numerical simulation of solitary wave and drifting ...

2nd Reading August 25, 2014

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Coastal Engineering Journal, Vol. 56, No. 3 (2014) 1450015 (24 pages) c World Scientific Publishing Company and Japan Society of Civil Engineers

Coast. Eng. J. 2014.56. Downloaded from www.worldscientific.com by SHANGHAI JIAOTONG UNIVERSITY on 10/08/15. For personal use only.

DOI: 10.1142/S0578563414500156

NUMERICAL SIMULATION OF SOLITARY WAVE AND DRIFTING OBJECT INTERACTING WITH BREAKWATER USING THE ALE METHOD

YUN-FENG LOU and XIAN-LONG JIN∗ State Key Laboratory of Mechanical System and Vibration, Shanghai Jiaotong University, No. 800 Dongchuan Road, Shanghai 200240, P. R. China ∗ [email protected] Received 1 April 2014 Accepted 7 July 2014 Published 27 August 2014 The solitary wave as well as objects drifted by run-up wave interacting with elastic breakwater was investigated using an Arbitrary Lagrangian Eulerian (ALE) method. A numerical breakwater-flume coupling model was developed in conjunction with experiments to best understand the wave impact. In the experiment, the solitary wave was generated and a physical breakwater model was used. The experimental data of the wave pressure and water-on-breakwater were then used to validate the established simulation model. Comparisons indicate good agreements between simulations and experiments. After the validation, the ALE method based simulation model was applied to practical engineering analyses. Effects of the design parameters of breakwaters on wave loads were investigated. In full-scale simulation for collision of the drifting object, two types of colliding objects were used. Effects of mass and initial position of the drifting object on collision force were discussed. It was found that the wave pressure and structural stress of rear wall increases significantly when the breakwater width decreases. In addition, the drifting object brought larger collision force compared to the solitary wave, and the mass and the initial position of drifting objects had great effects on the breakwater’s dynamic response. Keywords: Solitary wave; ALE method; object drift; elastic breakwater; collision force; wave load.

∗ Corresponding

author.

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Y.-F. Lou & X.-L. Jin


1. Introduction Breakwaters are important coastal defense structures for harbor and shore protections. They are vulnerable to harsh weather and so to very violent waves. Breakwaters may suffer from not only direct damages from tsunami waves triggered by ocean trench earthquakes such as the Indian Ocean earthquake reported in 2004, but also indirect damages caused by drifting objects, automobiles and debris of destroyed houses and other facilities [Madurapperuma and Wijeyewickrema, 2007]. Thus, the interactions between solitary wave, drifting object and breakwater should be discussed and its countermeasures should be prepared. In the past decades, the stability of breakwaters in coastal region has been extensively studied by many researchers. The developed methods can be roughly classified into two groups: the experimental approach and the numerical approach. For instance, a simple and intuitive set of prediction formulae have been proposed by Cuomo et al. [2010] based on physical model tests. The model is able to calculate quasi-static, impact forces and overturning moments. This method belongs to the experimental approach. Some other literatures have also been reported, which fall into the group of numerical approaches. For example, a two-dimensional Reynolds Averaged Navier–Stokes (RANS) model was developed to simulate the shoaling, breaking and overtopping of wave over the breakwater [Yeganeh-Bakhtiary et al., 2010]. Based on the RANS equations and the k–ε turbulence closure solver, a twodimensional volume of fluid (VOF) type model called the Cornell Breaking and Structure (COBRAS) model was validated by experimental data and then applied to investigate tsunami-like solitary waves impinging and overtopping an impermeable trapezoidal seawall [Hsiao & Lin, 2010]. The COBRAS-UC model, a new version of the COBRAS, has been used by Losada et al. [2008] and Guanche et al. [2009] to study the breakwater under wave impact, overtopping and stability of breakwater under wave actions. In most of the studies related to wave-structure interactions, the structure is assumed to be rigid without considering the effect of its elasticity. The elastic effect of structures, however, needs to be considered in some situations, such as large floating structures (VLFS) [Zhao and Hu, 2012], structures subjected to large wave impact (e.g. breaking wave impacts on ship hulls, sloshing impacts on the tank walls) [Sriram and Ma, 2012] and etc. For these situations, the “hydroelasticity” is very important. The wave and structural dynamics, therefore, need to be accounted for simultaneously. As alternatives of aforementioned methods, some other methods have been proposed for the numerical simulation of the interaction between an incompressible fluid and an elastic structure. For example, methods developed are based on the Particle Finite Element Method (PFEM) [Pin et al., 2007] and the Smoothed Particle Hydrodynamics (SPH) method [Antoci et al., 2007; Rafiee and Thiagarajan, 2009].

