CN113468667A - Structure state monitoring and load identification method based on inverse finite element and finite element method - Google Patents

Structure state monitoring and load identification method based on inverse finite element and finite element method Download PDF

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CN113468667A
CN113468667A CN202110806838.XA CN202110806838A CN113468667A CN 113468667 A CN113468667 A CN 113468667A CN 202110806838 A CN202110806838 A CN 202110806838A CN 113468667 A CN113468667 A CN 113468667A
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李腾腾
武湛君
徐浩
李建乐
杨雷
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Dalian University of Technology
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Abstract

The invention provides a structure state monitoring and load identification method based on an inverse finite element and finite element method, which comprises the following steps: selecting an adaptive inverse shell unit to disperse the plate shell structure; calculating the membrane strain, bending strain and shearing strain of each inverse shell unit based on a mindlin plate theory; selecting strain measurement points on the upper surface and the lower surface of the inverse shell unit, and pasting a strain sensor on the strain measurement points to measure strain in real time to obtain strain measurement data; calculating the membrane strain and curvature of each inverse shell element based on the strain measurement data; constructing a functional based on a least square method, deriving the freedom of the nodes to obtain a similar rigidity matrix and a load matrix of the inverse shell unit, assembling the similar rigidity matrix and the load matrix, endowing proper boundary conditions, and calculating a displacement field of the structure; the relation between a displacement field and a loading force vector is established by a finite element method, errors generated by strain acquisition and displacement reconstruction are reduced by Tikhonov, and monitoring of state information of the structure and identification of load information are achieved.

Description

Structure state monitoring and load identification method based on inverse finite element and finite element method
Technical Field
The invention relates to the technical fields of structure real-time deformation monitoring, load identification and the like in the fields of aerospace, vehicles, buildings and the like, in particular to a structure state monitoring and load identification method based on an inverse finite element and finite element method.
Background
The structure is inevitably damaged during long-term use. Damage can cause degradation of the stiffness of the structure and even a catastrophic effect on the load-bearing properties of the structure. In the fields of aerospace, vehicles, construction, etc., a great deal of effort is invested each year in detecting and maintaining structures to ensure their functionality. Therefore, the structure state is monitored in real time based on the structure health monitoring technology, and the method plays a great role in improving the structure performance, guaranteeing the structure safety, reducing the maintenance cost and the like. The structural health monitoring technology is a multidisciplinary frontier technology, and comprises various sensing and reconstruction methods. In various state sensing methods, an Inverse Finite Element Method (iFEM) fully considers the complexity of boundary conditions and structural topology, and the calculation does not need a plurality of prior information such as load conditions, material information and the like, so that the method is a strong online health monitoring tool. The state information of the structure such as deformation, strain and the like can be accurately obtained in real time by means of an inverse finite element method. In addition, by using the method in combination with a damage detection method such as a infinitesimal dynamic response method, damage information of the structure can be further clarified. The external load is used as an important ring for representing the structural state and has a crucial influence on the expansion of structural damage, but the influence of the external load is not considered in the calculation process of the inverse finite element method.
Disclosure of Invention
According to the technical problem, a structure state monitoring and load identification method based on an inverse finite element and a finite element method is provided. The method comprises the steps of measuring the surface strain of the plate shell structure in real time, reconstructing state information such as deformation of the structure by adopting an inverse finite element method, reversely calculating the external load of the structure by using the displacement data obtained by reconstruction as the input of the finite element method, and reducing the influence of measurement noise and reconstruction errors of the inverse finite element method by introducing a Tikhonov regularization method in the calculation process, so that the size of the external load of the structure is accurately identified, and further the state monitoring and the load identification of the structure are realized.
