Damage assessment of the target area of the island/reef under the attack of missile warhead

Cheng-lin Zhai ,Xiao-wei Chen

a State Key Laboratory of Explosion Science and Technology,Beijing Institute of Technology,Beijing 100081,China

b School of Mechatronical Engineering,Beijing Institute of Technology,Beijing 100081,China

c Advanced Research Institute of Multidisciplinary Science,Beijing Institute of Technology,Beijing 100081,China

Keywords:Island/reef facilities Shock wave Fragment Damage assessment Fuzzy comprehensive evaluation method

ABSTRACT Under the background of vigorously developing facilities of island/reef in the world,the economic value and strategic significance of the island/reef have become increasingly important.Unfortunately,they may easily become the attacking target of missiles in the war time.Therefore,aiming at the damage quantification of the targets in the island/reef under the missile attacking,a macroscopic damage assessment model for the target area is proposed in this paper.The model focuses on the construction of the assessment model framework.Firstly,the analytic hierarchy process and the grey relational analysis are applied to measure the importance of each target in the region through four indicators of the target hazard,striking urgency,damage advantage and mission relevance,respectively.Secondly,based on the damage mechanism of shock wave and fragments to target,the corresponding damage model is established,and the damage grade of each target in the evaluated area is obtained according to the damage criteria and grading standard.Finally,the model obtains the overall damage grade of the target area by employing the fuzzy comprehensive evaluation method,with synthesizing the importance and the damage grade of each target.Through the verification of an example,it demonstrates the certain feasibility of the model and provides a certain basis and reference for the subsequent research.©2020 China Ordnance Society.Production and hosting by Elsevier B.V.on behalf of KeAi Communications Co.This is an open access article under the CCBY-NC-ND license(http://creativecommons.org/licenses/by-nc-nd/4.0/).

1.Introduction

In recent years,as the competition for marine resources intensifies,the resulting disputes are becoming more frequent.In order to obtain more benefits and effectively defend their own interests,a lot of countries are vigorously developing and building related facilities on the island/reef.How ever,precisely because of the importance of the island/reef,it easily becomes as the destructed target in the event of war.Being as a long-range and extremely powerful attacking weapon,the missile poses a serious threat to the safety of the island/reef.Therefore,it is important to assess the damage of the facilities on the island/reef under the attacking of the missiles.

At present,there are scarce studies related to the damage evaluation of facilities on the island/reef.Sometimes it may be divided into a single target and a group targets,respectively.Regarding the damage assessment of single target,the theoretical calculation,system analysis or simulation are conducted to predict the damage degree of personnel,equipment and buildings[1,2].For the group targets damage assessment,the research gives either the damage evaluation algorithm corresponding to the same type of target,or the striking order according to the target importance[3,4],and rarely involves the assessment of the damage degree of different types of targets.In the actual situations,the damage caused by the missile warhead may involve multiple and different types of targets at the same time.Therefore,how to quantify the damage degree is of great significance for guiding and formulating the defense programs.

Aiming at the facilities on the island/reef,this paper quantitatively evaluates the targets in the area under missile attacking,and gives the final possible damage results,which can provide reference for the strike and protection of the island/reef.The relevant methods are also beneficial to the damage assessment of other targets.

2.Assessment of island/reef facilities

2.1.Island/reef facilities and their features

2.1.1.Island/reef facilities

Facilities are always limited by the conditions of island/reef and topographical factors,and thus the complex and large-scale construction is often impossible.In order to ensure its necessary functions,the facilities on the conventional island/reef include the following categories:

1)Detection and early warning devices:air traffic control radar,warning radar and search radar,etc.,mainly for the monitoring and control of offshore airspace;

2)Communication system: satellite ground station,highfrequency radio and command systems,etc.,mainly responsible for the real-time transmission and sharing of information;

3)Defense weapon:short-range air defense system,near-antiaircraft guns and motorized missile bunkers,etc.,mainly to carry out the necessary self-defense when the island/reef is attacked.

