Temperature-Related Containment Analysis and Optimal Design of Aluminum Honeycomb Sandwich Aero-Engine Casings

Abstract

Aero-engine casings with excellent impact resistance are a practical requirement for ensuring the safe operation of aero-engines. In this paper, we report on numerical simulations of broken rotating blades impacting aluminum honeycomb sandwich casings under different temperatures and optimization of structural parameters. Firstly, an impact test system with adjustable temperature was established. Restricted by the temperature range of the strain gauge, ballistic impact tests were carried out at 25 °C, 100 °C, and 200 °C. Secondly, a finite element (FE) model including a pointed bullet and an aluminum honeycomb sandwich plate was built using LS-DYNA. The corresponding simulations of the strain–time curve and damage conditions showed good agreement with the test results. Then, the containment capability of the aluminum honeycomb sandwich aero-engine casing at different temperatures was analyzed based on the kinetic energy loss of the blade, the internal energy increment of the casing, and the containment state of the blade. Finally, with the design objectives of minimizing the casing mass and maximizing the blade kinetic energy loss, the structural parameters of the casing were optimized using the multi-objective genetic algorithm (MOGA).

1. Introduction

An aero-engine generates external thrust by compressing and burning gas. During operation, the temperature of its turbine casing can reach 1770 °C, while that of the compressor casing reaches 400 °C [1,2]. When a broken rotating blade impacts the aero-engine casing, the casing is subjected to both thermal and impact loads. Moreover, the trailing blades may cause further damage to the casing upon impact. These high-energy fragments from broken rotating blades can penetrate the engine casing, damage other accessories, impair flight performance, and even lead to aircraft loss. Previously, several researchers have studied the impact of broken rotating blades on containment casings under various conditions. Kraus and Frischbier [3] analyzed the influence of broken rotating blades and adjacent blades on casing containment. Stamper [4] and Sinha [5] established an engine simulation model containing static and rotor components to study the dynamic response of the casing when impacted by broken blades.. Jain [6] and Carney [7] conducted an energy absorption study for the blade impact casing process according to the different structural casings. Duan [8] presented the casing containment under frontal impact. Wan [9] showed the effect of impact pitch angle on the casing containment. Cao [10] investigated the failure mechanisms of the bolted casing flange structure under impact loadings. Roy [11] presented a thorough comparative analysis of various single and bilayer targets composed of aluminum foam. He [12] carried out numerical simulation analysis on primary fan casing containment capability using LS-DYNA. Liu [13] proposed a method for evaluating casing containment capability by combining a real casing target test with simulation analysis. Yu [14] investigated the impact dynamics, deformation characteristics, and energy absorption mechanisms during blade detachment events, shedding light on the containment process. The shedding experiments were compared with high-resolution finite element simulations of a fully bladed fan disc of a realistic gas turbine engine [15].

Honeycomb structures are widely applied in engineering fields such as aerospace and vehicle protection due to their lightweight characteristics, high specific strength, high specific stiffness, excellent impact resistance, and strong energy absorption properties. Among them, hexagonal honeycomb structures have become typical structural forms in the field of impact load resistance and energy absorption due to their superior structural strength and wear resistance [16,17]. While extensive research has been conducted on the containment performance of aero-engine casings, few studies have explored the influence of temperature on their containment capability. In this paper, the aluminum honeycomb sandwich structure casing, which operates at a maximum temperature of 400 °C, is selected as the research object. Through experimental programs and simulation analyses, we investigate the impact of different temperatures on casing containment. This work is intended to provide a fundamental basis for the design of aero-engine casings.

