Review ArticleOptimal Simulation of Sandcastle Life in Dynamic EnvironmentBased on Stability Principle

Peng-Hui Yang Ya-Yu Jiang Qi Tang Yi-Fang Li and Jia-Ming Zhu

School of Statistics and Applied Mathematics Anhui University of Finance and Economics Bengbu 233030 China

Correspondence should be addressed to Jia-Ming Zhu zhujm1973163com

Received 21 August 2020 Revised 28 September 2020 Accepted 29 September 2020 Published 13 October 2020

Academic Editor Shaohui Wang

Copyright copy 2020 Peng-Hui Yang et al 0is is an open access article distributed under the Creative Commons AttributionLicense which permits unrestricted use distribution and reproduction in any medium provided the original work isproperly cited

0is paper mainly studies to explore a three-dimensional geometric model including three modules of a sandcastle foundationwith optimal stability Firstly based on the knowledge of streamline structure structural mechanics and fluidmechanics the moststable sand pile foundation model under the action of tidal current and wave is established Secondly by limiting the degree ofallowable aggregation the discrete global optimization algorithm based on the continuous descent method is adopted to find outthe optimal water-sand ratio Finally we apply the above results to verify the reliability of the optimal model by comprehensivelyconsidering the influence of rainfall factors on sandcastles

1 Introduction

Playing is the nature of humans but it is not easy to come toa kind of inspiration while playing 0ere are castles ofvarious shapes on the beach how to make our castles moredurable is a question that most people are curious about0ere are numerous factors which influence the firmness ofsandcastles [1] such as sand-to-water mixture proportionthe type of sand weather etc In this paper we attempt toexplore a three-dimensional geometric model of a sandcastlefoundation having the best stability

Our model is formulated on a certain theoretical basisAfter consulting a lot of literature we carefully selected theparameters of the model In this way we can make ourmodel as close to reality as possible [2] Based on thestreamlined structure and the knowledge of structuralmechanics and fluid mechanics the most stable sand pilefoundation model under the action of tides and waves isestablished 0e water-sand ratio is limited to a reasonablerange by limiting the degree of allowable polymerization0e discrete global optimization algorithm based on suc-cessive descent method was used to efficiently find theoptimal water-sand mix ratio [3] Meanwhile we divide theimpact of rain on sandcastles into two parts scour and

infiltration 0rough the calculation and analysis of themodel it is found that the sandcastle with streamlinedstructure is still themost durable which proves the reliabilityof the model

2 Basic Assumptions

In order to solve the problem we make assumptions asfollows (i) it is assumed that in the process of naturalerosion one should ignore the situation of huge waves andwinds sweeping the sandcastle far from the original location(ii) It is assumed that there is no significant difference in thesand quality of our sandcastle foundation (iii) It is assumedthat there is negligible chemical erosion of sandcastlefoundations (iv) 0ere are no unexpected factors affectingour assessment during the study period

3 Construction of the Best Sandcastle ShapeModel Based on Dynamics

31 Sandcastle-Erosion Model Based on the streamlinedstructure the most durable sand pile foundation modelunder the action of tides and waves is established by theoptimization model Based on the knowledge of structural

HindawiDiscrete Dynamics in Nature and SocietyVolume 2020 Article ID 8850110 7 pageshttpsdoiorg10115520208850110

mechanics and fluid mechanics this method can mitigatethe impact of water flow on sandcastle as much as possibleUnder certain conditions such as sandcastle base volumeand distance from the sea the ratio of the horizontal impactforce to the volume of sandcastle can be as small as feasiblewhich can reduce the loss of sand grains and ensure itsstability [4]

311 Model Preparation 0e damage of waves and tides onthe foundation of sandcastle is mainly manifested as thetangential impact force parallel to the beach and the impactforce close to the horizontal direction 0e smaller theprojected area of wave front influence is the smaller theimpact force of the entire foundation will be 0e waterflowed along the side of the sandcastle foundation 0esmaller the angle between the water flow and the contactsurface the smaller the local impact force 0e sand base issubject to the impact force of the water flow [5] For ex-ample when the tangential force of the sand element isgreater than the adhesion force between the sand grainsthe sand grains will go with the water flow and thesandcastle will be destroyed In order to guarantee thestability of the foundation we change the shape of thefoundation to reduce the resultant force of water flow in thetangential direction

312 Model Establishing and Solving Sand under certainwater flow under the force is divided into two kinds thefriction force and pressure difference the friction is causedby fluid viscosity close to flow around objects surfaceboundary layer within the scope of the boundary layerthickness flow around fluid velocity increased dramaticallyby objects close to the wall of a stationary speed where V0 isoutflow velocity thus the object surface has larger frictionshear stress

On the other hand when there is a relative motion andfluid flow field around the quantities that causes disturbancesurface pressure distribution of symmetry is broken theunbalanced pressure leads to flow around objects flow pe-riod and period of pressure difference exists in the objectmoving direction through the above analysis the frictionforce and differential pressure can be expressed as followsAs shown in Figure 1 ds is an area of an objectrsquos surface T isfriction shear stress and P is the compressive stress

Friction and differential pressure force respectively areas follows

Ff 1113946Sτ cos αds

FP 1113946SP sin αds

(1)

From formula (1) we get the total force

F Ff + FP 1113946Sτ cos αds + 1113946

SP sin αds (2)

It is assumed that the water flowing towards each part ofthe sand base at high tide is the same in nature and the

smaller the ratio between the resistance of the part in contactwith the sea water and the base volume is the smaller theamount of sand taken away by the sea water after the impactcan be regarded as the smallest damage caused by the seawater impact on the sand base and the most stable in nature[6] Since the streamlined structure is subject to less resis-tance in the water the streamlined structure of sand in-frastructure can mitigate the impact of waves and tides onthe sandcastle From the kinetic energy loss of water flow inthe process of high tide to low tide we preliminarilydesigned the structure of sandcastle as a semi-elliptic andparabolic streamlined semi-rotating body 0e top view isshown in Figure 2

0e streamline inlet section of the sand pile foundationclose to the sea water is a semi-ellipse which can beexpressed as follows

y plusmnK0

2Lz

L2z minus x

21113969

(3)

In formula (3) the variable K0 represents the maximumcross section diameter of the rotary body and the variable Lzrepresents the length of incoming flow segment the units aremm 0e seawater first passes through the incoming flowsection and then through the outgoing flow section 0eoutgoing flow section is a parabola which can be expressedas follows

y plusmnK0

21 minus

x2

L2Y

1113888 1113889 (4)

In formula (4) LY represents the length of the outletsection the units are mm Since only the influence of thebasic shape on its stability is considered we assume that thewater flow is laminar for the convenience of calculation Fora semi-rotary body it is necessary to integrate the marchsurface in the direction of its length when calculating itsfrictional resistance since the cross section of a rotary bodyis a semicircle it is only necessary to integrate the function ofthe length in the y direction [7ndash9]

F 2π 1113946l

0rτ0dx (5)

For incoming flow segments

r K0

2Lz

L2z minus x

21113969

(6)

0e friction shear force of the laminar flow on the plate isexpressed as follows

τ0 0343ρV20

1Re

1113970

(7)

In the formula the variable V0 represents the comingflow speed the units are ms and Re represents Reynoldsnumber it is a dimensionless quantity then the laminarfrictional resistance in the incoming flow section is

2 Discrete Dynamics in Nature and Society

FLZ0343πK0

μρV30

1113969

LZ

11139460

minusLZ

L2z minus x

2

x + LZ

dx

1113971

(8)

In the formula the Gauss three-point interpolationformula with fifth-degree algebraic precision was used for anapproximate calculation and the laminar frictional resis-tance in the incoming flow segment was obtained as follows

FLZ 05145πK0

μρV30LZ

1113969

(9)

0erefore the frictional resistance in the reaching flowsection is

FLY 2π 1113946

LY

00343

μρV30

xmiddot

1113971

K0

21 minus

x2

L2Y

1113888 1113889dx (10)

Finally from formula (5) to formula (10) the totalfrictional resistance of semi-elliptic and parabolic stream-lined rotating bodies under laminar flow is obtained asfollows

FL FLZ+ FLY

005145πK0

μρV30LZ

1113969

+ 05488πK0

μρV30LY

1113969

(11)

As the flow velocity Reynolds number and other pa-rameters are all certain and only the influence of thefoundation shape on stability is considered ρ μ V0 valuesare set to 1 and the total frictional resistance of the rotarybody is as follows

F 05145πK0LZ

1113968+ 05488πK0

LY

1113968 (12)

From formula (11) to formula (12) the optimal solutionis obtained by calculation X 063 Y 022

0erefore the function expression of the basic model isas follows

f(x y) 3087

x

radic+ 32928

1 minus x

radic

xy + 4y (13)

313 Result Analysis 0e effect of waves and tides on thestability of a sandcastle foundation is converted into a

functional relationship between the impact force and vol-ume ratio of sandcastle foundation and the three-dimensional shape parameters of sandcastle foundation andthe minimum ratio of impact force and volume is obtained[10]

0ree-dimensional shape parameters of sand fortfoundation in order to mitigate the effects of water flow onthe sand as best as possible to reduce the loss of sand on thebasis of sandcastle simple structural mechanics and fluidmechanics analysis the optimal model determines the best3D sand foundation model

32OptimalWater-to-SandMixtureProportion 0e ratio ofwater to sand will directly affect the antierosion ability ofsandcastle foundation By introducing the relation equationbetween water-sand polymerization degree and water-sandratio we limited the allowable range of polymerizationdegree to obtain a reasonable range of water-sand ratio andthen solved the optimal water-sand ratio through models[11 12]

321 Model Preparation We considered the aggregation ofwater and sand leading to the concept of the degree ofaggregation of water and sand [13] We defined the degree ofwater and sand polymerization according to the volumechange of water and sand specific gravity before and afterpolymerization

