CN107292040A - Finite Element Model Validation based on Kriging response surfaces - Google Patents
Finite Element Model Validation based on Kriging response surfaces Download PDFInfo
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- G06F30/00—Computer-aided design [CAD]
- G06F30/20—Design optimisation, verification or simulation
- G06F30/23—Design optimisation, verification or simulation using finite element methods [FEM] or finite difference methods [FDM]
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Abstract
The invention discloses a kind of finite Element Model Validation based on Kriging response surfaces.It includes carrying out system subdivision to pressure vessel model, set up FEM model, dependence on parameter and Parameter Sensitivity Analysis are carried out to model parameter, build the Kriging response surfaces of pressure vessel model maximum equivalent, random sampling is carried out to Kriging response surfaces, sampling samples probability distribution and confidential interval are analyzed, single result of finite element and FEM Numerical Simulation are contrasted, realizes that FEM model confirms.The present invention can provide facility for the work of the model validation in complication system computer sim- ulation, while structure design and health evaluating for product etc. provides foundation.
Description
Technical field
The invention belongs to model validation technical field, a kind of FEM model based on Kriging response surfaces is especially designed
Confirmation method.
Background technology
In the research and development of modern project structure and design process and Grand Equipments or job facilities stage under arms peace
In terms of full property and reliability assessment, computer sim- ulation just playing more and more important effect, particularly Design Stage and
Complication system is difficult that in the case of carrying out all systems test, must be just estimated by computer sim- ulation, therefore simulation model
Precision and confidence level are most important.Model validation is proposed in the initial concept of engineering circles by USDOE, is mainly used in war
Slightly in the reliability assessment and decision-making of weapon storage management, and computational fluid dynamics emulation mould is drafted by AIAA in 1998
The guide that type is verified and confirmed, Oberkampf has carried out system summary to this, and it is true to review mechanical engineering field simulation model
The development recognized.The layered system of model validation is applied in the exploitation of common engineering product by Jung.During computer sim- ulation,
Because various probabilistic presence cause predicting the outcome for computer sim- ulation often to there is very big difference between result of the test, because
This Model Updating Technique is also widely used, and influence and modification method of the emphasis to uncertainty to model prediction are carried out
Discuss.Model validation is also gradually taken seriously at home, and the concept of model validation is more incorporated into the country, Guo Qintao by an order earliest
It is applied to Deng by model validation in specific research, Wang Ruili and Deng little Gang are dynamic in computer program and fluid to model validation respectively
Application in mechanics is discussed.Liu Xinen discusses and simplified to the Bayesian frame in model validation.However, overall next
See, the basic procedure of model validation is still not clear, and is also in conceptual phase.
The transformational relation between input parameter and output individual features for labyrinth, can act on behalf of mould by building
Type replaces FEM model, can carry out a large amount of random samplings in the short time, greatly reduce amount of calculation.In recent years, response surface
The agent models such as model, Kriging models and neutral net are fast-developing.But various agent models have its respective excellent scarce
Point.Response surface model is easily achieved, but it is poor to approach nonlinear problem ability；Kriging models have to nonlinear problem compared with
The high degree of accuracy, but the acquisition of model and use difficulty are larger, and the precision and generalization ability of neural network model rely on its network
Structure and substantial amounts of learning sample, occasionally there are " cross and learn " phenomenon.Simultaneously as the uncertain parameters of complex model are many
It is many, how to choose to the parameter of target response correlation maximum be also research focus.
The content of the invention
The present invention goal of the invention be：In order to solve problem above, the present invention proposes a kind of based on Kriging response surfaces
Finite Element Model Validation, in complication system computer sim- ulation model validation work facility is provided, while being product
Structure design and health evaluating etc. provide foundation.
