CN113468667B - Structural state monitoring and load identification method based on inverse finite element and finite element method - Google Patents
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Abstract
The invention provides a structural state monitoring and load identification method based on an inverse finite element and finite element method, which comprises the following steps: dispersing the plate-shell structure by using an adaptive inverse shell unit; calculating the membrane strain, bending strain and shearing strain of each inverse shell unit based on mindlin plate theory; selecting strain measurement points on the upper surface and the lower surface of the inverse shell unit, and pasting strain sensors on the strain measurement points to measure strain in real time so as 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 degree of freedom of the node, obtaining a class stiffness matrix and a load matrix of the inverse shell unit, assembling, endowing proper boundary conditions, and calculating a displacement field of the structure; the relation between the displacement field and the loading force vector constructed by the finite element method is utilized, and Tikhonov is utilized to reduce errors generated by strain acquisition and displacement reconstruction, so that the monitoring of the state information of the structure and the identification of the load information are realized.
Description
Technical Field
The invention relates to the technical fields of real-time structural deformation monitoring, load identification and the like in the fields of aerospace, vehicles, buildings and the like, in particular to a structural 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. The damage may cause the rigidity of the structure to deteriorate or even have a devastating effect on the load carrying properties of the structure. In the fields of aerospace, vehicles, construction and the like, a great deal of effort is invested every year to detect and repair structures to ensure the functions of the structures. Therefore, the real-time monitoring of the state of the structure based on the structure health monitoring technology plays a great role in improving the structure performance, guaranteeing the structure safety, reducing the maintenance cost and the like. Structural health monitoring technology is a multidisciplinary leading edge technology that includes various sensing, reconstruction methods. In various state sensing methods, the Inverse Finite Element Method (iFEM) fully considers the complexity of boundary conditions and structural topology, and numerous prior information such as load conditions, material information and the like are not needed for calculation, so that the method is a powerful online health monitoring tool. The state information such as deformation, strain and the like of the structure can be accurately obtained in real time by means of an inverse finite element method. In addition, the structural damage information can be further clarified by being used in combination with damage detection methods such as a infinitesimal dynamic response method. While external load is an important ring for representing the structural state, the expansion of structural damage has a critical effect, and the inverse finite element method does not consider the effect of external load in the calculation process.
Disclosure of Invention
According to the technical problems, a structural state monitoring and load identification method based on an inverse finite element and finite element method is provided. According to the invention, the external load of the structure is reversely calculated by measuring the surface strain of the plate shell structure in real time and reconstructing the state information such as the deformation of the structure by adopting a reverse finite element method, and then the reconstructed displacement data is used as the input of the finite element method.
The invention adopts the following technical means:
A structural state monitoring and load identification method based on an inverse finite element and finite element method comprises the following steps:
S1, selecting an adaptive inverse shell unit to discrete the structure;
S2, calculating the membrane strain, bending strain and shearing strain of each inverse shell unit based on mindlin plate theory;
S3, selecting strain measurement points on the upper surface and the lower surface of the inverse shell unit, pasting strain sensors on the strain measurement points, and measuring strain in real time to obtain strain measurement data;
s4, calculating the membrane strain and the 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 degree of freedom of the node, obtaining a class stiffness matrix and a load matrix of the inverse shell unit, assembling, finally giving proper boundary conditions, and calculating a displacement field of the structure;
s6, constructing a classical finite element statics balance equation according to the reconstructed displacement information, and deducing to obtain a relation between the reconstructed node displacement and the unknown load;
S7, reducing errors in the strain acquisition and inverse finite element reconstruction process by using a Tikhonov regularization method, and reconstructing the size of the load outside the structure.
Further, the structure in the step S1 includes all complex geometric models formed by plate shells, and all the complex geometric models are equal-thickness plates.
Further, the specific implementation process of the 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
Where u e represents each inverse shell element node displacement vector, and B e represents a matrix containing derivative of the shape function;
S22, calculating the bending strain of each inverse shell unit, wherein the calculation formula is as follows:
k(ue)＝Bkue
Wherein B k represents a matrix containing derivatives of the shape function;
s23, calculating the shear strain of each inverse shell unit, wherein the calculation formula is as follows:
g(ue)＝Bgue
Wherein B g represents a matrix containing derivatives of the shape function.