4)Aircraft system:hangars,fighter jets and runway,etc.,mainly maintaining a certain counter-attack capability;

5)Replenishment system:oil and gas reservoirs and water source,etc.,mainly for the supply of various equipment to ensure their normal operation;

2.1.2.Features of island/reef facilities

Regardless of the personnel,the facilities on the island/reef are mainly divided into constructions and equipment.Most of the islands/reefs distributed in the ocean are based on rocks and accumulated by sand.Therefore,islands/reefs are usually small in size,simple in terrain,and in a humid,windy environment.These characteristics of the island/reef determine the features of the facilities on it.

For the constructions on the island/reef,the buildings built on it are generally low in height,simple in structure and small in scale due to the influence of the topography,geology and climate of the island/reef.For the equipment on the island/reef,due to the small area of the island/reef and the limited energy,the equipment that can be deployed is very limited.Usually,the equipment on the island/reef is relatively small,such as aircraft and air defense systems.At the same time,due to the location of the island/reef,radar and communication equipment will be deployed in large numbers.

In the following,the corresponding damage model is constructed according to the features of the facilities mentioned above.

2.2.Damage assessment process of targets on island/reef

In this paper,a quantitative assessment of missile warhead damage to the island/reef target area is carried out.Since the type and quantity of targets are not fixed in the area attacked by the missile warhead,the importance of targets in the area,the damage ability of damage elements to targets and the overall damage degree of missile warhead to the island/reef area should be taken into account in the quantitative evaluation of the area.In order to visually and clearly express the damage quantification method,the specific assessment process is given,as shown in Fig.1.

3.The importance of targets on island/reef

3.1.Measuring indicators of target importance

When quantifying the damage to the targets on island/reef,the importance of different targets in the assessed area indirectly affects the overall damage grade.At the same time,island/reef is generally limited in area and defensive depth,and the dimension of the facilities built on it are relatively limited.Therefore,when conducting an attack,comprehensive consideration should be given to different objectives according to the operational tasks,the importance of the targets,and the magnitude of the threat to the striker and the length of the response time.

In this paper,four indicators are proposed to measure the importance of different targets in terms of the features of targets and the purpose of attack.The four indicators are:

1)target hazard Xi(1):It refers to the threat degree of the evaluated target to other targets in terms of its own capabilities,which can be reflected from the direct threat and the indirect threat.The higher the threat degree,the greater the hazard of the target;

2)strike urgency Xi(2):It refers to the urgency of striking island/reef targets on time scale.It is mainly reflected by the length of the target reaction time.The shorter the reaction time,the higher the urgency;

3)damage benefit Xi(3):It refers to the strategic position of different targets and their impact,which is determined by the position and role of the targets in the deployment.The higher the strategic position and the greater the role,the higher the damage benefit;

4)task relevance Xi(4):It refers to the degree of correlation between the target being attacked and the mission.The correlation degree of different missions to the same target is different.

These four indicators that measure the importance of different targets may have certain relevance in some cases.How ever,in the measurement of the importance of the island/reef targets in this paper,the four indicators are considered to be independent of each other.

3.2.Target importance calculation

Since an evaluated target is often a complex with multiple factors,it is difficult to form a clear concept in the comparison and evaluation process,and it is as the facilities in the island/reef.The grey relational analysis[5]applies the grey theory,which is a comprehensive judgment method for comparing multiple indicators with respect to a certain benchmark in the case of incomplete system information.This method can solve such problems much better,and its weight solving process is as follow s:

There are reference series X0(k)and comparison series Xi(k),where k=1,2,…,n and i=1,2,…,m.n is the number of evaluation indicators,and m is the number of island/reef facility targets involved.The correlation coefficient is defined as:

where ρ is the resolution coefficient and ρ∈[0,1].The smaller the value of ρ,the larger the resolution,generally taking ρ=0.5.