2. Impact Test and Simulation Analysis

2.1. Impact Test

Figure 1 shows the impact test system with adjustable temperature. The impact test system mainly consists of a gas source control system, a high-pressure gas source, a launcher, a Hopkinson impact test bench, a synergy data collector, a pneumatic gate, an electric resistance furnace, an air compressor, a thermocouple, and a temperature control system. The resistance wire is made of high-temperature Cr15Ni60 alloy with a power of 5 kW and a chamber size of 160 mm × 160 mm. The furnace chamber with resistance wire can withstand high temperatures up to 1600 °C and has strong damage resistance, making it suitable for impact test research. A synergy data collector produced by Hi-Techniques is a system integrating the functions of a data acquisition system, digital oscilloscope, FFT analyzer, and transient recorder. Each channel has a sampling rate of up to 2 MS/s and a 1 MHz bandwidth with 16 M sample memory for high-speed trigger capture, and its maximum sampling rate can reach 200 kHz. It has the advantages of good system stability and strong anti-interference ability and is suitable for strain testing of specimens in impact tests.

In the present research, the test aluminum alloy plate is fixed to the mount base through bolts. The test aluminum alloy plate is heated to the specified temperature by the electric resistance furnace. The pneumatic gate is subsequently opened by the gas source control system. A bullet is launched from the Hopkinson impact test bench. The medium temperature strain foils are attached to the surface of the test aluminum alloy plate. A synergy data collector serves to collect stress–strain data. The temperature control system is shown in Figure 2. The aluminum honeycomb sandwich plate is composed of two aluminum alloy plates with an aluminum honeycomb core layer in the middle. The shape of the single aluminum honeycomb core is hexagonal, its inner diameter is 3 mm, and the thickness of the aluminum foil is 0.05 mm. Specific parameters of the pointed bullet and the aluminum honeycomb sandwich plate are given in

Table 1. Specific parameters of the specimen.

Name	Material Trademark	Size (mm)
Upper and lower aluminum alloy plates	7075-T651 aluminum alloy	140 × 90 × 1
Aluminum honeycomb core layer	5052 aluminum foil	120 × 80 × 3
pointed bullet	40 Cr steel	φ15.4 × 210

.

The medium-temperature strain foil BAB 350-3AA250(23) is affixed to the aluminum honeycomb sandwich plate surface. Medium-temperature strain foil can be used in the range of −269~250 °C. The strain bridge is connected to the synergy data collector by a quarter bridge. Since the melting point of the aluminum honeycomb sandwich panel used in this paper is 463~671 °C, ballistic impact tests are conducted only at temperatures of 25 °C, 100 °C, and 200 °C, which are far below its melting point. According to the factory calibration value table of the Hopkinson impact test bench, we know that the initial impact velocity of the pointed bullet is 60 m/s when the shock pressure is 0.7 MPa.

2.2. Impact Resistance Analysis

The impact process has the characteristics of rapid action time, large force change, and high strain rate. Nonlinear problems for material parameters, boundary conditions, and the FE model can occur during the impact process. We built the 3D impact model, including the pointed bullet and the aluminum honeycomb sandwich plate, based on Creo. The 3D model grid is refined because the grid requires high quality in the impact penetration process. The mesh size is closely related to simulation accuracy: the finer the mesh, the higher the accuracy. To maximize the fidelity of the dynamic response of aluminum honeycomb sandwich panels under impact, the mesh should theoretically be refined as much as possible. However, denser meshes lead to exponentially increased computation time. Thus, it is necessary to select an appropriate mesh size under the premise that computation time remains feasible, and the analysis results are not compromised. This section examines aluminum honeycomb sandwich panels and projectiles using different mesh scales.

Table 2. Mesh model size.

Number	Name	Mesh Type	Mesh Size/mm	Number of Elements	Calculation Time/h
1	Upper and lower aluminum alloy plates	solid 164	4 × 4 × 1	1576	0.5
	Aluminum honeycomb core panel	shell 163	3	6386
	Pointed bullet	solid 164	4 × 4 × 4	7896
2	Upper and lower aluminum alloy plates	solid 164	3 × 3 × 1	2738	2.5
	Aluminum honeycomb core panel	shell 163	2	9762
	Pointed bullet	solid 164	3 × 3 × 3	13,398
3	Upper and lower aluminum alloy plates	solid 164	1 × 2 × 2	6120	6.5
	Aluminum honeycomb core panel	shell 163	1.5	12,624
	Pointed bullet	solid 164	2 × 2 × 2	31,208
4	Upper and lower aluminum alloy plates	solid 164	1 × 1 × 1	15,624	26
	Aluminum honeycomb core panel	shell 163	1	37,464
	Pointed bullet	solid 164	1 × 1 × 1	152,760

lists the selected mesh scales for the models.