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Numerical Simulation of Solitary Wave and Drifting Object

Due to the fluid-structure interaction (FSI), highly nonlinear and large deformations are presented, which brings difficulties to defining the structure and fluid domain in a unified coordinate system. As a result of that, it is hard to study the nonlinear FSI and large deformation using Lagrangian or Eulerian description separately. The coupling of two media is usually obtained by an Arbitrary Lagrangian Eulerian (ALE) formulation for the fluid [Tallec and Mouro, 2001]. The method has been successfully applied to a wide range of problems. Bathe et al. [1999] studied the interaction between structure with large deformations and the fluid. The ALE method has also been applied to the study of the development and dissolution process of nonlinear waves [Zhu et al., 2012] and the highly nonlinear interactions between waves and rubble mound breakwaters [Yang et al., 2010]. Most recently, it was used to simulate highly nonlinear wave-structure interactions by Wu et al. [2013]. In the work of Wu et al. [2013], the ALE method was employed to accurately capture the configuration of free surface. Regarding the behavior of drifting body due to tsunami and its collision force, Mizutani et al. [2005, 2006] pointed out the water mass (added mass) behind the drifting container plays important roles on the collision. In their study, a formula was proposed to estimate the collision force due to a drifting container considering the effect of the added mass behind the container. Yeom et al. [2009] estimated the collision force using ALE method in finite-element (FE) models, but the collision model was only used during the collision phase. The turbulence models have not been supported in LS-DYNA version 971 [Hallquist, 2006; LSTC, 2007]. Currently, ALE is incorporated into LS-DYNA version 971 in an explicit framework, making it ideally suited for impact events. As in the paper of Pin et al. [2007] and Antoci et al. [2007], turbulence was not modeled in the numerical methods. The numerical results and the experimental results in their researches were in reasonable agreement with slight difference. The purpose of this paper is twofold: (1) to evaluate the wave impacting with consideration of the FSI and discuss effects of structural configurations on wave loads; (2) to estimate collision force of objects drifted by run-up wave. For the first purpose, physical model experiments are carried out to verify the numerical model and method. After the validation, a full-scale breakwater-flume coupling model is established. For the second purpose, a full-scale drifting objects-wave-breakwater collision coupled model is constructed. Then, the solitary wave generation and propagation of wave, wave run-up onto breakwater, object drift, collision with seawall and collision force estimation are presented. The penalty method is employed to handle the coupling between waves and dynamics of structures. Because of its high computational effort [Anghileri et al., 2005], the simulation is performed in LS-DYNA Version 971 MPP on a Dawning supercomputer. The paper is organized as follows: Section 2 describes the numerical model and gives a brief review of the ALE method, the explicit time integration scheme and the 1450015-3


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penalty method for the FSI. In Sec. 3, results obtained by numerical simulations are compared with the experiment data to validate the ALE method. Then the method is applied to the wave impacting analysis. Following that, the method is applied to the drifting objects collision analysis in Sec. 4. Conclusions are drawn in Sec. 5.

2. Numerical Method and Model


2.1. ALE kinematics Conservation of momentum and mass for incompressible Newtonian fluids in the ALE form are represented by the Navier–Stokes equations and the continuity equation, as follows: ∂σij ∂vi ∂vi + ρcj = + ρbi , (1) ρ ∂t χ ∂χj ∂χj ∂vi ∂ρ ∂ρ + ci +ρ = 0, (2) ∂t χ ∂χi ∂χi where bi represents the body force. vi = vi (xi , t) are the material velocity field in space coordinate system x, χ denotes the ALE coordinate system; ρ is the density. c is the ALE convective velocity. The free surface motion is tracked by the VOF technique [Hirt and Nichols, 1981; Shen and Chan, 2008]. In the VOF method, the free surface is described by the following volume function: 1 if fully occupied by water C= (3) 0 if fully occupied by air and the tracked interface exists in the partially-filled cells (with a volume fraction between 0 and 1). The governing equation for the volume fraction can be written in the conservative form as: ∂C + ∇ · (C · v) = 0. ∂t

(4)

The constitutive equation of Newtonian fluid is given by: σij = −pδij + τij .

(5)

Here deviatoric stress is given as τij = λδij skk + 2µsij , where sij = 1/2((∂vi /∂xj ) + (∂vj /∂xi )). The first term on the right-hand side of Eq. (5) is defined by the equation of state (EOS) in this simulation. For the weakly compressible Newtonian fluid used in the paper, the viscous stress λδij skk is approximately zero and thus negligible. 1450015-4


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The second term on the right-hand side of Eq. (5) only includes the shear stress 2µsij . Thus, Eq. (5) is rewritten as: σij = −pδij + 2µsij ,

(6)


where µ is kinematic viscosity and is a constant for Newtonian fluid. The constitutive equation of the fluid in the paper is composed of the EOS and the material model. The EOS defines the volumetric compression (or expansion) behavior of the fluid, and the material model defines the relationship between the shear stress and the shear strain rate. Here, the Gruneisen equation is chosen as the EOS of water [Shyue, 2001] and the polynomial EOS is used as the EOS of air [Alia and Souli, 2006]. 2.2. Fluid structure interaction In this work the structural dynamic responses are taken into consideration for the wave impact. Thus, the deformation of the breakwater structure is described by Eq. (7) with a constitutive equation of St.-Venant’s elastic bodies. ∂σij ∂ 2 u + fi , (7) ρs 2 = ∂t X ∂xi where X denotes the Lagrangian coordinate, ρs is the density of the structure, fi is the body force, u is the displacement of the structure. The approach used in the paper for FSI problems is penalty-based finite element method allied with ALE description. Equations (1), (2) and (7) are coupled under the following geometric compatibility and mechanical equilibrium conditions that should be satisfied on the FSI interfaces [Farhat et al., 2003]: ∂uf ∂us ·n= · n, ∂t ∂t σs · n = −p · n + σf · n,

