Research on Runway Blockage Probability through Simulation

Abstract

The damage and blockade of airport runways are important objectives in modern warfare. The minimum takeoff and landing window is regarded as the damage criterion of an airport runway. On the basis of Mathematica, a simulation is performed by judging whether the inequality group has a solution domain. An improved algorithm for judging whether an airport has a minimum takeoff and landing window is proposed. The algorithm is used to calculate and analyze the blockade probability of an ordinary missile to an airport and the influence of the performance of a guided submunition on the blockade of an airport runway. Results show that increasing the number of submunition, reducing the circular error probable of submunition, and increasing the number of subprojectiles can increase the blocking capability of submunition, while increasing the dispersion radius of subprojectiles will first increase the blocking probability and then decrease it. Through simulation, the inflection point of the blocking efficiency of submunitions is found. This article improves the judgment formula of takeoff and landing window, Compared with the window scanning method, the proposed method solves the problems of error caused by excessively large scanning step and extremely long calculation time caused by considerably small scanning step.

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

1. Introduction

In modern wars, the guarantee of air power plays an important role in national defense. As an important part of an airport, a runway is more obvious than other targets, which makes it the most vulnerable area. Another important issue is the availability of an area for aircraft to take off after the runway is attacked. Many criteria exist for evaluating missile damage to a target, which are generally based on the damage mechanism of a shock wave and debris caused by the missile to the target; the corresponding damage model is established, and the damage level is determined in accordance with damage criteria [1].

For the airport runway, the crater caused by the missile is the most important factor. Airport runways use the minimum takeoff and landing window as a criterion to judge whether an airport is blocked. The takeoff and landing window is the smallest intact rectangular area required for a fixed-wing aircraft to take off and land, which means that there should be no craters in this area. After an attack, if no area can be found within a certain period to meet the requirements for takeoff, then the runway is considered to have been blocked successfully. In the past, most research objects were ordinary missiles [2, 3]. With the development of technology, the damage effect of a guided submunition is enhanced [4], and the blockade efficiency is increased. Many factors influence the damage effect of a guided submunition on a target [5, 6], such as the shooting accuracy of the submunition, the dispersion radius of subprojectiles, and the size of a crater caused by subprojectiles. In particular, the dispersion radius of subprojectiles and shooting accuracy of submunition have a great impact on the airport blockade probability. Because airports have many sizes, the runway parameters also should be considered when analyzing the influencing factors of a guided submunition on airport damage [7]. At present, when analyzing the damage of a guided submunition to an airport, the Monte Carlo method [8] is often used to simulate and calculate the impact point [9, 10]. Three methods are adopted to judge whether a minimum takeoff and landing window exists; they are the pixel simulation method [11], random sampling method, and region search method [12]. Prabhakar [13] developed software to evaluate the damage of a guided submunition to an airport and other facilities. They considered many factors that affect the damage of a guided submunition to a target, but their analysis of the standard of airport damage and influencing factors is insufficient. Xiaomei Wang [11] used the area search method to study the impact point of a submunition and the blockade probability of an airport runway, but they did not research the situation in which a plane takes off with an angle to the runway. Junfei Xu [12] used inequality to judge whether the impact point of subprojectiles has an influence on the takeoff and landing window. However, an analysis of the probability of blocking the runway is lacking, especially about the dispersion radius of subprojectiles, the number of subprojectiles, the size of the runway, and the takeoff and landing window. The inequality they used can be further optimized, and the core method is still a region search method. At present, most region search methods have the problems of an excessively large scanning step size resulting in an error and an extremely small scanning step size resulting in long calculation time.

Because the past research about the existing form of takeoff and landing window has imperfection and the judgment formula also has some optimization space, this paper optimizes the judgment formula of takeoff and landing window. In order to solve the inaccurate calculation problem of window scanning method, in this article, finding the takeoff and landing window is transformed into judging whether the inequality system has a solution domain. This research solves the problems of region search methods, and the solution domain diagram is processed to quickly judge whether there is a solution domain by judging the image quality. In this way, it is more convenient to simulate the blocking probability. The calculation of blocking probability is no longer affected by scanning step size. The simulation software is Mathematica [14]. The influence of various factors on the blockade probability is systematically analyzed, and a formula is provided to judge rapidly whether an airport can be blocked. This method can effectively simulate and analyze the blockade probability of common missiles, guided submunitions, and other weapons, which lays a foundation for the damage simulation research of other weapons in the future.

