An approximate hessian matrix based multipoint approximation method for structural optimization

Abstract

The application provides a multi-point approximation method based on an approximate Hessian matrix for structure optimization, comprising the following steps: step 1, establishing a finite element model of a spacecraft based on an initial design of a spacecraft structure; step 2, performing mechanical prediction of the spacecraft, and the analysis mode comprises static analysis, modal analysis, buckling analysis and transient response; step 3, establishing a structure optimization mathematical model, namely, defining an optimization target, a constraint function and a design variable; step 4, establishing a sequence approximation problem considering second-order information to approximate the original problem; step 5, estimating the Hessian matrix of the multi-point approximation function by using a quasi-Newton method; step 6, selecting known points by using a neighborhood selection strategy based on the Euclidean distance; step 7, establishing convergence discrimination and an automatic optimization system; step 8, method comparison, analysis and iteration explicitness; and setting an optimization example to illustrate the flow from spacecraft design to analysis and then to optimization, and the effectiveness is verified by comparing the results of the optimization method with those of a commercial software.

Description

Technical Field

[0001] This invention relates to a multi-point approximation method based on the approximate Hessian matrix for structural optimization. The method is aimed at complex spacecraft structures composed of rods, beams, plates, shells and their combinations, and belongs to the field of spacecraft structure and mechanism design. Background Technology

[0002] Spacecraft structural design is a critical aspect of ensuring the success of space missions. It requires structures to meet the demands of the extreme space environment while also being lightweight enough to reduce launch costs. In particular, advanced optimization design techniques are needed to improve the performance of spacecraft structures in terms of lightweighting and adaptability to multiple environments.

[0003] Structural optimization design typically involves complex nonlinear behavior, multidisciplinary coupling, and a large number of design variables and complex constraints, making the solution of optimization problems challenging. With technological advancements and increasing performance requirements for spacecraft, traditional dimensional optimization design methods, especially first-order methods integrated into commercial software, are gradually failing to meet the needs of modern space missions. For spacecraft structures with large finite element scales, optimization iterations can lead to a large number of structural analyses, ultimately resulting in excessively long optimization times. Therefore, there is an urgent need for an efficient optimization design method (or optimization system) to improve the efficiency of spacecraft structural optimization.

[0004] This invention proposes a multi-point approximation method for structural optimization based on an approximate Hessian matrix. This method, for the first time, introduces approximate second-order information into the construction of the multi-point approximation function. Specifically, it uses a quasi-Newton method to calculate the Hessian matrix of the multi-point approximation function, thereby enhancing approximation accuracy and improving the efficiency of dimensional optimization design for complex structures. Furthermore, it proposes a neighborhood selection strategy based on Euclidean distance to select known points for constructing the approximation function. Finally, based on a sequential approximation optimization strategy, a spacecraft dimensional optimization program, jointly developed using Nastran and Matlab, is applied to solve dimensional optimization design problems for spacecraft structures such as space trusses and satellites under typical design conditions. Summary of the Invention

[0005] This invention addresses the problem of excessively long structural analysis time and low optimization efficiency caused by the large number of iterations in traditional size optimization methods for optimizing large aerospace structures. It proposes a novel multi-point approximation method for spacecraft structural optimization design. It is important to emphasize that the finite element model is a numerical representation established to accurately predict the physical behavior of a structure during design and analysis. The finite element model of the spacecraft structure described in this invention consists of rods, beams, plates, shells, and solid elements. The optimization object is the properties of the rod, beam, plate, and shell elements of the model, such as cross-sectional dimensions, plate thickness, and node coordinates.

[0006] The optimization method includes the following steps:

[0007] Step 1: Establish a finite element model of the spacecraft based on the initial design of the spacecraft structure.

[0008] Step 2: Perform mechanical prediction of the spacecraft under typical operating conditions. The analysis methods include static analysis, modal analysis, buckling analysis, transient response, etc.

[0009] Mechanical analysis is required to obtain the structural mechanical response values, which are then used to clarify the current state of the structural design and establish the objective function or constraints in the optimization problem. Static analysis refers to calculating the stress, strain, and displacement of a spacecraft structure under static loads. The fundamental finite element equations for solving this problem are:

[0010] KU=P

[0011] Where K is the global stiffness matrix, U is the nodal displacement vector, and P is the external load vector. The element stress and other response values ​​can be further obtained using the stress-strain relationship of the material and interpolation formulas.

[0012] Modal analysis is used to determine the natural frequencies and mode shapes of a structure, which is crucial for avoiding response resonance between the satellite and the launch vehicle. The fundamental finite element equations for solving this problem are:

[0013] KΦ-ω 2 MΦ=0

[0014] Where Φ is a matrix containing mode shape vectors, and ω is an eigenvalue (the square root of the natural frequency).

[0015] Linear buckling analysis is a type of analysis used to evaluate the buckling behavior of a structure under static loads. The buckling load factor and mode shape obtained from this analysis can be used to assess the deformation mode during structural buckling. The fundamental finite element equations solved are:

[0016] KU-λK u U=0

[0017] Among them, K u It is the geometric stiffness matrix of the structure under initial load, which is usually related to the nodal displacement vector U, and λ is the linear buckling load factor.