(8) (9)

where us and uf are respectively, the nodal displacement for the solid and fluid, p is the fluid pressure, σs and σf are the structure stress tensor and the fluid viscous stress tensor and n is the normal at a point to the fluid-structure interface boundary. The penalty method is chosen to calculate the interaction forces because it is the simplest and most efficient way. The interaction between the container and fluid is a strongly nonlinear process. The penalty method is able to handle nonlinear cases and can be easily used in explicit schemes. Moreover, it can guarantee the energy conservation. For these reasons, the penalty algorithm has been applied in the numerical simulation of FSI [Antoine et al., 2006]. The penalty-based coupling system is shown in Fig. 1. The basic idea of the penalty method is tracking the relative displacements between the corresponding coupling nodes on the structure surface and inside the ALE fluid elements. When the 1450015-5


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Time step: n Structure

Time step: n+1

Integral coupling point

d

ALE element


Fig. 1. Wave-structure coupling algorithm.

fluid and the structure are coupled with each other, the penalty method introduces a coupling force between them. The coupling force is defined to be proportional to the depth of the penetration and the contact rigidity. In order to restrain numerical oscillation, a viscous damper system is added to damp out the high frequency. The coupling force can be calculated by the following equation F =

dZ d2 Z + ω 2 Z, +ξ 2 dt dt

(10)

where F is the couple force, ξ is the damp coefficient, ω = k(ms + mf )/(ms − mf ), k is the contact rigidity, Z represents the penetration, which is iterated in the following equation c s − Vn+1/2 ) · ∆tn+1/2 , Zn+1 = Zn + (Vn+1/2

(11)

c s and Vn+1/2 are the velocities of the coupling points in the ALE and where Vn+1/2 Lagrangian body, respectively. The stiffness coefficient k in an explicit contact algorithm can be calculated by k = αKi A2i /Vi , in which Ki is the bulk modulus of the element, Ai is the element area, Vi is the element volume, α is the scale factor. For numerical stabilities, the scalar factor should satisfy 0 ≤ α ≤ 1. In this paper, we choose α = 0.5 to consider the dynamic coupling stability and leakage prevention. The computational accuracy of the penalty-based FSI largely depends on the leakage control. In order to obtain high computational accuracy and reduced leakage, several other parameters are introduced: (1) the number of coupling nodes distributed over each coupled Lagrangian surface segment is set to be 4; (2) the additional leakage control penalty factor (similar with α) is 0.1 and (3) the minimum volume fraction of fluid in an ALE element to activate coupling is 0.3.

2.3. Explicit dynamic analysis The equation of motion for the nonlinear case at time tn is given by: ¨n + C U˙ n + F int (Un ) = Pn , MU n 1450015-6

(12)


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where M is the mass matrix, C is the damping coefficient matrix, Pn accounts for ¨n , U˙ n and Un the external and body force vector, Fnint is the internal force vector, U are the acceleration, velocity and displacement vectors, respectively. Equation (12) can be integrated by the explicit central difference integration rule and is rewritten as follows: ¨n = M −1 [Pn − C U˙ n − Fnint (Un )]. U

(13)

Velocities and displacements are updated in each time-step as below:


¨n , U˙ n+1/2 = U˙ n−1/2 + ∆tn U Un+1 = Un + ∆tn+1/2 U˙ n+1/2 ,

(14) (15)

where ∆tn = (∆tn + ∆tn+1 )/2. The explicit integration scheme improves the computational efficiency by using diagonal mass matrix because the inversion of mass matrix used in Eq. (13) is trivial. One of the disadvantages of the explicit integration procedure is that the stability depends on the time-step size. For stability, the calculation time-step size must be smaller than the critical time-step ∆tcr , which is determined by the character length of the element and its material properties. For constant stain and rate independent materials, the critical time-step can be calculated by ∆tct ≤ min(le /ce ), where le is e the character length of the element and ce is the wave speed in the element. In order to account for the destabilizing effects of nonlinearities, a reduction factor Ts is introduced by ∆t = Ts ∆tct . This reduction factor should satisfy 0.67 ≤ Ts ≤ 0.90. In this work, we chose Ts = 0.80 to account for high contact nonlinearities in the wave impaction simulation. The set of ALE equations is usually solved using an operator-split procedure. Each time increment, from time t to t + ∆t, is divided into two successive steps. The first one is performed exactly in the same way as in the classical Lagrangian case. During this Lagrangian step, the mesh sticks to the material (ρ(∂vi /∂t)|x = (∂σij /∂xj ) + ρbi ) until an equilibrated Lagrangian configuration is obtained. The second step, which is called the Eulerian step, is divided into two substeps: the definition of an appropriate mesh velocity by relocating each node of the mesh to a more suitable position, followed by the data transfer from the old mesh configuration to the new one. This new configuration of the nodes is conveniently called the Eulerian configuration. 2.4. Numerical simulation model Experiments were conducted in the irregular wave flume at Zhejiang Institute of Hydraulics and Estuary. Figure 2 presents a photo of the experimental breakwater. The wave flume (70 m × 1.2 m × 1.2 m) was equipped with a push plate-type wave √ generator at the end. The Froude number VM / gLM and the Strouhal number VM TM /LM were used between the physical model and the actual engineering model. 1450015-7


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Fig. 2. Photograph of the breakwater model used in this study.