2. Blockade Criteria for Airport Runways

As the most important infrastructure of an airport, a runway undertakes the tasks of taxiing, taking off and landing, and parking. An airport runway is generally rectangular, and different layout designs are adopted in accordance with different types of airports. Airport runways are composed of main and auxiliary runways, and the length and width of runways should meet the requirements of safe takeoff and taxiing. The main factors affecting the probability of runway blockade are the characteristics of airport runways and damage weapons. The characteristics of airport runways include the strength and thickness of runway surfaces and the size of runways [15]. When the damage weapon is shrapnel, the characteristics of the weapon include the number of submunition, the crater area, the shooting accuracy of the submunition (circular error probable (CEP)), and the dispersion radius of subprojectiles. Generally, the takeoff and landing window is used as the criterion of airport damage, which is related to such factors as the taxiing distance and the track width of the aircraft.

Assuming that the runway length is l, the runway width is h, and the crater radius is r, the required length and width of the takeoff and landing window are m and n, respectively. Figure 1 shows the impact range of an effective antirunway projectile. S is the takeoff and landing window required by aircraft. The width of regions ②, ④, ⑤, and ⑦ is r, and regions ①, ③, ⑥, and ⑧ are a quarter circle of radius r. When a missile hits these areas, the takeoff and landing window will be affected by the projectile. At this time, the area shown by S will fail. Therefore, judging whether a runway is blocked can be realized by judging whether area S exists on the runway that makes all missiles hit outside the area as shown in Figure 1.

[figure(s) omitted; refer to PDF]

After a runway is damaged, an aircraft can take off via two ways: one is to take off along the runway length, and the other is to take off at a certain angle with the runway length, as shown in Figure 2. Thus, research on damage criteria should consider the existence of a certain angle between the takeoff and landing window and the runway length.

[figure(s) omitted; refer to PDF]

3. Simulation Principle

The established coordinate system is shown in Figure 3, and the coordinate axis is set at the end of the runway. The coordinates of projectile P_i are (e_i, f_i). In accordance with the above analysis, when the projectile falls on the rectangles EFIJ and KLGH and four quarter circles centered on A, B, C, and D, a takeoff and landing window exists.

[figure(s) omitted; refer to PDF]

In accordance with the vector rule, when the projectile falls outside the rectangle EFIJ, the condition is transformed into the system of inequalities, which is shown as follows:$$\begin{array}{}\text{(1)}& \begin{array}{c}\stackrel{⟶}{\text{JI}}\times \stackrel{⟶}{\text{JP}}\cdot \stackrel{⟶}{\text{FE}}\times \stackrel{⟶}{\text{FP}}\le 0\\ \text{or}\\ \stackrel{⟶}{\text{EJ}}\times \stackrel{⟶}{\text{EP}}\cdot \stackrel{⟶}{\text{IF}}\times I\stackrel{⟶}{P}\le 0\end{array}\mathrm{.}\end{array}$$

When the projectile falls outside the rectangle KLGH,$$\begin{array}{}\text{(2)}& \begin{array}{c}\stackrel{⟶}{\text{KH}}\times \stackrel{⟶}{\text{KP}}\cdot \stackrel{⟶}{\text{GL}}\times \stackrel{⟶}{\text{GP}}\le 0\\ \text{or}\\ \stackrel{⟶}{\text{LK}}\times \stackrel{⟶}{\text{LP}}\cdot \stackrel{⟶}{\text{HG}}\times \stackrel{⟶}{\text{HP}}\le 0\end{array}.\end{array}$$

For the convenience of calculation, when a projectile falls on the outside of four quarter circles, it can be regarded as falling on the outside of four circles with A, B, C, and D as the center. The redundant part coincides with the rectangles EFIJ and KLGH, and it does not affect the result. In this case, the system of inequalities is$$\begin{array}{}\text{(3)}& \begin{array}{c}\stackrel{⟶}{\text{PA}}\ge r,\\ \stackrel{⟶}{\text{PB}}\ge r,\\ \stackrel{⟶}{\text{PC}}\ge r,\\ \stackrel{⟶}{\text{PD}}\ge r.\end{array}\end{array}$$