The technical means adopted by the invention are as follows:
a structure state monitoring and load identification method based on an inverse finite element and finite element method comprises the following steps:
s1, dispersing the structure by selecting the adaptive inverse shell unit;
s2, calculating the membrane strain, the bending strain and the shearing strain of each inverse shell unit based on a mindlin plate theory;
s3, selecting strain measurement points on the upper surface and the lower surface of the inverse shell unit, pasting a strain sensor on the strain measurement points, and measuring strain in real time to obtain strain measurement data;
s4, calculating the membrane strain and curvature of each inverse shell unit based on the obtained strain measurement data;
s5, constructing a functional based on a least square method, deriving the freedom of the nodes to obtain a similar rigidity matrix and a load matrix of the inverse shell unit, assembling, endowing proper boundary conditions, and calculating a displacement field of the structure;
s6, constructing a classical finite element static equilibrium equation according to the reconstructed displacement information, and deducing to obtain the relationship between the reconstructed node displacement and the unknown load size;
s7, reducing errors existing in the process of strain acquisition and inverse finite element reconstruction by using a Tikhonov regularization method, and reconstructing the size of the external load of the structure.
Further, the structure in step S1 includes all complex geometric models composed of plate shells, and all the complex geometric models are equal-thickness plates.
Further, the specific implementation process of step S2 is as follows:
s21, calculating the membrane strain of each inverse shell unit, wherein the calculation formula is as follows:
e(ue)＝Beue
wherein u iseRepresenting the displacement vector of each inverse shell element node, BeRepresenting a matrix containing derivatives of a shape function;
s22, calculating the bending strain of each inverse shell unit, wherein the calculation formula is as follows:
k(ue)＝Bkue
wherein, BkRepresenting a matrix containing derivatives of a shape function;
s23, calculating the shear strain of each inverse shell unit, wherein the calculation formula is as follows:
g(ue)＝Bgue
wherein, BgRepresenting a matrix containing the derivatives of the shape function.
Further, the strains measured by the strain sensors in real time in step S3 are respectively:
Figure BDA0003166928920000031
Figure BDA0003166928920000032
wherein i represents the ith inverse shell unit, n represents the number of strain measurement points in the inverse shell unit, epsilon represents positive strain in the 1 and 2 directions, and gamma represents shear strain in the 1-2 directions.
Further, the strain sensor comprises one or more of a resistance strain gauge sensor, a fiber bragg grating sensor and a distributed fiber sensor.
Further, the strain sensors are arranged on the upper and lower surfaces of the inverted shell unit in the form of a single axis or a strain flower.
Further, the specific implementation process of step S4 is as follows:
s41, calculating the membrane strain of each inverse shell unit according to the obtained strain measurement data, wherein the calculation formula is as follows:
Figure BDA0003166928920000033
s42, calculating the curvature of each inverse shell unit according to the obtained strain measurement data, wherein the calculation formula is as follows:
Figure BDA0003166928920000034
where h represents half the cell thickness.
Further, the specific implementation process of step S5 is as follows:
s51, adopting a least square error function for a single inverse shell unit, wherein the function expression is as follows:
Φe(ue)＝we||e(ue)-eε||2+wk||k(ue)-kε||2+wg||g(ue)-gε||2
wherein we、wkAnd wgRespectively representing weight coefficients related to the membrane strain, the bending strain and the shear strain of each inverse shell unit, wherein if the membrane strain, the bending strain and the shear strain can be obtained through calculation of strain measurement values, the corresponding weight coefficients are 1; otherwise, the weight coefficient is adjusted to a small value;
s52, obtaining variation of the node degree of freedom of the structure to obtain a similar rigidity matrix and a load matrix of each inverse shell unit:
Figure BDA0003166928920000041
in the above formula, the first and second carbon atoms are,
Figure BDA0003166928920000042
a stiffness-like matrix representing the inverse shell element,
Figure BDA0003166928920000043
a class load matrix representing the inverse shell elements; wherein:
Figure BDA0003166928920000044
Figure BDA0003166928920000045
in the above formula, AeRepresents the area of the inverted shell element;
s53, assembling the quasi-rigidity matrix and the load matrix of the inverse shell unit according to a standard finite element program to obtain an integral linear equation set of the discrete structure, wherein the integral linear equation set comprises the following steps:
KiUi＝Fi
in the above formula, the first and second carbon atoms are,
Figure BDA0003166928920000046
wherein, TeRepresenting a coordinate transformation matrix, KiQuasi-stiffness matrix representing the inverse finite element method globaliRepresenting a load-like matrix of an inverse finite element method overall;
s54, applying proper boundary conditions, and calculating to obtain a displacement field of the structure, wherein the displacement field comprises the following steps:
KRiUi＝Fi
wherein, KRiThe quasi-stiffness matrix after the boundary condition is applied is shown as a positive definite matrix.
