GPU-SPH simulation of Tsunami-like wave interaction with a...

43

GPU-SPH simulation of Tsunami-like wave interaction

with a seawall associated with underwater

Safiyari, Omid Reza1; Akbarpour Jannat, Mahmood Reza

1*; Banijamali, Babak

2

1- Iranian National Institute for Oceanography and Atmospheric Science (INIOAS), Tehran, IR. Iran

2- Darya Bandar Engineering Consultant Company (DBC), Tehran, IR. Iran

Received: January 2016 Accepted: May 2016

© 2016 Journal of the Persian Gulf. All rights reserved.

Abstract Investigation of the waves generated by underwater disturbances gives precious insight into the

effect of man-made underwater explosions as well as natural phenomena, such as underwater

volcanoes or oceanic meteor impact. On the other hand, prediction of the effects of such waves

on the coastal installations and structures is required for preparation worthwhile criteria for

coastal engineers to prepare a reliable design. This study aimed to investigate the interactional

effects of water waves generated by underwater disturbances on sea walls through numerical

modeling using the Smoothed Particle Hydrodynamics (SPH) method. The simulation was

performed using the Dual-SPHysics numerical code. Comparison of the numerical results with

the experimental data extracted from case studies demonstrated the good capability of the SPH

algorithm used in the numerical code in the simulation of initial wave generation by the

underwater disturbance and its propagation over the body of water. To examine the wave force

exerted on the walls, the results of laboratory experiences on the effect of tsunami waves on

coastal structures were used to verify the numerical model. The study found that the phenomena

with such nonlinear behavior can be very well simulated with a calibrated SPH model. We also

explored the effects of this type of wave and temporal changes of its resultant force on the wall.

In this article, the explosion-generated water wave produced much stronger fluctuations in the

vicinity of the wall than did the solitary wave, thus naturally, it can be more destructive.

Keywords: Wave generation, Seawall, Numerical modeling, Wave force, Underwater explosion, Smoothed Particle Hydrodynamics (SPH), Graphics Processing Unit (GPU)

1. Introduction

The research on the likelihood and effects of

underwater explosions is of particular importance for

engineers in different fields of design or construction

because of: (1) the development and presence of

variety of marine installations and structures located

in coastal regions, (2) the potential of manmade

action near critical oil and gas assets in territorial

waters of countries or explosions for construction

purposes in shallow water, and (3) the possibility of

explosion of offshore hydrocarbon facilities and

pipelines. The review of research and experiences in

the field of underwater disturbance demonstrates that

the human quest to apply the waves generated by

manmade explosions as an underwater disturbance

generator since the nineteenth century. In the past

century, the occurrence of unintentional and

unexpected strong explosions in the oceanic area and

Journal of the Persian Gulf

(Marine Science)/Vol. 7/No. 24/ June 2016/23/43-66

* Email: [email protected]

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Safiyari et al. / GPU-SPH Simulation of Tsunami-like wave intraction…

44

related deep water tests and field experiments draw

the attention of famous scientists and engineers

working on marine research and wave theory to a

new field of research focused on direct and indirect

consequences of underwater explosions, and

especially wave generation and propagation during

this phenomenon (LeMe’haute’, 1971, 1976, 1993).

In 1917, the explosion of Mont Blanc warship in the

Bay of Halifax, Canada produced huge tidal surges

that affected coastal areas and ports near the area. The destructive water waves generated during this

accident had major implications for military and

marine engineering. The number of deaths and the

extent of damage to buildings and facilities following

this accident were so high that it was labeled a

human and national catastrophe (Greenberg and et

al.., 1993). In 1944 and 1945, Professor Lich at Auckland

University directed a series of field tests aimed at

producing tsunami waves by underwater

disturbances, as part of a common research operation

called “Project Seal” involving New Zealand, US

and British forces. The results of these surveys

indicate that man-made explosions, produced in

various depths of water, can generate waves of long

wavelengths capable of destroying coastal areas. In

the mid-decades of the twentieth century, many

researchers, including Van Dorn (1961-1968),

Whalin (1962-1970), and LeMehaute (1967-1998),

used the results of underwater explosion field tests

conducted by US Army to introduce and develop

some theories in this regard. Considering the difficulty and limitations of

laboratory modeling and the hazards of field

experiments on this subject, numerical simulation is

the preferred method of marine engineers and

researchers investigating the reaction of marine

environments to underwater explosions. Other

researchers, such as Crawford and Mader (1998) and

Simskin and Fiske (2003) have employed the results

and theoretical models of this field to develop a

series of concepts for studying the effects of similar

phenomena such as tsunami waves and waves

generated by explosion of submarine volcanoes and

for calculating the characteristics of waves generated

by the collision of celestial objects and meteorites. From the 1960s until the early 21st century, the

scholars working in this field used to employ the

Eulerian numerical methods to arrange governing

equations into discrete form and solve them

accordingly. But, because of reasons, such as the

high velocity of the shock wave, emergence of large

deformations, and separation of water particles from

each other, solving the explosion-generated wave

equations is a computationally complex process. With the development of meshless computational

methods, improvement of numerical calculation

techniques, and progress of hardware technology

since the 1990s, the Lagrangian-based techniques

have become the analysis method of choice in

several fields. The most common application of such

methods is in the simulation of shock propagation

problems wherein variables change drastically with

time. One of the notable Lagrangian methods for

solving fluid mechanics problems is the smoothed

particle hydrodynamics (SPH) method. This method

was developed by Monaghan and Gingold in 1977 to

simulate problems in various physical aspects. The

original application of this method was to analyze

the problems concerning astronomy and movements

of cosmic devices, but given the variety and

complexity of particular problems of computational

fluid mechanics, such as free surface problems in

coastal engineering, and also because of the extreme

difficulty of developing Eulerian models for these

problems, SPH is being increasingly used for the

analysis of marine phenomena.