The technical scheme is that：A kind of finite Element Model Validation based on Kriging response surfaces, including it is following
Step：
A, to pressure vessel model carry out system subdivision, set up FEM model；
B, the model parameter to FEM model in step A carry out dependence on parameter and Parameter Sensitivity Analysis；
C, selection and maximum equivalent correlation maximum parameter, are built using Latin hypercube experimental design method and pressed
The Kriging response surfaces of force container model maximum equivalent；
D, in step C Kriging response surfaces carry out random sampling, using Density Estimator method to sampling samples probability
Distribution and confidential interval are analyzed；
E, single result of finite element and FEM Numerical Simulation contrasted, realize that FEM model confirms.
Further, the step B carries out dependence on parameter and parametric sensitivity point to the model parameter of FEM model
Analysis is specially：
The model parameter of FEM model is divided into control parameter and geometry material parameter, calculated using experimental design module
Sensitivity of the geometry material parameter to maximum equivalent and maximum deformation quantity, obtains model parameter sensitivity block diagram；Using
Experimental design module computational geometry material parameter obtains model parameter correlation matrix to the correlation of maximum equivalent.
Further, in the step C Kriging response surfaces of pressure vessel model maximum equivalent mathematical modeling
Specially：
Wherein,The response prediction value for being Kriging response surfaces at sample point x,For the basic function system after estimation
Number, f (x) is the basic function of fitting function, and R is correlation matrix, yDFor the observation of sample, F is f (x) observation.
Further, the step D carries out random sampling to Kriging response surfaces, using Density Estimator method to taking out
All probability distribution and confidential interval are analyzed specially：
Using Monte Carlo and arbitrary sampling method, 5000 radius r, thickness th parameter sample combination are chosen, it is right
Kriging response surfaces, which are sampled, obtains 5000 groups of maximum equivalent amplitudes, and maximum equivalent is entered using histogram method
Row analysis；The probability density function of equivalent stress maximum is calculated using nonparametric probability method, it is different by setting
Level of confidence, obtain the confidential interval of the maximum equivalent of pressure vessel model.
Further, the step E is contrasted single result of finite element and FEM Numerical Simulation, and realization has
Limit meta-model confirms：
According to pressure vessel modelling low-level and high-level atmospheric pressure experiment, two groups of Kriging response surfaces are built,
Each Kriging response surfaces are sampled and obtain certification sample, two groups of obtained certification samples are utilized into Density Estimator side
Method is fitted, and obtains KDE curves；The maximum equivalent that finite element simulation under KDE curves and two kinds of experiment conditions is obtained should
Force value is compared, and realizes that FEM model confirms.
The beneficial effects of the invention are as follows：The present invention sets up parameter by carrying out hierarchical design to pressure vessel model system
Change FEM model, and dependence on parameter and Parameter Sensitivity Analysis are carried out to model parameter, choose and target response correlation
Maximum parameter builds Kriging response surfaces, realizes the prediction to target response to response surface random sampling, is estimated using cuclear density
Meter method is analyzed sampling samples probability distribution and confidential interval, and carries out contrast realization with single result of finite element
Model validation, so that facility is provided for model validation, while structure design and health evaluating for product etc. provides foundation.
Brief description of the drawings
Fig. 1 is the schematic flow sheet of the finite Element Model Validation based on Kriging response surfaces of the present invention.
Fig. 2 is pressure vessel model validation layering schematic diagram in the embodiment of the present invention.
Fig. 3 is pressure vessel model test design load application schematic diagram in the embodiment of the present invention.
Fig. 4 is parametric sensitivity block diagram in the embodiment of the present invention.
Fig. 5 is dependence on parameter matrix diagram in the embodiment of the present invention.
Fig. 6 is pressure vessel model maximum equivalent Kriging response surface schematic diagrames in the embodiment of the present invention.
Fig. 7 is pressure vessel model maximum equivalent Density Estimator schematic diagram in the embodiment of the present invention.
Fig. 8 is the Kriging response surface schematic diagrames of certification experiment one in the embodiment of the present invention.
Fig. 9 is the KDE curve synoptic diagrams of certification experiment one in the embodiment of the present invention.
Figure 10 is the CDF curve synoptic diagrams of certification experiment one in the embodiment of the present invention.
