Metal materials are widely used in weaponry, aerospace and transportation fields. The equation of state at high temperature and pressure is the basis for studying properties of metal materials. The equation of state is an equation that expresses the functional relationship between the independent parameters of the equilibrium macroscopic properties of a uniform system and other state parameters [1, 2]. It generally refers to the functional relationship between system pressure P, temperature T (or internal energy E) and volume (or specific volume) V. Metal materials will fail or suffer serious damage under high temperature and high pressure environment, which has a certain relationship with their performance. The performance research of metal materials under high temperature and pressure environment mainly depends on the equation of state to judge the law of performance change. The Grüneisen equation of state can be combined with finite element software to study the change of material properties under extreme conditions, so its application is very wide [3]. Therefore, it is of great significance to explore the identification method of parameters of Grüneisen equation of state.

GH4169 nickel-based superalloy was selected as the research object with two reasons. First, GH4169 alloy is mainly used in the hot end parts of industrial gas turbines and aero-engines, and its working environment is usually high temperature, high pressure and impact load. Its performance is related to the life cycle and safety of the product. Therefore, obtaining the parameters of the Grüneisen equation of state has important meaning for the application of the material in the aerospace field. Second, as an alloy material containing nearly ten kinds of elements, GH4169 nickel-based superalloy is extremely difficult to develop its potential function, and there are many precipitation phases in the alloy. Therefore, a Grüneisen state equation parameter identification method that can break through the restriction of the potential function is needed.

With the increasing demand for complex alloys with excellent properties in the fields of aerospace and weaponry, the circumstances or environment where we use metal materials has become more demanding. For example, an aero engine is always in a state of high temperature and high pressure during the working state. Therefore, the impact of alloy properties under extreme service environments must be considered when selecting materials.

Both isothermal and adiabatic equations of state have made great progress. There are two types of isothermal state equations: one is to draw a cold curve (0 K), and the other is at 300 K. Bridgman used a static compression device when studying the compression properties of materials under high pressure. Studies have shown that the heat generated inside the material can achieve complete exchange with the outside air due to the very slow compression process. Therefore, this compression process could be approximated as isothermal compression [4]. Many scholars have conducted high-pressure isothermal compression studies on materials [5,6,7,8]. Syassen and Holzapfel [9] used 12 GPa x-ray diffraction to study the isothermal compression process and curve changes of Ag and Al. In their research, the Birch equation was used as an isothermal state equation to compare with experimental data. Vočadlo et al. [10] and Shanker et al. [11] have explored issues of isothermal body modulus and first-order pressure derivative by comparing various isothermal state equations. Hrubiak et al. [12] explored the isotherm of Hf at room temperature. They obtained the P(E, V)-type equation of state for

Adiabatic equations of state are usually based on Hugoniot data. Hugoniot data comes from a flying debris impact experiment powered by a light airgun or explosive. Al’e Tshuler et al. [14] studied the equations of state of Al, Cu and Pb at high pressure environment. Bringa et al. [15] and Agarwal et al. [16] explored the linear relationship between the downward shock velocity and particle velocity of different crystals by analysing the non-equilibrium molecular dynamics (NEMD) of the adiabatic shock line of Mg and Cu single crystals. Mackenchery and Dongare [17] analysed the impact of four different atomic potentials on Ti single crystal at collision velocities and Hugoniot curves of approximately 0.5–2.0 km/s based on molecular dynamics (MD).

As a solid model under high pressure, Mie-Gruneisen equation of state has been widely used in various fields. In many simulation software, the parameters of Mie-Gruneisen equation of state could be directly inputted to define the material. Walsh and Christian [18] Walsh et al. [19] studied the equation of state of metals through a blasting system, a measuring volume dependence for Gruneisen coefficient

For isotherms, Hugoniot curve and Mie-Grüneisen equation of state have been studied extensively. The types of materials studied mainly focus on single element and binary alloys, while the research on multi-element alloys is relatively rare. As we all know, the physical properties of composite alloys are very different from the physical properties of their constituent elements. They are usually used in more complex environments. Therefore, it is of great significance to explore dynamic mechanical properties of complex alloys under extreme conditions. However, due to the lack of corresponding potential functions, parameters of Mie-Gruneisen equation of state for multi-element alloys are difficult to obtain through molecular dynamics simulations. This research proposes a parameter identification method for calculating parameters of Mie-Gruneisen equation of state for multi-element alloys according to Ji et al. [26] and Millett et al. [27].

Due to the addition of a variety of elements to improve various aspects of performance, multi-element alloys have a wider range of applications than general metal materials. Due to the excellent performance of multi-element alloys, the service environment is generally extreme. The study of the equation of state mainly relies on theoretical analysis, experiment and molecular dynamics simulation. However, the experiment will cost too much. Equilibrium dynamics is not suitable for dynamic analysis, and it takes a lot of time to calculate large-scale models for non-equilibrium dynamics. Reed et al. proposed Multi-scale Shock Technique (MSST) that is not only overcome these problems, but also reduce the model size while maintaining the simulation accuracy, thereby effectively reducing the calculation time [33].

