CN109885896B - Nonlinear structure finite element model correction method based on complex variation differential sensitivity - Google Patents
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Abstract
The invention discloses a nonlinear structure finite element model correction method based on complex variation differential sensitivity, which comprises the following steps of firstly, establishing a nonlinear finite element initial analysis model of a structure; secondly, carrying out complex step perturbation on parameters to be corrected in the structure, and calculating the nonlinear dynamic response sensitivity; thirdly, measuring response data of the structure, and establishing a target function corrected by the nonlinear model; and finally, performing least square optimization on the measured dynamic response and the analyzed dynamic response, thereby realizing the correction of the nonlinear finite element model. The invention can modify the structure parameters of the nonlinear finite element analysis model and improve the calculation precision of the analysis model.
Description
Technical Field
The invention belongs to the field of nonlinear structure finite element model correction, and particularly relates to a nonlinear structure finite element model correction method based on complex variable differential sensitivity.
Background
The maturity of finite element analysis methods provides a fast and effective numerical solution for the analysis of engineering structures. However, errors in the finite element modeling process often result in the initial finite element model not being able to accurately characterize the structural mechanical characteristics. Finite element model modification is used as a method for providing an accurate engineering structure analysis model, and the modeling parameters of the finite element analysis model are modified by using the measurement data of an analysis structure, so that the residual error between an analysis response and a measurement dynamic response is minimum, and an accurate analysis model is obtained.
The application of the existing model modification technology in linear structures is mature, but the theory of the finite element model modification method of the nonlinear structure is still further developed when the nonlinear characteristics of the structure are considered. In the existing nonlinear finite element model correction technology, the finite element model of the nonlinear structure is corrected by generally adopting frequency domain response data of the structure and a non-gradient-based optimization algorithm, so that the analysis time is increased. How to utilize the time domain response of the structure and the correction method based on the sensitivity, the optimization iteration times are reduced, the correction efficiency is improved, and the practical engineering problem to be solved urgently is formed.
Disclosure of Invention
The purpose of the invention is as follows: aiming at the problems, the invention provides a nonlinear structure finite element model correction method based on complex variation differential sensitivity, which comprises the steps of carrying out complex variation step perturbation on a parameter to be corrected of a nonlinear structure, expanding the calculation of nonlinear structure dynamic response from a real number domain to a complex number domain, and calculating the sensitivity of nonlinear structure dynamic response by extracting imaginary part data of a calculation result, thereby realizing the nonlinear structure finite element model correction based on the complex variation differential sensitivity and improving the calculation precision of a nonlinear finite element analysis model.
The technical scheme is as follows: in order to realize the purpose of the invention, the technical scheme adopted by the invention is as follows: a nonlinear structure finite element model correction method based on complex variation differential sensitivity comprises the following steps:
(1) establishing a nonlinear finite element initial analysis model with a nonlinear characteristic structure, and calculating the initial analysis dynamic response of the structure;
(2) carrying out complex step perturbation on parameters to be corrected in the structure, and calculating the nonlinear dynamic response sensitivity;
(3) measuring response data of the structure, and establishing a target function corrected by a nonlinear model;
(4) and performing least square optimization on the measured dynamic response and the analyzed dynamic response to realize nonlinear finite element model correction.