Further, the strains measured in real time by the strain sensor in the step S3 are respectively:
Wherein i represents the ith backshell unit, n represents the number of strain measurement points in the backshell unit, epsilon represents positive strain in the 1-2 direction, and gamma represents tangential strain in the 1-2 direction.
Further, the strain sensor comprises one or a combination of more than two of a resistance strain gauge sensor, a fiber Bragg grating sensor and a distributed fiber sensor.
Further, the strain sensor is arranged on the upper surface and the lower surface of the counter shell unit along a single shaft or in the form of strain flower.
Further, the specific implementation process of the 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:
S42, calculating the curvature of each inverse shell unit according to the obtained strain measurement data, wherein the calculation formula is as follows:
where h represents half the cell thickness.
Further, the specific implementation process of the step S5 is as follows:
S51, for a single inverse shell unit, a least square error function is adopted, and the function expression is as follows:
Φe(ue)＝we||e(ue)-eε||2+wk||k(ue)-kε||2+wg||g(ue)-gε||2
Wherein w e、wk and w g represent the weight coefficients related to the membrane strain, bending strain and shear strain of each inverse shell element, respectively, if the membrane strain, bending strain and shear strain can be calculated by strain measurement values, the corresponding weight coefficients are 1; otherwise, the weight coefficient is adjusted to a small value;
s52, taking variation of the node degrees of freedom of the structure to obtain a class stiffness matrix and a load matrix of each inverse shell unit:
In the above-mentioned method, the step of, Matrix of class stiffness representing inverse shell element,/>Representing a class load matrix of the inverse shell element; wherein:
in the above formula, A e represents the area of the inverse shell unit;
s53, assembling the class stiffness matrix and the load matrix of the inverse shell unit according to a standard finite element program to obtain an integral linear equation set with a discrete structure, wherein the integral linear equation set is as follows:
KiUi＝Fi
In the above-mentioned method, the step of, Wherein T e represents a coordinate transformation matrix, K i represents a global class stiffness matrix of the inverse finite element method, and F i represents a global class load matrix of the inverse finite element method;
s54, applying proper boundary conditions, and calculating to obtain a displacement field of the structure, wherein the displacement field is as follows:
KRiUi＝Fi
where K Ri denotes the stiffness-like matrix after applying the boundary conditions, which is a positive matrix.
Further, the specific implementation process of the step S6 is as follows:
s61, according to classical finite element theory, rewriting a static finite element solving equation into the following form:
KfUf＝Ff
Wherein K f represents a stiffness matrix in the finite element method, U f represents a displacement vector in the finite element method, and F f represents a load vector in the finite element method;
s62, expressing the equation into a block matrix according to different structural boundary conditions, wherein the equation is as follows:
Wherein, the subscript 1 represents a region applying displacement boundary conditions, the subscript 2 represents a displacement known region, the subscript 3 represents a region to be subjected to external load, and the subscript 4 represents a region with known external load or no internal external load; 1. the displacement of the node of the area 2 is known, the load of the area 4 is known, and the load area is generally avoided when the measuring point is selected, so that the load outside the area 2 is 0;
s63, taking a fixed boundary condition as an example, i.e. U 1 =0, the blocking matrix is simplified into the following form:
S64, calculating unknown acting force F c through U 2 obtained through inverse finite element calculation:
Wherein,
Further, the specific implementation process of the step S7 is as follows:
S71, according to a Tikhonov regularization method, the fitting error is expressed as follows:
To reduce fitting error Introducing a cost function:
Wherein λ represents a regularization parameter, and H represents an hermite transpose; to minimize the cost function, the first derivative of the force vector F c must be zero, calculated as:
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 structural state monitoring and load identification method based on the inverse finite element and finite element method, the strain sensor is adopted to measure the structural surface strain, the deformation of the structure is reconstructed in real time by adopting the inverse finite element method, and the load of the structure is reconstructed by adopting the finite element method.
2. The method can accurately indicate the load on the basis of accurately reconstructing the structure state, and provides a basis for damage expansion and service life assessment of the 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 deformation monitoring, load identification and the like of structures in the fields of aerospace, vehicles, buildings and the like.
Drawings
In order to more clearly illustrate the embodiments of the present invention or the technical solutions in the prior art, the drawings that are required in the embodiments or the description of the prior art will be briefly described, and it is obvious that the drawings in the following description are some embodiments of the present invention, and other drawings may be obtained according to the drawings without inventive effort to a person skilled in the art.
FIG. 1 is a flow chart of the method of the present invention.