W 1 minusM1C1( 1113857 + M2C2( 1113857( 1113857

M1 + M2( 11138571113888 1113889 times 100 (14)

where M1 and M2 represent the volume of sand and waterbefore mixing and C1 and C2 represent water absorptioncoefficient and water solubility coefficient respectively Letus use these two coefficients to express the polymerizationcapacity of water and sand

322 Model Establishing and Solving Let us use water-sandratio NR to simplify this equation

W 1 minusNRC1( 1113857 + 1C2( 1113857( 1113857

NR + 1( 11138571113888 1113889 times 100 (15)

From formula (13) to formula (15) MATLAB was usedto solve the equation and the relation curve between water-

y

x

P

z

ds

o

τα

Figure 1 Simplified diagram of sandcastle model section

Lz Ly

K0

Figure 2 Top view of sandcastle structure

Discrete Dynamics in Nature and Society 3

sand polymerization degree and water-sand ratio was ob-tained as shown in Figure 3

With the increase of water-sand ratio the polymeriza-tion degree of water-sand increases In order to ensure thelongest erosion resistance time of sandcastles not only therelatively larger adhesion but also the optimization oferosion resistance performance should be considered Incombination with the actual situation the two-point intervalwith a wide viscosity range as shown in the figure was se-lected as the optimization parameter [14 15]

Based on the optimal proportion of the best shape se-lected above and the initial value of water-sand ratio NR as avariable the target T was discretized in the same wayAccording to the variation of foundation ΔS and Δh therelationship between the damage index and time the con-straint is added

07ltNRlt 085Obtain the continuous schedule record ofNR equidistant

points as shown in Table 1Using the above model we can get that when the water-

sand ratio is NR 074 under the condition of good shapeadhesion the maximum service life of sandcastle foundationcan be achieved

33 Sandcastle-Rain-Erosion Model When consideringrainfall a sandcastle-rain-erosion model is established andthe influence of rainfall is directly superimposed with that ofseawater that is the rainfall directly affects the sandcastleseroded by seawater Suppose the raindrops were particles ofmass hitting the sandcastle vertically with the same force asthe waves [16ndash18]

331 Model Preparation To study the effect of rain onsandcastles we first need to understand some of the motioncharacteristics of rain Let us say the rain falls straight downRaindrops have a lot of kinetic energy [19] When they hit asandcastle they destroy the structure of the sandcastle andchange the water content of the sandcastle 0erefore we

divide the impact of rain on sandcastles into two parts scourand infiltration [20 21]

Formula for calculating the final velocity of raindrop is asfollows

If dle 3mm we have

Vmax

389v

d1113874 1113875

2+ 2400g d minus 389

v

d

1113971

(16)

If 3mmlt dle 6mm we have

Vmax d

(0113 + 0845 d) (17)

In the formula v is the viscosity coefficient of air motionWhen T 293K v 1810741555times10minus5 PamiddotS

0ink of raindrops as spheres According to the mo-mentum formula q mVmax and energy formulaE (mV2

max)2 the momentum and energy contained ineach raindrop can be obtained and then the relationshipbetween raindrop diameter and momentum and energy canbe obtained [22ndash24]

332 Model Establishing and Solving However in the ac-tual calculation of rainfall characteristics such as rainfallkinetic energy only the data of rainfall or rainfall intensity areusually available and the observed data of raindrop diameterare not available In order to facilitate production and appli-cation the final raindrop velocity can be expressed as a functionof rain intensity [25] According to the analysis of themeasureddata there is a power function relationship between themedianraindrop diameter and rainfall intensity

dm 252i032

(18)

In the formula dm represents median diameter ofraindrops the units are mm i represents rain intensity theunits are mmmin 0e final velocity of raindrop is deter-mined by rainfall intensity through substitution andcalculation

If ile 213 we have

Vmax

1544v

i0231113874 11138752

+ 6048gi023

1113971

minus 1544v

i023

(19)

If 213lt ile 4346 we have

Vmax i023

00448 + 00845i023

1113872 1113873 (20)

From formula (16) to formula (20) the relationshipbetween precipitation and raindrop velocity energy andmomentum is calculated as shown in Figures 4ndash6

After understanding the motion parameters of rain-drops we also need to analyze the influence of raindrops onsandcastles [26] 0e SartorndashBoyd scour model is mainly

0 01 02 03 04 05 06 07 08 09 1The ratio of water and sand

(0 036)

(07 0476)(085 0487)

034

036

038

04

042

044

046

048

05

052

Deg

ree o

f con

verg

ence

of w

ater

and

sand

Figure 3 Convergence of water and sand

Table 1 Different water-sand ratios correspond to durationWater-sand ratio 070 072 074 076 078 080 082 084Duration 47 49 52 48 51 50 46 44

4 Discrete Dynamics in Nature and Society

applicable to the rainfall process with initial scour effect asshown in the following equation

dVp

dt KeVpRp (21)

In formula (21) Ke represents the erosion coefficient theunits are mmminus1Vp represents the volume of sandcastle at thebeginning the units are m3 Rp represents rainwater runoffper unit time per unit area the units are mmmin t rep-resents time the unit is min In our model take Ke as 02 toget the change of sand grain quantity washed away byprecipitation as shown in Figure 7

333 Result Analysis When the sandcastle is affected byrainfall MATLAB is used to fit the relationship between theinternal friction angle and water content of the sandcastle asshown in Figure 8

0 5 10 15 20 25 30 35 40 45Rain fall (mmmin)

0

15times10ndash23

Rain

drop

fina

l ene

rgy

(J)

03

06

09

12

Figure 5 Relationship between precipitation and raindrop energy

0 5 10 15 20 25 3530 4540 5550 60Time (min)

0

Volu

me o

f san

d sc

oure

d (m

3 )

0005

001

0015

002

0025

003

0035

004

Figure 7 Changes in the amount of sand washed away byprecipitation

0 5 10 15 20 25 30 35 40 45Rain fall (mmmin)

0

3times10ndash9

Rain

drop

fina

l mom

entu

m (k

gmiddotm

s)

03

06

09

12

15

18

21

24

27

Figure 6 Relationship between precipitation and raindropmomentum

0 5 10 15 20 25 30 35 40 45Rain fall (mmmin)

7678

882848688

99294969810

Rain

drop

fina

l spe

ed (m

s)

Figure 4 Relationship between precipitation and raindrop speed

40 45 50 55 60 65 70 75 80Water content ()

70In

tern

al fr

icito

n an

gle ϕ

(ordm)

40

45

50

55

60

65

ϕ = 079468w + 900782

Figure 8 Relationship between water content and internal frictionangle caused by rainfall on sandcastle

Discrete Dynamics in Nature and Society 5

In the precipitation model we get the formula for cal-culating the final velocity of raindrop 0en the watercontent of sandcastle and the angle of internal frictionduring rainfall are calculated Finally compare the rela-tionship between the two and draw a conclusion 0e resultsshow that the rain has a certain effect on the structuralstability of the sandcastle 0rough the establishment andsolution of the above model we find that streamline is stillthe optimal geometric shape which verifies the reliability ofthe model

4 Conclusions

On the basis of the simulation results the destructive powerof seawater rainy days and other environments is sum-marized which is an important threat to the sandcastlefoundation [27ndash31] Several models established in the de-tailed mathematical analysis provided answers to the re-quired questions including a view of the exterior of thesandcastle with the longest period of stability in the naturalstate

In general a sandcastle foundation with a gentle slope isgood at resisting seawater erosion while a foundation with alarge top is good at resisting rain After many iterations allinitial sandcastle foundation transfers have similar struc-tures As mentioned earlier the optimal sand-water mixtureratio is about 074 Due to the trade-off between friction andfluidity three measures can be taken to strengthen thesandcastle foundation such as adjusting construction timeadding support structure and improving the building0oseare all practical measures one can take in order to obtain abetter sandcastle foundation

Data Availability

0e data in this paper come from Question B of the 2020American College Students Mathematical ModelingCompetition

Conflicts of Interest

0e authors declare that there are no conflicts of interestregarding the publication of this paper

Acknowledgments

0is study was funded by the Humanities and Social SciencesResearch Major Project of the Education Department ofAnhui Province (SK2017A0452) the Teaching and ResearchFund Project of the Education Department of AnhuiProvince (2018jyxm1305) and the Teaching and ResearchFund Project of the Anhui University of Finance andEconomics (acxkjsjy201803zd and acjyyb2020011)

References

[1] J P Bouchaud M E Cates J R Prakash and S F EdwardsldquoHysteresis and metastability in a continuum sandpilemodelrdquo Physical Review Letters vol 74 no 11 pp 1982ndash19851995

[2] Y L Chen G Y Liu N Li X Du S-R Wang and R AzzamldquoStability evaluation of slope subjected to seismic effectcombined with consequent rainfallrdquo Engineering Geologyvol 266 Article ID 105461 2020

[3] M Pakpour M Habibi P Moslashller and D Bonn ldquoHow toconstruct the perfect sandcastlerdquo Scientific Reports vol 2no 1 pp 1ndash3 2012

[4] S Dumont and I Noureddine ldquoOn a dual formulation for thegrowing sand pile problemrdquo European Journal of AppliedMathematics vol 20 no 2 pp 169ndash185 2009

[5] N Fraysse H 0ome and L Petit ldquoHumidity effects on thestability of a sand pilerdquo e European Physical JournalB-Condensed Matter and Complex Systems vol 11 no 4pp 615ndash619 1999

[6] T Groger U Tuzun and D M Heyes ldquoModelling andmeasuring of cohesion in wet granular materialsrdquo PowderTechnology vol 133 no 3 pp 203ndash215 2003

