# The Aero-Engine Component-Level Modelling Research Based on NSDE Hybrid Damping Newton Method

Authors: Chen Wang; Ximing Sun; Xian Du
Journal: International Journal of Aerospace Engineering
 (2022)

---

## Abstract

Advanced aero-engine component-level models are characterized by strong nonlinearity and multivariate, and traditional iterative algorithms cannot meet the requirements of convergence, real-time, and accuracy at the same time. To improve the convergence and alleviate the initial value dependence, a hybrid damped Newton algorithm based on the neighborhood based speciation differential evolution (NSDE) is proposed in this paper for solving the aero-engine component-level model. The computational efficiency and convergence of the hybrid damped Newton algorithm and NSDE hybrid damped Newton algorithm under four typical steady-state operating point conditions are analyzed, and then, the accuracy of the model is verified. It is demonstrated that the hybrid damped Newton method has the advantage of low initial value sensitivity and high computational efficiency under large deviation conditions. The hybrid damped Newton method is more efficient than the Broyden algorithm in terms of iterative efficiency, faster than the traditional N-R algorithm in terms of computation speed, and has the highest computational convergence rate under the four typical operating conditions, but it cannot eliminate the initial value dependence. The NSDE hybrid damped Newton method offers high simulation accuracy and greatly increases computational real-time performance under large deviation conditions, and the maximum error between the numerical simulation results and the experimental reference value is 8.1%. This study provides advanced theoretical support for component-level modelling and has certain engineering application value.


---

## Body

## 1. Introduction

At present, nonlinear component-level models of aero-engine are the most commonly utilized in research on analytical residuals, fault diagnostics, aircraft design, and control systems. It is well documented that engine modelling and knowledge of power system characteristics account for 80% of the work in aero-engine control and monitoring system research [1]. Modern advanced engine models have more operating modes and adjustable components, which require higher accuracy and convergence of component-level model solving. A compromise between accuracy and real-time is required in modelling while the increased demand for real-time models in engine semiphysical testing and control systems. Therefore, determining ways to increase the iteration algorithm’s convergence and real-time performance while fulfilling accuracy criteria has become a pressing issue.Iterative algorithms for aero-engine models have received a lot of attention from academics. Early iterative algorithms included parametric loop methods such as SPEEDY and CARPET and equilibrium loop methods such as AFQUIR and DSPOOL with nested loops [2], which were mainly used for engine simulation under steady-state conditions, but the iteration time was long and could not be applied to dynamic simulation. Gradient-based optimization algorithms, such as the steepest descent method, the N+1 residual method, the Newton-Raphson (N-R) method, and the Broyden method [3], were later used in the numerical calculation of aero-engine models because they do not require nested loops and are multiple iteration methods based on initial values and are widely used in steady-state and dynamic engine model calculations. The N-R method is characterized by strong second-order convergence and is widely utilized in the USAF general-purpose simulation programs SMOTE, GENENG, and DYNENG [4], but it needs repetitive Jacobi matrix computations and too many component calculations, resulting in poor real-time performance. The Broyden quasi Newton technique updates the correction function using a recursive formula, which eliminates the problem of recurrent Jacobian matrix calculation in the N-R method and considerably increases real-time performance. This approach is used in the primary simulation programs NASA’s NCP and NLR’s GSP and terts engine models [5, 6], although the convergence stability is weak. The N-R and Broyden techniques are locally convergent algorithms that rely heavily on the initial value of the iterations, restricting their use to “big deviation” dynamic engine modelling and simulation. Scholars have proposed initial value fitting, finite-domain optimization search, component characteristic expansion, and variable step size [7–12], as well as tried hybrid algorithm schemes of N-R and Broyden method [13, 14] to improve convergence and real-time performance from the perspective of improving initial conditions and algorithm update function, but cannot solve the problem of global convergence from the root.Many researchers have introduced advanced intelligent optimisation algorithms in recent years. Several scholars have used genetic algorithms [15, 16] (GA) and hybrid genetic algorithms [17] to solve engine models, which can improve global convergence due to the lack of dependence on initial values, but the GA has more algorithmic control parameters and cannot meet the requirements of real-time. Yang et al. [18] coupled the neural network algorithm with the N-R method, utilizing the neural network to approximate the inverse function of the model and the neural network solution result as the initial value of the N-R method, which increased real-time performance and convergence. Several scholars have employed the PSO [19] algorithm, but the computational efficiency of PSO depends on the choice of algorithm parameters and is prone to falling into local optimum solutions. The improved PSO [20] solves the engine model and improves convergence, although it still suffers from the problem of low solution rate. To increase the model’s real-time performance, the improved hybrid particle swarm [21] approach combines the advantages of fast calculation efficiency and superior global convergence. However, because the PSO method includes numerous changeable parameters and varied suitable values for different operating situations, it cannot adapt to the computation of a large envelope range. DE algorithm [22] is another evolutionary algorithm after GA with simple structure, few control parameters, and easy implementation using real number encoding and theoretically proven convergence [23]. The nondominated sorting difference algorithm [24] (NSDE) is a kind of evolutionary algorithm, which utilizes the strategy of neighborhood search in evolutionary programming to adjust the scaling factor, improve the global search capability, and maintain the ideal solution efficiency [24].For the past few years, research into engine modelling has mainly focused on improvements to the modelling of key components, including inlet ejector [25], multiple variable geometries, and combustion chamber design. Scholars have applied various KF-based methods [26–29] to estimate engine performance parameters and using various neural networks [30], support vector machine [31], extreme learning machine [32–34] to increase the accuracy and the real-time performance of the model in full life circle. Less analysis of the accuracy, convergence, and real-time performance of iterative algorithms for engine models, especially the application of hybrid intelligence algorithms. At this stage, NSDE is not used as an iterative algorithm in aero-engine intelligent algorithms. Due to the strong nonlinearity and multi-iterative variables of the aero-engine model, intelligent algorithms suffer from poor real-time performance and computational accuracy. Therefore, combining the traditional gradient-based optimisation algorithm with the intelligent algorithm can combine the advantages of both, while taking into account the problems of convergence, real-time, and computational accuracy.To resolve the challenges of convergence and real-time performance, a hybrid damped Newton algorithm based on NSDE has been designed to solve the aero-engine component level model. The effectiveness of the hybrid damped Newton algorithm is first validated by comparing it to traditional algorithms in terms of computational efficiency and convergence; second, on the basis of the hybrid damped Newton algorithm, the NSDE algorithm is integrated to further improve the convergence of the engine model; and finally, the simulation results of the engine model are compared to the experimental results to verify its effectiveness.


## 2. Method

### 2.1. Integrated Inlet/Engine Component Level Model

The principles of component-level model solution for aero-engine are described in this part. Aero-engine models are multivariable, nonlinear, time-varying systems that frequently employ component-level aero thermodynamic models. Conventional engine models of the inlet system are mostly calculated using idealised models and empirical formulas, which cannot match the calculation accuracy and fidelity criteria. Taking the shock structure and resistance calculation approach into consideration, the inlet is estimated utilizing quasi one-dimensional flow theory and the engine fundamental calculation procedure represented in Figure1. Flow continuity, static pressure balance, and power balance between components are all required when calculating off-design performance. Flow continuity means that the flow from the upstream component to the current component is the same as the flow calculated by the current component itself; static pressure balance means that when the two incoming flows are mixed, the static pressure should be equal; and power balance means that the power consumed on the same shaft should be equal to the power generated. The residuals of the balancing equations are expressed in terms of e. The iterative variables are selected and coworking equations are solved in association expressed as
(1)f1x0,x1,x2,⋯,x8=e1,f2x0,x1,x2,⋯,x8=e2,⋯⋯,f8x0,x1,x2,⋯,x8=e8.Figure 1
Calculation flow chart of inlet/engine integrated component level model.To satisfy the flow balance of the inlet and engine, the residuals of the inlet and fan flow are used as balance equations, and the ratio of the flow coefficient/total pressure recovery coefficient is chosen as the iteration’s starting guess value. If the number of iterative variables and residuals is both identical, the issue transforms into a nonlinear implicit set of equations with unknown independent variables, and the engine model is considered to have a reliable solution when the residuals tend to zero.


### 2.2. Basic Principles of the NSDE Hybrid Damped Newton Method

To solve the nonlinear implicit system of the aero-engine model, the hybrid damped Newton method and the adaptive NSDE hybrid damped Newton method are designed, and the principles of both methods are described in this section.