[0018] Transient response analysis is used to determine the dynamic response of a structure under time-varying loads. This analysis considers the variation of the load over time, as well as the inertia and damping characteristics of the structure; the fundamental finite element equations obtained are not given hereafter. Based on the above analysis, the mechanical response values ​​of interest can be obtained, which can be used to evaluate the rationality of the structural design and to establish subsequent optimization models.

[0019] Step 3: Establish a mathematical model for structural optimization, that is, clarify the optimization objective, constraint functions, and design variables.

[0020] The mathematical description of the primal problem of spacecraft structure optimization with respect to continuous variables is as follows:

[0021]

[0022] In the formula, X represents a continuously variable dimensional variable related to the structural member (such as the cross-sectional area of ​​a member, the thickness of a plate, and the coordinates of nodes), x k For each element, m is the number of continuous variables; These represent the upper and lower limits of the variable, respectively; f(X) is the structural quality, which is the objective function for optimization design; g j (X) represents the j-th structural morphological constraint, which is an implicit function of the design variables, such as stress constraints, displacement constraints, natural frequency constraints, and stability constraints. J0 is the number of constraints. To eliminate the influence of the response magnitude, the constraints corresponding to the structural response are normalized.

[0023] Step 4: Establish a sequence approximation problem that considers second-order information to approximate the original problem.

[0024] A multi-point approximation function is used, and intermediate variables are selected. We establish a sequential explicit approximation problem for the original optimization problem. The approximation problem for the p-th iteration is formulated as follows:

[0025]

[0026] In the formula, f (p) (X) and Let w be the objective and constraint functions established using the multi-point approximation technique in the p-th iteration, respectively, and uniformly represented by the function w. (p) (X) represents the number of critical constraints that are retained; J1 is the number of critical constraints that are retained. and It is x k At this point, the upper and lower limits, and It is x k The motion limit at the p-th iteration is given, and the width of the motion limit for each design variable at the current point is generally taken as 20% of the width of the variable interval for each variable. Using the accumulated information of known points, the following approximate function containing second-order derivative information for variable X is established using the multi-point approximation method considering approximate second-order information proposed in this invention:

[0027]

[0028] In the formula, H is the number of known points, and its upper limit is H. max If the number of known points is greater than H maxThen only the last H is retained. max Known points; X t For the t-th known design point; intermediate variables Y t (X t ) represents the value of the intermediate variable at a known point; w (p) (X) is the objective or constraint function established using multi-point approximation techniques in the p-th iteration; w(X) t () is the function value of the objective or constraint function; h is the Hessian matrix of the t-th known point at stage p; t (X) is the weighting function; It is an approximate function that contains first-order information about known points. t (X) and The calculation method is as follows:

[0029]

[0030]

[0031] In the formula, w(X) t )and These are the original function at a known point X. t The function value and partial derivatives at the given point are obtained through a semi-analytic method. X s And l is a substitution symbol in computation, with no actual meaning; r t It is an adaptive parameter that controls the degree of nonlinearity of the approximation function, obtained by solving the following least squares problem:

[0032]

[0033] In the formula, r t L ,r t U r t The upper and lower limits, generally, take r. t0 =-1,r t L =-5,r t U =5.

[0034] As the structural optimization process proceeds, the adaptive parameter r needs to be adjusted. tGiven the complexity of least squares problems after considering second-order information, this invention utilizes an interior-point algorithm for solution. The interior-point algorithm uses points within the feasible region to avoid handling boundary conditions and employs a barrier function to ensure the feasibility of the solution. The original optimization problem is explicitly transformed into an approximation problem using an approximation function. This approximation problem is a nonlinear constrained programming problem, which is also solved using the interior-point algorithm, which is effective for solving complex constraints and large-scale optimization problems.

[0035] Step 5: Estimate the Hessian matrix of the multi-point approximation function using the quasi-Newton method.

[0036] Specifically, using To approximate Each iteration only needs to use the function value and first-order information to make corrections. initial matrix In the absence of a better estimate, the identity matrix is ​​usually sufficient. It should be noted that the Hessian matrix is ​​only corrected for the current point (i.e., the H-th known design point). Immediately (When t = 1, 2, ..., H-1 and H ≥ 2). Based on the quasi-Newton conditions, the BFGS correction formula for the H-th known point is derived as follows:

[0037]

[0038] In the formula, and These are the approximate Hessian matrices for the H-th known design point in stage p and stage p-1, respectively; s (p -1) y represents the increment of the intermediate variable at adjacent iteration points; (p-1) The gradient change of adjacent iteration points with respect to intermediate variables has the following form:

[0039] s (p-1) =Y H (X (p) )-Y H-1 (X (p-1) )

[0040]

[0041] In the formula, X (p) Let r be the initial point of the p-th iteration, which is also the secondary advantage obtained in the (p-1)-th iteration; H and r H-1These are the adaptive parameters for the H-th and H-1-th known points, respectively; the conventions for other parameters are the same as before. Quasi-Newton correction is suitable for approximating the second derivative of frequency constraints (or eigenvalues) and nodal displacements. Since stress constraints are often accompanied by a large number of finite element stress response values, when constraint deletion is used, the finite element stress response values ​​may be retained or deleted in adjacent iterations. Therefore, stress constraints cannot be approximated using this method to approximate the Hessian matrix. In practice, the stress constraints can be made... It is a zero matrix.

