CN117057038B - Wing-oriented single-cycle reliability topology optimization design method - Google Patents

Wing-oriented single-cycle reliability topology optimization design method Download PDF

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CN117057038B
CN117057038B CN202311022266.1A CN202311022266A CN117057038B CN 117057038 B CN117057038 B CN 117057038B CN 202311022266 A CN202311022266 A CN 202311022266A CN 117057038 B CN117057038 B CN 117057038B
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CN117057038A (en
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孟增
杨刚
李锐
陈亮
李长泉
余波
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Hefei University of Technology
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    • G06—COMPUTING; CALCULATING OR COUNTING
    • G06F—ELECTRIC DIGITAL DATA PROCESSING
    • G06F30/00—Computer-aided design [CAD]
    • G06F30/10—Geometric CAD
    • G06F30/15—Vehicle, aircraft or watercraft design
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    • G06F—ELECTRIC DIGITAL DATA PROCESSING
    • G06F17/00—Digital computing or data processing equipment or methods, specially adapted for specific functions
    • G06F17/10—Complex mathematical operations
    • G06F17/11—Complex mathematical operations for solving equations, e.g. nonlinear equations, general mathematical optimization problems
    • G—PHYSICS
    • G06—COMPUTING; CALCULATING OR COUNTING
    • G06F—ELECTRIC DIGITAL DATA PROCESSING
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    • G—PHYSICS
    • G06—COMPUTING; CALCULATING OR COUNTING
    • G06F—ELECTRIC DIGITAL DATA PROCESSING
    • G06F30/00—Computer-aided design [CAD]
    • G06F30/20—Design optimisation, verification or simulation
    • G—PHYSICS
    • G06—COMPUTING; CALCULATING OR COUNTING
    • G06F—ELECTRIC DIGITAL DATA PROCESSING
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    • G06—COMPUTING; CALCULATING OR COUNTING
    • G06F—ELECTRIC DIGITAL DATA PROCESSING
    • G06F2119/00—Details relating to the type or aim of the analysis or the optimisation
    • G06F2119/02—Reliability analysis or reliability optimisation; Failure analysis, e.g. worst case scenario performance, failure mode and effects analysis [FMEA]
    • G—PHYSICS
    • G06—COMPUTING; CALCULATING OR COUNTING
    • G06F—ELECTRIC DIGITAL DATA PROCESSING
    • G06F2119/00—Details relating to the type or aim of the analysis or the optimisation
    • G06F2119/14—Force analysis or force optimisation, e.g. static or dynamic forces
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Abstract

The invention discloses a wing-oriented single-cycle reliability topology optimization design method, which comprises the following steps: performing discrete design on a wing structure design domain, and performing initial design variables and random variables; the sensitivity of the single characteristic frequency and the multi-characteristic frequency to the design variable and the random variable is respectively obtained; based on the sensitivity, constructing a wing reliability topology optimization model with uncertainty frequency band constraint; and iterating the wing reliability topology optimization model to obtain a wing structure topology model. The wing-oriented single-cycle reliability topology optimization design method provided by the invention can not only reduce the calculation cost, but also obviously reduce the optimization time, and simultaneously consider the multi-band constraint.

Description

Wing-oriented single-cycle reliability topology optimization design method
Technical Field
The invention belongs to the field of wing structure optimization design, and particularly relates to a wing-oriented single-cycle reliability topology optimization design method.