#### 2.2.1. Hybrid Damped Newton Method

The hybrid damped Newton method is an enhanced N-R technique with a damping factor in which the nonlinear equation is enlarged according to the Taylor series and first-order approximation is used to generate iterative general formulae of the independent variables:(2)Xk+1=Xk−αk·ΔX,(3)ΔX=∂FX∂X−1·FXk,(4)αk=minc·FXmkΔXmk,1,wherein ∂FXk/∂X partial derivative is a jacobi matrix of FXk, and the Jacobi matrix is nonsingular; αk is the damping factor; Xmk represents a variable that makes FXmk/ΔXmk largest in the independent variables. FXk is an error Ek determined by the coworking equation, and the differential term of the jacobi matrix is replaced by forward difference, i.e.,
(5)∂Ek∂X=ΔEkΔXk.The hybrid damped Newton method indicates that the damped Newton method is used as the main algorithm, and the Jacobi matrix is not adjusted immediately after each iteration, but the Broyden quasi-Newton technique is utilized for computation to minimize the quantity of iterative calculation. Two iterative techniques are utilized alternatively to effectively minimize the number of aero-thermodynamic iterations based on the indices of specified convergence speed and range. If the iteration result of thenth damped Newton’s method is Xn, the Broyden quasi-Newton method is used in the intermediate m steps, and the damped Newton’s method is used again from n+1 step, using the following iteration format:
(6)Xn,0=Xn,Xn,j=Xn,j−1−αj−1Bj−1−1FXn,j−1,Bj=Bj−1+βj−1·yj−1−Bj−1Sj−1·Sj−1TSj−1TSj−1,Xn+1=Xn,m.In the formula,Xn,j represents the jth use of the Broyden quasi-Newton method in the nth Newton iteration; yj−1=FXn,j−FXn,j−1; sj−1=Xn,j−Xn,j−1; βj−1 and αj−1 are damping factors; and B0 takes the jacobi matrix of Xn,0 point.
(7)Evak=∑n=18ΔeXnkΔXnk,(8)Evak≥λ1·Evak−1,(9)Evajk≥λ2·Evaj−1k.The specific switching conditions are shown in Eqs. (7)–(9), where Eva indicates the update efficiency at each step, and the Broyden algorithm can be used when Eva varies widely, while the N-R algorithm is used when Eva varies less. λ1 indicates the condition for switching the Broyden algorithm, and λ2 indicates the condition of whether to jump out of the broyden algorithm. The calculation process is shown in Figure 2.Figure 2
Basic flowchart of the adaptive NSDE hybrid damping Newton method.


#### 2.2.2. Adaptive NSDE Algorithm Method

The adaptive NSDE algorithm, which is based on the differential evolution algorithm, can adaptively adjust the scaling factor of the variation process and the crossover rate of the crossover process to improve global convergence.Initializing a population: determining the value range of each iteration variable in the adaptive differential evolution algorithm based on the characteristics of the engine components and working condition limitations, as the variable value rangesxj,iL and xj,iU of an initial population of the adaptive differential evolution algorithm; setting the initial population number NP of the differential evolution algorithm, and randomly generating the initial population.
(10)xi0xj,iL≤xj,i0≤xj,iU,i=1,2,⋯,NP;j=1,2,⋯,D,(11)xj,i0=xj,iL+rand0,1·xj,iU−xj,iL,wherein xj,i0 represents the jth gene of the ith individual in the 0 generation; NP represents the population size; and rand0,1 represents random numbers evenly distributed in an interval of 0,1; selecting a residual em of each coworking equation (the number of equations is m) of the engine model, and taking Of=1/m∑0m−1ej as a fitness function to serve as an optimization objective function.Adaptive mutation strategies: attempting to use two different mutation strategies, and introducing probabilityp to control the selection; conducting self-adapting by p according to the learning experience in the calculation process, and obtaining a scaling factor F based on a Gaussian distribution function; initializing p=0.5; after the population fully evolves in the current round, recording the number ns1 of individuals which enter the next generation and the number nf1 of individuals not entering the next generation under the condition of Ui0,1<p from vi, and the number ns2 of individuals which enter the next generation and the number nf2 of individuals not entering the next generation under the condition of Ui0,1≥p from vi, wherein xbest represents a current optimal individual, respectively, recording the two groups of numbers for 50 generations, called as a “learning cycle”; and when the probability p is updated after the learning cycle, resetting the values of ns1, ns2, nf1, and nf2, with the formulas of the adaptive mutation strategies shown as follows:
(12)vi=xi1+Fi·xi2−xi3,Ui0,1<p,xbest+Fi·xi1−xi2,(13)Fi=Ni0.5,0.3,(14)p=ns1·ns2+nf2ns2·ns1+nf1+ns1·ns2+nf2.Crossover operation: the adaptive evolution crossover operation is conducted for dimensions. The new individual has the probability ofCR to select the dimension in vij, and other dimensions select xij, wherein the adaptive crossover rate CR allocates a crossover rate CRi for each individual, CRm is initialized as 0.5, and CRi is updated every 5 generations. In each generation, the value of CRm and a subgeneration successfully enter the next generation; corresponding CRi enters an array CRrec and is updated every 25 generations; and after the update, CRrec is emptied. In Eq. (17), CRrec represents the length of the array, and when a better individual appears in each generation, the result is inserted into CRrec, and eventually, the mean is calculated by summing from 1 to CRrec, and the value of CRm is updated.
(15)uij=vij,Ui0,1<CRiorj=jrand,xij,(16)CRi=NiCRm,0.1(17)CRm=∑k=1CRrecCRreck.Selection operation: the selection operation is selection of a better individual from a mutated individualui and an old individual xi by using a greedy algorithm to generate a new individual xi′.
(18)x′i=ui,fui≤fxi,xi.


#### 2.2.3. The NSDE Hybrid Damping Newton Method

The NSDE hybrid damped Newton method combines the hybrid damped Newton method and the NSDE algorithm, with the number of iterations and the size of the residuals calculated in real time used as a basis for judgement, switching between the two algorithms and thus improving computational efficiency and solution accuracy. The NSDE hybrid damped Newton method’s computational flow is depicted in Figure2. If the hybrid damped Newton method fails to converge within the maximum number of iterations Nmax1, or if the iterations diverge, the calculation is passed to the NSDE algorithm until the equilibrium residuals are reduced to a certain error value Emax. Returning to the hybrid damped Newton method, the iteration ends when the equilibrium equation’s residuals fulfill the error criterion. Nloop1 represents the actual number of iterations of the hybrid damped Newton method, Nloop2 and Nmax2 denote the actual number of iterations and the maximum number of iterations of the NSDE algorithm, respectively.
(19)∑n=18ekn≥α·∑n=18ek−1norNloop1≥Nmax1,(20)maxek,k=1,2,⋯,8≤EmaxorNloop2≥Nmax2.


## 2.1. Integrated Inlet/Engine Component Level Model

The principles of component-level model solution for aero-engine are described in this part. Aero-engine models are multivariable, nonlinear, time-varying systems that frequently employ component-level aero thermodynamic models. Conventional engine models of the inlet system are mostly calculated using idealised models and empirical formulas, which cannot match the calculation accuracy and fidelity criteria. Taking the shock structure and resistance calculation approach into consideration, the inlet is estimated utilizing quasi one-dimensional flow theory and the engine fundamental calculation procedure represented in Figure1. Flow continuity, static pressure balance, and power balance between components are all required when calculating off-design performance. Flow continuity means that the flow from the upstream component to the current component is the same as the flow calculated by the current component itself; static pressure balance means that when the two incoming flows are mixed, the static pressure should be equal; and power balance means that the power consumed on the same shaft should be equal to the power generated. The residuals of the balancing equations are expressed in terms of e. The iterative variables are selected and coworking equations are solved in association expressed as
(1)f1x0,x1,x2,⋯,x8=e1,f2x0,x1,x2,⋯,x8=e2,⋯⋯,f8x0,x1,x2,⋯,x8=e8.Figure 1
Calculation flow chart of inlet/engine integrated component level model.To satisfy the flow balance of the inlet and engine, the residuals of the inlet and fan flow are used as balance equations, and the ratio of the flow coefficient/total pressure recovery coefficient is chosen as the iteration’s starting guess value. If the number of iterative variables and residuals is both identical, the issue transforms into a nonlinear implicit set of equations with unknown independent variables, and the engine model is considered to have a reliable solution when the residuals tend to zero.