MSST refers to the molecular dynamics simulation of a small part of the shock wave at a given transient time. Adjusting the applied stress and energy can be used to simulate the shock wave effect of this small molecule dynamics system. These stresses and energies satisfy the continuity theory describing the shock wave structure [30, 31]. This method ensures that the thermodynamic path is correctly captured, and the (P, T) thermodynamic state simulated by the shock wave is used to obtain a result consistent with the stable macroscopic shock wave. Compared with non-equilibrium molecular dynamics, MSST significantly reduces the number of particles and ensures that the simulation correctly converges to the thermodynamic state [32]. MSST makes it possible to simulate with fewer atomic shock waves, which is of great significance. According to Ji et al. [26] and Millett et al. [27], we propose a method for parameter identification of the Grüneisen equation of state for multi-element alloys.

The chemical composition of GH4169 alloy is a complex one, and the mass fraction of many trace metal elements is <1%. Therefore, we have simplified the composition of GH4169 to a certain extent. First, retain the metal elements with a mass fraction >1%. The elements that meet this condition are Ni, Cr, Fe, Nb, Mo and Ti. Secondly, because GH4169 nickel-based alloy has

Mass fraction and atomic fraction of each element of GH4169 alloy

Ni | 53.40 | 52.40 |

Cr | 18 | 20.80 |

Fe | 18.20 | 19.50 |

Mo | 3.50 | 1.40 |

Nb | 4.80 | 2.60 |

Ti | 1.50 | 1.90 |

Al | 0.60 | 1.40 |

Molecular dynamics (MD) is a numerical simulation method for studying scientific problems at the atomic scale [29]. In molecular dynamics simulation, the motion of all particles satisfies Newton's second law. By solving the Newtonian differential equation of motion, the velocity and position of any atom in the system can be obtained.

_{i}_{i}_{i}_{i}_{i}

Any particle in the system satisfies at time

After determining the potential function used by the research body, the trajectory of each atom can be obtained by analysing the above Newton equation, and the coordinates, acceleration, particle velocity and other information at any time can be obtained. Then, based on the obtained information, the macroscopic physical quantities of the system (such as pressure, energy, temperature, radial distribution function) can be calculated through statistics and averaging, and the relevant properties of the material can be explained.

The size of all models is 20a×20a×20a, where a is the lattice constant of a single unit cell. In each direction, periodic boundary conditions are used. Before loading the shock, relax the model in the NPT system by 40 ps at 300 K, making it in a state where the residual stress can be ignored. After relaxation, a shock wave is applied in the X-axis direction of the model through MSST. The wave speed of Ni is 11–15 km/s, the wave speed of Ni_{3}Nb is 10–18 km/s, the wave speed of Ni_{3}Ti is 6–10 km/s, the wave speed of Ni_{3}Al is in the range is 6–10 km/s, the wave speed range of Cr is 11–15 km/s, the wave speed range of Fe is 6–10 km/s and the wave speed range of Mo is 6–10 km/s. The particle's velocity, pressure, temperature and other physical quantities will stabilise when the simulation lasts for a long enough time to extract the required data from the data in the steady state.

_{0}and

In GH4169 alloy, the _{3}Nb, which is the main strengthening phase. The _{3}Al and Ni_{3}Ti, which are auxiliary strengthening phases; Ni is the _{0} and

Mass fraction and Hugoniot parameters of each phase in GH4169 alloy

_{0} |
|||
---|---|---|---|

Ni | 35 | 4.4133 | 1.1419 |

Cr | 20 | 4.9712 | 1.1196 |

19 | 3.8327 | 2.2667 | |

Mo | 5.00 | 5.9806 | 0.9444 |

_{3}Nb |
12.60 | 5.5252 | 2.6675 |

_{3}Ti |
4.80 | 5.6461 | 1.0397 |

_{3}Al |
3.60 | 5.3751 | 1.068 |

Table 3 shows the C_{0} and _{0} and _{0} and

Hugoniot parameters and relative error of GH4169 alloy

_{0} |
||
---|---|---|

Hugoniot parameters in this research | 4.532 | 1.62 |

Liu et al. [34] | 4.485 | 1.45 |

Relative error | 0.70% | 15.60% |

Figures 1 and 2 show GH4169 alloy's Hugoniot curve and internal energy curve, respectively. The research results of Liu et al. are consistent with the calculation results of this research, which shows that the calculation results of this paper are reliable.

Through molecular dynamics simulation of the heating process, substituting the obtained volume data into the related formula, the volume expansion coefficient ^{−6}K^{−1}. After that, the 0 K isotherms of Born-Meyer potential and Morse potential and their internal energy curves are obtained by calculation. The value of Q, q, A and B are 70.55 GPa, 11.55, 151.52 GPa and 4.45, respectively.

Figures 3 and 4 show the 0K isotherm _{c}_{c}_{c}_{c}_{0K}/V. The greater the V_{0K}/V, the greater the growth rate of _{c}_{c}_{0K}/V is <1.5, the isotherm and internal energy curves almost coincide, which is the same as the research result of Liu et al. When V_{0K}/V is >1.5, the isotherms and internal energy curves of the two potentials gradually separate. Whether it is isotherm or internal energy, Morse potential is closer to the results of Liu et al.'s research. This shows that when V_{0K}/V is <1.5, both Born-Meyer potential and Morse potential are more accurate. However, when V_{0K}/V is >1.5, the Morse potential is more accurate.