[0042] Step 6: Select known points using a neighborhood selection strategy based on Euclidean distance.

[0043] As the iteration progresses, the number of known points will gradually increase, requiring consideration of which point information is more reasonable for constructing a multi-point approximation function. First, it is necessary to set an upper limit H for the number of known points to be selected. max This is because using too many known points can introduce errors into the calculation of nonlinear adaptive parameters, and the reference value of information between relatively distant design points is limited. Therefore, this method proposes a neighborhood point selection strategy based on Euclidean distance. First, the concept of a boundary circle is introduced. For example... Figure 1 Example of an iterative process shown, X (9) This is the latest secondary advantage. Take the latest H. max =Construct a new approximate function using 5 points (i.e., select X) (5) ~X (9) ), where the distance X (9) The farthest point is X. (6) Therefore, establish with X (9) Let X be the center of the circle. (6) The boundary circle has a radius equal to the Euclidean distance. Clearly, the previous iteration point X... (2) X (4) Also within the boundary circle, a reasonable suggestion is to use X. (2) Replace X (6) We construct a multi-point approximation function because points closer to the current optimum are more valuable for reference.

[0044] For optimization problems involving 3-dimensional and higher-dimensional design variables, the boundary circle can be generalized to a 3-dimensional and higher-dimensional sphere. Based on the above description and analysis, the method for selecting known points in this invention is: H≤H max When, add the known points to be selected in sequence; H≥H max When doing so, follow these steps:

[0045] (1) Let the latest point be denoted as X. (p) Select the five newest known points to establish a set A of known points;

[0046] (2) Calculate the maximum envelope radius X t ∈A, and establish the boundary sphere and distance X (p) The farthest point is denoted as X. t,max (This point lies on the boundary sphere);

[0047] (3) Determine whether there are other known points within the sphere that do not belong to A. If they exist, denote the set of other known points as B and proceed to step (4); if they do not exist, proceed to step (5).

[0048] (4) Find the distance X in B. (p) The nearest known point is added to set A while X is added to set A. t,max Remove and proceed to step (2);

[0049] (5) Once all known points have been selected, output set A.

[0050] In the iterative process of the sequence approximation problem (step 4), the Hessian matrix of the multi-point approximation function is estimated using the quasi-Newton method (step 5), and known points are selected based on the neighborhood selection strategy using Euclidean distance (step 6).

[0051] Step 7: Convergence determination and establishment of automatic optimization system.

[0052] To improve the efficiency of spacecraft structural optimization design, the entire optimization iteration must be automated. Iteration convergence is considered achieved when the difference / relative difference of the objective function between adjacent iterations is less than the tolerance value, and all constraints are met. The operational logic of the optimization system is as follows: initial finite element modeling of the structure is completed in Patran, and Nastran is invoked for analysis by submitting a BDF file; the main program is built in Matlab, including modules for convergence determination, approximation function establishment, optimizer selection, and visualization of optimization results; additionally, an external file is needed to store the iteration results. Figure 2 The development framework for optimizing the platform is presented. Below are some core commands or functions in the program system, with detailed explanations of their input, output, and calling relationships.

[0053] (1)[optx,fval]=fmincon(@objfun,x0,[],[],[],[],move_lower,move_upper,@constraint,options);

[0054] This function is used for optimization of approximation problems, and defaults to the interior-point algorithm. Inputs include handle functions `@objfun` and `@constraint`, which are the objective and constraint functions, respectively, established by the multi-point approximation method proposed in this invention; `x0` is the initial value of the design variables; `move_lower` is the lower bound vector of the variable movement limits; `move_upper` is the upper bound vector of the variable movement limits; and `option` is a structure used to define the convergence tolerance or single-step display content. In the outputs, `optx` is the optimal design variable for the current approximation problem, and `fval` is the optimal objective function value for the current approximation problem. In each iteration of the approximation problem, the movement limit values ​​need to be updated and this function called.

[0055] (2)[rsp(LOOP,:),drsp(LOOP,:,:)]=Sensitivity_anaysis(optx,lower_limit,upper_limit);

[0056] After obtaining the optimized values, this function needs to be called for structural analysis. In the input, optx is the optimal design variable for the current approximation problem (as above). In the output, LOOP is the iteration number; rsp(LOOP,:) is the actual structural response value obtained after the LOOP-th iteration of the approximation problem optimization; similarly, drsp(LOOP,:,:) is the sensitivity value. An example of the call statement for commercial software is system('D:\MSCsoftware\Nastran2018\bin\nastran.exe"',bdf_name,'.bdf'],'-echo'), where the variable bdf_name is the file name.

