CN116205093A - Structural topology optimization design method considering load multi-peak uncertainty - Google Patents
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Abstract
The invention discloses a structural topology optimization design method considering load multi-peak uncertainty, which belongs to the field of uncertainty structural optimization design and mainly comprises three parts of load multi-peak uncertainty model establishment, random response solution and optimization column solution. According to the invention, load uncertainty is described through a Gaussian mixture model, gaussian mixture model coefficients are solved based on an EM algorithm, and a multimodal load distribution probability model is established; the method comprises the steps of solving a mean value, a standard deviation and sensitivity of response by adopting a sparse grid technology through decorrelation of random variables, and further solving a topological optimization model considering load multi-peak uncertainty. The method solves the problem of low reliability of structural products possibly occurring in structural design considering load multi-peak uncertainty by establishing an accurate probability model of multi-peak load uncertainty. The method is simple and feasible, is convenient for engineering application, and can improve the design efficiency of structural designers under the condition of considering complex loads.
Description
Technical Field
The invention relates to the field of aerospace structure optimization design, in particular to a structure topology optimization design method considering load multi-peak uncertainty.
Background
The structural topology optimization technology is a key technology in the lightweight design of the aerospace structure, the most widely applied topology optimization algorithm in engineering is a variable density method, and the topology optimization algorithm is integrated into various commercial optimization software, so that a large number of deterministic structure optimization problems are solved, and however, the problem of uncertainty structure optimization is solved. Uncertainties in engineering structures, such as geometry, materials, and loads, can affect structural performance and reliability, and structural failure may result if the effect of such uncertainties is ignored in the design.
In the existing structural optimization considering load uncertainty, if probability density distribution is unknown, an ellipsoid model is mostly adopted to envelop a random variable change boundary, so that optimization design is performed, the resistance of the structure to the uncertainty load is improved, however, for the problem of multi-peak load uncertainty, the method is too conservative, and the design cost is too high. In addition, the existing load uncertainty-considered structural optimization algorithm is complex through commercial finite elements, and is not beneficial to engineering application.
The invention discloses a structural topology optimization method considering load multi-peak uncertainty, which can reduce the sensitivity of a structure to multi-peak uncertainty load and improve the reliability of the structure; meanwhile, the method is convenient to integrate with commercial finite element software, engineering application can be realized, and design efficiency of structural designers considering complex load working conditions is improved.
Disclosure of Invention
The invention aims to provide a structural topology optimization design method considering load multi-peak uncertainty, and aims to solve the problems that in the existing structural optimization considering the multi-peak load uncertainty, structural design is too conservative, design cost is too high, an algorithm is unfavorable for engineering application and the like. The method is simple and feasible, is convenient for integration with commercial finite elements, realizes engineering application, and improves the design efficiency of a designer considering complex load working conditions. The invention is suitable for structural optimization design considering load uncertainty in the fields of aerospace, machinery and the like.
In order to achieve the above object, the present invention provides a structural topology optimization design method considering load multi-peak uncertainty, including: the method comprises the following specific steps of establishing a load multi-peak uncertainty model, solving random response and optimizing column type solution:
step one, building a structural topology optimization model considering multi-peak uncertainty, and giving an optimization solving expression;
step two, determining the number n of random variables and the number m of model Gaussian distribution components by observing initial data point distribution characteristics;
thirdly, establishing a Gaussian mixture model expression list, and determining coefficients to be solved;
solving coefficients in the Gaussian mixture model expression by adopting an EM algorithm;
step five, respectively decorrelating n random variables in m Gaussian distributions;
step six, according to the calculation result of the step five, determining the integration points under the sparse grid integration rule, wherein the integration precision generally takes 4 steps;
establishing a finite element model, and solving the objective function value and the sensitivity of each integral point in the step six according to the topological optimization column;
and step eight, solving the overall objective function and the sensitivity of the topological optimization model, and substituting the overall objective function and the sensitivity into an optimization algorithm to solve the optimization column.