Figure 11 is the Kriging response surface schematic diagrames of certification experiment two in the embodiment of the present invention.
Figure 12 is the KDE curve synoptic diagrams of certification experiment two in the embodiment of the present invention.
Figure 13 is the CDF curve synoptic diagrams of certification experiment two in the embodiment of the present invention.
Embodiment
In order to make the purpose , technical scheme and advantage of the present invention be clearer, it is right below in conjunction with drawings and Examples
The present invention is further elaborated.It should be appreciated that specific embodiment described herein is only to explain the present invention, not
For limiting the present invention.
As shown in figure 1, the flow signal of the finite Element Model Validation based on Kriging response surfaces for the present invention
Figure.A kind of finite Element Model Validation based on Kriging response surfaces, comprises the following steps：
A, to pressure vessel model carry out system subdivision, set up FEM model；
B, the model parameter to FEM model in step A carry out dependence on parameter and Parameter Sensitivity Analysis；
C, selection and maximum equivalent correlation maximum parameter, are built using Latin hypercube experimental design method and pressed
The Kriging response surfaces of force container model maximum equivalent；
D, in step C Kriging response surfaces carry out random sampling, using Density Estimator method to sampling samples probability
Distribution and confidential interval are analyzed；
E, single result of finite element and FEM Numerical Simulation contrasted, realize that FEM model confirms.
In step, because pressure vessel model has solid-liquid-gas coupled problem, uncertain parameter is more, therefore this hair
It is bright that system subdivision is carried out to pressure vessel model, and set up the FEM model of correlation；Subsystem after division possesses less
Uncertain parameter, can greatly reduce during analysis uncertain parameter error coupler produce influence.The present invention is logical
Cross and the real load and restraint condition of pressure vessel are analyzed, carried out in terms of environment complexity and system complexity two
Consider, system subdivision is carried out to simulation model.As shown in Fig. 2 being layered for pressure vessel model validation in the embodiment of the present invention
Schematic diagram.
Pressure vessel is a closed container in the present invention, is made up of a cylindrical housings and two hemispherical shells,
Apply the special liquid for having certain atmospheric pressure and certain altitude in fixed constraint, pressure vessel in the face of cylinder and sphere connection end
Body.Set up the parametrization three-dimensional entity model of pressure vessel in Pro/E softwares, and by with ANSYS workbench 14.0
Seamless link set up finite element analysis model.
The present invention confirms flow and layering thought according to FEM model, pressure vessel model is carried out system subdivision and
Experimental design.Experimental design is divided into calibration test, validation test and certification test, and the load for applying different levels respectively is tried
Test.As shown in figure 3, applying schematic diagram for pressure vessel model test design load in the embodiment of the present invention, wherein left for calibration
Experiment, in be validation test, the right side be certification test.
(1) calibration test：Normal pressure；
(2) validation test：Apply certain pressure intensity P load；
(3) certification test：Apply certain pressure intensity P and certain altitude H liquid combined loads.
Calibration test is by testing to sub-structure model parameter (such as elasticity modulus of materials, Poisson's ratio, density, proportion)
Directly or indirectly measured, the error to parameter is estimated；Validation test by building Kriging response surface models, and
Probability distribution and confidential interval to target response are analyzed；Certification test is on the basis of acknowledged subsystem model, structure
Certification test FEM model is built, and FEM model simulation result is predicted the outcome with response surface is compared.
In stepb, the model parameter of FEM model is divided into control parameter and geometry material parameter by the present invention.Control
Parameter includes pressure, liquid height, liquid specific gravity, and geometry material parameter includes length, radius, thickness, modulus of elasticity etc..
Using the experimental design module (DOE) in ANSYS workbench to the uncertain parameter L in validation test, r,
Th, E carry out sensitivity analysis, using parameter L, r, th, E as input parameter, using maximum equivalent and maximum deformation quantity as
Output parameter.Assuming that setting normal distribution of the parameter of structure design as X~N (μ, σ), μ is each design parameter initial value, and σ=μ α are change
Standard deviation is measured, α is the coefficient of variation of design parameter, and the coefficient of variation of five design parameters takes 5%.It is pressure as shown in table 1
Container model carries out the uncertain parameters of DOE analyses.