FIG. 2 is a schematic view of the aluminum alloy sheet of the present invention in size and loading position;
FIG. 3 is a diagram showing the position of a sensor posted on the upper and lower surfaces of a structure according to an embodiment of the present invention
In the figure: 1. loading a position 1; 2. loading a No. 2 position; 3. the sensor layout position on the upper surface of the structure; 4. sensor layout position of lower surface of structure
Detailed Description
In order that those skilled in the art will better understand the present invention, a technical solution in the embodiments of the present invention will be clearly and completely described below with reference to the accompanying drawings in which it is apparent that the described embodiments are only some embodiments of the present invention, not all embodiments. All other embodiments, which can be made by those skilled in the art based on the embodiments of the present invention without making any inventive effort, shall fall within the scope of the present invention.
It should be noted that the terms "first," "second," and the like in the description and the claims of the present invention and the above figures are used for distinguishing between similar objects and not necessarily for describing a particular sequential or chronological order. It is to be understood that the data so used may be interchanged where appropriate such that the embodiments of the invention described herein may be implemented in sequences other than those illustrated or otherwise 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 invention provides a structural state monitoring and load identification method based on an inverse finite element and finite element method, which comprises the following steps:
S1, selecting an adaptive inverse shell unit to discrete the structure;
S2, calculating the membrane strain, bending strain and shearing strain of each inverse shell unit based on mindlin plate theory;
S3, selecting strain measurement points on the upper surface and the lower surface of the inverse shell unit, pasting strain sensors on the strain measurement points, and measuring strain in real time to obtain strain measurement data;
s4, calculating the membrane strain and the 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 degree of freedom of the node, obtaining a class stiffness matrix and a load matrix of the inverse shell unit, assembling, finally giving proper boundary conditions, and calculating a displacement field of the structure;
s6, constructing a classical finite element statics balance equation according to the reconstructed displacement information, and deducing to obtain a relation between the reconstructed node displacement and the unknown load;
S7, reducing errors in the strain acquisition and inverse finite element reconstruction process by using a Tikhonov regularization method, and reconstructing the size of the load outside the structure.
In specific implementation, as a preferred embodiment of the present invention, the structure in the step S1 includes all complex geometric models formed by plate shells, and all the complex geometric models are equal-thickness plates. As shown in FIG. 2, the aluminum alloy plate size and loading position are schematically shown.
In specific implementation, as a preferred embodiment of the present invention, the specific implementation procedure of the 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
Where u e represents each inverse shell element node displacement vector, and B e represents a matrix containing derivative of the shape function;
S22, calculating the bending strain of each inverse shell unit, wherein the calculation formula is as follows:
k(ue)＝Bkue
Wherein B k represents a matrix containing derivatives of the shape function;
s23, calculating the shear strain of each inverse shell unit, wherein the calculation formula is as follows:
g(ue)＝Bgue
Wherein B g represents a matrix containing derivatives of the shape function.
In specific implementation, as a preferred embodiment of the present invention, the strain measured in real time by the strain sensor in step S3 is respectively:
Wherein i represents the ith backshell unit, n represents the number of strain measurement points in the backshell unit, epsilon represents positive strain in the 1-2 direction, and gamma represents tangential strain in the 1-2 direction.
In particular, the strain sensor includes one or a combination of two or more of a resistive strain gauge sensor, a fiber bragg grating sensor and a distributed fiber optic sensor as a preferred embodiment of the present invention.
In specific implementation, as a preferred embodiment of the present invention, as shown in fig. 3, the strain sensor is disposed on the upper and lower surfaces of the counter shell unit along a single axis or in the form of strain gauge.
In specific implementation, as a preferred embodiment of the present invention, the specific implementation procedure of the 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:
S42, calculating the curvature of each inverse shell unit according to the obtained strain measurement data, wherein the calculation formula is as follows:
where h represents half the cell thickness.