The results from simulation analyses of four meshes with varying levels of refinement are presented in Figure 3.

Figure 3 presents the time history curves of the kinetic energy of the pointed projectile and the internal energy of the aluminum honeycomb core panel. As can be seen from the figure, the simulation results for Mesh Model No. 1 differ significantly from those of the other models. The curves of the remaining three mesh models exhibit very similar trends, leading to basically consistent simulation conclusions. Considering both the accuracy of the simulation analysis and the computation time, Mesh Model No. 3 is selected. The grid model is shown in Figure 4. Solid element 164 is used in the grid type of pointed bullet and aluminum alloy plate. The pointed bullet is divided into 31,208 elements, and the alloy plate is divided into 6120 elements. The aluminum honeycomb core layer grid type adopts shell element 163 and is divided into 12,624 elements.

The upper and lower aluminum alloy plates’ material trademark is 7075-T651; the aluminum alloy plates are modeled as Johnson–Cook constitutive and cumulative damage criterion. In this model, the influences of material temperature softening, strain rate hardening, etc., are considered, and it is widely used in the fields of impact and material damage. Therefore, when conducting simulation analysis, the Johnson–Cook constitutive model and the Johnson–Cook cumulative damage criterion are adopted for the material, and their mathematical model expressions are as follows:

$$\sigma =\left(A+B{\epsilon }^n\right)\left(1+Cln{\displaystyle \frac{\stackrel{·}{\epsilon }}{{\stackrel{·}{\epsilon }}_0}}\right)\left[1-{\left({\displaystyle \frac{T-T_r}{T_m-T_r}}\right)}^m\right]$$

(1)

A is J–C yield stress; B is J–C strain hardening coefficient; n is J–C strain hardening exponent; C is J–C strain rate constant; m is J–C softening exponent; σ is the flow stress; ε is the equivalent plastic strain; $\stackrel{·}{\epsilon } \mathrm{and} \stackrel{·}{{\epsilon }_0}$ are the strain rate and reference strain rate of the material, respectively; T is the deformation temperature; T_m is the melting point; and T_r is the reference temperature. The three terms in the brackets of Equation (1) represent the influences of equivalent plastic strain, strain rate, and temperature on the flow stress, respectively [18,19].

The J–C cumulative damage criterion is adopted for the front and rear panels of aluminum alloy. Then the damage parameter D is

$$D={\displaystyle \int \left(1/{\epsilon }_f\right)}d\epsilon$$

(2)

In Equation (2)

$${\epsilon }_f=\left[D_1+D_2\exp \left(D_3{\displaystyle \frac{{\sigma }_m}{{\sigma }_e}}\right)\right]\left(1+D_{4}ln{\displaystyle \frac{\stackrel{·}{\epsilon }}{{\stackrel{·}{\epsilon }}_0}}\right)\left[1+D_5\left({\displaystyle \frac{T-T_r}{T_m-T_r}}\right)\right]$$

(3)

In the formula, is the mean stress, is the equivalent stress of the material, and D₁~D₅ are the parameters of the J–C failure model. When D = 1, it indicates that the material fails. The aluminum honeycomb core material trademark is 5052; the aluminum honeycomb is modeled as *MAT_PLASTIC_KINEMATIC model. Material model parameters are shown in

Table 3. Material model parameters of upper and lower aluminum alloy plates.

Material Trademark	Parameter	Numerical Value	Parameter	Numerical Value
7075-T651	ρ(kg/m3)	2810	n	0.52
	E/GPa	71.7	m	1.61
	ν	0.33	D1	0.096
	Tm/°C	620	D2	0.049
	A/MPa	520	D3	3.465
	B/MPa	477	D4	0.016
	C	0.0025	D5	1.099

and

Table 4. Aluminum honeycomb core layer material model parameters.

