CN109885896A - A kind of nonlinear organization correction method for finite element model based on multiple change differential sensitivity - Google Patents
A kind of nonlinear organization correction method for finite element model based on multiple change differential sensitivity Download PDFInfo
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Abstract
The present invention discloses a kind of nonlinear organization correction method for finite element model based on multiple change differential sensitivity, firstly, establishing the nonlinear finite element initial analysis model of structure；Secondly, carrying out multiple step-length perturbation to the parameter to be modified in structure, Nonlinear Dynamic response sensitivity is calculated；Again, the response data of measurement structure establishes nonlinear model modified objective function；Finally, measurement dynamic response and analysis dynamic response are made Least-squares minimization, to realize that nonlinear finite element model is corrected.The present invention can correct the structural parameters of nonlinear finite element analysis model, improve the computational accuracy of analysis model.
Description
Technical field
The invention belongs to nonlinear organization FEM updating fields, more particularly to one kind to be based on becoming differential sensitivity again
Nonlinear organization correction method for finite element model.
Background technique
The maturation of finite element method provides quickly and effectively numerical value solution for the analysis of engineering structure.But
Error in finite element modeling process frequently can lead to initial finite element model and be unable to accurate characterization structural mechanics feature.Finite element
Modifying model corrects finite element using the measurement data of analysis structure as the method for accurate Engineering Structure Analysis model is provided
The modeling parameters of analysis model, so that analysis response and measurement dynamic response residual error are minimum, to obtain accurate analysis model.
Application of the existing Model Updating Technique in linear structure have been relatively mature, but consider structural nonlinear feature
When nonlinear organization correction method for finite element model theory also in further development.And current nonlinear finite element model
In correction technique, generally uses the frequency domain response data of structure and had based on non-gradient optimization algorithm to nonlinear organization
Limit meta-model is modified, and it is time-consuming to increase analysis.How the time domain response of structure and the modification method based on sensitivity utilized,
Optimized Iterative number is reduced, amendment efficiency is improved, becomes Practical Project problem urgently to be resolved.
Summary of the invention
Goal of the invention: in view of the above problems, the present invention proposes that a kind of nonlinear organization based on multiple change differential sensitivity has
Meta-model modification method is limited, multiple variable step perturbation is carried out to the parameter to be modified of nonlinear organization, by nonlinear organization dynamic response
Calculating complex field is expanded to by real number field, and the imaginary data by extracting calculated result calculates nonlinear organization dynamic response
Sensitivity is realized on this basis based on the multiple nonlinear organization FEM updating for becoming differential sensitivity, is improved non-linear
The computational accuracy of finite element analysis model.
Technical solution: to achieve the purpose of the present invention, the technical scheme adopted by the invention is that: one kind is divided based on multiple be deteriorated
The nonlinear organization correction method for finite element model of sensitivity, method includes the following steps:
(1) the nonlinear finite element initial analysis model with nonlinear characteristic structure is established, initial point of structure is calculated
Analyse dynamic response；
(2) multiple step-length perturbation is carried out to the parameter to be modified in structure, calculates Nonlinear Dynamic response sensitivity；
(3) response data of measurement structure establishes nonlinear model modified objective function；
(4) measurement dynamic response and analysis dynamic response are made into Least-squares minimization, realizes nonlinear finite element model amendment.
Further, in step (1), the nonlinear finite element initial analysis model of structure is established, the non-thread of structure is calculated
Property response, the specific steps are as follows:
(1.1) discrete modeling is carried out to structure, obtains the nonlinear finite element initial analysis model of structure；
(1.2) numerical analysis method is utilized, the nonlinear finite element model established is carried out according to the initial value of structure
Dynamic response calculates, and obtains the nonlinear analysis dynamic response initial value r of structurea(θ1,t)。
Further, in step (2), multiple step-length perturbation is carried out to the parameter to be modified in structure, calculates the non-linear sound of something astir
Answer sensitivity, the specific steps are as follows:
(2.1) the multiple step-length perturbation of structural texture parameter to be modified, obtains the complex field format of structure parameter to be modified:
The complex field format of formula (1) expression structure parameter to be modified, wherein i indicates imaginary unit, has following relationship:
i2=-1；θ indicates the parameter to be modified of structure, hθIndicate the multiple step-length perturbation of structure parameter to be modified,Indicate multiple step-length perturbation
The parameter to be modified of structure afterwards；
(2.2) nonlinear finite element analysis model for being established the parameter to be modified replacement (1.1) after the perturbation of multiple step-length
In corresponding parameter to be modified, and using numerical analysis method to after perturbation nonlinear finite element analysis model carry out complex field
It solves, extracts the imaginary data of structure dynamic response, obtain corresponding to nonlinearity in parameters dynamic response sensitivity to be modified:
In formula, t indicates time, sθIndicate that structure dynamic response treats the sensitivity of corrected parameter, subscript displacement,
Velocity, acceleration respectively indicate displacement, speed, acceleration；Indicate the imaginary part of extraction structure dynamic response
Data；Displacement, velocity and acceleration response after respectively indicating Parameter Perturbation to be modified, hθIndicate knot
The multiple step-length of structure parameter to be modified perturbs；
(2.3) all parameters to be modified are calculated with dynamic respond sensitivity matrix S or speed responsive sensitivity after perturbation
Matrix S or acceleration responsive sensitivity matrix S.