[0057] (3)[ap,ap_dg]=Approximate_Function(in_x,x,g,dg,r);

[0058] The core of this function is the multi-point approximation method based on the approximate Hessian matrix proposed in this invention, used to establish approximate functions for the objective function and constraint functions. In the input, in_x represents the design variables in the optimization process within the approximation problem; x is a matrix containing the design variable values ​​at known points; g is a vector containing the constraint function values ​​at known points; dg is a matrix containing the sensitivity of the constraint function to the design variables at known points; and r stores the nonlinear adaptive parameter values ​​in the multi-point approximation function. In the output, ap is the objective / constraint function value obtained based on the approximation function when the design variable takes the value in_x; ap_dg is the approximate sensitivity value. By calling this function, the structural analysis program is avoided in each iteration of the optimization process, thereby reducing the overall optimization time and improving design efficiency.

[0059] Step 8: Method Comparison, Analysis, and Iteration Explicitization. An optimization example is set up to illustrate the general process from spacecraft design to analysis and optimization. The effectiveness of the method proposed in this invention is verified by comparing the results with those of optimization methods using typical commercial software. Using the multi-point approximation method based on the approximate Hessian matrix proposed in this invention, the optimal value of the objective function and the constraint function values ​​can be obtained, and the iterative curve of the objective function can be plotted.

[0060] The advantages and beneficial effects of this invention are as follows: The method / optimization strategy mentioned in this invention can improve the approximation accuracy of the approximation function, improve optimization efficiency, and is applicable to mixed optimization of size and shape. Based on the method mentioned in this invention, a practical optimization system has been established after secondary development to ensure that the entire optimization iteration is automatic. In particular, for large plates, beams, and their combined structures in the aerospace field, this method can significantly reduce the number of structural analyses in the optimization process, thereby reducing the time spent on the optimization design of engineering structures. This method can handle constraints under static and dynamic responses, and the optimization results obtained are true solutions that can be directly applied to actual structural design without particularly complex post-processing. Attached Figure Description

[0061] Figure 1 An example of optimizing the iterative process.

[0062] Figure 2 To optimize the platform's development framework.

[0063] Figure 3 It is a 72-bar truss structure.

[0064] Figure 4 This is the iteration history of the objective function in Example 1.

[0065] Figure 5 The initial structure is a 37-bar hybrid frame.

[0066] Figure 6 The optimal design for a 37-bar planar hybrid frame.

[0067] Figure 7a , Figure 7b , Figure 7c , Figure 7d The optimized configuration mode is obtained by the multi-point approximation method proposed in this invention.

[0068] Figure 8 This is the iteration history of the objective function in Example 2.

[0069] Figure 9a , Figure 9b These are the results of satellite modal analysis.

[0070] Figure 10For comparison of the overall satellite mass iteration curves.

[0071] Figure 11 For comparison of fundamental frequency iteration curves. Detailed Implementation

[0072] The following is in conjunction with the appendix Figure 1-11 The table below provides a detailed description of specific implementations of the multi-point approximation method based on the approximate Hessian matrix, with optimization targets being typical truss structures and satellite structures. The following description includes three embodiments, which are for illustrative purposes only and are not intended to limit the scope of the invention.

[0073] Example 1: Dimensional Optimization of a 72-Bar Truss

[0074] Truss structures are also a classic form of spacecraft structure. This example is a complex truss structure with seventy-two bars and four layers of frame, used to further test the approximation effect and reliability of the multi-point approximation method based on the approximate Hessian matrix.

[0075] Step 1: Establish a finite element model of the spacecraft based on the initial design of the spacecraft structure. The finite element model of the truss structure was established in the commercial software Patran and simulated using Rod elements. The external dimensions and member numbers of the truss are as follows: Figure 3 As shown. The material's elastic modulus E = 68.971 GPa and density ρ = 2767.991 kg / m³. 3 The structure can withstand two loading conditions, as shown in Table 1.

[0076] Table 1 Load Conditions for the 72-Bar Truss

[0077]

[0078] Note: 22241.1N is 5 kilops (kilopound-force, imperial unit).

[0079] Step 2: Perform mechanical prediction of the spacecraft under typical operating conditions. In this example, the analysis method includes static analysis. The maximum response displacement is u under operating condition 1. 1x =u 1y = 9.774 mm.

[0080] Step 3: Establish a mathematical model for structural optimization, i.e., define the optimization objective, constraint functions, and design variables. The size optimization problem is described as follows:

[0081] Objective function: Minimize the total mass of the 72-bar truss

[0082] Design variables: Considering the chaining relationship of variables, the truss section is divided into 16 groups, with the cross-sectional area of ​​the members in each group being the same. The chaining details are shown in Table 2. In the optimization design, the initial cross-sectional area is taken as 322.58 mm². 2 And Ai ≥64.516mm 2 i = 1, 2, ..., 16;

[0083] Constraints: The x and y displacements of the top nodes 1, 2, 3, and 4 of the structure are all less than ±6.35 mm; the allowable tensile and compressive stresses of all members are 172.4 MPa.

[0084] Step 4: Establish a sequence approximation problem considering second-order information to approximate the original problem. Take the upper bound H of the known points. max =4. Utilizing the accumulated information of known points, an approximate function containing approximate second-order derivative information about the variable X is established using the multi-point approximation method considering approximate second-order information proposed in this invention. The approximate function approximates the nodal displacements. At the known point X... t The function value w(X) at the location t ) and partial derivative values Obtained from Nastran analysis.

