[0134]

[0135] Where g(x) = (g1(x), g2(x), ..., g5(x)) T =(1,h) s ,θ,t u ,t s Let β = (β1, β2, ..., β5) be the vector formed by five linear regression functions of x. T Let z(x) represent the parameters of the regression function, where the mean is 0 and the variance is σ. 2 The stochastic process. Given any two design variables x within the design domain Γ. i and x j Define z(x) i ) and z(x j The covariance between them is:

[0136]

[0137] in, Represents a vector with correlation coefficients The relevant function is often based on the Gaussian model, and its expression is:

[0138]

[0139] in, and x i and x j The value of the k-th design variable, Correlation coefficient The component of the k-th design variable.

[0140] Polynomial coefficients caused by arbitrary design variable input x It can be viewed as a set of sample points ξ containing known design variables. k Caused by C 1,k The linear weighting of (k = 1, 2, ..., 30) leads to the following expression:

[0141]

[0142] C1=(C 1,1 C 1,2 ,...,C 1,30 ) T Let η be the first coefficient vector corresponding to the sample points of the design variable, where η = (η1, η2, ..., η). 30 ) T Let G be an r×p matrix representing the weight coefficient vector. ij =g j (ξ i )(i=1,...,30; j=1,...,5). Z=(z(ξ1),z(ξ2),...,z(ξ 30 )) T Let x be the error vector at the sample points of the design variable. Therefore, the prediction error at any design variable input x is calculated as follows:

[0143]

[0144] Among them, G T η is the basis function at the input x of the design variable, obtained by weighted summation of the sample points of the design variable. To ensure that the predicted value satisfies the unbiased estimation, the following equation relationship is obtained:

[0145] G T η = g(x);

[0146] Therefore, the mean square error at the design variable input x is calculated as follows:

[0147]

[0148] in Represents the correlation coefficient matrix of a stochastic process among sample points of design variables. The design variable sample point set ξ = {ξ1, ξ2, ..., ξ3} represents the set of sample points. 30 The correlation coefficient matrix between η and any design variable input x within the design domain Γ is given. Solve for the minimum mean square error. The optimization problem allows us to obtain the polynomial coefficients for any design variable input x. Mapping model:

[0149]

[0150] Similarly, we can obtain the Kriging model for all polynomial coefficients {C1(x), C2(x), ... C8(x)} with respect to the design variable x. Finally, we establish an embedded hybrid proxy model using the polynomial model and the Kriging model, with the following expression:

[0151]

[0152] Among them, {X (1) ,X (2) ,...X (8)}={constant term,λ,h,λ×α,E×h,λ 2 ,α 2 ,h 2 It is only related to the uncertain parameters. At this point, the embedded hybrid surrogate model of the thermal protection system response with respect to design variables and uncertain parameters is complete. This hybrid surrogate model can predict the system response for any set of inputs (x, a) within both the uncertain domain and the design domain.

[0153] Step 5: Using a hybrid proxy model, calculate the response range of the thermal protection system in a dual approximation quantization model, and determine the conservative and aggressive reliability indices based on the set response safety threshold.

[0154] Specifically, assuming the thermal protection system has q responses, using the established hybrid surrogate model, for any set of design variable values ​​within the design domain, the response intervals for the q responses under the upper and lower approximation sets can be calculated:

[0155]

[0156]

[0157] in, The set of response intervals obtained within the above approximate set. F (A) represents the set of response intervals obtained within the lower approximation set, where subscripts '1, 2, ..., q' indicate the order of the response intervals, and superscripts 'L' and 'U' represent the lower and upper bounds of the response intervals, respectively. Based on the given system response safety threshold {δ1, δ2, ..., δ...} q The response-threshold interference interval obtained within the upper and lower approximate sets is expressed as follows:

[0158]

[0159]

[0160] in, The response-threshold interference interval is obtained within the above approximation set. H (A) represents the response-threshold interference interval obtained within the lower approximation set. Taking the i-th response of the upper approximation set as an example, the boundary calculation formula for the response-threshold interference interval is as follows:

[0161]

[0162]

[0163] Meanwhile, assume that the response-threshold interference interval obtained within the original set of uncertain parameters A is:

[0164]

[0165] For the i-th response, the response-threshold interference interval boundaries obtained within the uncertain parameter set A and the upper and lower approximation sets satisfy the following inequality relationship:

[0166]

[0167] The radical reliability index η of the i-th response is respectively... 1,i and conservative reliability index η 2,i Defined as:

[0168]

[0169]

[0170] The two reliability indices satisfy the relationship η 1,i ≥η 2,i (i = 1, 2, ..., q).