Generally, Gruneisen equation of state is used to explore the behaviour of solids and to show the contribution of thermal vibration of the lattice. It's one of the most widely used equations of state in fields of high-pressure physics and explosion mechanics. Hugoniot curve is the most commonly used reference curve for the Gruneisen equation of State [28], and its classic form is:

_{c}_{c}_{H}_{H}

The Grüneisen coefficient is a dimensionless quantity, and the expression is

When _{0}, _{S}

When

We assume that the first derivative is equal to the second derivatives of _{S}_{H}_{0}. _{0}) and _{0}) can be directly obtained by experiments or molecular dynamics simulations. Then, we can get:

Substituting Eqs (8)–(10) into Eq. (4), we can get:

The Grüneisen coefficient

Based on the above formula, the Grüneisen equation of state of the complex alloy can be finally obtained.

The Ni, Cr, Fe and Mo phases are simple phases with simple crystal structures. Ni is FCC structure. Figure 5(a) shows its crystal structure. Figure 5(b) shows the BCC crystal structure of Cr, Fe and Mo. Figure 5(c) shows the Ni_{3}N_{b} crystal's structure, which is the PMMN space group in the orthorhombic system. As the P63/MMC space group in the hexagonal system, Figure 5(d) presents the structure of Ni_{3}Ti. Figure 5(e) shows the structure of Ni_{3}Al, which is the PM-3M space group in the orthorhombic system.

Using the PBE functions in the generalised gradient approximation (GGA), the unit cells of the seven stable phases in the GH4169 alloy are optimised. On the basis of optimising the unit cell, the energy under different volumes was calculated, and the E-V curve was obtained. Calculation of the Grüneisen coefficient

Mass fraction and Grüneisen coefficient of each phase in GH4169 alloy

Ni | 38 | 2.4454 |

Cr | 19 | 1.9647 |

18 | 2.2604 | |

Mo | 3.50 | 2.2604 |

_{3}Nb |
13.50 | 2.2112 |

_{3}Ti |
4.90 | 2.1526 |

_{3}Al |
3.40 | 2.1586 |

Table 5 shows the Grüneisen coefficients of GH4169 alloy obtained by fitting. By comparison, it is found that the Grüneisen coefficient obtained from the theoretical calculation proposed in this study is far from the Grüneisen coefficient obtained from the flat plate impact experiment of the first-level light gas gun. Its correctness will be verified in subsequent experiments and numerical simulations.

Grüneisen coefficient and relative error of GH4169 alloy

Grüneisen coefficient | |
---|---|

Grüneisen coefficient in this research | 2.4 |

Liu et al. [34] | 1.82 |

Relative error | 23.6% |

Figures 6 and 7 show the 3D surface of Grüneisen equation of state.

Figures 6 and 7 are the equations of state respectively obtained from Eqs (18) and (19) as Grüneisen coefficients. Although viewed from the front, the two Grüneisen equations of state in P-V-E space are both concave in shape and similar in shape, but it can be found that the maximum pressures are not the same. This is caused by the difference in the Grüneisen coefficient equation. Therefore, the Grüneisen coefficient is a factor that affects the pressure value of Grüneisen equation of state.

In order to further verify the correctness of this set of parameters, a plate collision experiment was carried out. By extracting the GH4169 target free surface particle speed history, the flying target and the peeling target are restored. Using the state parameter Grüneisen equation calculated in the previous section, the collision process of the plate collision experiment was numerically simulated and compared with the speed of free surface particles in the experiment. At the same time, the collision process corresponding to the speed history of the free surface particles in the numerical simulation was studied, and the morphology of the recovered flying piece and the target piece after peeling were compared. Through data comparison and morphological comparison, the accuracy of the Grüneisen state parameter equation calculated for GH4169 alloy is judged.

The plate impact experiment is mainly used to study the high-pressure state equation of the material and to obtain parameters through the high-speed impact of flying plate on the material sample. The shock wave reaches both the free surface of the flyer and the free surface of the target from the impact surface at the same time, so that the material is in a high pressure state. When the shock wave reaches the free surface, it will produce sparse waves in the opposite direction, so that the free surface maintains the ‘zero pressure’ boundary condition. When the sparse wave from the free surface of flyer meets that from target plate, tensile stress is generated.

After the stress meets certain conditions, cracks will be generated inside the target sheet, and the crack edges will become new ‘The free side.’ When the sparse wave encounters the edge of the crack, a compression wave is emitted in the reverse direction. After the compression wave reaches the free surface, a reverse sparse wave is generated again. With the repeated generation of sparse waves and compression waves, the speed of free surface particles exhibits an oscillation form of rising and falling. It also causes the cracks to expand continuously, and finally leads to the formation of layers in the sample material.

