Hydroelastic behaviours of laser-welded lightweight corrugated sandwich panels subjected to water impact: Experiments and simulations

Thin–Walled Structures 146 (2020) 106452

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Hydroelastic behaviours of laser-welded lightweight corrugated sandwich panels subjected to water impact: Experiments and simulations Hao Wang a, b, Yuansheng Cheng b, *, Jun Liu b, Pan Zhang b a b

China Institute of Marine Technology & Economy (CIMTEC), Peking, 100081, PR China School of Naval Architecture and Ocean Engineering, Huazhong University of Science and Technology, Wuhan, 430074, PR China

A R T I C L E I N F O

A B S T R A C T

Keywords: Laser-welded Corrugated-core Sandwich panel Water impact Hydroelastic

Few studies have focused on the dynamics of laser-welded corrugated-core (LASCOR) sandwich panels subjected to water impacts. In the present study, three series of water impact tests, using different velocities, are conducted to investigate the hydroelastic response of LASCOR sandwich panels. The water impact pressures and acceler­ ation characteristics are measured and compared with those from fluid-structure interaction (FSI) numerical analysis results. The simulation results agree well with the experimental data for the initial water impact stage, while the difference is relatively significant in the later decay stages. Both the experimental and computational results reveal that the water impact characteristics of the LASCOR sandwich panels are different from the general observations of the stiffened-plate water impact problem. The peak amplitude of the water-impact pressure rises proportionally with v1.0~1.1 for the LASCOR sandwich panel, while the peak water impact pressure rises pro­ portionally with v1.9~2.0 for the stiffened plate from previous experiments.

1. Introduction Compared to the conventional steel structures used in shipbuilding, the steel sandwich panels welded by lasers can offer higher structural rigidity, lower weights (~30–50%) and excellent anti-shock capacities. Early in the 1980s, the initial research on the applications of laserwelded sandwich panels in ship structures was reported by The Penn­ sylvania State University, supported by U. S. Navy [1]. More recently, the laser-welded corrugated (LASCOR) stainless steel panels have also been proposed for use in the secondary structures of naval ship. Allegheny Technologies Incorporated. reported that the ATI 2003® lean duplex stainless steel is being used to manufacture LASCOR sandwich panels for personnel safety barriers [2]. To date, most studies, including those on the static strength, buckling and anti-blast characteristics of LASCOR, are broadly focused [3–10,54–57]. Up to date, rare discussions about the LASCOR water impact problem have been reported. So, some related investigations are reviewed here, the firstof which is focused on the study of a flat panel water impact problem; the second topic concerns on the new test technologies used in water impact problems, and the third topic is focused on the investi­ gation of composite structures water impact problem. Ando, Fujita and Yamaguchi firstly studied the flat-ship-structure

water impact problem [11][. In this investigation, fine test pieces made of a hard aluminium alloy fell onto the water surface from a 2.5–20 cm height. An empirical formula was also proposed. For a flat stiffened-panel water-impact problem, a series of experimental results were obtained by Nagai [12–14], Maclean [15], Chuang [16,17] and Huang [18]. Chuang firstly discussed the important role of air cushion [16,17]. In his classic work, the experimental results revealed that the air cushion cannot be neglected if the deadrise angle of ship bottom structure is less than ~3deg. A two-dimensional (2D) water impact problem was analysed by adopting a finite volume method code SOLA-VOF in Ref. [19], which the rigid-air-water coupling effects is included. Gil and Michael [20] also discussed the water impact problem of ship flat-bottom structures. They found that the hydroelastic effect is significant when the water impact velocity is greater than 5 m/s. Chen, Xiao and Wang [21] investigated the flat panel water impact dynamics based on finite element code MSC. Dytran. Some empirical formulas were proposed. Most of above discussions concerned simple objects, such as the rigid/elastic flat-bottom panel. Some literature reviews are summarised by Kapsenberg [22], Faltinsen [23], Morabito [24], Kor­ obkin and Parau [25,26], Abrate [27]. To obtain precise fluid flow characteristics of water impact phe­ nomenon, some new technologies, such as particle image velocimetry (PIV), are used in the experimental tests. Based on the high-speed PIV,

* Corresponding author. School of Naval Architecture and Ocean Engineering, Huazhong University of Science and Technology, Wuhan, 430074, PR China. E-mail address: [email protected] (Y. Cheng). https://doi.org/10.1016/j.tws.2019.106452 Received 22 January 2019; Received in revised form 8 August 2019; Accepted 2 October 2019 Available online 1 November 2019 0263-8231/© 2019 Elsevier Ltd. All rights reserved.

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Nomenclature

σYD σY ε_

D, q tb tf tc Hc θ Dx DQx Df a, b v v0 h β α*, k* α, λ Pmax

Pmax, center Maximum of impact pressure impulse for centre Amax Peak acceleration Water impact duration 2τs ϕ Angle between flow vector and horizontal plane g Acceleration of gravity fs Penetration factor ux, uy, uz, θx, θy Degrees of freedom per node ρ0 Initial density of the fresh water Specific internal energy of water ew ρa Density of the air γ Heat capacity of the gas Specific internal energy of air ea lmin,f Mesh size of fluid lmin,s Mesh size of structures δ The amount of penetration k Contact stiffness in simulation K The bulk modulus of the fluid element C0 Speed of sound in the air a1, a2, a3,b0,b1, b2 Constants for the fluid X; Y Non-dimensional variables for X, Y r Radius of the circular plate

Dynamic yield stress Static yield stress Strain rate Constants of Material Thickness of bottom face sheet Thickness of top face sheet Thickness of core web Thickness of core Inclination angle of core web Bending stiffness in the X-direction Transverse shear stiffness in the X-direction Bending stiffness of face sheet The length and breadth of the plate Water impact velocity Initial water impact velocity Dropping height The impact angle Parameters of water impact Constants for empirical equations Maximum of impact pressure for whole plate

Fig. 1. The characteristics of the LASCOR and the distribution of measuring points: (a) specimen No.1; (b) specimen No.2; (c) specimen No.3; (d) geometric description.

