CN104915478A - Product design model equivalent simplifying method based on multi-parameter uncertainty analysis - Google Patents

Abstract

The invention discloses a product design model equivalent simplifying method based on multi-parameter uncertainty analysis. The method comprises the steps that sampling points are selected through a Latin optimization method, and a Kriging equivalent simplifying model is established; the influences of the uncertainty of design variables, the uncertainty of system parameters and the uncertainty of the equivalent simplifying model on the performance are calculated, and the mean value and a standard difference calculation function of performance targets are established; newly-added design variable sampling points are selected based on a performance prediction interval, newly-added system parameter sampling points are selected based on mean square errors of the equivalent simplifying model, and then the established Kriging equivalent simplifying model is updated. Through the method, the performance of uncertainties of multiple parameters is solved based on the established equivalent simplifying model; the influences of the uncertainties of various kinds of parameters on the performance are reduced, and the robustness of the design result is improved.

Description

Based on the product design model equivalent-simplification method of multiparameter uncertainty analysis

Technical field

The present invention relates to a kind of product model short-cut method, relate to a kind of product design model equivalent-simplification method based on multiparameter uncertainty analysis.

Background technology

Simulation analysis (finite element analysis, Fluid Computation analysis etc.) is the effective ways of product design.Due to the impact of many factors, based on existing in the based Robust Design of simulation analysis, equivalent simplified model uncertainty, design variable uncertainty and systematic parameter are uncertain.In order to reduce the impact of all kinds of uncertain factor on performance, there has been proposed a series of method.Such as, Chen.W proposed a kind of for the effective of based Robust Design and accurate overall sensitivity analysis method in 2005 in the paper at " Journal of Mechanical Design " (127 (2): 184-195) " Analytical variance-based global sensitivity analysis in simulation-based design under uncertainty ".Gu.X proposed in the paper at " Journal of Mechanical Design " (128 (4): 1001-1013) " Implicit uncertainty propagation for robust collaborative optimization " in 2006 and considers the uncertain and variable the worst probabilistic possible uncertainty propagation analytical approach of approximate model.Yao.W proposed one and considers the uncertain and probabilistic closed loop Uncertainty Analysis Method of noise parameter of approximate model in the paper in " Progress in Aerospace Sciences " (47 (6), 450-479) " Review of uncertainty-based multidisciplinary design optimization methods for aerospace vehicles " in 2011.Zhang.S in 2013 at " Structural and Multidisciplinary Design " (47 (1), propose one in paper " Concurrent treatment of parametric uncertainty and metamodeling uncertainty in robust design " 63-76) and consider model uncertainty and the probabilistic robust design method of design variable, and the method is applied to the crashworthiness lightweight structure based Robust Design of automobile case.Although these methods improve the robustness of design result, all do not have to build equivalent simplified model for multiparameter uncertain factor, between based Robust Design result and exact value, error is larger.

Summary of the invention

In order to improve the accuracy of design result, the object of the present invention is to provide a kind of product design model equivalent-simplification method based on multiparameter uncertainty analysis, consider equivalent simplified model uncertainty, design variable is uncertain and systematic parameter is uncertain, reduce the uncertain impact on performance of multiparameter, improve the accuracy of design result.

The technical solution used in the present invention comprises the following steps, as shown in Figure 1:

1) the initial samples point optimized Latin method acquisition k and include design variable X and systematic parameter W is used;

2) Kriging equivalent simplified model G is built _k(x, W), k represents sampled point number;

3) uncertain according to design variable, systematic parameter is uncertain and the uncertain impact on performance objective y of equivalent simplified model, builds mean value computation function mu (x) and the variance computing function σ of performance objective y ²x (), calculates based on G _kthe optimal design variate-value x of (x, W) _{min, k};

4) set up performance prediction interval, increase new design variable sampled point x _k+1;

5) computation model square error, increases new systematic parameter sampled point W _k+1;

6) step 2 is repeated) ~ 5) carry out iterative computation, according to existing sampled point and newly-increased sampled point (x in calculating each time _k+1, W _k+1) build new equivalent simplified model G _k+1(x, W), until the error between the equivalent simplified model that obtains of sampled point quantity and adjacent iterative computation meets stop technology condition;

7) using the equivalent simplified model that calculates for the last time as best equivalence simplified model, realize the simplification of product design model.