[0171] In the example of the corrugated sandwich thermal protection system, the maximum temperature of the lower panel, the maximum displacement of the upper panel, and the maximum stress of the web are selected as the response, with a set of design variables {h} within the design domain Γ. s ,θ,t u ,t s Two types of reliability indices are defined and calculated for the set {10mm, 80°, 2mm, 2mm}. Using a hybrid surrogate model, the response intervals of the upper and lower approximation sets with respect to the three system responses are obtained within the upper and lower approximation sets:

[0172]

[0173] F (A)={[330.61,367.24],[1.78,2.16],[0.79,0.95]};

[0174] Based on the given response security threshold {δ T ,δ D ,δ S}={353K,2mm,1GPa}, the response-threshold interference interval obtained within the upper and lower approximate sets is:

[0175]

[0176] H (A)={[-14.24,22.39],[-0.15,0.20],[0.0423,0.1890]};

[0177] Taking temperature response as an example, the calculation results can verify that the boundary values ​​of the response-threshold interference interval obtained within the upper and lower approximation sets satisfy the following inequality relationship:

[0178] -19.59≤-14.24≤22.39≤24.66;

[0179] Based on the obtained response-threshold interference range and H (A) will use the aggressive reliability index η of the response. 1,i and conservative reliability η 2,i Defined as:

[0180]

[0181]

[0182] Specifically, the aggressive and conservative reliability indices for temperature response, displacement response, and stress response were calculated separately, and the results are as follows:

[0183] η 1,i ={η1,T ,η 1,D ,η 1,S}={0.6339,0.6374,1};

[0184] η 2,i ={η 2,T ,η 2,D ,η 2,S}={0.5335,0.4710,1};

[0185] The results can verify η 1,i ≥η 2,i (i = 1, 2, 3).

[0186] Step Six: Using conservative and aggressive reliability indices as constraints and the quality of the thermal protection system as the optimization objective, establish corresponding conservative and aggressive optimization models, and optimize the conservative and aggressive optimization models respectively to obtain conservative and aggressive lightweight design schemes.

[0187] Specifically, the goal is to achieve lightweight design of the corrugated sandwich thermal protection system, i.e., to optimize the quality of the thermal protection system, with the objective function being M. Since different responses have different requirements for reliability indicators, the threshold values ​​for the reliability indicators are determined to be {λ1, λ2, ..., λ...}. q Using a radical reliability index as a constraint, a corresponding radical optimization model is established. This radical optimization model is as follows:

[0188] min M

[0189] stη 1,i ≤λ i (i = 1, 2, ..., q);

[0190] x∈Γ

[0191] Simultaneously, using a conservative reliability index as a constraint, a corresponding conservative optimization model is established. This conservative optimization model is as follows:

[0192] min M

[0193] stη 2,i ≤λ i (i = 1, 2, ..., q);

[0194] x∈Γ

[0195] The two optimization models were solved using optimization algorithms, resulting in two lightweight design schemes for the corrugated sandwich thermal protection system: a conservative scheme and an aggressive scheme.

[0196] Based on the examples above of obtaining aggressive and conservative reliability indices for temperature response, displacement response, and stress response, the areal density ρ of the corrugated sandwich thermal protection system is used as an example. area The objective function is:

[0197]

[0198] Where, ρ l ,ρ u ,ρ s ,ρ w The material densities of the bottom panel, top panel, sandwich insulation layer, and web, in order, are as follows: t l ,t u ,h s ,t s The thicknesses are, in order: bottom panel thickness, top panel thickness, core insulation layer height, and web thickness, where θ is the inclination angle of the web. Figure 3 As shown.

[0199] Because temperature response, displacement response, and stress response have different requirements for reliability indicators, it is assumed that the reliability indicator thresholds for the three responses are {λ}. T ,λ D ,λ S}={0.85,0.9,0.95}. The aggressive reliability optimization model, constrained by the aggressive reliability index, is as follows:

[0200] min ρ area

[0201] stη 1,T ≤λ T η 1,D ≤λ D η 1,S ≤λ S ;

[0202] x∈Γ

[0203] An optimization algorithm is used to optimize and solve the radical reliability optimization model, and the radical reliability optimization results are obtained (see Table 2), thereby obtaining a radical lightweight design scheme.

[0204] Table 2 Results of Aggressive Reliability Optimization

[0205] Optimize variables <![CDATA[h s =11.0960mm;θ=85°;t u =1mm;t s =1mm;]]> Reliability constraints <![CDATA[η 1,T =1;h 1,D =0.9267;h 1,S =0.9503;]]>

[0206] The conservative reliability optimization model, constrained by a conservative reliability index, is as follows:

[0207] min ρ area

[0208] stη2,T ≤λ T η 2,D ≤λ D η 2,S ≤λ S ;

[0209] x∈Γ

[0210] An optimization algorithm is used to optimize and solve the conservative reliability optimization model, and the conservative reliability optimization results are obtained (see Table 3), thereby obtaining a conservative lightweight design scheme.

[0211] Table 3. Conservative Reliability Optimization Results

[0212] Optimize variables <![CDATA[h s =11.7268mm;θ=84.98°;t u =1mm;t s =1mm;]]> Reliability constraints <![CDATA[η 2,T =1;h 2,D =0.9489;h 2,S =0.9501;]]>

[0213] Both optimization results can be used as the final design scheme for the corrugated sandwich thermal protection system. The aggressive reliability optimization design has higher load-bearing efficiency, but it carries a certain risk of failure compared to the conservative reliability optimization design. The conservative reliability optimization result has some design redundancy, but it offers higher reliability and safety compared to the aggressive reliability optimization design.

