# Inversion for Constitutive Model Parameters of Bird in Case of Bird Striking

Authors: Peng Yu; Siliu Yu; Sheng He; Xinheng Huang; Weijing Yun
Journal: Shock and Vibration
 (2022)

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## Abstract

The strike between flying birds and airplanes results in the unacceptable losses of aircraft structures. The normal approach to evaluate the bird impact resistance is the combined full-scale experimental-numerical study. However, the simulation results from the current available bird constitutive models are usually not in good agreement with the experimental data. Establishing a reasonable bird constitutive model is difficult and significant to the simulation of the bird striking process. In this paper, based on the displacement measurements of an aluminum plate subjected to soft body impact, an inversion of the bird constitutive model is conducted by using backpropagation (BP) neural network. A comparative evaluation of this inversion model and other constitutive models is carried out, indicating that the proposed inversion model is more reasonable.


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## Body

## 1. Introduction

The strike between flying birds and airplanes frequently results in catastrophic damage to the aircraft. According to the International Bird Strike Committee, 55 fatal accidents, in which 108 aircraft were damaged and 277 people died, were reported between 1912 and 2009 [1]. The serious events of birds striking can lead to an unacceptable loss of aircraft and the death of crewmembers. In early studies, the common method to determine the capability against bird strikes is to carry out a full-scale bird-strike experiment [2]. However, various uncertainties may lead to diversities of experiments, and the experiments cost much. The development of advanced numerical techniques and computing technologies makes it possible to perform the coupling analysis and simulate the whole process of bird strike problems. However, the numerical results of current models are not in good agreement with the experimental data. Establishing a reasonable bird constitutive model is significant to the finite element analysis of bird strikes on aircraft structures.The anatomic structure of birds includes several internal cavities like pneumatic bones, lungs, and peculiar air sacs, which is a complex nonlinear system. Moreover, the identification of a bird constitutive model and the corresponding material parameters is difficult. Several authors used an elastic-plastic bird constitutive model with a defined failure strain [3–5], while others highlighted the limitations of this simplified material law [6]. It was observed that no fluid-like flow response can be achieved with such an elastic-plastic model, only if the shear modulus G is set very low [7]. In Ref. [8] a rubber-like hyper-elastic Mooney–Rivlin model was adopted to simulate the bird's behavior. A more common method is an equation of state (EoS) for the bird constitutive model, by defining the pressure-volume relationship with parameters of water at room temperature. A polynomial form of the EoS and a simpler Murnaghan EoS were used in most studies [2, 9–14]. A further approach of the Grüneisen EoS was adopted in Refs. [15, 16]. However, in Ref. [17] it was reported that the Grüneisen EoS is only valid for solid materials that remained in the solid-state during the impact process and should therefore be carefully used for bird strike simulations and the usage of Grüneisen EoS dependent on the finite element code, since there are only a few of the abovementioned algorithms available in most commercial software codes. The material constants of all EoS have to be defined, which cannot be measured directly.A regular technique is the parameter calibrations, for example, the elastic, elastic-plastic, or nonrevolving flow model is assumed to be the bird constitute model, and then the plate response during the bird-plate impact experiment is used to confirm the parameters of the constitutive model. This methodology was performed in studies such as [16, 18] using optimization software to determine the parameters in conjunction with bird strike experiments on instrumented plates. Nishikawa et al. [19] impacted gelatin birds on rigid targets and confirmed that the deformation of the bird model did behave like fluid and numerical simulation results correlated well with experiment data.In recent years, the artificial neural network (ANN) has provided a fundamentally practical and powerful approach to material modeling. The basic advantage of ANN is that it does not explicitly need physical knowledge of the deformation mechanism and mathematical model. Darryl et al. [20] used a contact algorithm based on the Lagrange multiplier method to predict appropriate impact force, this method solved the severe contact-induced deformation due to a large contact force and highly deformable projectile. Previous studies usually adopted the Lagrangian formulation to model bird strikes. However, due to element instability, refining mesh elements and narrowing the time-step method had to be used to alleviate the problem, to solve this problem, nowadays more researchers use arbitrary Lagrange–Euler (ALE) [19, 21–23] and smoothed particle hydrodynamics (SPH) [19, 24–27] method in bird strike problems [25]. However, there are few reports about the application of ANN to model the bird constitutive model by using SPH Formulation during the impact process. In this paper, based on the displacement measurement results of bird impact on the aluminum plate, an ANN model with an error backpropagation learning algorithm has been applied to an inversion on bird constitutive model parameters and the parameters are obtained by inversion. The validity and reliability of the inversion model are discussed. Furthermore, a comparative evaluation of three other constitutive equations and the trained network model is carried out.