The pressure vessel model uncertainty parameter value of table 1
25 groups of design points are produced using DOE modules, and each design point are solved, 25 groups of pressure vessels are obtained
The sample point of maximum equivalent and maximum total deformation.As shown in figure 4, being parametric sensitivity column in the embodiment of the present invention
Figure, it can be seen that the sensitivity of radius r and thickness th to maximum stress is highest, the length L of model and elastic modulus E
Sensitivity is then fairly small, can be ignored substantially；And sensitivity of the material parameter elastic modulus E to maximum strain amount is maximum
's.
The present invention can obtain detailed input parameter and output using Workbench DOE modules progress experimental design
The particularly relevant property numerical value of parameter.Among relation analysis of parameter, if input parameter is more than to the degree of correlation of output parameter
80%, then it represents that correlation is higher, if conversely, the degree of correlation is less than 10%, then it represents that correlation is relatively low.It is illustrated in figure 5 this hair
Dependence on parameter matrix diagram in bright embodiment, is as shown in table 2 dependence on parameter matrix table, it can be found that the pressure in input parameter
The thickness th of force container and the degree of correlation of maximum equivalent have reached 81.83%, the degree of correlation of radius r and maximum equivalent
For 54.93%, remaining input parameter is respectively less than 10% to the degree of correlation of maximum equivalent, it is believed that the radius of pressure vessel
Of a relatively high to the correlation of maximum equivalent with thickness, the length and elasticity modulus of materials of pressure vessel should to maximum equivalent
The correlation of power is very low.The parameter with care amount (maximum equivalent) correlation maximum can be chosen using correlation analysis
(radius r and thickness th) builds response surface.
The dependence on parameter matrix table of table 2
In step C, the present invention is using Latin hypercube experimental design method (Latin hypercube Design, LD)
To build Kriging response surfaces.25 groups of design sample points are have chosen, are pressure vessel radius r, input by the x-axis of input parameter
The y-axis of parameter is pressure shell body thickness th, and z-axis is maximum equivalent (Max Eqv.Stress).It is illustrated in figure 6 this
Pressure vessel model maximum equivalent Kriging response surface schematic diagrames in inventive embodiments
The Kriging response surface models of the present invention are approximately retouched in the way of approximation by polynomi-als to phantom
State, its expression formula is：
Wherein, fj(x) be fitting function basic function, βjIt is the coefficient of basic function, z (x) is the deviation letter for fitting
Number.Kriging interpolation methods are generally acknowledged that the fitness bias amount at different interpolation points is not separate, and assume deviation letter
Number is a kind of random process Z (x), and the average of random process is 0, and variance is σ2, and covariance is not 0.Random two point t and u
The covariance function at place is defined as：
Cov [Z (t), Z (u)]=σ2ρ(t,u；θ)
Wherein, ρ (t, u；It is θ) correlation function of point-to-point transmission；θ is the parameter of correlation function, and the parameter is used for weighing two
Correlation between sample point t and u is with two increased doughs softening of sample point spacing, and this parameter is smaller, then constructed sound
Answer face more smooth.
It is determined that the observation y of correlation function ρ and sampleD=[y (x1),y(x2),…y(xn)]TAfterwards, will also be according to sample
Observed value calculates random process variances sigma2, basic function factor beta and correlation function parameter θ, three parameters after estimation are denoted as The as hyper parameter (Hyperparameters) of Kriging models.Built and observed using maximum likelihood estimate
Joint probability distribution function p (the y of sample valueD), specifically counted by the way that joint probability distribution function is maximized to obtain hyper parameter
Value：
L(β,σ2, θ) and=p (yD)
Wherein,
F=[f (x1),f(x2),…f(xn)]T
R is correlation matrix, Rij=ρ [xi,xi；θ],(1≤i,j≤n).