Material Trademark	Parameter	Numerical Value	Parameter	Numerical Value
5052 aluminum foil	ρ (kg/m3)	2680	σ0/MPa	215
	E/GPa	70	Etan/MPa	450
	v	0.33	β	0.5

[20,21,22].

Metal materials have different thermophysical parameters at different temperatures. The process of a projectile impacting an aluminum honeycomb sandwich panel is rapid, and heat conduction plays a dominant role in the heat transfer during contact and collision, while the influences of thermal radiation and heat convection are relatively small. Therefore, the heat generation is determined by the specific heat capacity and thermal conductivity of the material. Since both 5052 and 7075-T651 are aluminum alloys with little difference in thermophysical parameters, the same set of thermophysical parameters is used. The thermophysical parameters of the pointed projectile and aluminum honeycomb sandwich panel are shown in

Table 5. Thermal properties of upper and lower aluminum alloy sheet materials.

Material Trademark	Temperature/°C	Thermal Conductivity (W/m°C)	Heat Capacity (J/Kg°C)
7075-T651	25	124	796
	100	142	921
	200	170	1005
	300	185	1020
	400	193	1120

and

Table 6. Thermal properties of pointed bullet materials.

Material Trademark	Temperature/°C	Thermal Conductivity (W/m°C)	Heat Capacity (J/Kg°C)
40Cr	100	23.87	465
	200	25.12	515
	300	25.96	548
	400	26.80	557
	500	27.63	567

. The material density changes slightly with temperature, which is considered unchanged in this paper. Full restraint is set at both ends of the aluminum honeycomb sandwich plate. The pointed bullet is subjected to an initial velocity of 60 m/s in the Z-axis.

2.3. Comparison of Experimental Test and Numerical Simulation

2.3.1. Damage Comparison

Figure 5 shows the experimental results of damage to the aluminum honeycomb sandwich plate by a pointed bullet impacting at 60 m/s under different temperatures. The figure also shows the corresponding results of a numerical simulation to facilitate comparisons with our test results.

Figure 5a shows the deformation and damage to the aluminum honeycomb core plate at 25 °C. It is found that the upper aluminum alloy plate exhibits complex failure patterns, containing the petal deformation caused by bending and local necking as well as the plugging deformation caused by high strain rate, large deformation, and adiabatic shear. The lower aluminum alloy plate shows crack damage, and the crack extension is more significant in the test. The aluminum honeycomb core layer shows shear damage, and the damaged aluminum honeycomb structure is deformed all around. Figure 5b,c shows the deformation and damage to the aluminum honeycomb core plate at 100 °C and 200 °C. It is observed that the aluminum honeycomb core plate deformation at 100 °C and 200 °C is mainly caused by the ductile hole enlargement. The front of the aluminum honeycomb core plate is pushed by the projectile, and metal deformation gradually occurs. This phenomenon was also observed by Senthil [23]. By comparing the deformation of aluminum honeycomb core plates at 200 °C and 25 °C, it is found that at 200 °C, the deformation of metal caused by bending is more obvious, and the plugging deformation caused by shear is smaller.

2.3.2. Strain–Time Curve Comparison

The position of the strain foil and the corresponding element are shown in Figure 6. Figure 7 gives the comparisons of the strain–time curve between test and simulation at different temperatures. Comparisons of maximum strain values between test and simulation are given in

Table 7. Comparison of maximum strain values between test and simulation.

Temperature/°C	Test Value	Simulation Value	Relative Error %
25	5797.5	5462.2	5.78
100	4156.7	4228.2	−1.72
200	4209.4	3879.1	7.85

.