The high-voltage device should meet two basic requirements. First, it should be able to adjust the load pressure range to obtain the Hugoniot curve from low pressure to high pressure. Second, its shock wave must have a certain degree of flatness.

In this experiment, GH4169 alloy's free surface velocity was measured by all-fibre displacement interferometer, as well as the internal state of the sample during impact, the Hugoniot elastic limit (HEL), and spallation.

When measuring the impact insulation line, the flat impact test piece needs to meet two conditions: (1) the shock wave must be a plane wave and (2) the shock wave should be uniform, that is, the parameters behind the wave front do not change with time and distance.

The experimental material is GH4169 nickel-based superalloy. It adopts a 37 mm calibre first-class light gas gun. The designed flyer diameter is 35 mm, flyer's thickness is 2 mm, the diameter of target specimen is 30 mm and the designed thickness of the target plate is 3.75 mm.

This experiment mainly adopts a symmetrical collision configuration, and the optical fibre probe and AFDISAR are used to directly measure free surface's velocity distribution on the back of the target sample.

A total of one plate impact experiment was carried out, and the effective data of the free surface particle velocity curve was one time. The symmetrical collision configuration was adopted, the thickness of the flyer was about 2.00 mm, and the actual impact velocity range was 545 m/s. Under the condition of plate impact, the specimen is in a one-dimensional strain state, and there is circumferential hydrostatic pressure. Therefore, the relationship between the Hugoniot elastic limit (HEL) _{H}_{0} of the material under the one-dimensional stress state is given by the equation
_{H}_{H}_{fm}_{HEL}

_{H}

For GH4169 alloy, a total of one test was performed, and one valid data was obtained. The analysis results are shown in Table 6, and the impact velocity is 535 m/s. Figure 8 shows the free surface particle velocity results of one test. It shows that it's a ‘elastic precursor wave-plastic wave’ for the shock response of the GH4169 alloy.

Experimental results

Ni-1 | 535 | 518 | 9.916 | 80 | 38 | 1.498 |

Based on the experimental curve, and using formula _{H}_{HEL} YH is calculated, as shown in Table 6.

The nonlinear dynamics programme AUTODYN is used for simulation analysis. Set the material model with the density of GH4169 alloy as 8.24 g/cm^{3}. The J-C constitutive model is used as the constitutive model in this paper. Tables 7 and 8 show the parameters of the state equation and constitutive equation, respectively. The parameters

Grüneisen equation of state parameters of GH4169 alloy

4526 | 1.46 | 0 | 0 | 2.2 | 0.46 |

Johnson-Cook constitutive model parameters of GH4169 alloy

1306 | 1008 | 0.01 | 0.460 | 1.08 |

The numerical analysis uses a two-dimensional plane model. A model with a 1:1 ratio of flyer to sample was established. In order to give the initial speed of the flyer, the material models of the flyer and the target were established, respectively. The parameters set in the two material models (flyer and target) are given in Tables 7 and 8, respectively. It can be seen from the experimental results that the flyer did not fail or corrode. Therefore, in order to save calculation time, only failure criteria and erosion are defined in the target model.

After setting the simulation conditions, add Gaussian point 1 (Figure 9) to the target model, and the position is at the centre of the back of the target, which is same as the position measured by AFDISAR in experiment. After that, the free surface particle velocity at the Gauss point can be obtained, which can be used to compare and analyse the velocity curve of free surface particle in the experiment, so as to verify parameters’ accuracy.

The numerical simulation of the plate impact experiment provides the historical curve of the Gauss point 1 of the target model. Compare the curve obtained by numerical simulation with the curve obtained by experiment (Figure 10). The black curve in Figure 10 represents the speed of the free surface particles obtained through the plate collision experiment; the red curve represents the speed of the free surface particles obtained through the numerical simulation. It can be seen that the two curves have platforms at the first wave crest. Both curves have five complete waveforms in a time of 5 μm. From the last four waveforms, the particle velocity at the trough is basically the same. The velocity of the peak particles in the plate impact experiment dropped rapidly, while the velocity of the peak particles in the numerical simulation decreased slowly, causing the gap between the peak velocities to gradually increase. In addition, the disparity between the time points at which the velocity reaches the trough of the two curves keeps increasing, indicating that in the numerical simulation, the time of shock wave to propagate from free surface to spalling surface and then bounce back to the free surface is longer than in the actual plate impact experiment. Compared with the actual test results, the thickness of the slab obtained by the numerical simulation has a certain error.

The numerical simulation results of the spallation process of the plate collision experiment are shown in Figure 11. During the impact, the target was shot down. A new lamella was split on the lamina on the impact side. In the sample recovered after the plate impact experiment, flakes peeled off from the inside of the sample were also found. The experimentally recovered flyers and the non-impact side slabs have a concave centre and a conical shape as a whole. In the numerical simulation, the flyers and the non-impact side slabs also have the same deformation, which is consistent with the experimental phenomenon.

The comparison of the above data and the morphology shows that the numerical simulation results of the flyer impact experiment are consistent with experimental results. It proves that parameters of Grüneisen equation of state calculated above are consistent with the actual situation.