Parameter beta, θ expression formulas are
Obtain：
Coefficient correlation θ is solved using numerical method, asks maximum likelihood function extreme value to obtain the size of its numerical value.Note
R (x)=[ρ (x, x1),ρ(x,x2),…,ρ(x,xn),]T, then Kriging response surface models are at any design sample point x
Response prediction expression formula is：
In order to contrast influence of the different response surface experimental design methods to the precision of prediction of response surface, in identical sample point
Under conditions of number (25 groups), built using other two kinds of experimental design methods (horizontal total divisor design and complex centre are designed)
Kriging response surface models.Meanwhile, using response surface accuracy test index, three kinds of constructed response surface precision are examined
Test.The present invention is Kriging response surface accuracy test results by the use of 5 random sample points as check point, as shown in table 3.
Table 3Kriging response surface accuracy test results
The Kriging response surface accuracy test indexs R that three kinds of test design methods of contrast are built2, RMSE it can be found that draw
Fourth hypercube design method can build more accurate Kriging response surface models under the conditions of the sample point of identical quantity.
To sum up analyze, the present invention builds pressure vessel maximum equivalent using Latin hypercube experimental design method
Kriging response surfaces are rationally effective.
In step D, Kriging response surfaces of the invention are based on experimental design parameter sample point, to non-test
The combination of design parameter sample pushes force container surface of shell maximum equivalent amplitude and is predicted.With reference to Monte Carlo and at random
The methods of sampling is sampled to response surface, selects 5000 radius r, thickness th parameter sample combination, response surface is sampled
Obtain 5000 groups of maximum equivalent amplitudes.Maximum equivalent is analyzed first with histogram method, its distribution is observed
Situation.
Because the distribution pattern of pressure vessel maximum equivalent is unknown, using nonparametric probability method
(Kernel Density Estimation, KDE) calculates the probability density function for obtaining equivalent stress maximum, as shown in Figure 7
For pressure vessel model maximum equivalent Density Estimator schematic diagram in the embodiment of the present invention；And by setting different confidences
Degree level, obtains the confidential interval of the maximum equivalent of pressure vessel, as shown in table 4 for equivalent stress amplitude in different confidences
Confidential interval under degree, it can be found that under the conditions of different confidence levels, the bound of equivalent stress maximum is not with confidence level
Decline occur significant changes.
Confidential interval of the equivalent stress amplitude of table 4 under different confidence levels
In step E, the present invention adds experiment special case in the FEM model of foundation and the FEM model of foundation is entered
Row certification is assessed.The present invention designs the experiment of low-level and high-level atmospheric pressure validation test is authenticated to assess, choosing
Two groups of data in pressure force container test data, separately design certification experiment one：P=36.725psi, χ=0.5, H=
13in；Certification experiment two：P=52.884psi, χ=0.7, H=45in are tested, and the flow according to validation test is obtained
Kriging response surfaces under two groups of control parameters, frequency accumulation Nogata is obtained after carrying out Monte Carlo random sampling to response surface
Figure, Density Estimator curve and cumulative distribution function curve.
Experimental design is carried out from acknowledged minor structure, under two groups of experimental conditions, two groups of Kriging response surfaces are built,
Each response surface agent model is sampled 5000 times, 5000 certification samples are just obtained, by two groups of obtained certification samples
It is fitted using Density Estimator method, obtains KDE curves.Meanwhile, the finite element simulation under two kinds of experiment conditions is obtained
Maximum equivalent value contrasted with KDE curves.It is illustrated in figure 8 certification experiment one in the embodiment of the present invention
Kriging response surface schematic diagrames.It is illustrated in figure 9 the KDE curve synoptic diagrams of certification experiment one in the embodiment of the present invention.Such as Figure 10
It show the CDF curve synoptic diagrams of certification experiment one in the embodiment of the present invention.It is certification in the embodiment of the present invention as shown in figure 11
The Kriging response surface schematic diagrames of experiment two.It is the KDE curves signal of certification experiment two in the embodiment of the present invention as shown in figure 12
Figure.It is the CDF curve synoptic diagrams of certification experiment two in the embodiment of the present invention as shown in figure 13.It is imitative by KDE curves and finite element
The maximum equivalent value really obtained, which is compared, to be understood, response surface prediction data and emulation data have certain deviation, but phase
Difference is less big.Most response surface predicts the outcome will be smaller than FEM Numerical Simulation, therefore utilizes Kriging response surfaces
The result that agent model is predicted is too conservative, so as to realize that FEM model confirms.