## 2.2. Basic Principles of the NSDE Hybrid Damped Newton Method

To solve the nonlinear implicit system of the aero-engine model, the hybrid damped Newton method and the adaptive NSDE hybrid damped Newton method are designed, and the principles of both methods are described in this section.


### 2.2.1. Hybrid Damped Newton Method

The hybrid damped Newton method is an enhanced N-R technique with a damping factor in which the nonlinear equation is enlarged according to the Taylor series and first-order approximation is used to generate iterative general formulae of the independent variables:(2)Xk+1=Xk−αk·ΔX,(3)ΔX=∂FX∂X−1·FXk,(4)αk=minc·FXmkΔXmk,1,wherein ∂FXk/∂X partial derivative is a jacobi matrix of FXk, and the Jacobi matrix is nonsingular; αk is the damping factor; Xmk represents a variable that makes FXmk/ΔXmk largest in the independent variables. FXk is an error Ek determined by the coworking equation, and the differential term of the jacobi matrix is replaced by forward difference, i.e.,
(5)∂Ek∂X=ΔEkΔXk.The hybrid damped Newton method indicates that the damped Newton method is used as the main algorithm, and the Jacobi matrix is not adjusted immediately after each iteration, but the Broyden quasi-Newton technique is utilized for computation to minimize the quantity of iterative calculation. Two iterative techniques are utilized alternatively to effectively minimize the number of aero-thermodynamic iterations based on the indices of specified convergence speed and range. If the iteration result of thenth damped Newton’s method is Xn, the Broyden quasi-Newton method is used in the intermediate m steps, and the damped Newton’s method is used again from n+1 step, using the following iteration format:
(6)Xn,0=Xn,Xn,j=Xn,j−1−αj−1Bj−1−1FXn,j−1,Bj=Bj−1+βj−1·yj−1−Bj−1Sj−1·Sj−1TSj−1TSj−1,Xn+1=Xn,m.In the formula,Xn,j represents the jth use of the Broyden quasi-Newton method in the nth Newton iteration; yj−1=FXn,j−FXn,j−1; sj−1=Xn,j−Xn,j−1; βj−1 and αj−1 are damping factors; and B0 takes the jacobi matrix of Xn,0 point.
(7)Evak=∑n=18ΔeXnkΔXnk,(8)Evak≥λ1·Evak−1,(9)Evajk≥λ2·Evaj−1k.The specific switching conditions are shown in Eqs. (7)–(9), where Eva indicates the update efficiency at each step, and the Broyden algorithm can be used when Eva varies widely, while the N-R algorithm is used when Eva varies less. λ1 indicates the condition for switching the Broyden algorithm, and λ2 indicates the condition of whether to jump out of the broyden algorithm. The calculation process is shown in Figure 2.Figure 2
Basic flowchart of the adaptive NSDE hybrid damping Newton method.


### 2.2.2. Adaptive NSDE Algorithm Method

The adaptive NSDE algorithm, which is based on the differential evolution algorithm, can adaptively adjust the scaling factor of the variation process and the crossover rate of the crossover process to improve global convergence.Initializing a population: determining the value range of each iteration variable in the adaptive differential evolution algorithm based on the characteristics of the engine components and working condition limitations, as the variable value rangesxj,iL and xj,iU of an initial population of the adaptive differential evolution algorithm; setting the initial population number NP of the differential evolution algorithm, and randomly generating the initial population.
(10)xi0xj,iL≤xj,i0≤xj,iU,i=1,2,⋯,NP;j=1,2,⋯,D,(11)xj,i0=xj,iL+rand0,1·xj,iU−xj,iL,wherein xj,i0 represents the jth gene of the ith individual in the 0 generation; NP represents the population size; and rand0,1 represents random numbers evenly distributed in an interval of 0,1; selecting a residual em of each coworking equation (the number of equations is m) of the engine model, and taking Of=1/m∑0m−1ej as a fitness function to serve as an optimization objective function.Adaptive mutation strategies: attempting to use two different mutation strategies, and introducing probabilityp to control the selection; conducting self-adapting by p according to the learning experience in the calculation process, and obtaining a scaling factor F based on a Gaussian distribution function; initializing p=0.5; after the population fully evolves in the current round, recording the number ns1 of individuals which enter the next generation and the number nf1 of individuals not entering the next generation under the condition of Ui0,1<p from vi, and the number ns2 of individuals which enter the next generation and the number nf2 of individuals not entering the next generation under the condition of Ui0,1≥p from vi, wherein xbest represents a current optimal individual, respectively, recording the two groups of numbers for 50 generations, called as a “learning cycle”; and when the probability p is updated after the learning cycle, resetting the values of ns1, ns2, nf1, and nf2, with the formulas of the adaptive mutation strategies shown as follows:
(12)vi=xi1+Fi·xi2−xi3,Ui0,1<p,xbest+Fi·xi1−xi2,(13)Fi=Ni0.5,0.3,(14)p=ns1·ns2+nf2ns2·ns1+nf1+ns1·ns2+nf2.Crossover operation: the adaptive evolution crossover operation is conducted for dimensions. The new individual has the probability ofCR to select the dimension in vij, and other dimensions select xij, wherein the adaptive crossover rate CR allocates a crossover rate CRi for each individual, CRm is initialized as 0.5, and CRi is updated every 5 generations. In each generation, the value of CRm and a subgeneration successfully enter the next generation; corresponding CRi enters an array CRrec and is updated every 25 generations; and after the update, CRrec is emptied. In Eq. (17), CRrec represents the length of the array, and when a better individual appears in each generation, the result is inserted into CRrec, and eventually, the mean is calculated by summing from 1 to CRrec, and the value of CRm is updated.
(15)uij=vij,Ui0,1<CRiorj=jrand,xij,(16)CRi=NiCRm,0.1(17)CRm=∑k=1CRrecCRreck.Selection operation: the selection operation is selection of a better individual from a mutated individualui and an old individual xi by using a greedy algorithm to generate a new individual xi′.
(18)x′i=ui,fui≤fxi,xi.


### 2.2.3. The NSDE Hybrid Damping Newton Method

The NSDE hybrid damped Newton method combines the hybrid damped Newton method and the NSDE algorithm, with the number of iterations and the size of the residuals calculated in real time used as a basis for judgement, switching between the two algorithms and thus improving computational efficiency and solution accuracy. The NSDE hybrid damped Newton method’s computational flow is depicted in Figure2. If the hybrid damped Newton method fails to converge within the maximum number of iterations Nmax1, or if the iterations diverge, the calculation is passed to the NSDE algorithm until the equilibrium residuals are reduced to a certain error value Emax. Returning to the hybrid damped Newton method, the iteration ends when the equilibrium equation’s residuals fulfill the error criterion. Nloop1 represents the actual number of iterations of the hybrid damped Newton method, Nloop2 and Nmax2 denote the actual number of iterations and the maximum number of iterations of the NSDE algorithm, respectively.
(19)∑n=18ekn≥α·∑n=18ek−1norNloop1≥Nmax1,(20)maxek,k=1,2,⋯,8≤EmaxorNloop2≥Nmax2.


## 2.2.1. Hybrid Damped Newton Method

The hybrid damped Newton method is an enhanced N-R technique with a damping factor in which the nonlinear equation is enlarged according to the Taylor series and first-order approximation is used to generate iterative general formulae of the independent variables:(2)Xk+1=Xk−αk·ΔX,(3)ΔX=∂FX∂X−1·FXk,(4)αk=minc·FXmkΔXmk,1,wherein ∂FXk/∂X partial derivative is a jacobi matrix of FXk, and the Jacobi matrix is nonsingular; αk is the damping factor; Xmk represents a variable that makes FXmk/ΔXmk largest in the independent variables. FXk is an error Ek determined by the coworking equation, and the differential term of the jacobi matrix is replaced by forward difference, i.e.,
(5)∂Ek∂X=ΔEkΔXk.The hybrid damped Newton method indicates that the damped Newton method is used as the main algorithm, and the Jacobi matrix is not adjusted immediately after each iteration, but the Broyden quasi-Newton technique is utilized for computation to minimize the quantity of iterative calculation. Two iterative techniques are utilized alternatively to effectively minimize the number of aero-thermodynamic iterations based on the indices of specified convergence speed and range. If the iteration result of thenth damped Newton’s method is Xn, the Broyden quasi-Newton method is used in the intermediate m steps, and the damped Newton’s method is used again from n+1 step, using the following iteration format:
(6)Xn,0=Xn,Xn,j=Xn,j−1−αj−1Bj−1−1FXn,j−1,Bj=Bj−1+βj−1·yj−1−Bj−1Sj−1·Sj−1TSj−1TSj−1,Xn+1=Xn,m.In the formula,Xn,j represents the jth use of the Broyden quasi-Newton method in the nth Newton iteration; yj−1=FXn,j−FXn,j−1; sj−1=Xn,j−Xn,j−1; βj−1 and αj−1 are damping factors; and B0 takes the jacobi matrix of Xn,0 point.
(7)Evak=∑n=18ΔeXnkΔXnk,(8)Evak≥λ1·Evak−1,(9)Evajk≥λ2·Evaj−1k.The specific switching conditions are shown in Eqs. (7)–(9), where Eva indicates the update efficiency at each step, and the Broyden algorithm can be used when Eva varies widely, while the N-R algorithm is used when Eva varies less. λ1 indicates the condition for switching the Broyden algorithm, and λ2 indicates the condition of whether to jump out of the broyden algorithm. The calculation process is shown in Figure 2.Figure 2
Basic flowchart of the adaptive NSDE hybrid damping Newton method.