Background
Resonance of the wing structure is a key technical problem to be faced in the development stage of modern wings. Vibration of the wing structure can affect normal operation of the airborne equipment and even cause fatigue failure of the structure, thereby resulting in reduced safety and reliability of the aircraft. Because the frequency distribution of the external exciting force is often similar to the natural frequency distribution of the wing structure, the structure resonates, and finally the damage and the failure of the wing structure can be caused. Therefore, in the design stage of the wing structure, the natural frequency of the structure is usually required to be far away from the frequency and the frequency band of the external exciting force so as to avoid resonance. Topology optimization has been a powerful tool for wing structural design over the past decades. In the field of eigenvalue topology optimization, many efforts have focused on the natural frequency of the structure as a design goal, and the material layout of the structure is designed by topology optimization so that its eigenvalue is far from the excitation frequency to avoid resonance. In particular, existing work mostly focuses on maximizing the fundamental characteristic frequency of a structure or the interval between two adjacent characteristic frequencies. However, these methods do not guarantee that the natural frequency of the optimization results does not fall within the operating frequency. Thus, eigenvalue topology optimization models with band constraints were developed and applied to wing structural designs with multiband constraints. In addition, the natural frequency of the wing structure is extremely sensitive to material properties and structural dimensions. Even small manufacturing tolerances, wear and tear can result in changes in the natural frequency of the wing structure, thereby causing resonance. Therefore, the influence of uncertainty factors on the topological optimization of the wing structure should be considered in the design process so as to obtain a more safe and reliable structural design. Reliability-based wing topology optimization introduces reliability analysis into the topology optimization to address the impact of uncertainty factors on wing structural performance. Moreover, for environmental or external excitation with specific operating frequencies, it is of practical significance to perform wing reliability topology optimization with band constraints.
The existing topological optimization design mainly has the following problems when the wing structure is optimized:
for the wing structure optimization design, the reliability topology optimization with the band constraint needs to process the upper and lower bounds of the band simultaneously, which brings great computational burden to the reliability analysis of the band constraint. In addition, reliability analysis of multiple frequency band constraints can generate multiple minimum function target points, and the selection of the minimum function target points in the subsequent process can also bring great calculation cost. Furthermore, eigenvalue topology optimization may present local eigenvector and modal switching problems during the iteration process, which will lead to irreducible objective or constraint functions and to convergent oscillations such that a reasonable topology optimization model is not obtained.
Disclosure of Invention
In order to solve the technical problems, the invention provides a wing-oriented single-cycle reliability topology optimization design method, which not only can reduce the calculation cost, but also can obviously reduce the optimization time and simultaneously consider the multi-band constraint.
In order to achieve the above purpose, the present invention provides a wing-oriented single-cycle reliability topology optimization design method, which comprises:
performing discrete design on a wing structure design domain;
initializing design parameters, wherein the design parameters comprise: filter radius for density and sensitivity filtering, allowable reliability index for measuring allowable failure probability, initial design variable, initial random variable, penalty factor for SIMP interpolation model, lower and upper band bounds for band constraint, lower and upper functional displacement scalar for measuring uncertainty;
obtaining the characteristic frequency of the wing structure by solving a structural characteristic value equation; wherein the characteristic frequency comprises: single and multiple characteristic frequencies;
the sensitivity of the single characteristic frequency and the multi-characteristic frequency to the initial design variable and the initial random variable is respectively obtained; scalar Δs based on lower bound function displacement i And an upper bound function displacement scalar Δq l Establishing a Heaviside function for representing the band constraint and obtaining the sensitivity of the Heaviside function to the design variable; based on the sensitivity, constructing a wing reliability topology optimization model with uncertainty frequency band constraint;
and iterating the wing reliability topology optimization model to obtain a wing structure topology model.
Optionally, constructing the wing reliability topology optimization model includes:
performing a Heaviside filter on the sensitivity;
based on the filtered sensitivity, a minimum function target pointAnd->
Based on the minimum function target pointAnd->Acquiring new lower bound function displacement scalar and upper bound function displacement scalar, and constructing a wing reliability topology optimization model with uncertainty band constraint based on the function displacement scalar;
wherein the expression of the uncertainty band constraint is:
wherein ρ is (k) Representation ofDesign variable, ω, of the kth cycle j Represents the characteristic frequency of the j th order, mu x Mean value of random variable, V and V * Respectively represent the structural volume and volume factors, J 0 Represents the number of natural frequencies used to calculate the fundamental frequency, J represents the number of natural frequencies used to calculate the band constraint, N e The number of units is represented by the number of units,and v e Representing the density and volume of the cell, respectively. ζ represents the steepness of the Heaviside function, P represents the index factor of the P-norm in the Heaviside function, which is used to calculate the band constraint. />And->Intermediate frequency value and half bandwidth, respectively, representing frequency bands, < >>And->Respectively represent the lower boundary of the mth frequency band and the upper boundary of the mth frequency band,/for>And->Respectively representing an mth lower bound functional displacement scalar and an mth upper bound functional displacement scalar.