Further, the specific implementation process of step S6 is as follows:
s61, according to the classical finite element theory, the static finite element solution equation is rewritten into the following form:
KfUf＝Ff
wherein, KfRepresenting the stiffness matrix, U, in a finite element methodfRepresenting displacement vectors in a finite element method, FfRepresenting a load vector in a finite element method;
s62, according to different structural boundary conditions, expressing the equation into a form of a block matrix, as follows:
Figure BDA0003166928920000051
wherein, subscript 1 represents applied displacement boundary condition area, subscript 2 represents displacement known area, subscript 3 represents external load area to be obtained, and subscript 4 represents known external load or internal no external load area; 1. the displacement of the node of the 2 area is known, the load of the 4 area is known, and the loading area is generally avoided when measuring points are selected, so that the load outside the 2 area is 0;
s63, taking fixed boundary conditions as an example, U1The partitioning matrix is simplified to the following form, 0:
Figure BDA0003166928920000052
S64U obtained through inverse finite element calculation2Calculating the unknown force Fc：
Figure BDA0003166928920000053
Wherein the content of the first and second substances,
Figure BDA0003166928920000054
Figure BDA0003166928920000055
further, the specific implementation process of step S7 is as follows:
s71, according to the Tikhonov regularization method, representing the fitting error as the following form:
Figure BDA0003166928920000056
to reduce fitting errors
Figure BDA0003166928920000057
Introducing a cost function:
Figure BDA0003166928920000058
wherein λ represents a regularization parameter, and H represents a hermite transpose; to minimize the cost function, for the force vector FcMust be zero, calculated to yield:
Figure BDA0003166928920000061
and S72, determining the value of the regularization parameter lambda by using a common cross-validation method, a generalized cross-validation method or an L curve method, and finally calculating to obtain the magnitude of the external load.
Compared with the prior art, the invention has the following advantages:
1. according to the structure state monitoring and load identification method based on the inverse finite element and finite element method, the strain sensor is used for measuring to obtain the structure surface strain, the inverse finite element method is used for reconstructing the deformation of the structure in real time, and the finite element method is used for reconstructing the load size of the structure.
2. The method can accurately indicate the size of the load on the basis of accurately reconstructing the structure state, and provides a basis for damage expansion and service life evaluation of a subsequent structure.
3. Compared with the existing damage identification method, the method has the advantages of high precision, accuracy, strong noise resistance and the like.
Based on the reasons, the method can be widely popularized in the fields of real-time structural deformation monitoring, load identification and the like in the fields of aerospace, vehicles, buildings and the like.
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In order to more clearly illustrate the embodiments of the present invention or the technical solutions in the prior art, the drawings needed to be used in the description of the embodiments or the prior art will be briefly introduced below, and it is obvious that the drawings in the following description are some embodiments of the present invention, and for those skilled in the art, other drawings can be obtained according to these drawings without creative efforts.
FIG. 1 is a flow chart of the method of the present invention.
FIG. 2 is a schematic view showing the size and loading position of an aluminum alloy sheet according to the present invention;
FIG. 3 is a diagram of sensor locations for upper and lower surface posts of a structure, according to an embodiment of the present invention
In the figure: 1. loading position No. 1; 2. loading position No. 2; 3. the arrangement position of the surface sensor on the structure; 4. arrangement position of surface sensor under structure
Detailed Description
In order to make the technical solutions of the present invention better understood, the technical solutions in the embodiments of the present invention will be clearly and completely described below with reference to the drawings in the embodiments of the present invention, and it is obvious that the described embodiments are only a part of the embodiments of the present invention, and not all of the embodiments. All other embodiments, which can be derived by a person skilled in the art from the embodiments given herein without making any creative effort, shall fall within the protection scope of the present invention.
