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

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.

This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

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.$$\begin{array}{}\text{(1)}& fx=\frac{1}{1+\exp -x}.\end{array}$$

[figure omitted; refer to PDF]

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 Figure 2.

[figure omitted; refer to PDF]

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 Table 5).

Table 5

The value range of the four parameters.

Optimize parameters	Label	Ranges of values	Units
PC	X1	[−0.05, −1.25]	MPa
MU	X2	[0, 1.5]	N·ms/mm2
TEROD	X3	[0.05, 1.25]	—
CEROD	X4	[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 Table 6.

Table 6

Parameters in sample groups.

Sample groups	X1	X2	X3	X4
1	−0.05	0	0.05	1.25
2	−0.1	0.1	0.1	1.2
3	−0.15	0.2	0.15	1.15
4	-0.2	0.3	0.2	1.1
5	−0.25	0.4	0.25	1.05
……..	……..	……..	……..	……..

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 Table 7.

Table 7

Sample groups.

Time/ms	Sample groups (Displacement/mm)
	1	2	3	4	5	……..
0.34	0.123064	0.123165	0.123333	0.122845	0.121246	……..
0.59	1.220337	1.216867	1.216224	1.213989	1.232132	……..
0.71	2.043114	2.048179	2.048194	2.048631	2.060089	……..
0.98	4.100285	4.112417	4.128776	4.153035	4.19781	……..
1.09	4.668175	4.686835	4.689432	4.709893	4.751519	……..
1.34	5.355844	5.376423	5.394958	5.417252	5.453059	……..
1.53	5.576561	5.608154	5.628895	5.652006	5.696968	……..
1.60	5.537479	5.571992	5.605718	5.646124	5.67513	……..
1.79	4.84706	4.895124	4.93118	4.952708	4.973954	……..
1.91	4.184824	4.227517	4.255795	4.27969	4.277913	……..
2.11	3.242367	3.260473	3.273862	3.262928	3.225929	……..
2.35	2.13491	2.12319	2.114001	2.076703	2.046814	……..

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 Table 8.

Table 8

The setting of training parameters for network.

Name of parameters	Contents
Network	Backpropagation
Training function	Trainlm
Performance function	MSE
Training epoch	2,000
Goal	0.001

4. Results and Discussion

The performance of ANN in the training stage is shown in Figure 5. 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 omitted; refer to PDF]

The optimized parameters can be obtained, as can be seen in Table 9. 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 parameters	Label	Range of values	Optimized results
PC	X1	[−0.05, −1.5]	−0.3705
MU	X2	[0, 1.5]	0.641
TEROD	X3	[0.05, 1.25]	0.34
CEROD	X4	[0.05, 1.25]	0.9295

[figure omitted; refer to PDF]

Table 10

Parameters of elastic model.

Elastic modulus/Pa	Poisson ratio	Density/kg/m3
6.89 × 107	0.49	938

Table 11

Parameters of plastic kinematic model.

Elastic modulus/Pa	Poisson ratio	Density/kg/m3	Yield stress/Pa	Failure strain
6.89 × 107	0.49	938	6894	1.25

Table 12

Parameters of elastic-plastic hydrodynamic model.

Density/kg/m3	Shear modulus/MPa	Hardening modulus /MPa	Pressure cutoff/MPa	Yield stress/MPa	Failure strain
938	10.2	0.13	–0.9	0.03	0.95

[figures omitted; refer to PDF]

Table 13

Eos linear polynomial.

C0	C1	C2	C3	C4	C5	C6	E0	V0
0	2072	6217	10362	0	0	0	0	1

Unit: ($C_0\sim C_6$ , N/mm²; $E_0$, J; $V_0$, no unit).

[figure omitted; refer to PDF]

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. Figure 9 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 omitted; refer to PDF]

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.

Acknowledgments

This study was funded by National Natural Science Foundation (12102095), the Scientific Research Project for Young Innovative Talents of Guangxi Province (20200312), Research grant for Talent of Guangxi Plan (100), the starting research grant for High-level Talents from Guangxi University, the Science and Technology Major Project of Guangxi Province (AA18118055), the Guangxi Natural Science Foundation (20181096), and application of key technology in building construction of prefabricated steel structure (20190528).

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