## 2.2.2. Adaptive NSDE Algorithm Method

The adaptive NSDE algorithm, which is based on the differential evolution algorithm, can adaptively adjust the scaling factor of the variation process and the crossover rate of the crossover process to improve global convergence.Initializing a population: determining the value range of each iteration variable in the adaptive differential evolution algorithm based on the characteristics of the engine components and working condition limitations, as the variable value rangesxj,iL and xj,iU of an initial population of the adaptive differential evolution algorithm; setting the initial population number NP of the differential evolution algorithm, and randomly generating the initial population.
(10)xi0xj,iL≤xj,i0≤xj,iU,i=1,2,⋯,NP;j=1,2,⋯,D,(11)xj,i0=xj,iL+rand0,1·xj,iU−xj,iL,wherein xj,i0 represents the jth gene of the ith individual in the 0 generation; NP represents the population size; and rand0,1 represents random numbers evenly distributed in an interval of 0,1; selecting a residual em of each coworking equation (the number of equations is m) of the engine model, and taking Of=1/m∑0m−1ej as a fitness function to serve as an optimization objective function.Adaptive mutation strategies: attempting to use two different mutation strategies, and introducing probabilityp to control the selection; conducting self-adapting by p according to the learning experience in the calculation process, and obtaining a scaling factor F based on a Gaussian distribution function; initializing p=0.5; after the population fully evolves in the current round, recording the number ns1 of individuals which enter the next generation and the number nf1 of individuals not entering the next generation under the condition of Ui0,1<p from vi, and the number ns2 of individuals which enter the next generation and the number nf2 of individuals not entering the next generation under the condition of Ui0,1≥p from vi, wherein xbest represents a current optimal individual, respectively, recording the two groups of numbers for 50 generations, called as a “learning cycle”; and when the probability p is updated after the learning cycle, resetting the values of ns1, ns2, nf1, and nf2, with the formulas of the adaptive mutation strategies shown as follows:
(12)vi=xi1+Fi·xi2−xi3,Ui0,1<p,xbest+Fi·xi1−xi2,(13)Fi=Ni0.5,0.3,(14)p=ns1·ns2+nf2ns2·ns1+nf1+ns1·ns2+nf2.Crossover operation: the adaptive evolution crossover operation is conducted for dimensions. The new individual has the probability ofCR to select the dimension in vij, and other dimensions select xij, wherein the adaptive crossover rate CR allocates a crossover rate CRi for each individual, CRm is initialized as 0.5, and CRi is updated every 5 generations. In each generation, the value of CRm and a subgeneration successfully enter the next generation; corresponding CRi enters an array CRrec and is updated every 25 generations; and after the update, CRrec is emptied. In Eq. (17), CRrec represents the length of the array, and when a better individual appears in each generation, the result is inserted into CRrec, and eventually, the mean is calculated by summing from 1 to CRrec, and the value of CRm is updated.
(15)uij=vij,Ui0,1<CRiorj=jrand,xij,(16)CRi=NiCRm,0.1(17)CRm=∑k=1CRrecCRreck.Selection operation: the selection operation is selection of a better individual from a mutated individualui and an old individual xi by using a greedy algorithm to generate a new individual xi′.
(18)x′i=ui,fui≤fxi,xi.


## 2.2.3. The NSDE Hybrid Damping Newton Method

The NSDE hybrid damped Newton method combines the hybrid damped Newton method and the NSDE algorithm, with the number of iterations and the size of the residuals calculated in real time used as a basis for judgement, switching between the two algorithms and thus improving computational efficiency and solution accuracy. The NSDE hybrid damped Newton method’s computational flow is depicted in Figure2. If the hybrid damped Newton method fails to converge within the maximum number of iterations Nmax1, or if the iterations diverge, the calculation is passed to the NSDE algorithm until the equilibrium residuals are reduced to a certain error value Emax. Returning to the hybrid damped Newton method, the iteration ends when the equilibrium equation’s residuals fulfill the error criterion. Nloop1 represents the actual number of iterations of the hybrid damped Newton method, Nloop2 and Nmax2 denote the actual number of iterations and the maximum number of iterations of the NSDE algorithm, respectively.
(19)∑n=18ekn≥α·∑n=18ek−1norNloop1≥Nmax1,(20)maxek,k=1,2,⋯,8≤EmaxorNloop2≥Nmax2.


## 3. Results and Discussion

### 3.1. Computational Efficiency and Convergence Analysis of the Hybrid Damped Newton Method

#### 3.1.1. Hybrid Damped Newton Method for Computational Efficiency Analysis

This paper compares the hybrid damped Newton algorithm with the N-R method and Broyden method under different computational initial value conditions and analyzes the computational efficiency by comprehensively analyzing the number of iteration steps and component calculations. In this section, a typical supersonic condition (Alt=12.47, Ma=2.0) is selected, the initial deviation conditions are determined by the deviation rate, and the algorithm iterations’ default step size is set to 0.5. Figures 3–5 depict the fluctuation of the iterative equations’ equilibrium residuals with the number of component computations and iteration steps for different deviation condition. This section has defined initial deviation conditions of 0.1, 0.4, and 0.5 as small deviations, medium deviations, and large deviations, respectively. The values of eight variables are determined when the equilibrium equations are calculated to converge, and the value deviation is set as the rate of deviation done relative to the initial values of the eight variables taken. The formula for initial value deviation should be
(21)∑k=18xk,ini−xk,finalxk,final×100%.Figure 3
The variation of residuals with the number of model calculations and iteration steps under the initial deviation of 0.1.
(a)(b)Figure 4
The variation of residuals with the number of model calculations and iteration steps under the initial deviation of 0.4.
(a)(b)Figure 5
The variation of residuals with the number of model calculations and iteration steps under the initial deviation of 0.5.
(a)(b)When shown in Figure3, as the number of component calculations increases, the equilibrium residuals of the three iterative algorithms decline to zero and all achieve a state of convergence. The N-R method requires more component computations than the others since it requires iterative solution of the differential terms of the Jacobi matrix at each iteration step. The Broyden method has the fewest model back iterations calculated, with just 28. As seen in Figure 3(b), the hybrid damped Newton method has the highest number of iterations at 53, whereas the N-R and Broyden methods have a comparable amount of iterations. Under the small deviation condition, all three iterative methods may converge in a limited number of iteration steps, with the Broyden method possessing the fastest computational speed and the N-R and Broyden algorithms being highly efficient in terms of actual iterations.Figure4(a) shows that the three algorithms still converge for an initial deviation of 0.4, but the number of component computations grows dramatically, with the Broyden method having the lowest number. Figure 4(b) shows that the Broyden method has the highest number of iteration steps, 107, while the hybrid damped Newton method has the lowest number of iteration steps, 37. The Broyden method has the quickest computing speed but the lowest iteration efficiency under the condition of considerable departure from the initial value, while the hybrid damped Newton method has the maximum iteration efficiency.Figure5(a) shows that the Broyden method fails to converge for an initial deviation of 0.5, whereas the N-R method converges although the equation residuals are slow to reduce. The number of iterations for the hybrid damped Newton method is less than that of the N-R method, as seen in Figure 5(b). When the initial deviation is large, the Broyden algorithm can no longer guarantee convergence, and the N-R algorithm can converge but at a slow rate. The hybrid damped Newton method’s iteration efficiency and computational speed remain steady at the optimal level, and the computational effect is the best.To examine the computational speed and iteration efficiency of the three different algorithms under different initial deviation and step size settings, four different flight conditions (Case1:Alt=0km, Ma=0, Case2: Alt=0.06km, Ma=0.3, Case3: Alt=9.96km, Ma=1.5, Case4: Alt=12.47km, Ma=2.0) were used, as shown in Figure 6.Figure 6
Box plots of the computational speed and iteration efficiency of the three algorithms for different initial deviation and step size settings.
(a)(b)(c)(d)The initial deviations were set to 0-0.9, and the step sizes were set to 0.1-1.0, with each box-line plot reflecting the set of data for each flight condition. As shown in Figure6(a), the hybrid damped Newton method has the lowest mean values for the number of model returns and iteration steps in Case1, 85.778 and 17.667, indicating that the algorithm has the fastest average computation speed and the highest iteration efficiency. The hybrid damped Newton method has the smallest box line plot area for the number of model returns and iteration steps compared to the other two algorithms, implying that the median and mean results of the hybrid damped Newton method are close to each other. The mean values of the other two algorithms are larger than the median, which is caused by the high number of outliers in the Broyden algorithm and the excessive number of iterations back in the N-R method in the large deviation condition.The results for the Case2-4 conditions are shown in Figures6(b)–6(d). The mean values of the number of component calculations and iteration steps of the three algorithms gradually increase as the flight altitude and Mach number increase, and the difference between the average and the median value gradually increases, indicating that the overall computational efficiency decreases and stability become less stable. In terms of computing speed, the Broyden method is comparable to the hybrid damped Newton method, but its iterative efficiency is lower and the number of outliers is highest. In summary, the hybrid damped Newton method can show the ideal calculation speed and the best iteration efficiency in the four operating conditions.The above analysis demonstrates that the hybrid damped Newton method may maintain high computing efficiency and convergence even under significant deviation conditions, demonstrating that the algorithm is less sensitive to different initial conditions. The computational speed of the hybrid damped Newton method is comparable to the Broyden method, but the iteration efficiency is higher; when compared to the traditional N-R method, the computational speed of the hybrid damped Newton method is significantly improved, and the iteration efficiency is also optimized.