It should be noted that the terms "first," "second," and the like in the description and claims of the present invention and in the drawings described above are used for distinguishing between similar elements and not necessarily for describing a particular sequential or chronological order. It is to be understood that the data so used is interchangeable under appropriate circumstances such that the embodiments of the invention described herein are capable of operation in sequences other than those illustrated or described herein. Furthermore, the terms "comprises," "comprising," and "having," and any variations thereof, are intended to cover a non-exclusive inclusion, such that a process, method, system, article, or apparatus that comprises a list of steps or elements is not necessarily limited to those steps or elements expressly listed, but may include other steps or elements not expressly listed or inherent to such process, method, article, or apparatus.
As shown in fig. 1, the present invention provides a structure state monitoring and load identification method based on inverse finite element and finite element method, comprising the following steps:
s1, dispersing the structure by selecting the adaptive inverse shell unit;
s2, calculating the membrane strain, the bending strain and the shearing strain of each inverse shell unit based on a mindlin plate theory;
s3, selecting strain measurement points on the upper surface and the lower surface of the inverse shell unit, pasting a strain sensor on the strain measurement points, and measuring strain in real time to obtain strain measurement data;
s4, calculating the membrane strain and curvature of each inverse shell unit based on the obtained strain measurement data;
s5, constructing a functional based on a least square method, deriving the freedom of the nodes to obtain a similar rigidity matrix and a load matrix of the inverse shell unit, assembling, endowing proper boundary conditions, and calculating a displacement field of the structure;
s6, constructing a classical finite element static equilibrium equation according to the reconstructed displacement information, and deducing to obtain the relationship between the reconstructed node displacement and the unknown load size;
s7, reducing errors existing in the process of strain acquisition and inverse finite element reconstruction by using a Tikhonov regularization method, and reconstructing the size of the external load of the structure.
In specific implementation, as a preferred embodiment of the present invention, the structure in step S1 includes all complex geometric models composed of plate shells, and all the complex geometric models are equal-thickness plates. FIG. 2 is a schematic view showing the size and the loading position of the aluminum alloy sheet.
In specific implementation, as a preferred embodiment of the present invention, the specific implementation process of step S2 is as follows:
s21, calculating the membrane strain of each inverse shell unit, wherein the calculation formula is as follows:
e(ue)＝Beue
wherein u iseRepresenting the displacement vector of each inverse shell element node, BeRepresenting a matrix containing derivatives of a shape function;
s22, calculating the bending strain of each inverse shell unit, wherein the calculation formula is as follows:
k(ue)＝Bkue
wherein, BkRepresenting a matrix containing derivatives of a shape function;
s23, calculating the shear strain of each inverse shell unit, wherein the calculation formula is as follows:
g(ue)＝Bgue
wherein, BgRepresenting a matrix containing the derivatives of the shape function.
In a specific implementation, as a preferred embodiment of the present invention, the strains measured by the strain sensors in real time in step S3 are respectively:
Figure BDA0003166928920000081
Figure BDA0003166928920000082
wherein i represents the ith inverse shell unit, n represents the number of strain measurement points in the inverse shell unit, epsilon represents positive strain in the 1 and 2 directions, and gamma represents shear strain in the 1-2 directions.
In specific implementation, as a preferred embodiment of the present invention, the strain sensor includes one or a combination of two or more of a resistance strain gauge sensor, a fiber bragg grating sensor, and a distributed optical fiber sensor.
In specific implementation, as a preferred embodiment of the present invention, as shown in fig. 3, the strain sensors are arranged on the upper and lower surfaces of the inverted shell unit along a single axis or in a strain flower form.