[7] T C Halsey and A J Levine ldquoHow sandcastles fallrdquo PhysicalReview Letters vol 80 no 14 pp 31ndash41 1998

[8] G L George ldquoMagic sand modeling the hydrophobic effectand reversed-phase liquid chromatographyrdquo Journal ofChemical Education vol 67 no 6 pp 512ndash515 1990

[9] M Scheel R Seemann M Brinkmann et al ldquoMorphologicalclues to wet granular pile stabilityrdquo Nature Materials vol 7no 3 pp 189ndash193 2008

[10] S Dumontl and N Igbidal ldquoOn the collapsing sandpileproblemrdquo Communications on Pure and Applied Analysisvol 10 no 2 pp 625ndash638 2010

[11] C Mejia and J A Montoya ldquoOn the complexity of sand pilecritical avalanchesrdquo eoretical Computer Science vol 412no 30 pp 3964ndash3974 2011

[12] S Qiao S Qin J Chen X Hu and Z Ma ldquo0e application ofa three-dimensional deterministic model in the study ofdebris flow prediction based on the rainfall-unstable soilcoupling mechanismrdquo Processes vol 7 no 2 2019

[13] N Igbida and N Noureddine ldquoA partial integrodifferentialequation in granular matter and its connection with a sto-chastic modelrdquo Siam Journal on Mathematical Analysisvol 44 no 3 pp 1950ndash1975 2012

[14] N Igbida ldquoA generalized collapsing sandpile modelrdquo ArchivDer Mathematik vol 94 no 2 pp 193ndash200 2010

[15] A Sotirov and S H Yu ldquoOn the solution of a Boltzmannsystem for gas mixturesrdquo Archive for Rational Mechanics ampAnalysis vol 195 no 2 pp 675ndash700 2010

[16] G Sorbino and M V Nicotera ldquoUnsaturated soil mechanicsin rainfall-induced flow landslidesrdquo Engineering Geologyvol 165 pp 105ndash132 2013

[17] M Burylo C Hudek and F Rey ldquoSoil reinforcement by theroots of six dominant species on eroded mountainous marlyslopes (Southern Alps France)rdquo CATENA vol 84 no 1-2pp 70ndash78 2011

[18] Y Hong R Chen C Wu and J Chen ldquoShaking table testsand stability analysis of steep nailed slopesrdquo CanadianGeotechnical Journal vol 42 no 5 pp 1264ndash1279 2011

[19] H Moriwaki T Inokuchi T Hattanji K Sassa H Ochiaiand G Wang ldquoFailure processes in a full-scale landslideexperiment using a rainfall simulatorrdquo Landslides vol 1no 4 pp 277ndash288 2004

[20] M Lin and K Wang ldquoSeismic slope behavior in a large-scaleshaking table model testrdquo Engineering Geology vol 86 no 2-3 pp 118ndash133 2006

[21] K Sasahara and N Sakai ldquoDevelopment of shear deformationdue to the increase of pore pressure in a sandy model slope

6 Discrete Dynamics in Nature and Society

during rainfallrdquo Engineering Geology vol 170 pp 43ndash512014

[22] L Pantelidis and D V Griffiths ldquoStability of earth slopes PartII three dimensional analysis in closed-formrdquo InternationalJournal for Numerical and Analytical Methods in Geo-mechanics vol 37 no 13 pp 1987ndash2004 2013

[23] N Iwata R Yoshinaka and T Sasaki ldquoApplicability of theseismic response analysis using multiple yield model fordiscontinuous rockrdquo International Journal of Rock Mechanicsand Mining Sciences vol 60 pp 196ndash207 2013

[24] H Yuehua Z Cheng and L I Hongmei ldquoCombined effectsof trees and macropores on slope stability subjected torainfallrdquo Journal of Water Resources amp Architectural Engi-neering 2018

[25] Z Yu Y L Kun G Z Zheng et al ldquoEvaluation of the stabilityof Maliulin landslide under the reservoir water level fluctu-ation combined with rainfallrdquo Geological Science and Tech-nology Information 2019

[26] A Chinkulkijniwat T Tirametatiparat C Supotayan et alldquoStability characteristics of shallow landslide triggered by rainfallrdquoJournal of Mountain Science vol 16 pp 2171ndash2183 2019

[27] J B Liu J Zhao J Cao and J Min ldquoOn the Hosoya index ofgraphs formed by a fractal graphrdquo Fractals-Complex GeometryPatterns and Scaling in Nature and Society vol 27 no 8pp 19ndash35 2019

[28] J B Liu J Zhao H He and Z Shao ldquoValency-based To-pological descriptors and structural property of the gener-alized Sierpiacute nski networksrdquo Journal of Statistical Physicsvol 177 no 6 pp 1131ndash1147 2019

[29] J B Liu J Zhao and Z Cai ldquoOn the generalized adjacencyLaplacian and signless Laplacian spectra of the weighted edgecorona networksrdquo Physica A Statistical Mechanics and itsApplications vol 540 pp 12ndash30 2020

[30] J M Zhu W Y Xia J J Sun J-B Liu and F-H Yu ldquo0espread pattern on Ebola and the control schemesrdquo Interna-tional Journal of Innovative Computing and Applicationsvol 9 no 2 pp 77ndash89 2018

[31] J-M Zhu L Wang and J-B Liu ldquoEradication of Ebola basedon dynamic programmingrdquo Computational and Mathemat-ical Methods in Medicine vol 2016 p 9 Article ID 15809172016

Discrete Dynamics in Nature and Society 7

FLZ0343πK0

μρV30

1113969

LZ

11139460

minusLZ

L2z minus x

2

x + LZ

dx

1113971

(8)

In the formula the Gauss three-point interpolationformula with fifth-degree algebraic precision was used for anapproximate calculation and the laminar frictional resis-tance in the incoming flow segment was obtained as follows

FLZ 05145πK0

μρV30LZ

1113969

(9)

0erefore the frictional resistance in the reaching flowsection is

FLY 2π 1113946

LY

00343

μρV30

xmiddot

1113971

K0

21 minus

x2

L2Y

1113888 1113889dx (10)

Finally from formula (5) to formula (10) the totalfrictional resistance of semi-elliptic and parabolic stream-lined rotating bodies under laminar flow is obtained asfollows

FL FLZ+ FLY

005145πK0

μρV30LZ

1113969

+ 05488πK0

μρV30LY

1113969

(11)

As the flow velocity Reynolds number and other pa-rameters are all certain and only the influence of thefoundation shape on stability is considered ρ μ V0 valuesare set to 1 and the total frictional resistance of the rotarybody is as follows

F 05145πK0LZ

1113968+ 05488πK0

LY

1113968 (12)

From formula (11) to formula (12) the optimal solutionis obtained by calculation X 063 Y 022

0erefore the function expression of the basic model isas follows

f(x y) 3087

x

radic+ 32928

1 minus x

radic

xy + 4y (13)

313 Result Analysis 0e effect of waves and tides on thestability of a sandcastle foundation is converted into a

functional relationship between the impact force and vol-ume ratio of sandcastle foundation and the three-dimensional shape parameters of sandcastle foundation andthe minimum ratio of impact force and volume is obtained[10]

0ree-dimensional shape parameters of sand fortfoundation in order to mitigate the effects of water flow onthe sand as best as possible to reduce the loss of sand on thebasis of sandcastle simple structural mechanics and fluidmechanics analysis the optimal model determines the best3D sand foundation model

32OptimalWater-to-SandMixtureProportion 0e ratio ofwater to sand will directly affect the antierosion ability ofsandcastle foundation By introducing the relation equationbetween water-sand polymerization degree and water-sandratio we limited the allowable range of polymerizationdegree to obtain a reasonable range of water-sand ratio andthen solved the optimal water-sand ratio through models[11 12]

321 Model Preparation We considered the aggregation ofwater and sand leading to the concept of the degree ofaggregation of water and sand [13] We defined the degree ofwater and sand polymerization according to the volumechange of water and sand specific gravity before and afterpolymerization

W 1 minusM1C1( 1113857 + M2C2( 1113857( 1113857

M1 + M2( 11138571113888 1113889 times 100 (14)

where M1 and M2 represent the volume of sand and waterbefore mixing and C1 and C2 represent water absorptioncoefficient and water solubility coefficient respectively Letus use these two coefficients to express the polymerizationcapacity of water and sand

322 Model Establishing and Solving Let us use water-sandratio NR to simplify this equation

W 1 minusNRC1( 1113857 + 1C2( 1113857( 1113857

NR + 1( 11138571113888 1113889 times 100 (15)

From formula (13) to formula (15) MATLAB was usedto solve the equation and the relation curve between water-

y

x

P

z

ds

o

τα

Figure 1 Simplified diagram of sandcastle model section

Lz Ly

K0

Figure 2 Top view of sandcastle structure

Discrete Dynamics in Nature and Society 3

sand polymerization degree and water-sand ratio was ob-tained as shown in Figure 3

With the increase of water-sand ratio the polymeriza-tion degree of water-sand increases In order to ensure thelongest erosion resistance time of sandcastles not only therelatively larger adhesion but also the optimization oferosion resistance performance should be considered Incombination with the actual situation the two-point intervalwith a wide viscosity range as shown in the figure was se-lected as the optimization parameter [14 15]

Based on the optimal proportion of the best shape se-lected above and the initial value of water-sand ratio NR as avariable the target T was discretized in the same wayAccording to the variation of foundation ΔS and Δh therelationship between the damage index and time the con-straint is added

07ltNRlt 085Obtain the continuous schedule record ofNR equidistant

points as shown in Table 1Using the above model we can get that when the water-

sand ratio is NR 074 under the condition of good shapeadhesion the maximum service life of sandcastle foundationcan be achieved

