CN105912809A - Structure steady design method with uncertain load action point position - Google Patents

Abstract

The invention relates to a structure steady design method with an uncertain load action point position. The structure steady design method with the uncertain load action point position comprises the following steps: S1: establishing a continuous body structure steady optimization model; S2: according to a distribution form of the load action point position borne by the structure, and a Gaussian quadrature node position, selecting a plurality of foundation working conditions; S3: initializing a design variable; S4: carrying out finite element analysis on each foundation working condition to obtain a corresponding structure displacement vector; S5: calculating the mean value and the standard deviation of structural flexibility; S6: carrying out sensitivity analysis on the structure; S7: utilizing an asymptote method to update the design variable; and S8: judging whether a termination condition can be met or not, stopping iteration if the termination condition can be met, and otherwise, repeating S4 to S7 until an iteration termination condition is met. The structure steady design method has the advantages of being small in calculation amount, high in relative precision, and simple in implementation on the basis of a traditional topological optimization foundation.

Description

A kind of load position of action point uncertain structure robust design method

Technical field

The present invention relates to structural Topology Optimization technical field, particularly relate to a kind of uncertain structure of load position of action point Robust design method.

Background technology

Under normal circumstances, structural Topology Optimization completes under certain conditions, but this optimum structure is outside opposing Being likely to be comparison during boundary's disturbance fragile, therefore the structural Topology Optimization problem under research condition of uncertainty is necessary 's.There is all kinds of uncertainty in actual design process, as load is uncertain, material is uncertain, and manufacture process produces Physical dimension and border are uncertain.Load is uncertain is divided into again that magnitude of load is uncertain, loading direction is uncertain and load is made Uncertain with a position.The uncertain impact on structural system of load position of action point is relatively big, designs relative to deterministic optimization Method, it is considered to load position of action point probabilistic Robust Optimal Design method can obtain more stable structure.Load The technological difficulties of position of action point uncertain structure robust design method are that the calculating application point of how precise and high efficiency is uncertain The average of structural compliance and standard deviation and its sensitivity to design variable under load.

The optimization aim of random optimization design method of the prior art is that quality minimizes, and is constrained to load position the most true Fix the probability decision degree of flexibility.In prior art, probabilistic type optimum topology design method mainly processes load position of action point and is The structure of the continuous type under random case, its Optimized model is to minimize discrete topology weight under the constraint on flexibility and border Weighted sum.Above two method be mainly used in load position of action point uncertain under non-individual body topology based on reliability excellent Changing, and optimization is to be minimised as target with architecture quality, the probability that flexibility meets is constraint.

Prior art lacks based on load position of action point uncertain robustness topological optimization.But, Practical Project Middle load position of action point is uncertain the most common, and research load position of action point uncertain robustness topological optimization has very much must Want.

Summary of the invention

The technical problem to be solved is: how to provide a kind of uncertain robustness of load position of action point to open up Flutter optimization method.

For solving above-mentioned technical problem, the present invention proposes a kind of load position of action point uncertain structure based Robust Design Method, the method load position of action point uncertain structure robust design method includes:

S1: set up the sane Optimized model of Continuum Structure:

S2: the distribution form of the load position of action point born according to structure and Gaussian quadrature node location, choose Multiple bases operating mode；

S3: initialization design variable；

S4: each basis operating mode is carried out finite element analysis and obtains corresponding displacement structure vector；

S5: the average of computation structure flexibility and standard deviation；

S6: structure is carried out sensitivity analysis；

S7: utilize asymptote method to update design variable；

S8: judge whether to meet end condition, if meeting, then stops iteration；Otherwise repeat S4 to S7 until meeting iteration End condition.

Alternatively, the sane Optimized model of described Continuum Structure includes:

S.t.:Ku (ω)=f (ω) (ω ∈ Θ)

0≤ρ≤1

Wherein, J is that object function, μ (c) and σ (c) are respectively the average of structural compliance c and standard deviation, α and β be two non- Bear real number and meet alpha+beta=1；K is structure Bulk stiffness matrix, f (ω) and u (ω) be respectively load and the displacement of structure to Amount, ω is for representing the uncertainty of load position of action point；M is discrete number of unit, v_eAnd ρ_eIt is respectively the volume of unit And density, g is the inequality constraints of the sane Optimized model of Continuum Structure, and V is to calculate the volumetric usage obtained, V every time_maxFor Given material volume consumption；ρ is by ρ_e(e=1 ..., vector m) formed.