In specific implementation, as a preferred embodiment of the present invention, the specific implementation process of step S4 is as follows:
s41, calculating the membrane strain of each inverse shell unit according to the obtained strain measurement data, wherein the calculation formula is as follows:
Figure BDA0003166928920000091
s42, calculating the curvature of each inverse shell unit according to the obtained strain measurement data, wherein the calculation formula is as follows:
Figure BDA0003166928920000092
where h represents half the cell thickness.
In specific implementation, as a preferred embodiment of the present invention, the specific implementation process of step S5 is as follows:
s51, adopting a least square error function for a single inverse shell unit, wherein the function expression is as follows:
Φe(ue)＝we||e(ue)-eε||2+wk||k(ue)-kε||2+wg||g(ue)-gε||2
wherein we、wkAnd wgRespectively representing weight coefficients related to the membrane strain, the bending strain and the shear strain of each inverse shell unit, wherein if the membrane strain, the bending strain and the shear strain can be obtained through calculation of strain measurement values, the corresponding weight coefficients are 1; otherwise, the weight coefficient is adjusted to a small value;
s52, obtaining variation of the node degree of freedom of the structure to obtain a similar rigidity matrix and a load matrix of each inverse shell unit:
Figure BDA0003166928920000093
in the above formula, the first and second carbon atoms are,
Figure BDA0003166928920000094
a stiffness-like matrix representing the inverse shell element,
Figure BDA0003166928920000095
a class load matrix representing the inverse shell elements; wherein:
Figure BDA0003166928920000101
Figure BDA0003166928920000102
in the above formula, AeRepresents the area of the inverted shell element;
s53, assembling the quasi-rigidity matrix and the load matrix of the inverse shell unit according to a standard finite element program to obtain an integral linear equation set of the discrete structure, wherein the integral linear equation set comprises the following steps:
KiUi＝Fi
in the above formula, the first and second carbon atoms are,
Figure BDA0003166928920000103
wherein, TeRepresenting a coordinate transformation matrix, KiQuasi-stiffness matrix representing the inverse finite element method globaliRepresenting a load-like matrix of an inverse finite element method overall;
s54, applying proper boundary conditions, and calculating to obtain a displacement field of the structure, wherein the displacement field comprises the following steps:
KRiUi＝Fi
wherein, KRiThe quasi-stiffness matrix after the boundary condition is applied is shown as a positive definite matrix.
In specific implementation, as a preferred embodiment of the present invention, the specific implementation process of step S6 is as follows:
s61, according to the classical finite element theory, the static finite element solution equation is rewritten into the following form:
KfUf＝Ff
wherein, KfRepresenting the stiffness matrix, U, in a finite element methodfRepresenting displacement vectors in a finite element method, FfRepresenting a load vector in a finite element method;
s62, according to different structural boundary conditions, expressing the equation into a form of a block matrix, as follows:
Figure BDA0003166928920000104
wherein, subscript 1 represents applied displacement boundary condition area, subscript 2 represents displacement known area, subscript 3 represents external load area to be obtained, and subscript 4 represents known external load or internal no external load area; 1. the displacement of the node of the 2 area is known, the load of the 4 area is known, and the loading area is generally avoided when measuring points are selected, so that the load outside the 2 area is 0;
s63, taking fixed boundary conditions as an example, U1The partitioning matrix is simplified to the following form, 0:
Figure BDA0003166928920000111
S64U obtained through inverse finite element calculation2Calculating the unknown force Fc：
Figure BDA0003166928920000112
Wherein the content of the first and second substances,
Figure BDA0003166928920000113
Figure BDA0003166928920000114
in specific implementation, as a preferred embodiment of the present invention, the specific implementation process of step S7 is as follows:
s71, according to the Tikhonov regularization method, representing the fitting error as the following form:
Figure BDA0003166928920000115
to reduce fitting errors
Figure BDA0003166928920000116
Introducing a cost function:
Figure BDA0003166928920000117
wherein λ represents a regularization parameter, and H represents a hermite transpose; to minimize the cost function, for the force vector FcMust be zero, calculated to yield:
Figure BDA0003166928920000118
and S72, determining the value of the regularization parameter lambda by using a common cross-validation method, a generalized cross-validation method or an L curve method, and finally calculating to obtain the magnitude of the external load. The resulting load and applied load results were reconstructed as shown in table 1:
TABLE 1 results of calculation
Figure BDA0003166928920000119
Finally, it should be noted that: the above embodiments are only used to illustrate the technical solution of the present invention, and not to limit the same; while the invention has been described in detail and with reference to the foregoing embodiments, it will be understood by those skilled in the art that: the technical solutions described in the foregoing embodiments may still be modified, or some or all of the technical features may be equivalently replaced; and the modifications or the substitutions do not make the essence of the corresponding technical solutions depart from the scope of the technical solutions of the embodiments of the present invention.