At the same time, the end points A, B, C, and D of the takeoff and landing window should be within the runway. Combined with the above analysis, let the coordinates of point A be (x, y) and the angle between AB and X axis be z such that the coordinates of the other points are B (x + m·cosz, y + m·sinz), C (x + m·cosz-n·sinz, y + m·sinz + n·cosz), and D (x-n·sinz, y + n·cosz). Each end point should satisfy the system of inequalities, which is shown as follows:$$\begin{array}{}\text{(4)}& \begin{array}{c}0\le x\le l,\\ 0\le y\le h,\\ 0\le x+m\cdot \cos z\le l,\\ 0\le y+m\cdot \sin z\le h,\\ 0\le x+m\cdot \cos z-n\cdot \sin z\le l,\\ 0\le y+m\cdot \sin z+n\cdot \cos z\le h,\\ 0\le x-n\cdot \sin z\le l,\\ 0\le y+n\cdot \cos z\le h.\end{array}\end{array}$$

We transform the problem into whether the solution domains of x, y, and z satisfying the abovementioned inequality system exist. The data are integrated into the inequality system to obtain the solution domain satisfying the inequality system.

When the inequality system has a solution domain, the runway blockade fails. If no rectangular region satisfies the condition, then the runway blockade is successful. Figure 4 shows the flow of analyzing whether the takeoff and landing window exists. Different numbers of inequalities will be generated in accordance with the number of projectiles, which can be processed using Mathematica.

[figure(s) omitted; refer to PDF]

Mathematica is scientific computing software, which combines numerical and symbolic computing engine, graphics system, programming language, text system, and advanced connection with other applications [14]. In accordance with the actual situation, the airport runway and missile data are substituted into the inequality, then the solution domain of the inequality system is obtained. The solution domain of the inequality system is output in the form of image by using RegionPlot3D code to determine rapidly whether the runway is blocked successfully and save analysis time, as shown in Figure 5.

[figure(s) omitted; refer to PDF]

When the blocking probability needs to be simulated for many times, ParallelTable code is used to complete the simulation work. An image recognition method is used for statistics to improve the work efficiency. The axis is removed from the solution domain diagram and processed into the form shown in Figure 6. The quality of each image is judged using the ImageMeasurements code. When the system of inequalities has a solution domain, that is, the runway has a takeoff and landing window, the image quality is not 1. When the system of inequalities does not have a solution domain, that is, no takeoff and landing window exists in the runway, the image is blocked and the image quality is 1. The runway blockade probability can be rapidly calculated through the image quality.

[figure(s) omitted; refer to PDF]

4. Simulation of Damage to Airport Runways by Conventional Missiles

Tables 1 and 2 show the typical data of airport runways and the takeoff and landing windows for some aircraft.

Table 1

Typical data of foreign airport runways.

Runway category	Aircraft type	Runway length l (m)	Runway width h (m)
Level 1	Heavy bomber	2500–5000	60–100
Level 2	Medium bomber	2500	45–60
	Fighter-bomber	2000	45
Level 3	Fighter	1800–2000	40
Level 4	Trainer	<1800	30

Table 2

Takeoff and landing windows for some airplanes.

Aircraft type	IDF	Mirage 2000-5	F-16
Length of a takeoff and landing window m (m)	800	457	350
Width of a takeoff and landing window n (m)	20	20	20

An airport adopts a runway with a length of 3000 m and a width of 60 m, and the minimum takeoff and landing window is 400 m in length and 20 m in width. Given that the crater radius is related to the charge quantity of a missile and the strength and thickness of each layer of an airport, the simulated crater radius is randomly distributed in the range of 4–6 m. The missiles are 10, 20, and 30, and the drop points are randomly and evenly distributed within the airport runway. The simulation times are 100.

The missile landing point (e_i, f_i) is$$\begin{array}{}\text{(5)}& \begin{array}{c}{e}_i=l\cdot u_i,\\ f_i=h\cdot u_i,\end{array}\end{array}$$where u is a uniformly distributed random number between (0, 1).

Table 3 presents that the blockage probability of ordinary missiles to an airport is low, and a large number of missiles are needed to complete the blockade of the airport. The simulation results show that the uniform distribution of impact points on the airport is an ideal state, but, in fact, it is difficult to achieve due to the problem of missile accuracy. Therefore, to save the economic cost and achieve the tactical purpose rapidly and efficiently, countries began to develop submunition and other weapons to block airports.

Table 3

Simulation result.