33 Sandcastle-Rain-Erosion Model When consideringrainfall a sandcastle-rain-erosion model is established andthe influence of rainfall is directly superimposed with that ofseawater that is the rainfall directly affects the sandcastleseroded by seawater Suppose the raindrops were particles ofmass hitting the sandcastle vertically with the same force asthe waves [16ndash18]

331 Model Preparation To study the effect of rain onsandcastles we first need to understand some of the motioncharacteristics of rain Let us say the rain falls straight downRaindrops have a lot of kinetic energy [19] When they hit asandcastle they destroy the structure of the sandcastle andchange the water content of the sandcastle 0erefore we

divide the impact of rain on sandcastles into two parts scourand infiltration [20 21]

Formula for calculating the final velocity of raindrop is asfollows

If dle 3mm we have

Vmax

389v

d1113874 1113875

2+ 2400g d minus 389

v

d

1113971

(16)

If 3mmlt dle 6mm we have

Vmax d

(0113 + 0845 d) (17)

In the formula v is the viscosity coefficient of air motionWhen T 293K v 1810741555times10minus5 PamiddotS

0ink of raindrops as spheres According to the mo-mentum formula q mVmax and energy formulaE (mV2

max)2 the momentum and energy contained ineach raindrop can be obtained and then the relationshipbetween raindrop diameter and momentum and energy canbe obtained [22ndash24]

332 Model Establishing and Solving However in the ac-tual calculation of rainfall characteristics such as rainfallkinetic energy only the data of rainfall or rainfall intensity areusually available and the observed data of raindrop diameterare not available In order to facilitate production and appli-cation the final raindrop velocity can be expressed as a functionof rain intensity [25] According to the analysis of themeasureddata there is a power function relationship between themedianraindrop diameter and rainfall intensity

dm 252i032

(18)

In the formula dm represents median diameter ofraindrops the units are mm i represents rain intensity theunits are mmmin 0e final velocity of raindrop is deter-mined by rainfall intensity through substitution andcalculation

If ile 213 we have

Vmax

1544v

i0231113874 11138752

+ 6048gi023

1113971

minus 1544v

i023

(19)

If 213lt ile 4346 we have

Vmax i023

00448 + 00845i023

1113872 1113873 (20)

From formula (16) to formula (20) the relationshipbetween precipitation and raindrop velocity energy andmomentum is calculated as shown in Figures 4ndash6

After understanding the motion parameters of rain-drops we also need to analyze the influence of raindrops onsandcastles [26] 0e SartorndashBoyd scour model is mainly

0 01 02 03 04 05 06 07 08 09 1The ratio of water and sand

(0 036)

(07 0476)(085 0487)

034

036

038

04

042

044

046

048

05

052

Deg

ree o

f con

verg

ence

of w

ater

and

sand

Figure 3 Convergence of water and sand

Table 1 Different water-sand ratios correspond to durationWater-sand ratio 070 072 074 076 078 080 082 084Duration 47 49 52 48 51 50 46 44

4 Discrete Dynamics in Nature and Society

applicable to the rainfall process with initial scour effect asshown in the following equation

dVp

dt KeVpRp (21)

In formula (21) Ke represents the erosion coefficient theunits are mmminus1Vp represents the volume of sandcastle at thebeginning the units are m3 Rp represents rainwater runoffper unit time per unit area the units are mmmin t rep-resents time the unit is min In our model take Ke as 02 toget the change of sand grain quantity washed away byprecipitation as shown in Figure 7

333 Result Analysis When the sandcastle is affected byrainfall MATLAB is used to fit the relationship between theinternal friction angle and water content of the sandcastle asshown in Figure 8

0 5 10 15 20 25 30 35 40 45Rain fall (mmmin)

0

15times10ndash23

Rain

drop

fina

l ene

rgy

(J)

03

06

09

12

Figure 5 Relationship between precipitation and raindrop energy

0 5 10 15 20 25 3530 4540 5550 60Time (min)

0

Volu

me o

f san

d sc

oure

d (m

3 )

0005

001

0015

002

0025

003

0035

004

Figure 7 Changes in the amount of sand washed away byprecipitation

0 5 10 15 20 25 30 35 40 45Rain fall (mmmin)

0

3times10ndash9

Rain

drop

fina

l mom

entu

m (k

gmiddotm

s)

03

06

09

12

15

18

21

24

27

Figure 6 Relationship between precipitation and raindropmomentum

0 5 10 15 20 25 30 35 40 45Rain fall (mmmin)

7678

882848688

99294969810

Rain

drop

fina

l spe

ed (m

s)

Figure 4 Relationship between precipitation and raindrop speed

40 45 50 55 60 65 70 75 80Water content ()

70In

tern

al fr

icito

n an

gle ϕ

(ordm)

40

45

50

55

60

65

ϕ = 079468w + 900782

Figure 8 Relationship between water content and internal frictionangle caused by rainfall on sandcastle

Discrete Dynamics in Nature and Society 5

In the precipitation model we get the formula for cal-culating the final velocity of raindrop 0en the watercontent of sandcastle and the angle of internal frictionduring rainfall are calculated Finally compare the rela-tionship between the two and draw a conclusion 0e resultsshow that the rain has a certain effect on the structuralstability of the sandcastle 0rough the establishment andsolution of the above model we find that streamline is stillthe optimal geometric shape which verifies the reliability ofthe model

4 Conclusions

On the basis of the simulation results the destructive powerof seawater rainy days and other environments is sum-marized which is an important threat to the sandcastlefoundation [27ndash31] Several models established in the de-tailed mathematical analysis provided answers to the re-quired questions including a view of the exterior of thesandcastle with the longest period of stability in the naturalstate

In general a sandcastle foundation with a gentle slope isgood at resisting seawater erosion while a foundation with alarge top is good at resisting rain After many iterations allinitial sandcastle foundation transfers have similar struc-tures As mentioned earlier the optimal sand-water mixtureratio is about 074 Due to the trade-off between friction andfluidity three measures can be taken to strengthen thesandcastle foundation such as adjusting construction timeadding support structure and improving the building0oseare all practical measures one can take in order to obtain abetter sandcastle foundation

Data Availability

0e data in this paper come from Question B of the 2020American College Students Mathematical ModelingCompetition

Conflicts of Interest

0e authors declare that there are no conflicts of interestregarding the publication of this paper

Acknowledgments

0is study was funded by the Humanities and Social SciencesResearch Major Project of the Education Department ofAnhui Province (SK2017A0452) the Teaching and ResearchFund Project of the Education Department of AnhuiProvince (2018jyxm1305) and the Teaching and ResearchFund Project of the Anhui University of Finance andEconomics (acxkjsjy201803zd and acjyyb2020011)

References

[1] J P Bouchaud M E Cates J R Prakash and S F EdwardsldquoHysteresis and metastability in a continuum sandpilemodelrdquo Physical Review Letters vol 74 no 11 pp 1982ndash19851995

[2] Y L Chen G Y Liu N Li X Du S-R Wang and R AzzamldquoStability evaluation of slope subjected to seismic effectcombined with consequent rainfallrdquo Engineering Geologyvol 266 Article ID 105461 2020

[3] M Pakpour M Habibi P Moslashller and D Bonn ldquoHow toconstruct the perfect sandcastlerdquo Scientific Reports vol 2no 1 pp 1ndash3 2012

[4] S Dumont and I Noureddine ldquoOn a dual formulation for thegrowing sand pile problemrdquo European Journal of AppliedMathematics vol 20 no 2 pp 169ndash185 2009

[5] N Fraysse H 0ome and L Petit ldquoHumidity effects on thestability of a sand pilerdquo e European Physical JournalB-Condensed Matter and Complex Systems vol 11 no 4pp 615ndash619 1999

[6] T Groger U Tuzun and D M Heyes ldquoModelling andmeasuring of cohesion in wet granular materialsrdquo PowderTechnology vol 133 no 3 pp 203ndash215 2003

[7] T C Halsey and A J Levine ldquoHow sandcastles fallrdquo PhysicalReview Letters vol 80 no 14 pp 31ndash41 1998

[8] G L George ldquoMagic sand modeling the hydrophobic effectand reversed-phase liquid chromatographyrdquo Journal ofChemical Education vol 67 no 6 pp 512ndash515 1990

[9] M Scheel R Seemann M Brinkmann et al ldquoMorphologicalclues to wet granular pile stabilityrdquo Nature Materials vol 7no 3 pp 189ndash193 2008

[10] S Dumontl and N Igbidal ldquoOn the collapsing sandpileproblemrdquo Communications on Pure and Applied Analysisvol 10 no 2 pp 625ndash638 2010

[11] C Mejia and J A Montoya ldquoOn the complexity of sand pilecritical avalanchesrdquo eoretical Computer Science vol 412no 30 pp 3964ndash3974 2011

[12] S Qiao S Qin J Chen X Hu and Z Ma ldquo0e application ofa three-dimensional deterministic model in the study ofdebris flow prediction based on the rainfall-unstable soilcoupling mechanismrdquo Processes vol 7 no 2 2019

[13] N Igbida and N Noureddine ldquoA partial integrodifferentialequation in granular matter and its connection with a sto-chastic modelrdquo Siam Journal on Mathematical Analysisvol 44 no 3 pp 1950ndash1975 2012

[14] N Igbida ldquoA generalized collapsing sandpile modelrdquo ArchivDer Mathematik vol 94 no 2 pp 193ndash200 2010

[15] A Sotirov and S H Yu ldquoOn the solution of a Boltzmannsystem for gas mixturesrdquo Archive for Rational Mechanics ampAnalysis vol 195 no 2 pp 675ndash700 2010

[16] G Sorbino and M V Nicotera ldquoUnsaturated soil mechanicsin rainfall-induced flow landslidesrdquo Engineering Geologyvol 165 pp 105ndash132 2013