Described step 3) calculating adopt following steps:

3.1) design variable X be decomposed into determinacy part x and characterize the uncertain part d of variable perturbations, the decomposition of design variable X as shown in Equation 1:

X＝x+d (1)

I-th design variable X in sampled point _iuncertain part d _inormal Distribution wherein represent d _iand X _ivariance; I-th systematic parameter W in sampled point _inormal Distribution wherein i represents the ordinal number of design variable in sampled point, w _iwith w respectively _iaverage and variance;

3.2) prediction mean value function μ (x) of performance objective y and the prediction variance function σ of performance objective y that following formula 2 and formula 3 represent respectively is set up ²(x), and calculate:

$\mathrm{\μ}\left(x\right)={\∫}_W{\∫}_d{y}_{G_k}\left(X,W\right)p\left(d\right)p\left(W\right)dddW---\left(2\right)$

$\begin{array}{c}{\mathrm{\σ}}^2\left(x\right)={\∫}_W{\∫}_d{\[y_{G_k}\left(X,W\right)\]}^{2}p\left(d\right)p\left(W\right)dddW-{\[{\∫}_W{\∫}_d{\[y_{G_k}\left(X,W\right)\]}^{2}p\left(d\right)p\left(W\right)dddW\]}^2\\ +{\∫}_W{\∫}_d{\[{\mathrm{\ϵ}}_y\left(X,W\right)\]}^{2}p\left(d\right)p\left(W\right)dddW\\ +2{\∫}_W{\∫}_d{\[y_{G_k}\left(X,W\right)\×{\mathrm{\ϵ}}_y\left(X,W\right)\]}^{2}p\left(d\right)p\left(W\right)dddW\end{array}---\left(3\right)$

Wherein, the probability density function of the uncertain part d that p (d) is design variable X, the probability density function that p (W) is systematic parameter W, ε _y(X, W) is Kriging equivalent simplified model G _kthe model variance function of (x, W) prediction average, for using Kriging equivalent simplified model G _kthe predicted value of the performance objective y that (x, W) calculates.

3.3) the based Robust Design function that following formula 4 represents is built carry out the miniaturized design of performance objective y:

f(x)＝μ(x)+cσ(x) (4)

Wherein, c represents and steadily and surely spends constant, and μ (x) represents the prediction average of performance objective y, and σ (x) represents the prediction standard deviation of performance objective y.

3.4) adopt optimization method to carry out optimal design to design variable X with based Robust Design function f (x) for target, obtain based on Kriging equivalent simplified model G _kthe optimal design variate-value x that (x, W) calculates _{min, k}.

Described step 3.4) in optimization method adopt the optimization method such as genetic algorithm, ant group algorithm, particle cluster algorithm.

Described step 4) specifically comprise:

4.1) according to step 3) prediction average μ (x) of performance objective y that obtains and prediction standard deviation sigma (x) of performance objective y, set up the forecast interval [μ (x)-c σ (x), μ (x)+c σ (x)] of performance objective y;

4.2) the forecast interval function h that following formula 5 represents is built ₁x (), with forecast interval function h ₁design variable value x corresponding when () gets maximal value is x as new design variable sampled point x _k+1be increased in existing design variable sampled point:

h ₁(x) _＝(μ(x _min,k)+cσ(x _min,k))-(μ(x)-cσ(x)) (5)

Wherein, μ ( _{xmin, k}), σ ( _{xmin, k}) represent that performance objective y is at optimal design variate-value x respectively _{min, k}the average at place and standard deviation.