According to the definition,the number of the correlation coefficients ξi（k）has a total of n×m.Since the information is too scattered,the correlation degree of the comparison series Xi(k)to the reference series X0(k)is defined as:

Fig.1.Flow chart of damage assessment for the target area of the island/reef.

where ωkis the weight of the corresponding correlation coefficient,and if it is the same,take 1/n.

As shown in Section 3.1,the number of indicators is 4,and thus n=4.Since the indicators describing the target importance are subjective concepts and ambiguous,it needs to be quantified when performing the calculation.In this paper,the interval value method is used,and the index is divided into five level intervals based on Ref.[5],as shown in Table 1.

As the whiting value of the indicator is larger,the strategic value of the target is greater,so the reference series is taken as X0=｛max X0（1），max X0（2），max X0（3），max X0（4）｝=｛1.0，1.0，1.0，1.0｝.Based on the expert opinions and data information,the indicator values of the evaluated m targets on the island/reef are obtained.Then,the correlation coefficient of each target corresponding to the reference target is calculated by Eq.(1)and composed into a matrix.

Since the w eights of different indicators are generally different,it is necessary to consider the w eights of different indicators when calculating the correlation degree.In this paper,Analytic Hierarchy Process(AHP)[6]is used to determine the weight effect of different indicators on the target importance.Analytic Hierarchy Process is a multi-objective decision analysis method that combines qualitative analysis with quantitative calculation.The steps are as follow s:

1)Constructing the judgment matrix.According to Table 2,the domain experts should make a pairwise comparison judgment on the importance of each indicator,and then establish a judgment matrix H=(hij)n×n,where hji=1/hij,i,j=1,2,…,n and n is the number of indicators;

2)Calculating the maximum eigenvalue and eigenvector of the matrix.The maximum eigenvalue is represented by λmax,and the normalized eigenvector is the weight value;

3)The consistency checking test.Since the judgment matrix may have inconsistency,it is necessary to check the consistency of the judgment matrix.The consistency test CR=CI/RI,where CI=(λmax-n)/(n-1),and RI is the average random consistency index(RI is 0.90 when n=4).The smaller the CR,the better theconsistency of the judgment matrix.Generally,when CR＜0.1,the judgment matrix satisfies the consistency requirement,and the corresponding normalized feature vector is the index weight vector.Otherwise,the judgment matrix needs to be corrected until satisfied consistency is obtained.

Table 2 Implication table of importance scale.

After obtaining the indicator weight value,the correlation degree of each target is calculated by Eq.(2).The correlation degree can be expressed as the weight of the target in a certain sense,so the target weight value wiof each facility is obtained by normalization:

4.Power field of damage elements

The destruction of the warhead to the target is mainly through impact and explosion,etc.It converts its own energy into damage elements which cause the target damage and disable.At present,common damage elements include shock wave,fragment,jet,penetrator,temperature,sound,light,magnetism,and so on.Due to the tendency of different damage elements to target damage,the damage assessment of conventional warhead to multiple(constructions,equipment,etc.,excluding personnel)target areas is studied in this paper.According to the damage tendency of the damage elements,in this paper we focus on only shock wave and fragment.In which,the shock wave mainly damages the construction-type target,and the fragments mainly damage the equipment-type target.

4.1.Shock wave damage

Regarding the shock wave that is the main damage element of the conventional warhead,we will focus on the construction target made of most brittle materials.Since the target parameters are mainly based on the satellite images and the height scale of the island/reef buildings is usually lower or far lower than the area scale,the height information of the buildings can be ignored in the damage evaluation caused by the shock wave.In a single damage process,the shock wave is generated from a landing point explosion of the warhead and appears as a hemispherical shape with a negative correlation gradient.The damage diagram is shown in Fig.2.The rectangular area is the building equivalent area,the circular area is the shock wave damage area,and the overlapping area is the area of the actual damage building.