[0085] Step 5: Estimate the Hessian matrix of the multi-point approximation function using the quasi-Newton method. The second derivative of the nodal displacement is approximated using the BFGS correction formula for the H-th known point given in this invention.

[0086] Step 6: Select known points using a neighborhood selection strategy based on Euclidean distance. It should be noted that only the Hessian matrix of the current point (i.e., the H-th known design point) is corrected. Immediately (When t = 1, 2, ..., H-1 and H ≥ 2). In the iterative process of the sequence approximation problem (step 4), the Hessian matrix of the multi-point approximation function needs to be estimated using the quasi-Newton method (step 5), and known points are selected based on a neighborhood selection strategy using Euclidean distance (step 6), with a maximum number of selected known points being H. max .

[0087] Step 7: Convergence Judgment and Establishment of the Automatic Optimization System. Set the maximum number of iterations to 20, and utilize... Figure 2 The provided optimization platform automatically updates the design variables.

[0088] Step 8: Method Comparison, Analysis, and Iterative Explicitization. The truss mass at the optimized initial point is 193.48 kg. Under load case 1, the initial point violates all displacement constraints, and the maximum response displacement u... 1x =u 1y=9.774mm, therefore the initial point is infeasible. The method of correcting the Hessian matrix using the quasi-Newton method is compared with Ray optimization, Adaptation of the Method of Inscribed Hypersphere (AIP), and Nastran's built-in optimization algorithm.

[0089] Table 2. Variable chaining of the 72-bar truss

[0090]

[0091] Figure 4 Table 3 shows the iteration history of the objective function, and the statistics of the optimal design values ​​for this truss under different optimization methods. From the optimization results of the objective function, the method proposed in this invention has higher computational efficiency, requires the fewest structural analyses, and produces the lightest results.

[0092] Table 3. Statistics on the Optimal Design Values ​​of Seventy-Two-Bar Truss

[0093]

[0094] Note: Design variable A in the table i Unit: mm 2 No.S represents the number of structural analyses; the displacement constraint results from Nastran's built-in optimization slightly deviate from the expected values.

[0095] Example 2: Shape and Size Hybrid Optimization of Planar Hybrid Framework

[0096] The 37-bar planar hybrid framework is a standard problem for many scholars studying shape optimization. Structural optimization with frequency constraints is a challenging problem because it involves a highly nonlinear, discontinuous, and nonconvex search space composed of multiple local optima. This example studies the performance of the multi-point approximation method based on the quasi-Newton corrected Hessian matrix in hybrid shape and size optimization.

[0097] Step 1: Establish a finite element model of the spacecraft based on the initial design of the spacecraft structure. The geometry and support conditions of the 37-bar planar hybrid frame are as follows: Figure 5 As shown, node 1 is a hinged support, and node 20 is a sliding hinged support. The lower chord members are modeled using rectangular beam elements with a cross-section of B = 8cm and H = 5cm, with an additional 10kg non-structural mass added at each node; the upper chord members are modeled using simple bar elements, with an initial cross-sectional area of ​​A = 1cm². 2 The material's elastic modulus E = 201 GPa, and its density ρ = 7800 kg / m³. 3 Poisson's ratio ν = 0.3.

[0098] Step 2: Perform mechanical prediction of the spacecraft under typical operating conditions. In this example, the analysis method is modal analysis.

[0099] Step 3: Establish a mathematical model for structural optimization, i.e., define the optimization objective, constraint functions, and design variables. The size optimization problem is described as follows:

[0100] The objective function is to minimize the mass of the hybrid architecture, with an initial mass of 336.3 kg;

[0101] The design variables include 14 dimensional variables and 5 shape variables, and symmetry is taken into account. The dimensional variable is the cross-sectional area of ​​the top chord, with a value ranging from 1 × 10⁻⁶. -4 ~10×10 -4 m 2 And satisfy A1=A 27 A2 = A 26 A3 = A 24 A4 = A 25 A5 = A 23 A6 = A 21 A7 = A 22 A8 = A 20 A9 = A 18 A 10 =A 19 A 11 =A 17 A 12 =A 15 A 13 =A 16 The shape variable is the y-coordinate of the upper chord member, ranging from 1.0 to 3.0 m, and satisfying y3 = y 19 y5=y 17 y7=y 15 y9=y 13 ;

[0102] The constraints are the first three natural frequencies: f1≥20Hz, f2≥40Hz, f3≥60Hz;

[0103] Step 4: Establish a sequence approximation problem that considers second-order information to approximate the original problem. Similarly, take the upper bound H of the known points. max =4. Utilizing the accumulated information from known points, an approximate function containing approximate second-order derivative information about the variable X is established using the multi-point approximation method considering approximate second-order information proposed in this invention. The approximate function approximates the first three frequencies. At the known point X... t The function value w(X) at the location t ) and partial derivative values Obtained from Nastran analysis.

[0104] Step 5: Estimate the Hessian matrix of the multi-point approximation function using the quasi-Newton method. The second derivatives of each frequency are approximated using the BFGS correction formula for the H-th known point given in this invention.