Optionally, iterating the wing reliability topology optimization model includes:
calculating the wing reliability topology optimization model, and updating design variables according to a moving gradual-in line method;
judging whether the design variable is converged or not, and stopping iteration if the design variable is converged; otherwise, setting k=k+1, and re-acquiring the sensitivity;
and obtaining a wing structure topology model according to the converged design variables.
Optionally, solving the structural eigenvalue equation:
wherein,representing the j-th order eigenvector, K and M representing the overall stiffness matrix and the overall mass matrix, the unit stiffness matrix K e And a cell quality matrix M e Combining to obtain; the unit stiffness matrix and the unit mass matrix are calculated by adopting an improved SIMP interpolation model:
wherein r and q represent penalty factors for the SIMP interpolation model, ρ e The design variables are represented by the values of the design variables,and->Representing a matrix of cell stiffness of two materials, +.>And->Representing a matrix of cell masses of two materials.
Optionally, the functional displacement scalar includes: lower boundary ofScalar of functional displacementAnd upper bound function displacement scalar
The lower bound function displacement scalarThe method comprises the following steps:
wherein,representing a performance function>Representing equivalent deterministic constraints ρ (k) Design variable representing the kth cycle, +.>Representing a kth cycle original space lower bound minimum performance target point;
the function displacement scalarThe method comprises the following steps:
wherein,representing a performance function>Represents equivalent deterministic constraints->Representing the upper bound minimum performance target point in the original space of the kth cycle.
Optionally, acquiring the sensitivities of the single and multiple characteristic frequencies to the design variable and the random variable, respectively, includes:
if the jth characteristic frequency is a single characteristic frequency, the characteristic frequencies of adjacent orders satisfy ω j-1 ＜ω j ＜ω j+1 The method comprises the steps of carrying out a first treatment on the surface of the Obtaining the sensitivity of the single characteristic frequency to the design variable and the random variable:
wherein x is i Representing random variables, taking into account the uncertainty of the material properties and structural dimensions of the wing structure,representing differentiation;
if the characteristic frequency is satisfiedThen->Is a group of multi-characteristic frequencies, the weight of which is n 2 -n 1 +1; acquiring the sensitivity of the multi-characteristic frequency to the design variable:
wherein,and->Representing the feature vector, I representing the cell matrix; solving to obtain characteristic value->Representing the sensitivity of the multiple characteristic frequencies to the design variable>
Acquiring the sensitivity of the multi-characteristic frequency to the random variable:
solving to obtain a characteristic valueRepresenting the sensitivity of multiple characteristic frequencies to random variables
Using a functional displacement scalarAnd->A Heaviside function is established for representing the band constraint, and the purpose of the Heaviside function is to optimize the wing structure for obtaining the characteristic frequency avoiding working frequency band:
wherein,and->Respectively representing the intermediate frequency value and half bandwidth of the frequency band, and xi represents the steepness of the Heaviside function; p-norm agglomerating the above band constraints:
wherein J represents the number of characteristic frequencies used for calculating the band constraint, and P represents an index factor of P-norm. Obtaining sensitivity of band constraints to the random variables:
optionally, the method of performing a Heaviside filter on the sensitivity is:
wherein eta (d) ，η (i) ，η (e) Representing three predictive cut-off thresholds, θ represents the degree of nonlinearity of the mapping function,representing the physical field of expansion>Representing the intermediate physical field +.>Representing the erosive physical field.
Compared with the prior art, the invention has the following advantages and technical effects:
reliability-based wing topology optimization with band constraints requires the simultaneous processing of upper and lower bounds, which places a heavy computational burden on band reliability constraints. However, eigenvalue topology optimization may have local eigenvalue modes and modal switching, which may lead to non-differentiable objective or constraint functions and converging oscillations. For this purpose, a reliability-based wing topology optimization model of the uncertain frequency band constraint is first established.