Gómez-Gesteira et al. (2004) and Gómez-Gesteira

et al. (2012) studied wave impact on wall structures

by 3D SPH method. Rogers et al.. (2010) simulated

the movement of caisson breakwaters due to the

impact of waves using a 2D SPH model. Robertson

et al.. (2011) tried to understand the response of

coastal walls under tsunami forces by laboratory

investigations. Pu and Shao (2012) investigated

overtopping characteristics of wave impact on

coastal structures by SPH technique. El-Solh (2012)

studied the impact of solitary wave and its

hydrodynamics effects on sea walls. Barreiro et al.

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Journal of the Persian Gulf (Marine Science)/Vol .7/No .24/June 2016/23/43-66

45

(2013) tried to solve wave behavior and interaction

by SPH method. Altomare et al. (2014) analyzed

interaction of waves on armor type breakwaters,

Altomare et al. (2015) investigated wave impact on

coastal structures by SPH.

Using parallel processing methods in combination

with properties of graphical processing units of

computers, a new generation of high performance

computing techniques was developed to apply SPH

method in CFD and oceanic computations.

Domínguez et al. (2013), studied and developed new

computational techniques for applying GPU to

numerical modeling by SPH. Crespo et al. (2015)

developed traditional SPH model on CPU into new

form by applying GPU. Altomare et al. (2015)

studied applicability of smoothed particle

hydrodynamics for estimation of sea wave impact on

coastal structures. Altomare et al. (2017),

investigated modeling of long-crested wave

generation and absorption by SPH on GPU.

Previous studies have shown that the use of this

new methodology on GPU, provides better

computational speed and precision to study complex

marine phenomena that require long analysis time.

Besides, for problems with complicated and large

deformation on free water surface, a Lagrangian

point of view is more practical and relevant.

In this study, we use the SPH method to simulate

the waves generated by the underwater disturbance,

such as explosion and examine its capability in this

application. One of the main goals of this research is

to study the impacting forces on the coastal wall in

front of an explosion-generated water wave. Having

the pressure distribution, the size of force applied to

the wall could be easily determined. Naturally,

knowing the load induced by explosion-generated

water waves, engineers will be able to design the

walls and coastal installations with better measures

to counteract such effects. Another issue implicitly

addressed in this study is the difference between the

time histories of the force produced by the tsunami

wave and the explosion-generated water wave on the

vertical coastal walls.

2. Materials and Methods

The smoothed-particle hydrodynamics (SPH)

method is a Lagrangian approach based on the

simulation of the path of particle motion in an

analytical domain. The major advantages of the SPH

method are the independence of its simulations from

the computational grid. This feature allows the SPH to

bypass the limitation of the finite element or other

mesh-based methods for computational problems and

avoid the complexity imposed by re-meshing in the

problems that require the simulation of particle

segregation. Besides, with this numerical analysis

method, problems of varying complexities can be

numerically solved with significantly higher speed and

reduced simulation run time.

Given the high complexity of calculations required

for simulation of explosion-generated water waves,

utilization of high-performance computing systems for

this purpose is essential. For this purpose, we used a

GPU-based parallel processing tool called

DualSPHysics. The DualSPHysics is an open source

numerical code capable of using GPU’s processing

power, which has been developed in a joint research

project involving several European universities. In this

study, DualSPHysics and its capabilities in the

Lagrangian reference frame were used for modeling,

and the simulation results were compared with field

and experimental data to ensure the validity of the

numerical code. The hardware specifications of the

simulation system are listed in Table (1).

The uncertainties that exist in field and laboratory

observations of this phenomenon have caused

numerous and complicated errors in the results.

LeMehaute (1971) has said that the amount of

laboratory errors is between 30% and 100%. To

simulate this problem, a numerical model should be

used to minimize the effects on the accuracy and results

of such errors.

The SPH model has the ability to provide conditions

for separating water particles from each other and take

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Table 1: General Specification of the applied hardware and analytical parameters of various simulations

Executive

Parameters

Hardware Specifications

Machine No. 1

Hardware Specifications

Machine No. 2

Particle size (m) 10-3 ~ 5×10-1 CPU type AMD A8-

3870 APU CPU type

Intel®

Core™ i7-

4700MQ

Total

Particle 105 ~ 4×106 GPU type

GeForce

GTX 680 GPU type

GeForce

GTX 780M

GPU Run Time (sec) 1200 ~ 3400 CUDA

Core No. 1536

CUDA

Core No. 1536

CPU Run Time (sec) 3×103 ~

40×103

into account the high displacement of the particles in

the disturbance origin. Also, when waves impact on a

coastal wall, it is possible to examine the collision of

water particles with structures more precisely. These

advantages make it possible to simulate the generation

and propagation of waves from underwater explosion

with adequate reality in comparison with the Eulerian

type models that previously used by researchers (Mader

1998, Mader 2003, LeMehaute et al. 1986, LeMehaute

et al. 1998). Also, by applying the parallel processing

technique on GPUs, huge analyses could be performed

with cheaper prices in a shorter computational time.