A parameter identification method of the Grüneisen equation of state for multi-element alloys is proposed. Taking GH4169 alloy as an example, parameters of Grüneisen equation of state are calculated and compared with the existing literature. When the parameters of numerical simulation and actual experiment are compared, it verified the accuracy of parameters of Grüneisen equation of state of GH4169 alloy calculated by the proposed method.

According to molecular dynamics, Hugoniot parameters C_{0} and

Although this research has achieved certain results, the application scope of the Grüneisen equation of state parameter calculation method for multi-element alloys proposed in this paper is still unclear as we have done only for GH4169 alloy. The parameter identification of Grüneisen equation of state for other metal materials needs to be explored in the future to further expand the scope of application of this method.

#### Grüneisen coefficient and relative error of GH4169 alloy

Grüneisen coefficient | |
---|---|

Grüneisen coefficient in this research | 2.4 |

Liu et al. [ |
1.82 |

Relative error | 23.6% |

#### Grüneisen equation of state parameters of GH4169 alloy

4526 | 1.46 | 0 | 0 | 2.2 | 0.46 |

#### Mass fraction and Hugoniot parameters of each phase in GH4169 alloy

_{0} |
|||
---|---|---|---|

Ni | 35 | 4.4133 | 1.1419 |

Cr | 20 | 4.9712 | 1.1196 |

19 | 3.8327 | 2.2667 | |

Mo | 5.00 | 5.9806 | 0.9444 |

_{3}Nb |
12.60 | 5.5252 | 2.6675 |

_{3}Ti |
4.80 | 5.6461 | 1.0397 |

_{3}Al |
3.60 | 5.3751 | 1.068 |

#### Mass fraction and Grüneisen coefficient of each phase in GH4169 alloy

Ni | 38 | 2.4454 |

Cr | 19 | 1.9647 |

18 | 2.2604 | |

Mo | 3.50 | 2.2604 |

_{3}Nb |
13.50 | 2.2112 |

_{3}Ti |
4.90 | 2.1526 |

_{3}Al |
3.40 | 2.1586 |

#### Hugoniot parameters and relative error of GH4169 alloy

_{0} |
||
---|---|---|

Hugoniot parameters in this research | 4.532 | 1.62 |

Liu et al. [ |
4.485 | 1.45 |

Relative error | 0.70% | 15.60% |

#### Mass fraction and atomic fraction of each element of GH4169 alloy

Ni | 53.40 | 52.40 |

Cr | 18 | 20.80 |

Fe | 18.20 | 19.50 |

Mo | 3.50 | 1.40 |

Nb | 4.80 | 2.60 |

Ti | 1.50 | 1.90 |

Al | 0.60 | 1.40 |

#### Johnson-Cook constitutive model parameters of GH4169 alloy

1306 | 1008 | 0.01 | 0.460 | 1.08 |

#### Experimental results

Ni-1 | 535 | 518 | 9.916 | 80 | 38 | 1.498 |

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Collaborative Filtering Personalized Recommendation System Based on Linear Regression Equation Study on Interactive Relations between Enterprise Social Media and Decision Style Based on a vector Autoregressive Model Research on Dynamics of Flexible Multibody System with Deployable Antenna Based on Static Lagrangian Function The Optimization of Mathematics Teaching Models in Colleges and Universities Based on Nonlinear Differential Equations Study on spatial planning and design of learning commons in university libraries based on fuzzy matrix model The Stability Model of Piano Tone Tuning Based on Ordinary Differential Equations Construction and application of automobile user portrait based on k-mean clustering model The Homework Model of Screening English Teaching Courses Based on Fractional Differential Equations Research on a method of completeness index based on complex model Fake online review recognition algorithm and optimisation research based on deep learning Research on the sustainable development and renewal of Macao inner harbour under the background of digitisation Support design of main retracement passage in fully mechanised coal mining face based on numerical simulation Study on the crushing mechanism and parameters of the two-flow crusher Interaction design of financial insurance products under the Era of AIoT Modeling the pathway of breast cancer in the Middle East ARMA analysis of the green innovation technology of core enterprises under the ecosystem – Time series data Image design and interaction technology based on Fourier inverse transform Calculus Logic Function in Integrated Manufacturing Automation of Single Chip Microcomputer Football Offense Training Strategy Based on Fractional Differential Mathematical Modeling Research on China interregional industrial transformation slowdown and influencing factors of industrial transformation based on numerical simulation The medical health venture capital network community structure, information dissemination and the cognitive proximity The Mathematical Analysis Model of Educational System in Music Courses in Colleges and Universities Continuing Education Network Data Center Model Based on Fractional Differential Mathematical Equations A generative design method of building layout generated by path Analysis of the causes of the influence of the industrial economy on the social economy based on multiple linear regression equation Research of neural network for weld penetration control Computer Art Design Model Based on Nonlinear Fractional Differential Equations The Optimization Model of Public Space Design Teaching Reform Based on Fractional Differential Equations The Approximate Solution of Nonlinear Vibration of Tennis Based on Nonlinear Vibration Differential Equation Graphical Modular Power Technology of Distribution Network Based on Machine Learning Statistical Mathematical Equation Employment and Professional Education Training System of College Graduates Based on the Law of Large Numbers Economic Research on Multiple Linear Regression in Fruit Market inspection and Management Nonlinear Differential Equations in Preventing Financial Risks Lagrange’s Mathematical Equations in the Sports Training of College Students Simulation Research of Electrostatic Precipitator Power Supply Voltage Control System Based on Finite Element Differential Equation Research on the effect of generative adversarial network based on wavelet transform hidden Markov model on face creation and classification Research on Lightweight Injection Molding (CAE) and Numerical Simulation Calculate of New Energy Vehicle Power Flow Calculation in Smart Distribution Network Based on Power Machine Learning Based on Fractional