In specific implementation, as a preferred embodiment of the present invention, the specific implementation procedure of the step S5 is as follows:
S51, for a single inverse shell unit, a least square error function is adopted, and the function expression is as follows:
Φe(ue)＝we||e(ue)-eε||2+wk||k(ue)-kε||2+wg||g(ue)-gε||2
Wherein w e、wk and w g represent the weight coefficients related to the membrane strain, bending strain and shear strain of each inverse shell element, respectively, if the membrane strain, bending strain and shear strain can be calculated by strain measurement values, the corresponding weight coefficients are 1; otherwise, the weight coefficient is adjusted to a small value;
s52, taking variation of the node degrees of freedom of the structure to obtain a class stiffness matrix and a load matrix of each inverse shell unit:
In the above-mentioned method, the step of, Matrix of class stiffness representing inverse shell element,/>Representing a class load matrix of the inverse shell element; wherein:
in the above formula, A e represents the area of the inverse shell unit;
s53, assembling the class stiffness matrix and the load matrix of the inverse shell unit according to a standard finite element program to obtain an integral linear equation set with a discrete structure, wherein the integral linear equation set is as follows:
KiUi＝Fi
In the above-mentioned method, the step of, Wherein T e represents a coordinate transformation matrix, K i represents a global class stiffness matrix of the inverse finite element method, and F i represents a global class load matrix of the inverse finite element method;
s54, applying proper boundary conditions, and calculating to obtain a displacement field of the structure, wherein the displacement field is as follows:
KRiUi＝Fi
where K Ri denotes the stiffness-like matrix after applying the boundary conditions, which is a positive matrix.
In specific implementation, as a preferred embodiment of the present invention, the specific implementation procedure of the step S6 is as follows:
s61, according to classical finite element theory, rewriting a static finite element solving equation into the following form:
KfUf＝Ff
Wherein K f represents a stiffness matrix in the finite element method, U f represents a displacement vector in the finite element method, and F f represents a load vector in the finite element method;
s62, expressing the equation into a block matrix according to different structural boundary conditions, wherein the equation is as follows:
Wherein, the subscript 1 represents a region applying displacement boundary conditions, the subscript 2 represents a displacement known region, the subscript 3 represents a region to be subjected to external load, and the subscript 4 represents a region with known external load or no internal external load; 1. the displacement of the node of the area 2 is known, the load of the area 4 is known, and the load area is generally avoided when the measuring point is selected, so that the load outside the area 2 is 0;
s63, taking a fixed boundary condition as an example, i.e. U 1 =0, the blocking matrix is simplified into the following form:
S64, calculating unknown acting force F c through U 2 obtained through inverse finite element calculation:
Wherein,
In specific implementation, as a preferred embodiment of the present invention, the specific implementation procedure of the step S7 is as follows:
S71, according to a Tikhonov regularization method, the fitting error is expressed as follows:
To reduce fitting error Introducing a cost function:
Wherein λ represents a regularization parameter, and H represents an hermite transpose; to minimize the cost function, the first derivative of the force vector F c must be zero, calculated as:
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 load and applied load results obtained by the reconstruction are shown in table 1:
Table 1 calculation results
Finally, it should be noted that: the above embodiments are only for illustrating the technical solution of the present invention, and not for limiting the same; although the invention has been described in detail with reference to the foregoing embodiments, it will be understood by those of ordinary skill in the art that: the technical scheme described in the foregoing embodiments can be modified or some or all of the technical features thereof can be replaced by equivalents; such modifications and substitutions do not depart from the spirit of the invention.
Claims (8)
1. The structural state monitoring and load identifying method based on the inverse finite element and finite element method is characterized by comprising the following steps:
S1, selecting an adaptive inverse shell unit to discrete the structure;
S2, calculating the membrane strain, bending strain and shearing strain of each inverse shell unit based on mindlin plate theory;
S3, selecting strain measurement points on the upper surface and the lower surface of the inverse shell unit, pasting strain sensors on the strain measurement points, and measuring strain in real time to obtain strain measurement data;
s4, calculating the membrane strain and the 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 degree of freedom of the node, obtaining a class stiffness matrix and a load matrix of the inverse shell unit, assembling, finally giving proper boundary conditions, and calculating a displacement field of the structure;
s6, constructing a classical finite element statics balance equation according to the reconstructed displacement information, and deducing to obtain a relation between the reconstructed node displacement and the unknown load;
the specific implementation process of the step S6 is as follows:
s61, according to classical finite element theory, rewriting a static finite element solving equation into the following form:
KfUf＝Ff
Wherein K f represents a stiffness matrix in the finite element method, U f represents a displacement vector in the finite element method, and F f represents a load vector in the finite element method;
s62, expressing the equation into a block matrix according to different structural boundary conditions, wherein the equation is as follows:
Wherein, the subscript 1 represents a region applying displacement boundary conditions, the subscript 2 represents a displacement known region, the subscript 3 represents a region to be subjected to external load, and the subscript 4 represents a region with known external load or no internal external load; 1. the displacement of the node of the area 2 is known, the load of the area 4 is known, and the load area is generally avoided when the measuring point is selected, so that the load outside the area 2 is 0;
s63, taking a fixed boundary condition as an example, i.e. U 1 =0, the blocking matrix is simplified into the following form:
S64, calculating unknown acting force F c through U 2 obtained through inverse finite element calculation:
Wherein,
S7, reducing errors in the strain acquisition and inverse finite element reconstruction processes by using a Tikhonov regularization method, and reconstructing the size of the load outside the structure;
The specific implementation process of the step S7 is as follows:
S71, according to a Tikhonov regularization method, the fitting error is expressed as follows:
To reduce fitting error Introducing a cost function:
Wherein λ represents a regularization parameter, and H represents an hermite transpose; to minimize the cost function, the first derivative of the force vector F c must be zero, calculated as:
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.