[0105] Step 6: Select known points using a neighborhood selection strategy based on Euclidean distance. Similarly, only correct the Hessian matrix of the current point (i.e., the H-th known design point). Immediately (When t = 1, 2, ..., H-1 and H ≥ 2). In the iterative process of the sequence approximation problem (step 4), the Hessian matrix of the multi-point approximation function needs to be estimated using the quasi-Newton method (step 5), and known points are selected based on a neighborhood selection strategy using Euclidean distance (step 6), with a maximum number of selected known points being H. max .

[0106] Step 7: Convergence Judgment and Establishment of the Automatic Optimization System. Set the maximum number of iterations to 40, and utilize... Figure 2 The provided optimization platform automatically updates the design variables.

[0107] Step 8: Method Comparison, Analysis, and Iterative Explicitization. Optimization methods include HS (HarmonySearch algorithm based on metaheuristics), HSCFA (Hybrid Sine and Cosine Firefly Algorithm), and the multi-point approximation method based on the quasi-Newton corrected Hessian matrix proposed in this invention. The optimal design obtained by the method proposed in this invention is as follows: Figure 6 As shown, the optimized first four modes are as follows: Figure 7a , Figure 7b , Figure 7c , Figure 7d As shown, where, Figure 7a In the above, f1 = 19.992 Hz Figure 7b In the above, f2 = 40.004 Hz Figure 7c In the above, f3 = 60.001Hz Figure 7d In the above, f4 = 72.507 Hz.

[0108] Various metaheuristic algorithms were repeatedly calculated 20 times, and the best and average values ​​for each calculation are recorded in Table 4. It is clear that the metaheuristic algorithms require tens of thousands of structural analyses, which is time-consuming. The method proposed in this invention has extremely high solution efficiency, converging with only 27 structural analyses. The optimized structure quality is significantly better than the results calculated by HS, but slightly better than the results of HSCFA. Table 5 shows a comparison of the natural frequencies of this planar hybrid framework after optimization by various algorithms. Figure 8 The iteration history of the objective function of the method proposed in this invention is given.

[0109] Table 4 Comparison of Optimization Designs for 37-Pole Planar Hybrid Frames

[0110]

[0111]

[0112] Note: The unit of A is cm. 2 The unit of y is m; N / A indicates no corresponding result; No.S represents the number of structural analyses.

[0113] Table 5 Comparison of natural frequencies of 37-bar planar hybrid frames

[0114]

[0115]

[0116] Example 3: Optimized Design of Box-Type Satellite

[0117] The initial mass of a certain satellite is 173.2 kg. Its side plates, top plate, bottom plate, and internal partitions all adopt a honeycomb sandwich structure. The satellite size optimization design using the method proposed in this invention can demonstrate the engineering value of the method.

[0118] Step 1: Establish a finite element model of the spacecraft based on the initial design of the spacecraft structure. Using Patran, a whole-satellite finite element model mainly composed of plate and beam elements is established, containing a total of 142,010 nodes and 141,430 elements.

[0119] Step 2: Perform mechanical prediction of the spacecraft under typical operating conditions. In this example, modal analysis is used. The modal analysis adopts the boundary condition of the bottom adapter's intermediate node being fixed. The satellite's first-order frequency is 49.779 Hz (fundamental frequency), and the second-order frequency is 54.615 Hz. Figure 9a , Figure 9b The results show the mode shapes of the first two modes of the satellite.

[0120] Step 3: Establish a mathematical model for structural optimization, i.e., define the optimization objective, constraint functions, and design variables. The description of the whole-satellite size optimization problem is as follows:

[0121] Objective function: Minimize the total satellite mass;

[0122] Design variables: Core thickness T2 of satellite side panels (4), top panel, bottom panel, internal X-axis partition, and internal Z-axis partition; thicknesses T1 and T3 of upper and lower panels, totaling 24 continuous dimensional variables. The value range is 0.1≤T1, T3≤1.0mm, 10.0≤T2≤40.0mm. Variable names and initial optimization values ​​are shown in Table 6.

[0123] Constraints: The fundamental frequency of the entire satellite must be ≥50Hz;

[0124] Step 4: Establish a sequence approximation problem that considers second-order information to approximate the original problem. Similarly, take the upper bound H of the known points. max =4. Utilizing the accumulated information from known points, an approximate function containing approximate second-order derivative information about the variable X is established using the multi-point approximation method considering approximate second-order information proposed in this invention. The approximate function approximates the frequency of the satellite structure. At the known point X... t The function value w(X) at the location t ) and partial derivative values Obtained from Nastran analysis.

[0125] Step 5: Estimate the Hessian matrix of the multi-point approximation function using the quasi-Newton method. Use the BFGS correction formula for the Hth known point given in this invention to approximate the second derivative of the entire satellite's fundamental frequency.

[0126] Step 6: Select known points using a neighborhood selection strategy based on Euclidean distance. Similarly, only correct the Hessian matrix of the current point (i.e., the H-th known design point). Immediately (When t = 1, 2, ..., H-1 and H ≥ 2). In the iterative process of the sequence approximation problem (step 4), the Hessian matrix of the multi-point approximation function needs to be estimated using the quasi-Newton method (step 5), and known points are selected based on a neighborhood selection strategy using Euclidean distance (step 6), with a maximum number of selected known points being H. max .