#### 3.1.2. Hybrid Damped Newton Method for Convergence Analysis

This section examines the convergence of the three iterative algorithms under four typical operating settings with varying initial conditions. The same data set as in Figure6 is still used in Figure 7 to characterize the convergence of the various iterative algorithms using hotspot plots. The heat map shows the iteration residuals from tiny to large, with red indicating that the residuals are too large to converge. The horizontal coordinate represents the initial deviation rate, while the vertical coordinate is the maximum step size for just a single iteration.Figure 7
Hotspot diagram for convergence analysis of different iterative algorithms.As the initial deviation rate and maximum iteration step size increase, the algorithm eventually converges to a nonconverging state, as seen in Figure7. Case 1 indicates that the experimental deviation rate of the Broyden method is more than 0.2 and that it does not converge at all when the deviation rate is greater than 0.7. The convergence of the hybrid damped Newton method and the N-R method is similar, but the N-R method shows partial nonconvergence at a maximum step size of 0.1 in a single iteration.This is due to the fact that the maximum number of iterations is exceeded. For the four operating conditions, the iterative convergence patterns of the three algorithms were similar, with convergence rates of 75.31%, 75.31%, 76.54%, and 90.12% for the hybrid damped Newton method; 67.90%, 70.37%, 72.84%, and 81.48% for the N-R method; and 49.38%, 49.38%, 59.26%, 55.56%, and 46.91% for the Broyden method, indicating that the convergence rates of the hybrid damped Newton method were the highest, and the convergence rates of the Broyden method were the worst. 59.26%, 55.56%, and 46.91% indicate that the mixed damped Newton method has the highest convergence rate, and the Broyden method has the lowest. All three iterative algorithms did not converge when the initial value deviation rate was at 0.9.The preceding analysis demonstrates that the initial deviation rate and the maximum step size of a single iteration have a significant impact on the convergence of the iterative algorithm, and too large a parameter will lead to nonconvergence. Under the four typical operating conditions, the hybrid damped Newton method has the highest convergence rate, indicating strong convergence and good stability, but it cannot completely eliminate the dependence on the initial value of the iteration.


#### 3.1.3. Dynamic Process Analysis of the Hybrid Damped Newton Method

The simulation verifies the variation of the number of component calculations and dynamic residuals during a typical acceleration process, as shown in Figures8 and 9, to affirm the effectiveness of the three iterative algorithms in the dynamic calculation. Due to the slow change in the computational deviation of the iterative algorithm, the acceleration process is in a small deviation state and all three algorithms converge. The N-R method consistently has a higher number of component computations than the Broyden and hybrid damped Newton methods, suggesting a slower computational time, as seen in Figure 8. The computing speed and equilibrium times of the Broyden and hybrid damped Newton methods are comparable. Shown in Figure 9, the dynamic residuals of the N-R method fluctuate markedly, taking 13 s to reach equilibrium, whereas the dynamic residuals of the Broyden and hybrid damped Newton methods are similar, taking 8 s to reach equilibrium, indicating that both have a clear advantage in terms of dynamic computational speed.Figure 8
The variation pattern of the number of component calculations during the acceleration process.Figure 9
The variation pattern of equation residual during the acceleration process.The results show that the hybrid damped Newton method, which incorporates the concept of damping update and the integration of Broyden and traditional N-R algorithms, has ideal performance in dynamic process simulation while ensuring better convergence in large deviation states, providing a good theoretical basis and application value for engine modelling techniques.


### 3.2. Analysis of the NSDE Hybrid Damped Newton Method

The preceding study validates the benefits of the hybrid damped Newton method in terms of computing efficiency and stability, although the gradient algorithm is not totally free of the reliance of iterative beginning values. As a heuristic algorithm, the swarm intelligence algorithm may avoid the dependency on beginning values. To validate the effectiveness of the model calculation, the NSDE algorithm, which has high global convergence and computational efficiency, is integrated with the hybrid damped Newton method in this paper.The population size of NSDE is 30, wherens1, ns2, nf1, and nf2 are dynamically changing data for each generation. Figure 10 depicts the fluctuation of the computational residuals of the NSDE algorithm, the hybrid damped Newton method, and the NSDE hybrid damped Newton method with the number of model iterations under Alt=12.47km, Ma=2.0, and an initial deviation rate of 0.9. The Alt=12.47km, Ma=2.0 condition was chosen because it provided the best convergence of the hybrid damped Newton method. The initial residuals of the hybrid damped Newton method are larger, while the residuals of the NSDE algorithm are lower in the first iteration due to the initial population update in the first step, indicating that the NSDE algorithm optimises better in the first iteration; the hybrid damped Newton method model terminates after 10 iterations when the calculation is scattered, whereas the NSDE algorithm can calculate until the model converges. The NSDE algorithm requires 81 model iterations for computational convergence, and the optimization efficiency at the late iterative stage is quite sluggish, failing to fulfill the real-time requirement. It is demonstrated that the NSDE damped Newton method can calculate convergence in the presence of a large deviation of the initial deviation rate of 0.1, while being free of initial value sensitivity and maintaining the ideal computational speed.Figure 10
Variation of equation residuals with the number of component calculations for different iterative algorithms underAlt=12.47km and Ma=2.0.Figure11 represents the variation pattern of the calculated residuals of the NSDE algorithm, the hybrid damped Newton method and the NSDE hybrid damped Newton method with the number of component calculations for Alt=9.96km, Ma=1.5 working conditions and an initial deviation rate of 0.5. The hybrid damped Newton method was able to sustain convergence under typical large deviation conditions with 47 model iterations, but the NSDE algorithm required 97 model iterations, demonstrating that the NSDE algorithm is computationally sluggish. In addition, the number of NSDE component calculations is 2910. The NSDE hybrid damped Newton method requires 23 component calculations, which is fewer than the hybrid damped Newton method, indicating that this method has the quickest computational speed. The above analysis shows that in the large deviation state (initial deviation rate of 0.5), both the NSDE hybrid damped Newton method and the hybrid damped Newton method can ensure computational convergence, but the NSDE hybrid damped Newton method is faster. The NSDE hybrid damped Newton method triggers the NSDE algorithm only under conditions where the initial conditions are unknown or deviate significantly, more suitable for applications in steady-state models and large deviations from dynamic calculations to improve convergence.Figure 11
Variation of equation residuals with the number of component calculations for different iterative algorithms underAlt=9.96km and Ma=1.5.In summary, the NSDE hybrid damped Newton method designed in this paper can get rid of the dependence on the initial value of the iterations, maintain the model convergence, and guarantee the computational speed under the condition that the initial deviation rate is 0.9. The simple NSDE method is not suitable for the solving of the coworking equation because every individual needs to call the model to calculate its fitness. Furthermore, under the large deviation condition (initial deviation rate of 0.5), the computational speed of the NSDE hybrid damped Newton method is faster than that of the hybrid damped Newton method, greatly improving the real-time performance, which is useful for the development of real-time aero-engine models.