4.1.1.Parameters calculation of shock wave

The damage of the shock wave to the target is mainly due to the overpressure and impulse.By referring the literature,the calculation methods of shock wave overpressure and impulse are introduced.

1)Overpressure calculation[7].

Fig.2.Top view of shock wave damage.

When the warhead explodes on the ground,the shock wave can only propagate in a semi-in finite space,and the ground reflection makes the shock wave energy strengthened.According to the explosion theory and test results,the overpressure calculation formula can be obtained:

where W is the TNT equivalent of the explosive,kg;mwis the mass of the explosive,kg;Qviis the explosive heat of explosive,k J/kg;Qv TNTis the explosive heat of TNT,k J/kg,and here is 4187 k J/kg.

where R is the scaled distance,and its available range in the following text is based on the calculated value when α=1;r is the distance from the center of the explosion,m;α is the correction factor.Regarding the explosion in infinite air and the shock wave propagates as spherical uniform,α takes as 1.Regarding the explosion on rigid ground and the shock wave propagates as hemispherical uniform(concrete,rock,etc.),α takes as 2.How ever,regarding the explosion on ordinary ground(clay,sand,etc.)and the shock wave still propagates as hemispherical form,α takes as 1.8.

where Δ pmis the peak value of shock wave overpressure,MPa;

2)Scaled impulse calculation.

The scaled impulse I' can be calculated according to the following empirical formula[8]:

where I' is the scaled impulse when the air shock wave generated by the explosion acts on the target,Pa·s;μ is the coefficient determined by the test.Regarding the explosion occurred in the air,on the rigid ground and on the ordinary ground,μ takes 22.5,35.7 and 33.3,respectively.

4.1.2.Shock wave damage criterion

At present,the damage criterion of shock wave to target mainly includes overpressure criterion, impulse criterion and overpressure-impulse criterion[9].The overpressure criterion takes the overpressure as the sole condition of the shock wave,and it has a strict scope of application as the overpressure duration is not considered.Although the impulse criterion considers overpressure,duration and waveform simultaneously,and it is more comprehensive than the overpressure criterion,it ignores the fact that critical overpressure exists in target damage,and there are still strict application limits when applied.

The overpressure-impulse criterion believes that the damage effect of the shock wave on the target should be determined by the combination of overpressure and impulse.Only when both of them reach a certain critical value can they cause a certain degree of damage to the target.Obviously,the overpressure-impulse criterion is more scientific and universal than the overpressure criterion and the impulse criterion,and thus the judgment result will be much more applicable.

There are many factors affecting the degree of damage caused by shock wave in buildings,which are related not only to the peak value of overpressure,the time of positive pressure action and the waveform of shock wave,but also to the natural frequency and static strength of the building.Therefore,in order to comprehensively and intuitively express the damage relationship of shock wave to buildings,it is an important way to f i t the building failure critical curve by data.By comprehensively processing the data of several explosion experiments in China,the overpressure-impulse(Δp-I′)curves of explosion damage are obtained[10],as shown in Fig.3.

Fig.3.Δp-I′curves of building explosion damage.

The four curves in Fig.3 are the overpressure-impulse(Δp-I′)curves under different degrees of damage,and the corresponding curve equations are:

where Δ p is the overpressure of shock wave,MPa;I′is the scaled impulse,Pa·s.

According to the requirements of different degrees of damage to the target,the corresponding damage criterion is selected.

4.2.Fragment damage

When a warhead explodes,a large number of high-speed fragments are formed after the shell collapsed,and scatter around to form a fragment field.The damage of fragments to targets is mainly due to the breakdown effect of fragments on targets at high speed.Limited by the parameters acquisition and the small difference of equipment dimensions in all directions,this paper simplifies the device into a sphere(the top view area is consistent with the side view area).The warhead explodes perpendicularly to the ground,and the fragments are radiated from the landing point to the surroundings in a cylindrical manner.The damage diagram is shown in Fig.4,in which the damage field is a circular area composed of the lethal fragments.The effective damage field is a circular area formed by direct damage to the target,and the target surface directly contacting the fragments is called as the damaged surface.