In the above structural topology optimization design method considering load multi-peak uncertainty, in the first step, the structural topology optimization model considering multi-peak uncertainty optimizes the following formula:
min:μ(c(x,ξ))+ασ(c(x,ξ))
subjectto:V c ≤V 0
KU(ξ)＝P(ξ)
0＜x min ≤x e ≤1
wherein c (x, ζ) is structural flexibility, ζ= [ ζ ] 1 ,ξ 2 ,ξ 3 ...ξ n ] T As a load random variable, x= [ x ] 1 ,x 2 ,x 3 ...x N ] T N is the total unit number, mu (c (x, xi)) is the flexibility mean value, alpha is the weight parameter, sigma (c (x, xi)) is the flexibility standard deviation, V 0 At the upper limit of the volume percentage, V c Is the material volume percentage, K is the overall stiffness matrix, U (ζ) is the structural overall displacement vector, P (ζ) is the uncertainty load vector, x e For unit density, p=3 is penalty coefficient, k 0 U is a unit stiffness array e For the unit node displacement vector, x min Is the minimum cell density.
The structural topology optimization design method considering load multi-peak uncertainty comprises the following steps:
wherein m is the number of terms of the Gaussian mixture model, and θ= [ mu ] is calculated by using the method 1 ,Σ 1 ；μ 2 ,Σ 2 ；μ 3 ,Σ 3 ...μ m ,Σ m ]Matrix consisting of m Gaussian distribution mean value and standard deviation omega k Is a Gaussian mixture model weight coefficient
θ k For the kth term Gaussian distribution parameter, Σ k For the kth term gaussian distribution covariance matrix, μk is the k-th gaussian distribution mean vector.
The structural topology optimization design method considering load multi-peak uncertainty, wherein in the third step, the coefficient to be solved is required to be determined to have omega k 、 μk 、Σ k 。
In the fifth step, the decorrelation finds the orthogonal matrix Q to obtain the covariance matrix Σ k Converting into a similar diagonal matrix, wherein the formula is as follows:
Σ k ′＝Q T Σ k Q
μ k ′＝Q T μ k
wherein ,Σk ' and mu k ' is the covariance matrix and the mean vector after decorrelation, respectively.
In the above structural topology optimization design method considering load multi-peak uncertainty, in the eighth step, the overall objective function solving expression is as follows:
wherein ,nq the number of integration points, ζ i Is a random variable value at the integration point, ω (ζ) i ) And the weight corresponding to each integration point of the ith.
In the above structural topology optimization design method considering load multi-peak uncertainty, in the eighth step, the sensitivity may be converted into an integral point to solve:
wherein ,
the sensitivity at each integration point of the ith is obtained by the seventh step.
Compared with the prior art, the invention has the technical advantages that:
(1) The method considers the topological optimization design method of the multi-peak load uncertainty structure, can reduce the sensitivity of the structure to the multi-peak uncertainty load, and improves the reliability of the structure design;
(2) The method establishes a multimodal load probability description model based on the Gaussian mixture model, avoids the problem that structural design is too conservative due to the adoption of a traditional method, and reduces structural design cost;
(3) The method of the invention obtains the optimized objective function and the sensitivity thereof through the decorrelation of the random variable and the sparse grid integration technology, and has high calculation efficiency and low cost.
(4) The method can be integrated with the existing commercial finite element software, is convenient for engineering application, and can improve the design efficiency of a designer under the consideration of complex working conditions.
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The invention provides a structural topology optimization design method considering load multi-peak uncertainty, which is given by the following embodiment and the attached drawing.
FIG. 1 is an analytical flow chart of the present invention;
FIG. 2 is a graph of a multimodal uncertainty load distribution in accordance with an embodiment of the invention;
FIG. 3 is a probability density function of a Gaussian mixture model according to an embodiment of the invention;
FIG. 4 is a diagram of a four corner clamped beam design domain according to an embodiment of the present invention;
fig. 5 is an optimal topology of a four-corner clamped beam according to an embodiment of the present invention.
Detailed Description
The following describes the structural topology optimization design method taking the load multi-peak uncertainty into consideration in further detail.