[17] M Burylo C Hudek and F Rey ldquoSoil reinforcement by theroots of six dominant species on eroded mountainous marlyslopes (Southern Alps France)rdquo CATENA vol 84 no 1-2pp 70ndash78 2011

[18] Y Hong R Chen C Wu and J Chen ldquoShaking table testsand stability analysis of steep nailed slopesrdquo CanadianGeotechnical Journal vol 42 no 5 pp 1264ndash1279 2011

[19] H Moriwaki T Inokuchi T Hattanji K Sassa H Ochiaiand G Wang ldquoFailure processes in a full-scale landslideexperiment using a rainfall simulatorrdquo Landslides vol 1no 4 pp 277ndash288 2004

[20] M Lin and K Wang ldquoSeismic slope behavior in a large-scaleshaking table model testrdquo Engineering Geology vol 86 no 2-3 pp 118ndash133 2006

[21] K Sasahara and N Sakai ldquoDevelopment of shear deformationdue to the increase of pore pressure in a sandy model slope

6 Discrete Dynamics in Nature and Society

during rainfallrdquo Engineering Geology vol 170 pp 43ndash512014

[22] L Pantelidis and D V Griffiths ldquoStability of earth slopes PartII three dimensional analysis in closed-formrdquo InternationalJournal for Numerical and Analytical Methods in Geo-mechanics vol 37 no 13 pp 1987ndash2004 2013

[23] N Iwata R Yoshinaka and T Sasaki ldquoApplicability of theseismic response analysis using multiple yield model fordiscontinuous rockrdquo International Journal of Rock Mechanicsand Mining Sciences vol 60 pp 196ndash207 2013

[24] H Yuehua Z Cheng and L I Hongmei ldquoCombined effectsof trees and macropores on slope stability subjected torainfallrdquo Journal of Water Resources amp Architectural Engi-neering 2018

[25] Z Yu Y L Kun G Z Zheng et al ldquoEvaluation of the stabilityof Maliulin landslide under the reservoir water level fluctu-ation combined with rainfallrdquo Geological Science and Tech-nology Information 2019

[26] A Chinkulkijniwat T Tirametatiparat C Supotayan et alldquoStability characteristics of shallow landslide triggered by rainfallrdquoJournal of Mountain Science vol 16 pp 2171ndash2183 2019

[27] J B Liu J Zhao J Cao and J Min ldquoOn the Hosoya index ofgraphs formed by a fractal graphrdquo Fractals-Complex GeometryPatterns and Scaling in Nature and Society vol 27 no 8pp 19ndash35 2019

[28] J B Liu J Zhao H He and Z Shao ldquoValency-based To-pological descriptors and structural property of the gener-alized Sierpiacute nski networksrdquo Journal of Statistical Physicsvol 177 no 6 pp 1131ndash1147 2019

[29] J B Liu J Zhao and Z Cai ldquoOn the generalized adjacencyLaplacian and signless Laplacian spectra of the weighted edgecorona networksrdquo Physica A Statistical Mechanics and itsApplications vol 540 pp 12ndash30 2020

[30] J M Zhu W Y Xia J J Sun J-B Liu and F-H Yu ldquo0espread pattern on Ebola and the control schemesrdquo Interna-tional Journal of Innovative Computing and Applicationsvol 9 no 2 pp 77ndash89 2018

[31] J-M Zhu L Wang and J-B Liu ldquoEradication of Ebola basedon dynamic programmingrdquo Computational and Mathemat-ical Methods in Medicine vol 2016 p 9 Article ID 15809172016

Discrete Dynamics in Nature and Society 7

sand polymerization degree and water-sand ratio was ob-tained as shown in Figure 3

With the increase of water-sand ratio the polymeriza-tion degree of water-sand increases In order to ensure thelongest erosion resistance time of sandcastles not only therelatively larger adhesion but also the optimization oferosion resistance performance should be considered Incombination with the actual situation the two-point intervalwith a wide viscosity range as shown in the figure was se-lected as the optimization parameter [14 15]

Based on the optimal proportion of the best shape se-lected above and the initial value of water-sand ratio NR as avariable the target T was discretized in the same wayAccording to the variation of foundation ΔS and Δh therelationship between the damage index and time the con-straint is added

07ltNRlt 085Obtain the continuous schedule record ofNR equidistant

points as shown in Table 1Using the above model we can get that when the water-

sand ratio is NR 074 under the condition of good shapeadhesion the maximum service life of sandcastle foundationcan be achieved

33 Sandcastle-Rain-Erosion Model When consideringrainfall a sandcastle-rain-erosion model is established andthe influence of rainfall is directly superimposed with that ofseawater that is the rainfall directly affects the sandcastleseroded by seawater Suppose the raindrops were particles ofmass hitting the sandcastle vertically with the same force asthe waves [16ndash18]

331 Model Preparation To study the effect of rain onsandcastles we first need to understand some of the motioncharacteristics of rain Let us say the rain falls straight downRaindrops have a lot of kinetic energy [19] When they hit asandcastle they destroy the structure of the sandcastle andchange the water content of the sandcastle 0erefore we

divide the impact of rain on sandcastles into two parts scourand infiltration [20 21]

Formula for calculating the final velocity of raindrop is asfollows

If dle 3mm we have

Vmax

389v

d1113874 1113875

2+ 2400g d minus 389

v

d

1113971

(16)

If 3mmlt dle 6mm we have

Vmax d

(0113 + 0845 d) (17)

In the formula v is the viscosity coefficient of air motionWhen T 293K v 1810741555times10minus5 PamiddotS

0ink of raindrops as spheres According to the mo-mentum formula q mVmax and energy formulaE (mV2

max)2 the momentum and energy contained ineach raindrop can be obtained and then the relationshipbetween raindrop diameter and momentum and energy canbe obtained [22ndash24]

332 Model Establishing and Solving However in the ac-tual calculation of rainfall characteristics such as rainfallkinetic energy only the data of rainfall or rainfall intensity areusually available and the observed data of raindrop diameterare not available In order to facilitate production and appli-cation the final raindrop velocity can be expressed as a functionof rain intensity [25] According to the analysis of themeasureddata there is a power function relationship between themedianraindrop diameter and rainfall intensity

dm 252i032

(18)

In the formula dm represents median diameter ofraindrops the units are mm i represents rain intensity theunits are mmmin 0e final velocity of raindrop is deter-mined by rainfall intensity through substitution andcalculation

If ile 213 we have

Vmax

1544v

i0231113874 11138752

+ 6048gi023

1113971

minus 1544v

i023

(19)

If 213lt ile 4346 we have

Vmax i023

00448 + 00845i023

1113872 1113873 (20)

From formula (16) to formula (20) the relationshipbetween precipitation and raindrop velocity energy andmomentum is calculated as shown in Figures 4ndash6

After understanding the motion parameters of rain-drops we also need to analyze the influence of raindrops onsandcastles [26] 0e SartorndashBoyd scour model is mainly

0 01 02 03 04 05 06 07 08 09 1The ratio of water and sand

(0 036)

(07 0476)(085 0487)

034

036

038

04

042

044

046

048

05

052

Deg

ree o

f con

verg

ence

of w

ater

and

sand

Figure 3 Convergence of water and sand

Table 1 Different water-sand ratios correspond to durationWater-sand ratio 070 072 074 076 078 080 082 084Duration 47 49 52 48 51 50 46 44

4 Discrete Dynamics in Nature and Society

applicable to the rainfall process with initial scour effect asshown in the following equation

dVp

dt KeVpRp (21)

In formula (21) Ke represents the erosion coefficient theunits are mmminus1Vp represents the volume of sandcastle at thebeginning the units are m3 Rp represents rainwater runoffper unit time per unit area the units are mmmin t rep-resents time the unit is min In our model take Ke as 02 toget the change of sand grain quantity washed away byprecipitation as shown in Figure 7

333 Result Analysis When the sandcastle is affected byrainfall MATLAB is used to fit the relationship between theinternal friction angle and water content of the sandcastle asshown in Figure 8

0 5 10 15 20 25 30 35 40 45Rain fall (mmmin)

0

15times10ndash23

Rain

drop

fina

l ene

rgy

(J)

03

06

09

12

Figure 5 Relationship between precipitation and raindrop energy

0 5 10 15 20 25 3530 4540 5550 60Time (min)

0

Volu

me o

f san

d sc

oure

d (m

3 )

0005

001

0015

002

0025

003

0035

004

Figure 7 Changes in the amount of sand washed away byprecipitation

0 5 10 15 20 25 30 35 40 45Rain fall (mmmin)

0

3times10ndash9

Rain

drop

fina

l mom

entu

m (k

gmiddotm

s)

03

06

09

12

15

18

21

24

27

Figure 6 Relationship between precipitation and raindropmomentum

0 5 10 15 20 25 30 35 40 45Rain fall (mmmin)

7678

882848688

99294969810

Rain

drop

fina

l spe

ed (m

s)

Figure 4 Relationship between precipitation and raindrop speed

40 45 50 55 60 65 70 75 80Water content ()

70In

tern

al fr

icito

n an

gle ϕ

(ordm)

40

45

50

55

60

65

ϕ = 079468w + 900782

Figure 8 Relationship between water content and internal frictionangle caused by rainfall on sandcastle

Discrete Dynamics in Nature and Society 5

In the precipitation model we get the formula for cal-culating the final velocity of raindrop 0en the watercontent of sandcastle and the angle of internal frictionduring rainfall are calculated Finally compare the rela-tionship between the two and draw a conclusion 0e resultsshow that the rain has a certain effect on the structuralstability of the sandcastle 0rough the establishment andsolution of the above model we find that streamline is stillthe optimal geometric shape which verifies the reliability ofthe model

