Dual physically-driven and data-driven method for reconstructing internal response of bridge

A dual physically-driven and data-driven method for reconstructing internal response of a bridge, the method comprising:

• step 1, arranging sensors on m nodes of the bridge, respectively, wherein the sensors are configured to collect acceleration response of the bridge, wherein m is positive integers, and m denotes number of external node;
• step 2, modeling the bridge by utilizing a finite element model to obtain the finite element model of the bridge, applying a random load to the finite element model of the bridge, and obtaining an acceleration response matrix {umlaut over (X)}k=[u1, u2, . . . , ui, . . . , um]T of m external nodes and a frequency response function matrix Hk=[H1, H2, . . . , Hi, . . . , Hm]T thereof, and an acceleration response matrix {umlaut over (X)}u=[u1′, u2′, . . . uj′, . . . un′]T of n internal nodes of the bridge and a frequency response function matrix Hu=[H1′, H2′, . . . Hj′, . . . Hn′]T thereof, wherein ui denotes acceleration response with a length of w at an ith external node, uj′ denotes acceleration response with a length of w at a jth internal node, Hi denotes frequency response with a length of w at an mth external node, Hj denotes frequency response with a length of w at a nth internal node, n denotes number of internal node, w denotes length, n and w are positive integers, let a total acceleration response set be {umlaut over (X)}={{umlaut over (X)}k, {umlaut over (X)}u}, and let a total frequency response function set be H={Hk, Hu};
• step 3, constructing a convolutional neural network (CNN) model sequentially, the CNN model comprising an input layer, convolutional layers with a-layer, maximum pooling layers with (a−1)-layer, fully connected layers with e-layer, wherein a and e are positive integers, and an output layer, adding a maximum pooling layer between two adjacent convolutional layers, and utilizing a relu activation function between the fully connected layers;
• inputting {umlaut over (X)}k as an input set into the CNN model, sequentially performing convolution and pooling by the convolutional layers and the maximum pooling layers that are cross-connected, and outputting a feature vector with a dimension of b×c×d from an ath convolutional layer;
• wherein b, c, and d are positive integers, b denotes a number of channels for signal input, c denotes a number of channels for convolution generation, and d denotes a size of a convolution kernel;
• flattening a feature vector set with the dimension of b×c×d, and obtaining a feature vector set with a dimension of b×c;
• and
• sequentially processing the feature vector set with the dimension of b×c by the fully connected layers, and obtaining an internal node response prediction vector with a dimension of w×n expressed as {umlaut over (X)}net;
• step 4, constructing a difference function;
• step 4.1:
• performing dimensionality reduction on acceleration responses in {umlaut over (X)}k, performing Fourier transform, and obtaining acceleration response matrices {umlaut over (X)}k(w)=[u1(w), u2(w), . . . , ui(w), . . . um(w)]T of m external nodes on a frequency domain with a dimension of m×w, wherein ui(w) denotes acceleration response on a frequency domain with a length of w at an ith node;
• step 4.2, performing Fourier transform on {umlaut over (X)}net, and obtaining a prediction vector {umlaut over (X)}net(w)=[u1′(w), u2′(w), . . . , uj′(w), . . . , un′(w)]T of n internal response nodes on a frequency domain with a dimension of n×w, wherein uj′(w) denotes acceleration response on a frequency domain with a length of w at a jth internal node predicted by the CNN mode;
• step 4.3:
• constructing a relation formula between external node response and internal node response in structural dynamics by formula (1):
• {
• X
• ..
• net
• (
• w
• )
• =
• T
• aku
• (
• ω
• )
• ⁢
• X
• ..
• k
• (
• ω
• )
• T
• aku
• (
• ω
• )
• =
• H
• u
• (
• H
• k
• )
• +
• (
• 1
• )
• wherein in formula (1), Taku(ω) denotes a transfer matrix, performing pseudo-inverse multiplication on an acceleration frequency response function matrix Hu of an internal node and an acceleration frequency response function matrix Hk of an external node, obtaining a true acceleration response transfer matrix Taku(w), and computing a difference f by formula (2):
• f={umlaut over (X)}net(w)−Taku(ω){umlaut over (X)}k(w) (2)
• step 4.4, performing pseudo-inverse multiplication on {umlaut over (X)}net(w) and {umlaut over (X)}k(w), and obtaining a predicted acceleration response transfer matrix Tnet(w)=[a1(w), a2(w), . . . , aj(w), am(w)];
• wherein aj(w) denotes an acceleration response transfer function with a length n at a jth external node;
• and
• computing a difference T by formula (3):
• T=Tnet(w)−Taku(ω) (3)
• step 5:
• generating a total loss La11 of the CNN model by formula (4):
• La11=L1+L2+L3+L4 (4)
• wherein in formula (4), L1 denotes a first loss and is obtained by formula (5), L2 denotes a second loss and is obtained by formula (6), L3 denotes a third loss and is obtained by formula (7), and L4 denotes a fourth loss and is obtained by formula (8);
• L1=(f−f0)T(f−f0) (5)
• L2=(T−f0)T(T−f0) (6)
• L3=({umlaut over (X)}net0−{umlaut over (X)}0)T({umlaut over (X)}net0−{umlaut over (X)}0) (7)
• L4=({umlaut over (X)}net−{umlaut over (X)}u′)T({umlaut over (X)}net−{umlaut over (X)}u′) (8)
• wherein in formulas (5)-(8), f0 denotes an all-zero matrix with a dimension of n×w, {umlaut over (X)}net0 denotes predicted response output by an acceleration response matrix under an initial condition through the CNN model, let the acceleration response matrix under the initial condition be an all-zero matrix with a dimension of w×1×m, and {umlaut over (X)}0 denotes acceleration response of {umlaut over (X)}u under the initial condition;
• and
• step 6:
• based on the total acceleration response set {umlaut over (X)}={{umlaut over (X)}k, {umlaut over (X)}u} and the total frequency response function set H={Hk, Hu}, training the CNN model with an Adam optimizer, computing a total loss La11 and updating parameters of the CNN model until a maximum training time is reached or the total loss La11 converges, and obtaining a trained CNN model for reconstructing response of any internal node of the bridge.