[0127] Step 7: Convergence Judgment and Establishment of the Automatic Optimization System. Set the maximum number of iterations to 50, and utilize... Figure 2 The provided optimization platform automatically updates the design variables.

[0128] Step 8: Method Comparison, Analysis, and Iterative Explicitization. The above problem is solved using two methods: Nastran's built-in optimization method and the multi-point approximation based on the quasi-Newton corrected Hessian matrix proposed in this invention. Figure 10 and Figure 11 The iterative curves for the total satellite mass and fundamental frequency are shown in Table 6, and the optimization results are also shown in Table 6. The Nastran solution converged after 40 iterations, but the structural analysis required a large number of iterations and exhibited obvious jagged edges during the iteration process. The multi-point approximation method converged after 13 iterations, while having a slightly lower total satellite mass, indicating that the method proposed in this invention has better convergence performance.

[0129] The satellite structure was designed and adjusted using the optimization results obtained from the multi-point approximation method based on the quasi-Newton corrected Hessian matrix proposed in this invention. After a ground-based full-satellite system test, the satellite's fundamental frequency experienced only a slight drift, and the typical positional characteristic curves on the main structure coincided well, indicating that the satellite's mechanical performance did not change significantly. The satellite has been successfully launched and has undergone mechanical environmental testing during the active phase.

[0130] Table 6 Physical meaning, initial values ​​and optimization results of optimization variables.

[0131]

[0132]

Claims

1. A multi-point approximation method based on an approximate Hessian matrix for structural optimization, characterized in that: The steps include the following: Step 1: Establish a finite element model of the spacecraft based on the initial design of the spacecraft structure; Step 2: Perform mechanical prediction of the spacecraft under typical operating conditions. The analysis methods include static analysis, modal analysis, buckling analysis, and transient response. Step 3: Establish a structural optimization mathematical model, that is, clarify the optimization objective, constraint functions, and design variables; Step 4: Establish a sequence approximation problem that considers second-order information to approximate the original problem; Step 5: Estimate the Hessian matrix of the multi-point approximation function using the quasi-Newton method; Step 6: Select known points using a neighborhood selection strategy based on Euclidean distance; Step 7: Convergence determination and establishment of the automatic optimization system; Step 8: Method comparison, analysis, and explicit iteration; An optimization example is set up to illustrate the process from spacecraft design to analysis and optimization, and the effectiveness is verified by comparing the results with those of commercial software optimization methods; In step 5, the Hessian matrix of the multi-point approximation function is estimated using the quasi-Newton method; use To approximate Each iteration only needs to use the function value and first-order information to make corrections. Initial matrix Correct only the Hessian matrix at the current point , that is to say Based on quasi-Newtonian conditions, the following is derived for the first... H The BFGS correction formula for a known point is: ； In the formula, and They are the first H Known design points at p Stages and p -1 stage approximate Hessian matrix; The increment of the intermediate variable at adjacent iteration points; The gradient change of adjacent iteration points with respect to intermediate variables has the following form: ； In the formula, For the first The initial point of the nth iteration, that is, the nth... The secondary advantage obtained in the next iteration; and They are the first H The known point and the first H -1 adaptive parameters for known points; In step 6, known points are selected using a neighborhood selection strategy based on Euclidean distance. Set the upper limit for the number of known points to select. , It is the latest obtained secondary advantage; take the latest one. Construct a new approximate function from points, that is, select... Among them, distance The farthest point is Therefore, establish based on With the center as the boundary, and its distance from the center as the boundary. The boundary circle with Euclidean distance as its radius; previous iteration points. , Also within the boundary circle, using replace To construct a multi-point approximation function.

2. The multi-point approximation method for structural optimization based on the approximate Hessian matrix according to claim 1, characterized in that: In step 2, static analysis refers to calculating the stress, strain, and displacement of the spacecraft structure under static loads. The fundamental finite element equations to be solved are: ； in, It is the global stiffness matrix. It is a nodal displacement vector. It is the external load vector; the element stress response value is obtained through the stress-strain relationship of the material and the interpolation formula; Modal analysis is used to determine the natural frequencies and mode shapes of a structure. The fundamental equations of the finite element method are: ； in, It is a matrix containing mode shape vectors. It is an eigenvalue; Buckling analysis is used to evaluate the buckling behavior of a structure under static loads, obtaining buckling load factors and mode shapes, and assessing the deformation modes during structural buckling. The fundamental finite element equations to be solved are: ； in, It is the geometric stiffness matrix of the structure under initial load, and the nodal displacement vectors. Related, It is the linear buckling load factor.

3. The multi-point approximation method based on the approximate Hessian matrix for structural optimization according to claim 1, characterized in that: In step 3, the mathematical description of the primal problem for spacecraft structure optimization with respect to continuous variables is as follows: ； In the formula, For dimensional variables that can take continuous values ​​related to structural components, For the elements, The number of continuous variables; , These are the upper and lower limits of the variable, respectively; Structural quality is the objective function for optimal design; For the first Structural trait constraints, The number of constraints.

