CN116956428A - Bridge response calculation method under moving load based on physical information neural network - Google Patents
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Abstract
The invention discloses a bridge response calculation method under a mobile load based on a physical information neural network, which comprises the following steps: 1. extracting residual point coordinates from the calculation region, and extracting boundary point coordinates at the boundary in the calculation region; 2. performing Fourier feature mapping on the coordinates of the sampling nodes; 3. inputting the mapped data into a multi-scale deep neural network MDNN, and deriving an output result; 4. substituting the derived data into a partial differential equation and boundary conditions to obtain a loss function value, and updating MDNN parameters; 5. and solving the bridge response under the moving load by using the trained MDNN. According to the method, the bridge response under the moving load is calculated by using the physical information neural network, so that the problems of complex calculation process and difficult grid division in the traditional finite element dynamic response calculation numerical simulation method are solved, and the calculation efficiency and accuracy of the beam bridge dynamic response are improved by using the physical information neural network.
Description
Technical Field
The invention relates to the field of bridge engineering, in particular to a bridge response calculation method under a moving load based on a physical information neural network, and the result can be used for evaluating the structural safety state.
Background
The problem of moving load is common in bridge engineering, examples of which include vehicles moving on a bridge, trains moving on rails, and the like, and bridge dynamics analysis is a process of researching and analyzing the response and behavior of a bridge structure under the action of external load. The dynamic analysis of the bridge is crucial to ensuring the safety and reliability of the bridge structure, and the bearing capacity, stability and vibration characteristics of the bridge can be evaluated by simulating the response of the bridge under different load conditions, so that the safety evaluation and design optimization can be performed. In practice, the dynamic response of the bridge is usually calculated by using a finite element method, however, for traffic infrastructures such as railways, the track infrastructures are hundreds of kilometers, the grid division of the bridge is difficult (especially for bridges with complex bridges), and the problems of huge calculation amount, low calculation efficiency and the like are caused by too many division grids.
Disclosure of Invention
The invention aims to overcome the defects of the existing dynamic analysis method, and provides a bridge response calculation method under the moving load based on a physical information neural network, so that complicated grid division steps can be avoided, the problems of difficult grid division and huge calculation amount in the traditional finite element dynamic analysis are solved, accurate bridge dynamic response can be obtained rapidly, and the calculation efficiency and accuracy of beam bridge dynamic response are improved.
In order to achieve the aim of the invention, the invention adopts the following technical scheme:
the invention discloses a bridge response calculation method under a moving load based on a physical information neural network, which is characterized by comprising the following steps of:
step 1: determining bridge length asThe height is h, the width is b, the density is rho, the elastic modulus is EI, and the mass per linear meter is m=rho×b×h;
let the space coordinates beTime coordinate is +.>The moving load is +.>Wherein P is 0 V is the load moving speed, delta is the Dirac function;
the load bridge crossing time is set asAnd set x epsilon 0, L]Is a dimensionless space coordinate, t is [0, T ]]Is a nondimensional time coordinate, wherein l=1 is the nondimensional beam length, T is the transformation duration, and +.>Characteristic parameters representing the vibration frequency of the beam;
step 2: n internal point coordinates { (x) are extracted from the calculation region omega epsilon (x, t) by using the super Latin sampling method in ,t in ) I in=1, 2, …, N }, and extracts M boundary point coordinates { (x) on the boundary region B e (x, t) bc ,t bc ) |bc=1, 2, …, M }, where x in And x bc Representing respectively the in-th beam space coordinate of the calculation region Ω and the bc-th beam space coordinate of the boundary region B, t in And t bc Respectively representing the time coordinate of the in-th beam vibration of the calculation region omega and the time coordinate of the bc-th beam vibration of the boundary region B, wherein N represents the number of internal point coordinates, and M represents the number of boundary point coordinates;
step 3: generating P random numbers { omega } using normal distribution i I=1, 2, …, P }, where ω i Represents the ith random number and coordinates (x) of the in-th internal point according to equations (1 a) and (1 b) in ,t in ) And the bc boundary point coordinate (x bc ,t bc ) Respectively performing Fourier feature mapping to obtain the in-th internal training data D after mapping in And the bc boundary training data D bc Thereby obtaining a training data set D= [ D ] in ,D bc ]：
Step 4: sequentially building a plurality of input layers and K linear layers { F k |k=1, 2, …, K } and K-1 function activation layers { σ } k A multi-scale deep neural network consisting of i k=1, 2, …, K-1, a multi-scale feature layer and an output layer, wherein the number of neural units of a linear layer is s, and a function activation layer is selected as a tanh activation function;