Unit:m Model scale:25

Wave absorbing device



5

a1 a2 a3 a4 F 5.26

Hf D 0.24

Wave maker

1:2

1

b1 Wave pressure gauges: b2 Hr a1:0.44 ; a2:0.4 ; a3:0.36 ; a4:0.32 b3 b4 b1:0.498 ; b2:0.472 ; b3:0.448 ; b4:0.418 The bottom of front wall: F:0.308 0.3 0.4 0.3 0.56 0.62 0.82 0.456

SWL 0.4 0.12

1:3

0.22 1:3

0.408

0.508 1:2. 5

0.16

0.0

Fig. 3. Schematic of the breakwater cross-section.

Considering the height of the breakwater, wave characteristics, water depth and flume size, etc., the model scaling was taken as 25. Figure 3 shows the cross-section of the experimental breakwater. As shown in Fig. 3, eight pressure gauges were placed on the seawalls. The sampling frequency of the pressure gauges was set to be 1000 Hz to record the time history of the water impact pressure. The experiment was repeated six times to eliminate the uncertain factors, and the pressure values were the average of reasonable data. The experiments were recorded by a high-speed video camera for a qualitative understanding of the water-on-breakwater occurrence. A three-dimensional simulation model of the breakwater-flume coupling system was established based on the physical model. Reliability and accuracy of numerical results depend on the FE models and other crucial parameters for the analysis. Thus the elements of the models were carefully meshed in the work. The ratio of the element length and width is no more than 4. The breakwater was built with eight-node hexahedron solid elements. The minimum grid spacing of the structure was 5 mm. The friction between structures was modeled according to the Coulomb friction law. The static and dynamic friction coefficients were 0.3 and 0.25, respectively. Table 1 1450015-8


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Numerical Simulation of Solitary Wave and Drifting Object Table 1. Material parameters of model. Part Seawall Surface protection structure Rubble mound


Water

Density (kg/m3 )

Elastic modulus (Pa)

Poisson ratio

2.40 × 103 2.40 × 103

2.00 × 1010 2.00 × 1010

0.15 0.15

2.00 × 103

5.00 × 1010

0.30

Initial destiny

Coefficient of kinematic viscosity

Sound speed in water

1.00 × 103 kg/m3

8.68 × 10−4 Pa·s

1.48 × 103 m/s

FSI: Fluid-structure interaction boundary condition; SSI: Soil-structure interaction boundary condition; IW: Incident wave boundary condition; NC: Normal constraint boundary condition; NS: No-slip boundary condition; 7.5 NC Elevation view Z IW o x NS NC Plane view Y o

IW

Unit:m Model scale:25

2.5

FSI FSI SSI NC

NC 1.2 FSI

NC NC 1.2

x NC

NC

Fig. 4. Experimental model of breakwater — flume coupling system.

gives material parameters of the coupling model. The fluid domain (water and air) was discretized with eight-node hexahedron elements in the ALE description. The ALE elements around the breakwater and the free surface were intensively meshed. In order to save the CPU time, a smaller numerical tank other than the physical wave tank was used. Finally, as shown in Fig. 4, a 10 m × 1.2 m × 1.2 m numerical Table 2. Solution times.

Test series

Ts

Minimum grid spacing of ALE/Structure [m]

Experimental model

0.8

0.004/0.005

32

1.07e−6

4

93.33

Full-scale model

0.8

0.004/0.005 0.1/0.125 0.1/0.125

48 32 48

1.07e−6 3.33e−5 3.33e−5

4 20 20

76.67 24.67 20.89

Number of CPUs

Time step (s)

Termination time (s)

Solution CPU time (h)

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wave flume model was established. The total numbers of nodes and elements were 592, 645 and 544, 272, respectively. The boundary conditions are indicated in Fig. 4 as well. The computational cost of regular calculations using explicit algorithms such as LS-DYNA depends on the element size and wave speed. Table 2 summarizes the solution time for various mesh sizes (de) and calculation parameters.