2.1. Hydrodynamics of explosion-generated water

waves - variations of the initial conditions of the free

surface

The governing equations for an underwater explosion

consist of a set of conservation equations (mass

continuity, momentum conservation, and energy

conservation), the equation of state of detonation

products, the equation of state of water, and the

combustion equation, which is used to analyze the main

process of an explosion in the water environment. For

an explosion-generated surface wave to form, the gas

contained bubble produced at the moment of explosion

migrates toward water surface, and then, collides with

the free surface, rupturing its wall. This is followed by

the release of gas and fume constituents, which push

the water body outward while throwing the water

particles upward, thus generating the primary wave. Hence, the wave generation process can be divided into

two phases. In the first phase, first, the nucleus of the

bubble is formed, then, starts its evolution from a small

high-pressure dome at the depth of water to a large

bubble pulsating toward the free surface. The second

phase starts when the bubble reaches the water surface,

where it forms an initial deformation.

The initial deformation equation expressed by

Eq.(1), which presents a second order parabolic curve,

is based on the theoretical proposals of Poisson (1816)

and Kranzer and Keller (1959):

𝜂(𝑟) = {𝜂0 [2 (

r

R0)

2− 1] ; r ≤ R0

0 ; r > R0

(1)

Where 𝜂0 represents the maximum initial depth

and the initial height of the edges of the explosion

crater, R is the maximum span, "𝑟" represents the

distance to the center of the explosion location at

the water level and 𝜂(𝑟) is the function of water

surface variations at "𝑟" . Normally, the

characteristic parameters of crater shape, 𝜂0 and

R0, can be evaluated by empirical equations or

experimental results as a function of charge weight,

water depth and depth of explosion (Van Dorn,

1961, LeMehaute et al.. 1989). The initial geometry

of the free surface as characterized by Eq.(1) is

shown in Figure 1:

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Fig.1: Initial Parabolic with lip deformation of water level

The field studies conducted by Penney (1945 and

1946) and Jordan (1965) have proven that only 40% of

the energy released by an underwater explosion through

the release of chemical energy stored in an explosive

charge may generate water waves at free surface, as the

rest of the explosion energy will be spent in generation

of pressure waves in the water body and increase of

water temperature. The effective part of the released

chemical energy will be rapidly transformed into the

potential energy stored in the form of the initial

deformation of the water surface around the site of the

explosion. This potential energy is the generator of base

water wave.

Efficiency of an explosion as a wave producing

mechanism is one of the important issues of the

physical phenomenon. Amount of energy related to

water waves produced by underwater disturbances was

proposed by Whalin (1967) for different types of initial

water surface deformation on the basis of Sakurai

(1965) studies. The potential energy of deformed water

elevation can be represented by Eq.(2):

𝐸 = 2𝜋𝜌𝑔 ∫𝜂0

2(𝑟)

2𝑟𝑑𝑟

∞

0 (2)

By insertion of water level formula such as shown

by Eq.(1) the energy equation for quadratic surface

deformation with lip is determined as following by

Eq.(3):

𝐸 =𝜋𝜌𝑔𝜂0

2𝑅02

6 (3)

According to Figure 1, Van Dorn, LeMehaute and

Hwang (1968) by performing physical laboratory tests

show that for a parabolic crater, the radius of crater at

edge of lip is related to the yield of the explosive charge

by an empirical formula as per Eq.(4):

𝑅0 = 9.6𝑊0.3 (4)

Where the charge yield "𝑊" has units of pounds of

TNT, 𝑅0 and 𝜂0 have units of feet. 𝜂0 can be

determined for the parabolic crater by Eq.(5)

𝜂0 × 𝑅0 = 0.81𝐻𝑚𝑎𝑥 × 𝑟 (5)

Which was resulted from experiments. 𝐻𝑚𝑎𝑥 is the

maximum wave height in a wave group at a distance of

𝑟 from the location of the explosion. The value of

𝐻𝑚𝑎𝑥 × 𝑟 is dependent upon the depth of submergence

of the charge and can be determined for any depth of

submergence that was presented by Whalin, Pace and

Lane (1970) using curve fitting of field data. At upper

critical depth they proposed Eq.(6):

𝐻𝑚𝑎𝑥×𝑟

𝑊0.54 ≅ 34 (6)

Also, LeMehaute (1980) presents slightly greater

value by Eq.(7) on the basis of test results carried out at

Waterways Experimental Station:

𝜼𝟎

𝜼𝟎

𝑹𝟎

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𝐻𝑚𝑎𝑥×𝑟

𝑊0.54 ≅ 36 (7)

In this study, equations 3 to 7 are used to calculate

and align the energy of waves generated by an

explosion with a solitary wave in the further studies.

2.2. Smoothed Particle Hydrodynamics (SPH)

method

SPH is a Lagrangian method whereby the

continuous medium can be discretized into a set of

disordered points. SPH allows a function to be

expressed in terms of its values at a set of particles by

interpolation without needing any grid to calculate

spatial derivatives. In this way, physical properties of

each particle, such as acceleration, density, etc. are

quantified as an interpolation of the same values in

neighbor nodes. The SPH technique approximates a

scalar function 𝐴(𝒓) at any point with r vector of

position, as follows:

𝐴(𝒓) ≅ ∫ 𝐴(𝒓′). 𝑊(𝒓 − 𝒓′, ℎ)𝑑𝒓′ (8)

In Eq.(8), ℎ is called the smoothing length and

represents the influence of the nearest particles in a

neighboring domain (see Figure 2), and it is therefore

weighted in proportion to the distance between

particles. Kernel function 𝑊 is used to estimate the

amount of participation through the parameter ℎ.