the training data set D is input into an input layer of the multi-scale deep neural network for processing to obtain predicted bridge displacement response
Step 5: constructing a loss function of the multi-scale deep neural network;
step 5.1: determining the mean value as mu=v×t, the variance as sigma, and carrying out original moving load according to a Gaussian function shown in a formula (5)After dimensionless characterization, a normalized mobile load F (x, t) is obtained:
step 5.2: constructing a dimensionless Euler beam vibration equation by using the formula (6):
in the formula (6), the amino acid sequence of the compound,is the second partial derivative of the displacement response u (x, t) of the bridge to the time coordinate t of the beam vibration,the fourth-order partial derivative of the displacement response u (x, t) of the bridge to the Liang Kongjian coordinate x;
step 5.3: constructing a loss function loss according to (6) PDE ：
In the formula (7), the amino acid sequence of the compound,for input, the in-th internal point coordinate (x in ,t in ) Predicted in-th bridge displacementResponse->Second partial derivative of the time coordinate t of the beam vibration,/, for>For predicted displacement response of bridge +.>Fourth order partial derivative of Liang Kongjian coordinate x;
step 5.4: based on predicted bridge displacement responseDetermining left endpoint boundary displacement condition->And right endpoint boundary displacement condition->Left endpoint boundary bending condition->And right endpoint boundary bending conditionInitial Displacement Condition->Initial speed Condition->Thereby constructing a loss function loss using equation (8) BC ：
Step 5.5: establishing a total loss functionL total ＝loss pde +loss bc ；
Step 6: training a multi-scale deep neural network by using an Adam optimizer, and simultaneously calculating a total loss function L total Up to the loss function L total Is smaller than the set threshold value, thereby obtaining a trained multi-scale depth neural model, and is used for calculating the optimal displacement response u of the bridge at any position x under the action of the moving load F (x, t) at any time t * (x,t)。
The bridge response calculation method based on the physical information neural network under the moving load is also characterized in that the multi-scale deep neural network in the step 4 processes the training data set D according to the following steps:
the training data set D is input into an input layer of the multi-scale deep neural network, and after the K linear layers and the K-1 function activation layers are transformed, a scale information data set D with the dimension s and 1 is obtained by using a formula (2):
in the formula (2), the amino acid sequence of the compound,representing a matrix multiplication operator;
the multi-scale feature layer converts the scale information data set d into a data set consisting of Q scale functions { C }, according to (3) q Set of scaling functions C consisting of q=1, 2, …, Q }, where C q Representing the q-th scale function in the scale function set C;
C＝W sacle ·d+b scale (3)
in the formula (3), W sacle For dimension [ Q, s ]]Weight matrix of b) sacle For dimension [ Q,1]Is a bias matrix of (a);
the output layer will have Q scale functions { C q When q=1, 2, …, Q } are combined, a predicted bridge displacement response is calculated according to equation (4)
In the formula (4), W output Is of dimension [1, Q]Is a weight matrix of (a).
The electronic device of the invention comprises a memory and a processor, wherein the memory is used for storing a program for supporting the processor to execute the bridge response calculation method under the moving load, and the processor is configured to execute the program stored in the memory.
The invention relates to a computer readable storage medium, wherein a computer program is stored on the computer readable storage medium, and the computer program is executed by a processor to execute the steps of the bridge response calculation method under the moving load.
Compared with the prior art, the invention has the beneficial effects that:
1. according to the invention, the neural network model with physical information is constructed to calculate the bridge response under the moving load, so that the problems that the traditional finite element dynamic analysis needs to carry out grid division and the complete deflection section is difficult to obtain for a long bridge are solved, the dynamic response of any point of the bridge can be obtained, the calculation difficulty caused by grid division is avoided, the dynamic response calculation efficiency for a complex long bridge is improved, the real engineering situation is met, and the calculation cost is reduced.
2. According to the invention, by fitting the partial differential equation and the boundary condition, the unsupervised training of the neural network is realized, and compared with other neural network models, relatively accurate response calculation results can be obtained without training data.
3. The method for solving the partial differential equation by the neural network based on the physical information can effectively avoid meshing, is different from the traditional neural network, utilizes the partial differential equation and the boundary condition as a loss function, realizes the unsupervised training of the neural network without collecting data, and can greatly improve the calculation efficiency of response to a complex model.
