Dynamic constitutive identification of concrete based on improved dung beetle algorithm to optimize long short-term memory model

In order to improve the accuracy of concrete dynamic principal identification, a concrete dynamic principal identification model based on Improved Dung Beetle Algorithm (IDBO) optimized Long Short-Term Memory (LSTM) network is proposed. Firstly, the apparent stress–strain curves of concrete containing damage evolution were measured by Split Hopkinson Pressure Bar (SHPB) test to decouple and separate the damage and rheology, and this system was modeled by using LSTM network. Secondly, for the problem of low convergence accuracy and easy to fall into local optimum of Dung Beetle Algorithm (DBO), the greedy lens imaging reverse learning initialization population strategy, the embedded curve adaptive weighting factor and the PID control optimal solution perturbation strategy are introduced, and the superiority of IDBO algorithm is proved through the comparison of optimization test with DBO, Harris Hawk Optimization Algorithm, Gray Wolf Algorithm, and Fruit Fly Algorithm and the combination of LSTM is built to construct the IDBO-LSTM dynamic homeostasis identification model. The final results show that the IDBO-LSTM model can recognize the concrete material damage without considering the damage; in the case of considering the damage, the IDBO-LSTM prediction curves basically match the SHPB test curves, which proves the feasibility and excellence of the proposed method.

convex lens imaging strategy and introducing random difference change strategy, the relationship between search diversity and convergence accuracy of the algorithm is balanced, and the convergence speed of DBO is improved.
Based on the above analysis, an improved Dung Beetle Algorithm (IDBO) optimized Long short-term memory (LSTM) neural network dynamic constitutive identification model of concrete is proposed in this paper.First, the apparent stress-strain curve of concrete containing damage evolution was measured by split Hopkinson pressure bar test, the damage and rheology were decouple, and the system was modeled by LSTM neural network.Secondly, based on the principle of dung beetle algorithm, improved strategies (greedy lens imaging reverse learning initializing population strategy, curve adaptive weight factor and PID control optimal solution perturbation strategy) are introduced, and their optimization performance is verified by CEC2005 test set.Finally, four LSTM hyperparameters (number of hidden units, maximum training period, initial learning rate and L2 regularization parameter) are optimized using the improved Dung Beetle algorithm.The results show that the proposed IDBO algorithm has good optimization effect and the IDBO-LSTM model has high precision in identifying concrete dynamic constitutive.

Dung Beetle optimization algorithm
Dung Beetle Optimizer (DBO) was inspired by dung beetle behaviors such as ball rolling, dancing, foraging, stealing and reproduction 16 .Thus, the dung beetle population consists of rolling dung beetles, breeding dung beetles, young dung beetles, and stealing dung beetles.

Rolling dung Beetle
In nature, dung beetles' habit is to shape animal dung into balls, which is conducive to fast and efficient movement of dung and prevent its peers from robbing them.Dung beetles need to navigate through celestial cues (sun orientation, polarized light, etc.) as they roll to keep the dung ball rolling on a straight path.As it rolls, the beetle's position updates as, In the formula, t represents the current number of iterations, x i (t) is where the i beetle is in the t iteration.α is the natural coefficient and is assigned − 1 or 1 using algorithm 1. k ∈ (0, 0.2] stands for the perturbation coefficient, set to 0.1 for a fixed value.Set the value of b ∈ (0, 1) to 0.3,X ω represents the worst position in the world, x is used to simulate the change of strong light.
When dung beetles hit an obstacle, they usually climb onto the dung ball and "dance" (a series of spins and pauses) to reorient themselves and gain a new route.So the beetle's position update is, In the formula, θ ∈ [0, π ] is the interference Angle, when θ = 0, π/2, π , the beetle's position does not update.|x i (t) − x i (t − 1)| is the difference between where the i beetle was on the t iteration and where it was on the t − 1 iteration.