The new wing topology optimization model based on reliability is established according to the proposed frequency band constraint displacement method, and the uncertainty of material characteristics and structural dimensions is considered. The model provides more optimized layouts than deterministic topology optimization. Since the computational cost of the band reliability constraint is high, a band constraint shift method is proposed to reduce the computational burden with excellent accuracy and to handle the multiband constraint. In addition, the reliability-based wing topology optimization problem is solved by adopting a moving asymptote method, and the sensitivity of single characteristic values and multiple characteristic values to design and random variables is determined. Local eigenmodes are a common problem in eigenvalue topology optimization, and typically occur in sub-regions where material density values are small. However, wing topology optimization of a bi-material structure can avoid local eigenmodes by filling the void with another material. The mode switching results in a target or constraint function that is not differentiable, resulting in a non-convergence problem. To mitigate the effects of mode conversion, a "boundary formula" is employed. Furthermore, a "robust formula" is applied to avoid checkerboard phenomena and gray elements.
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The accompanying drawings, which are included to provide a further understanding of the application, illustrate and explain the application and are not to be construed as limiting the application. In the drawings:
FIG. 1 is a schematic diagram of an implementation flow of a wing design of a topology optimization model based on single cycle reliability in accordance with an embodiment of the present invention;
FIG. 2 is a schematic diagram of a case design domain and boundary conditions according to an embodiment of the present invention;
FIG. 3 is a schematic diagram of an optimization result according to an embodiment of the present invention; wherein (a) is an optimization result of case 1, (b) is an optimization result of case 2, (c) is an optimization result of case 3, (d) is an optimization result of case 4, (e) is an optimization result of case 5, (f) is an optimization result of case 6, (g) is an optimization result of case 7, and (h) is an optimization result of case 8;
FIG. 4 is a schematic diagram of frequency iteration curves for cases 1,4,5, and 8 according to an embodiment of the present invention; wherein, (a) is a frequency iteration curve of case 1, (b) is a frequency iteration curve of case 2, (c) is a frequency iteration curve of case 5, and (d) is a frequency iteration curve of case 6.
Detailed Description
It should be noted that, in the case of no conflict, the embodiments and features in the embodiments may be combined with each other. The present application will be described in detail below with reference to the accompanying drawings in conjunction with embodiments.
It should be noted that the steps illustrated in the flowcharts of the figures may be performed in a computer system such as a set of computer executable instructions, and that although a logical order is illustrated in the flowcharts, in some cases the steps illustrated or described may be performed in an order other than that illustrated herein.
As shown in fig. 1, the invention provides a wing-oriented single-cycle reliability topology optimization design method,
the method comprises the following steps:
step 1: and carrying out discrete design on the wing structure design domain according to boundary conditions. The calculation cost can be reduced through discrete design, and the continuous variable can be converted into discrete variable to facilitate design parameter definition.
Step 2: parameter definition including filter radius r min Reliability index beta i t Initial design variable ρ e Initial random variable x (k) Penalty factors r and q, frequencyWith constraint lower bound omega low And an upper boundary omega upp Lower bound function displacement scalarAnd upper bound function displacement scalar ++>
Step 3: obtaining the characteristic frequency of the wing structure by solving a structural characteristic value equation; wherein the characteristic frequency comprises: single and multiple characteristic frequencies;
solving a structural eigenvalue equation:
wherein omega j Representing the characteristic frequency of the j-th order,representing the j-th order eigenvector, K and M representing the overall stiffness matrix and the overall mass matrix, the unit stiffness matrix K e And a cell quality matrix M e Combining to obtain; the unit stiffness matrix and the unit mass matrix are calculated by adopting an improved SIMP interpolation model:
wherein r and q represent penalty factors for the SIMP interpolation model, ρ e The design variables are represented by the values of the design variables,and->Representing a matrix of cell stiffness of two materials, +.>And->Representing a matrix of cell masses of two materials.
Step 4: the sensitivity of the single and multiple characteristic frequencies to the design variable and the random variable are calculated separately. In a complex structure having many degrees of freedom, the vibration structure often exhibits a phenomenon of multiple eigenvalues with partially identical eigenvalues. In contrast, the sensitivity of the single characteristic frequency and the multi-characteristic frequency corresponding to the single characteristic frequency and the multi-characteristic frequency to the design variable and the random variable (namely the initial design variable and the initial random variable) is respectively obtained; a Heaviside function is built based on the lower and upper functional displacement scalars for representing the band constraints and obtaining its sensitivity to the design variables.