Alternatively, choose multiple bases operating mode described in include:

According to the probability-distribution function of load position of action point, find the Gaussian quadrature node of its correspondence, each point is executed Add load and set up a kind of operating mode, select n basis operating mode f₁,…,f_n；

Wherein, n is the quadrature nodes of Gauss integration.

Alternatively, described initialization design variable includes:

Identical density is given by all of unit

Alternatively, described each basis operating mode is carried out finite element analysis obtain corresponding displacement structure vector and include:

To each basis operating mode f₁,…,f_nCarry out finite element finite meta analysis and obtain corresponding displacement structure vector u₁,…, u_n。

Alternatively, the average of described computation structure flexibility includes with standard deviation:

The average of structural compliance is:

The variance of structural compliance is:

Wherein, c (x_i, ρ) and=f (x_i)u(x_i) represent load f (x_i) at displacement components u (x_i) upper power, x_iRepresent i-th Operating loading position of action point, A_iRepresent Gaussian quadrature coefficient.

Alternatively, described structure carried out sensitivity analysis include:

According to the expression formula of structural compliance average Yu variance, object function is for each unit density p_ePartial derivative be:

Retraining the sensitivity for each unit density is:

Alternatively, described utilize asymptote method update design variable include:

Approximate function according to the sensitivity information structure object function that sensitivity analysis obtains:

Wherein, ρ^(k)For current iteration point,WithFor mobile asymptote, its Iteration is:

For design variable ρ_eCurrency,WithIt is respectively ρ_eBound；

This approximate function is utilized to replace the object function in former optimization problem to form optimization subproblem；

Design variable after utilizing Dual Method to solve this optimization subproblem and being updated.

Alternatively, described in judge whether to meet end condition, if meeting, then stop iteration, otherwise repeat S4 to S7 until Meet stopping criterion for iteration to include:

If iterations has reached maximum times or twice iteration variable density amount of various discrete unit is respectively less than given Value, then stop iteration；Otherwise continue iterative process.

The load position of action point uncertain structure robust design method that the present invention provides, based on known load effect The point regularity of distribution, regards by structural compliance as the function of position of action point as, uses displacement superposed principle and the Gauss of linear structure The flexibility average of quadrature formula computation structure and standard deviation, and object function is carried out sensitivity analysis.The method amount of calculation is little And relative accuracy is higher, implement the simplest on the basis of existing topological optimization.

Accompanying drawing explanation

By being more clearly understood from the features and advantages of the present invention with reference to accompanying drawing, accompanying drawing is schematic and should not manage Solve as the present invention is carried out any restriction, in the accompanying drawings:

Fig. 1 shows the stream of the load position of action point uncertain structure robust design method of one embodiment of the invention Journey schematic diagram；

Fig. 2 shows the foundation structure of one embodiment of the invention and border and load-up condition；

Fig. 3 a, Fig. 3 b, Fig. 3 c show three kinds of load position of action point distribution schematic diagrams of one embodiment of the invention；

Fig. 4 shows and does not considers the probabilistic optimum results of load position of action point；

Fig. 5 shows the consideration load position of action point sane optimum results of uncertain structure.

Detailed description of the invention

Below in conjunction with accompanying drawing, embodiments of the present invention is described in detail.

Fig. 1 shows the stream of the load position of action point uncertain structure robust design method of one embodiment of the invention Journey schematic diagram.As it is shown in figure 1, the load position of action point uncertain structure robust design method of this embodiment, including:

S1: set up the sane Optimized model of Continuum Structure:

S2: the distribution form of the load position of action point born according to structure and Gaussian quadrature node location, choose Multiple bases operating mode；

S3: initialization design variable；

S4: each basis operating mode is carried out finite element analysis and obtains corresponding displacement structure vector；

S5: the average of computation structure flexibility and standard deviation；

S6: structure is carried out sensitivity analysis；

S7: utilize asymptote method to update design variable；

S8: judge whether to meet end condition, if meeting, then stops iteration；Otherwise repeat S4 to S7 until meeting iteration End condition.

The load position of action point uncertain structure robust design method of the present embodiment, based on known load application point The regularity of distribution, regards by structural compliance as the function of position of action point as, uses the displacement superposed principle of linear structure and Gauss to ask The flexibility average of long-pending formula computation structure and standard deviation, and object function is carried out sensitivity analysis.The method amount of calculation little and Relative accuracy is higher, implements the simplest on the basis of existing topological optimization.