Described step 5) specifically comprise:

5.1) according to step 2) the Kriging equivalent simplified model G that obtains _k(x, W), calculates the square error of this model

5.2) the sampled point Selection of Function h that following formula 6 represents is built ₂(W), with sampled point Selection of Function h ₂(W) system parameter values W corresponding when getting maximal value is as new systematic parameter sampled point W _k+1, be increased in existing systematic parameter sampled point:

$h_2\left(W\right)={\mathrm{\σ}}_{G_k}\left(x_{k+1},W\right)p\left(W\right)---\left(6\right)$

Wherein, p (W) probability density function that is system parameter values W.

Described step 6) in the calculating that meets stop technology condition of error between the equivalent simplified model that obtains of sampled point quantity and adjacent iterative computation with judge in the following ways:

6.1) following formula 7 is adopted to calculate the equivalent simplified model y comprising k sampled point _k(x, W) and comprise the equivalent simplified model y of k+1 sampled point _k+1error function h between two models of (x, W) ₃(x, W):

h ₃(x,W)＝y _k+1(x,W)-y _k(x,W) (7)

6.2) following formula 8 is adopted to calculate gap function Z (x) comprised between the equivalent simplified model of k sampled point and the equivalent simplified model comprising k+1 sampled point again;

Z(x)＝∫ _wh ₃(x,W)p(W)dW (8)

6.3) when sampled point quantity k meets k<k _maxand the prediction average E (Z (x)) of gap function Z (x) meets E (Z (x))≤E _maxwhen, k _maxrepresent sampled point quantity maximal value, then repeat step 2) ~ 5);

When sampled point quantity k meets k=k _maxor the prediction average E (Z (x)) of gap function Z (x) meets E (Z (x)) >E _maxwhen, E _maxrepresent error threshold, then stop iterative computation.

Described step 7) finally obtain best equivalence simplified model and design for the Robust Performance of product, calculate the optimal design variate-value meeting multiparameter uncertainty and require.

The beneficial effect that the present invention has is:

1, calculate design variable uncertainty, systematic parameter is uncertain and equivalent simplified model is uncertain on the probabilistic impact of performance, construct and consider that the probabilistic performance objective uncertainty of multiparameter characterizes function.

2, propose the nonuniform sampling point choosing method based on performance objective forecast interval and model square error, carried out equivalent simplified model structure, decreased the uncertainty of the equivalent simplified model of structure, improve the accuracy of performance metric.

3, the performance equivalent simplified model of structure being applied to complex product solves, and decreases the uncertain impact on performance of multiparameter, improves the robustness of design result.

Accompanying drawing explanation

Fig. 1 is process flow diagram of the present invention.

Fig. 2 is the true response model between design object and design variable in embodiment 1, systematic parameter.

Fig. 3 is the prediction average of initial Kriging equivalent simplified model in embodiment 1.

Fig. 4 is the prediction square error of initial Kriging equivalent simplified model in embodiment 1.

Fig. 5 considers the probabilistic based Robust Design function of multiparameter in embodiment 1.

Fig. 6 is the prediction average of final equivalent simplified model in embodiment 1.

Fig. 7 is the prediction square error of final equivalent simplified model in embodiment 1.

Fig. 8 be in embodiment 1 evaluation function with the change procedure of sampled point quantity.

Fig. 9 considers the probabilistic based Robust Design function of multiparameter in embodiment 1.

Embodiment

Below in conjunction with specific embodiments and the drawings the present invention be further explained and illustrate.

As shown in Figure 1, specific embodiment is as follows for the inventive method:

This designs a model as minimizing y (x, w)=80x ^-2+ 2xw+x ²w.The design space of design variable x is x ∈ [0.8,2.5], and design variable uncertainty is x ~ N (x, 0.07 ²); Systematic parameter is w=10, and its uncertainty meets normal distribution w ~ N (10,2).In the present embodiment, design object y and the true response model between design variable x, systematic parameter w adopt polygon curved surface as shown in Figure 2.