Differential Equations Demonstration of application program of logistics public information management platform based on fuzzy constrained programming mathematical model Basketball Shooting Rate Based on Multiple Regression Logical-Mathematical Algorithm The Optimal Application of Lagrangian Mathematical Equations in Computer Data Analysis Similarity Solutions of the Surface Waves Equation in (2+1) Dimensions and Bifurcation Research on predictive control of students’ performance in PE classes based on the mathematical model of multiple linear regression equation Beam control method for multi-array antennas based on improved genetic algorithm The influence of X fuzzy mathematical method on basketball tactics scoring Optimal decisions and channel coordination of a green supply chain with marketing effort and fairness concerns Application of regression function model based on panel data in bank resource allocation financial risk management Game theoretic model for low carbon supply chain under carbon emissions reduction sensitive random demand Research on aerobics training posture motion capture based on mathematical similarity matching statistical analysis Limit cycles of a generalised Mathieu differential system Application of Sobolev-Volterra projection and finite element numerical analysis of integral differential equations in modern art design Influence of displacement ventilation on the distribution of pollutant concentrations in livestock housing Research on motion capture of dance training pose based on statistical analysis of mathematical similarity matching Application of data mining in basketball statistics Application of B-theory for numerical method of functional differential equations in the analysis of fair value in financial accounting Research on the influence of fuzzy mathematics simulation model in the development of Wushu market Study on audio-visual family restoration of children with mental disorders based on the mathematical model of fuzzy comprehensive evaluation of differential equation Difference-in-differences test for micro effect of technological finance cooperation pilot in China Application of multi-attribute decision-making methods based on normal random variables in supply chain risk management 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Exploration of Geological and Mineral Construction AdaBoost Algorithm in Trustworthy Network for Anomaly Intrusion Detection Burnout of front-line city administrative law-enforcing personnel in new urban development areas: An empirical research in China Enterprise Financial Risk Early Warning System Based on Structural Equation Model A Study on the Application of Quantile Regression Equation in Forecasting Financial Value at Risk in Financial Markets Fractional Differential Equations in the Model of Vocational Education and Teaching Practice Environment Information transmission simulation of Internet of things communication nodes under collision free probability equation Image denoising model based on improved fractional calculus mathematical equation Random Fourier Approximation of the Kernel Function in Programmable Networks The Complexity of Virtual Reality Technology in the Simulation and Modeling of Civil Mathematical Models University Library Lending System Model Based on Fractional Differential Equations Calculation and Performance Evaluation of Text Similarity Based on Strong Classification Features Intelligent Matching System of Clauses in International Investment Arbitration Cases Based on Big Data Statistical Model Evaluation and Verification of Patent Value Based on Combination Forecasting Model Financial Institution Prevention Financial Risk Monitoring System Under the Fusion of Partial Differential Equations Prediction and Analysis of ChiNext Stock Price Based on Linear and Non-linear Composite Model Calculus Logic Function in Tax Risk Avoidance in Different Stages of Enterprises The Psychological Memory Forgetting Model Based on the Analysis of Linear Differential Equations Optimization Simulation System of University Science Education Based on Finite Differential Equations The Law of Large Numbers in Children's Education Optimization System of Strength and Flexibility Training in Aerobics Course Based on Lagrangian Mathematical Equation Data structure simulation for the reform of the teaching process of university computer courses Calculating university education model based on finite element fractional differential equations and macro-control analysis Educational research on mathematics differential equation to simulate the model of children's mental health prevention and control system Analysis of enterprise management technology and innovation based on multilinear regression model Verifying the validity of the whole person model of mental health education activities in colleges based on differential equation RETRACTION NOTE Research on the mining of ideological and political knowledge elements in college courses based on the combination of LDA model and Apriori algorithm Research on non-linear visual matching model under inherent constraints of images Good congruences on weakly U-abundant semigroups Research on the processing method of multi-source heterogeneous data in the intelligent agriculture cloud platform Internal control index and enterprise growth: An empirical study of Chinese listed-companies in the automobile manufacturing industry Determination of the minimum distance between vibration source and fibre under existing optical vibration signals: a study Nonlinear differential equations based on the B-S-M model in the pricing of derivatives in financial markets Fed-UserPro: A user profile construction method based on federated learning A multi-factor Regression Equation-based Test of Fitness Maximal Aerobic Capacity in Athletes Garment Image Retrieval based on Grab Cut Auto Segmentation and Dominate Color Method Financial Risk Prediction and Analysis Based on Nonlinear Differential Equations Constructivist Learning Method of Ordinary Differential Equations in College Mathematics Teaching Multiple Effects Analysis of Hangzhou Issuing