Claims (10)

1. A structure state monitoring and load identification method based on an inverse finite element and finite element method is characterized by comprising the following steps:
s1, dispersing the structure by selecting the adaptive inverse shell unit;
s2, calculating the membrane strain, the bending strain and the shearing strain of each inverse shell unit based on a mindlin plate theory;
s3, selecting strain measurement points on the upper surface and the lower surface of the inverse shell unit, pasting a strain sensor on the strain measurement points, and measuring strain in real time to obtain strain measurement data;
s4, calculating the membrane strain and curvature of each inverse shell unit based on the obtained strain measurement data;
s5, constructing a functional based on a least square method, deriving the freedom of the nodes to obtain a similar rigidity matrix and a load matrix of the inverse shell unit, assembling, endowing proper boundary conditions, and calculating a displacement field of the structure;
s6, constructing a classical finite element static equilibrium equation according to the reconstructed displacement information, and deducing to obtain the relationship between the reconstructed node displacement and the unknown load size;
s7, reducing errors existing in the process of strain acquisition and inverse finite element reconstruction by using a Tikhonov regularization method, and reconstructing the size of the external load of the structure.
2. The method for structural condition monitoring and load identification based on inverse finite element and finite element method as claimed in claim 1, wherein the structure in step S1 includes all complex geometric models composed of plate shells, and all are equal thickness plates.
3. The method for structural condition monitoring and load identification based on inverse finite element and finite element method according to claim 1, wherein the step S2 is implemented as follows:
s21, calculating the membrane strain of each inverse shell unit, wherein the calculation formula is as follows:
e(ue)＝Beue
wherein u iseRepresenting the displacement vector of each inverse shell element node, BeRepresenting a matrix containing derivatives of a shape function;
s22, calculating the bending strain of each inverse shell unit, wherein the calculation formula is as follows:
k(ue)＝Bkue
wherein, BkRepresenting a matrix containing derivatives of a shape function;
s23, calculating the shear strain of each inverse shell unit, wherein the calculation formula is as follows:
g(ue)＝Bgue
wherein, BgRepresenting a matrix containing the derivatives of the shape function.
4. The method for structural condition monitoring and load identification based on inverse finite element and finite element method as claimed in claim 1, wherein the strains measured by the strain sensors in real time in step S3 are respectively:
Figure FDA0003166928910000021
Figure FDA0003166928910000022
wherein i represents the ith inverse shell unit, n represents the number of strain measurement points in the inverse shell unit, epsilon represents positive strain in the 1 and 2 directions, and gamma represents shear strain in the 1-2 directions.
5. The method of claim 4, wherein the strain sensor comprises one or more of a resistive strain gauge sensor, a fiber Bragg grating sensor and a distributed fiber sensor.
6. The method for structural condition monitoring and load identification based on inverse finite element and finite element method as claimed in claim 5, wherein the strain sensors are arranged on the upper and lower surfaces of the inverse shell unit in the form of a single axis or a strain rosette.
7. The method for structural condition monitoring and load identification based on inverse finite element and finite element method according to claim 1, wherein the step S4 is implemented as follows:
s41, calculating the membrane strain of each inverse shell unit according to the obtained strain measurement data, wherein the calculation formula is as follows:
Figure FDA0003166928910000031
s42, calculating the curvature of each inverse shell unit according to the obtained strain measurement data, wherein the calculation formula is as follows:
Figure FDA0003166928910000032
where h represents half the cell thickness.