Fig.2: Affected area and neighbor particles of ith

particle

In Eq.(9), the summation is extended to all particles

within the neighboring distance of particle "a", and the

volume associated with the particle "b" is 𝑚b

𝜌b, where

𝑚b and 𝜌b denote respectively the mass and the density

of this neighbor particle. The kernel functions shall

have several properties, including: positivity inside the

area of interaction, compact support, normalization, and

monotonic decrease with distance. Among notable

options for the kernel is the quintic kernel proposed by

Wendland (1995), where the weighting function

vanishes for inter-particle distances greater than 2ℎ.

This kernel has been defined as (Altomare et al., 2014,

2015) by Eq.(10):

𝑊(𝑞) = 𝛼𝐷 (1 −𝑞

2)

4(2𝑞 + 1) ; 0 ≤ 𝑞 ≤ 2 (10)

Where 𝑞 =|𝒓|

ℎ is the ratio of particle distance to

smoothing length, and 𝛼D is a normalization constant,

which is equal to 7

14𝜋.ℎ2 in two dimensions and 21

16𝜋.ℎ3

in three dimensions.

The kernel functions have been used to transform the

conservation laws of continuum fluid dynamics from

their differential form into particle forms. The

momentum equation proposed by Monaghan (1992)

has been used to determine the acceleration of particle

"a" as a function of its interaction with a neighbor

particle, such as "b":

𝑑𝒗𝑎

𝑑𝑡= − ∑ 𝑚𝑏 (

𝑃𝑏

𝜌𝑏2 +

𝑃𝑎

𝜌𝑎2 + 𝛱𝑎𝑏) 𝜵𝑎𝑊𝑎𝑏 + 𝒈𝑏 (11)

In Eq.(11), 𝒗 is the particle velocity, 𝑃 is the particle

pressure, 𝒈 is the gravitational acceleration vector, and

𝑊𝑎𝑏 is the kernel function, which depends on the

distance between particles "a" and "b". 𝛱𝑎𝑏 is the

viscous term according to the artificial viscosity

definition proposed by Monaghan (1992):

𝛱𝑎𝑏 = {−

𝛼𝑣 . 𝑐�̅�𝑏 . 𝜇𝑎𝑏

�̅�𝑎𝑏 ; 𝒗𝑎𝑏 . 𝒓𝑎𝑏 < 0

0 ; 𝒗𝑎𝑏 . 𝒓𝑎𝑏 ≥ 0 (12)

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In Eq.(12),𝒓ab = 𝒓a − 𝒓b and 𝒗ab = 𝒗a − 𝒗b are

the particle position and velocity, respectively. 𝛼𝑣 is a

coefficient that needs to be tuned to introduce the

proper attenuation. In this study, we have used

𝛼𝑣 = 0.01 , as it is the minimum value that prevents

instability and spurious oscillations in the numerical

scheme. 𝑐a̅b = 𝑐a + 𝑐b

2 is the mean speed of sound of

particles, �̅�ab = 𝜌a + 𝜌b

2 is the mean density of particles

and 𝜇ab =ℎ .𝒗ab . 𝒓ab

𝒓ab . 𝒓ab+ 𝜂2 where 𝜂2 = 0.01 × ℎ2.

The equation of states for ideal gases are:

𝑃 = 𝐵 [(𝜌

𝜌0)

𝛾− 1] (13)

𝐵 =𝑐0

2 . 𝜌0

𝛾 (14)

𝑐0 = 𝑐(𝜌0) = √𝜕𝑃

𝜕𝜌|

𝜌0

(15)

Where 𝐵 is a constant related to the fluid

compressibility modulus, 𝜌0 = 1000 kg/m3 is the

reference density, chosen as the density at the free

surface, 𝛾 is a constant, normally between 1 and 7, and

𝑐0 is the speed of sound at the reference density.

3. Results

3.1 Simulation of Tsunami-like wave interaction

with a seawall

Most of the previous researches about the effects of

water waves generated by underwater explosion were

focused on wave propagation and inundation effects.

Besides, some scientists studied the near field or

offshore effects of these waves on ships or underwater

infrastructures. So far, very limited field or laboratory

investigations have provided observational and

qualitative information about the effects and magnitude

of the impact of explosion-generated water waves on

coastal structures. The lack of quantitative recorded

data of interaction forces or pressure distribution on

coastal structures due to underwater explosion induced

wave, cause limitation for verification of numerical

models of this phenomenon. Thus, in this article, before

estimating the effects of these waves on coastal walls

by numerical SPH analysis, the validity of the

DualSPHysics code was checked with the help of

variously available field datasets in different stages

such as wave generation, propagation, and interaction.

First, the available experimental data about the

generation of the initial wave by the underwater

explosion and its propagation in the surrounding

environment were used to verify the ability of the SPH

method to simulate this part of the phenomenon.

In the second step, the ability of the numerical code

to simulate the effects of the wave collision with the

coastal structures was measured with the help of

existing experimental data about the impulse applied by

recorded laboratory data of artificially generated

tsunami (solitary) wave on a vertical coastal wall.

In the final step, the water waves generated by the

underwater explosion were simulated to estimate the

force applied to the wall. On the other words, an

explosion generated a wave with the equal energy of

solitary waves of the second step was reproduced and

simulated in the same numerical flume. The results of

these waves with different types of origin are compared

with each other.

In order to obtain the appropriate results in each of

the models, the number of water particles was increased

to the extent that the error results were less than 5% in

two successive sequences of the numerical model (see

tables 3,4,5). Subsequent simulations were executed in

each case using the number of particles fixed for each

example to assure the results.

3.1.1 Calibration and validation of the SPH Model

for generation and propagation of explosion-

generated water wave

We used two laboratory test cases to validate the

DualSPHysics code and the SPH method incorporated

therein. For generation of primary waves at the region

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50

of explosion, we first calculate the amount of radius of

the crater lip, 𝑅0, by insertion of the charge weight "𝑊"

in the Eq.(4). Then, with the help of equations 6 and 7,

the range of expression 𝐻𝑚𝑎𝑥 × 𝑟 could be evaluated.