### 3.3. Experimental Validation of Model Calculation Results Based on the NSDE Hybrid Damped Newton Method

The accuracy of the model must be validated in order to further evaluate the validity of the simulation results based on the NSDE hybrid damped Newton method. The numerical simulation results of the component-level model are compared to the steady-state reference data of the engine test at four typical operating points. Figure12 represents the error analysis graph of the numerical calculation results of the model and the test reference data, comparing the parameters including engine thrust Fn, low pressure rotor speed Nl, high pressure rotor speed Nh, fan outlet pressure P3, outlet temperature T3, low-pressure turbine outlet pressure P6, and temperature T6. It can be seen that the maximum error between the numerical simulation results of the four operating conditions and the test reference value is 8.1%. The rest of the errors are controlled within 7.5%.Figure 12
Error analysis diagram of model numerical calculation results and experimental reference data.The model calculation errors are mainly caused by the accuracy of the component characteristics and the convergence residuals of the model equations. The overall model errors in this paper are maintained within a reasonable range, which indicates that the NSDE hybrid damped Newton method based model can maintain a good simulation accuracy because the gradient method is still used in the later iteration to convergence. Therefore, the NSDE hybrid damped Newton method can be applied to the simulation of aero-engine component-level models.


## 3.1. Computational Efficiency and Convergence Analysis of the Hybrid Damped Newton Method

### 3.1.1. Hybrid Damped Newton Method for Computational Efficiency Analysis

This paper compares the hybrid damped Newton algorithm with the N-R method and Broyden method under different computational initial value conditions and analyzes the computational efficiency by comprehensively analyzing the number of iteration steps and component calculations. In this section, a typical supersonic condition (Alt=12.47, Ma=2.0) is selected, the initial deviation conditions are determined by the deviation rate, and the algorithm iterations’ default step size is set to 0.5. Figures 3–5 depict the fluctuation of the iterative equations’ equilibrium residuals with the number of component computations and iteration steps for different deviation condition. This section has defined initial deviation conditions of 0.1, 0.4, and 0.5 as small deviations, medium deviations, and large deviations, respectively. The values of eight variables are determined when the equilibrium equations are calculated to converge, and the value deviation is set as the rate of deviation done relative to the initial values of the eight variables taken. The formula for initial value deviation should be
(21)∑k=18xk,ini−xk,finalxk,final×100%.Figure 3
The variation of residuals with the number of model calculations and iteration steps under the initial deviation of 0.1.
(a)(b)Figure 4
The variation of residuals with the number of model calculations and iteration steps under the initial deviation of 0.4.
(a)(b)Figure 5
The variation of residuals with the number of model calculations and iteration steps under the initial deviation of 0.5.
(a)(b)When shown in Figure3, as the number of component calculations increases, the equilibrium residuals of the three iterative algorithms decline to zero and all achieve a state of convergence. The N-R method requires more component computations than the others since it requires iterative solution of the differential terms of the Jacobi matrix at each iteration step. The Broyden method has the fewest model back iterations calculated, with just 28. As seen in Figure 3(b), the hybrid damped Newton method has the highest number of iterations at 53, whereas the N-R and Broyden methods have a comparable amount of iterations. Under the small deviation condition, all three iterative methods may converge in a limited number of iteration steps, with the Broyden method possessing the fastest computational speed and the N-R and Broyden algorithms being highly efficient in terms of actual iterations.Figure4(a) shows that the three algorithms still converge for an initial deviation of 0.4, but the number of component computations grows dramatically, with the Broyden method having the lowest number. Figure 4(b) shows that the Broyden method has the highest number of iteration steps, 107, while the hybrid damped Newton method has the lowest number of iteration steps, 37. The Broyden method has the quickest computing speed but the lowest iteration efficiency under the condition of considerable departure from the initial value, while the hybrid damped Newton method has the maximum iteration efficiency.Figure5(a) shows that the Broyden method fails to converge for an initial deviation of 0.5, whereas the N-R method converges although the equation residuals are slow to reduce. The number of iterations for the hybrid damped Newton method is less than that of the N-R method, as seen in Figure 5(b). When the initial deviation is large, the Broyden algorithm can no longer guarantee convergence, and the N-R algorithm can converge but at a slow rate. The hybrid damped Newton method’s iteration efficiency and computational speed remain steady at the optimal level, and the computational effect is the best.To examine the computational speed and iteration efficiency of the three different algorithms under different initial deviation and step size settings, four different flight conditions (Case1:Alt=0km, Ma=0, Case2: Alt=0.06km, Ma=0.3, Case3: Alt=9.96km, Ma=1.5, Case4: Alt=12.47km, Ma=2.0) were used, as shown in Figure 6.Figure 6
Box plots of the computational speed and iteration efficiency of the three algorithms for different initial deviation and step size settings.
(a)(b)(c)(d)The initial deviations were set to 0-0.9, and the step sizes were set to 0.1-1.0, with each box-line plot reflecting the set of data for each flight condition. As shown in Figure6(a), the hybrid damped Newton method has the lowest mean values for the number of model returns and iteration steps in Case1, 85.778 and 17.667, indicating that the algorithm has the fastest average computation speed and the highest iteration efficiency. The hybrid damped Newton method has the smallest box line plot area for the number of model returns and iteration steps compared to the other two algorithms, implying that the median and mean results of the hybrid damped Newton method are close to each other. The mean values of the other two algorithms are larger than the median, which is caused by the high number of outliers in the Broyden algorithm and the excessive number of iterations back in the N-R method in the large deviation condition.The results for the Case2-4 conditions are shown in Figures6(b)–6(d). The mean values of the number of component calculations and iteration steps of the three algorithms gradually increase as the flight altitude and Mach number increase, and the difference between the average and the median value gradually increases, indicating that the overall computational efficiency decreases and stability become less stable. In terms of computing speed, the Broyden method is comparable to the hybrid damped Newton method, but its iterative efficiency is lower and the number of outliers is highest. In summary, the hybrid damped Newton method can show the ideal calculation speed and the best iteration efficiency in the four operating conditions.The above analysis demonstrates that the hybrid damped Newton method may maintain high computing efficiency and convergence even under significant deviation conditions, demonstrating that the algorithm is less sensitive to different initial conditions. The computational speed of the hybrid damped Newton method is comparable to the Broyden method, but the iteration efficiency is higher; when compared to the traditional N-R method, the computational speed of the hybrid damped Newton method is significantly improved, and the iteration efficiency is also optimized.


### 3.1.2. Hybrid Damped Newton Method for Convergence Analysis

This section examines the convergence of the three iterative algorithms under four typical operating settings with varying initial conditions. The same data set as in Figure6 is still used in Figure 7 to characterize the convergence of the various iterative algorithms using hotspot plots. The heat map shows the iteration residuals from tiny to large, with red indicating that the residuals are too large to converge. The horizontal coordinate represents the initial deviation rate, while the vertical coordinate is the maximum step size for just a single iteration.Figure 7
Hotspot diagram for convergence analysis of different iterative algorithms.As the initial deviation rate and maximum iteration step size increase, the algorithm eventually converges to a nonconverging state, as seen in Figure7. Case 1 indicates that the experimental deviation rate of the Broyden method is more than 0.2 and that it does not converge at all when the deviation rate is greater than 0.7. The convergence of the hybrid damped Newton method and the N-R method is similar, but the N-R method shows partial nonconvergence at a maximum step size of 0.1 in a single iteration.This is due to the fact that the maximum number of iterations is exceeded. For the four operating conditions, the iterative convergence patterns of the three algorithms were similar, with convergence rates of 75.31%, 75.31%, 76.54%, and 90.12% for the hybrid damped Newton method; 67.90%, 70.37%, 72.84%, and 81.48% for the N-R method; and 49.38%, 49.38%, 59.26%, 55.56%, and 46.91% for the Broyden method, indicating that the convergence rates of the hybrid damped Newton method were the highest, and the convergence rates of the Broyden method were the worst. 59.26%, 55.56%, and 46.91% indicate that the mixed damped Newton method has the highest convergence rate, and the Broyden method has the lowest. All three iterative algorithms did not converge when the initial value deviation rate was at 0.9.The preceding analysis demonstrates that the initial deviation rate and the maximum step size of a single iteration have a significant impact on the convergence of the iterative algorithm, and too large a parameter will lead to nonconvergence. Under the four typical operating conditions, the hybrid damped Newton method has the highest convergence rate, indicating strong convergence and good stability, but it cannot completely eliminate the dependence on the initial value of the iteration.


