A structural topology optimization design method considering load multi-peak uncertainty

Abstract

This invention discloses a structural topology optimization design method considering the uncertainty of multi-peak loads, belonging to the field of uncertain structural optimization design. It mainly includes three parts: establishing a multi-peak load uncertainty model, solving for the random response, and solving for the optimization formula. This invention describes load uncertainty using a Gaussian mixture model, solves for the Gaussian mixture model coefficients based on the EM algorithm, and establishes a multi-peak load distribution probability model. By decorrelating random variables, sparse grid technology is used to solve for the mean, standard deviation, and sensitivity of the response, thereby solving for the topology optimization model considering the multi-peak load uncertainty. This method addresses the problem of low structural product reliability that may occur in structural designs considering multi-peak load uncertainty by establishing an accurate probabilistic model of multi-peak load uncertainty. This method is simple and easy to implement, readily applicable in engineering, and can improve the design efficiency of structural designers considering complex load conditions.

Description

Technical Field

[0001] This invention relates to the field of aerospace structural optimization design, and more specifically to a structural topology optimization design method that considers the uncertainty of multi-peak loads. Background Technology

[0002] Structural topology optimization is a key technology in lightweight aerospace structural design. The most widely used topology optimization algorithm in engineering is the variable density method, which has been integrated into various commercial optimization software programs and has solved numerous deterministic structural optimization problems. However, it has encountered obstacles in solving uncertain structural optimization problems. Uncertainties in the geometry, materials, and loads of engineering structures all affect structural performance and reliability. Ignoring these uncertainties in the design process can lead to structural failure.

[0003] In existing structural optimization methods that consider load uncertainties, if the probability density distribution is unknown, ellipsoidal models are often used to enclose the boundary of the random variables to improve the structure's resistance to uncertain loads. However, for multi-peak load uncertainty problems, such methods are too conservative and the design cost is too high. In addition, implementing existing structural optimization algorithms that consider load uncertainties using commercial finite element methods is quite complex and not conducive to engineering applications.

[0004] This invention discloses a structural topology optimization method that considers the uncertainty of multi-peak loads. This method can reduce the sensitivity of the structure to multi-peak uncertain loads and improve the structural reliability. At the same time, this method is easy to integrate with commercial finite element software, can realize engineering applications, and improve the design efficiency of structural designers considering complex load conditions. Summary of the Invention

[0005] The purpose of this invention is to provide a structural topology optimization design method considering the uncertainty of multi-peak loads. Addressing the problems of overly conservative structural design, high design costs, and unsuitable engineering applications in existing structural optimization methods considering multi-peak load uncertainties, this invention is based on a probabilistic description method using a Gaussian mixture model and applies sparse mesh integration techniques to obtain the objective function value, thus realizing the optimization process. This method is simple and easy to implement, readily integrated with commercial finite element methods, enabling engineering applications and improving the design efficiency for designers considering complex load conditions. This invention is applicable to structural optimization design considering load uncertainties in aerospace, mechanical, and other fields.

[0006] To achieve the above objectives, this invention provides a structural topology optimization design method considering multi-peak load uncertainty, comprising: establishing a multi-peak load uncertainty model, solving for the random response, and solving for the optimization formula. The specific steps are as follows:

[0007] Step 1: Establish a structural topology optimization model that considers multi-peak uncertainty and provide the optimization solution expression;

[0008] Step 2: By observing the distribution characteristics of the initial data points, determine the number of random variables n and the number of Gaussian distribution components m in the model;

[0009] Step 3: Establish the Gaussian mixture model expression and determine the coefficients that need to be solved;

[0010] Step 4: Use the EM algorithm to solve for the coefficients in the expression of the Gaussian mixture model;

[0011] Step 5: Perform decorrelation on each of the n random variables in the m Gaussian distributions;

[0012] Step 6: Based on the calculation results in Step 5, determine the integration points under the sparse grid integration rule, and take the integration accuracy as 4th order;

[0013] Step 7: Establish a finite element model, and based on the topology optimization formula, solve for the objective function value and its sensitivity at each integration point in Step 6.

[0014] Step 8: Solve for the overall objective function and sensitivity of the topology optimization model, and substitute them into the optimization algorithm to solve for the optimization formula.

[0015] The above-mentioned structural topology optimization design method considering multi-peak load uncertainty, wherein, in step one, the optimization formula of the structural topology optimization model considering multi-peak uncertainty is as follows:

[0016]

[0017] in, For structural flexibility, Let the load be a random variable. For design variables, This represents the total number of units. The mean of compliance. For weight parameters, The standard deviation of compliance, This represents the upper limit of the body fraction ratio. The material volume ratio, For the overall stiffness matrix, Let be the overall displacement vector of the structure. For uncertain load vectors, For unit density, The penalty coefficient is... For the element stiffness matrix, The element node displacement vector. This represents the minimum unit density.