## 2. Introduction of BP Neural Network

The artificial neural network is an intelligent information-treatment system with the characteristics of adaptive learning and treating complex relationships [28]. An artificial neural network (ANN) consists of a large number of highly connected artificial units (neurons or nodes). Each neural network comprises an input layer, an output layer, and one or more hidden layers, which are connected by the processing units, as can be seen in Figure 1. The input layer is used to receive data, while the output layer sends the output values to users. The role of the hidden layer is to provide the necessary complexity (which denotes to abstract the characteristics of the input data to another dimension space to show its more abstract features) for nonlinear problems. In the present model, sigmoid functions are employed due to the processing units for computational convenience.(1)fx=11+exp−x.Figure 1
The architecture of the BP ANN model.Among various types of ANN models, an iterative gradient algorithm called backpropagation (BP) algorithm is popular in the field in engineering applications and it is used for the present work. The training of a network by backpropagation involves three stages: the feedforward of the input training pattern, the training and backpropagation with the associated error, and the adjustment of the weights. When the artificial neural network is established for the soft body constitutive model, a feedforward backpropagation algorithm is selected to train the network.


## 3. An Example

### 3.1. Flow of Soft Body Parameter Optimization

First of all, determine the soft body parameters to be optimized, and then according to the experience to determine the range of parameters, the values determined according to the experimental data of the soft body impact aluminum sample points. And then the sample parameters in LS-DYNA in the calculation, can get aluminum midpoint displacement curve of time, choose corresponding time point in the experimental data of the displacement, training sample is completed. After the improved BP neural network training, the required soft body model parameters are output and the stable network unit connection weights are obtained. Then network training is carried out to obtain the soft body model parameters. Lastly, the obtained parameters are put into LS-DYNA for calculation, and the calculation results are compared with the experimental results. The flow chart of soft body parameter optimization as shown in Figure2.Figure 2
Flow chart of soft body parameter optimization.


### 3.2. Construction of Bird Body Constitutive Model

Newtonian fluid is selected to describe the constitutive model of the bird body, since the real bird body is a structure with flesh and blood [29]. Newtonian fluid is a viscous fluid with a linear proportional relationship between shear stress and deformation gradient, and its stress-strain relationship is determined by the following equation:(2)σij=−pδij+DijklVij,where σij is the stress tensor, Vij is the deformation rate tensor, Dijkl is the viscosity coefficient tensor of fluid, and P is the static pressure. The term pδij represents the state of stress in the fluid at rest (when Vij = 0). Assume that static pressure P depends on the density and temperature of the fluid according to some equation of state. For Newtonian fluids, we assume that the elements of the tensor Dijkl can be temperature-dependent and the stress or deformation rate is independent. Tensor Dijkl is of order 4 with 34 = 81 elements.If the fluid is isotropic, that is, the tensorDijkl has the same component matrix in any Cartesian rectangular system, then Dijkl can be expressed by two independent constants λ and μ as the following equation:(3)Dijkl=λδijδkl+μδikδjl+δilδjk.Then,(4)σij=−pδij+λVkkδij+2μVij.Combining equations(5)σkk=−3p+3λ+2μVkk.If the mean normal stress 1/3σkk is assumed to be independent of the volume expansion rate Vkk, then there must be(6)3λ+2μ=0.Thus, the constitutive equation becomes(7)σij=−pδij+2μVij−23μVkkδij.This formula was put forward by Stokes, so the fluid subject to the above equation is called Stokes fluid, for this kind of fluid only a material constantμ (i.e., viscosity coefficient) is enough to determine its properties.If the fluid is incompressible, thenVkk = 0, and the constitutive equation of incompressible viscous fluid can be obtained:(8)σij=−pδij+2μVij.Ifμ = 0, the constitutive equation of inviscid fluid is obtained:(9)σij=−pδij.The appearance of hydrostatic pressureP marks the fundamental difference between hydrodynamics and elasticity. To accommodate the needs of this new variable, it is usually assumed that there exists some equation of state, which gives the relationship between density ρ, pressure P, and absolute temperature T, that is(10)fp,ρ,T=0.