### 3.1.3. Dynamic Process Analysis of the Hybrid Damped Newton Method

The simulation verifies the variation of the number of component calculations and dynamic residuals during a typical acceleration process, as shown in Figures8 and 9, to affirm the effectiveness of the three iterative algorithms in the dynamic calculation. Due to the slow change in the computational deviation of the iterative algorithm, the acceleration process is in a small deviation state and all three algorithms converge. The N-R method consistently has a higher number of component computations than the Broyden and hybrid damped Newton methods, suggesting a slower computational time, as seen in Figure 8. The computing speed and equilibrium times of the Broyden and hybrid damped Newton methods are comparable. Shown in Figure 9, the dynamic residuals of the N-R method fluctuate markedly, taking 13 s to reach equilibrium, whereas the dynamic residuals of the Broyden and hybrid damped Newton methods are similar, taking 8 s to reach equilibrium, indicating that both have a clear advantage in terms of dynamic computational speed.Figure 8
The variation pattern of the number of component calculations during the acceleration process.Figure 9
The variation pattern of equation residual during the acceleration process.The results show that the hybrid damped Newton method, which incorporates the concept of damping update and the integration of Broyden and traditional N-R algorithms, has ideal performance in dynamic process simulation while ensuring better convergence in large deviation states, providing a good theoretical basis and application value for engine modelling techniques.


## 3.1.1. Hybrid Damped Newton Method for Computational Efficiency Analysis

This paper compares the hybrid damped Newton algorithm with the N-R method and Broyden method under different computational initial value conditions and analyzes the computational efficiency by comprehensively analyzing the number of iteration steps and component calculations. In this section, a typical supersonic condition (Alt=12.47, Ma=2.0) is selected, the initial deviation conditions are determined by the deviation rate, and the algorithm iterations’ default step size is set to 0.5. Figures 3–5 depict the fluctuation of the iterative equations’ equilibrium residuals with the number of component computations and iteration steps for different deviation condition. This section has defined initial deviation conditions of 0.1, 0.4, and 0.5 as small deviations, medium deviations, and large deviations, respectively. The values of eight variables are determined when the equilibrium equations are calculated to converge, and the value deviation is set as the rate of deviation done relative to the initial values of the eight variables taken. The formula for initial value deviation should be
(21)∑k=18xk,ini−xk,finalxk,final×100%.Figure 3
The variation of residuals with the number of model calculations and iteration steps under the initial deviation of 0.1.
(a)(b)Figure 4
The variation of residuals with the number of model calculations and iteration steps under the initial deviation of 0.4.
(a)(b)Figure 5
The variation of residuals with the number of model calculations and iteration steps under the initial deviation of 0.5.
(a)(b)When shown in Figure3, as the number of component calculations increases, the equilibrium residuals of the three iterative algorithms decline to zero and all achieve a state of convergence. The N-R method requires more component computations than the others since it requires iterative solution of the differential terms of the Jacobi matrix at each iteration step. The Broyden method has the fewest model back iterations calculated, with just 28. As seen in Figure 3(b), the hybrid damped Newton method has the highest number of iterations at 53, whereas the N-R and Broyden methods have a comparable amount of iterations. Under the small deviation condition, all three iterative methods may converge in a limited number of iteration steps, with the Broyden method possessing the fastest computational speed and the N-R and Broyden algorithms being highly efficient in terms of actual iterations.Figure4(a) shows that the three algorithms still converge for an initial deviation of 0.4, but the number of component computations grows dramatically, with the Broyden method having the lowest number. Figure 4(b) shows that the Broyden method has the highest number of iteration steps, 107, while the hybrid damped Newton method has the lowest number of iteration steps, 37. The Broyden method has the quickest computing speed but the lowest iteration efficiency under the condition of considerable departure from the initial value, while the hybrid damped Newton method has the maximum iteration efficiency.Figure5(a) shows that the Broyden method fails to converge for an initial deviation of 0.5, whereas the N-R method converges although the equation residuals are slow to reduce. The number of iterations for the hybrid damped Newton method is less than that of the N-R method, as seen in Figure 5(b). When the initial deviation is large, the Broyden algorithm can no longer guarantee convergence, and the N-R algorithm can converge but at a slow rate. The hybrid damped Newton method’s iteration efficiency and computational speed remain steady at the optimal level, and the computational effect is the best.To examine the computational speed and iteration efficiency of the three different algorithms under different initial deviation and step size settings, four different flight conditions (Case1:Alt=0km, Ma=0, Case2: Alt=0.06km, Ma=0.3, Case3: Alt=9.96km, Ma=1.5, Case4: Alt=12.47km, Ma=2.0) were used, as shown in Figure 6.Figure 6
Box plots of the computational speed and iteration efficiency of the three algorithms for different initial deviation and step size settings.
(a)(b)(c)(d)The initial deviations were set to 0-0.9, and the step sizes were set to 0.1-1.0, with each box-line plot reflecting the set of data for each flight condition. As shown in Figure6(a), the hybrid damped Newton method has the lowest mean values for the number of model returns and iteration steps in Case1, 85.778 and 17.667, indicating that the algorithm has the fastest average computation speed and the highest iteration efficiency. The hybrid damped Newton method has the smallest box line plot area for the number of model returns and iteration steps compared to the other two algorithms, implying that the median and mean results of the hybrid damped Newton method are close to each other. The mean values of the other two algorithms are larger than the median, which is caused by the high number of outliers in the Broyden algorithm and the excessive number of iterations back in the N-R method in the large deviation condition.The results for the Case2-4 conditions are shown in Figures6(b)–6(d). The mean values of the number of component calculations and iteration steps of the three algorithms gradually increase as the flight altitude and Mach number increase, and the difference between the average and the median value gradually increases, indicating that the overall computational efficiency decreases and stability become less stable. In terms of computing speed, the Broyden method is comparable to the hybrid damped Newton method, but its iterative efficiency is lower and the number of outliers is highest. In summary, the hybrid damped Newton method can show the ideal calculation speed and the best iteration efficiency in the four operating conditions.The above analysis demonstrates that the hybrid damped Newton method may maintain high computing efficiency and convergence even under significant deviation conditions, demonstrating that the algorithm is less sensitive to different initial conditions. The computational speed of the hybrid damped Newton method is comparable to the Broyden method, but the iteration efficiency is higher; when compared to the traditional N-R method, the computational speed of the hybrid damped Newton method is significantly improved, and the iteration efficiency is also optimized.


## 3.1.2. Hybrid Damped Newton Method for Convergence Analysis

This section examines the convergence of the three iterative algorithms under four typical operating settings with varying initial conditions. The same data set as in Figure6 is still used in Figure 7 to characterize the convergence of the various iterative algorithms using hotspot plots. The heat map shows the iteration residuals from tiny to large, with red indicating that the residuals are too large to converge. The horizontal coordinate represents the initial deviation rate, while the vertical coordinate is the maximum step size for just a single iteration.Figure 7
Hotspot diagram for convergence analysis of different iterative algorithms.As the initial deviation rate and maximum iteration step size increase, the algorithm eventually converges to a nonconverging state, as seen in Figure7. Case 1 indicates that the experimental deviation rate of the Broyden method is more than 0.2 and that it does not converge at all when the deviation rate is greater than 0.7. The convergence of the hybrid damped Newton method and the N-R method is similar, but the N-R method shows partial nonconvergence at a maximum step size of 0.1 in a single iteration.This is due to the fact that the maximum number of iterations is exceeded. For the four operating conditions, the iterative convergence patterns of the three algorithms were similar, with convergence rates of 75.31%, 75.31%, 76.54%, and 90.12% for the hybrid damped Newton method; 67.90%, 70.37%, 72.84%, and 81.48% for the N-R method; and 49.38%, 49.38%, 59.26%, 55.56%, and 46.91% for the Broyden method, indicating that the convergence rates of the hybrid damped Newton method were the highest, and the convergence rates of the Broyden method were the worst. 59.26%, 55.56%, and 46.91% indicate that the mixed damped Newton method has the highest convergence rate, and the Broyden method has the lowest. All three iterative algorithms did not converge when the initial value deviation rate was at 0.9.The preceding analysis demonstrates that the initial deviation rate and the maximum step size of a single iteration have a significant impact on the convergence of the iterative algorithm, and too large a parameter will lead to nonconvergence. Under the four typical operating conditions, the hybrid damped Newton method has the highest convergence rate, indicating strong convergence and good stability, but it cannot completely eliminate the dependence on the initial value of the iteration.