Further, in step (3), the response data of measurement structure establishes nonlinear model modified objective function, tool
Steps are as follows for body:
(3.1) with the accurate parameter value of the setting to be modified of structure, model the adding in test point of foundation is calculated
Speed or speed or dynamic respond are as measurement dynamic response value re(t)；
(3.2) dynamic analysis is carried out to the nonlinear finite element model of foundation, using numerical analysis method, obtained sound
It should be used as analysis dynamic response ra(θ, t), by measuring, the dynamic response jointly constructs that dynamic response and finite element analysis model obtain are non-thread
The objective function of property Modifying model, the objective function of construction are as follows:
In formula, t indicates time, re(t) measurement dynamic response is indicated, to determine value；ra(θ, t) is analysis dynamic response, to calculate
Value, θ indicate the parameter vector to be modified being made of multiple parameters to be modified,θWithRespectively indicate parameter to be modified lower bound and
The upper bound controls the constant interval of corrected parameter.
Further, in step (4), measurement dynamic response and analysis dynamic response are made into Least-squares minimization, to realize non-
Linear finite Modifying model, the specific steps are as follows:
Using single object optimization mode, by establishing restriction on the parameters boundary, it is excellent that jth step iteration is acquired using sensitivity method
Parameters variation amount during change:
In formula, t indicates time, θjThe parameter vector to be modified of j-th of iteration step is indicated, when j=1, using set
Initial value；SjIt indicates in j-th of iteration optimization step by the obtained structural nonlinear acceleration responsive sensitivity matrix of step 2；re
(t) measurement dynamic response is indicated, to determine value；ra(θj, t) and indicate that the nonlinear analysis of j-th of iteration step obtains dynamic response；θj+1Table
Show+1 iteration step of the jth parameter vector to be modified being calculated by formula (6), using least square optimization, to ginseng to be modified
Number θ is constantly corrected, and when analysis dynamic response meets convergent requirement, be can be realized and is become the non-linear of differential sensitivity based on multiple
Structural model amendment.
The utility model has the advantages that compared with prior art, technical solution of the present invention has following advantageous effects:
The present invention provides a kind of based on the multiple nonlinear organization correction method for finite element model for becoming differential sensitivity, passes through
It treats corrected parameter and carries out multiple variable step perturbation, the dynamic response for calculating nonlinear organization becomes differential sensitivity again, realizes and be based on
The FEM updating of the nonlinear organization of sensitivity, this method has to improve for the Dynamic Modeling of nonlinear organization builds
The beneficial effect of mould precision.
Detailed description of the invention
Fig. 1 is algorithm flow chart of the invention；
Fig. 2 is the nonlinear flexible supports cantilever beam structure schematic diagram of the embodiment of the present invention；
Fig. 3 is that present invention amendment front-end geometry I point acceleration dynamic response compares figure；
Fig. 4 is that present invention amendment front-end geometry Section II point acceleration dynamic response compares figure；
Fig. 5 is the iteration convergence curve graph of parameter to be modified in makeover process of the present invention.
Specific embodiment
Further description of the technical solution of the present invention with reference to the accompanying drawings and examples.
As shown in Figure 1, it is a kind of based on the multiple nonlinear organization correction method for finite element model for becoming differential sensitivity, including such as
Lower step:
(1) the nonlinear finite element initial analysis model with the structure of nonlinear characteristic is established, the initial of structure is calculated
Analyze dynamic response；
(2) multiple step-length perturbation is carried out to the parameter to be modified in structure, calculates Nonlinear Dynamic response sensitivity；
(3) response data of measurement structure establishes nonlinear model modified objective function；
(4) measurement dynamic response and analysis dynamic response are made into Least-squares minimization, to realize that nonlinear finite element model is repaired
Just.