8. The method for structural condition monitoring and load identification based on inverse finite element and finite element method according to claim 1, wherein the step S5 is implemented as follows:
s51, adopting a least square error function for a single inverse shell unit, wherein the function expression is as follows:
Φe(ue)＝we||e(ue)-eε||2+wk||k(ue)-kε||2+wg||g(ue)-gε||2
wherein we、wkAnd wgRespectively representing weight coefficients related to the membrane strain, the bending strain and the shear strain of each inverse shell unit, wherein if the membrane strain, the bending strain and the shear strain can be obtained through calculation of strain measurement values, the corresponding weight coefficients are 1; otherwise, the weight coefficient is adjusted to a small value;
s52, obtaining variation of the node degree of freedom of the structure to obtain a similar rigidity matrix and a load matrix of each inverse shell unit:
Figure FDA0003166928910000033
in the above formula, the first and second carbon atoms are,
Figure FDA0003166928910000034
a stiffness-like matrix representing the inverse shell element, fi eA class load matrix representing the inverse shell elements; wherein:
Figure FDA0003166928910000035
Figure FDA0003166928910000036
in the above formula, AeRepresents the area of the inverted shell element;
s53, assembling the quasi-rigidity matrix and the load matrix of the inverse shell unit according to a standard finite element program to obtain an integral linear equation set of the discrete structure, wherein the integral linear equation set comprises the following steps:
KiUi＝Fi
in the above formula, the first and second carbon atoms are,
Figure FDA0003166928910000041
wherein, TeA coordinate transformation matrix is represented by a matrix of coordinates,Kiquasi-stiffness matrix representing the inverse finite element method globaliRepresenting a load-like matrix of an inverse finite element method overall;
s54, applying proper boundary conditions, and calculating the obtained structure displacement field as follows:
KRiUi＝Fi
wherein, KRiThe quasi-stiffness matrix after the boundary condition is applied is shown as a positive definite matrix.
9. The method for structural condition monitoring and load identification based on inverse finite element and finite element method according to claim 1, wherein the step S6 is implemented as follows:
s61, according to the classical finite element theory, the static finite element solution equation is rewritten into the following form:
KfUf＝Ff
wherein, KfRepresenting the stiffness matrix, U, in a finite element methodfRepresenting displacement vectors in a finite element method, FfRepresenting a load vector in a finite element method;
s62, according to different structural boundary conditions, expressing the equation into a form of a block matrix, as follows:
Figure FDA0003166928910000042
wherein, subscript 1 represents applied displacement boundary condition area, subscript 2 represents displacement known area, subscript 3 represents external load area to be obtained, and subscript 4 represents known external load or internal no external load area; 1. the displacement of the node of the 2 area is known, the load of the 4 area is known, and the loading area is generally avoided when measuring points are selected, so that the load outside the 2 area is 0;
s63, taking fixed boundary conditions as an example, U1The partitioning matrix is simplified to the following form, 0:
Figure FDA0003166928910000043
S64U obtained through inverse finite element calculation2Calculating the unknown force Fc：
Figure FDA0003166928910000051
Wherein the content of the first and second substances,
Figure FDA0003166928910000052
Figure FDA0003166928910000053
10. the method for structural condition monitoring and load identification based on inverse finite element and finite element method according to claim 1, wherein the step S7 is implemented as follows:
s71, according to the Tikhonov regularization method, representing the fitting error as the following form:
Figure FDA0003166928910000054
to reduce fitting errors
Figure FDA0003166928910000055
Introducing a cost function:
Figure FDA0003166928910000056
wherein λ represents a regularization parameter, and H represents a hermite transpose; to minimize the cost function, for the force vector FcMust be zero, calculated to yield:
Figure FDA0003166928910000057
and S72, determining the value of the regularization parameter lambda by using a common cross-validation method, a generalized cross-validation method or an L curve method, and finally calculating to obtain the magnitude of the external load.
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