Concrete steps of the present invention are:

1) use optimization Latin hypercube method to choose 15 sampled points, the simulation analysis result based on these 15 sampled points builds initial Kriging simplified model.As shown in Figure 3, square error as shown in Figure 4 for the average of this initial Kriging simplified model at each design point place.

2) on initial Kriging simplified model basis, structure considers design variable uncertainty, systematic parameter uncertainty and the probabilistic Robust Performance design function of equivalent simplified model, as shown in Figure 5, the optimal design variate-value calculated is x=1.4511.

3) design variable is carried out and systematic parameter sampled point is chosen.Final equivalent simplified model comprises 29 sampled points, and as shown in Figure 6, model prediction variance as shown in Figure 7 for the average of this simplified model.In simplified model building process, along with the increase of sampled point, the Change in Mean process of Z (x) as shown in Figure 8.Based on final simplified model, the performance of structure solves design function as shown in Figure 9, and the optimal design variate-value obtained is x=1.4239.

The optimal design variate-value using traditional simplified model to obtain is x=1.5072.The true optimal design value of this example is x=1.3542.Result of calculation of the present invention and existing methods result of calculation, exact computation results contrast as shown in table 1, and comparing result shows result of calculation of the present invention closer to actual value.

Table 1

	Result of the present invention	Existing methods and results	Exact value
				Sane desired value	113.05	115.99	109.10
Performance mean value	88.42	94.36	88.16
				Performance standard difference	8.21	7.21	6.98
Optimal design variate-value	1.4239	1.5072	1.3542

As can be seen here, the present invention characterizes the mode equivalent-simplifications such as function product model by performance objective uncertainty, decreases the uncertainty of the equivalent simplified model of structure, improves the accuracy of performance metric and the robustness of product design result.

Claims (7)

1., based on a product design model equivalent-simplification method for multiparameter uncertainty analysis, it is characterized in that comprising the following steps:

1) the initial samples point optimized Latin method acquisition k and include design variable X and systematic parameter W is used;

2) Kriging equivalent simplified model G is built _k(x, W), k represents sampled point number;

3) uncertain according to design variable, systematic parameter is uncertain and the uncertain impact on performance objective y of equivalent simplified model, builds mean value computation function mu (x) and the variance computing function σ of performance objective y ²x (), calculates based on G _kthe optimal design variate-value x of (x, W) _{min, k};

4) set up performance prediction interval, increase new design variable sampled point x _k+1;

5) computation model square error, increases new systematic parameter sampled point W _k+1;

6) step 2 is repeated) ~ 5) carry out iterative computation, according to existing sampled point and newly-increased sampled point (x in calculating each time _k+1, W _k+1) build new equivalent simplified model G _k+1(x, W), until the error between the equivalent simplified model that obtains of sampled point quantity and adjacent iterative computation meets stop technology condition;

7) using the equivalent simplified model that calculates for the last time as best equivalence simplified model, realize the simplification of product design model.

2. a kind of product design model equivalent-simplification method based on multiparameter uncertainty analysis according to claim 1, is characterized in that:

Described step 3) calculating adopt following steps:

3.1) design variable X be decomposed into determinacy part x and characterize the uncertain part d of variable perturbations, the decomposition of design variable X as shown in Equation 1:

X＝x+d (1)

I-th design variable X in sampled point _iuncertain part d _inormal Distribution wherein represent d _iand X _ivariance; I-th systematic parameter W in sampled point _inormal Distribution wherein i represents the ordinal number of design variable in sampled point, w _iwith w respectively _iaverage and variance;

3.2) prediction mean value function μ (x) of performance objective y and the prediction variance function σ of performance objective y that following formula 2 and formula 3 represent respectively is set up ²(x), and calculate:

Wherein, the probability density function of the uncertain part d that p (d) is design variable X, the probability density function that p (W) is systematic parameter W, ε _y(X, W) is Kriging equivalent simplified model G _kthe model variance function of (x, W) prediction average, for using Kriging equivalent simplified model G _kthe predicted value of the performance objective y that (x, W) calculates;