Drawings
FIG. 1 is a schematic illustration of the calculation process of the method of the present invention;
FIG. 2 is a numerical simulation constant section simply supported beam bridge diagram of the present invention;
FIG. 3 is a graph comparing mid-span displacement response calculated by the method of the present invention with actual displacement response;
fig. 4 is a graph comparing a displacement response frequency domain calculated by the method of the present invention with an actual displacement response frequency domain.
Detailed Description
In this embodiment, as shown in fig. 2, the elastic modulus of the bridge is an uncertain parameter, the mass per linear meter of the bridge is p=62.5 kg/m, the height of the cross section is 1.6 m, the width is 0.5 m, and finite element software is used to build a finite element model of the bridge. The Newmark-beta method is used for calculating the sampling frequency of 1000Hz of the dynamic response of the bridge, as shown in fig. 1, and the bridge response calculation method under the moving load based on the physical information neural network comprises the following steps:
step 1: determining bridge length asHeight h=0.3 m, width b=0.2 m, density ρ=2500 kg/m 3 The elastic modulus is ei=2.68e+07, and the mass per linear meter is m=62.5 kg/m;
let the space coordinates beTime coordinate is +.>The moving load is +.>Wherein P is 0 10000N is the load size, v=10m/s is the load moving speed, and the load bridge crossing time length is +.>The actual displacement response of the bridge is +.>
And set x epsilon 0, L]T is non-dimensionalized space coordinates, t is E [0, T]Is a non-dimensionalized time coordinate, wherein L=1 is a non-dimensionalized beam length,for the transition duration.
Step 2: extracting n=1600 internal point coordinates { (x) from within the calculation region Ω e (x, t) using the super latin sampling method in ,t in ) I in=1, 2, …, N }, and extracts m=600 boundary point coordinates { (x) on the boundary region B e (x, t) bc ,t bc ) |bc=1, 2, …, M }, where x in And x bc Representing respectively the in-th beam space coordinate of the calculation region Ω and the bc-th beam space coordinate of the boundary region B, t in And t bc Respectively representing the time coordinate of the in-th beam vibration of the calculation region omega and the time coordinate of the bc-th beam vibration of the boundary region B, wherein N represents a total of N internal point coordinates, and M represents a total of M boundary point coordinates;
step 3: generating p=200 random numbers { ω using normal distribution i I=1, 2, …, P }, where ω i Represents the ith random number and coordinates (x) of the in-th internal point according to equations (1 a) and (1 b) in ,t in ) And the bc boundary point coordinate (x bc ,t bc ) Respectively performing Fourier feature mapping to obtain mapped in-th training data D in And the bc training data D bc Thereby obtaining a training data set D= [ D ] in ,D bc ]：
Step 4: sequentially building a linear layer { F formed by an input layer and K=3 linear layers k |k=1, 2, …, K } and K-1 function activation layers { σ } k A multi-scale deep neural network consisting of i k=1, 2, …, K-1, a multi-scale feature layer and an output layer, wherein the number of neural units of a linear layer is s=200, and a function activation layer is selected as a tanh activation function;
the training data set D is input into an input layer of the multi-scale deep neural network, and after transformation of K linear layers and K-1 function activation layers, a scale information data set D is obtained by utilizing a formula (2), wherein the dimension of D is [ s,1]:
in the formula (2), the amino acid sequence of the compound,representing the matrix multiplication operator.
The multi-scale feature layer converts the scale information data set data d into a scale function set C according to the step (3), wherein the C is formed by Q scale functions { C q Q=1, 2, …, Q } where Q represents the Q-th scale function in the set of scale functions C
C＝W sacle ·d+b scale (3)
In the formula (3), W sacle For dimension [ Q, s ]]Weight matrix of b) sacle For dimension [ Q,1]Is included in the bias matrix of (a).