At last, with the help of the Eq.(5) the amount of

𝜂0 × 𝑅0 is calculated, and then 𝜂0 may be estimated. In

this way, the main parameters of Eq.(1) are determined

and the deformed surface of water level as the initial

boundary condition is applicable to the numerical

model.

In order to measure the results of the numerical

model in comparison with the experimental results,

using the RMSE formula as Eq. (16), the output values

of the numerical results were evaluated.

𝑅𝑀𝑆𝐸 = √∑ (𝑋𝑚−𝑋𝑒)2𝑛

𝑖=1

𝑛 (16)

where 𝑋𝑚 is numerical model output, 𝑋𝑒 is recorded

data from experiments and 𝑛 is number of data. In the following, we first describe experimental

cases and then compare the results of numerical

modeling with the experimental data:

A. The first set of experimental data is the

result of studies conducted by Charlesworth (1945)

at the London Road Research Laboratory. In this

test, an underwater explosion was created in a 55 ft

(16.76 m) wide and 70 ft (21.34 m) long rectangular

gulf surrounded by the coast in the south and the

west. The test area had an almost uniform water

depth ranging from 15 to 18 feet (4.57 to 5.49

meters). The wave amplitude was recorded at a

point 56.5 ft (17.22 m) away from the explosion at

different times. A sample of the surface waves

produced following the explosion of a 32-pound

(14.5 kg) charge at a depth of 8 ft (2.44 m) was

recorded. The initial condition of this case is

considered on the basis of Eq.(1) as a parabolic

crater with a lip. The characteristics values of this

initial condition are: 𝜂0 = 2.07 m, R0 = 8.28 m.

This initial crater is defined in the midpoint of the

simulated basin. In Figure 3, the result of this test

case is shown with the red curve. The results of sthe

imulation with the SPH method were obtained by

adjustment of different parameters for this sample.

Fig. 3: Comparison of SPH model results with Charlesworth test results (1945)

-0.6

-0.4

-0.2

0

0.2

0.4

0.6

0 1 2 3 4 5 6 7 8 9

Wav

e A

mp

litu

de

(m)

Time (sec)

Charlesworth's experimental curve

SPH 3D,vis=0.00,D=50m



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In Figure 3, these results are plotted in the form of

dash-line and dot-line curves. The range of variation of

the parameters examined in this verification is

presented in Table (2). The plotted results demonstrate

the ability of SPH method to accurately simulate the

initial waves generated by the explosion at the desired

point. The maximum root mean square error (deviation

of simulation results from the corresponding

experimental values) was 11.4% and the minimum

coefficient of correlation between simulation results

and experimental data was estimated to 0.93.

Table 2. Characteristics of the main input parameters in the 3D SPH model used for simulation

of the Charlesworth’s test (1945)

Model Executive

Parameters Simulation

Parameters

Step Scheme Symplectic dP (m) 0.2 ~ 0.5

No. Particles 1383984

Kernel Wendland Model Size (m) 100 × 100

Viscosity

Treatment Artificial CFL

number 0.2

Viscosity 0.01 ~ 0.25 Coefficient

of Sound 10 ~ 15

Shepard Steps 20 ~ 30 Smooting

Coefficient 0.8 ~ 1.0

The second set of experimental data that was used to

measure the efficiency of the SPH model is the data

from the Prins' experiments (1956 and 1957) in the

Wave Research Laboratory at the UC Berkeley’s

Engineering and Technology Research Institute. These

experiments were carried out in a flume with a width of

one foot (0.3048 m) and length of 60 feet (18.288 m).

To produce an initial shock at water level (similar to an

underwater explosion event), an air blowing-suction

system was used to lower and raise the water level at

the flume’s upstream. The initial water wave was

released with a positive or negative height (compared to

the flume’s normal water level) to generate a shock in

the flume’s longitudinal direction. In this study, we

used the test results reported for the uniform water

depth of 2.3 ft (0.7 m), positive initial wave height of

0.4 feet (0.122 m), and length of 2 feet (0.61 m) at the

left side of the numerical flume. The initial wave is

considered at the center of the basin model. The

changes in the water elevation were measured at a point

15 ft (4.57 m) away from the shock initiation point. In

Figure 4, these measurements are displayed by the blue

curve. In Figure 4, the results of the simulation with the

SPH method following the adjustment of different

parameters for this test are shown in the form of dash-

line and dot-line curves. The range of variation of the

parameters examined in this verification is presented in

Table (3). As the presented results indicate, the SPH

method can accurately simulate the initial waves

generated by the explosion at the desired point.

In this case, the maximum root mean square error

(deviation of simulation results from the corresponding

experimental values) was 12.4% and the minimum

coefficient of correlation between simulation and

experimental results was 0.89.

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Fig. 4: Comparison of Two-Dimensional SPH model results with the Prins’ Test Results (1956)

Table 3. Characteristics of the main input parameters in the 2D SPH model used

for simulation of the Prins’ test (1945)

Model Executive

Parameters

Simulation

Parameters

Step Scheme Symplectic dP (m) 6.096×10

-3 ~ 1.016×10-2

No. Particles 141075

Kernel Wendland Model Size

(m) 20

Viscosity

Treatment

Laminar

SPS Artificial

CFL

number 0.2

Viscosity 10-6 0.01 ~ 0.25 Coefficient

of Sound 10 ~ 20

Shepard

Steps 20

Smoothing

Coefficient 0.87

3.1.2 Calibration and validation of the SPH Model

for estimation of laboratory generated tsunami

(solitary) wave induced force on the coastal wall

A large number of experiments have been carried

out to measure the force exerted on the sea walls by

tsunami waves. In this part of the study, we use two

laboratory experiences to assess the capability of the

SPH model for estimating the wave force on the coastal

walls.