In the optional embodiment of one, the sane Optimized model of described Continuum Structure includes:

S.t.:Ku (ω)=f (ω) (ω ∈ Θ)

0≤ρ≤1

Wherein, J is that object function, μ (c) and σ (c) are respectively the average of structural compliance c and standard deviation, α and β be two non- Bear real number and meet alpha+beta=1；K is structure Bulk stiffness matrix, f (ω) and u (ω) be respectively load and the displacement of structure to Amount, ω is for representing the uncertainty of load position of action point；M is discrete number of unit, v_eAnd ρ_eIt is respectively the volume of unit And density, g is the inequality constraints of the sane Optimized model of Continuum Structure, and V is to calculate the volumetric usage obtained, V every time_maxFor Given material volume consumption；ρ is by ρ_e(e=1 ..., vector m) formed.

Foundation and the method for " the setting up the sane Optimized model of Continuum Structure " of the embodiment of the present invention be: in certain load Under the conditions of, the flexibility of structure is the least, and its ability bearing external applied load is the strongest.Under load position of action point condition of uncertainty, knot The flexibility of structure can change along with the change of load position, the mean μ (c) of structural compliance to be reduced during design structure, also The fluctuation that structural compliance to be reduced causes with load change, i.e. reduces the standard deviation sigma (c) of structural compliance.The present invention is with continuously The minimum target of weighted sum J=α μ (c)+β σ (c) of body structural compliance mean μ (c) and standard deviation sigma (c) sets up Optimized model, number Learn and express simple and physical significance definitely.

Further, choose multiple bases operating mode described in include:

According to the probability-distribution function of load position of action point, find the Gaussian quadrature node of its correspondence, each point is executed Add load and set up a kind of operating mode, select n basis operating mode f₁,…,f_n；

Wherein, n is the quadrature nodes of Gauss integration.

Further, described initialization design variable includes:

Identical density is given by all of unit

Described each basis operating mode is carried out finite element analysis obtain corresponding displacement structure vector and include:

To each basis operating mode f₁,…,f_nCarry out finite element finite meta analysis and obtain corresponding displacement structure vector u₁,…, u_n。

Specifically, the average of described computation structure flexibility includes with standard deviation:

The average of structural compliance is:

The variance of structural compliance is:

Wherein, c (x_i, ρ) and=f (x_i)u(x_i) represent load f (x_i) at displacement components u (x_i) upper power, x_iRepresent i-th Operating loading position of action point, A_iRepresent Gaussian quadrature coefficient.

Described structure carried out sensitivity analysis include:

According to the expression formula of structural compliance average Yu variance, object function is for each unit density p_ePartial derivative be:

Retraining the sensitivity for each unit density is:

Described utilize asymptote method update design variable include:

Approximate function according to the sensitivity information structure object function that sensitivity analysis obtains:

Wherein, ρ^(k)For current iteration point,WithFor mobile asymptote, its Iteration is:

For design variable ρ_eCurrency,WithIt is respectively ρ_eBound；

This approximate function is utilized to replace the object function in former optimization problem to form optimization subproblem；

Design variable after utilizing Dual Method to solve this optimization subproblem and being updated.

Specifically, described in judge whether to meet end condition, if meeting, then stop iteration, otherwise repeat S4 to S7 until Meet stopping criterion for iteration to include:

If iterations has reached maximum times or twice iteration variable density amount of various discrete unit is respectively less than given Value, then stop iteration；Otherwise continue iterative process.

The load position of action point uncertain structure robust design method of the present embodiment, based on Gauss quadrature formula, will carry The uncertain topology optimization problem of lotus position of action point is converted into multi-state certainty topological optimization form, to a small amount of load working condition Carry out structural analysis and i.e. can get high precision solution；Theory of algorithm basis is simple herein, it is easy to implement and the scope of application is wider, can Being applicable to load position of action point is the uncertainty under various distribution.

It should be noted that the essence of the load position of action point uncertain structure robust design method of the embodiment of the present invention Degree, amount of calculation all quantity to Gaussian quadrature node is directly proportional；Generally, a small amount of quadrature node i.e. can reach precision and wants Ask.The present invention can simply be generalized to the based Robust Design of three-dimensional structure, therefore has higher to structure design in Practical Project Applicability.