Fig. 1 shows an analysis flow chart of an embodiment of the present invention, and fig. 2 shows a four-corner clamped beam design domain model of the embodiment. The invention provides a structural topology optimization method considering load multi-peak uncertainty, which mainly comprises three parts of load multi-peak uncertainty model establishment, stochastic response solving and optimization column solving, and more particularly comprises the following steps:
s1: building a structural topology optimization model considering multi-peak uncertainty, and giving an optimization solving expression:
min:μ(c(x,ξ))+ασ(c(x,ξ))
subject to:V c ≤V 0
KU(ξ)＝P(ξ)
0＜x min ≤x e ≤1
wherein c (x, ζ) is structural flexibility, ζ= [ ζ ] 1 ,ξ 2 ,ξ 3 ...ξ n ] T As a load random variable, x= [ x ] 1 ,x 2 ,x 3 ...x N ] T As design variables, N is the total unit number, μ (c (x, ζ)) is the compliance mean, α is the weight parameter, in this example α=1, σ (c (x, ζ)) is the compliance standard deviation, V 0 At the upper limit of the volume percentage, V c As a percentage of the material by volume, in this case V c =0.4, k is the global stiffness matrix, U (ζ) is the structural global displacement vector, P (ζ) is the uncertainty load vector, x e For unit density, p=3 is penalty coefficient, k 0 U is a unit stiffness array e For the unit node displacement vector, x min For minimum cell density, x in the examples min ＝0.01。
S2: by observing the distribution characteristics of initial data points, the initial distribution in the example is shown in figure 3, and the number n of random variables and the number m of model Gaussian distribution components are determined; in the example, the random variable n=2, and the number of Gaussian distribution components is 2;
s3: establishing a Gaussian mixture model expression list, and determining a coefficient omega to be solved k 、 μk 、Σ k ：
Wherein m is the number of terms of the Gaussian mixture model, and θ= [ mu ] is calculated by using the method 1 ,Σ 1 ；μ 2 ,Σ 2 ；μ 3 ,Σ 3 ...μ m ,Σ m ]Matrix consisting of m Gaussian distribution mean value and standard deviation omega k Is a Gaussian mixture model weight coefficient
θ k For the kth term Gaussian distribution parameter, Σ k For the kth term gaussian distribution covariance matrix, μk is the k-th gaussian distribution mean vector.
S4: solving coefficients in the Gaussian mixture model expression by adopting an EM algorithm; the result of the Gaussian mixture model obtained by solving is shown in FIG. 4.
S5, respectively performing decorrelation on n random variables in m Gaussian distributions;
decorrelation combines a covariance matrix Σ by finding a quadrature matrix Q k Converting into a similar diagonal matrix, wherein the formula is as follows:
Σ k ′＝Q T Σ k Q
μ k ′＝Q T μ k
wherein ,Σk ' and mu k ' is the covariance matrix and the mean vector after decorrelation, respectively.
S6: according to the calculation result of the step S5, determining an integration point under a sparse grid integration rule, and taking 4 th order of integration precision;
s7: establishing a finite element model, wherein a 300X50 grid is adopted by a finite element, and the objective function value and the sensitivity of each integral point in the step S6 are solved according to a topological optimization column;
and S8, solving the overall objective function and the sensitivity of the topological optimization model, and substituting the overall objective function and the sensitivity into an optimization algorithm to solve the optimization list. The overall objective function solution expression is as follows:
wherein ,nq the number of integration points, ζ i Is a random variable value at the integration point, ω (ζ) i ) And the weight corresponding to each integration point of the ith.
Sensitivity can be translated into solution at integration points:
wherein ,
the sensitivity at each integration point of the i-th is obtained by step S7. The optimization is realized by adopting an MMA algorithm, and the optimization result is shown in figure 5.
According to the invention, the problem of load multi-peak uncertainty structure optimization is solved by introducing a Gaussian mixture model and a sparse grid integration technology, and a solution idea is provided for the design of a space flight and aviation structure for solving the complex load working condition. The above embodiment is only one application case of the present invention, and is not intended to limit the scope of the application of the present invention.