It is observed that the experiment strain–time curve deviates from the simulation because the pointed bullet is affected by external resistance during firing. However, from the perspective of the overall change trend of the strain–time curve, the experiment is basically consistent with the simulation conclusion. It can be seen from Table 7 that the relative error of the maximum strain between the test and the simulation is less than 8%. The simulation results are in good agreement with the experimental ones. So, this fitted model is accurate enough to describe the dynamic behavior of aluminum honeycomb core plates at high strain rates and elevated temperatures.

3. Containment Capability Analysis Under Different Temperatures

In this section, the numerical simulations of a broken rotating blade impacting an aluminum honeycomb sandwich casing at different temperatures are conducted. The numerical simulations are performed using the nonlinear finite element code LS-DYNA. The explicit finite element software of LS-DYNA R13.0 can be used in conjunction with the Johnson–Cook material model to solve the penetration problems involving fracture.

The total thickness of the aluminum honeycomb sandwich casing is 5 mm, the thickness of the upper and lower casing shells is 1 mm, the length of the single aluminum honeycomb core is 6 mm, the inner diameter of the single aluminum honeycomb core is 3 mm, the thickness of the aluminum foil is 0.05 mm, and the height of the aluminum honeycomb core is 3 mm. Appropriate stiffness and viscous form of hourglass controls are implemented to reduce hourglass energy. The mesh details and material for the blade and the casing are compiled in

Table 8. Grid model parameters of the vane aluminum honeycomb sandwich casing.

Name	Material	Grid Type	Units Number
Casing shell	7075-T651	Solid164	18,550
Aluminum honeycomb core	5052	Solid163	47,130
Blade	TC4	Solid164	12,812

. The grid model is shown in Figure 8. The material constants, parameters of the state equation, and thermophysical parameters are listed in

Table 9. Johnson–Cook model parameters of the blade.

Material Trademark	Essential Parameter
TC4	E/GPa	v				ρ (Kg/m3)			Tm/K				Tr/K				Cp (J/Kg·K)
	114	0.31				4428			1878				293				611
	Johnson–Cook parameter
	A/MPa			B/MPa				C				n				m
	1098			1092				0.0014				0.93				1.1
	Johnson–Cook parameter
	d1		d2				d3				d4				d5
	−0.09		0.27				0.48				0.014				3.87
	Equation of state parameter
	C				S1					γ0				V0
	5.13				1.028					1.23				0

and

Table 10. Thermal physical parameters of the blade.

TC4 Thermophysical Parameter	Temperature/°C
	20	100	200	300	400	500
Thermal conductivity (W/(m·°C))	6.8	7.4	8.7	9.3	10.3	11.8
Heat capacity (J/Kg °C)	611	624	653	674	691	703

.

The connection part for the upper shell and the aero-engine wing in the z direction is set to a fixed constraint. The blade suddenly fails and breaks when the aero-engine is running, and the broken rotating blade’s movement trajectory will be offset under the influence of rotational inertia. We can decompose the initial velocity of the broken rotating blade into the centroid velocity v and the angular velocity ω, as shown in Figure 9. The arrows represent the velocity components, and the dotted lines are the center lines in Figure 9.

In this paper, the centroid velocity v of the broken blade is 300 m/s, and the angular velocity ω of the broken blade is 19,108 r/min. A constant rotation rate is applied to the blade by the VELOCITY_GENERATION keyword in LS-DYNA. The same temperature load is applied to the blade and to the aluminum honeycomb sandwich casing by the INITIAL_TEMPERATURE_SET keyword. The simulation data output interval is 10 μs, the image output interval is 20 μs, and the total solution time is 2000 μs. The contact settings and other thermal analysis solution conditions are the same as in Section 2.2. The equivalent stress nephogram and local magnification diagram of the blade impact casing at 25 °C, 100 °C, 200 °C, 300 °C, and 400 °C are shown in Figure 10.