The present invention is described below as a example by the robust design method of a concrete Two-dimension Continuum.As in figure 2 it is shown, F is concentrfated load size and Orientation.The length and width of this structure a size of 100 × 100, wherein structure lower end is fixed, and holds in upper end By uncertain load F=10N of position of action point.The uncertainty of load is represented by load position of action point x, and obeys uniformly Distribution, distributed area is [-8 ,+8].The Young's modulus of bar material is 1, and Poisson's ratio is 0.3, and it is 10000 that all materials amasss, and adopts With consistent per-unit system.This structure is carried out Robust Design Optimization so that it is at specified load position of action point condition of uncertainty Under the weighted sum of structural compliance average and standard deviation minimum.

The method specifically comprises the following steps that

S1: set up the sane Optimized model of structure:

S.t.:Ku (ω)=f (ω) (ω ∈ Θ)

0≤ρ≤1

Wherein J is average and standard deviation, α=0.8, β=0.2 that object function, μ (c) and σ (c) are respectively structural compliance； K is structure Bulk stiffness matrix, and f (ω) and u (ω) is respectively load and the motion vector of structure, and ω is used for representing load effect The uncertainty of some position；V_max=2000 is given material volume consumption；ρ is the vector being made up of individual cell density.

S2: according to Gaussian integrating formula, chooses three quadrature nodes, i.e. load position of action point is respectively 43.80, and 50, 56.20, form three basic operating modes, such as Fig. 3 a, Fig. 3 b, Fig. 3 c, wherein, sized by x and direction determine but position of action point not Determine load f_xActive position；f_xFor the size of load at application point x.

S3: initialization design variable.The density taking each unit is 0.2.

S4: three basic operating modes are carried out finite element analysis and obtains corresponding displacement structure vector u₁,u₂,u₃；

The average of S5: structural compliance isThe variance of structural compliance is:

S6: carry out STRUCTURAL SENSITIVITY ANALYSIS INDESIGN.Object function is for ρ_ePartial derivative be:

Retraining the sensitivity for design variable is:

S7: utilize asymptote method to update design variable；First the sensitivity obtained according to step 6 medium sensitivity analysis The approximate function of information structuring object function:

Wherein, x^(k)For current iteration point,WithFor mobile asymptote, its Iteration is:

For design variable x_eCurrency,WithIt is respectively x_eBound；

Then utilize this near The object function in former optimization problem is replaced to form optimization subproblem like function；Dual Method is finally utilized to solve this optimization Problem and design variable after being updated.

S8: judge whether to meet end condition, if meeting, then stops iteration；Otherwise repeat step 4 to step 7 until Iteration ends.In this example, the condition of iteration ends is that the adjacent all cell densities of twice iteration are all less than 0.01 or iteration time Number is more than 300.

Having carried out the structure optimization under the conditions of two kinds in this example, wherein the first does not consider that uncertainty, i.e. certainty are excellent Change；The second utilizes the load position of action point uncertain structure robust design method of the present invention, wherein α=0.8, β=0.2. In the case of two kinds, the result of structure design is the most as shown in Figure 4 and Figure 5.As seen from the figure, it is considered to load position of action point is uncertain Property the structure lower end bifurcated that optimizes, the stability of this structure is apparently higher than the optimum results under the conditions of determining.This example changes every time In generation, has only to 3 basic operating modes are carried out structural analysis, and the average of structural compliance and variance calculate and needed for sensitivity analysis The motion vector wanted all obtains by the motion vector of above 3 basic operating modes is carried out linear operation, the most extra calculating Measuring the least, computational efficiency is the highest.It addition, the present invention is theoretical simple, it is convenient to implement.Therefore, the application present invention obtains optimum results Engineering practice is simple, workable, and optimum results is preferable.

The load position of action point uncertain structure robust design method that the present invention provides, based on known load effect The point regularity of distribution, regards by structural compliance as the function of position of action point as, uses displacement superposed principle and the Gauss of linear structure The flexibility average of quadrature formula computation structure and standard deviation, and object function is carried out sensitivity analysis.The method amount of calculation is little And relative accuracy is higher, implement the simplest on the basis of existing topological optimization.

Although being described in conjunction with the accompanying embodiments of the present invention, but those skilled in the art can be without departing from this Making various modifications and variations in the case of bright spirit and scope, such amendment and modification each fall within by claims Within limited range.