Number of missiles	10	20	30
Blockage probability	3 (%)	39 (%)	88 (%)

5. Damage of Submunitions to Airport Runways

For submunitions, the main factors affecting the probability of blocking runways are the characteristics of runways and submunitions. The factors that influence the blocking probability of submunitions include the number of submunitions, the crater area, the shooting accuracy of submunitions, and the dispersion radius of subprojectiles [15]. Generally, CEP is used to express the accuracy, which is the sum of the systematic error and the dispersion error of the impact point. CEP is defined as [12]$$\begin{array}{}\text{(6)}& P=\frac{1}{2\pi {\sigma }_x{\sigma }_y}{\displaystyle \int {\displaystyle {\int }_{x^2+y^2\le R_1^2}\exp -\frac{1}{2}\frac{{x-{\mu }_x}^2}{{\sigma }_x^2}-\frac{{y-{\mu }_y}^2}{{\sigma }_y^2}\text{d}x\text{d}y}}=\mathrm{0.5.}\end{array}$$

In the formula, P is the hit probability of shrapnel, R₁ is the radius of the dispersion circle; μ_x and μ_y are the estimation of systematic error; and σ_x and σ_y are the estimation of random error.

The actual coordinates of a submunition are as follows [16]:$$\begin{array}{}\text{(7)}& \begin{array}{c}X=X_m+u\cdot \text{CEP}/1.1774,\\ Y=Y_m+v\cdot \text{CEP}/1.1774,\end{array}\end{array}$$where (x, y) are the actual impact points of the submunition, u and $v$ are two random numbers that are independently subject to normal distribution in (0, 1), and (X_m, Y_m) are the aiming points of the submunition.

The coordinates of a subprojectile (e_i, f_i) are as follows:$$\begin{array}{}\text{(8)}& \begin{array}{c}{e}_i=X+R\cdot \sqrt{j}\cdot \cos k\cdot 2\pi ,\\ f_i=Y+R\cdot \sqrt{j}\cdot \sin k\cdot 2\pi ,\end{array}\end{array}$$where j and k are uniformly distributed random numbers between (0, 1) and R is the scattering radius.

As shown in Figure 7, the aiming points of the submunition are evenly distributed along the center line of the runway, d is the distance among the aiming points of the submunition, and the distance between the aiming points of the submunition at both ends of the runway and the boundary of the runway is d/2.

[figure(s) omitted; refer to PDF]

Therefore, the coordinates of the preaiming points of the submunition should be $$\begin{array}{}\text{(9)}& \begin{array}{c}{X}_i=\frac{d}{2}+i-1\cdot d,\\ Y_i=\frac{h}{2},\\ d=\frac{l}{S}.\end{array}\end{array}$$

In the formula, l is the length of the runway, h is the width of the runway, and S is the number of submunitions. Figure 8 shows the simulation of the drop point of subprojectiles.

[figure(s) omitted; refer to PDF]

Some subprojectiles will be dispersed out of the runway during the process of submunition dispersion, and these subprojectiles will not affect the runway. A schematic of the impact point of subprojectiles is shown in Figure 9. The subprojectiles should be filtered to accelerate the calculation and improve the simulation efficiency. The coordinate system in Figure 3 is used when the subprojectile coordinates (e_i, f_i) satisfy the following formula:$$\begin{array}{}\text{(10)}& \begin{array}{c}-r\le e_i\le l+r,\\ -r\le f_i\le h+r.\end{array}\end{array}$$

[figure(s) omitted; refer to PDF]

At this time, the subprojectiles will damage the runway, which are called effective subprojectiles. All subprojectiles are screened, and all valid ones are recorded. The coordinates of effective subprojectiles are integrated into the inequality system to judge whether a solution domain exists, then the blockage probability is obtained.

In accordance with the analysis, the simulation process is as follows:

(1) The runway data and the minimum takeoff and landing window size are input

(2) The number of submunition, CEP, the dispersion radius of subprojectiles, and the size of the subprojectile crater are input

(3) The impact points of submunitions and subprojectiles are generated, and the invalid subprojectiles are filtered out

(4) A solution domain graph of the inequality system is generated, the existence of a solution domain is judged using image quality, and the blockage probability is recorded

The specific simulation process is shown in Figure 10.

[figure(s) omitted; refer to PDF]

The same airport and submunition data as those used by Xiaomei Wang [11] are used to verify the accuracy of the simulation. Suppose that the CEP of a submunition is 100 m, the dispersion radius R is 150 m, every submunition contains 120 subprojectiles, and the radius of the subprojectile crater is 2 m. The submunition dispersion method presents a uniform distribution in the circle. A runway with a length of 3000 m and a width of 60 m is adopted for the target airport. The minimum takeoff and landing window has a size of 400 m × 20 m. The simulation times is 1000.