It is a kind of based on the multiple nonlinear organization correction method for finite element model for becoming differential sensitivity, using containing nonlinear elasticity
Property bearing cantilever beam structure verify, the parameter of structure is respectively as follows: the length l=0.2m of cantilever beam structure, and cross section is circle
Shape, section radius r=0.005m, is modeled using aluminium, elasticity modulus of materials E=70GPa, and density p=2.7 ×
103kg/m3, Poisson's ratio μ=0.3.External drive power active position is as shown in Fig. 2, its value is F (t)=10sin (60 π t).
Parameter to be modified is the stiffness coefficient and damped coefficient of five nonlinear flexible supports, totally ten parameters to be modified,
This example is simulation example, it is assumed that the exact value of corrected parameter is as listed in table 1.Five nonlinear flexible supports generate non-linear
Restoring force expression formula isWherein,WithThe rigidity system of respectively i-th nonlinear flexible supports
Several and damped coefficient, as shown in table 1.ΔxiWithRespectively indicate the displacement difference and speed difference of i-th of nonlinear flexible supports.
The parameter exact value to be modified of 1 nonlinear flexible supports of table
Wherein, rigidity unit N/m3, damp unit N/ (m/s)3；
Concrete operations are as follows:
(1.1) in this example, the girder construction containing nonlinear characteristic is modeled, obtains the nonlinear finite element of structure
Initial analysis model.The initial value of parameter to be modified in nonlinear finite element model as listed in table 2, obtained parameter to be modified
Vector isParameter to be modified in this example is intended only as illustrating
It is bright, the method, which is also referred to, about the other parameters of girder construction is calculated.Nonlinear characteristic include: material nonlinearity,
Geometrical non-linearity and boundary nonlinear, the nonlinear characteristic of this example are boundary nonlinear.
The initial parameter value to be modified of 2 nonlinear flexible supports of table
Wherein, rigidity unit N/m3, damp unit N/ (m/s)3；
(1.2) dynamic response calculating is carried out to the nonlinear finite element model established using numerical analysis method, commonly
Numerical analysis method includes: central difference method, runge kutta method, Newmark- β method and Newton method etc., is used in this example
Method is Newmark- β method and Newton method.The nonlinear finite element model obtained to initial parameter in table 2 solves, and obtains
To the analysis dynamic response of structure, as the analysis dynamic response initial value r in step (3)a(θ1, t), wherein dynamic response initial value can be
Dynamic respond x (t), speed responsiveAcceleration responsiveAny one of.The corner that I, II two o'clock are chosen in this example adds
Speed responsive is as target response, as shown in Figure 2.
In step (2), the multiple point perturbation that is deteriorated is carried out to the parameter in nonlinear finite element analysis model, calculates non-linear sound
Answer sensitivity, the specific steps are as follows:
(2.1) the multiple step-length perturbation of structural texture parameter to be modified, obtains the complex field format of structure parameter to be modified, withParameter complex field format to be modified for parameter, after multiple step-length perturbation are as follows:
The complex field format of formula (1) expression structure parameter to be modified, wherein i indicates imaginary unit, has following relationship:
i2=-1；Indicate the parameter to be modified of structure,The multiple step-length perturbation for indicating structure parameter to be modified, is taken asI.e. 8.5 × 102,Structure parameter to be modified after indicating multiple step-length perturbation.Damped coefficient uses this step
Same calculation method is calculated.
(2.2) nonlinear finite element analysis model for being established the parameter to be modified replacement (1.1) after the perturbation of multiple step-length
In corresponding parameter, using numerical analysis method to after perturbation nonlinear finite element model carry out dynamic response calculating, commonly
Numerical analysis method includes: central difference method, runge kutta method, Newmark- β method and Newton method etc., is used in this example
Method is Newmark- β method and Newton method.Nonlinear Dynamic response after the perturbation being calculated is only used for the non-thread of this step
Property dynamic response sensitivity analysis.The dynamic response imaginary data for extracting model after perturbing, obtains corresponding to nonlinearity in parameters to be modified
Dynamic response sensitivity.This example obtains parameter to be modified by taking acceleration dynamic response sensitivity solution as an exampleThe acceleration sound of something astir
Answer sensitivity are as follows:
In formula,Indicate that structure acceleration dynamic response treats corrected parameterSensitivity, subscript acceleration
Indicate acceleration；Indicate the imaginary data of extraction structure dynamic response；Indicate parameter to be modifiedAdding after perturbation
Speed responsive is solved to obtain using numerical method to the model after perturbation；Indicate structure parameter to be modifiedAnswer
Step-length perturbation；Remaining 9 parameter in corrected parameter vector θ, which are treated, using the same procedure of step (2) carries out the non-linear sound of something astir
It answers sensitivity to solve, finally obtains non-linear acceleration responsive sensitivity matrix S.