[0018] The above-mentioned structural topology optimization design method considering multi-peak load uncertainty, wherein, in step three, the Gaussian mixture model expression is:

[0019]

[0020] in, Let be the number of terms in the Gaussian mixture model. Let m be the matrix composed of the mean and standard deviation of the Gaussian distribution. The weight coefficients of the Gaussian mixture model and , Let k be the Gaussian distribution parameter. Let k be the covariance matrix of the Gaussian distribution. Let be the mean vector of the k-th Gaussian distribution.

[0021] The above-mentioned structural topology optimization design method considering the uncertainty of multi-peak loads, wherein in step three, the coefficients to be solved need to be determined are: , , .

[0022] The above-mentioned structural topology optimization design method considering the uncertainty of multi-peak loads, wherein in step five, decorrelation is achieved by finding an orthogonal matrix. The covariance matrix The formula for converting to a similar diagonal matrix is ​​as follows:

[0023]

[0024] in, and These are the covariance matrix and mean vector after decorrelation, respectively.

[0025] The above-mentioned structural topology optimization design method considering the uncertainty of multi-peak loads, wherein, in step eight, the overall objective function is solved as follows:

[0026] in, The number of integration points. Let be the value of the random variable at the integration point. Let be the weights corresponding to the i-th integration point.

[0027] The above-mentioned structural topology optimization design method considering the uncertainty of multi-peak loads, wherein in step eight, the sensitivity can be transformed into solving at the integration point:

[0028]

[0029] in, The sensitivity at each integration point i is obtained through step seven.

[0030] Compared with the prior art, the beneficial effects of the present invention are:

[0031] (1) The method of the present invention is a topology optimization design method for structures considering the uncertainty of multi-peak loads, which can reduce the sensitivity of the structure to multi-peak uncertain loads and improve the reliability of the structural design;

[0032] (2) The method of the present invention establishes a multi-peak load probability description model based on Gaussian mixture model, which avoids the problem of overly conservative structural design caused by traditional methods and reduces structural design cost;

[0033] (3) The method of the present invention obtains the optimization objective function and its sensitivity through random variable decorrelation and sparse grid integration techniques, which has high computational efficiency and low cost.

[0034] (4) The method of the present invention can be integrated with existing commercial finite element software, which facilitates engineering applications and can improve the design efficiency of designers considering complex working conditions. Attached Figure Description

[0035] The structural topology optimization design method of the present invention, which considers the uncertainty of multiple peak loads, is given by the following embodiments and figures.

[0036] Figure 1 This is a flowchart illustrating the analysis process of the present invention;

[0037] Figure 2 This is a multi-peak uncertainty load distribution according to an embodiment of the present invention;

[0038] Figure 3 The probability density function of the Gaussian mixture model in this embodiment of the invention;

[0039] Figure 4 This is the design domain of the four corner fixed beams in an embodiment of the present invention;

[0040] Figure 5 This is the optimal topology for the four-corner fixed-support beam in an embodiment of the present invention. Detailed Implementation

[0041] The following provides a more detailed description of a structural topology optimization design method that considers the uncertainty of multi-peak loads according to the present invention.

[0042] like Figure 1 An analysis flowchart of an embodiment of the present invention is provided. Figure 2The design domain model of the four-corner fixed beam in this embodiment is given. This invention provides a structural topology optimization method considering multi-peak load uncertainty, mainly including three parts: establishing a multi-peak load uncertainty model, solving for the stochastic response, and solving for the optimization formula. More detailed, it includes the following steps:

[0043] S1: Establish a structural topology optimization model considering multi-peak uncertainty, and provide the optimization solution expression:

[0044]

[0045] in, For structural flexibility, Let the load be a random variable. For design variables, This represents the total number of units. The mean of compliance. For weight parameters, in this example , The standard deviation of compliance, This represents the upper limit of the body fraction ratio. In this example, the material volume ratio is... , For the overall stiffness matrix, Let be the overall displacement vector of the structure. For uncertain load vectors, For unit density, The penalty coefficient is... For the element stiffness matrix, The element node displacement vector. For the minimum unit density, in the examples .

[0046] S2: By observing the distribution characteristics of the initial data points, in this example, the initial distribution is as follows: Figure 3 As shown, determine the number of random variables n and the number of Gaussian distributions m in the model; in this example, the random variable n=2, and the number of Gaussian distributions m is 2.

[0047] S3: Establish the Gaussian mixture model expression and determine the coefficients that need to be solved. , , :

[0048]

[0049] in, Let be the number of terms in the Gaussian mixture model. Let m be the matrix composed of the mean and standard deviation of the Gaussian distribution. The weight coefficients of the Gaussian mixture model and , Let k be the Gaussian distribution parameter. Let k be the covariance matrix of the Gaussian distribution. Let be the mean vector of the k-th Gaussian distribution.