Table 4 shows the comparison of simulation results. The blockage probability obtained using the method in this article is generally smaller than that obtained using the method of Xiaomei Wang [11]. This paper considers the situation in which the takeoff and landing window has a certain angle with the runway, and no omission exists due to the problem of scanning step size. Consequently, the result calculated in this paper is smaller than that in Reference [11].

Table 4

Comparison of simulation results.

Simulation method	Number of submunitions
	5 (%)	6 (%)	7 (%)	8 (%)	9 (%)
Method of this article	1.6	10.1	20.1	31.2	42.3
Method of Xiaomei Wang [11]	1.8	11	25.7	36	46

Figure 11 clearly shows that in the case of five or six submunitions damaging the runway, the difference in blockage probability between the two methods is minimal. The main reason is that the distance among the aiming points is excessively large and the blockage probability is extremely small. When the number of submunitions is increased to 7, 8, and 9, the difference in blockage probability between the two methods becomes evident. The reason is that with the decrease in the distance among the aiming points, the blockage probability decreases due to the existence of an oblique takeoff and landing window. Hence, the gap between the two methods increases.

[figure(s) omitted; refer to PDF]

5.1. Simulation Analysis of Submunition Blockage Probability

5.1.1. Influence of the Number of Submunitions on the Blockage Probability

To study the influence of the number of submunitions on the airport blockage probability, in accordance with the above data, the influence of the number of submunitions on the airport blockage probability is studied by changing the number of submunitions. The simulation results are shown in Figure 12.

[figure(s) omitted; refer to PDF]

From the figure, increasing the number of submunitions can increase the blockage probability, which is approximately linear at the beginning. When the number of submunitions increases to a certain extent, the effect of increasing the number of submunitions on improving the blockage probability decreases. The turning point is at the place where the number of submunitions is 11. At this time, the dispersion radius of subprojectiles is 150 m, which is close to 136 m half of the distance among the preaiming points of the submunitions. The coverage area of subprojectiles has overlapped. This condition is speculated as the reason for the decline in blockage efficiency, which will be discussed later.

5.1.2. Influence of the Takeoff and Landing Window Length and Runway Length on the Blockage Probability

Owing to aircraft performance and other reasons, different aircraft need diverse takeoff and landing window lengths. Simulation analysis is carried out to study the influence of the takeoff and landing window length on the blockage probability. The runway length is 3000 m, and the number of submunitions is 6. Other data are the same as above. The simulation results are shown in Figure 13.

[figure(s) omitted; refer to PDF]

When the length of the takeoff and landing window increases, the blockage probability of the same number of submunitions increases. When the length of the takeoff and landing window is 250–400 m, the blocking probability increases more gently than that when the length is 400–800 m. Given that few subprojectiles landed on the runway and the probability of blockade is small, the difference caused by the change in takeoff and landing windows is minimal.

Although the blockage probability of the same number of submunitions under different runway lengths is different, when the number of submunitions is changed such that the preaiming points of the submunitions are the same, the blockade probability is the same. To verify the analysis, the runway length is set to 4000 m, and other data are the same as above. The simulation results are shown in Table 5.

Table 5

Blockage probability simulation at 4000 m runway length.

Number of submunitions (pieces)	8	9	10	11	12
Blockage probability (%)	10.1	21.2	23.4	35.9	38.8

The simulation results are transformed into variables with the preaiming point spacing of submunitions. The data in Figure 11 are processed and plotted in Figure 14.

[figure(s) omitted; refer to PDF]

When the preaiming points of submunitions are evenly distributed along the runway axis, the damage probability of different numbers of submunitions to a 4000 m runway falls near the damage probability curve of submunitions to a 3000 m runway. Therefore, the effect of the number of submunitions on the blockage probability of runways is essentially the distance among the preaiming points of submunitions. When we obtain the blockage probability curve of a certain type of submunition to an airport with different numbers, we can apply the curve to judge the blockage probability of this type of submunition to the airport with other lengths by conversion. Then, we can simplify the calculation of the blockage probability of the submunition to the airport runway.

5.1.3. Influence of the Number of Subprojectiles on the Blockage Probability

Six submunitions and different numbers of subprojectiles are used to simulate the impact of the number of subprojectiles on the blockage probability. Other data are the same as above. The simulation results are shown in Table 6.