In step (3), the response data of measurement structure establishes nonlinear model modified objective function, and specific steps are such as
Under:
This example is simulation example, and the acceleration of I, II two o'clock obtained with the model that accurate parameter listed by table 1 is established is rung
It should be used as measurement dynamic response value re(t).Dynamic analysis is carried out to the nonlinear finite element model of foundation, utilizes numerical analysis side
Method, common numerical analysis method include: central difference method, runge kutta method, Newmark- β method and Newton method etc., this calculation
The method used in example is Newmark- β method and Newton method.Obtained response is as analysis dynamic response ra(θ, t) will be measured
The dynamic response jointly constructs nonlinear model modified objective function that dynamic response and finite element analysis model obtain, the target of construction
Function are as follows:
In formula, re(t) measurement dynamic response is indicated, to determine value；ra(θ, t) is analysis dynamic response, is calculated value.This example
It is middle to use acceleration analysis dynamic response and acceleration analysis dynamic response, it can promote dynamic using Displacement Analysis dynamic response and velocity analysis
Response, the corresponding sound of something astir that measures should be displacement measurement dynamic response and tachometric survey dynamic response.θ is indicated by multiple parameter groups to be modified
At parameter vector to be modified,θWithLower bound and the upper bound for respectively indicating parameter to be modified, control the constant interval of corrected parameter.
In step (4), measurement dynamic response and analysis dynamic response are made into Least-squares minimization, to realize nonlinear finite element
Modifying model, the specific steps are as follows:
Using single object optimization mode, by establishing restriction on the parameters boundary, iteration optimization is walked to jth using sensitivity method
Parameters variation amount in the process is solved, the Optimal Parameters θ after obtaining jth step iterationj+1:
In formula, θjIndicate the parameter vector to be modified of j-th of iteration step；When j=1, using initial value listed by table 2；SjIt indicates
By the obtained structural nonlinear acceleration responsive sensitivity matrix of step 2 in j-th of iteration optimization step；re(t) measurement is indicated
Dynamic response, to determine value；ra(θj, t) indicate j-th of iteration step nonlinear analysis acceleration dynamic response.θj+1It indicates by formula (4)
+ 1 iteration step of the jth parameter vector to be modified being calculated.Initial parameter value to be modified is as listed in table 2, excellent using least square
Change method is treated corrected parameter θ and is constantly corrected, and when analysis dynamic response meets convergent requirement, can be realized and is become based on multiple
The nonlinear organization Modifying model of differential sensitivity.Fig. 3 and Fig. 4 respectively indicates the nonlinear organization finite element model of amendment front and back
The iteration convergence curve of obtained acceleration dynamic response and 10 parameters to be modified.Table 3 is the mistake before and after parameters revision to be modified
Difference analysis.
Error before and after the parameters revision to be modified of table 3
Wherein, stiffness coefficient unit N/m3, damped coefficient unit N/ (m/s)3。
Claims (5)
1. a kind of based on the multiple nonlinear organization correction method for finite element model for becoming differential sensitivity, which is characterized in that this method
The following steps are included:
(1) the nonlinear finite element initial analysis model with nonlinear characteristic structure is established, the initial analysis for calculating structure is dynamic
Response；
(2) multiple step-length perturbation is carried out to the parameter to be modified in structure, calculates Nonlinear Dynamic response sensitivity；
(3) response data of measurement structure establishes nonlinear model modified objective function；
(4) measurement dynamic response and analysis dynamic response are made into Least-squares minimization, realizes nonlinear finite element model amendment.