The fragment damage generated by the missile warhead's explosion on the target is affected by many factors,such as the ballistic angle and the convected velocity,etc.We should emphasize that,this paper only focuses on the construction of damage

Fig.4.Top view of the fragment damage.

assessment model,and do not want to discuss carefully the fragment damage model.Therefore,only the traditional simplified fragment damage power field is employed in this paper,which can be directly replaced by the other fragment damage model.The simplified ideas of the fragment damage model are as follow s:

1)The ballistic angle is set as 90°(vertical incidence)in the fragment damage model.By following this assumption,the distribution density of the fragments is the same in all directions,and thus the amount of effective fragments reaching the target is converted into the angle ratio of the target;

2)The fragment damage of missile warhead striking target should belong to dynamic explosion condition.The convected velocity of warhead may have an impact on the initial velocity and the dispersion angle of fragments.How ever,considering the large difference between the convected velocity of warhead and the initial velocity of fragments and the complexity of the model setting,the convected velocity of the warhead is ignored,and the process is approximated as a static explosion condition.

4.2.1.Parameters calculation of fragment

The kinetic energy and quantity of the fragments are the basic parameters of the damage effect of targets,and a reasonable and accurate description is essential for the assessment results.Therefore,under the premise of comprehensively comparing various methods,Mott formula[11]is selected to conduct the fragmentation analysis,which has a simple form,only a few parameters,wide application and good consistency.The specific calculation is as follow s:

1)Total number of fragments N0:

where msis the mass of the shell,kg;mais the average mass of the fragment,kg;t0is the wall thickness of the shell,m;diis the inner diameter of the shell,m;K is the correlation coefficient of explosive,kg1/2/m7/6,which taking 3.81.

2)The cumulative number N(mp)of fragments that mass greater than mp:

where mpis the minimum fragment mass that causes target damage,kg.

3)There are many factors affecting the initial velocity of the frag

ments.To simplify the problem,the following assumptions are made:the detonation is instantaneous,the warhead shell has an equal wall thickness,and the explosive products and the pressure acting on the inner surface of the shell are evenly distributed.Under this premise,the fragments have the same speed.In order to ensure the simplicity,practicality and extensiveness of the calculation,the initial velocity V0of the fragment is calculated by the Gurney formula[12]for the preset cylindrical fragment distribution:

4)During the flight,the fragment is affected by the air resistance and its speed is attenuated.The attenuation law is determined by the following formula,i.e.,the residual velocity Vsin the fragment flight:

where CDis the drag coefficient of the fragment,which depends on the shape and speed of the fragment and takes 1.5;x is the flight distance of the fragment,m;ρais the local air density,taking 1.292 kg/m3;mfis the fragment mass,kg;A is the fragment shape factor,m2·kg-2/3,and takes 3.09×10-3.

5)The terminal kinetic energy of fragment Ek:

4.2.2.Fragment damage criterion

The damage criterion of a fragment acting on a target depends mainly on whether the parameters of the fragment impacting the target exceed the maximum threshold that the target can withstand.Generally,the damage criteria based on mass,velocity,kinetic energy and scaled kinetic energy of the fragment are commonly applied,respectively.Among those criteria,the kinetic energy criterion is widely used for its simplicity and accuracy.Here the kinetic energy criterion is chosen as the damage criterion for fragments,combined with the characteristics of the assessed fragments.

It considers that the kinetic energy of the fragment acting on the target determines the damage degree.When the kinetic energy of the fragment reaches or exceeds the target's damage threshold,the fragment is regarded as an effective one for the target.The damage thresholds of fragment kinetic energy for different targets[13]are shown in Table 3.