3. Wave Impacting Analysis


3.1. Experimental and simulation results comparison Since a push plate wave maker was used for the wave generation, it is necessary to obtain the motion of the wavemaker. Theory solutions of a solitary wave of finite amplitude propagating without change of shape are given by (16) η = H sec h2 [ 3H/4d3 (x − Ct)], (17) C = g(H + d). At the wavemaker surface, it satisfies the following equation: dS/dt = u(x, t),

x = S,

u(x, t) = Cη(x, t)/[d + η(x, t)],

(18) (19)

where S is the displacement of wave maker, u(x, t) is the average speed in the x-direction under shallow water condition. By integrating Eq. (18), we obtain the actual wavemaker motion as follows: 4H 3H d tanh (Ct − S) (20) S(t) = 3d 4d3 in which H is the height of the wave with respect to the unperturbed surface, d is the depth from the bottom to the unperturbed surface, g is the acceleration of gravity, η is the elevation of the free surface. The quasi-analytical solution will be used for comparison with numerical results. Table 3 summarizes different conditions of applications, which include water depth Hw , breakwater width D, front seawall elevation relative to the breakwater top Hf , rear seawall elevation relative to the breakwater top Hr , and wave height Hm . Figure 5 shows the impact pressure at point a2 for various mesh sizes (de). It indicates that by using the coupling-damping factor, the simulated time history of the pressure is smooth. Using large elements near the coupling region has a favorable influence on the size of the time-step of the explicit time integration. Nevertheless, these large elements are unsuitable for calculating wave impact pressures. A better agreement between results may be achieved by increasing the ALE mesh density near the breakwater. In this procedure, the coupling-damping factor is introduced, 1450015-10


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Numerical Simulation of Solitary Wave and Drifting Object Table 3. Test conditions. Test series Experimental model Full-scale model Elevation discussion

Width discussion Collision Analysis

Hf = 0.048 m, Hr = 0.1 m, Hw = 0.4 m, D = 0.56 m

0.144

Hf = 1.2 m; 0.8m; 0.4 m; 0 m, Hr = 2.5 m; 2.75 m; 3.0 m, D = 14 m, Hw = 10 m Hf = 1.2 m, Hr = 2.5 m, D = 10 m; 14 m; 18 m, Hw = 10 m Hf = 1.2 m, Hr = 2.5 m, D = 14 m, Hw = 10 m

3.6

3.6 3.6

6.4

Pressure_a2 (KPa)


Wave height Hm (m)

Configuration

exp de=4mm; coupling-damping factor=0 de=4mm; 4.8 coupling-damping factor=0.3 de=8mm; coupling-damping factor=0.3 3.2

1.6

0.0 0.8

1.2

1.6

2.0 Time (s)

2.4

2.8

3.2

Fig. 5. Comparison of impact pressure for different mesh size and coupling coefficient.

leading to a relatively smoother pressure time history, which is crucial for a successful simulation of the wave-structure interaction. Before performing simulations with the breakwater, simulations are implemented without the structure. The purpose of doing so is to compare the undisturbed simulated wave elevation with the theoretical wave elevation. Figure 6(a) shows a 3D wave shape of numerical simulation. Figure 6 also depicts wave profiles at the same instant obtained from the analytical method (Fig. 6(b)) and the simulation method (Fig. 6(c)). It illustrates that the simulation results match well with the analytical solutions for the range of values, in which the analytical solutions are valid. The compared simulation results include the wave height and shape. Figure 7 shows spatial snapshots of waves during its evolutionary courses. The presented waves include wave on the sloping beach, water impact the front wall and water impact the rear wall and the overtopping. Laboratory images and numerical results are also plotted on the figure for comparison. The figures show that the wave 1450015-11


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3-D wave shape

Fig. 6. Comparison between numerical solutions and analytical solution for the solitary wave propagation.

approaching the breakwater does not break. Because of the front vertical seawall, the wave is deviated upward and is evolved into an overtopping flow. Then, the upper fluids directly cross the rear wall. This generates a transient splash-up for a reflected jet not fully developed. Finally, the fluid violently impact with the rear wall, and then overtop. Despite of this very violent phenomenon, the simulation results show good agreement with the experimental results. Figure 8 gives the comparison of time histories obtained from simulations and experiments for the impact pressure along the weatherside seawall surface. It is apparent that numerical results agree well with laboratory data. The hydrostatic pressure values at points a3 and a4 are 380 Pa, 785 Pa, respectively. They are very close to the theoretical results.

3.2. Engineering application and discussion After the validation, the ALE method is applied to practical engineering analyses. A full-scale breakwater-wave coupling model, which is proportional to the prototype of physical model, is used. It is obtained by enlarging the experimental model as presented in Fig. 4. Based on the full-scale coupling model, the pressure distribution on the wall and effects of structural configurations on wave loads are studied. 1450015-12


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Fig. 7. Qualitative comparisons of wave evolution between simulation results (left column) and laboratory images (right column).

Fig. 8. Comparison between numerical results and experimental results for time history of the impact pressure along the windward seawall surface.

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The different conditions of applications are summarized in Table 3 (Full-scale model: Elevation discussion and width discussion).