Step 5: the sensitivity was subjected to a Heaviside filter.
Wherein eta (d) ，η (i) ，η (e) Three prediction cut-off thresholds are set to 0.3,0.5,0.7, respectively. θ represents the degree of nonlinearity of the mapping function, and θ=0 means that it satisfies a linear relationship. Calculating a filtered density field using convolution density filtering:
wherein n is j And the coordinate vector representing the center point of the j-th element. w (n) j )＝r min -||n j -n e I is a linear weight function. |·| denotes 2 norms and Σ e ＝{j|||n j -n e ||≤r min Expressed in terms of filter radius r min The designated circular area is a neighborhood set of cells centered.
Step 5: calculating a minimum function target pointAnd->(minimum functional target point in the kth cycle).
Step 6: calculating new lower bound function displacement scalarAnd upper bound function displacement scalar ++>And a new wing reliability topology optimization model with uncertainty band constraint is established.
Step 6-1: further, the new wing reliability topology optimization model with uncertainty band constraint is:
solving the cell density maximizing the characteristic frequency under the constraint condition, wherein the constraint condition is as follows:
wherein ρ is (k) Is the design variable for the kth cycle, μ x Is the mean value of random variables, V and V * Respectively represent the structural volume and volume factors, J 0 Represents the number of natural frequencies used to calculate the fundamental frequency, J represents the number of natural frequencies used to calculate the band constraint, N e The number of units is represented by the number of units,and v e Representing the density and volume of the cell, respectively. ζ represents the steepness of the Heaviside function, P represents the index factor of the P-norm in the Heaviside function, which is used to calculate the band constraint.And->Intermediate frequency value and half bandwidth, respectively, representing frequency bands, < >>And->Respectively represent the lower boundary of the mth frequency band and the upper boundary of the mth frequency band,/for>And->Respectively representing an mth lower bound functional displacement scalar and an mth upper bound functional displacement scalar.
Step 6-2: further, the function displacement scalarAnd->Can be calculated by respectively
Wherein the method comprises the steps ofAnd->Respectively representing a lower bound minimum function target point and an upper bound minimum function target point in the original space in the kth cycle.
Step 7: and calculating an airfoil optimization model, and updating design variables according to a moving gradual-in line method.
Step 8: it is determined whether the design variables converge. If they converge, the iteration is stopped. Otherwise, set k=k+1, return to step 3.
Step 9: and giving a wing structure topological model according to the converged design variables.
The wing design of the single-cycle reliability topology optimization model provided by the embodiment can reduce the calculation cost, can obviously reduce the optimization time, and can simultaneously consider the multiband constraint.
As shown in fig. 2, the embodiment further uses the NACA0018 airfoil as a research object, and further describes a wing-oriented single-cycle reliability topology optimization design method, and the model size is 5m×1m. The parameters of the two materials used optimally are shown in table 1. The model takes into account the uncertainties of the Young's modulus and the material density of the structural material, and both obey Gumbel distribution. Mean and branchThe cloth and variability coefficients are listed in Table 2, E 1 And E is 2 Represents Young's modulus of two materials, M 1 And M 2 Indicating their material density. The filter radius is chosen to be five times the cell size. Allowable volume fraction V of material 1 in optimization model * =0.3. Consider a single frequency band [8501150 ]]rad/s and dual band [550750 ]]∪[16501750]Optimizing model under rad/s constraint and considering reliability index respectively
TABLE 1 Material Properties
TABLE 2 random variable
The sensitivity of the single characteristic frequency to cell density is then calculated:
wherein the method comprises the steps ofAnd->Calculated by the following formula:
calculating the sensitivity of the multi-characteristic frequency to the unit density:
wherein,and->Representing the feature vector, I representing the cell matrix; solving to obtain characteristic value->Representing the sensitivity of multiple characteristic frequencies to design variables
The sensitivity of the single characteristic frequency to random variables (elastic modulus and material density) was calculated:
and->Calculated by the following formula:
the sensitivity of the multi-characteristic frequency to random variables (elastic modulus and material density) was calculated:
solving to obtain a characteristic valueRepresenting the sensitivity of multiple characteristic frequencies to random variables
Using a functional displacement scalarAnd->A Heaviside function is established for representing the band constraint, and the purpose of the Heaviside function is to optimize the wing structure for obtaining the characteristic frequency avoiding working frequency band:
wherein,and->Respectively representing the intermediate frequency value and half bandwidth of the frequency band, and xi represents the steepness of the Heaviside function; p-norm agglomerating the above band constraints:
wherein J represents the number of characteristic frequencies used for calculating the band constraint, and P represents an index factor of P-norm. Sensitivity of band constraint to cell density:
wherein the method comprises the steps ofCalculated by the following formula:
the sensitivity was subjected to a Heaviside filter.