### 3.3. Finite Element Model

The bird strike setup is conducted based on the experiment of the bird-impact plate in Figure3 by using LS-DYNA element code.Figure 3
The finite element model: (a) top view; (b) front view.
(a)(b)The flat plate fixed along the four edges is impacted by a soft body at the center of the flat plate at 91.1 m/s with an impact angle of 30°. The dimensions of the flat plate are 525 mm × 430 mm × 10 mm. The geometric model of a bird is simulated by a cylinder with a hemisphere at the two ends, the diameter of the hemisphere is 53 mm and the length of the cylinder is 156 mm. The mass of the bird is 1.8 kg.In the finite element model, the 8-node solid and 164-element type in the LS-DYNA element library is employed for the bird and flat plate. In addition, hourglass control and single-point integration are also used. The numbers of the elements for the bird and flat plate model are 4504 and 2000.


### 3.4. Materials Properties

#### 3.4.1. Materials Properties of Flat Plate

The plate is made from LY12-CZ aluminum alloy. The material parameters of the flat plate are mentioned in Table1. The material properties of the flat plate are modeled by the MAT_PLASTIC_KINEMATIC in LS-DYNA code where the values are set in Table 1.Table 1
Material parameters of flat plate.
Young’s modulus (GPa)Poisson ratioDensity (kg/m3)Yield stress (MPa)Failure strain710.327806900.3


#### 3.4.2. Material Properties of Bird

Generally speaking, a trend is visible towards the utilization of SPH models with an equation of state for the material of bird using the properties of water–air-mixture at room temperature. A very good correlation to experimental results could be obtained with these models. For the fluid model, in LS-DYNA, the material model of NULL (empty material) is used to describe its material properties. The parameters to be input are shown in Table2.Table 2
Parameters of NULL model.
ROMass densityPCPressure cutoffMUViscosity coefficientTERODRelative vol. in tensionCERODRelative vol. in compressionYMYoung’s modulusPRPoisson’s ratioAccording to the property of the bird in the experiment, using fluid constitutive and Gruneisen state equation to describe soft body is closer to the reality. The bird model is defined by MAT_NULL and the equation of state (EoS) is specified. The inputs of parameters are shown in Table3.Table 3
Parameters of Gruneisen model.
CS1S2S3GAMA0AE0 (J)V016702.48000001During the impact process, the deformation of the bird is described by the equation of state. Elastic modulus and passion ratio are 0, and the density is 938 kg/m3. The four parameters of the constitutive model of the bird to be determined are shown in Table 4.Table 4
Parameters to be determined.
PCPressure cutoffMUViscosity coefficientTERODRelative vol. in tensionCERODRelative vol. in compression