4 Conclusions

On the basis of the simulation results the destructive powerof seawater rainy days and other environments is sum-marized which is an important threat to the sandcastlefoundation [27ndash31] Several models established in the de-tailed mathematical analysis provided answers to the re-quired questions including a view of the exterior of thesandcastle with the longest period of stability in the naturalstate

In general a sandcastle foundation with a gentle slope isgood at resisting seawater erosion while a foundation with alarge top is good at resisting rain After many iterations allinitial sandcastle foundation transfers have similar struc-tures As mentioned earlier the optimal sand-water mixtureratio is about 074 Due to the trade-off between friction andfluidity three measures can be taken to strengthen thesandcastle foundation such as adjusting construction timeadding support structure and improving the building0oseare all practical measures one can take in order to obtain abetter sandcastle foundation

Data Availability

0e data in this paper come from Question B of the 2020American College Students Mathematical ModelingCompetition

Conflicts of Interest

0e authors declare that there are no conflicts of interestregarding the publication of this paper

Acknowledgments

0is study was funded by the Humanities and Social SciencesResearch Major Project of the Education Department ofAnhui Province (SK2017A0452) the Teaching and ResearchFund Project of the Education Department of AnhuiProvince (2018jyxm1305) and the Teaching and ResearchFund Project of the Anhui University of Finance andEconomics (acxkjsjy201803zd and acjyyb2020011)

References

[1] J P Bouchaud M E Cates J R Prakash and S F EdwardsldquoHysteresis and metastability in a continuum sandpilemodelrdquo Physical Review Letters vol 74 no 11 pp 1982ndash19851995

[2] Y L Chen G Y Liu N Li X Du S-R Wang and R AzzamldquoStability evaluation of slope subjected to seismic effectcombined with consequent rainfallrdquo Engineering Geologyvol 266 Article ID 105461 2020

[3] M Pakpour M Habibi P Moslashller and D Bonn ldquoHow toconstruct the perfect sandcastlerdquo Scientific Reports vol 2no 1 pp 1ndash3 2012

[4] S Dumont and I Noureddine ldquoOn a dual formulation for thegrowing sand pile problemrdquo European Journal of AppliedMathematics vol 20 no 2 pp 169ndash185 2009

[5] N Fraysse H 0ome and L Petit ldquoHumidity effects on thestability of a sand pilerdquo e European Physical JournalB-Condensed Matter and Complex Systems vol 11 no 4pp 615ndash619 1999

[6] T Groger U Tuzun and D M Heyes ldquoModelling andmeasuring of cohesion in wet granular materialsrdquo PowderTechnology vol 133 no 3 pp 203ndash215 2003

[7] T C Halsey and A J Levine ldquoHow sandcastles fallrdquo PhysicalReview Letters vol 80 no 14 pp 31ndash41 1998

[8] G L George ldquoMagic sand modeling the hydrophobic effectand reversed-phase liquid chromatographyrdquo Journal ofChemical Education vol 67 no 6 pp 512ndash515 1990

[9] M Scheel R Seemann M Brinkmann et al ldquoMorphologicalclues to wet granular pile stabilityrdquo Nature Materials vol 7no 3 pp 189ndash193 2008

[10] S Dumontl and N Igbidal ldquoOn the collapsing sandpileproblemrdquo Communications on Pure and Applied Analysisvol 10 no 2 pp 625ndash638 2010

[11] C Mejia and J A Montoya ldquoOn the complexity of sand pilecritical avalanchesrdquo eoretical Computer Science vol 412no 30 pp 3964ndash3974 2011

[12] S Qiao S Qin J Chen X Hu and Z Ma ldquo0e application ofa three-dimensional deterministic model in the study ofdebris flow prediction based on the rainfall-unstable soilcoupling mechanismrdquo Processes vol 7 no 2 2019

[13] N Igbida and N Noureddine ldquoA partial integrodifferentialequation in granular matter and its connection with a sto-chastic modelrdquo Siam Journal on Mathematical Analysisvol 44 no 3 pp 1950ndash1975 2012

[14] N Igbida ldquoA generalized collapsing sandpile modelrdquo ArchivDer Mathematik vol 94 no 2 pp 193ndash200 2010

[15] A Sotirov and S H Yu ldquoOn the solution of a Boltzmannsystem for gas mixturesrdquo Archive for Rational Mechanics ampAnalysis vol 195 no 2 pp 675ndash700 2010

[16] G Sorbino and M V Nicotera ldquoUnsaturated soil mechanicsin rainfall-induced flow landslidesrdquo Engineering Geologyvol 165 pp 105ndash132 2013

[17] M Burylo C Hudek and F Rey ldquoSoil reinforcement by theroots of six dominant species on eroded mountainous marlyslopes (Southern Alps France)rdquo CATENA vol 84 no 1-2pp 70ndash78 2011

[18] Y Hong R Chen C Wu and J Chen ldquoShaking table testsand stability analysis of steep nailed slopesrdquo CanadianGeotechnical Journal vol 42 no 5 pp 1264ndash1279 2011

[19] H Moriwaki T Inokuchi T Hattanji K Sassa H Ochiaiand G Wang ldquoFailure processes in a full-scale landslideexperiment using a rainfall simulatorrdquo Landslides vol 1no 4 pp 277ndash288 2004

[20] M Lin and K Wang ldquoSeismic slope behavior in a large-scaleshaking table model testrdquo Engineering Geology vol 86 no 2-3 pp 118ndash133 2006

[21] K Sasahara and N Sakai ldquoDevelopment of shear deformationdue to the increase of pore pressure in a sandy model slope

6 Discrete Dynamics in Nature and Society

during rainfallrdquo Engineering Geology vol 170 pp 43ndash512014

[22] L Pantelidis and D V Griffiths ldquoStability of earth slopes PartII three dimensional analysis in closed-formrdquo InternationalJournal for Numerical and Analytical Methods in Geo-mechanics vol 37 no 13 pp 1987ndash2004 2013

[23] N Iwata R Yoshinaka and T Sasaki ldquoApplicability of theseismic response analysis using multiple yield model fordiscontinuous rockrdquo International Journal of Rock Mechanicsand Mining Sciences vol 60 pp 196ndash207 2013

[24] H Yuehua Z Cheng and L I Hongmei ldquoCombined effectsof trees and macropores on slope stability subjected torainfallrdquo Journal of Water Resources amp Architectural Engi-neering 2018

[25] Z Yu Y L Kun G Z Zheng et al ldquoEvaluation of the stabilityof Maliulin landslide under the reservoir water level fluctu-ation combined with rainfallrdquo Geological Science and Tech-nology Information 2019

[26] A Chinkulkijniwat T Tirametatiparat C Supotayan et alldquoStability characteristics of shallow landslide triggered by rainfallrdquoJournal of Mountain Science vol 16 pp 2171ndash2183 2019

[27] J B Liu J Zhao J Cao and J Min ldquoOn the Hosoya index ofgraphs formed by a fractal graphrdquo Fractals-Complex GeometryPatterns and Scaling in Nature and Society vol 27 no 8pp 19ndash35 2019

[28] J B Liu J Zhao H He and Z Shao ldquoValency-based To-pological descriptors and structural property of the gener-alized Sierpiacute nski networksrdquo Journal of Statistical Physicsvol 177 no 6 pp 1131ndash1147 2019

[29] J B Liu J Zhao and Z Cai ldquoOn the generalized adjacencyLaplacian and signless Laplacian spectra of the weighted edgecorona networksrdquo Physica A Statistical Mechanics and itsApplications vol 540 pp 12ndash30 2020

[30] J M Zhu W Y Xia J J Sun J-B Liu and F-H Yu ldquo0espread pattern on Ebola and the control schemesrdquo Interna-tional Journal of Innovative Computing and Applicationsvol 9 no 2 pp 77ndash89 2018

[31] J-M Zhu L Wang and J-B Liu ldquoEradication of Ebola basedon dynamic programmingrdquo Computational and Mathemat-ical Methods in Medicine vol 2016 p 9 Article ID 15809172016

Discrete Dynamics in Nature and Society 7

applicable to the rainfall process with initial scour effect asshown in the following equation

dVp

dt KeVpRp (21)

In formula (21) Ke represents the erosion coefficient theunits are mmminus1Vp represents the volume of sandcastle at thebeginning the units are m3 Rp represents rainwater runoffper unit time per unit area the units are mmmin t rep-resents time the unit is min In our model take Ke as 02 toget the change of sand grain quantity washed away byprecipitation as shown in Figure 7

333 Result Analysis When the sandcastle is affected byrainfall MATLAB is used to fit the relationship between theinternal friction angle and water content of the sandcastle asshown in Figure 8

0 5 10 15 20 25 30 35 40 45Rain fall (mmmin)

0

15times10ndash23

Rain

drop

fina

l ene

rgy

(J)

03

06

09

12

Figure 5 Relationship between precipitation and raindrop energy

0 5 10 15 20 25 3530 4540 5550 60Time (min)

0

Volu

me o

f san

d sc

oure

d (m

3 )

0005

001

0015

002

0025

003

0035

004

Figure 7 Changes in the amount of sand washed away byprecipitation

0 5 10 15 20 25 30 35 40 45Rain fall (mmmin)

0

3times10ndash9

Rain

drop

fina

l mom

entu

m (k

gmiddotm

s)

03

06

09

12

15

18

21

24

27

Figure 6 Relationship between precipitation and raindropmomentum

0 5 10 15 20 25 30 35 40 45Rain fall (mmmin)

7678

882848688

99294969810

Rain

drop

fina

l spe

ed (m

s)

Figure 4 Relationship between precipitation and raindrop speed

40 45 50 55 60 65 70 75 80Water content ()

70In

tern

al fr

icito

n an

gle ϕ

(ordm)