Wherein eta (d) ，η (i) ，η (e) Three prediction cut-off thresholds are set to 0.3,0.5,0.7, respectively. θ represents the degree of nonlinearity of the mapping function, and θ=0 means that it satisfies a linear relationship. Calculating a filtered density field using convolution density filtering:
wherein n is j And the coordinate vector representing the center point of the j-th element. w (n) j )＝r min -||n j -n e I is a linear weight function. |·| denotes 2 norms and Σ e ＝{j|||n j -n e ||≤r min Expressed in terms of filter radius r min The designated circular area is a neighborhood set of cells centered.
Calculating a functional displacement scalarAnd->
Wherein the method comprises the steps ofAnd->Respectively representing a lower bound minimum function target point and an upper bound minimum function target point in the original space in the kth cycle
Then a new wing reliability topology optimization model with uncertainty band constraint is established:
solving the cell density maximizing the characteristic frequency under the condition that constraint conditions are satisfied, wherein the constraint conditions are as follows:
wherein ρ is (k) Is the design variable for the kth cycle, μ x Is the mean of the random variables.
And then calculating an airfoil optimization model, and updating design variables according to a moving gradual-in line method.
It is determined whether the design variable (cell density) converges. If they converge, the iteration is stopped. Otherwise, setting k=k+1, and returning to calculate the sensitivity of the single characteristic frequency and the multi-characteristic frequency to the design variable and the random variable.
And finally, giving a wing structure topological optimization model according to the converged design variables.
FIG. 3 shows wing optimization results under different constraints, black and grey representing materials 1 and 2, respectively; wherein fig. 3 (a) is an optimization result of case 1, fig. 3 (b) is an optimization result of case 2, fig. 3 (c) is an optimization result of case 3, fig. 3 (d) is an optimization result of case 4, fig. 3 (e) is an optimization result of case 5, fig. 3 (f) is an optimization result of case 6, fig. 3 (g) is an optimization result of case 7, and fig. 3 (h) is an optimization result of case 8. Table 3 lists the first 8 characteristic frequencies. And a comparison with the results of the optimization of the two-cycle process is given in table 4. Fig. 4 shows the frequency iteration curves for cases 1, 2, 5, 6. The method comprises the following steps: fig. 4 (a) is a frequency iteration curve of case 1, fig. 4 (b) is a frequency iteration curve of case 2, fig. 4 (c) is a frequency iteration curve of case 5, and fig. 4 (d) is a frequency iteration curve of case 6.
TABLE 3 optimization results
TABLE 4 comparison of optimization efficiency
The optimization result shows that the deterministic topology optimization and the material layout based on the reliable topology optimization have larger difference, which indicates that the uncertainty of the structural parameters can have larger influence on the optimized structure. Compared with deterministic topology optimization under single-band constraint, the reliability-based topology optimization result has wider frequency band, and safer design result is ensured. In the reliability index beta i t At=2, 3, 4, the optimized single-band bandwidths show a tendency to increase gradually, 322,391, and 462rad/s, respectively. Similar to the single band, the reliability topology optimization result for the dual band has a wider band and gradually increases in bandwidth. Such a trend of variation ensures safer design results, indicating the importance of considering uncertain parameters in the optimization process. Table 4 gives a comparison of the optimization efficiencies, wherein the calculation times for the single-cycle method are 15136s, 15662s, 15006s, 17342s, 17344s and 17602s, respectively, and the calculation times for the double-cycle method are 29174s, 31856s, 30650s, 44140s, 44602s and 45172s, respectively. And the calculation result of the proposed single-cycle method is consistent with the calculation result of double-cycle optimization. However, the computational efficiency of the present invention is approximately twice that of the two-cycle optimization, which suggests that the proposed wing design based on reliability topology optimization can significantly reduce the computational cost of the wing optimization design process.