In this section, the effect of seawall elevation on the wave pressure is investigated by changing the elevations of the front and rear seawalls. Figure 9 shows the results, which include the wave pressure on both the front and rear seawalls (in Figs. 9(a) and 9(b), respectively) and the horizontal force on the rear seawall (in Fig. 9(c)) (Zf and Zr are the positions up the front seawall and rear seawall, respectively). Calculations are made for the same wave condition with various elevations of front seawall Hf = 1.2, 0.8, 0.4, 0. Figure 9(a) shows the maximum dimensionless pressure impulses on the front seawall. It can be seen that the impact pressures at the SWL increase Hm=3.6m;Hw=10m;D=14m Hr=2.5m;Hf=1.2m Hr=2.5m;Hf=0.8m Hr=2.5m;Hf=0.4m Hr=2.5m;Hf=0m 1.2

2.500 Hm=3.6m;Hw=10m;D=14m Hr=2.5m;Hf=1.2m Hr=2.5m;Hf=0.8m Hr=2.5m;Hf=0.4m Hr=2.5m;Hf=0m 1.875

Zr(m)

Zf /Hw

1.6

0.8

0.4

1.250

0.625

0.0 1.0

1.4

1.8 2.2 Pmax ρgH m

0.000 65

2.6

(a)

135 170 Pmax (KPa)

D=10m D=14m D=18m

Zr(m)

1.875

300

Hr=2.50m;The front seawall Hr=2.50m;The rear seawall Hr=2.75m;The rear seawall Hr=3.00m;The rear seawall

200

0.0

0.4 0.8 Hf (m)

205

2.500 Hm=3.6m;Hw=10m;Hr=2.5m;Hf=1.2m

400

100

100

(b)

500 Hm=3.6m;Hw=10m;D=14m

The total horizontal forces (kN)


3.2.1. The effect of seawall elevation

1.2

1.250

0.625

0.000 60.0 90.0 120.0 150.0 180.0 210.0 Pmax (KPa)

(c)

(d)

Fig. 9. Pressure impulses and thrust distribution up the front and rear walls. 1450015-14


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when the front seawall elevation decreases. On the contrary, the impact pressures at the bottom decrease when the front seawall elevation decreases. Figure 9(b) shows the maximum pressure impulses on the rear seawall. Compared with the pressure at the top of rear seawall, the results show that the pressure at the bottom is raised significantly by decreasing the elevation of the front seawall. Figure 9(c) shows the horizontal force on the seawall for different elevations of seawalls. In this study, the total horizontal forces (Fh ) on the wall are given by Fh =

4

Pk · ∆Z,

(21)

where Pk are the numerical pressures at the key points (a1, a2, a3, a4, b1, b2, b3 and b4), ∆Z is the distance up the wall between two key points. It is obvious that by decreasing the elevation of front seawall, the maximum horizontal forces on the front seawall decrease from 480.86 kN to 334.37 kN. Nevertheless, the maximum horizontal forces on the rear seawall increase from 251.08 kN to 326.41 kN. Under the same elevation of the front seawall, the maximum horizontal forces on the rear seawall increase when the elevation of rear seawall increases from 2.5 m to 3.0 m. In general, the front seawall has a protective effect against the rear seawall. Additionally, to consider the security of rear seawall structure but not wave overtopping, the elevation of rear seawall should be as low as possible. 3.2.2. The effect of breakwater width In this section, effects of breakwater width on the wave pressure are studied. The maximum pressure impulses on the rear seawall for different breakwater widths are plotted in Fig. 9(d). It can be seen that the maximum impact pressure on rear wall decreases about 10% when the breakwater width increases from 14 m to 18 m. 12.5 The maximum impact stress (MPa)


k=1

10.0

Hm=3.6m;Hw=10m Hr=2.5m;Hf=1.2m D=10m D=14m D=18m

7.5 5.0 2.5 0.0 0

5

10

15

Time (s)

Fig. 10. Stress response of rear wall structural. 1450015-15

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On the other hand, the max impact pressure on rear wall will increase about 64% when the breakwater width decreases from 14 m to 10 m. Figure 10 shows the maximum stress curves of the breakwater structure for different breakwater widths. It indicates that the structures reach their maximum stresses at different time. The figure clearly shows the quivering behavior of the structures owing to the impact of the fluid and as expected the stress of the structure decreases by increasing the breakwater width. 4. Collision Analysis Coast. Eng. J. 2014.56. Downloaded from www.worldscientific.com by SHANGHAI JIAOTONG UNIVERSITY on 10/08/15. For personal use only.

4.1. Drifting objects For the full-scale collision analysis, as shown in Fig. 11, two types of colliding object were used. The finite element models of objects were established in Hypermesh, meanwhile, each component density was set up. Then, the gravity center and inertia tensor of objects were calculated by the function “ctr of gravity” and “mom of inertia” of Hypermesh, respectively. The mass distribution of the objects was described by inertia tensor. Table 4 gives the inertia tensor of colliding objects (using center of gravity as center). In this drift simulation, coefficients of static and dynamic friction were determined at 0.4 and 0.3 based on the coefficient of friction between stone and steel. The stiffness of colliding objects is much higher than that of the breakwater. In addition, Ikeno et al. [2001] pointed out that collision force can be obtained if the momentum of a drifting body is identical, regardless of shapes of photograph

photograph

FE-Model

FE-Model

1.86m 2.54m m

6.058

2.54m

1.95m

(a)

6.03m (b)

Fig. 11. Drifting object models. (a) Modeling of 20 ft ISO fright container and (b) modeling of C2500 truck. 1450015-16


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Numerical Simulation of Solitary Wave and Drifting Object Table 4. Inertia tensors of the object drifted by run-up tsunami.