3.3) the based Robust Design function that following formula 4 represents is built carry out the miniaturized design of performance objective y:

f(x)＝μ(x)+cσ(x) (4)

Wherein, c represents and steadily and surely spends constant, and μ (x) represents the prediction average of performance objective y, and σ (x) represents the prediction standard deviation of performance objective y;

3.4) adopt optimization method to carry out optimal design to design variable X with based Robust Design function f (x) for target, obtain based on Kriging equivalent simplified model G _kthe optimal design variate-value x that (x, W) calculates _{min, k}.

3. a kind of product design model equivalent-simplification method based on multiparameter uncertainty analysis according to claim 1, is characterized in that: described step 3.4) in optimization method adopt the optimization method of genetic algorithm, ant group algorithm or particle cluster algorithm.

4. a kind of product design model equivalent-simplification method based on multiparameter uncertainty analysis according to claim 1, is characterized in that: described step 4) specifically comprise:

4.1) according to step 3) prediction average μ (x) of performance objective y that obtains and prediction standard deviation sigma (x) of performance objective y, set up the forecast interval [μ (x)-c σ (x), μ (x)+c σ (x)] of performance objective y;

4.2) the forecast interval function h that following formula 5 represents is built ₁x (), with forecast interval function h ₁design variable value x corresponding when () gets maximal value is x as new design variable sampled point x _k+1be increased in existing design variable sampled point:

h ₁(x)＝(μ(x _min,k)+cσ(x _min,k))-(μ(x)-cσ(x)) (5)

Wherein, μ (x _{min, k}), σ (x _{min, k}) represent that performance objective y is at optimal design variate-value x respectively _{min, k}the average at place and standard deviation.

5. a kind of product design model equivalent-simplification method based on multiparameter uncertainty analysis according to claim 1, is characterized in that: described step 5) specifically comprise:

5.1) according to step 2) the Kriging equivalent simplified model G that obtains _k(x, W), calculates the square error of this model

5.2) the sampled point Selection of Function h that following formula 6 represents is built ₂(W), with sampled point Selection of Function h ₂(W) system parameter values W corresponding when getting maximal value is as new systematic parameter sampled point W _k+1, be increased in existing systematic parameter sampled point:

Wherein, p (W) probability density function that is system parameter values W.

6. a kind of product design model equivalent-simplification method based on multiparameter uncertainty analysis according to claim 1, is characterized in that: described step 6) in the calculating that meets stop technology condition of error between the equivalent simplified model that obtains of sampled point quantity and adjacent iterative computation with judge in the following ways:

6.1) following formula 7 is adopted to calculate the equivalent simplified model y comprising k sampled point _k(x, W) and comprise the equivalent simplified model y of k+1 sampled point _k+1error function h between two models of (x, W) ₃(x, W):

h ₃(x,W)＝y _k+1(x,W)-y _k(x,W) (7)

6.2) following formula 8 is adopted to calculate gap function Z (x) comprised between the equivalent simplified model of k sampled point and the equivalent simplified model comprising k+1 sampled point again;

Z(x) _＝∫ _wh ₃(x,W)p(W)dW (8)

6.3) when sampled point quantity k meets k<k _maxand the prediction average E (Z (x)) of gap function Z (x) meets E (Z (x))≤E _maxwhen, k _maxrepresent sampled point quantity maximal value, then repeat step 2) ~ 5);

When sampled point quantity k meets k=k _maxor the prediction average E (Z (x)) of gap function Z (x) meets E (Z (x)) >E _maxwhen, E _maxrepresent error threshold, then stop iterative computation.

7. a kind of product design model equivalent-simplification method based on multiparameter uncertainty analysis according to claim 1, it is characterized in that: described step 7) finally obtain best equivalence simplified model and design for the Robust Performance of product, for calculating the optimal design variate-value meeting multiparameter uncertainty and require.