2. according to claim 1 a kind of based on the multiple nonlinear organization FEM updating side for becoming differential sensitivity
Method, which is characterized in that in step (1), establish the nonlinear finite element initial analysis model of structure, calculate the non-linear sound of structure
It answers, the specific steps are as follows:
(1.1) discrete modeling is carried out to structure, obtains the nonlinear finite element initial analysis model of structure；
(1.2) numerical analysis method is utilized, the sound of something astir is carried out according to the initial value of structure to the nonlinear finite element model established
It should calculate, obtain the nonlinear analysis dynamic response initial value r of structurea(θ1,t)。
3. a kind of multiple be deteriorated according to claim 1 or 2 divides the nonlinear finite element model amendment side of nonlinear-sensitivity
Method, which is characterized in that in step (2), multiple step-length perturbation is carried out to the parameter to be modified in structure, calculates Nonlinear Dynamic response spirit
Sensitivity, the specific steps are as follows:
(2.1) the multiple step-length perturbation of structural texture parameter to be modified, obtains the complex field format of structure parameter to be modified:
The complex field format of formula (1) expression structure parameter to be modified, wherein i indicates imaginary unit, has following relationship: i2=-
1；θ indicates the parameter to be modified of structure, hθIndicate the multiple step-length perturbation of structure parameter to be modified,After indicating multiple step-length perturbation
Structure parameter to be modified；
(2.2) right in the nonlinear finite element analysis model for being established the parameter to be modified replacement (1.1) after the perturbation of multiple step-length
The parameter to be modified answered, and complex field is carried out to the nonlinear finite element analysis model after perturbation using numerical analysis method and is asked
Solution extracts the imaginary data of structure dynamic response, obtains corresponding to nonlinearity in parameters dynamic response sensitivity to be modified:
In formula, t indicates time, sθIndicate that structure dynamic response treats the sensitivity of corrected parameter, subscript displacement,
Velocity, acceleration respectively indicate displacement, speed, acceleration；Indicate the imaginary part of extraction structure dynamic response
Data；Displacement, velocity and acceleration response after respectively indicating Parameter Perturbation to be modified, hθIndicate knot
The multiple step-length of structure parameter to be modified perturbs；
(2.3) all parameters to be modified are calculated with dynamic respond sensitivity matrix S or speed responsive sensitivity matrix S after perturbation
Or acceleration responsive sensitivity matrix S.
4. a kind of multiple be deteriorated according to claim 1 or 2 divides the nonlinear finite element model amendment side of nonlinear-sensitivity
Method, which is characterized in that in step (3), the response data of measurement structure establishes nonlinear model modified objective function, specifically
Steps are as follows:
(3.1) with the accurate parameter value of the setting to be modified of structure, acceleration of the model in test point of foundation is calculated
Or speed or dynamic respond are as measurement dynamic response value re(t)；
(3.2) dynamic analysis is carried out to the nonlinear finite element model of foundation, using numerical analysis method, obtained response is made
To analyze dynamic response ra(θ, t), the dynamic response jointly constructs nonlinear model that measurement dynamic response and finite element analysis model are obtained
Type modified objective function, the objective function of construction are as follows:
In formula, t indicates time, re(t) measurement dynamic response is indicated, to determine value；ra(θ, t) is analysis dynamic response, is calculated value, θ
Indicate the parameter vector to be modified being made of multiple parameters to be modified,θWithLower bound and the upper bound of parameter to be modified are respectively indicated,
Control the constant interval of corrected parameter.
5. a kind of multiple be deteriorated according to claim 4 divides the nonlinear finite element model modification method of nonlinear-sensitivity,
It is characterized in that, in step (4), measurement dynamic response and analysis dynamic response are made into Least-squares minimization, to realize non-linear have
Limit meta-model amendment, the specific steps are as follows:
Using single object optimization mode, by establishing restriction on the parameters boundary, jth step iteration optimization mistake is acquired using sensitivity method
Parameters variation amount in journey:
In formula, t indicates time, θjThe parameter vector to be modified of j-th of iteration step is indicated, when j=1, using set initial
Value；SjIt indicates in j-th of iteration optimization step by the obtained structural nonlinear acceleration responsive sensitivity matrix of step 2；re(t)
Measurement dynamic response is indicated, to determine value；ra(θj, t) and indicate that the nonlinear analysis of j-th of iteration step obtains dynamic response；θj+1It indicates
+ 1 iteration step of the jth parameter vector to be modified being calculated by formula (6), using least square optimization, to ginseng to be modified
Number θ is constantly corrected, and when analysis dynamic response meets convergent requirement, be can be realized and is become the non-linear of differential sensitivity based on multiple
Structural model amendment.
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