One of ordinary skill in the art will be appreciated that embodiment described here is to aid in reader and understands this hair
Bright principle, it should be understood that protection scope of the present invention is not limited to such especially statement and embodiment.This area
Those of ordinary skill can make according to these technical inspirations disclosed by the invention various does not depart from the other each of essence of the invention
Plant specific deformation and combine, these deformations and combination are still within the scope of the present invention.
Claims (5)
1. a kind of finite Element Model Validation based on Kriging response surfaces, it is characterised in that comprise the following steps：
A, to pressure vessel model carry out system subdivision, set up FEM model；
B, the model parameter to FEM model in step A carry out dependence on parameter and Parameter Sensitivity Analysis；
C, selection and maximum equivalent correlation maximum parameter, build pressure using Latin hypercube experimental design method and hold
The Kriging response surfaces of device model maximum equivalent；
D, in step C Kriging response surfaces carry out random sampling, using Density Estimator method to sampling samples probability distribution
Analyzed with confidential interval；
E, single result of finite element and FEM Numerical Simulation contrasted, realize that FEM model confirms.
2. the finite Element Model Validation as claimed in claim 1 based on Kriging response surfaces, it is characterised in that described
Step B carries out dependence on parameter to the model parameter of FEM model and Parameter Sensitivity Analysis is specially：
The model parameter of FEM model is divided into control parameter and geometry material parameter, using experimental design module computational geometry
Sensitivity of the material parameter to maximum equivalent and maximum deformation quantity, obtains model parameter sensitivity block diagram；Using experiment
Module computational geometry material parameter is designed to the correlation of maximum equivalent, model parameter correlation matrix is obtained.
3. the finite Element Model Validation as claimed in claim 2 based on Kriging response surfaces, it is characterised in that described
The mathematical modeling of the Kriging response surfaces of pressure vessel model maximum equivalent is specially in step C：
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Wherein,The response prediction value for being Kriging response surfaces at sample point x,For the basic function coefficient after estimation, f
(x) it is the basic function of fitting function, R is correlation matrix, yDFor the observation of sample, F is f (x) observation.
4. the finite Element Model Validation as claimed in claim 3 based on Kriging response surfaces, it is characterised in that described
Step D carries out random sampling to Kriging response surfaces, using Density Estimator method to sampling samples probability distribution and confidence area
Between analyzed specially：
Using Monte Carlo and arbitrary sampling method, 5000 radius r, thickness th parameter sample combination are chosen, to Kriging
Response surface, which is sampled, obtains 5000 groups of maximum equivalent amplitudes, and maximum equivalent is analyzed using histogram method；
The probability density function of equivalent stress maximum, the different confidence level by setting are calculated using nonparametric probability method
Level, obtains the confidential interval of the maximum equivalent of pressure vessel model.
5. the finite Element Model Validation as claimed in claim 4 based on Kriging response surfaces, it is characterised in that described
Step E is contrasted single result of finite element and FEM Numerical Simulation, realizes that FEM model confirms to be specially：
According to pressure vessel modelling low-level and high-level atmospheric pressure experiment, two groups of Kriging response surfaces are built, to every
Individual Kriging response surfaces, which are sampled, obtains certification sample, and two groups of obtained certification samples are entered using Density Estimator method
Row fitting, obtains KDE curves；The maximum equivalent value that finite element simulation under KDE curves and two kinds of experiment conditions is obtained
It is compared, realizes that FEM model confirms.
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