Digital Consumer Coupons Based on Simultaneous Equations of CDM Model Response Model of Teachers’ Psychological Education in Colleges and Universities Based on Nonlinear Finite Element Equations A Hybrid Computational Intelligence Method of Newton's Method and Genetic Algorithm for Solving Compatible Nonlinear Equations Pressure Image Recognition of Lying Positions Based on Multi-feature value Regularized Extreme Learning Algorithm English Intelligent Question Answering System Based on elliptic fitting equation Precision Machining Technology of Jewelry on CNC Machine Tool Based on Mathematical Modeling Application Research of Mathematica Software in Calculus Teaching Computer Vision Communication Technology in Mathematical Modeling Skills of Music Creation Based on Homogeneous First-Order Linear Partial Differential Equations Mathematical Statistics Technology in the Educational Grading System of Preschool Students Music Recommendation Index Evaluation Based on Logistic Distribution Fitting Transition Probability Function Children's Educational Curriculum Evaluation Management System in Mathematical Equation Model Query Translation Optimization and Mathematical Modeling for English-Chinese Cross-Language Information Retrieval The Effect of Children’s Innovative Education Courses Based on Fractional Differential Equations Fractional Differential Equations in the Standard Construction Model of the Educational Application of the Internet of Things Optimization research on prefabricated concrete frame buildings based on the dynamic equation of eccentric structure and horizontal-torsional coupling Optimization in Mathematics Modeling and Processing of New Type Silicate Glass Ceramics Green building considering image processing technology combined with CFD numerical simulation MCM of Student’s Physical Health Based on Mathematical Cone Sports health quantification method and system implementation based on multiple thermal physiology simulation Research on visual optimization design of machine–machine interface for mechanical industrial equipment based on nonlinear partial equations Research on identifying psychological health problems of college students by logistic regression model based on data mining Abnormal Behavior of Fractional Differential Equations in Processing Computer Big Data Mathematical Modeling Thoughts and Methods Based on Fractional Differential Equations in Teaching A mathematical model of PCNN for image fusion with non-sampled contourlet transform Nonlinear Differential Equations in Computer-Aided Modeling of Big Data Technology The Uniqueness of Solutions of Fractional Differential Equations in University Mathematics Teaching Based on the Principle of Compression Mapping Financial customer classification by combined model Influence of displacement ventilation on the distribution of pollutant concentrations in livestock housing Recognition of Electrical Control System of Flexible Manipulator Based on Transfer Function Estimation Method Automatic Knowledge Integration Method of English Translation Corpus Based on Kmeans Algorithm Real Estate Economic Development Based on Logarithmic Growth Function Model Design of Tennis Mobile Teaching Assistant System Based on Ordinary Differential Equations Financial Crisis Early Warning Model of Listed Companies Based on Fisher Linear Discriminant Analysis High Simulation Reconstruction of Crowd Animation Based on Optical Flow Constraint Equation Construction of Intelligent Search Engine for Big Data Multimedia Resource Subjects Based on Partial Least Squares Structural Equation 3D Animation Simulation of Computer Fractal and Fractal Technology Combined with Diamond-Square Algorithm Analysis of the Teaching Quality of Physical Education Class by Using the Method of Gradient Difference The Summation of Series Based on the Laplace Transformation Method in Mathematics Teaching Optimal Solution of the Fractional Differential Equation to Solve the Bending Performance Test of Corroded Reinforced Concrete Beams under Prestressed Fatigue Load Animation VR scene mosaic modeling based on generalized Laplacian equation Radial Basis Function Neural Network in Vibration Control of Civil Engineering Structure Optimal Model Combination of Cross-border E-commerce Platform Operation Based on Fractional Differential Equations The influence of accounting computer information processing technology on enterprise internal control under panel data simultaneous equation Research on Stability of Time-delay Force Feedback Teleoperation System Based on Scattering Matrix BIM Building HVAC Energy Saving Technology Based on Fractional Differential Equation Construction of comprehensive evaluation index system of water-saving irrigation project integrating penman Montei the quation Human Resource Management Model of Large Companies Based on Mathematical Statistics Equations Data Forecasting of Air-Conditioning Load in Large Shopping Malls Based on Multiple Nonlinear Regression Analysis of technical statistical indexes of college tennis players under the win-lose regression function equation Automatic extraction and discrimination of vocal main melody based on quadratic wave equation Analysis of wireless English multimedia communication based on spatial state model equation Optimization of Linear Algebra Core Function Framework on Multicore Processors Optimisation of Modelling of Finite Element Differential Equations with Modern Art Design Theory Mathematical function data model analysis and synthesis system based on short-term human movement Human gait modelling and tracking based on motion functionalisation The Control Relationship Between the Enterprise's Electrical Equipment and Mechanical Equipment Based on Graph Theory Financial Accounting Measurement Model Based on Numerical Analysis of Rigid Normal Differential Equation and Rigid Functional Equation Mathematical Modeling and Forecasting of Economic Variables Based on Linear Regression Statistics Nonlinear Differential Equations in Cross-border E-commerce Controlling Return Rate Differential equation model of financial market stability based on Internet big data 3D