Breeding dung beetles
In order to provide a safe environment for their young, dung beetles roll their balls to a safe place to hide.Therefore, the female dung beetle's spawning zone boundary strategy is expressed as: In the formula, X * represents the current local best position, LB * and Ub * are the lower and upper boundaries of the spawning area, respectively.R = 1 − t/T max ,T max is the maximum number of iterations, Lb and Ub represent the lower and upper bounds of the optimization problem, respectively.
After identifying the spawning area, the female dung beetle chooses that area for laying eggs.Formula (3) represents the dynamic spawning area, therefore, the position update of the oosphere in the iteration process is dynamic, which is expressed as, (1)

Little dung Beetle
When young dung beetles grow up after hatching eggs, they need to establish an optimal feeding area to guide them to be fed.The optimal feeding area boundary strategy is expressed as, In the formula, X b represents the global optimal feeding position, Lb b and Ub b are the lower and upper bounds of the optimal feeding region, respectively.The beetle's position update is represented by, In the formula,x i (t) is where the i beetle is on the t iteration,C 1 is a random number with a normal distribution, C 2 ∈ (0, 1) is a random vector.

Dung Beetle stealing
There are some misbehaving dung beetles (thieving dung beetles) in the dung beetle population, stealing the fruits of other people's labor.According to formula (5), it is the optimal feeding position, so the setting is the best place for dung beetles to compete for food.During the iteration, the position of the thieving beetle is updated to, In the formula, x i (t) is where the i beetle is on iteration t, g is a random vector of 1 × D-dimension size that fol- lows a normal distribution, S is constant.

Improved dung beetle optimization algorithm
In the optimization problem, Dung Beetle optimization algorithm (DBO) has the advantages of high precision, faster convergence and stronger stability.According to the No Free Lunch theorem 20 , no algorithm can perform optimally in any domain.Therefore, three DBO strengthening strategies are proposed in this section to accelerate the convergence speed and enhance the global search capability of the algorithm.IDBO enhances the optimization ability of DBO through greedy lens imaging reverse learning, fusion PID control optimal solution perturbation strategy and introduction of curve adaptive factors.The pseudo-code for the improved dung beetle optimization algorithm is shown in Algorithm 3.

Greedy lens imaging reverse learning to initialize the population
Ideally, a good algorithm should have the final optimal solution independent of the initial position, but for almost all random algorithms, the reality is the opposite, if the initial solution is established at the optimal position in the population, the probability of the population convergence to the optimal position is very high, and determines the convergence speed and accuracy of the future algorithm 21 .Therefore, based on the uniform random initialization of the population, the introduction of lens imaging reverse learning to generate new populations, and the use of greedy idea to screen new populations from the combined population according to the fitness value, is conducive to reducing the optimization time in the algorithm iteration process.The lens imaging reverse learning strategy is mathematically expressed as, (4) In the formula, x j (i) is the ith individual of dimension j, ub j and lb j are the J-dimensional components of the upper and lower bounds of the decision variable, respectively.k is the scaling factor.

PID control optimal solution perturbation strategy
In the process of iteration, the individuals in the initial population are updated, and the diversity of the population is lost.Researchers used variation-perturbation strategies to increase population diversity to obtain more search information.For example, Guo et al. 22 integrated the follower position update mechanism in Sparrow search algorithm to perturb the algorithm, and combined Cauchy-Gauss variation strategy to help the algorithm jump out of the local optimal solution.Pan et al. 23 introduced adaptive Gauss-Cauchy hybrid mutation perturbation to enhance the ability of dung beetle algorithm to coordinate its local development and global exploration.Wang et al. 24 proposed a decreasing control strategy for the convergence factor of disturbance index to achieve a good coordination between the exploration and development capabilities of Gray Wolf algorithm.Chen et al. 25 designed a perturbation strategy for wavelet optimal solutions to improve population diversity and avoid the algorithm falling into local optimality.Vu-Huu et al. 26 proposed a push-process technique to improve the effectiveness of the BA algorithm by reducing the wide distribution of the optimal global solution of its to produce an intervention in the BA algorithm, so as to achieve a true global optimal that can be exposed in several generations without many computational times.In this paper, PID control is designed to disturb the optimal solution to get new individuals, so that individuals in the population can be optimized in multiple directions, increase the diversity of the population and improve the search ability of the algorithm.PID algorithm has excellent performance in the field of control, combining proportional control, integral control and differential control, aiming at fast and stable output of setpoint 27 .In the fitting regression problem, the algorithm is optimized according to the value of the adaptation function.Therefore, the optimal individual is fine-tuned by PID controlling the optimal fitness function value, which helps the algorithm to jump out of the local extreme value and avoid premature maturity.The mathematical expression of PID control is:

Curve adaptive weight factor
In order to better coordinate the global search and local exploration capabilities, and enhance the optimization and later development capabilities of the algorithm iteration, Cuong-Le et al. 28 proposed a new cuckoo search algorithm (NMS-CS) based on the Levy flight, which uses the Levy distribution to calculate the random step size, and randomly selects the newly created functions (convex function, concave function, linear function, etc.) to control the parameters in the CS algorithm, so as to expand the search space in the early stage of algorithm iteration and improve the development ability in the late iteration stage.Inspired by the idea of inertial weighting in the improved particle swarm optimization 29 , this paper adds curve adaptive weights to the update formula (2) of the rolling ball dung beetle, and the weight factor is guided by the cosine function, and with the increase of the number of iterations, the curve changes similar to the cosine function (0-π range), so the weight factor keeps decreasing slowly in the early stage of iteration, the decline rate accelerates in the middle of iteration, and slowly decreases again in the late iteration, which can ensure that the algorithm slows down the global search performance in the early stage and improves the local optimal in the later stage.The expression is, In the formula, t is the current number of iterations, T max is the maximum number of iterations, a is the adjust- ment factor,w max and w min are the maximum and minimum values of the factor, respectively.In formula (2), the adaptive weight of the curve is added and modified as follows:

Simulation experiment
In order to test the optimization performance of IDBO algorithm, 9 benchmark test functions in CEC2005 dataset 30 are selected in this paper, as shown in Table 1.There are 4 categories: unimodal problem f 1 − f 4 , Basic multimodal problem f 7 , f 10 , Extended multimodal and hybrid composite problems f 13 , f 15 , f 20 .

Analysis of algorithm test results
IDBO algorithm is compared with standard Dung Beetle algorithm (DBO), Grey Wolf algorithm (GWO), Firefly algorithm (FA) and Harris Eagle algorithm (HHO) to optimize test functions.In order to reduce the chance of the experiment, set the same experimental parameters, The population size of each algorithm N = 30, maximum number of iterations T max = 1000.Perform 30 independent experiments on 9 test functions respectively.Figure 1 draws the fitness iteration convergence comparison curve, and evaluates and compares it by convergence speed and iteration number.The experimental results are shown in Table 2.
CEC2005 test set, f 1 − f 4 function structure is relatively simple, test algorithm convergence performance; The f 7 , f 10 function has a local optimal solution and tests the algorithm's ability to balance global development and local exploration in the search space.Function f 13 , f 15 and f 20 tests the ability of the algorithm to deal with mixed complex problems.From Fig. 1(a-d), it can be clearly seen that IDBO algorithm converges first and has a faster convergence speed as the number of iterations increases.It shows that the introduction of greedy lens imaging reverse learning to initialize the population can effectively improve the population quality and accelerate the convergence speed.From Fig. 1e, f, IDBO can quickly jump out of the local extreme solution at the early stage of iteration, so as to achieve the global optimal explanation, and the introduction of curve adaptive weight factor and PID control optimal solution perturbation strategy can help the algorithm get rid of local extrema and enhance the global optimization ability.As can be seen from Fig. 1g-i, IDBO also has excellent optimization ability in dealing with mixed complex problems.It can be seen that the improved strategy of dung beetle optimization algorithm is effective.
In Table 2, the IDBO algorithm and the evaluation indexes of the four algorithms are the best value, the worst value, the standard deviation and the average value respectively.By observing the optimization results in Table 2, we can see that except for f 7 , f 10 and f 13 , IDBO can find theoretical optimal values in other benchmark functions.Standard DBO can find the theoretical optimal value of the f 1 function, the other algorithms failed to find the theoretical optimal value.IDBO, standard DBO, and HHO have the same standard deviation on the f 1 , f 3 function, the mean and standard deviation of IDBO optimization results on unimodal function are 0. The comprehensive performance of IDBO is obviously better than DBO, HHO, GWO and FA in terms of optimization accuracy and stability.For multi-modal functions, the optimization results of f 7 function show    For the mixed composite function, the optimization results of f 13 function show that the optimal value of DBO is 10 -32 , while IDBO is not good.The optimization results of function f 15 and f 20 show that the optimal values of the five algorithms are close to the theoretical optimal values, indicating that IDBO, DBO, HHO, GWO and FA algorithms have the ability to deal with this complex problem.