For this part of the study, the amount of energy for

solitary waves is also calculated, so that in the next

-0.04

-0.02

0

0.02

0.04

0.06

0.08

0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5

Wav

e am

pli

tud

e (m

)

Time (sec)

Prins' experimental curve

SPH 2D, dp=0.01016,Vis.=0.01,dt=0.0005

SPH 2D, dp=0.01016,Vis.=0.05,dt=0.001

SPH 2D, dp=0.01016,Vis.=0.15,dt=0.001

𝑅𝑀𝑆𝐸𝑚𝑎𝑥 = 12.4% 𝑅𝑚in = 0.89

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section of the paper, the surface waves due to

underwater explosion with their equivalent energy are

generated.

Longuet-Higgins (1974), reviewed a number of

successful models for prediction of solitary waves. His

studies showed that using the Eq.(17) may represent

excellent results to predict the waveform.

{𝜂 = 𝐴𝑒−𝜇|𝑥| + 𝐵𝑒−𝜈|𝑥| 𝐴 = 1.5389 ; 𝐵 = −0.7093𝜇 = 1.0495 ; 𝜈 = 1.4630

(17)

He also assumed the equations 18 and 19 to calculate

the kinetic "𝑇" and potential "𝑉" energy of the solitary

wave. Then total energy "𝐸" of wave could be

calculated by 𝐸 = 𝑇 + 𝑉.

𝑇 = ∫ ∫1

2(𝑢2 + 𝑣2)

𝜂

−ℎ𝑑𝑦 𝑑𝑥

∞

−∞ (18)

𝑉 = ∫ 1

2𝑔𝜂2𝑑𝑥

∞

−∞ (19)

Where 𝑢 and 𝑣 are velocity component of water

particle in horizontal and vertical directions,

respectively. Also 𝜂 characterizes water elevation and ℎ

is water depth.

By applying equation 17 into equations 18 and 19,

he was able to provide a very good approximation for

estimating the energy values of the solitary wave,

which is considered in equations 20 and 21, properly.

𝑇 = 0.527 𝑔ℎ3 (20)

𝑉 = 0.431 𝑔ℎ3 (21)

Finally, the total energy of a solitary wave can be

approximated by Eq.(22) as follows:

𝐸 = 0.958 𝑔ℎ3 (22)

Besides, in this section, the corresponding simulation

results obtained with the DualSPHysics numerical code

are presented correspondingly for each case.

A. The results of laboratory-based physical

modeling performed by Robertson et al.

(2011). This test set was constructed in a

laboratory flume with a length of 83.13 m,

the width of 3.66 m and depth of 4.57 m in

the laboratory of Oregon State University

(see Figs. 5 and 6).

Fig. 5: Side view of the Laboratory Flume of Robertson et al. (2011) at the University of Oregon State Laboratory

Fig. 6: SPH environment of Robertson et al. Test case flume

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To reproduce the experimental data in simulation, a

solitary wave (representing the tsunami wave) with a

wave height of 1.064 meters and wavelength of 30

meters was generated in the water depth of 2.66 m. The

propagation of the generated solitary wave in the

numerical simulation of experiments in deep water

region is displayed in Figure 7. Time history variations

of pressure distribution on a vertical wall are shown in

Figure 8. In Figure 9, time history variations of the

force applied to the wall are compared with the

corresponding experimental data. Also, variations of

impulse with time are plotted in Figure 10. The

maximum sum of squares of relative errors of

simulation is 5.03% for force values and 1.41% for

impulse values. The minimum coefficient of correlation

between simulation results and experimental data is

98.16 for force values and 99.96 for impulse. The total

energy of this wave is estimated to be 𝐸 ≅ 177 kJouls.

Fig. 7: Snapshots of solitary wave generation (by paddle)

and propagation along the numerical SPH flume for Robertson’s experiments

t=2.0 sec

t=4.0 sec

t=6.0 sec

t=8.0 sec

t=10.0 sec

t=12.0 sec

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Fig. 8: Time history of pressure distribution on the vertical wall

Fig. 9: Time history of the force on the wall during the wave impact to the wall, (a) laboratory (circle) and (b) simulation of

the SPH (star)

0

0.2

0.4

0.6

0.8

1

0 5 10 15 20

Hei

gth

(m

)

Pressure (kPa)

t=21 sec t=24 sec

t= 28 sec t= 30 sec

0

5

10

15

20

25

30

35

40

0 2 4 6 8 10 12

Fo

rce

(kN

)

Time (sec)

Experiment data SPH results

𝑅𝑀𝑆𝐸𝑚𝑎𝑥 = 5.03% 𝑅𝑚in = 98.16

20 22 24 26 28 30

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Fig. 10: Time history of the impact on the wall by Robertson (2011), laboratory results (dash) and SPH results (line)

B. Results of laboratory experiments performed

by Yu Yao et al. (2017)

This case study was carried out in a laboratory flume

with a length of 36 m, the width of 0.55 m and depth of

0.60 m in Nanyang Technological University,

Singapore. A 1:5.7 steep fore slope starting from

16.33m away from the wave paddle was constructed

and a rectangular seawall model with 6 cm long, 5.5 cm

wide, and 5 cm high, was placed on the top point of the

slope to represent a vertical caisson type seawall (see

Figure 11).