From Figure 10, it can be observed that there are differences in the damage to blades at different temperatures. The damage degrees in the aluminum honeycomb sandwich casing are similar, and the broken blades are all contained in the casing after impacting at 25 °C, 100 °C, and 200 °C. The damage degree of the casing under 300 °C is obviously increased, but the broken blade is still contained in the casing. The damage degree of the casing under 400 °C is significantly increased, and the broken blade penetrates the casing and flies out. So, the critical containment temperature is between 300 °C and 400 °C. Figure 11 shows the time history curves of the blade’s kinetic energy and the casing’s internal energy.

From Figure 11a, it is found that the variation trend of blade kinetic energy with time is generally consistent at different temperatures. In the period of 0~1000 μs, the broken blade and the casing impact violently, and the kinetic energy of the blade decreases rapidly. In the period of 1000~2000 μs, the impact process is over between the broken blade and the casing, but the broken blade continues to move depending on the remaining kinetic energy. Therefore, the kinetic energy change of the broken blade tends to be smooth. From Figure 11b, it is shown that the internal energy time course curve of the casing first increases rapidly to reach a relatively flat status and then increases slightly again from 0 °C to 300 °C. At 400 °C, the casing’s internal energy rapidly increases to a certain value and then tends to be stable. Combined with Figure 11, we know that the blade coil deformation occurs after the blade impacts the casing between 0 °C and 300 °C. Because the broken blade does not penetrate the casing, the coiled blade and the inner surface of the casing collide twice, and the casing’s internal energy continues to grow after leveling off.

At 400 °C, the penetration phenomenon occurs when the broken blade impacts the casing; the broken blade fails to make secondary contact with the casing, so the casing’s internal energy tends to be stable. Figure 12 shows the curve of the broken blade’s kinetic energy loss and the casing’s internal energy increment at various temperatures.

The overall trend of the blade’s kinetic energy loss and the casing’s energy increment shows an increase in Figure 12. Considering the damage and energy variation of aluminum honeycomb sandwich casing, it can be seen that the damage area of the casing increases with the increase in temperature. This shows that the overall containment capability of the casing is declining.

4. Optimizing Design

4.1. Response Surface Modeling

To achieve higher performance, aero-engines require a lightweight design for their overall structure, aiming to reduce their total mass as much as possible while ensuring structural strength. As the protective casing of an aero-engine, the optimal design of the casing structure is to realize its lightweight design on the premise of ensuring safety. The aluminum honeycomb sandwich thickness has a great influence on the containment capability of the casing. In order to investigate the casing shell thickness and honeycomb core size effects on the containment capability of casing, simulations were carried out for different casing shell thickness and honeycomb core size combinations. The internal and external shells of the casing were generally the same material and the same thickness for the sake of avoiding the pull–bending coupling effect and the warping deformation of the casing. Four design variables were selected: aluminum honeycomb sandwich casing inner shell thickness ×1, aluminum honeycomb core height ×2, single aluminum honeycomb core edge length ×3, and single aluminum honeycomb core thickness ×4. The design variables are shown in Figure 13. According to the specification and dimension parameter range of aluminum honeycomb sandwich casing, the range of design variables was determined as shown in

Table 11. Design variables and their value ranges.

Design Variable	Initial Conditions	Lower Limit Value	Upper Limit Value
x1/mm	1	0.5	1.5
x2/mm	4	3	5
x3/mm	6	3	8
x4/mm	0.05	0.01	0.08

.

Common experimental design methods include Central Composite Design (CCD), Orthogonal Experimental Design (OED), and Latin Hypercube Design (LHD), etc. This paper adopts the Central Composite Design method. For the case of n factors and two levels, the experimental points consist of three parts: central points, axial points, and factorial points, and a schematic diagram of their composition is shown in Figure 14.

The relationship among the designed test points z, design variables n, and factorial coefficient ζ is shown in Equation (4).

$$z=2^{n-\zeta }+2n+1$$

(4)

The vertices of the quadrilateral in Figure 14 are 2^n−ζ factorial design points, which are used to estimate the first-order terms and interaction terms; the asterisk points are 2n axial points, which are used to estimate the pure quadratic terms; the center point of the quadrilateral is used to evaluate the accuracy and error of the design.