Model principle and parameter optimization process Long short-term memory regression prediction model
Long short-term memery (LSTM) is an improvement of recurrent neural networks (RNN), widely used in machine translation 31 , speech recognition 32 , and image description.The LSTM network structure newly establishes a memory unit with feedback connections in the direction of time, which is reflected in the addition of three gate structures, namely, forget gate, input gate and output gate.At time step t, input data is x t , Then the hid- den state of the previous moment is h t−1 , and the cell state is c t−1 .The LSTM model structure is shown in Fig. 2. Forgetting gate f t , control the forgetting degree of the unit state at the previous moment: www.nature.com/scientificreports/ In the formula, W f is the weight matrix of the input x t ,U f is the weight matrix of the hidden state h t−1 at the previous time, b f is the biased variable, σ is the sigmoid function.
Enter gate i t to control the input of new information: In the formula, W i is the weight matrix of the input x t ,U i is the weight matrix of the hidden state h t−1 at the previ- ous time, b i is the biased variable, σ is the sigmoid function.
Output gate A, control output degree: In the formula, W o is the weight matrix of the input x t ,U o is the weight matrix of the hidden state h t−1 at the previous time, b o is the biased variable, σ is the sigmoid function.New cell status ct , update the current cell status: In the formula, W c is the weight matrix of the input x t ,U c is the weight matrix of the hidden state h t−1 at the previous time, b c is the biased variable,tanh is a hyperbolic tangent function.Calculate the cell state A at the current moment and update the hidden state S: In the formula,⊙ stands for element-by-element product.

Parameter optimization based on IDBO
The objective of IDBO algorithm is to help LSTM model select appropriate parameters, which are LSTM hidden unit number, maximum training period, initial learning rate and L2 regularization parameter.So the fitness function of dung beetles is defined as, In the formula, N is the number of samples, ŷi is the predicted value for sample i, y i is the actual value for sample i.
Set the total number of dung beetles SearchAgent-n = 30, among them, the proportion of rolling dung beetles, laying dung beetles, small dung beetles and thieving dung beetles was 20%, 40%, 20% and 20% respectively.The maximum number of iterations Max_iter = 10, the number of variables dim = 4, and the search range of variables are determined.The parameter optimization process of LSTM model based on IDBO algorithm is shown in Fig. 3.

Data sources
The data comes from a series of experiments designed by our group, and the concrete with a wide range of strength grades C40 is selected as the experimental specimen, and the newly constructed model is verified by using this experimental data.(12)

Concrete specimen preparation
The test specimen is a cylinder with a diameter of 70 mm and a height of 35 mm.The cement in the concrete material composition is ordinary Portland cement with a strength grade of 42.5, the coarse aggregate is made of pebbles and gravel with a particle size of 5-10 mm, and the fine aggregate is made of medium coarse river sand, with a large particle size of 5 mm, a fineness modulus of 2.8-3.0, and a mud content of less than 1%.The admixture is polycarboxylic acid high-efficiency superplasticizer mother liquor.The mix ratio of steel fiber concrete specimens is: cement: 425 kg/m 3 , sand: 600 kg/m 3 , stone: 1132 kg/m 3 , water: 184 kg/m 3 , water reducer: 8 kg/m 3 , Steel fiber: 39 kg/m 3 .The concrete specimen is shown in Fig. 4.

Experimental results of separated Hopkinson bar (SHPB)
In order to verify the accuracy of the test results, we performed 6-10 repeated tests for each strain rate at room temperature, and took the average of the multiple tests as the test results for the loading condition.The concrete specimen after loading is shown in Fig. 5.

recessive modulus constitutive equation of concrete material
The dynamic failure of materials is a process in which different forms of micro-damage (micro-cracks, microvoids, micro-shear bands, etc.) accumulate at a finite rate over time.The macroscopic continuous damage D is defined as follows: In the formula, σ 0 is non-damaging material stress, σ is the apparent stress of the damaged material.In general, material damage evolves with the rheological process, so damage D is ε function of strain ε .How- ever, A large number of dynamic tests show that the evolution of material damage under impact load depends on both strain and strain rate, i.e.D = D(ε, ε) .From a macroscopic point of view, from the perspective of systems science, the constitutive relation is equivalent to the relationship between the cause (input) and the effect (output) of a system, that is, the system identification problem.Therefore, the one-dimensional constitutive relationship of steel fiber reinforced concrete under different strain rates can be expressed as, In the formula, ε th is the damage threshold (0.75 times the peak strain), measured by the "damage freezing method".The damage value D cannot be directly determined in the test.Considering that the damage value D is a function of time, the inverse function of time with respect to the damage value D is taken as the damage.