4. A multi-point approximation method for structural optimization based on an approximate Hessian matrix according to claim 3, characterized in that: In step 4, a multi-point approximation function is used, and intermediate variables are selected. Establish a sequential explicit approximation problem for the original optimization problem; the first... The approximate problem for the next iteration is formulated as follows: ； ； ； In the formula, and The first In the next iteration, the objective and constraint functions are established using a multi-point approximation method, and then uniformly represented by the function. To indicate; The number of critical constraints that are retained; and yes At this point, the upper and lower limits, and yes In the The motion limit at the next iteration, and the width of the motion limit for each design variable at the current point is taken as 20% of the width of the variable range of each variable.

5. A multi-point approximation method for structural optimization based on an approximate Hessian matrix according to claim 4, characterized in that: Using the accumulated information of known points, establish the following information about the variables. An approximate function containing information about the second derivative: In the formula, H Let be the number of known points, with an upper limit of . H max If the number of known points is greater than H max Then only the last one will be retained. H max One known point; For the first t One known design point; intermediate variables ; It is the value of the intermediate variable at the known point; This is the first The objective or constraint function is established using a multi-point approximation method in the next iteration; It is the function value of the objective or constraint function; It is the first t The known point is at the th p Hessian matrix of the phase; It is a weighting function; It is an approximate function that contains first-order information about known points; and The calculation method is as follows: ； ； ； In the formula, and These are the original function at the known points. The function value and partial derivative at the point are obtained through a semi-analytical method; , as well as It is a substitution symbol used in calculations and has no actual meaning.

6. A multi-point approximation method for structural optimization based on an approximate Hessian matrix according to claim 5, characterized in that: It is an adaptive parameter that controls the degree of nonlinearity of the approximation function, obtained by solving the following least squares problem: ； In the formula, , They are respectively The upper and lower limits, take , , ; As the structural optimization process proceeds, the adaptive parameters need to be corrected. Given that the least squares problem is quite complex after considering second-order information, an interior-point algorithm is used to solve it.

7. A multi-point approximation method for structural optimization based on an approximate Hessian matrix according to claim 1, characterized in that: For optimization problems involving 3-dimensional or higher-dimensional design variables, the method for selecting known points is as follows: At that time, the known points to be selected are added in sequence. The steps are as follows: (1) Record the latest point as Select the latest 5 known points to establish a set of known points. ; (2) Calculate the maximum envelope radius , And establish the boundary sphere and the distance The farthest point is denoted as ; (3) Determine whether there are other objects within the sphere that do not belong to the sphere. The set of known points is denoted as ; if it exists, then the set of other known points is denoted as . If it does not exist, proceed to step (4); if it does not exist, proceed to step (5). (4) Find Mid-range Add the nearest known point to the set And Remove and proceed to step (2); (5) Once all known points have been selected, output the set. .

8. A multi-point approximation method for structural optimization based on an approximate Hessian matrix according to claim 1, characterized in that: In step 7, the initial finite element modeling of the structure is completed in Patran, and Nastran is called for analysis by submitting the bdf file; the main program is built in Matlab, including modules for convergence judgment, establishment of approximate functions, optimization of optimizer, and visualization of optimization results; in addition, an external file is needed to store the iteration results; some core commands or functions in the program system are given below, and the input, output and calling relationships are explained in detail. (1)[optx,fval] = fmincon(@objfun,x0,[],[],[],[],move_lower,move_upper,@constraint, options); This function is used for optimization of approximation problems, and the default is to use the interior-point algorithm. The inputs include handle functions `@objfun` and `@constraint`, which are the objective and constraint functions, respectively; `x0` is the initial value of the design variables; `move_lower` is the lower bound vector of the variable movement limits; `move_upper` is the upper bound vector of the variable movement limits; and `option` is a structure used to define the convergence tolerance or the content displayed in a single step. In the outputs, `optx` is the optimal design variable for the current approximation problem, and `fval` is the optimal objective function value for the current approximation problem. In each iteration of the approximation problem, the movement limit values ​​need to be updated and this function called. (2) [rsp(LOOP,:),drsp(LOOP,:,:)] = Sensitivity_anaysis(optx,lower_limit,upper_limit); After obtaining the optimized values, this function needs to be called for structural analysis; in the input, optx is the optimal design variable for the current approximate problem; in the output, LOOP is the number of iteration steps. rsp(LOOP,:) is the actual structural response value obtained after the LOOPth iteration of the approximate problem optimization, and drsp(LOOP,:,:) is the sensitivity value; an example of the call statement for commercial software is system('D:\MSCsoftware\Nastran2018\bin\nastran.exe" ',bdf_name,'.bdf'],'-echo'), where the variable bdf_name is the file name; (3)[ap,ap_dg] = Approximate_Function(in_x,x,g, dg,r); The approximation function is used to establish the objective function and constraint function. In the input, in_x is the design variable in the optimization process within the approximation problem, and x is a matrix containing the design variable values ​​at known points. g is a vector containing the constraint function values ​​at known points. dg is a matrix containing the sensitivity of the constraint function to the design variables at known points. r stores the nonlinear adaptive parameter values ​​in the multi-point approximation function. In the output, ap is the objective / constraint function value obtained based on the approximation function when the design variable takes the value in_x. ap_dg is the approximation sensitivity value.

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