According to the composition formula of test points in Equation (4), with the optimized design variable n = 4 and factorial coefficient ζ = 0, the total number of tests is 25. Considering the requirements of casing containment and lightweight design, the kinetic energy loss k_b caused by the broken blade piercing the casing and the casing mass m_c are taken as response values. The calculation results are shown in

Table 12. Test scheme and response value.

Number	x1	x2	x3	x4	kb/J	mc/kg
1	1	4	5.5	0.0489	313.77	18.49
2	1	4	3	0.0489	289.81	18.41
3	1	4	8	0.0489	244.48	18.56
4	1	3	5.5	0.0489	300.78	18.44
5	1	5	5.5	0.0489	248.03	18.53
6	1	4	5.5	0.01	272.18	18.38
7	1	4	5.5	0.08	357.81	18.59
8	0.5	4	5.5	0.0489	70.56	15.60
9	1.50	4	5.5	0.0489	418.41	21.37
10	0.6479	3.2958	3.7395	0.0270	146.85	16.34
11	0.6479	3.2958	7.2605	0.0270	144.76	16.39
12	0.6479	4.7042	3.7395	0.0270	180.42	16.37
13	0.6479	4.7042	7.2605	0.0270	184.20	16.43
14	0.6479	3.2958	3.7395	0.0708	131.39	16.42
15	0.6479	3.2958	7.2605	0.0708	153.77	16.55
16	0.6479	4.7042	3.7395	0.0708	162.61	16.48
17	0.6479	4.7042	7.2605	0.0708	227.49	16.66
18	1.3521	3.2958	3.7395	0.027	280.08	20.40
19	1.3521	3.2958	7.2605	0.027	256.49	20.45
20	1.3521	4.7042	3.7395	0.027	252.53	20.42
21	1.3521	4.7042	7.2605	0.027	260.63	20.49
22	1.3521	3.2958	3.7395	0.0708	304.54	20.48
23	1.3521	3.2958	7.2605	0.0708	290.51	20.61
24	1.3521	4.7042	3.7395	0.0708	244.50	20.54
25	1.3521	4.7042	7.2605	0.0708	317.41	20.72

.

In light of the optimization analysis module, we can obtain the determination coefficient R² and the adjustment determination coefficient R²_adj in variance analysis. The determination coefficient R² and the adjustment determination coefficient R²_adj can verify the fitting degree of the response surface. These two parameters are closer to 1, so the predictive power of the response surface model is stronger, and these two parameters are defined as follows:

$${{\displaystyle R}}^2={\displaystyle \frac{{{\displaystyle \sum _{i=1}^n\left({\stackrel{\wedge }{y}}_i-\stackrel{-}{y_i}\right)}}^2}{{{\displaystyle \sum _{i=1}^n\left(y_i-\stackrel{-}{y_i}\right)}}^2}}$$

(5)

$${{\displaystyle R}}_{adj}^2=1-{\displaystyle \frac{{{\displaystyle \sum _{i=1}^n\left({\stackrel{\wedge }{y}}_i-\stackrel{-}{y_i}\right)}}^2(n-1)}{{{\displaystyle \sum _{i=1}^n\left(y_i-\stackrel{-}{y_i}\right)}}^2(n-k-1)}}$$

(6)

In the formula, n is the design point number, k is the degree of freedom, and $y_i、{\stackrel{\wedge }{y}}_i、{\stackrel{-}{y}}_i$ represent the measured value, predicted value, and average measured value, respectively. The determination coefficient R² and the adjustment determination coefficient R²_adj of the response surface model are shown in

Table 13. Fitting values.

Response Values	R2/%	R2adj/%
kb	99.325	99.082
mc	99.831	99.793

.

According to Table 12, the response surface model has strong prediction ability, and the fitting precision of the response surface model meets the optimal design standard.