The output layer divides the Q scale functions C according to the formula (4) q After combination, the predicted bridge displacement response is calculated
In the formula (4), W output Is of dimension [1, Q]Weight matrix of (2)；
Step 5: constructing a loss function of the multi-scale deep neural network;
step 5.1: determining the mean value as mu=v×t, the variance as sigma, and carrying out original moving load according to a Gaussian function shown in a formula (5)After dimensionless characterization, a normalized mobile load F (x, t) is obtained:
step 5.2: constructing a dimensionless Euler beam vibration equation by using the formula (5):
in the formula (5), the amino acid sequence of the compound,is the second partial derivative of the displacement response u (x, t) of the bridge to the time coordinate t of the beam vibration,the fourth-order partial derivative of the displacement response u (x, t) of the bridge to the Liang Kongjian coordinate x;
step 5.3: constructing a loss function loss according to (6) PDE ：
In the formula (6), the amino acid sequence of the compound,for input, the in-th internal point coordinate (x in ,t in ) Predicted in bridge displacement response +.>Second partial derivative of the time coordinate t of the beam vibration,/, for>For predicted displacement response of bridge +.>Fourth order partial derivative of Liang Kongjian coordinate x;
step 5.4: based on predicted bridge displacement responseDetermining left endpoint boundary displacement condition->And right endpoint boundary displacement condition->Left endpoint boundary bending condition->And right endpoint boundary bending conditionInitial Displacement Condition->Initial speed Condition->Thereby constructing a loss function loss using equation (8) BC ：
Step 5.5: establishing a total loss function L total ＝loss pde +loss bc ；
Step 6: training a multi-scale deep neural network by using an Adam optimizer, and simultaneously calculating a total loss function L total Up to the loss function L total Is smaller than the set threshold value, thereby obtaining a trained multi-scale depth neural model, and is used for calculating the optimal displacement response u of the bridge at any position x under the action of the moving load F (x, t) at any time t * (x,t)。
Step 7: response u to optimal displacement * (x, t) inverse dimensionless to obtain a predicted displacement responseWill predict the displacement response +.>Response to actual displacement is +.>In contrast, as shown in fig. 3.
Step 8: will predict displacement responseResponse to actual displacement is +.>A fast fourier transform is performed and the spectra of the two resulting displacement responses are compared as shown in fig. 4.
In this embodiment, an electronic device includes a memory for storing a program supporting the processor to execute the above method, and a processor configured to execute the program stored in the memory.
In this embodiment, a computer-readable storage medium stores a computer program that, when executed by a processor, performs the steps of the method described above.
In summary, by constructing the neural network model with physical information, the bridge response under the moving load is calculated, and the full-bridge dynamic response is obtained. The invention considers the beam bridge parameters and the moving load size, and blends the partial differential equation into the neural network for non-supervision training, solves the problems that the traditional finite element power analysis needs to carry out grid division and the complete deflection section is difficult to obtain for a long bridge, can obtain the power response of any point of the bridge, avoids the calculation difficulty caused by grid division, improves the power response calculation efficiency for a complex long bridge, accords with the actual engineering condition and reduces the calculation cost.
Claims (4)
1. The bridge response calculation method under the moving load based on the physical information neural network is characterized by comprising the following steps of:
step 1: determining bridge length asThe height is h, the width is b, the density is rho, the elastic modulus is EI, and the mass per linear meter is m=rho×b×h;
let the space coordinates beTime coordinate is +.>The moving load is +.>Wherein P is 0 V is the load moving speed, delta is the Dirac function;
the load bridge crossing time is set asAnd set x epsilon 0, L]Is a dimensionless space coordinate, t is [0, T ]]Is a nondimensional time coordinate, wherein l=1 is the nondimensional beam length, T is the transformation duration, and +.>Characteristic parameters representing the vibration frequency of the beam;
step 2: n internal point coordinates { (x) are extracted from the calculation region omega epsilon (x, t) by using the super Latin sampling method in ,t in ) I in=1, 2, …, N }, and extracts M boundary point coordinates { (x) on the boundary region B e (x, t) bc ,t bc ) |bc=1, 2, …, M }, where x in And x bc Representing respectively the in-th beam space coordinate of the calculation region Ω and the bc-th beam space coordinate of the boundary region B, t in And t bc Respectively representing the time coordinate of the in-th beam vibration of the calculation region omega and the time coordinate of the bc-th beam vibration of the boundary region B, wherein N represents the number of internal point coordinates, and M represents the number of boundary point coordinates;