2. The method for monitoring structural state and identifying load based on inverse finite element and finite element method according to claim 1, wherein the structure in step S1 comprises all complex geometric models composed of plate shells, and all the complex geometric models are equal-thickness plates.
3. The method for monitoring structural state and identifying load based on inverse finite element and finite element method according to claim 1, wherein 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
Where u e represents each inverse shell element node displacement vector, and B e represents a matrix containing derivative of the shape function;
S22, calculating the bending strain of each inverse shell unit, wherein the calculation formula is as follows:
k(ue)＝Bkue
Wherein B k represents a matrix containing derivatives of the shape function;
s23, calculating the shear strain of each inverse shell unit, wherein the calculation formula is as follows:
g(ue)＝Bgue
Wherein B g represents a matrix containing derivatives of the shape function.
4. The method for monitoring structural state and identifying load based on inverse finite element and finite element method according to claim 1, wherein the strain measured in real time by the strain sensor in step S3 is:
Wherein i represents the ith backshell unit, n represents the number of strain measurement points in the backshell unit, epsilon represents positive strain in the 1-2 direction, and gamma represents tangential strain in the 1-2 direction.
5. The method for monitoring structural state and identifying load based on inverse finite element and finite element method according to claim 4, wherein the strain sensor comprises one or a combination of more than two of a resistive strain gauge sensor, a fiber bragg grating sensor and a distributed fiber optic sensor.
6. The method for monitoring structural state and identifying load based on inverse finite element and finite element method according to claim 5, wherein the strain sensor is arranged on the upper and lower surfaces of the inverse shell element along a single axis or in the form of strain gauge.
7. The method for monitoring structural state and identifying load based on inverse finite element and finite element method according to claim 1, wherein 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:
S42, calculating the curvature of each inverse shell unit according to the obtained strain measurement data, wherein the calculation formula is as follows:
where h represents half the cell thickness.
8. The method for monitoring structural state and identifying load based on inverse finite element and finite element method according to claim 1, wherein the specific implementation process of step S5 is as follows:
S51, for a single inverse shell unit, a least square error function is adopted, and the function expression is as follows:
Φe(ue)＝we||e(ue)-eε||2+wk||k(ue)-kε||2+wg||g(ue)-gε||2
Wherein w e、wk and w g represent the weight coefficients related to the membrane strain, bending strain and shear strain of each inverse shell element, respectively, if the membrane strain, bending strain and shear strain can be calculated by strain measurement values, the corresponding weight coefficients are 1; otherwise, the weight coefficient is adjusted to a small value;
s52, taking variation of the node degrees of freedom of the structure to obtain a class stiffness matrix and a load matrix of each inverse shell unit:
In the above-mentioned method, the step of, Representing a class stiffness matrix of the inverse shell unit, and f i e represents a class load matrix of the inverse shell unit; wherein:
in the above formula, A e represents the area of the inverse shell unit;
s53, assembling the class stiffness matrix and the load matrix of the inverse shell unit according to a standard finite element program to obtain an integral linear equation set with a discrete structure, wherein the integral linear equation set is as follows:
KiUi＝Fi
In the above-mentioned method, the step of, Wherein T e represents a coordinate transformation matrix, K i represents a global class stiffness matrix of the inverse finite element method, and F i represents a global class load matrix of the inverse finite element method;
s54, applying proper boundary conditions, and calculating a structural displacement field as follows:
KRiUi＝Fi
where K Ri denotes the stiffness-like matrix after applying the boundary conditions, which is a positive matrix.
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