Table 6

Influence of the number of subprojectiles on the blockage probability.

Subprojectile number (pieces)	40	60	80	100	120	140	160	180	200
Blockage probability (%)	0.3	3	5.6	7.8	10.1	12.7	15.1	18.7	22.2
Subprojectile number (pieces)	210	220	240	260	280	300	320	340	360
Blockage probability (%)	23.7	25.4	28.3	29.8	30.9	31.3	31.5	31.8	32.2

Figure 15 depicts that when the number of subprojectiles is 20–40, the blockage probability increases slowly. Given that the density of subprojectiles in the dispersion range is small, only a few subprojectiles fall on the runway, and increasing the number of subprojectiles cannot produce evident changes in the blockade probability. When the number of subprojectiles is 80–240, the impact of the number of subprojectiles on the blockage probability tends to increase. When the number of subprojectiles is more than 240, the curve tends to be flat. The analysis shows that when a large number of subprojectiles fall on the runway, no takeoff and landing window exists in the coverage area of the submunition, and the window exists in the area not covered by the submunition. Thus, the number of bullets should be reasonably selected to save the economic cost and ensure the blockade efficiency.

[figure(s) omitted; refer to PDF]

5.1.4. Influence of the Dispersion Radius of Subprojectiles on the Blockage Probability

Six submunitions are used, the number of subprojectiles is 120, and different dispersion radii of subprojectiles are chosen to simulate. The simulation results are shown in Table 7.

Table 7

Influence of the dispersion radius of subprojectiles on the blockage probability.

Dispersion radius of subprojectiles (m)	25	50	75	100	125	150	175	200	225	250	275	300
Blockage probability (%)	0	0	0.4	2.6	7.3	10.1	19.5	27	33.5	37	37.8	38.9
Dispersion radius of subprojectiles (m)	325	350	375	400	425	450	475	500	525	550	575	600
Blockage probability (%)	39.6	38.2	37.3	35.1	32.7	29.8	27.9	26.4	23.6	21.8	18.8	16.7

Figure 15 depicts that when the dispersion radius of subprojectiles increases from 75 m to 250 m, the blocking probability increases rapidly. The probability of the subprojectiles blocking the runway is affected by the coverage area. With the increase in the dispersion radius of subprojectiles, the probability of the subprojectiles landing on the airport runway increases. The coverage area also increases, which makes the blockade probability increase rapidly. When the radius increases from 250 m to 350 m, the change in the blockage probability is relatively gentle. At this time, the dispersion radius of subprojectiles begins to overlap, and the blocking efficiency decreases. When the radius increases from 350 m to 600 m, the density of subprojectiles and the number of subprojectiles falling on the runway decrease, which reduces the blockage probability. As a result, the influence of the dispersion radius of subprojectiles on the blockage probability has an extreme value. When optimizing the blocking performance, we should consider the influence of the dispersion radius of subprojectiles, and the dispersion radius should not be increased blindly. A certain subprojectile density should be ensured.

5.1.5. Influence of CEP and Crater Size of Subprojectiles on the Blockage Probability

We select 10 submunitions to study the impact of the CEP and the size of the crater produced by subprojectiles on the blockage probability. Other data are the same as above. The simulation data are shown in Table 8.

Table 8

Influence of the CEP and crater size of subprojectiles on the blockage probability.

Radius of crater/m	CEP
	100 (%)	90 (%)	80 (%)	70 (%)	60 (%)	50 (%)	40 (%)
r = 1	49.3	63.5	75.4	89	94.2	99.1	100
r = 2	54	64.7	78.7	89.3	96.3	99.2	100
r = 3	57.2	70.6	81.5	91.8	97.6	99.3	100
r = 4	60	72.6	83.6	92.7	97.7	99.4	100

The dispersion of submunitions at the preview point will produce the largest impact area on the runway. The CEP accuracy affects the probability of the submunitions falling at the preview point. In accordance with the data, with the decrease in CEP, the blockage probability of submunitions increases remarkably. Increasing the crater size of subprojectiles will also increase the blockage probability of submunitions, but the effect is unobvious compared with the effect when reducing CEP. With the increase in accuracy, when the CEP of submunitions is reduced to a certain range, the change in the influence area of submunitions on the runway at the actual dispersion point is reduced. The change in the blockage probability will also be reduced because the dispersion range of submunitions is circular. Although improving the accuracy of submunitions is more effective than increasing the explosive capability of subprojectiles, with the development of today’s airport runway repair technology, the repair time of a large crater is much longer than that of a small crater. Subprojectiles still need a certain power to ensure that the damage can produce sufficient blocking time to meet the tactical requirements.