Claims (7)
1. The structural topology optimization design method considering load multi-peak uncertainty is characterized by comprising the following steps of: the method comprises the following specific steps of establishing a load multi-peak uncertainty model, solving random response and optimizing column type solution:
step one, building a structural topology optimization model considering multi-peak uncertainty, and giving an optimization solving expression;
step two, determining the number n of random variables and the number m of model Gaussian distribution components by observing initial data point distribution characteristics;
thirdly, establishing a Gaussian mixture model expression list, and determining coefficients to be solved;
solving coefficients in the Gaussian mixture model expression by adopting an EM algorithm;
step five, respectively decorrelating n random variables in m Gaussian distributions;
step six, according to the calculation result of the step five, determining the integration points under the sparse grid integration rule, wherein the integration precision generally takes 4 steps;
establishing a finite element model, and solving the objective function value and the sensitivity of each integral point in the step six according to the topological optimization column;
and step eight, solving the overall objective function and the sensitivity of the topological optimization model, and substituting the overall objective function and the sensitivity into an optimization algorithm to solve the optimization column.
2. The structural topology optimization design method considering load multi-peak uncertainty as recited in claim 1, wherein in the first step, the structural topology optimization model considering multi-peak uncertainty optimizes the following formula:
min:μ(c(x,ξ))+ασ(c(x,ξ))
subjectto:V c ≤V 0
KU(ξ)＝P(ξ)
0＜x min ≤x e ≤1
wherein c (x, ζ) is structural flexibility, ζ= [ ζ ] 1 ,ξ 2 ,ξ 3 ...ξ n ] T As a load random variable, x= [ x ] 1 ,x 2 ,x 3 ...x N ] T N is the total unit number, mu (c (x, xi)) is the flexibility mean value, alpha is the weight parameter, sigma (c (x, xi)) is the flexibility standard deviation, V 0 At the upper limit of the volume percentage, V c Is the material volume percentage, K is the overall rigidity matrix, U (ζ) isStructural overall displacement vector, P (ζ) is uncertainty load vector, x e For unit density, p=3 is penalty coefficient, k 0 U is a unit stiffness array e For the unit node displacement vector, x min Is the minimum cell density.
3. The structural topology optimization design method considering load multi-peak uncertainty as claimed in claim 2, wherein in the third step, a gaussian mixture model expression is as follows:
wherein m is the number of terms of the Gaussian mixture model, and θ= [ mu ] is calculated by using the method 1 ,Σ 1 ；μ 2 ,Σ 2 ；μ 3 ,Σ 3 ...μ m ,Σ m ]Matrix consisting of m Gaussian distribution mean value and standard deviation omega k Is a Gaussian mixture model weight coefficient
θ k For the kth term Gaussian distribution parameter, Σ k For the kth term gaussian distribution covariance matrix, μk is the k-th gaussian distribution mean vector.
4. A structural topology optimization design method taking into account load multi-peak uncertainty as recited in claim 3, wherein in said step three, the coefficients to be solved are determined to have ω k 、 μk 、Σ k 。
5. The structural topology optimization design method of claim 4, wherein in said step five, decorrelation calculates covariance matrix Σ by finding quadrature matrix Q k Converting into a similar diagonal matrix, wherein the formula is as follows:
Σ k ′＝Q T Σ k Q
μ k ′＝Q T μ k
wherein ,Σk ' and mu k ' is the covariance matrix and the mean vector after decorrelation, respectively.
6. The structural topology optimization design method considering load multi-peak uncertainty as set forth in claim 5, wherein in the eighth step, the overall objective function solving expression is as follows:
wherein ,nq the number of integration points, ζ i Is a random variable value at the integration point, ω (ζ) i ) And the weight corresponding to each integration point of the ith.
7. The structural topology optimization design method considering load multi-peak uncertainty as recited in claim 6, wherein in said step eight, the sensitivity can be converted into solution at integral points:
wherein ,
the sensitivity at each integration point of the ith is obtained by the seventh step. />
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