Experimental parameter settings
See Table 3.

Analysis of simulation results
In order to verify the effectiveness of the proposed concrete dynamic constitutive identification model, a steel fiber reinforced concrete specimen with a strain rate of 113.05 s −1 is taken as an example.The experimental data   were input into the trained LSTM, DBO-LSTM, GWO-LSTM and IDBO-LSTM models for identification, and the identification results of the four models are shown in Fig. 6.
As can be seen from Fig. 6, the LSTM model has the lowest recognition ability among the four constitutive recognition models.LSTM can define macroscopic continuous damage.However, after taking into account the damage data, the curve predicted by the LSTM model does not agree with the test curve, so it cannot be verified to be accurate in defining the damage.Figure 6b, c show that the LSTM model optimized with GWO or DBO has improved the accuracy of its definition of damage, and the verification ability of DBO optimization is relatively high.In Fig. 6e, within the scope of ε ≤ ε th ,the IDBO-LSTM prediction curve is in good agreement with the test curve, but after the deformation of the concrete specimen exceeds the limit of model learning, the test curve and  www.nature.com/scientificreports/ the prediction curve deviate, and we believe that it is the occurrence of damage that causes the deviation of the curve, as shown in Fig. 6d red dotted line.After considering the damage evolution, that is, adding time as the inverse function of damage, the prediction curve of the IDBO-LSTM model is in good agreement with the test curve in the whole strain range, as shown in Fig. 6d blue underlined.
In order to further verify the universality of the IDBO-LSTM model, a steel fiber reinforced concrete specimen with a strain rate of 143.13 s −1 was selected, and the sample test data was input into the IDBO-LSTM model, and the identification results are shown in Fig. 7.
From the above identification, the continuous damage is determined as a function of strain and strain rate according to Eq. ( 18), in the form of the evolution of damage D with strain for different constant strain rates as shown in Fig. 8.

Conclusions
In this paper, a concrete dynamic constitutive identification model (IDBO-LSTM identification model) based on improved dung beetle algorithm and optimized long short-term memory neural network is proposed.Based on the thermal activation damage evolution model, the damage and rheology are separated by the apparent stress-strain curve of concrete with damage evolution, and the LSTM method is used to model the system.The greedy lens imaging reverse learning strategy, curve adaptive weight factor and PID control optimal solution disturbance strategy are introduced to improve the original shortcomings of the Dung Beetle algorithm, and the improved Dung Beetle algorithm is combined with LSTM method to identify the dynamic constitutive of concrete, and the conclusions are as follows: • The greedy lens imaging reverse learning strategy was introduced to initialize the population, which improved the uneven position of the initial dung beetles, and the greedy idea was used to greatly reduce the optimiza- https://doi.org/10.1038/s41598-024-56960-zwww.nature.com/scientificreports/In the formula, B i (t) is the position of the i egg ball at the t iteration, b 1 and b 2 represent two independent ran- dom variables of 1 × D , D represents the dimension of the optimization problem.Therefore, the position of the oocytes is strictly controlled within a certain range.Breeding dung beetle position update Algorithm 2 shows. https://doi.org/10.1038/s41598-024-56960-zwww.nature.com/scientificreports/

Figure 1 .
Figure 1.Convergence curve of the test function.

4 Figure 6 .
Figure 6.Comparison of model identification results.Annotation*: the damage-free curve (red dashed line) is obtained by using strain and strain rate as inputs and stress as output.The damage curve (blue line) is obtained by using strain, strain rate, and time as inputs, and stress as output.

Figure 7 .
Figure 7. Identification results of specimens with a strain rate of 143.13 s −1 .

Table 2 .
Comparison of test function results.

Table 3 .
The main parameters of the algorithm.