40

45

50

55

60

65

ϕ = 079468w + 900782

Figure 8 Relationship between water content and internal frictionangle caused by rainfall on sandcastle

Discrete Dynamics in Nature and Society 5

In the precipitation model we get the formula for cal-culating the final velocity of raindrop 0en the watercontent of sandcastle and the angle of internal frictionduring rainfall are calculated Finally compare the rela-tionship between the two and draw a conclusion 0e resultsshow that the rain has a certain effect on the structuralstability of the sandcastle 0rough the establishment andsolution of the above model we find that streamline is stillthe optimal geometric shape which verifies the reliability ofthe model

4 Conclusions

On the basis of the simulation results the destructive powerof seawater rainy days and other environments is sum-marized which is an important threat to the sandcastlefoundation [27ndash31] Several models established in the de-tailed mathematical analysis provided answers to the re-quired questions including a view of the exterior of thesandcastle with the longest period of stability in the naturalstate

In general a sandcastle foundation with a gentle slope isgood at resisting seawater erosion while a foundation with alarge top is good at resisting rain After many iterations allinitial sandcastle foundation transfers have similar struc-tures As mentioned earlier the optimal sand-water mixtureratio is about 074 Due to the trade-off between friction andfluidity three measures can be taken to strengthen thesandcastle foundation such as adjusting construction timeadding support structure and improving the building0oseare all practical measures one can take in order to obtain abetter sandcastle foundation

Data Availability

0e data in this paper come from Question B of the 2020American College Students Mathematical ModelingCompetition

Conflicts of Interest

0e authors declare that there are no conflicts of interestregarding the publication of this paper

Acknowledgments

0is study was funded by the Humanities and Social SciencesResearch Major Project of the Education Department ofAnhui Province (SK2017A0452) the Teaching and ResearchFund Project of the Education Department of AnhuiProvince (2018jyxm1305) and the Teaching and ResearchFund Project of the Anhui University of Finance andEconomics (acxkjsjy201803zd and acjyyb2020011)

References

[1] J P Bouchaud M E Cates J R Prakash and S F EdwardsldquoHysteresis and metastability in a continuum sandpilemodelrdquo Physical Review Letters vol 74 no 11 pp 1982ndash19851995

[2] Y L Chen G Y Liu N Li X Du S-R Wang and R AzzamldquoStability evaluation of slope subjected to seismic effectcombined with consequent rainfallrdquo Engineering Geologyvol 266 Article ID 105461 2020

[3] M Pakpour M Habibi P Moslashller and D Bonn ldquoHow toconstruct the perfect sandcastlerdquo Scientific Reports vol 2no 1 pp 1ndash3 2012

[4] S Dumont and I Noureddine ldquoOn a dual formulation for thegrowing sand pile problemrdquo European Journal of AppliedMathematics vol 20 no 2 pp 169ndash185 2009

[5] N Fraysse H 0ome and L Petit ldquoHumidity effects on thestability of a sand pilerdquo e European Physical JournalB-Condensed Matter and Complex Systems vol 11 no 4pp 615ndash619 1999

[6] T Groger U Tuzun and D M Heyes ldquoModelling andmeasuring of cohesion in wet granular materialsrdquo PowderTechnology vol 133 no 3 pp 203ndash215 2003

[7] T C Halsey and A J Levine ldquoHow sandcastles fallrdquo PhysicalReview Letters vol 80 no 14 pp 31ndash41 1998

[8] G L George ldquoMagic sand modeling the hydrophobic effectand reversed-phase liquid chromatographyrdquo Journal ofChemical Education vol 67 no 6 pp 512ndash515 1990

[9] M Scheel R Seemann M Brinkmann et al ldquoMorphologicalclues to wet granular pile stabilityrdquo Nature Materials vol 7no 3 pp 189ndash193 2008

[10] S Dumontl and N Igbidal ldquoOn the collapsing sandpileproblemrdquo Communications on Pure and Applied Analysisvol 10 no 2 pp 625ndash638 2010

[11] C Mejia and J A Montoya ldquoOn the complexity of sand pilecritical avalanchesrdquo eoretical Computer Science vol 412no 30 pp 3964ndash3974 2011

[12] S Qiao S Qin J Chen X Hu and Z Ma ldquo0e application ofa three-dimensional deterministic model in the study ofdebris flow prediction based on the rainfall-unstable soilcoupling mechanismrdquo Processes vol 7 no 2 2019

[13] N Igbida and N Noureddine ldquoA partial integrodifferentialequation in granular matter and its connection with a sto-chastic modelrdquo Siam Journal on Mathematical Analysisvol 44 no 3 pp 1950ndash1975 2012

[14] N Igbida ldquoA generalized collapsing sandpile modelrdquo ArchivDer Mathematik vol 94 no 2 pp 193ndash200 2010

[15] A Sotirov and S H Yu ldquoOn the solution of a Boltzmannsystem for gas mixturesrdquo Archive for Rational Mechanics ampAnalysis vol 195 no 2 pp 675ndash700 2010

[16] G Sorbino and M V Nicotera ldquoUnsaturated soil mechanicsin rainfall-induced flow landslidesrdquo Engineering Geologyvol 165 pp 105ndash132 2013

[17] M Burylo C Hudek and F Rey ldquoSoil reinforcement by theroots of six dominant species on eroded mountainous marlyslopes (Southern Alps France)rdquo CATENA vol 84 no 1-2pp 70ndash78 2011

[18] Y Hong R Chen C Wu and J Chen ldquoShaking table testsand stability analysis of steep nailed slopesrdquo CanadianGeotechnical Journal vol 42 no 5 pp 1264ndash1279 2011

[19] H Moriwaki T Inokuchi T Hattanji K Sassa H Ochiaiand G Wang ldquoFailure processes in a full-scale landslideexperiment using a rainfall simulatorrdquo Landslides vol 1no 4 pp 277ndash288 2004

[20] M Lin and K Wang ldquoSeismic slope behavior in a large-scaleshaking table model testrdquo Engineering Geology vol 86 no 2-3 pp 118ndash133 2006

[21] K Sasahara and N Sakai ldquoDevelopment of shear deformationdue to the increase of pore pressure in a sandy model slope

6 Discrete Dynamics in Nature and Society

during rainfallrdquo Engineering Geology vol 170 pp 43ndash512014

[22] L Pantelidis and D V Griffiths ldquoStability of earth slopes PartII three dimensional analysis in closed-formrdquo InternationalJournal for Numerical and Analytical Methods in Geo-mechanics vol 37 no 13 pp 1987ndash2004 2013

[23] N Iwata R Yoshinaka and T Sasaki ldquoApplicability of theseismic response analysis using multiple yield model fordiscontinuous rockrdquo International Journal of Rock Mechanicsand Mining Sciences vol 60 pp 196ndash207 2013

[24] H Yuehua Z Cheng and L I Hongmei ldquoCombined effectsof trees and macropores on slope stability subjected torainfallrdquo Journal of Water Resources amp Architectural Engi-neering 2018

[25] Z Yu Y L Kun G Z Zheng et al ldquoEvaluation of the stabilityof Maliulin landslide under the reservoir water level fluctu-ation combined with rainfallrdquo Geological Science and Tech-nology Information 2019

[26] A Chinkulkijniwat T Tirametatiparat C Supotayan et alldquoStability characteristics of shallow landslide triggered by rainfallrdquoJournal of Mountain Science vol 16 pp 2171ndash2183 2019

[27] J B Liu J Zhao J Cao and J Min ldquoOn the Hosoya index ofgraphs formed by a fractal graphrdquo Fractals-Complex GeometryPatterns and Scaling in Nature and Society vol 27 no 8pp 19ndash35 2019

[28] J B Liu J Zhao H He and Z Shao ldquoValency-based To-pological descriptors and structural property of the gener-alized Sierpiacute nski networksrdquo Journal of Statistical Physicsvol 177 no 6 pp 1131ndash1147 2019

[29] J B Liu J Zhao and Z Cai ldquoOn the generalized adjacencyLaplacian and signless Laplacian spectra of the weighted edgecorona networksrdquo Physica A Statistical Mechanics and itsApplications vol 540 pp 12ndash30 2020

[30] J M Zhu W Y Xia J J Sun J-B Liu and F-H Yu ldquo0espread pattern on Ebola and the control schemesrdquo Interna-tional Journal of Innovative Computing and Applicationsvol 9 no 2 pp 77ndash89 2018

[31] J-M Zhu L Wang and J-B Liu ldquoEradication of Ebola basedon dynamic programmingrdquo Computational and Mathemat-ical Methods in Medicine vol 2016 p 9 Article ID 15809172016

Discrete Dynamics in Nature and Society 7

In the precipitation model we get the formula for cal-culating the final velocity of raindrop 0en the watercontent of sandcastle and the angle of internal frictionduring rainfall are calculated Finally compare the rela-tionship between the two and draw a conclusion 0e resultsshow that the rain has a certain effect on the structuralstability of the sandcastle 0rough the establishment andsolution of the above model we find that streamline is stillthe optimal geometric shape which verifies the reliability ofthe model

4 Conclusions

On the basis of the simulation results the destructive powerof seawater rainy days and other environments is sum-marized which is an important threat to the sandcastlefoundation [27ndash31] Several models established in the de-tailed mathematical analysis provided answers to the re-quired questions including a view of the exterior of thesandcastle with the longest period of stability in the naturalstate

In general a sandcastle foundation with a gentle slope isgood at resisting seawater erosion while a foundation with alarge top is good at resisting rain After many iterations allinitial sandcastle foundation transfers have similar struc-tures As mentioned earlier the optimal sand-water mixtureratio is about 074 Due to the trade-off between friction andfluidity three measures can be taken to strengthen thesandcastle foundation such as adjusting construction timeadding support structure and improving the building0oseare all practical measures one can take in order to obtain abetter sandcastle foundation