Claims (9)

1. a load position of action point uncertain structure robust design method, it is characterised in that including:

S1: set up the sane Optimized model of Continuum Structure:

S2: the distribution form of the load position of action point born according to structure and Gaussian quadrature node location, choose multiple Basis operating mode；

S3: initialization design variable；

S4: each basis operating mode is carried out finite element analysis and obtains corresponding displacement structure vector；

S5: the average of computation structure flexibility and standard deviation；

S6: structure is carried out sensitivity analysis；

S7: utilize asymptote method to update design variable；

S8: judge whether to meet end condition, if meeting, then stops iteration；Otherwise repeat S4 to S7 until meeting iteration ends Condition.

Load position of action point the most according to claim 1 uncertain structure robust design method, it is characterised in that institute State the sane Optimized model of Continuum Structure to include:

$\underset{x}{Min}:J=\mathrm{\α}\mathrm{\μ}\left(c\right)+\mathrm{\β}\mathrm{\σ}\left(c\right)$

S.t.:Ku (ω)=f (ω) (ω ∈ Θ)

$\begin{array}{c}g=V-V_{max}=\underset{e=1}{\overset{m}{\mathrm{\Σ}}}{v}_e{\mathrm{\ρ}}_e-V_{max}\≤0\\ 0\≤\mathrm{\ρ}\≤1\end{array}$

Wherein, J is average and the standard deviation that object function, μ (c) and σ (c) are respectively structural compliance c, α and β is that two non-negative are real Count and meet alpha+beta=1；K is structure Bulk stiffness matrix, and f (ω) and u (ω) is respectively load and motion vector, the ω of structure For representing the uncertainty of load position of action point；M is discrete number of unit, v_eAnd ρ_eIt is respectively the volume of unit and close Degree, g is the inequality constraints of the sane Optimized model of Continuum Structure, and V is to calculate the volumetric usage obtained, V every time_maxIt is given Material volume consumption；ρ is by ρ_e(e=1 ..., vector m) formed.

Load position of action point the most according to claim 2 uncertain structure robust design method, it is characterised in that institute State choose multiple basis operating mode include:

According to the probability-distribution function of load position of action point, find the Gaussian quadrature node of its correspondence, each applying is carried Lotus sets up a kind of operating mode, selects n basis operating mode f₁,…,f_n；

Wherein, n is the quadrature nodes of Gauss integration.

Load position of action point the most according to claim 2 uncertain structure robust design method, it is characterised in that institute State initialization design variable to include:

Identical density is given by all of unit

Load position of action point the most according to claim 2 uncertain structure robust design method, it is characterised in that institute State and each basis operating mode is carried out finite element analysis obtain corresponding displacement structure vector and include:

To each basis operating mode f₁,…,f_nCarry out finite element finite meta analysis and obtain corresponding displacement structure vector u₁,…,u_n。

Load position of action point the most according to claim 2 uncertain structure robust design method, it is characterised in that institute The average stating computation structure flexibility includes with standard deviation:

The average of structural compliance is:

$\mathrm{\μ}\left(c\right)=\∫c(x,\mathrm{\ρ})f\left(x\right)dx=\underset{i}{\overset{n}{\Σ}}{A}_{i}c(x_i,\mathrm{\ρ});$

The variance of structural compliance is:

${\mathrm{\σ}}^2\left(c\right)=\∫c^2(x,\mathrm{\ρ})f\left(x\right)dx-{\mathrm{\μ}}^2\left(c\right)=\∫c^2(x,\mathrm{\ρ})f\left(x\right)dx-{\mathrm{\μ}}^2\left(c\right)=\underset{i}{\overset{n}{\mathrm{\Σ}}}{A}_i{c}^2(x_i,\mathrm{\ρ})-{\mathrm{\μ}}^2\left(c\right)$

Wherein, c (x_i, ρ) and=f (x_i)u(x_i) represent load f (x_i) at displacement components u (x_i) upper power, x_iRepresent i-th operating mode Load position of action point, A_iRepresent Gaussian quadrature coefficient.