### 3.5. Identification of the Bird Model and the Inversion of the Parameters

The displacements at 12 different times are chosen as the inputs, and the outputs of the model are four constitutive parameters of the soft body. The 12 different times are based on the experimental displacement-time data of the center of the flat plate. The experimental displacement-time data of the center of the flat plate is given in Figure4.Figure 4
Experimental displacement-time data of the center of the flat plate.Before the network training, both input and output variables are normalized within the range 0 to 1 in order to obtain a reliable form for the neural network[31]. If the number of hidden layers is too small, the trained network might not have sufficient ability to learn the process correctly. As a result, in order to choose the appropriate number of hidden layers, the trial-and-error procedure is started with seven neurons in the hidden layer and further carried out with up to fourteen neurons. It is found that when the number of neurons is 9, the mean square error reaches the minimum.In the bird constitutive model, the values of the four parameters have their ranges based on the prior knowledge, specific ranges (see Table5).Table 5
The value range of the four parameters.
Optimize parametersLabelRanges of valuesUnitsPCX1[−0.05, −1.25]MPaMUX2[0, 1.5]N·ms/mm2TERODX3[0.05, 1.25]—CERODX4[0.05, 1.25]—The parameters to be optimized are uniformly divided into a number of values with the space. According to the orthogonal design and uniform design principle, these values are divided into 25 groups, and each sample group contains four parameters, which is shown in Table6.Table 6
Parameters in sample groups.
Sample groupsX1X2X3X41−0.0500.051.252−0.10.10.11.23−0.150.20.151.154-0.20.30.21.15−0.250.40.251.05……..……..……..……..……..Substituting these parameters of 25 groups into the bird constitutive model, 25 displacement-time curves of the center of the flat plate can be obtained in LS-DYNA simulation. Selecting the displacements of the center of the flat plate, the training of the network can be finished, which is shown in Table7.Table 7
Sample groups.
Time/msSample groups (Displacement/mm)12345……..0.340.1230640.1231650.1233330.1228450.121246……..0.591.2203371.2168671.2162241.2139891.232132……..0.712.0431142.0481792.0481942.0486312.060089……..0.984.1002854.1124174.1287764.1530354.19781……..1.094.6681754.6868354.6894324.7098934.751519……..1.345.3558445.3764235.3949585.4172525.453059……..1.535.5765615.6081545.6288955.6520065.696968……..1.605.5374795.5719925.6057185.6461245.67513……..1.794.847064.8951244.931184.9527084.973954……..1.914.1848244.2275174.2557954.279694.277913……..2.113.2423673.2604733.2738623.2629283.225929……..2.352.134912.123192.1140012.0767032.046814……..When developing the ANN model, the samples of 1, 5, 10, 15, 20, and 25 are removed to test the generalization capability of the network. The optimized ANN model consists of 12 input neurons, 4 output neurons, and a single hidden layer (with 9 neurons) and the transfer functions of the optimized ANN model are ‘tan sigmoid’ and ‘pure linear’. A feedforward backpropagation algorithm is selected to train the network. The setting of other training parameters for the neural network is listed in Table8.Table 8
The setting of training parameters for network.
Name of parametersContentsNetworkBackpropagationTraining functionTrainlmPerformance functionMSETraining epoch2,000Goal0.001


## 3.1. Flow of Soft Body Parameter Optimization

First of all, determine the soft body parameters to be optimized, and then according to the experience to determine the range of parameters, the values determined according to the experimental data of the soft body impact aluminum sample points. And then the sample parameters in LS-DYNA in the calculation, can get aluminum midpoint displacement curve of time, choose corresponding time point in the experimental data of the displacement, training sample is completed. After the improved BP neural network training, the required soft body model parameters are output and the stable network unit connection weights are obtained. Then network training is carried out to obtain the soft body model parameters. Lastly, the obtained parameters are put into LS-DYNA for calculation, and the calculation results are compared with the experimental results. The flow chart of soft body parameter optimization as shown in Figure2.Figure 2
Flow chart of soft body parameter optimization.


## 3.2. Construction of Bird Body Constitutive Model

Newtonian fluid is selected to describe the constitutive model of the bird body, since the real bird body is a structure with flesh and blood [29]. Newtonian fluid is a viscous fluid with a linear proportional relationship between shear stress and deformation gradient, and its stress-strain relationship is determined by the following equation:(2)σij=−pδij+DijklVij,where σij is the stress tensor, Vij is the deformation rate tensor, Dijkl is the viscosity coefficient tensor of fluid, and P is the static pressure. The term pδij represents the state of stress in the fluid at rest (when Vij = 0). Assume that static pressure P depends on the density and temperature of the fluid according to some equation of state. For Newtonian fluids, we assume that the elements of the tensor Dijkl can be temperature-dependent and the stress or deformation rate is independent. Tensor Dijkl is of order 4 with 34 = 81 elements.If the fluid is isotropic, that is, the tensorDijkl has the same component matrix in any Cartesian rectangular system, then Dijkl can be expressed by two independent constants λ and μ as the following equation:(3)Dijkl=λδijδkl+μδikδjl+δilδjk.Then,(4)σij=−pδij+λVkkδij+2μVij.Combining equations(5)σkk=−3p+3λ+2μVkk.If the mean normal stress 1/3σkk is assumed to be independent of the volume expansion rate Vkk, then there must be(6)3λ+2μ=0.Thus, the constitutive equation becomes(7)σij=−pδij+2μVij−23μVkkδij.This formula was put forward by Stokes, so the fluid subject to the above equation is called Stokes fluid, for this kind of fluid only a material constantμ (i.e., viscosity coefficient) is enough to determine its properties.If the fluid is incompressible, thenVkk = 0, and the constitutive equation of incompressible viscous fluid can be obtained:(8)σij=−pδij+2μVij.Ifμ = 0, the constitutive equation of inviscid fluid is obtained:(9)σij=−pδij.The appearance of hydrostatic pressureP marks the fundamental difference between hydrodynamics and elasticity. To accommodate the needs of this new variable, it is usually assumed that there exists some equation of state, which gives the relationship between density ρ, pressure P, and absolute temperature T, that is(10)fp,ρ,T=0.