step 3: generating P random numbers { omega } using normal distribution i I=1, 2, …, P }, where ω i Represents the ith random number and coordinates (x) of the in-th internal point according to equations (1 a) and (1 b) in ,t in ) And the bc boundary point coordinate (x bc ,t bc ) Respectively performing Fourier feature mapping to obtain the in-th internal training data D after mapping in And the bc boundary training data D bc Thereby obtaining a training data set D= [ D ] in ,D bc ]：
Step 4: sequentially building a plurality of input layers and K linear layers { F k |k=1, 2, …, K } and K-1 function activation layers { σ } k A multi-scale deep neural network consisting of i k=1, 2, …, K-1, a multi-scale feature layer and an output layer, wherein the number of neural units of a linear layer is s, and a function activation layer is selected as a tanh activation function;
the training data set D is input into an input layer of the multi-scale deep neural network for processing to obtain predicted bridge displacement response
Step 5: constructing a loss function of the multi-scale deep neural network;
step 5.1: determining the mean value as mu=v×t, the variance as sigma, and carrying out original moving load according to a Gaussian function shown in a formula (5)After dimensionless characterization, a normalized mobile load F (x, t) is obtained:
step 5.2: constructing a dimensionless Euler beam vibration equation by using the formula (6):
in the formula (6), the amino acid sequence of the compound,is the second partial derivative of the displacement response u (x, t) of the bridge to the time coordinate t of the beam vibration,the fourth-order partial derivative of the displacement response u (x, t) of the bridge to the Liang Kongjian coordinate x;
step 5.3: constructing a loss function loss according to (6) PDE ：
In the formula (7), the amino acid sequence of the compound,for input, the in-th internal point coordinate (x in ,t in ) Predicted in bridge displacement response +.>Second partial derivative of the time coordinate t of the beam vibration,/, for>For predicted displacement response of bridge +.>Fourth order partial derivative of Liang Kongjian coordinate x;
step 5.4: based on predicted bridge displacement responseDetermining left endpoint boundary displacement condition->And right endpoint boundary displacement condition->Left endpoint boundary bending condition->And right endpoint boundary bending conditionInitial Displacement Condition->Initial speed Condition->Thereby constructing a loss function loss using equation (8) BC ：
Step 5.5: establishing a total loss function L total ＝loss pde +loss bc ；
Step 6: training a multi-scale deep neural network by using an Adam optimizer, and simultaneously calculating a total loss function L total Up to the loss function L total Is smaller than the set threshold value, thereby obtaining a trained multi-scale depth neural model, and is used for calculating the optimal displacement response u of the bridge at any position x under the action of the moving load F (x, t) at any time t * (x,t)。
2. The method for calculating bridge response under mobile load based on physical information neural network according to claim 1, wherein the multi-scale deep neural network in step 4 processes the training data set D according to the following steps:
the training data set D is input into an input layer of the multi-scale deep neural network, and after the K linear layers and the K-1 function activation layers are transformed, a scale information data set D with the dimension s and 1 is obtained by using a formula (2):
in the formula (2), the amino acid sequence of the compound,representing a matrix multiplication operator;
the multi-scale feature layer converts the scale information data set d into a data set consisting of Q scale functions { C }, according to (3) q Scale of q=1, 2, …, Q }A function set C, wherein C q Representing the q-th scale function in the scale function set C;
C＝W sacle ·d+b scale (3)
in the formula (3), W sacle For dimension [ Q, s ]]Weight matrix of b) sacle For dimension [ Q,1]Is a bias matrix of (a);
the output layer will have Q scale functions { C q When q=1, 2, …, Q } are combined, a predicted bridge displacement response is calculated according to equation (4)
In the formula (4), W output Is of dimension [1, Q]Is a weight matrix of (a).
3. An electronic device comprising a memory and a processor, wherein the memory is configured to store a program for supporting the processor to execute the bridge response under moving load calculation method according to claim 1 or 2, the processor being configured to execute the program stored in the memory.
4. A computer readable storage medium having a computer program stored thereon, which when executed by a processor performs the steps of the bridge response calculation method under a moving load according to claim 1 or 2.
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|---|---|---|---|---|
| CN117272570A (en) * | 2023-11-07 | 2023-12-22 | 福州大学 | Fourier characterization PINN-based flow pipeline vibration analysis method |
| CN117272570B (en) * | 2023-11-07 | 2024-05-24 | 福州大学 | Fourier characterization PINN-based flow pipeline vibration analysis method |
| CN117807854A (en) * | 2024-02-29 | 2024-04-02 | 四川华腾公路试验检测有限责任公司 | Bridge monitoring deflection and temperature separation method based on physical constraint neural network |
| CN117807854B (en) * | 2024-02-29 | 2024-05-28 | 四川华腾公路试验检测有限责任公司 | Bridge monitoring deflection and temperature separation method based on physical constraint neural network |
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