Drifting object

Mass (Kg)

Ixx (Kg · m2 ) Iyy (Kg · m2 ) Izz (Kg · m2 ) Iyz (Kg · m2 )

Container (20ft) 2230 (empty) 13115 (half-loaded) 24000 (full-loaded)

1.43 × 104 4.52 × 104 8.08 × 104

1.43 × 104 4.86 × 104 8.33 × 104

4.71 × 103 1.17 × 104 2.39 × 104

0 0 0

Truck (C2500)

5.93 × 103 1.01 × 104 1.40 × 104

6.54 × 103 1.11 × 104 1.55 × 104

1.47 × 103 2.49 × 103 3.47 × 103

8.37 × 10 1.42 × 102 1.98 × 102

2306 (empty) 3901 (standard-loaded) 5496 (over-loaded)

(1) t=10.00s

(

) t=11.70s

(2) t=11.95s

(

) t=13.35s

(3) t=12.85s

(

) t=13.85s

(4) t=13.65s

(

) t=14.20s

(a)

(b)

Fig. 12. Representative objects drift collision with breakwater. (a) Representative full-loaded container drift collision and (b) representative full-loaded truck drift collision.

the drifting body. Thus, the colliding objects were defined as a rigid body. In the full-scale simulation, the solitary wave was generated as shown in Table 3 (Full-scale model: Collision analysis). 4.2. Drifting containers The container was initially placed 0.73 m seaward from the front seawall. Under the action of gravity and the ALE fluid-structure coupling force, the container floats at the surface of the water, as shown in Fig. 12(1). The settlement of containers in numerical simulation is compared with the theoretical solution as shown in Fig. 13. Due to the penetration caused by penalty function, the settlement value of simulation is slightly larger than that of the theoretical calculation. It demonstrates the feasibility of the wave-container coupling model. 1450015-17


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The settlement of containers (m)

Y.-F. Lou & X.-L. Jin 0.0 empty-Num

empty-Thero

-0.5

-1.0

half loaded-Num

half loaded-Thero

-1.5 full loaded-Num

-2.0 0

2

full loaded-Thero

4

6

8

10

Time (s)

Figure 12(a) gives representative views of full-loaded container drift collision with the front seawall. The run-up wave behind the container and the wave passing over the container sides are shown in the figure as well. From the results, it is concluded that the FSI between the container and wave is accomplished well, which verified the feasibility of the drift collision coupled model. Figure 14 shows the time-varying collision forces and wave forces obtained numerical experiments under various-loaded container collisions. The figure shows that for cases of half-loaded and full-loaded containers, wave forces reach their local minimums at 12.5 s and the drifting object forces (collision forces) reach their global maximums at the same time. The reason for this phenomenon is containers of halfloaded and full-loaded have deeper draft. When containers collide with the front

Wave forces (E+06N)

1.6

full-loaded

1.2 0.8

solitary wave

0.8

solitary wave empty-loaded half-loaded full-loaded

half-loaded

0.4 0.0

Collision forces (E+06N)


Fig. 13. The settlement curves of container.

empty-loaded 8

10

12

14

16

18

empty-loaded half-loaded full-loaded

0.6 0.4

full-loaded

0.2

half-loaded empty-loaded

0.0

8

10

12

14

16

18

Time (s) Fig. 14. The time-varying collision forces and wave forces under various-loaded container collisions. 1450015-18


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Solitary wave


Solitary wave

Empty container

Empty truck

Half-loaded container

Standard-loaded truck

Over-loaded truck

Full-loaded container

(a)

(b)

Fig. 15. Stress distribution of the front (a) and rear (b) walls. (a) Stress distribution of front seawall for various loaded containers and (b) stress distribution of rear seawall for various loaded trucks.

seawall, the contact area of run-up wave and front seawall decreases, which results in the diminution of wave force. As the containers gradually cross the front seawall, the wave force goes back to its normal level. As can be seen in Fig. 14, this phenomenon does not occur for the empty condition because the empty container draws shallower. It does not affect the size of the contact area of run-up wave and the front seawall. The stress distributions of the front seawall are given in Fig. 15(a). It can be seen that the collision positions change with the load of container. Figure 16 (solid line) shows the maximum stresses of the front seawall for various-loaded containers. Although we have no experimental data to validate the accuracy of numerical simulation results, the general trend of collision force and wave pressure will be presented in the numerical simulation. Based on the analysis, it is found that the mass of colliding container have significant effects on the collision force, the wave pressure and the breakwater’s dynamic response. 4.3. Drifting trucks As shown in Fig. 12(I), the truck was initially placed 1.5 m seaward from the rear seawall. Figure 12(b) shows representative views of the over-load truck drift collision with the rear seawall. The truck drifted by the run-up wave and the truck collided the rear seawall are shown in the figure as well. It is concluded that the FSI between 1450015-19


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front seawall rear seawall

28

21

14

7

0

0

5

10

15

20

25

Mass of colliding object (T)

Fig. 16. Maximum 1st principal stress of the seawall.