The corresponding target damage thresholds are selected in the evaluation process of specific island/reef targets according to Table 3.Due to the complexity of the target facilities on the island/reef,it is difficult to list them all in Table 3.Therefore,the damage thresholds of the different thickness of the armor plates are used as the equivalent values of the island/reef facilities.For example,the different radars in island/reef are equivalent to 7 mm thick armor.For the target whose damage value is within a certain range,the minimum value is selected as the damage threshold.

5.Damage assessment model

The damage degree of target is calculated by the damage models of the shock wave and the fragment.Combined with the gradingstandard of target damage and the weight of different facilities on the island/reef,the damage assessment model in this paper is constructed by the method of fuzzy comprehensive evaluation.

Table 3 Damage thresholds of fragment kinetic energy to targets.

5.1.Damage grading standard

The damage grading standard is a hierarchical quantification of the damage degree of the target during the assessment process.Taking into account the refinement of the target damage degree and the complexity of the assessment calculation,this paper sets the target damage grading standard to four.

1)Shock wave grading standard

The damage area of the building target can be calculated by the shock wave damage model.Because the functions of a building will be reduced when it is physically damaged,this paper simplifies that the building functions are evenly distributed on the physical structure,and there is no functional concentration area.Refer to the grading standard for damage degree of general buildings[14],shock wave damage is graded according to the ratio of serious damage area as damage criterion.Detailed information is shown in Table 4.

2)Fragment grading standard

When the fragments are radiated in a cylindrical manner,more than 90%of the total fragments are distributed in the upper and low er 10°of the vertical axis plane[15],that is,under the premise that the ballistic angle is 90°,the dispersion angle of the fragments is within±10°.Therefore,it can be approximated that the fragments scatter only in the horizontal direction within the damage radius.As the convected velocity of the warhead is not considered,the initial velocity of the fragments can be determined by the static explosion.It is assumed that the damaged surface of the target completely receives all the effective fragments in the direction,and thus the effective fragment amount of the target is:

where NTis the amount of effective fragments to reach the target;θ is the angle corresponding to the target damaged surface.

When a collision occurs,different numbers of fragments will cause different physical damage to the target,and thus it causes different functional reduction of the target.Therefore,by means of the idea of area elimination and superposition[16]and the fragment density distribution criterion[17]aimed at improving the hit rate,the target damage grading standard is established based on the effective fragment amount per unit area hitting the target.The amount of fragments per unit area is obtained through dividing the total amount of fragments by the damaged area.The grading standard is shown in Table 5.

5.2.Regional damage grade calculation

According to the damage model of shock wave and fragments,the damage grade of different targets can be obtained by referring to the target damage grading standard of Section 5.1.Subsequently,how to comprehensively analyze the damage grade of the area is then converted into the problem of membership degree,and the most common method to solve the membership problem is the fuzzy comprehensive evaluation method[18].This method is based on the theory of membership degree in fuzzy mathematics,which transforms qualitative judgment into quantitative evaluation.Combined with the research questions,the specific grade assessment process is as follow s:

1)Determine the factor universe of discourse U of the evaluated object:

where uiis the target within the island/reef area;m is the number of facility targets involved in the island/reef.

2)Determine the universe of discourse of rating V:

where vjis the j-th evaluation result of the island/reef target.There are z evaluation results,and it is know n from Section 5.1 that z=4.3)Establish the fuzzy relation matrix R:

where rijindicates the membership degree of the evaluated object uito the fuzzy subset vj,that is,the membership degree of a certain target in the island/reef area to a certain damage grade.Because of=1,the membership degree corresponding to the damage grade of the target is 1,and the remaining grades are 0.

4)Determine the weight vector W of the evaluation factors:

where wiis the weight of the i-th factor,i.e.the weight of the i-thtarget,calculated from Section 3.2.

Table 4 Grading standard of shock wave damage.

Table 5 Grading standard of fragment damage.

5)Multi-factor comprehensive evaluation

where bjindicates the membership degree of the evaluated object to the fuzzy subset vjof the evaluation grade,that is,the membership degree of the overall damage grade of the island/reef target area.