4.2. Parameter Optimization

The design goals are the lightest mass for the casing and maximum kinetic energy loss for the broken blade. The mathematical description of this optimization problem is given as follows:

$$\begin{gathered} \mathrm{m}\mathrm{i}\mathrm{n} m_c \\ \mathrm{m}\mathrm{a}\mathrm{x} k_b \\ s.t.\left\{\begin{array}{l}{x}_{\min }\le x_i\le x_{\max }\\ m_c\le m_{initial}\\ k_b\le k_{initial}\end{array}\right. \end{gathered}$$

(7)

In the equation, $x$_i is the design variable, $x$_min and $x$_max are the lower and upper limits of the design variables, mc is the mass of aluminum honeycomb sandwich casing, kb is the kinetic energy loss of the broken blades, m initial is the mass of the initial aluminum honeycomb sandwich casing model, and k initial is the kinetic energy loss of the initial broken blades. The MOGA algorithm used an initial population of 100, a crossover probability of 0.7, and a mutation rate of 0.3. After 50 iterations, three optimal solution sets were derived for optimizing the aluminum honeycomb sandwich structure casing. The specific parameters are shown in

Table 14. Candidate optimization scheme.

	1	2	3	Original Parameter
x1/mm	0.946	0.933	0.921	1
x2/mm	3.761	3.729	4.135	4
x3/mm	6.318	5.646	6.426	6
x4/mm	0.0202	0.0209	0.0198	0.05
kb/J	372.26	352.09	339.07	326.85
mc/kg	18.08	18.003	17.95	19.22

.

Compared with the original parameters of the aluminum honeycomb sandwich casing, it can be seen that the three optimized solution sets all achieve the purpose of lightweight casing and containment capability improvement. The casing mass of Scheme 3 is the lowest: 6.6% less than the original mass. The casing mass in Scheme 1 and Scheme 2 is reduced by 6.2% and 5.9% of the original mass, respectively. The broken blade kinetic energy loss of Scheme 1 is the largest, at 13.89% more than the original kinetic energy loss. The broken blade kinetic energy loss in Scheme 2 and Scheme 3 is increased by 7.72% and 3.74%, respectively. Considering the optimization degree of the two parameters of casing mass and broken blade kinetic energy loss, Scheme 1 is selected as the final casing optimization design scheme.

In order to reflect the containment capability improvement of aluminum honeycomb sandwich casing after optimization, the optimized casing model is simulated. The time course curves of the blade kinetic energy and casing internal energy before and after optimization are analyzed as shown in Figure 15.

As can be seen from Figure 15, the kinetic energy loss of the broken blade in the pre-optimized model is 326.85 J, and the casing internal energy increment is 59.83 J. After optimization, the kinetic energy loss of the broken blade is 372.26 J, and the casing internal energy increment is 72.03 J. By comparing the conclusions before and after optimization, it can be seen that the broken blade contact collisions take longer, and the broken blade kinetic energy loss and casing energy increase are larger, which indicates that the containment capability of the optimized aluminum honeycomb sandwich casing is significantly improved.

5. Conclusions

Impact tests on aluminum honeycomb sandwich panels were conducted using a temperature-adjustable impact test system. A finite element model of the aluminum honeycomb sandwich panel under impact by a pointed bullet was established using Creo, Hypermesh, and LS-DYNA. Strain–time curve analyses of the aluminum honeycomb sandwich panel at 25 °C, 100 °C, and 200 °C were performed. The reliability of the finite element model was verified by comparing the strain–time curves from the test and the simulation.

Based on the finite element model of an aero-engine compressor casing, analyses of the casing’s containment capability at different temperatures were conducted according to the blade’s kinetic energy loss, the casing’s internal energy increment, and containment status, among other factors.

The parameters, including the thickness of the casing’s inner shell, the height of the aluminum honeycomb core layer, the edge length of a single aluminum honeycomb core, and the thickness of a single aluminum honeycomb core, were optimized. Compared with the original model, the mass of the optimized casing was reduced by 6.2%, while the kinetic energy loss of the broken blade was increased by 13.89%. The optimized casing not only achieves the goal of lightweight but also possesses better containment capability.