Further, in the step (1), a nonlinear finite element initial analysis model of the structure is established, and nonlinear response of the structure is calculated, specifically comprising the following steps:
(1.1) carrying out discrete modeling on the structure to obtain a nonlinear finite element initial analysis model of the structure;
(1.2) using a numerical analysis method onThe established nonlinear finite element model carries out dynamic response calculation according to the initial value of the structure to obtain the initial value r of the dynamic response of nonlinear analysis of the structurea(θ1,t)。
Further, in the step (2), the parameters to be corrected in the structure are subjected to complex step perturbation, and the nonlinear dynamic response sensitivity is calculated, which specifically comprises the following steps:
(2.1) constructing a complex step perturbation of the parameters to be corrected of the structure to obtain a complex field format of the parameters to be corrected of the structure:
formula (1) represents a complex field format of the structural parameter to be corrected, wherein i represents an imaginary unit and has the following relationship: i.e. i2-1; theta denotes the parameter to be modified of the structure, hθRepresenting a complex step perturbation of a parameter to be modified of the structure,
representing a parameter to be corrected of the structure after the perturbation of the complex step length;
(2.2) replacing the parameters to be corrected in the established nonlinear finite element analysis model in step (1.1) with the parameters to be corrected after perturbation of the complex step length, performing complex field solution on the perturbed nonlinear finite element analysis model by using a numerical analysis method, extracting imaginary part data of structural dynamic response, and obtaining the nonlinear dynamic response sensitivity corresponding to the parameters to be corrected:
wherein t represents time, sθRepresentation structure dynamic response treatmentCorrecting the sensitivity of the parameters, and respectively representing displacement, speed and acceleration by superscript displacement, velocity and acceleration;
imaginary data representing the extracted structural dynamic response;
respectively representing the responses of the displacement, the speed and the acceleration after the perturbation of the parameter to be corrected, hθRepresenting a complex step perturbation of a parameter to be modified of the structure;
and (2.3) calculating a perturbed displacement response sensitivity matrix S or a speed response sensitivity matrix S or an acceleration response sensitivity matrix S for all the parameters to be corrected.
Further, in the step (3), response data of the structure is measured, and a nonlinear model modified objective function is established, specifically including the following steps:
(3.1) calculating the acceleration or speed or displacement response of the established model at the test point by using the set accurate parameter value to be corrected of the structure as the measured dynamic response value re(t)；
(3.2) carrying out dynamics analysis on the established nonlinear finite element model, and using a numerical analysis method to obtain a response as an analysis dynamic response ra(theta, t), constructing a nonlinear model modified objective function by combining the measured dynamic response and the dynamic response obtained by the finite element analysis model, wherein the constructed objective function is as follows:
wherein t represents time, re(t) represents a measured dynamic response, which is a deterministic value; r isa(theta, t) is an analysis dynamic response and is a calculated value, theta represents a parameter vector to be corrected consisting of a plurality of parameters to be corrected,θand
respectively representing the lower bound and the upper bound of the parameter to be corrected, and controlling the change of the correction parameterAnd (4) forming an interval.
Further, in the step (4), the least square optimization is performed on the measured dynamic response and the analyzed dynamic response, so that the nonlinear finite element model correction is realized, and the specific steps are as follows:
and (3) adopting a single-target optimization mode, and solving the parameter variation in the iterative optimization process of the step j by establishing a parameter constraint boundary and utilizing a sensitivity method:
wherein t represents time, θjWhen j is 1, the set initial value is adopted; sjRepresenting a structural nonlinear acceleration response sensitivity matrix obtained in the step 2 in the jth iterative optimization step; r ise(t) represents a measured dynamic response, which is a deterministic value; r isa(θjT) represents the dynamic response obtained by the nonlinear analysis of the jth iteration step; thetaj+1And (3) representing the to-be-corrected parameter vector of the (j + 1) th iteration step calculated by the formula (6), continuously correcting the to-be-corrected parameter theta by using a least square optimization method, and realizing the correction of the nonlinear structure model based on the complex variation differential sensitivity when the dynamic response is analyzed to meet the convergence requirement.
Has the advantages that: compared with the prior art, the technical scheme of the invention has the following beneficial technical effects:
the invention provides a nonlinear structure finite element model correction method based on complex variable differential sensitivity, which realizes the correction of a nonlinear structure finite element model based on sensitivity by carrying out complex variable step perturbation on a parameter to be corrected and calculating the dynamic response complex variable differential sensitivity of the nonlinear structure.
Drawings
FIG. 1 is a flow chart of the algorithm of the present invention;
FIG. 2 is a schematic view of a non-linear resiliently supported cantilever beam configuration according to an embodiment of the present invention;
FIG. 3 is a graph comparing the acceleration dynamic response at point I of the front and rear structure after correction according to the present invention;
FIG. 4 is a graph comparing acceleration dynamic responses at point II of the front and rear corrected structures according to the present invention;
FIG. 5 is a graph illustrating the iterative convergence of the parameter to be corrected during the correction process according to the present invention.
Detailed Description
The technical solution of the present invention is further described below with reference to the accompanying drawings and examples.