Data Availability

0e data in this paper come from Question B of the 2020American College Students Mathematical ModelingCompetition

Conflicts of Interest

0e authors declare that there are no conflicts of interestregarding the publication of this paper

Acknowledgments

0is study was funded by the Humanities and Social SciencesResearch Major Project of the Education Department ofAnhui Province (SK2017A0452) the Teaching and ResearchFund Project of the Education Department of AnhuiProvince (2018jyxm1305) and the Teaching and ResearchFund Project of the Anhui University of Finance andEconomics (acxkjsjy201803zd and acjyyb2020011)

References

[1] J P Bouchaud M E Cates J R Prakash and S F EdwardsldquoHysteresis and metastability in a continuum sandpilemodelrdquo Physical Review Letters vol 74 no 11 pp 1982ndash19851995

[2] Y L Chen G Y Liu N Li X Du S-R Wang and R AzzamldquoStability evaluation of slope subjected to seismic effectcombined with consequent rainfallrdquo Engineering Geologyvol 266 Article ID 105461 2020

[3] M Pakpour M Habibi P Moslashller and D Bonn ldquoHow toconstruct the perfect sandcastlerdquo Scientific Reports vol 2no 1 pp 1ndash3 2012

[4] S Dumont and I Noureddine ldquoOn a dual formulation for thegrowing sand pile problemrdquo European Journal of AppliedMathematics vol 20 no 2 pp 169ndash185 2009

[5] N Fraysse H 0ome and L Petit ldquoHumidity effects on thestability of a sand pilerdquo e European Physical JournalB-Condensed Matter and Complex Systems vol 11 no 4pp 615ndash619 1999

[6] T Groger U Tuzun and D M Heyes ldquoModelling andmeasuring of cohesion in wet granular materialsrdquo PowderTechnology vol 133 no 3 pp 203ndash215 2003

[7] T C Halsey and A J Levine ldquoHow sandcastles fallrdquo PhysicalReview Letters vol 80 no 14 pp 31ndash41 1998

[8] G L George ldquoMagic sand modeling the hydrophobic effectand reversed-phase liquid chromatographyrdquo Journal ofChemical Education vol 67 no 6 pp 512ndash515 1990

[9] M Scheel R Seemann M Brinkmann et al ldquoMorphologicalclues to wet granular pile stabilityrdquo Nature Materials vol 7no 3 pp 189ndash193 2008

[10] S Dumontl and N Igbidal ldquoOn the collapsing sandpileproblemrdquo Communications on Pure and Applied Analysisvol 10 no 2 pp 625ndash638 2010

[11] C Mejia and J A Montoya ldquoOn the complexity of sand pilecritical avalanchesrdquo eoretical Computer Science vol 412no 30 pp 3964ndash3974 2011

[12] S Qiao S Qin J Chen X Hu and Z Ma ldquo0e application ofa three-dimensional deterministic model in the study ofdebris flow prediction based on the rainfall-unstable soilcoupling mechanismrdquo Processes vol 7 no 2 2019

[13] N Igbida and N Noureddine ldquoA partial integrodifferentialequation in granular matter and its connection with a sto-chastic modelrdquo Siam Journal on Mathematical Analysisvol 44 no 3 pp 1950ndash1975 2012

[14] N Igbida ldquoA generalized collapsing sandpile modelrdquo ArchivDer Mathematik vol 94 no 2 pp 193ndash200 2010

[15] A Sotirov and S H Yu ldquoOn the solution of a Boltzmannsystem for gas mixturesrdquo Archive for Rational Mechanics ampAnalysis vol 195 no 2 pp 675ndash700 2010

[16] G Sorbino and M V Nicotera ldquoUnsaturated soil mechanicsin rainfall-induced flow landslidesrdquo Engineering Geologyvol 165 pp 105ndash132 2013

[17] M Burylo C Hudek and F Rey ldquoSoil reinforcement by theroots of six dominant species on eroded mountainous marlyslopes (Southern Alps France)rdquo CATENA vol 84 no 1-2pp 70ndash78 2011

[18] Y Hong R Chen C Wu and J Chen ldquoShaking table testsand stability analysis of steep nailed slopesrdquo CanadianGeotechnical Journal vol 42 no 5 pp 1264ndash1279 2011

[19] H Moriwaki T Inokuchi T Hattanji K Sassa H Ochiaiand G Wang ldquoFailure processes in a full-scale landslideexperiment using a rainfall simulatorrdquo Landslides vol 1no 4 pp 277ndash288 2004

[20] M Lin and K Wang ldquoSeismic slope behavior in a large-scaleshaking table model testrdquo Engineering Geology vol 86 no 2-3 pp 118ndash133 2006

[21] K Sasahara and N Sakai ldquoDevelopment of shear deformationdue to the increase of pore pressure in a sandy model slope

6 Discrete Dynamics in Nature and Society

during rainfallrdquo Engineering Geology vol 170 pp 43ndash512014

[22] L Pantelidis and D V Griffiths ldquoStability of earth slopes PartII three dimensional analysis in closed-formrdquo InternationalJournal for Numerical and Analytical Methods in Geo-mechanics vol 37 no 13 pp 1987ndash2004 2013

[23] N Iwata R Yoshinaka and T Sasaki ldquoApplicability of theseismic response analysis using multiple yield model fordiscontinuous rockrdquo International Journal of Rock Mechanicsand Mining Sciences vol 60 pp 196ndash207 2013

[24] H Yuehua Z Cheng and L I Hongmei ldquoCombined effectsof trees and macropores on slope stability subjected torainfallrdquo Journal of Water Resources amp Architectural Engi-neering 2018

[25] Z Yu Y L Kun G Z Zheng et al ldquoEvaluation of the stabilityof Maliulin landslide under the reservoir water level fluctu-ation combined with rainfallrdquo Geological Science and Tech-nology Information 2019

[26] A Chinkulkijniwat T Tirametatiparat C Supotayan et alldquoStability characteristics of shallow landslide triggered by rainfallrdquoJournal of Mountain Science vol 16 pp 2171ndash2183 2019

[27] J B Liu J Zhao J Cao and J Min ldquoOn the Hosoya index ofgraphs formed by a fractal graphrdquo Fractals-Complex GeometryPatterns and Scaling in Nature and Society vol 27 no 8pp 19ndash35 2019

[28] J B Liu J Zhao H He and Z Shao ldquoValency-based To-pological descriptors and structural property of the gener-alized Sierpiacute nski networksrdquo Journal of Statistical Physicsvol 177 no 6 pp 1131ndash1147 2019

[29] J B Liu J Zhao and Z Cai ldquoOn the generalized adjacencyLaplacian and signless Laplacian spectra of the weighted edgecorona networksrdquo Physica A Statistical Mechanics and itsApplications vol 540 pp 12ndash30 2020

[30] J M Zhu W Y Xia J J Sun J-B Liu and F-H Yu ldquo0espread pattern on Ebola and the control schemesrdquo Interna-tional Journal of Innovative Computing and Applicationsvol 9 no 2 pp 77ndash89 2018

[31] J-M Zhu L Wang and J-B Liu ldquoEradication of Ebola basedon dynamic programmingrdquo Computational and Mathemat-ical Methods in Medicine vol 2016 p 9 Article ID 15809172016

Discrete Dynamics in Nature and Society 7

during rainfallrdquo Engineering Geology vol 170 pp 43ndash512014

[22] L Pantelidis and D V Griffiths ldquoStability of earth slopes PartII three dimensional analysis in closed-formrdquo InternationalJournal for Numerical and Analytical Methods in Geo-mechanics vol 37 no 13 pp 1987ndash2004 2013

[23] N Iwata R Yoshinaka and T Sasaki ldquoApplicability of theseismic response analysis using multiple yield model fordiscontinuous rockrdquo International Journal of Rock Mechanicsand Mining Sciences vol 60 pp 196ndash207 2013

[24] H Yuehua Z Cheng and L I Hongmei ldquoCombined effectsof trees and macropores on slope stability subjected torainfallrdquo Journal of Water Resources amp Architectural Engi-neering 2018

[25] Z Yu Y L Kun G Z Zheng et al ldquoEvaluation of the stabilityof Maliulin landslide under the reservoir water level fluctu-ation combined with rainfallrdquo Geological Science and Tech-nology Information 2019

[26] A Chinkulkijniwat T Tirametatiparat C Supotayan et alldquoStability characteristics of shallow landslide triggered by rainfallrdquoJournal of Mountain Science vol 16 pp 2171ndash2183 2019

[27] J B Liu J Zhao J Cao and J Min ldquoOn the Hosoya index ofgraphs formed by a fractal graphrdquo Fractals-Complex GeometryPatterns and Scaling in Nature and Society vol 27 no 8pp 19ndash35 2019

[28] J B Liu J Zhao H He and Z Shao ldquoValency-based To-pological descriptors and structural property of the gener-alized Sierpiacute nski networksrdquo Journal of Statistical Physicsvol 177 no 6 pp 1131ndash1147 2019

[29] J B Liu J Zhao and Z Cai ldquoOn the generalized adjacencyLaplacian and signless Laplacian spectra of the weighted edgecorona networksrdquo Physica A Statistical Mechanics and itsApplications vol 540 pp 12ndash30 2020

[30] J M Zhu W Y Xia J J Sun J-B Liu and F-H Yu ldquo0espread pattern on Ebola and the control schemesrdquo Interna-tional Journal of Innovative Computing and Applicationsvol 9 no 2 pp 77ndash89 2018

[31] J-M Zhu L Wang and J-B Liu ldquoEradication of Ebola basedon dynamic programmingrdquo Computational and Mathemat-ical Methods in Medicine vol 2016 p 9 Article ID 15809172016

Discrete Dynamics in Nature and Society 7