6)Analysis of fuzzy comprehensive evaluation result vector.

The multi-factor comprehensive evaluation result is analyzed by using the principle of maximum membership degree,and the grade of the object to be evaluated is equal to the grade corresponding to the maximum membership degree,that is,the grade of the object to be evaluated is:

Through the fuzzy comprehensive evaluation method,the overall damage grade brof the target area of the island/reef is finally obtained,and according to the assessed damage grade results,it can provide the basis for the measurements and plans for the island/reef combat strike and defense.

6.Example analyses

Selecting a certain area of an island/reef,under the attack of a hypothetical missile warhead(the main parameters are shown in Table 6),the distribution and size of various facilities on the island/reef area and the actual location and relative position of the warhead are shown in Fig.5.

6.1.Weight calculation

1)Calculating the indicator weight

According to the four indicators proposed in Section 3.1 to evaluate the importance of different objectives,the Analytic Hierarchy Process is used to compare the importance of each indicator on the basis of the evaluations of experts and scholars in the field.The judgment matrix is:

By computing the matrix,the weight vector is ω=（0.604 0.211 0.119 0.065）T.The consistency ratio CR=0.003＜0.1,which indicates that the judgment matrix meets the consistency requirement.Therefore,the weight of each indicator is obtained from the vector.

2)Weight of each target in island/reef area

The targets involved in the island/reef area include a command center(X1),a radar(X2),and three hangars of the same type 1,2,and 3(X3、X4、X5).Corresponding to the indicators of the target importance in Section 3.1,the whiting value of the five targets being attacked are determined in combination with the firepower striking principle and expert opinions,as shown in Table 7

Since the four indicators are all positive indicators,the larger the indicator value,the better the evaluation.According to Table 7,the comparison series can be determined.Assuming the reference series X0=｛1.0，1.0，1.0，1.0｝and resolution coefficient ρ=0.5,the correlation coefficient matrix with reference series is calculated by using Eq.(1):

Substituting the w eights of the indicators and the correlation coefficient into Eq.(2),the correlation degree is:γi=（0.545 0.679 0.618 0.618 0.618）

Normalize the correlation degree to obtain the weight values of each target in the region:

6.2.Damage degree of each target in the area

There are 5 targets in the area of the island/reef.The command center and three hangars are construction-type targets,which are mainly damaged by shock wave.The radar is an equipment-type target and is mainly damaged by fragments.Based on the parameters of the missile warhead,the damage grades of each target may be calculated by the damage model of the shock wave and the fragment.The specific calculation process is as follows:

1)Construction-type target damage

Taking the command center as an example,the damage grade of shock wave is calculated.As the ground of the explosion is a rigid ground,we have α=2 and μ=35.7.At the same time,the damage criterion of target building is seriously damaged,so the serious damage criterion is chosen for the shock wave.

Substituting the explosive mass and explosive heat into Eq.(4),the equivalent of TNT is 1354.908 kg.Using r as an unknown number and substituting Eq.(5),Eq.(6)and Eq.(7)into Eq.(11)based on serious damage criterion,the damage radius centered on the placement is 15.985 m.Finally,through the geometric calculation,the damaged area of the command center is 54.996 m2,which accounting for 3.06%of the total target area.In the same way,the damaged area of the hangars 1,2 and 3 is 139.938 m2,0 m2and 0 m2,respectively,accounting for 34.98%,0%and 0%of the corresponding target area ratios.

Table 6 Main parameters of a hypothetical warhead.

Fig.5.Sketch map of placement of warhead and target distribution in the island/reef area.

Table 7 Whiting value of target indicators in the island/reef area.

2)Equipment-type target damage

The equipment-type target in the island/reef area contains only one radar,so damage calculation of radar is carried out directly.Since the target radar is a sheet metal material,it is equivalent to a 7 mm thick armor with a damage threshold of 2160 J.