5.1.6. Influence of the Dispersion Radius of Subprojectiles on the Blockage Probability under Different Numbers of Subprojectiles

Six submunitions are used, and the numbers of subprojectiles are 120, 210, and 300. The remaining data are the same as above to study the influence of the dispersion radius of subprojectiles on the blockage probability under different numbers of subprojectiles. The simulation results are shown in Figure 16.

[figure(s) omitted; refer to PDF]

In accordance with the simulation results, when the dispersion radius is 50 m and the number of subprojectiles is 300, the blockage probability is 0.3%. When the number of other subprojectiles is 0, increasing the number of subprojectiles can produce blocking at a lower dispersion radius. In the case of different numbers of subprojectiles, the gentle change range of the blockage probability of submunitions is different. Increasing the number of subprojectiles can increase the gentle change range, but the starting point of the gentle rise range is the same. The analysis shows that increasing the number of subprojectiles can ensure that a certain number of subprojectiles falls on the runway such that the subprojectile density can meet the requirements of maintaining or increasing the blockage probability. When the number of subprojectiles is extremely low, the blockage probability caused by increasing the dispersion radius of subprojectiles will decrease even if it does not reach the flat change range. Therefore, to ensure the stability of the blockage of the airport runway and reduce the impact caused by the fluctuation in the dispersion radius, we can select the appropriate number of submunitions to produce a certain gentle change range and choose a dispersion radius in the gentle change range to ensure the blockage of the airport runway.

5.1.7. Influence of the Dispersion Radius of Subprojectiles on the Blockage Probability under Different CEP

To research the influence of the dispersion radius of subprojectiles on the blockage probability under different CEP, four submunitions are used, the number of subprojectiles is 150, and CEP is set to 1, 25, and 50 m. Other data are the same as above. The simulation results are shown in Figure 17.

[figure(s) omitted; refer to PDF]

From the simulation results, changing the CEP of submunitions cannot change the trend of the effect exerted by the dispersion radius of subprojectiles on the blockage probability. The same gentle change range occurs under different CEP, and changing CEP only changes the blockage probability. This result shows that the influences of CEP and projectile radius are independent of each other. The analysis of Figures 14, 18, and 19 indicates that the starting point of the gentle change interval in Figures 14 and 18 is 250 m; the starting point of the gentle rise interval in Figure 19 is 375 m, which is half of the distance among the preaiming points of shrapnel (d/2). Accordingly, the next research is to determine the reason through simulation.

[figure(s) omitted; refer to PDF]

5.1.8. Influence of the Dispersion Radius of Subprojectiles on the Blockage Probability under Different Preaiming Point Spacing of Submunitions

Four, six, and eight submunitions are selected for simulation. In accordance with the previous analysis, CEP does not affect the range of gentle change in the blockage probability. Given that the blockage probability of four submunitions is extremely small when CEP = 100 m, four submunitions are studied under the condition of CEP = 50 m, and six and eight submunitions are studied under the condition of CEP = 100 m. The simulation results are shown in Figure 20.

[figure(s) omitted; refer to PDF]

The starting point of the range at which the blockage probability of submunitions changes gently is at half of the distance among the preaiming points of submunitions. The analysis shows that when the dispersion radius of submunitions is half of the distance among the preaiming points of submunitions, the dispersion ranges of subprojectiles at the preaiming points begin to overlap. As the dispersion radius continues to rise, the overlapping ranges continue to increase, the blockage efficiency begins to decline, and the changing trend of the blockage probability varies from rapid rise to slow rise. From the above analysis, we can conclude that the influence of the dispersion radius of subprojectiles on the starting point of the flat change interval is only related to the preaiming points of submunitions and has no connection with CEP and the number of subprojectiles.

5.2. Fitting Formula for Blocking Probability

In wars, judging whether airports can be blocked rapidly is sometime necessary. Therefore, a formula should be provided to calculate the probability of airport blockade accurately. When the dispersion radius of subprojectiles is greater than half of the distance among the preaiming points of submunitions (d/2), the blocking efficiency will decrease. Hence, the fitting range is selected in the area when the dispersion radius of subprojectiles is less than d/2. The runway length can be solved by converting it into the distance among the preaiming points of submunitions.