Mathematical Modeling Technology in Visualized Aerobics Dance Rehearsal System Children’s cognitive function and mental health based on finite element nonlinear mathematical model Fractional Differential Equations in Electronic Information Models BIM Engineering Management Oriented to Curve Equation Model Leakage control of urban water supply network and mathematical analysis and location of leakage points based on machine learning Analysis of higher education management strategy based on entropy and dissipative structure theory Prediction of corporate financial distress based on digital signal processing and multiple regression analysis Mathematical Method to Construct the Linear Programming of Football Training Multimedia sensor image detection based on constrained underdetermined equation The Size of Children's Strollers of Different Ages Based on Ergonomic Mathematics Design Application of Numerical Computation of Partial Differential Equations in Interactive Design of Virtual Reality Media Stiffness Calculation of Gear Hydraulic System Based on the Modeling of Nonlinear Dynamics Differential Equations in the Progressive Method Knowledge Analysis of Charged Particle Motion in Uniform Electromagnetic Field Based on Maxwell Equation Relationship Between Enterprise Talent Management and Performance Based on the Structural Equation Model Method Term structure of economic management rate based on parameter analysis of estimation model of ordinary differential equation Influence analysis of piano music immersion virtual reality cooperation based on mapping equation Chinese painting and calligraphy image recognition technology based on pseudo linear directional diffusion equation Label big data compression in Internet of things based on piecewise linear regression Animation character recognition and character intelligence analysis based on semantic ontology and Poisson equation Design of language assisted learning model and online learning system under the background of artificial intelligence Application of machine learning in stock selection Comparative analysis of CR of ideological and political education in different regions based on improved fuzzy clustering Action of Aut( G ) on the set of maximal subgroups ofp -groupsThe internal mechanism of corporate social responsibility fulfillment affecting debt risk in China: analysis of intermediary transmission effect based on degree of debt concentration and product market competitive advantage Study on transmission characteristics in three kinds of deformed finlines based on edge-based finite element method Asymptotic stability problem of predator–prey system with linear diffusion Research on loyalty prediction of e-commerce customer based on data mining Algebraic Equations in Educational Model of College Physical Education Course Education Professional English Translation Corpus Under the Binomial Theorem Coefficient Geometric Tolerance Control Method for Precision Machinery Based on Image Modeling and Novel Saturation Function Retrieval and Characteristic Analysis of Multimedia Tester Based on Bragg Equation Semiparametric Spatial Econometric Analysis of Household Consumption Based on Ordinary Linear Regression Model Video adaptive watermark embedding and detection algorithm based on phase function equation English Learning Motivation of College Students Based on probability Distribution Scientific Model of Vocational Education Teaching Method in Differential Nonlinearity Research on mobile Awareness service and data privacy Protection based on Linear Equations computing protocol Vocal Music Teaching Model Based on Finite Element Differential Mathematical Equations Research on threat assessment problems of island air defence system based on the leader-follower model Studying a matching method combining distance proximity and buffer constraints The trend and influence of media information Propagation based on nonlinear Differential equation Research on the construction of early warning model of customer churn on e-commerce platform Study on inefficient land use determination method for cities and towns from a city examination perspective A sentiment analysis method based on bidirectional long short-term memory networks Evaluation of ecosystem health in Futian mangrove wetland based on the PSR-AHP model A study of local smoothness-informed convolutional neural network models for image inpainting Towards more efficient control of the ironmaking blast furnace: modelling gaseous reduction of iron ores in H _{2}-N_{2}atmosphereAlgorithm of overfitting avoidance in CNN based on maximum pooled and weight decay Mathematical Calculus Modeling in Improving the Teaching Performance of Shot Put Application of Nonlinear Differential Equation in Electric Automation Control System Higher Mathematics Teaching Curriculum Model Based on Lagrangian Mathematical Model Decisions of competing supply chain with altruistic retailer under risk aversion Optimization of Color Matching Technology in Cultural Industry by Fractional Differential Equations The Marketing of Cross-border E-commerce Enterprises in Foreign Trade Based on the Statistics of Mathematical Probability Theory Application of Linear Partial Differential Equation Theory in Guiding Football Scientific Training The Evolution Model of Regional Tourism Economic Development Difference Based on Spatial Variation Function System Model of Shipping Enterprise Safety Culture Based on Dynamic Calculation Matrix Model The Inner Relationship between Students' Psychological Factors and Physical Exercise Based on Structural Equation Model (SEM) Analysis and Research on Influencing Factors of Ideological and Political Education Teaching Effectiveness Based on Linear Equation Fractional Differential Equations in Sports Training in Universities Examination and Countermeasures of Network Education in Colleges and Universities Based on Ordinary Differential Equation Model Higher Education Agglomeration Promoting Innovation and Entrepreneurship Based on Spatial Dubin Model Chinese-English Contrastive Translation System Based on Lagrangian Search Mathematical Algorithm Model