The parameters of shell mass,wall thickness and inner diameter are substituted into Eq.(12)and Eq.(13)to obtain a total of 14123 fragments.The explosive mass,detonation velocity and shell mass are substituted into Eq.(15)to obtain the initial velocity of the fragment of 2205.69 m/s.It can be seen from Eq.(16)and Eq.(17)that the larger the fragment mass,the smaller the velocity attenuation,and the corresponding kinetic energy is larger.As the failure criterion threshold is 2160 J,the threshold and initial maximum flight distance are substituted into Eq.(16)and Eq.(17)to obtain the minimum fragment speed of 1235.97 m/s,and the minimum mass of effective fragment is 2.828 g.Thus according to Eq.(14),the number of fragments with mass greater than the minimum mass of the effective fragment is 5778 pieces,i.e.,the effective fragments are 5778 pieces.

Since the target completely subjected to the number of fragments in its direction and the fragments are evenly distributed at 360°,the effective fragment amount of 5778 pieces and the θ angle of 33°are substituted into Eq.(18),and the effective fragment amount striking directly on the target is 530 pieces.At the same time,based on the premise that the equipment-type target is a sphere,it can be know n that the damaged surface is a spherical crow n,and the area of damaged surface is 289.247 m2by geometric operation.Combined with the effective fragment amount and the area of damaged surface,the number of effective fragments per unit area is 1.832.

3)Damage grade of each target

Based on the results of damage calculation and the distribution of each target in the target area of the island/reef,the illustration of damage calculation results for the target area of island/reef is obtained.As shown in Fig.6.

Referring to the damage grading standards of shock wave damage and fragments damage in Section 5.1,the damage grades of targets in the island/reef area under the attack of a warhead are obtained,as shown in Table 8.

Fig.6.Illustration of damage calculation results for the target area of island/reef.

6.3.Overall damage grade of the target area

Damage grade of five targets in the area is determined according to the grade standard in Section 5.1.Then,based on the obtained weight values of different targets,the damage of the target area is calculated by the fuzzy comprehensive evaluation method.The evaluation process is as follows:

According to the damage grade information of different targets in the area in Table 8,the fuzzy matrix value corresponding to the actual damage grade of the target is 1 and the rest is 0.The fuzzy judgment matrix is:

The weight value of each target is W=（0.177，0.220，0.201，0.201，0.201）.Therefore,the comprehensive evaluation result obtained by Eq.(22)is:

Table 8 Damage grades of the targets in the area.

B=W·R=（0.579，0.421，0，0）

Ultimately,the damage of the stroked area of the island/reef and the membership degree of each damage grade are obtained.Among them,the fourth membership degree is the maximum,but its value is not significantly larger than the value of the third-level membership degree.According to the principle of maximum membership degree,the overall damage grade of the target area is fourth.How ever,in the case of changing conditions,the overall damage grade is easily converted to the third-level,and thus the third-level still needs to be considered.

7.Conclusions

By studying the damage evaluation caused by the missile warhead to the target area of the island/reef,this paper draw s the following conclusions:

1)This paper proposes the damage models of shock wave and fragment to targets,and an assessment model for target area damage is further constructed by using the grey relational analysis,analytic hierarchy process and fuzzy comprehensive evaluation method.The model can flexibly adjust the evaluation scale and the evaluation object according to the specific situation.

2)The damage assessment model focuses on the construction of the assessment framework and the method development,which belongs to a relatively macro-scale assessment model.Since the requirements of the assessment model are met,the shock wave and fragment damage model involved in it are not thoroughly and in-depth explored.

3)The feasibility of the damage model is validated by taking the damage of targets in an island/reef under the missile attacking as an example.The assessment indicates that the damage grades of the command center,radar and hangars 1,2,and 3 are fourth,third,third,fourth,and fourth,respectively,and the overall damage grade of the area is fourth.

4)The damage assessment model proposed in this paper has certain reference and application value for the damage assessment of the target area,and the required data acquisition is relatively easy,which can provide certain guiding significance for subsequent related research.