In the above analysis, a coupling effect exists between the number of subprojectiles and the dispersion radius of subprojectiles. The number of subprojectiles can affect the flat range of the dispersion radius of subprojectiles in the blockage probability. Therefore, the following research is performed to verify whether the number of subprojectiles has an impact on the area before the starting point of the interval (d/2).

The blockage probability when the dispersion radius of subprojectiles is less than 250 m (d/2) in Figure 16 is selected, and the processed data are presented in Table 9.

Table 9

Processed data.

Dispersion radius of subprojectiles (m)	Blockage probability (%)
	120 pieces/210 pieces	210 pieces/300 pieces	120 pieces/300 pieces
75	0.250	0.762	0.190
100	0.500	0.754	0.377
125	0.646	0.706	0.456
150	0.391	0.787	0.308
175	0.515	0.758	0.390
200	0.503	0.851	0.428
225	0.531	0.841	0.447
250	0.489	0.900	0.440

When the dispersion radius of subprojectiles is less than 250 m (d/2), a minimal difference in the ratio of the blockage probability of the dispersion radius of subprojectiles exists for each number of subprojectiles. The coupling effect between the number of subprojectiles and the dispersion radius of subprojectiles is small. In accordance with the previous analysis, the influence of various factors on the blocking probability can be expressed in the form of power function. Accordingly, the fitting formula has the following form:$$\begin{array}{}\text{(11)}& P=A\cdot S^{{\alpha }_1}\cdot C^{{\alpha }_2}\cdot R^{{\alpha }_3}\cdot s^{{\alpha }_4}\cdot r^{{\alpha }_5}\cdot m^{{\alpha }_6}+B.\end{array}$$

In the formula, P is the blockage probability, S is the number of submunitions, C is the CEP, R is the dispersion radius of subprojectiles, s is the number of bullets, r is the radius of crater, m is the length of the takeoff and landing window, and A, B, a₁, a₂, a₃, a₄, and a₅ are the coefficients.

The fitting results are as follows:$$\begin{array}{}\text{(12)}& P=3.9175\times {10}^{-6}\cdot S^{1.6}\cdot C^{-0.05}\cdot R^{1.0582}\cdot s^{0.3786}\cdot r^{0.05}\cdot m^{1.11}-\mathrm{33.598.}\end{array}$$

The correlation coefficient is 0.93. Given that the blockage probability of submunitions to airport runways is affected by the characteristics of runways and submunitions, various influencing factors exist, and the fitting formula has a certain error compared with simulation. Thus, in accordance with the needs of judging time and accuracy, the fitting formula or simulation times should be selected reasonably to achieve the corresponding demands.

6. Conclusion

The proposed method has the advantages of fast judgment speed and accurate results. This article considers the existence of an oblique takeoff and landing window. Compared with the window scanning method, the proposed method solves the problems of error caused by excessively large scanning step and extremely long calculation time caused by considerably small scanning step. When studying the attack capability of submunitions to airports, we should consider various factors, such as the dispersion radius of subprojectiles, CEP, the number of submunitions, the crater area, and the economic cost. In accordance with the simulation analysis, the following conclusions are drawn:

(1) The starting point of the smooth range of the radius of subprojectile scattering coverage circle in the blockage probability is 1/2 of the distance among the preaiming points of submunitions. It is not influenced by the CEP of submunitions and the number of subprojectiles. When the distance is less than this value, the blocking efficiency will decrease.

(2) Through simulation, the inflection point of the blocking efficiency of submunitions is found. The research results provide a reference for the design of submunition and the optimization of strike scheme. When the number of subprojectiles is excessively low, the blockage probability caused by increasing the dispersion radius of subprojectiles will decrease even if it does not reach the inflection point. The dispersion radius and the number of submunitions should be considered comprehensively. The subprojectiles should be designed with reasonable dispersion density.

(3) This article improves the judgment formula of takeoff and landing window. The proposed method has a wide range of applicability and feasibility. It can judge the impact of ordinary missiles, submunitions, and other warheads on airport runway damage. A fitting formula for the probability of submunition blocking the airport is given. The proposed method and the formula can provide help for the study of the best depth of explosion, the best number of missiles, the dispersion radius of subprojectiles, CEP, and the best size of bullet craters.[17–24]

Acknowledgments

Financial supports from the National Natural Science Foundation of China (Grant nos. 12032004 and 11872157) are sincerely appreciated.

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