## 3.1.3. Dynamic Process Analysis of the Hybrid Damped Newton Method

The simulation verifies the variation of the number of component calculations and dynamic residuals during a typical acceleration process, as shown in Figures8 and 9, to affirm the effectiveness of the three iterative algorithms in the dynamic calculation. Due to the slow change in the computational deviation of the iterative algorithm, the acceleration process is in a small deviation state and all three algorithms converge. The N-R method consistently has a higher number of component computations than the Broyden and hybrid damped Newton methods, suggesting a slower computational time, as seen in Figure 8. The computing speed and equilibrium times of the Broyden and hybrid damped Newton methods are comparable. Shown in Figure 9, the dynamic residuals of the N-R method fluctuate markedly, taking 13 s to reach equilibrium, whereas the dynamic residuals of the Broyden and hybrid damped Newton methods are similar, taking 8 s to reach equilibrium, indicating that both have a clear advantage in terms of dynamic computational speed.Figure 8
The variation pattern of the number of component calculations during the acceleration process.Figure 9
The variation pattern of equation residual during the acceleration process.The results show that the hybrid damped Newton method, which incorporates the concept of damping update and the integration of Broyden and traditional N-R algorithms, has ideal performance in dynamic process simulation while ensuring better convergence in large deviation states, providing a good theoretical basis and application value for engine modelling techniques.


## 3.2. Analysis of the NSDE Hybrid Damped Newton Method

The preceding study validates the benefits of the hybrid damped Newton method in terms of computing efficiency and stability, although the gradient algorithm is not totally free of the reliance of iterative beginning values. As a heuristic algorithm, the swarm intelligence algorithm may avoid the dependency on beginning values. To validate the effectiveness of the model calculation, the NSDE algorithm, which has high global convergence and computational efficiency, is integrated with the hybrid damped Newton method in this paper.The population size of NSDE is 30, wherens1, ns2, nf1, and nf2 are dynamically changing data for each generation. Figure 10 depicts the fluctuation of the computational residuals of the NSDE algorithm, the hybrid damped Newton method, and the NSDE hybrid damped Newton method with the number of model iterations under Alt=12.47km, Ma=2.0, and an initial deviation rate of 0.9. The Alt=12.47km, Ma=2.0 condition was chosen because it provided the best convergence of the hybrid damped Newton method. The initial residuals of the hybrid damped Newton method are larger, while the residuals of the NSDE algorithm are lower in the first iteration due to the initial population update in the first step, indicating that the NSDE algorithm optimises better in the first iteration; the hybrid damped Newton method model terminates after 10 iterations when the calculation is scattered, whereas the NSDE algorithm can calculate until the model converges. The NSDE algorithm requires 81 model iterations for computational convergence, and the optimization efficiency at the late iterative stage is quite sluggish, failing to fulfill the real-time requirement. It is demonstrated that the NSDE damped Newton method can calculate convergence in the presence of a large deviation of the initial deviation rate of 0.1, while being free of initial value sensitivity and maintaining the ideal computational speed.Figure 10
Variation of equation residuals with the number of component calculations for different iterative algorithms underAlt=12.47km and Ma=2.0.Figure11 represents the variation pattern of the calculated residuals of the NSDE algorithm, the hybrid damped Newton method and the NSDE hybrid damped Newton method with the number of component calculations for Alt=9.96km, Ma=1.5 working conditions and an initial deviation rate of 0.5. The hybrid damped Newton method was able to sustain convergence under typical large deviation conditions with 47 model iterations, but the NSDE algorithm required 97 model iterations, demonstrating that the NSDE algorithm is computationally sluggish. In addition, the number of NSDE component calculations is 2910. The NSDE hybrid damped Newton method requires 23 component calculations, which is fewer than the hybrid damped Newton method, indicating that this method has the quickest computational speed. The above analysis shows that in the large deviation state (initial deviation rate of 0.5), both the NSDE hybrid damped Newton method and the hybrid damped Newton method can ensure computational convergence, but the NSDE hybrid damped Newton method is faster. The NSDE hybrid damped Newton method triggers the NSDE algorithm only under conditions where the initial conditions are unknown or deviate significantly, more suitable for applications in steady-state models and large deviations from dynamic calculations to improve convergence.Figure 11
Variation of equation residuals with the number of component calculations for different iterative algorithms underAlt=9.96km and Ma=1.5.In summary, the NSDE hybrid damped Newton method designed in this paper can get rid of the dependence on the initial value of the iterations, maintain the model convergence, and guarantee the computational speed under the condition that the initial deviation rate is 0.9. The simple NSDE method is not suitable for the solving of the coworking equation because every individual needs to call the model to calculate its fitness. Furthermore, under the large deviation condition (initial deviation rate of 0.5), the computational speed of the NSDE hybrid damped Newton method is faster than that of the hybrid damped Newton method, greatly improving the real-time performance, which is useful for the development of real-time aero-engine models.


## 3.3. Experimental Validation of Model Calculation Results Based on the NSDE Hybrid Damped Newton Method

The accuracy of the model must be validated in order to further evaluate the validity of the simulation results based on the NSDE hybrid damped Newton method. The numerical simulation results of the component-level model are compared to the steady-state reference data of the engine test at four typical operating points. Figure12 represents the error analysis graph of the numerical calculation results of the model and the test reference data, comparing the parameters including engine thrust Fn, low pressure rotor speed Nl, high pressure rotor speed Nh, fan outlet pressure P3, outlet temperature T3, low-pressure turbine outlet pressure P6, and temperature T6. It can be seen that the maximum error between the numerical simulation results of the four operating conditions and the test reference value is 8.1%. The rest of the errors are controlled within 7.5%.Figure 12
Error analysis diagram of model numerical calculation results and experimental reference data.The model calculation errors are mainly caused by the accuracy of the component characteristics and the convergence residuals of the model equations. The overall model errors in this paper are maintained within a reasonable range, which indicates that the NSDE hybrid damped Newton method based model can maintain a good simulation accuracy because the gradient method is still used in the later iteration to convergence. Therefore, the NSDE hybrid damped Newton method can be applied to the simulation of aero-engine component-level models.


## 4. Conclusion

(1)
The hybrid damped Newton method offers significant advantages in terms of iteration efficiency and computational speed; it is less sensitive to initial conditions and may keep strong computing efficiency and convergence, particularly in the high deviation state. Under the four typical operating conditions, the hybrid damped Newton method improves the comprehensive computational speed and iteration efficiency compared to the Broyden method and improves the comprehensive computational speed and iteration efficiency compared to the N-R method(2)
The initial deviation rate and the maximum step size of a single iteration have a substantial influence on the iterative method’s convergence, and too large a parameter in both situations leads to nonconvergence. For the four typical operating conditions, the hybrid damped Newton method has a substantially greater convergence rate than the N-R and Broyden methods, and it has the best convergence performance. The hybrid damped Newton method considerably improves convergence, but cannot avoid the dependence on beginning values(3)
The NSDE hybrid damped Newton method is able to further get rid of the dependence, maintaining the convergence and guarantee the computational speed under the initial deviation rate of 0.9; under the large deviation condition, the computational speed of the NSDE hybrid damped Newton method is better than that of the hybrid damped Newton method, which greatly improves the real-time performance and provides engineering application value. The maximum error between the NSDE hybrid damping Newton method numerical simulation results and the experimental reference value is 8.1%, demonstrating positive simulation accuracy and engineering application value



---

*Source: 8212150-2022-09-10.xml*