## 3.3. Finite Element Model

The bird strike setup is conducted based on the experiment of the bird-impact plate in Figure3 by using LS-DYNA element code.Figure 3
The finite element model: (a) top view; (b) front view.
(a)(b)The flat plate fixed along the four edges is impacted by a soft body at the center of the flat plate at 91.1 m/s with an impact angle of 30°. The dimensions of the flat plate are 525 mm × 430 mm × 10 mm. The geometric model of a bird is simulated by a cylinder with a hemisphere at the two ends, the diameter of the hemisphere is 53 mm and the length of the cylinder is 156 mm. The mass of the bird is 1.8 kg.In the finite element model, the 8-node solid and 164-element type in the LS-DYNA element library is employed for the bird and flat plate. In addition, hourglass control and single-point integration are also used. The numbers of the elements for the bird and flat plate model are 4504 and 2000.


## 3.4. Materials Properties

### 3.4.1. Materials Properties of Flat Plate

The plate is made from LY12-CZ aluminum alloy. The material parameters of the flat plate are mentioned in Table1. The material properties of the flat plate are modeled by the MAT_PLASTIC_KINEMATIC in LS-DYNA code where the values are set in Table 1.Table 1
Material parameters of flat plate.
Young’s modulus (GPa)Poisson ratioDensity (kg/m3)Yield stress (MPa)Failure strain710.327806900.3


### 3.4.2. Material Properties of Bird

Generally speaking, a trend is visible towards the utilization of SPH models with an equation of state for the material of bird using the properties of water–air-mixture at room temperature. A very good correlation to experimental results could be obtained with these models. For the fluid model, in LS-DYNA, the material model of NULL (empty material) is used to describe its material properties. The parameters to be input are shown in Table2.Table 2
Parameters of NULL model.
ROMass densityPCPressure cutoffMUViscosity coefficientTERODRelative vol. in tensionCERODRelative vol. in compressionYMYoung’s modulusPRPoisson’s ratioAccording to the property of the bird in the experiment, using fluid constitutive and Gruneisen state equation to describe soft body is closer to the reality. The bird model is defined by MAT_NULL and the equation of state (EoS) is specified. The inputs of parameters are shown in Table3.Table 3
Parameters of Gruneisen model.
CS1S2S3GAMA0AE0 (J)V016702.48000001During the impact process, the deformation of the bird is described by the equation of state. Elastic modulus and passion ratio are 0, and the density is 938 kg/m3. The four parameters of the constitutive model of the bird to be determined are shown in Table 4.Table 4
Parameters to be determined.
PCPressure cutoffMUViscosity coefficientTERODRelative vol. in tensionCERODRelative vol. in compression


## 3.4.1. Materials Properties of Flat Plate

The plate is made from LY12-CZ aluminum alloy. The material parameters of the flat plate are mentioned in Table1. The material properties of the flat plate are modeled by the MAT_PLASTIC_KINEMATIC in LS-DYNA code where the values are set in Table 1.Table 1
Material parameters of flat plate.
Young’s modulus (GPa)Poisson ratioDensity (kg/m3)Yield stress (MPa)Failure strain710.327806900.3