[0214] In summary, this invention fully considers the irregular boundary characteristics of the uncertain parameter set caused by the strong correlation between uncertain parameters, and uses a double approximation quantization model to quantify the uncertain set. Using conservative and aggressive reliability indices as constraints, a hybrid surrogate model is used to optimize the reliability design of the thermal protection system, ultimately resulting in two lightweight design schemes. While ensuring design safety, this effectively improves the structural efficiency and computational efficiency of reliability optimization in the thermal protection system.

[0215] Other embodiments of this application will readily occur to those skilled in the art upon consideration of the specification and practice of the invention disclosed herein. This application is intended to cover any variations, uses, or adaptations of this application that follow the general principles of this application and include common knowledge or customary techniques in the art not disclosed herein. The specification and examples are to be considered exemplary only, and the true scope and spirit of this application are indicated by the following claims.

[0216] It should be understood that this application is not limited to the precise structure described above and shown in the accompanying drawings, and various modifications and changes can be made without departing from its scope. The scope of this application is limited only by the appended claims.

Claims

1. A reliability optimization design method for a thermal protection system based on a dual approximation quantization model, characterized in that, include: Step 1: Determine the universe of discourse for the uncertain parameters of the thermal protection system, and based on the obtained experimental data, establish a double approximation quantization model for the set of uncertain parameters with irregular boundaries within the universe of discourse; Step 2: Determine the design domain based on the maximum and minimum values ​​of the design variables. Use the Latin hypercube sampling method to sample uncertain parameter points and design variable points in the domain of discourse and the design domain respectively to form a sample point matrix. Then, obtain the response value of the thermal protection system at each element in the sample point matrix through finite element calculation to form a response value matrix. Step 3: Based on the sample point matrix and the response value matrix, establish a polynomial model of the response with respect to the uncertain parameters; Step 4: Based on the polynomial model and the sample point matrix, use the Kriging model to determine the mapping relationship between the design variables and the polynomial coefficients in the polynomial model, and obtain the hybrid surrogate model of the response, the uncertain parameters, and the design variables; Step 5: Using the hybrid proxy model, solve the response range of the thermal protection system in the double approximation quantization model, and determine the conservative and aggressive reliability indices based on the set response safety threshold. Step Six: Using the conservative and aggressive reliability indices as constraints and the quality of the thermal protection system as the optimization objective, establish corresponding conservative and aggressive optimization models, and optimize the conservative and aggressive optimization models respectively to obtain conservative and aggressive lightweight design schemes.

2. The reliability optimization design method for a thermal protection system based on a dual approximation quantization model according to claim 1, characterized in that, Based on the sample point matrix and the response value matrix, a polynomial model of the response with respect to uncertain parameters is established, including: Based on the sample point matrix and the corresponding response value matrix, a complete quadratic polynomial of the response with respect to the uncertain parameter is established for each of the design variable sample points. The significance of the fitted terms in the complete quadratic polynomial is verified by the t-test method. The fitted terms that meet the significance requirements are retained. The fitted terms include constant terms, linear terms, cross terms, and quadratic terms. Based on the retained fitting terms, the least squares method is used to re-establish the polynomial model of the response with respect to the uncertain parameters, and the polynomial coefficients corresponding to the retained fitting terms are obtained.

3. The reliability optimization design method for a thermal protection system based on a dual approximation quantization model according to claim 2, characterized in that, Based on the polynomial model and the sample point matrix, the mapping relationship between the design variables and the polynomial coefficients in the polynomial model is determined using a Kriging model, thereby obtaining a hybrid surrogate model of the response, the uncertain parameters, and the design variables, including: Using the polynomial coefficients as output and the design variable sample points as input, a Kriging model of the polynomial coefficients with respect to the design variables is established. By nesting the Kriging model with the polynomial model, a hybrid surrogate model of the response with respect to the design variables and the uncertain parameters is obtained.

4. The reliability optimization design method for a thermal protection system based on a dual approximation quantization model according to claim 3, characterized in that, Using the hybrid proxy model, the response range of the thermal protection system is solved in the dual approximation quantization model, and conservative and aggressive reliability indices are determined based on the set response safety threshold, including: Based on the hybrid proxy model, the response range of the thermal protection system is calculated within the upper approximation set and the lower approximation set in the dual approximation quantization model, respectively. Based on the response safety threshold and the response interval, the response-threshold interference interval of the upper approximation set is obtained. and the response-threshold interference interval of the lower approximation set Where the subscript i represents the i-th response; Based on the response-threshold interference intervals of the upper approximation set and the lower approximation set, an aggressive reliability index and a conservative reliability index for the response are obtained, wherein the aggressive reliability index η 1,i and the conservative reliability index η 2,i They are respectively:

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