The beneficial effects brought by the embodiment are as follows:
reliability-based wing topology optimization with band constraints requires the simultaneous processing of upper and lower bounds, which places a heavy computational burden on band reliability constraints. However, eigenvalue topology optimization may have local eigenvalue modes and modal switching, which may lead to non-differentiable objective or constraint functions and converging oscillations. For this purpose, a reliability-based wing topology optimization model of the uncertain frequency band constraint is first established.
The new wing topology optimization model based on reliability is established according to the proposed frequency band constraint displacement method, and the uncertainty of material characteristics and structural dimensions is considered. The model provides more optimized layouts than deterministic topology optimization. Since the computational cost of the band reliability constraint is high, a band constraint shift method is proposed to reduce the computational burden with excellent accuracy and to handle the multiband constraint. In addition, the reliability-based wing topology optimization problem is solved by adopting a moving asymptote method, and the sensitivity of single characteristic values and multiple characteristic values to design and random variables is determined. Local eigenmodes are a common problem in eigenvalue topology optimization, and typically occur in sub-regions where material density values are small. However, wing topology optimization of a bi-material structure can avoid local eigenmodes by filling the void with another material. The mode switching results in a target or constraint function that is not differentiable, resulting in a non-convergence problem. To mitigate the effects of mode conversion, a "boundary formula" is employed. Furthermore, a "robust formula" is applied to avoid checkerboard phenomena and gray elements.
The foregoing is merely a preferred embodiment of the present application, but the scope of the present application is not limited thereto, and any changes or substitutions easily conceivable by those skilled in the art within the technical scope of the present application should be covered in the scope of the present application. Therefore, the protection scope of the present application shall be subject to the protection scope of the claims.

Claims (6)

1. A wing-oriented single-cycle reliability topology optimization design method is characterized by comprising the following steps:
performing discrete design on a wing structure design domain;
initializing design parameters, wherein the design parameters comprise: filter radius for density and sensitivity filtering, allowable reliability index for measuring allowable failure probability, initial design variable, initial random variable, penalty factor for SIMP interpolation model, band lower bound ω for band constraint low And upper boundary ω of frequency band upp Lower bound functional displacement scalar for measuring uncertaintyAnd upper bound function displacement scalar ++>
Obtaining the characteristic frequency of the wing structure by solving a structural characteristic value equation; wherein the characteristic frequency comprises: single and multiple characteristic frequencies;
the sensitivity of the single characteristic frequency and the multi-characteristic frequency to the initial design variable and the initial random variable is respectively obtained; establishing a Heaviside function based on the lower-bound function displacement scalar and the upper-bound function displacement scalar, wherein the Heaviside function is used for representing the band constraint and acquiring the sensitivity of the Heaviside function to the design variable; based on the sensitivity, constructing a wing reliability topology optimization model with uncertainty frequency band constraint;
iterating the wing reliability topology optimization model to obtain a wing structure topology model;
the building of the wing reliability topology optimization model comprises the following steps:
performing a Heaviside filter on the sensitivity;
based on the filtered sensitivity, a minimum function target point is acquiredAnd->
Based on the minimum function target pointAnd->Acquiring a new lower bound function displacement scalar +.>And upper bound function displacement scalar ++>Constructing a wing reliability topology optimization model with uncertainty band constraint based on the functional displacement scalar;
wherein the expression of the uncertainty band constraint is:
wherein ρ is (k) Design variable, ω, representing the kth cycle j Represents the characteristic frequency of the j th order, mu x Mean value of random variable, V and V * Respectively represent the structural volume and volume factors, J 0 Represents the number of natural frequencies used to calculate the fundamental frequency, J represents the number of natural frequencies used to calculate the band constraint, N e The number of units is represented by the number of units,and v e Representing the density and volume of the cell, respectively, ζ represents the steepness of the Heaviside function, P represents the index factor of the P-norm in the Heaviside function, which is used to calculate the band constraint,/>And->Intermediate frequency value and half bandwidth, respectively, representing frequency bands, < >>And->Respectively represent the lower boundary of the mth frequency band and the upper boundary of the mth frequency band,/for>And->Respectively representing an mth lower bound functional displacement scalar and an mth upper bound functional displacement scalar.