## 3.4.2. Material Properties of Bird

Generally speaking, a trend is visible towards the utilization of SPH models with an equation of state for the material of bird using the properties of water–air-mixture at room temperature. A very good correlation to experimental results could be obtained with these models. For the fluid model, in LS-DYNA, the material model of NULL (empty material) is used to describe its material properties. The parameters to be input are shown in Table2.Table 2
Parameters of NULL model.
ROMass densityPCPressure cutoffMUViscosity coefficientTERODRelative vol. in tensionCERODRelative vol. in compressionYMYoung’s modulusPRPoisson’s ratioAccording to the property of the bird in the experiment, using fluid constitutive and Gruneisen state equation to describe soft body is closer to the reality. The bird model is defined by MAT_NULL and the equation of state (EoS) is specified. The inputs of parameters are shown in Table3.Table 3
Parameters of Gruneisen model.
CS1S2S3GAMA0AE0 (J)V016702.48000001During the impact process, the deformation of the bird is described by the equation of state. Elastic modulus and passion ratio are 0, and the density is 938 kg/m3. The four parameters of the constitutive model of the bird to be determined are shown in Table 4.Table 4
Parameters to be determined.
PCPressure cutoffMUViscosity coefficientTERODRelative vol. in tensionCERODRelative vol. in compression


## 3.5. Identification of the Bird Model and the Inversion of the Parameters

The displacements at 12 different times are chosen as the inputs, and the outputs of the model are four constitutive parameters of the soft body. The 12 different times are based on the experimental displacement-time data of the center of the flat plate. The experimental displacement-time data of the center of the flat plate is given in Figure4.Figure 4
Experimental displacement-time data of the center of the flat plate.Before the network training, both input and output variables are normalized within the range 0 to 1 in order to obtain a reliable form for the neural network[31]. If the number of hidden layers is too small, the trained network might not have sufficient ability to learn the process correctly. As a result, in order to choose the appropriate number of hidden layers, the trial-and-error procedure is started with seven neurons in the hidden layer and further carried out with up to fourteen neurons. It is found that when the number of neurons is 9, the mean square error reaches the minimum.In the bird constitutive model, the values of the four parameters have their ranges based on the prior knowledge, specific ranges (see Table5).Table 5
The value range of the four parameters.
Optimize parametersLabelRanges of valuesUnitsPCX1[−0.05, −1.25]MPaMUX2[0, 1.5]N·ms/mm2TERODX3[0.05, 1.25]—CERODX4[0.05, 1.25]—The parameters to be optimized are uniformly divided into a number of values with the space. According to the orthogonal design and uniform design principle, these values are divided into 25 groups, and each sample group contains four parameters, which is shown in Table6.Table 6
Parameters in sample groups.
Sample groupsX1X2X3X41−0.0500.051.252−0.10.10.11.23−0.150.20.151.154-0.20.30.21.15−0.250.40.251.05……..……..……..……..……..Substituting these parameters of 25 groups into the bird constitutive model, 25 displacement-time curves of the center of the flat plate can be obtained in LS-DYNA simulation. Selecting the displacements of the center of the flat plate, the training of the network can be finished, which is shown in Table7.Table 7
Sample groups.
Time/msSample groups (Displacement/mm)12345……..0.340.1230640.1231650.1233330.1228450.121246……..0.591.2203371.2168671.2162241.2139891.232132……..0.712.0431142.0481792.0481942.0486312.060089……..0.984.1002854.1124174.1287764.1530354.19781……..1.094.6681754.6868354.6894324.7098934.751519……..1.345.3558445.3764235.3949585.4172525.453059……..1.535.5765615.6081545.6288955.6520065.696968……..1.605.5374795.5719925.6057185.6461245.67513……..1.794.847064.8951244.931184.9527084.973954……..1.914.1848244.2275174.2557954.279694.277913……..2.113.2423673.2604733.2738623.2629283.225929……..2.352.134912.123192.1140012.0767032.046814……..When developing the ANN model, the samples of 1, 5, 10, 15, 20, and 25 are removed to test the generalization capability of the network. The optimized ANN model consists of 12 input neurons, 4 output neurons, and a single hidden layer (with 9 neurons) and the transfer functions of the optimized ANN model are ‘tan sigmoid’ and ‘pure linear’. A feedforward backpropagation algorithm is selected to train the network. The setting of other training parameters for the neural network is listed in Table8.Table 8
The setting of training parameters for network.
Name of parametersContentsNetworkBackpropagationTraining functionTrainlmPerformance functionMSETraining epoch2,000Goal0.001