Wave forces (E+06N)

1.8



Max 1st principal stress (MPa)

35

1.5

solitary wave empty-loaded standard-loaded over-loaded

solitary wave empty-loaded

standard-loaded

1.2

over-loaded

0.9 0.6 0.3 0.0 12 1.0

13

0.8

14

15

16

over-loaded

0.6

standard-loaded

0.4

empty-loaded

17

18

empty-loaded standard-loaded over-loaded

0.2 0.0 12

13

14

15 Time (s)

16

17

18

Fig. 17. The time-varying collision forces and wave forces under various-loaded truck collisions.

the truck and wave is accomplished well. The feasibility of the drift collision coupled model is demonstrated effectively. Figure 17 shows the time-varying collision forces and wave forces under variousloaded truck collisions. The results show the collision forces of the empty, standardloaded and over-loaded trucks are 549.10, 700.89 and 826.06 KN, respectively. When the truck mass increases, on the rear seawall, the collision force of the drifting object increases while that of the wave decreases. The effect of the truck mass on the wave force, however, is very small. The stress distributions of rear seawall are shown in Fig. 15(b). The comparison of the maximum stresses of the rear seawall for variousloaded trucks is shown in Fig. 16 (broken line). It illustrates that the drifting trucks significantly enhance the destruction of run-up wave. 1450015-20


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1.6 Initial distance (d) 1.5m 3.0m 4.5m

d=1.5m d=3.0m

1.2 0.8

d=4.5m

0.4 0.0 12 1.2

13

14

15

16

d=4.5m 0.4

17

18

Initial distance (d) 1.5m 3.0m 4.5m

0.8

d=3.0m d=1.5m

0.0 12

13

14

15

16

17

18

Time (s)

Fig. 18. The time-varying collision forces and wave forces under various-initial position truck collisions.

In the following, the influence of the initial position on the collision force is studied in detail. Figure 18 shows the time varying collision forces and wave forces in numerical experiments. It can be seen that the collision forces increase when the distance between the drifting truck and rear seawall (d) decreases. On the contrary, the wave impaction pressures decrease when the distance (d) decreases. The reason for this is the effect of the wave force is weakened by the resistance effect of the drifting object. At the same time, the drifting objects get their kinetic energy through the wave energy transfer. When the distance becomes longer, the interaction time becomes longer as well and the drifting objects get more energy. Figure 19 shows the comparison of the maximum stresses of rear seawall for various initial positions. 18

Max 1st principal stress (MPa)


Wave forces (E+06N)


16

rear seawall

14 12 10 8 6 1

2

3

4

Initial distance (m)

Fig. 19. Max 1st principal stress of rear seawall. 1450015-21

5


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It indicates that the initial position of the colliding truck has a great effect on the breakwater’s dynamic response.


5. Conclusions ALE method is applied to the simulation of the solitary wave as well as the objects drifted by run-up wave interacting with the elastic breakwater. The fluid and structure are described by ALE method and Lagrangian method, respectively. Combined with the explicit integration scheme, in the paper, the Newtonian fluid is assumed as weakly compressible. The interaction between the fluid and the breakwater as well as the dynamic contact between structures is studied using the penalty method. A numerical breakwater-flume coupling model, which is similar as the physical model, and a full-scale breakwater-wave coupling model are established in Hypermesh. The validation of the numerical approaches and the model are performed based on the experimental and theoretical data. In the full-scale simulation for impacting of solitary wave and collision of the drifting object, the propagation of solitary wave, wave run-up onto breakwater, interaction with the seawall, object drift, collision with seawall are reproduced using LS-DYNA Version 971 MPP on the dawning supercomputer. Some concluding remarks are made as follows: (1) The approaches and models employed in the paper can provide an effective way for predictions of the wave pressure and the dynamic responses of breakwaters. (2) Based on the full-scale breakwater-wave coupling model, effects of seawall elevation and breakwater width have been studied. The front seawall and the breakwater top have a protective effect against the rear seawall. The wave pressure and structural stress of rear wall are increased significantly when the breakwater width decreases. Additionally, to consider the security of rear seawall structure but not wave overtopping, the elevation of rear seawall should be as low as possible. (3) In the full-scale simulation for the collision of the drifting object, the effects of mass and initial position of the drifting object on collision force are discussed. The results show that the drifting objects bring larger collision forces than the solitary wave. The collision positions and collision force change with the load of drifting object. With the increase of the distance between drifting object and rear seawall, the drifting object can get more energy, and thus, it will cause a larger collision. The methods and conclusion in collision analysis can provide reference for engineering analysis. Acknowledgments This work is supported by National 863 Program (No. 2012AA01AA307). We gratefully acknowledge the State Key Laboratory Mechanical System and Vibration 1450015-22


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(Shanghai Jiao Tong University), the Shanghai Nuclear Engineering Research and Design Institute, and the Zhejiang Institute of Hydraulics and Estuary and the Shanghai Supercomputer Center.


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