2. The wing-oriented single-loop reliability topology optimization design method of claim 1, wherein iterating the wing reliability topology optimization model comprises:
calculating the wing reliability topology optimization model, and updating design variables according to a moving gradual-in line method;
judging whether the design variable is converged or not, and stopping iteration if the design variable is converged; otherwise, setting k=k+1, and re-acquiring the sensitivity;
and obtaining a wing structure topology model according to the converged design variables.
3. The wing-oriented single-cycle reliability topology optimization design method of claim 1, wherein the structural eigenvalue equation is solved:
wherein the method comprises the steps of，Represents the j-th order eigenvector, ω j Represents the j-th characteristic frequency, K and M represent the overall stiffness matrix and the overall mass matrix, and the unit stiffness matrix K e And a cell quality matrix M e Combining to obtain; the unit stiffness matrix and the unit mass matrix are calculated by adopting an improved SIMP interpolation model:
wherein r and q represent penalty factors for the SIMP interpolation model, ρ e Representing the design variable (cell density),and->Representing a matrix of cell stiffness of two materials, +.>And->Representing a matrix of cell masses of two materials.
4. The wing-oriented single cycle reliability topology optimization design method of claim 1, wherein said functional displacement scalar comprises: lower bound function displacement scalarAnd upper bound function displacement scalar ++>
The lower bound function displacement scalarThe method comprises the following steps:
wherein,representing a performance function>Representing equivalent deterministic constraints ρ (k) Design variable representing the kth cycle, +.>Representing a kth cycle original space lower bound minimum performance target point;
the function displacement scalarThe method comprises the following steps:
wherein,representing a performance function>Represents equivalent deterministic constraints->Representing the upper bound minimum performance target point in the original space of the kth cycle.
5. The wing-oriented single-cycle reliability topology optimization design method of claim 1, wherein respectively obtaining the sensitivity of the single characteristic frequency and the multiple characteristic frequencies to the design variable and the random variable comprises:
if the jth characteristic frequency is a single characteristic frequency, the characteristic frequencies of adjacent orders satisfy ω j-1 ＜ω j ＜ω j+1 The method comprises the steps of carrying out a first treatment on the surface of the Obtaining the sensitivity of the single characteristic frequency to the design variable and the random variable:
wherein x is i Representing random variables, taking into account the uncertainty of the material properties and structural dimensions of the wing structure,representing differentiation;
if the characteristic frequency is satisfiedThen->Is a group of multi-characteristic frequencies, the weight of which is n 2 -n 1 +1; acquiring the sensitivity of the multi-characteristic frequency to the design variable:
wherein,and->Representing the feature vector, I representing the cell matrix; solving to obtain characteristic value->Representing the sensitivity of the multiple characteristic frequencies to the design variable>
Acquiring the sensitivity of the multi-characteristic frequency to the random variable:
solving to obtain a characteristic valueRepresenting the sensitivity of multiple characteristic frequencies to random variables
Using a functional displacement scalarAnd->The Heaviside function is built to represent the band constraints for the purpose ofOptimizing to obtain a wing structure with characteristic frequency avoiding working frequency band:
wherein,and->Respectively representing the intermediate frequency value and half bandwidth of the frequency band, and xi represents the steepness of the Heaviside function; p-norm agglomerating the above band constraints:
wherein J represents the number of characteristic frequencies used for calculating the band constraint, and P represents an index factor of P-norm; obtaining sensitivity of band constraints to the random variables:
6. the wing-oriented single-cycle reliability topology optimization design method of claim 1, wherein the method of performing a Heaviside filtering on the sensitivity is:
wherein eta (d) ，η (i) ，η (e) Representing three predictive cut-off thresholds, θ represents the degree of nonlinearity of the mapping function,representing the physical field of expansion>Representing the intermediate physical field +.>Representing the erosive physical field.
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