The recorded data by G1 and G2 wave gauges,

which were located 5.24 m and 0.14 m upstream from

the seaside of the wall, respectively. Besides, induced

dynamic pressure by wave impinging on the wall was

recorded by a pressure sensor installed on the center of

the seawall (see Figure 12).

To reproduce the experimental data in the

simulation, a solitary wave with a wave height of 0.05

m and wavelength of 2.05 m was generated in the water

depth of 0.36 m. The numerical model is considered

symmetrical with sufficient far sloped beach from point

of initial wave generation in the middle of the

numerical flume for canceling of reflecting waves.

Geometry and initial generated solitary wave in the

numerical flume are presented in Figures 13 and 14.

Fig.11 Sketch of the experimental setup (reported by Yu Yao et al., 2017)

0

20

40

60

80

100

120

140

160

0 2 4 6 8 10 12

Imp

uls

e (k

N-m

)

Time (sec)

Experiment data

SPH results

𝑅𝑀𝑆𝐸𝑚𝑎𝑥 = 1.41% 𝑅𝑚in = 99.96

20 22 24 26 28 30

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Fig.12 Seawall model and pressure gauge location on the wall (reported by Yu Yao et al., 2017)

Fig. 13: SPH environment of Yu Yao et al. Test case flume (up)

and initial solitary wave generated in the model (down)

Fig. 14: Enlargement of solitary wave generated in SPH model

(height of generated wave is approximately 5 cm)

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Comparing of measured free surface elevations with

the numerical model at different wave sampling

locations (by G1 and G2 sensors) are presented in

Figures 15 and 16. In addition time history of pressure

at the midpoint of the seawall is plotted in Figure 17.

Both of free surface and pressure graphs are presented

in non-dimensional scale as per following pictures and

show good agreement between experimental and SPH

results. The total energy of this wave is estimated to be

𝐸 ≅ 0.44 kJouls.

Fig. 15 Comparison between SPH and Experimental time history of non-dimensional solitary wave

surface at gauge G1 (Experimental results are reported by Yu Yao et al., 2017)

Fig. 16 Comparison between SPH and Experimental time history of non-dimensional solitary wave

surface at gauge G2 (Experimental results are reported by Yu Yao et al., 2017)

Fig. 17 Comparison between SPH and Experimental time history of non-dimensional dynamic

pressure of solitary wave (Experimental results are reported by Yu Yao et al., 2017)

-1

0

1

2

3

0 5 10 15 20 25 30 35 40 45

η/h

t/(h/g)

WG1 (SPH) WG1 (EXP)

-1

0

1

2

3

0 5 10 15 20 25 30 35 40 45

η/h

t/(h/g)

WG2 (SPH) WG2 (EXP)

-0.1

0

0.1

0.2

0.3

0 5 10 15 20 25 30 35 40 45

P/r

gh

t/(h/g)

SPH EXP

𝑅𝑀𝑆𝐸𝑚𝑎𝑥 = 1.6%

𝑅𝑚in = 0.98

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3.1.3 Simulation of the force applied by interaction

of explosion-generated water wave and wall

As mentioned earlier, in this section, we characterize

the surface wave produced by a fictitious explosion

with equivalent energy to the solitary waves of Section

(3.1.2) in the numerical flume. We compared the

characteristics of wave propagation and interaction

force with previous results in order to measure and

describe variations in the amount of force on the wall

and water level in similar locations. For quantifying the

energy of reproduced tsunami-like waves due to

explosion, by referring to Eq.(3) the value of 𝜂0 × 𝑅0

should be estimated for each case. Then, by use of

Eq.(5) simply 𝐻𝑚𝑎𝑥 × 𝑟 could be evaluated and if this

value be inserted in the equations 6 and 7, the margin of

charge weight "𝑊" may be calculated. At last, by

applying charge weight value into Eq.(4), the radius of

crater at edge of lip 𝑅0 and then 𝜂0 are found easily

(see Table 4). Now the initial boundary condition of

water free surface may be defined by Eq.(1) and

considered in numerical model.

For applying this procedure to the laboratory setup

which was performed by Yu Yao et al., an explosion is

assumed at the middle of the flume that produces a

wave with crater depth and lip height of 𝜂0 = 0.23 𝑚

and crater span of 𝑅0 = 1.3 𝑚 at the maximum height

of the lips, which propagates upstream and downstream

along the flume’s length (see Figures 18 and 19). All

other properties such as the geometry of flume, water

depth, etc. are considered same as previous.

Table 4: Characteristics of initial boundary condition of artificial explosion generated waves

Case Study Energy

(kJouls)

𝜼𝟎 × 𝑹𝟎

(m2)

Eq.(3)

𝑯𝒎𝒂𝒙 × 𝒓

(m2)

Eq.(5)

Average

Charge Weight

(kg)

𝑹𝟎

(m)

Eq.(4)

𝜼𝟎

(m)

A

Robertson et al. (2011) 177 5.9 7.25 3.83 5.4 1.5

B

Yu Yao et al. (2017) 0.44 0.6 0.75 0.03 1.3 0.23

Fig. 18: The Initial geometry of the explosion generated water wave for Yu Yao et al. Test case

Fig. 19: Enlargement of primary surface wave generated by underwater explosion in

SPH model for Yu Yao et al. Test case

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Figure 20 shows the time history of water level

variations at two gauges in deep and shallow regions of

flume. Figures 21 and 22 display the time history of

force on the vertical wall and the snap shots of wave

location during the insertion of peak forces at 25-meter

and 50-meter, respectively. The major modeling and

simulation parameters are listed and described in Table

(5).