## 4. Results and Discussion

The performance of ANN in the training stage is shown in Figure5. It can be seen that the trend of error reduction remains unchanged after 2000 training cycles. After being trained, the ANN model can map the nonlinear relationship between the parameters of the bird constitutive model and the displacements of the center of the flat plate.Figure 5
Training convergence curve of the neural network model.The optimized parameters can be obtained, as can be seen in Table9. The comparison of the numerical and experimental displacement of the center of the flat plate using the optimized constitutive model of bird is shown in Figure 6, indicating the simulation result using the parameters obtained from the developed network is in good agreement with experimental data. In order to verify the identification of the constitutive model after optimization, a comparative evaluation of three other constitutive equations and the present bird constitutive model is carried out. The parameters and stress distributing graphs of elastic model, plastic kinematic model, and elastic-plastic hydrodynamic Model are shown in Tables 10–12 and Figure 7, respectively, and the parameters of EOS_LINEAR_POLYNOMIA are shown in Table 13. As can be seen in Figure 8, the model of the bird remains in good condition while the flat plate with obvious deformation in the elastic model. The result of the plastic kinematic model and the elastic-plastic hydrodynamic model show a similar phenomenon that the model of bird has large deformation while the flat plate with little, and the fringe levels drop significantly.Table 9
The results of parametric inversion.
Optimized parametersLabelRange of valuesOptimized resultsPCX1[−0.05, −1.5]−0.3705MUX2[0, 1.5]0.641TERODX3[0.05, 1.25]0.34CERODX4[0.05, 1.25]0.9295Figure 6
Comparisons of the numerical displacements among different constitutive models.Table 10
Parameters of elastic model.
Elastic modulus/PaPoisson ratioDensity/kg/m36.89 × 1070.49938Table 11
Parameters of plastic kinematic model.
Elastic modulus/PaPoisson ratioDensity/kg/m3Yield stress/PaFailure strain6.89 × 1070.4993868941.25Table 12
Parameters of elastic-plastic hydrodynamic model.
Density/kg/m3Shear modulus/MPaHardening modulus /MPaPressure cutoff/MPaYield stress/MPaFailure strain93810.20.13–0.90.030.95Figure 7
Stress distributing graphs of the top view and front view at 1.5 ms time of impact: (a) elastic model; (b) plastic kinematic model; (c) elastic-plastic hydrodynamic model.
(a)(b)(c)Table 13
Eos linear polynomial.
C0C1C2C3C4C5C6E0V00207262171036200001Unit: (C0∼C6 , N/mm2; E0, J; V0, no unit).Figure 8
The comparison of numerical and experimental displacement of the center of the flat plate.During the impact process, different bird constitutive models result in different displacement and force-time histories of the center of the flat plate. The maximum displacement and the force of the center of the flat plate of the elastic constitutive model are the largest among the four constitutive models, in which the deviations from experimental results are also the largest. Figure9 shows that the present bird constitutive model developed from the present network is in better agreement with experimental data than the other three constitutive models.Figure 9
Comparisons of the force-time curves among different models.


## 5. Conclusion

In this paper, the artificial neural network model with a backpropagation learning algorithm was used to develop the constitutive relationship for the soft body during the impact process. The feasibility and practicability of this identification method were verified through a specific example. From the results of this present study, the main conclusions are drawn as follows:(1)
Different flexible body constitutive models are used, and the impact of the soft body presents different modes, and the displacement curve and impact force curve of the aluminum plate center is also different. Among them, the displacement and impact force of the linear elastic constitutive model is the largest, but the deviation from experimental results is also the largest.(2)
The calculated results of the soft body constitutive parameters optimized by the improved BP neural network algorithm are in good agreement with the experimental results. At the same time, it can be seen that the calculation curve optimized by BP neural network algorithm almost passes all the test points, which proves the accuracy and reliability of the optimized flexible body parameters, and the calculation convergence accuracy also meets the requirements.(3)
As a calculation method, BP neural network can be used in the optimization calculation of constitutive parameters of soft body impact. However, it should be pointed out that each constitutive model needs to build its own neural network when it is used for calculation, and the optimization results and errors will be different due to the difference of network structure.



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