As shown in fig. 1, a method for modifying a finite element model of a nonlinear structure based on complex-varying differential sensitivity includes the following steps:
(1) establishing a nonlinear finite element initial analysis model of a structure with nonlinear characteristics, and calculating the initial analysis dynamic response of the structure;
(2) carrying out complex step perturbation on parameters to be corrected in the structure, and calculating the nonlinear dynamic response sensitivity;
(3) measuring response data of the structure, and establishing a target function corrected by a nonlinear model;
(4) and performing least square optimization on the measured dynamic response and the analyzed dynamic response so as to realize nonlinear finite element model correction.
A nonlinear structure finite element model correction method based on complex variation differential sensitivity is verified by adopting a cantilever beam structure with a nonlinear elastic support, and the parameters of the structure are respectively as follows: the cantilever beam structure has length of 0.2m, circular cross section and section radius of 0.005m, and is modeled with aluminum, elastic modulus of material E of 70GPa and density rho of 2.7 × 103kg/m3The poisson ratio μ is 0.3. The position of the external excitation force is shown in fig. 2, and has a value of f (t) ═ 10sin (60 tt).
The parameters to be corrected are the stiffness coefficient and the damping coefficient of five nonlinear elastic supports, and the total number of the parameters to be corrected is ten, the example is a simulation example, and the accurate values of the correction parameters are assumed to be listed in table 1. The expression of the nonlinear restoring force generated by the five nonlinear elastic supports is
Wherein,
and
the stiffness coefficient and the damping coefficient of the ith nonlinear elastic support are shown in table 1.Δ xiAnd
respectively representing the displacement difference and the speed difference of the ith nonlinear elastic bearing.
TABLE 1 exact values of the parameters to be corrected for the non-linear elastic bearing
Wherein the unit of stiffness is N/m3Damping unit N/(m/s)3；
The specific operation is as follows:
(1.1) in this example, a beam structure containing nonlinear features was modeled to obtain a nonlinear finite element initial analysis model of the structure. The initial values of the parameters to be corrected in the nonlinear finite element model are listed in Table 2, and the obtained parameter vector to be corrected is
The parameters to be modified in this example are only given as examples, and other parameters related to the beam structure may be calculated with reference to this method. The non-linear characteristics include: material non-linearity, geometric non-linearity, and boundary non-linearity, the non-linearity characteristic of this example is boundary non-linearity.
TABLE 2 initial values of parameters to be corrected for a non-linear elastic bearing
Wherein the unit of stiffness is N/m3Damping unit N/(m/s)3；
(1.2) utilizing a numerical analysis method to calculate the dynamic response of the established nonlinear finite element model, wherein the commonly used numerical analysis method comprises a center difference method, a Runge Kutta method, a Newmark- β method, a Newton method and the like, the methods adopted in the embodiment are the Newmark- β method and the Newton method, the nonlinear finite element model obtained by the initial parameters in the table 2 is solved to obtain the analytical dynamic response of the structure, and the analytical dynamic response is used as the initial value r of the analytical dynamic response in the step (3)a(θ1T), wherein the initial value of the dynamic response can be the displacement response x (t), the velocity response
Acceleration response
Any one of the above. In the present example, a target response is selected as the corner acceleration response of I, II two points, as shown in fig. 2.
In the step (2), the parameters in the nonlinear finite element analysis model are subjected to complex variation differential perturbation, and nonlinear response sensitivity is calculated, and the specific steps are as follows:
(2.1) constructing a complex step perturbation of the parameters to be corrected of the structure to obtain a complex field format of the parameters to be corrected of the structure so as to
For example, the complex field format of the parameter to be corrected after the complex step perturbation is:
formula (1) represents a complex field format of the structural parameter to be corrected, wherein i represents an imaginary unit havingThe following relationships: i.e. i2＝-1；
A parameter to be modified representing a structure,
a complex step perturbation representing a parameter to be modified of the structure is taken as
I.e. 8.5 x 102，
And representing the parameters to be corrected of the structure after the perturbation of the complex step length. The damping coefficient is calculated by the same calculation method in the step.