Fig. 20: Time history of the water level due to underwater explosion at gauge G1 (line) and gauge G2 (dash line)

Fig. 21: Time history of the wave force on the vertical coastal wall generated

by the explosion (line) and the solitary wave (dash line)

-0.5

-0.25

0

0.25

0.5

0 10 20 30 40 50

η/h

t/(h/g)

gauge G1

gauge G2

-10.00

0.00

10.00

20.00

30.00

40.00

50.00

4.0 5.0 6.0 7.0 8.0 9.0 10.0

Forc

e (N

)

Time (sec)

Force due to underwater explosion

Force due to solitary wave

(a

(b)

(c)

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Fig. 22: Snap shots of wave impact on wall at (a) t=6.74 sec, (b) t=7.5 sec, (c) t= 8.78 sec

Table 5: Characteristics of the main input parameters in the 2D SPH model used for comparison of simulations

Executive Parameters Modeling Parameters

Symplectic Time step

algorithm 0.1

Particle size

(m)

Wendland Kernel type 200 Model overall

size (m)

Explosion

model

Tsunami

model GPU run time

(sec.)

Explosion

model

Tsunami

model Total Number

of Particles 1422 1224 45374 46702

Explosion

model

Tsunami

model CPU run time

(sec.) 0.2 CFL number

3560 3062

According to the plotted water level variation

diagrams, solitary wave resulted in conservation of

depth at the points close to the wave initiation point, but

explosion-generated water wave led to depth reduction

at these points. In other words, by moving away from

the source of the wave generator, the explosion waves

which was simulated by an initial parabolic water level,

in comparison with a solitary wave as a representative

of the tsunami wave, show attenuation of wave

amplitude and force, when they are propagated on a

flatbed.

4. Conclusions

In this research, we used the DualSPHysics

numerical code to simulate and examine the forces

induced by the waves generated by an underwater

explosion on a vertical coastal wall. To study the

behavior and effects of the explosion-generated water

wave, the results were investigated in comparison with

the effects of a solitary wave of equal energy. The

following conclusions are deduced from the results:

(a) (b)

(c)

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(a) The validation results show the ability of the

DualSPHysics code and the SPH method to

accurately simulate the free surface

variations due to an underwater explosion. In

this study, we used two sets of laboratory

data to assess the ability of the SPH method

in the simulation of phenomena for which

recorded field or laboratory data have

variations and inherently these types of

experiments may record with large error

tolerances. As shown in the validation part of

the study, the matching of numerical

simulation results with laboratory values

demonstrated the ability of the numerical

model to capture the phenomenon with ill-

conditioned data.

(b) Although the SPH model used in this study

has an accuracy order of 1 ≤ 𝒪(r) ≤ 2,

which is more inclined toward 1, the model

verification phase revealed the ability of this

method to predict a complicated nonlinear

phenomenon of high complexity with proper

precision. On the other hand, this method can

reproduce physical characteristics of

complex CFD problems in a short analysis

time, which is a valuable benefit for

engineers to understand the impacts of such a

dangerous condition for coastal structures.

(c) Due to the behavior of the explosion-

generated water waves, caused by the

lowering of the free surface of the water at

the “ground zero” (i.e. the location of the

center of an explosion), the water elevation

was decreased and then increased by the

progressive negative wave in the vicinity of

the obstacle. These statuses produced

reduction and increase of pressure and force

on the wall respectively.

(d) In fact, this effect is different from the forces

resulting from the collision of the tsunami

waves (solitary) on the coastal walls, in the

opposite way. Thus, in identifying the risk of

surface waves generated by underwater

explosion than the tsunami waves, it can be

inferred that this type of waves initially show

a decrease in water level, resulting in a

reduction of pressure and force on the wall,

and then these values are an uptrend.

(e) Obviously, both time history of forces have

approximately the same peak value, but total

impact time for tsunami waves is longer than

those related to explosion generated waves.

This result shows that the explosion-induced

waves are very dissipative and their effects

are important for near field objects.

(f) The force exerted on the vertical wall by

explosion-generated water waves has a

periodic nature and attenuates with time. In

contrast, the force of a solitary wave with an

equal peak acts on the structure in a short

time period and then settles down.

(g) The explosion-generated water wave has a

stronger perturbation character and remains

imposed on the structure for a longer period

of time. Obviously, repeating the simulation

by taking into account the effect of water

particles released into the surrounding area

and their return to the water surface (which

may produce secondary waves) will result in

longer duration of turbulence in the water

level and more fluctuations in the force

exerted on the wall.

(h) For the 2-D models, utilization of GPU-

based method yields a better computational

speed than the parallel computation in multi-

core CPU. In the case of this study, using the

former approach reduced the computational

time by at least 60%. This feature could be

particularly advantageous for the processing

of more complex models, especially those

involving the effects of sudden and

impulsive loads, such as explosion generated

water waves, and save the engineers precious

time in design as well as damage estimation.

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(i) For the 3-D models, the use of GPU-based

method led to 75-90% reduction in the

execution time, which demonstrates the

greater efficiency of simulation with GPU

and the SPH technique for more complex

problems. This significant reduction of run

time also signifies the nonlinear relationship

of processing time with the model

dimensions and the effect of large scale of

analyses that have a greater number of

particles on the computational time and the

performance of the optimization simulation

algorithm in the SPH method.

Acknowledgements

Funding for this research was provided by the

Iranian National Center for Ocean Hazards (INCOH),

affiliated to Iranian National Institute for

Oceanography and Atmospheric Science (INIOAS).

The authors appreciate this support.

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