[0050] S4: The coefficients in the expression of the Gaussian mixture model are solved using the EM algorithm; the resulting Gaussian mixture model is as follows. Figure 4 As shown.

[0051] S5: Decorrelate the n random variables from the m Gaussian distributions;

[0052] Decorrelation is achieved by finding orthogonal matrices. The covariance matrix The formula for converting to a similar diagonal matrix is ​​as follows:

[0053]

[0054] in, and These are the covariance matrix and mean vector after decorrelation, respectively.

[0055] S6: Based on the calculation results of step S5, determine the integration points under the sparse grid integration rule, and take the integration accuracy as 4th order;

[0056] S7: Establish a finite element model using a 300x50 mesh. Based on the topology optimization formula, solve for the objective function value and its sensitivity at each integration point in step S6.

[0057] S8: Solve for the overall objective function and sensitivity of the topology optimization model, and substitute them into the optimization algorithm to solve for the optimization formula. The expression for the overall objective function is as follows:

[0058] in, The number of integration points. Let be the value of the random variable at the integration point. Let be the weights corresponding to the i-th integration point.

[0059] Sensitivity can be transformed into solving at the integration point:

[0060]

[0061] in, The sensitivity at each integration point i is obtained through step S7. Optimization is implemented using the MMA algorithm, and the optimization results are as follows: Figure 5 .

[0062] This invention solves the structural optimization problem under multi-peak uncertainty loads by introducing a Gaussian mixture model and sparse mesh integration technology, providing a solution for aerospace structural design under complex load conditions. The above embodiment is merely an application example of this invention and should not be construed as limiting the scope of application of this patent.

Claims

1. A structural topology optimization design method considering the uncertainty of multi-peak loads, characterized in that, include: The specific steps for establishing the multi-peak uncertainty model of the load, solving the random response, and optimizing the formulation are as follows: Step 1: Establish a structural topology optimization model that considers multi-peak uncertainty and provide the optimization solution expression; In step one, the optimization formula for the structural topology optimization model considering multi-peak uncertainty is as follows: in, For structural flexibility, Let the load be a random variable. For design variables, This represents the total number of units. The mean of compliance. For weight parameters, The standard deviation of compliance, This represents the upper limit of the body fraction ratio. The material volume ratio, For the overall stiffness matrix, Let be the overall displacement vector of the structure. For uncertain load vectors, For unit density, The penalty coefficient is... For the element stiffness matrix, The element node displacement vector. Minimum unit density; Step 2: By observing the distribution characteristics of the initial data points, determine the number of random variables n and the number of Gaussian distribution components m in the model; Step 3: Establish the Gaussian mixture model expression and determine the coefficients that need to be solved; In step three, the Gaussian mixture model expression is: in, Let be the number of terms in the Gaussian mixture model. Let m be the matrix composed of the mean and standard deviation of the Gaussian distribution. The weight coefficients of the Gaussian mixture model and , Let k be the Gaussian distribution parameter. Let k be the covariance matrix of the Gaussian distribution. Let k be the mean vector of the Gaussian distribution. In step three, the coefficients to be solved need to be determined. , , ; Step 4: Use the EM algorithm to solve for the coefficients in the expression of the Gaussian mixture model; Step 5: Perform decorrelation on each of the n random variables in the m Gaussian distributions; Step 6: Based on the calculation results in Step 5, determine the integration points under the sparse grid integration rule, and take the integration accuracy as 4th order; Step 7: Establish a finite element model, and based on the topology optimization formula, solve for the objective function value and its sensitivity at each integration point in Step 6. Step 8: Solve for the overall objective function and sensitivity of the topology optimization model, and substitute them into the optimization algorithm to solve for the optimization formula.

2. The structural topology optimization design method considering the uncertainty of multi-peak loads as described in claim 1, characterized in that, In step five, the decorrelation is achieved by finding an orthogonal matrix. The covariance matrix The formula for converting to a similar diagonal matrix is ​​as follows: in, and These are the covariance matrix and mean vector after decorrelation, respectively.

3. The structural topology optimization design method considering the uncertainty of multiple peak loads as described in claim 2, characterized in that, In step eight, the overall objective function is expressed as follows: in, The number of integration points. Let be the value of the random variable at the integration point. Let be the weights corresponding to the i-th integration point.

4. The structural topology optimization design method considering the uncertainty of multiple peak loads as described in claim 3, characterized in that, In step eight, the sensitivity can be transformed into solving at the integration point: in, The sensitivity at each integration point i is obtained through step seven.

Images