(2.2) replacing the parameters to be corrected after perturbation of the complex step length with the corresponding parameters in the established nonlinear finite element analysis model (1.1), and performing dynamic response calculation on the perturbed nonlinear finite element model by using a numerical analysis method, wherein the commonly used numerical analysis method comprises a center difference method, a Runge Kutta method, a Newmark- β method, a Newton method and the like, the methods adopted in the embodiment are a Newmark- β method and a Newton method, the perturbed nonlinear dynamic response obtained by calculation is only used for analyzing the nonlinear dynamic response sensitivity of the step, the dynamic response imaginary part data of the perturbed model is extracted, and the nonlinear dynamic response sensitivity corresponding to the parameters to be corrected is obtained
The acceleration response sensitivity of (a) is:
in the formula,
representing structural acceleration dynamic response to-be-corrected parameters
The sensitivity of (1) is shown by the superscript acceleration;
imaginary data representing the extracted structural dynamic response;
representing parameters to be modified
Solving the perturbed model by using a numerical method to obtain the perturbed acceleration response;
representing structural parameters to be corrected
Perturbation of complex step length; and (3) carrying out nonlinear dynamic response sensitivity solution on the rest 9 parameters in the parameter vector theta to be corrected by using the same method in the step (2), and finally obtaining a nonlinear acceleration response sensitivity matrix S.
In the step (3), response data of the structure is measured, and a target function corrected by the nonlinear model is established, and the specific steps are as follows:
the present example is a simulation example, and the acceleration response of I, II two points obtained by the model established by the accurate parameters listed in Table 1 is used as the measured dynamic response value re(t) carrying out dynamic analysis on the established nonlinear finite element model by using a numerical analysis method, wherein the commonly used numerical analysis method comprises a central difference method, a Runge Kutta method, a Newmark- β method, a Newton method and the like, and the methods adopted in the embodiment are the Newmark- β method and the Newton method, and the obtained response is used as an analysis dynamic response ra(theta, t), constructing a nonlinear model modified objective function by combining the measured dynamic response and the dynamic response obtained by the finite element analysis model, wherein the constructed objective function is as follows:
in the formula, re(t) represents a measured dynamic response, which is a deterministic value; r isa(θ, t) is the analytical dynamic response and is the calculated value. In the embodiment, the acceleration measurement dynamic response and the acceleration analysis dynamic response are adopted, the displacement analysis dynamic response and the speed analysis dynamic response can be popularized and adopted, and the corresponding measurement dynamic response is the displacement measurement dynamic response and the speed measurement dynamic response. Theta denotes a parameter vector to be corrected composed of a plurality of parameters to be corrected,θand
respectively representing the lower bound and the upper bound of the parameter to be corrected, and controlling the change interval of the correction parameter.
In the step (4), least square optimization is carried out on the measured dynamic response and the analyzed dynamic response, so that the nonlinear finite element model is corrected, and the method specifically comprises the following steps:
adopting a single-target optimization mode, establishing a parameter constraint boundary, and solving the parameter variation in the iterative optimization process of the jth step by using a sensitivity method to obtain the optimized parameter theta after the iteration of the jth stepj+1：
In the formula, thetajRepresenting the parameter vector to be corrected of the jth iteration step; when j is 1, the initial values listed in table 2 are adopted; sjRepresenting a structural nonlinear acceleration response sensitivity matrix obtained in the step 2 in the jth iterative optimization step; r ise(t) represents a measured dynamic response, which is a deterministic value; r isa(θjAnd t) represents the nonlinear analysis acceleration response of the jth iteration step. Thetaj+1Represents the j +1 th iteration to-be-corrected parameter vector calculated by the formula (4). Initial values of parameters to be corrected are listed in table 2, the parameters theta to be corrected are continuously corrected by using a least square optimization method, and when the dynamic response is analyzed to meet the convergence requirement, the nonlinear structure based on the complex variation differential sensitivity can be realizedAnd (6) correcting the model. Fig. 3 and 4 show the acceleration dynamic response obtained by the finite element model of the nonlinear structure before and after correction and the iterative convergence curves of 10 parameters to be corrected respectively. Table 3 shows the error analysis before and after the parameter to be corrected is corrected.