Load position of action point the most according to claim 2 uncertain structure robust design method, it is characterised in that institute State and structure is carried out sensitivity analysis include:

According to the expression formula of structural compliance average Yu variance, object function is for each unit density p_ePartial derivative be:

$\begin{array}{c}\frac{\∂J}{\∂{\mathrm{\ρ}}_e}=\mathrm{\α}\underset{i}{\overset{n}{\mathrm{\Σ}}}{A}_i\frac{\∂c\left(x_i,\mathrm{\ρ}\right)}{\∂{\mathrm{\ρ}}_e}+\mathrm{\β}\frac{1}{2\sqrt{{\mathrm{\σ}}^2}}\frac{\∂{\mathrm{\σ}}^2}{\∂{\mathrm{\ρ}}_e}\\ =\mathrm{\α}\underset{i}{\overset{n}{\mathrm{\Σ}}}{A}_i\frac{\∂c\left(x_i,\mathrm{\ρ}\right)}{\∂{\mathrm{\ρ}}_e}+\mathrm{\β}\frac{1}{2\sqrt{{\mathrm{\σ}}^2}}\left(\frac{\∂\left(\underset{i}{\overset{n}{\mathrm{\Σ}}}{A}_i{c}^2\left(x_i,\mathrm{\ρ}\right)-{\mathrm{\μ}}^2\left(c\right)\right)}{\∂{\mathrm{\ρ}}_e}\right)\\ =\mathrm{\α}\underset{i}{\overset{n}{\mathrm{\Σ}}}{A}_i\frac{\∂c\left(x_i,\mathrm{\ρ}\right)}{\∂{\mathrm{\ρ}}_e}+\mathrm{\β}\frac{1}{\sqrt{{\mathrm{\σ}}^2}}\left(\underset{i}{\overset{n}{\mathrm{\Σ}}}{A}_{i}c\left(x_i,\mathrm{\ρ}\right)\frac{\∂c\left(x_i,\mathrm{\ρ}\right)}{\∂{\mathrm{\ρ}}_e}-\mathrm{\μ}\left(c\right)\frac{\∂\mathrm{\μ}\left(c\right)}{\∂{\mathrm{\ρ}}_e}\right)\end{array}$

Retraining the sensitivity for each unit density is:

$\frac{\∂g}{\∂{\mathrm{\ρ}}_e}=v_e.$

Load position of action point the most according to claim 2 uncertain structure robust design method, it is characterised in that institute State and utilize asymptote method renewal design variable to include:

Approximate function according to the sensitivity information structure object function that sensitivity analysis obtains:

$f\left(x\right)\≈f\left({\mathrm{\ρ}}^{\left(k\right)}\right)+\underset{e=1}{\overset{m}{\Σ}}{p}_e^{\left(k\right)}(\frac{1}{U_e^{\left(k\right)}-{\mathrm{\ρ}}_e}-\frac{1}{U_e^{\left(k\right)}-{\mathrm{\ρ}}_e^{\left(k\right)}})+\underset{e=1}{\overset{m}{\Σ}}{q}_e^{\left(k\right)}(\frac{1}{{\mathrm{\ρ}}_e-L_e^{\left(k\right)}}-\frac{1}{{\mathrm{\ρ}}_e^{\left(k\right)}-L_e^{\left(k\right)}})$

Wherein, ρ^(k)For current iteration point,WithFor mobile asymptote, its Iteration is:

$L_e^{\left(k\right)}={\mathrm{\ρ}}_e^{\left(k\right)}-0.1({\mathrm{\ρ}}_e^{\max }-{\mathrm{\ρ}}_e^{\min })$

$U_e^{\left(k\right)}={\mathrm{\ρ}}_e^{\left(k\right)}+0.1({\mathrm{\ρ}}_e^{\max }-{\mathrm{\ρ}}_e^{\min })$

For design variable ρ_eCurrency,WithIt is respectively ρ_eBound；

$p_e^{\left(k\right)}=max({\left(U_e^{\left(k\right)}-{\mathrm{\ρ}}_e^{\left(k\right)}\right)}^2\∂f/\∂{\mathrm{\ρ}}_e,0),q_e^{\left(k\right)}=max(-{\left({\mathrm{\ρ}}_e^{\left(k\right)}-L_e^{\left(k\right)}\right)}^2\∂f/\∂{\mathrm{\ρ}}_e,0);$

This approximate function is utilized to replace the object function in former optimization problem to form optimization subproblem；

Design variable after utilizing Dual Method to solve this optimization subproblem and being updated.

Load position of action point the most according to claim 2 uncertain structure robust design method, it is characterised in that institute State and judge whether to meet end condition, if meeting, then stopping iteration, otherwise repeating S4 to S7 until meeting stopping criterion for iteration bag Include:

If iterations has reached maximum times or twice iteration variable density amount of various discrete unit is respectively less than set-point, then Stop iteration；Otherwise continue iterative process.