TABLE 3 error before and after correction of parameter to be corrected
Wherein the unit of stiffness coefficient is N/m3Damping coefficient unit N/(m/s)3。
Claims (2)
1. A nonlinear structure finite element model correction method based on complex variation differential sensitivity is characterized by comprising the following steps:
(1) establishing a nonlinear finite element initial analysis model of a cantilever beam structure with a nonlinear elastic support, and calculating the initial analysis dynamic response of the structure, wherein the specific method comprises the following steps:
(1.1) carrying out discrete modeling on the structure to obtain a nonlinear finite element initial analysis model of the structure;
(1.2) performing dynamic response calculation on the established nonlinear finite element model according to the initial value of the structure by using a numerical analysis method to obtain the initial value r of the dynamic response of the nonlinear analysis of the structurea(θ1,t)；
(2) The method comprises the following steps of performing complex step perturbation on parameters to be corrected in a structure, and calculating the nonlinear dynamic response sensitivity:
(2.1) constructing a complex step perturbation of the parameters to be corrected of the structure to obtain a complex field format of the parameters to be corrected of the structure:
formula (1) represents a complex field format of the structural parameter to be corrected, wherein i represents an imaginary unit and has the following relationship: i.e. i2-1; theta denotes the parameter to be modified of the structure, hθComplex step representing parameters to be modified of structureThe long-time perturbation is realized,
representing a parameter to be corrected of the structure after the perturbation of the complex step length;
(2.2) replacing the parameters to be corrected in the established nonlinear finite element analysis model in step (1.1) with the parameters to be corrected after perturbation of the complex step length, performing complex field solution on the perturbed nonlinear finite element analysis model by using a numerical analysis method, extracting imaginary part data of structural dynamic response, and obtaining the nonlinear dynamic response sensitivity corresponding to the parameters to be corrected:
wherein t represents time, sθThe sensitivity of the structure dynamic response to the parameter to be corrected is represented, and displacement, velocity and acceleration are represented by superscript displacement, velocity and acceleration respectively;
imaginary data representing the extracted structural dynamic response;
respectively representing the responses of the displacement, the speed and the acceleration after the perturbation of the parameter to be corrected, hθRepresenting a complex step perturbation of a parameter to be modified of the structure;
(2.3) calculating a perturbed displacement response sensitivity matrix S or velocity response sensitivity matrix S or acceleration response sensitivity matrix S for all parameters to be corrected;
(3) measuring response data of a structure, and establishing a nonlinear model modified objective function, wherein the specific method comprises the following steps:
(3.1) calculating the acceleration or speed or displacement response of the established model at the test point by using the set accurate parameter value to be corrected of the structure as the measured dynamic response value re(t)；
(3.2) carrying out dynamics analysis on the established nonlinear finite element model, and using a numerical analysis method to obtain a response as an analysis dynamic response ra(theta, t), constructing a nonlinear model modified objective function by combining the measured dynamic response and the dynamic response obtained by the finite element analysis model, wherein the constructed objective function is as follows:
wherein t represents time, re(t) represents a measured dynamic response, which is a deterministic value; r isa(theta, t) is an analysis dynamic response and is a calculated value, theta represents a parameter vector to be corrected consisting of a plurality of parameters to be corrected,θand
respectively representing the lower bound and the upper bound of the parameter to be corrected, and controlling the change interval of the correction parameter;
(4) and performing least square optimization on the measured dynamic response and the analyzed dynamic response to realize nonlinear finite element model correction.
2. The method for modifying a finite element model of a nonlinear structure based on complex-varying differential sensitivity as claimed in claim 1, wherein in the step (4), least squares optimization is performed on the measured dynamic response and the analyzed dynamic response, so as to modify the nonlinear finite element model, and the specific steps are as follows:
and (3) adopting a single-target optimization mode, and solving the parameter variation in the iterative optimization process of the step j by establishing a parameter constraint boundary and utilizing a sensitivity method:
wherein t represents time, θjWhen j is 1, the set initial value is adopted; sjRepresenting a structural nonlinear acceleration response sensitivity matrix obtained in the step 2 in the jth iterative optimization step; r ise(t) represents a measured dynamic response, which is a deterministic value; r isa(θjT) represents the dynamic response obtained by the nonlinear analysis of the jth iteration step; thetaj+1And (3) representing the to-be-corrected parameter vector of the (j + 1) th iteration step calculated by the formula (6), continuously correcting the to-be-corrected parameter theta by using a least square optimization method, and realizing the correction of the nonlinear structure model based on the complex variation differential sensitivity when the dynamic response is analyzed to meet the convergence requirement.
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