Nila, Vanlanduit and Paepegem [28] proposed a method to determine the impact loads on rigid bodies during water impact. The proposed method has several advantages over the previous classical methods and eliminates the challenges of using the classical pressure sensors. In a series of experimental tests [29–33], Porfiriet al. evaluated the pressure field of the water-impact problem for a rigid V-type structure using PIV. And the experimental results showed that the impact configuration significantly influences both the velocity and the pressure field,

ultimately regulating the hydrodynamic loading on the wedge. Specif­ ically, Battley and Allen conducted experimental tests on marine com­ posite sandwich panels (not LASCOR) based on a novel panel-slamming test system [34–36]. The local large strains due to the high residual pressure combined with the moving peak pressure may cause core shear failure. A conclusion is that the hydroelastic effect of sandwich panels is far more significant than that of flat panels. Series comprehensive experimental and numerical investigations concerned on composite 2

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Fig. 2. The laser welding process.

structures water impact problem are conducted by Hassoon [41]. In their studies, the experimental investigation of the V-type flexible laminate and sandwich panels (panels with different stiffnesses) and different velocities of impact were implemented to characterise the response of structures subjected to water impacts [41]. And a novel intralaminar damage model using the VUMAT subroutine and inter­ laminar damage using the cohesive zone model is also given to simulate the progressive damage. The Coupled Eulerian Lagrangian (CEL) approach by using ABAQUS was employed to deal with water impact FSI progress. In present study, the main objective of present study is to investigate the hydro-elastic dynamic response characteristics of laser-welded lightweight corrugated sandwich panels used in ship bottom struc­ tures, which is a novel protection structural type. And how to design laser-welded lightweight corrugated sandwich panels subject to slam­ ming water impact is a new topic. Thus, in present study, the water impact fluid-structure interaction (FSI) dynamic characteristics of flat LASCOR sandwich panels are concerned. Based on the experiments and numerical simulations，the mathematical expressions of water impact pressure and the accelerations responses of structures are given. And the findings can give an initial guide and method for this structural design. And the outline of the study is organized as follows. Firstly, the geometric characteristics of LASCOR sandwich panels and the material model are conducted. Then, the manufacturing tech­ nology of LASCOR sandwich panels is briefly illustrated. In Sec. 3, the three series water impact tests of LASCOR sandwich panels are con­ ducted at different impact velocities. In addition, by using the nonlinear FEM code LS-DYNA, the water impact pressure and acceleration char­ acteristics are compared with experimental results. Finally, as a com­ parison with stiffened panels, some remarks about the water impact problem of LASCOR sandwich panels are also discussed.

Fig. 3. The water impact experimental system.

2. Manufacture of LASCOR sandwich panels Referring to a practical stiffened ship panel, a LASCOR sandwich panel consists of two thin face sheets attached to both sides of a corru­ gated core with an exposed area of a*b ¼ 1000.0 mm*780.0 mm.All of three specimens are shown in Fig. 1(a)-(c). The detailed geometric de­ scriptions of a LASCOR sandwich panel are plotted in Fig. 1(d). Considering the steel used in ship building industry, the ASTM low alloy A36 steel and S304 stainless steel are used as base metal. The entire core is made using hydraulic plate bender equipment. In manufacturing process, the quality of core test sample must be carefully checked. During the laser welding process, to prevent welding leakage, the welding parameters should be confirmed. These welding parameters include the power of the laser equipment, the velocity of the laser welding and the speed of the protective air bowing. In this study, the IPG optic fibre laser system is used. The laser welding process is illustrated in Fig. 2. According to the coordinates of face sheets and core, which are both fixed to the clamped system, the starting welding point of the welding gun is located on the face sheet. Then, the face sheet and core are joined using local thermal energy (see Fig. 2), and the joint type is similar to a ‘T’. The welding tests prove that the special fixture tech­ nological equipment can meet the core laser welding requirements. The main differences between all three specimens are the bending stiffness of a sandwich panel and bending stiffness of face sheet (Table 1), which can be estimated from Ref. [42].

Table 1 Parameters and test cases for three specimens. (a) Parameters for three specimens No.

θ

tc(mm)

tf (mm)

tb(mm)

Dx(Nm)

DQx(Nm)

Db(Nm)

1 2 3

π/4 π/4 π/4

1.00 1.50 1.00

1.50 1.50 3.00

1.50 1.50 3.00

20.92 � 103 21.74 � 103 40.22 � 103

28.52 � 106 41.46 � 106 29.55 � 106

64.90 64.90 519.23

(b) Test cases for three specimens No.

a(m)

b(m)

θ

tc(mm)

tf(mm)

tb(mm)

Height(m)

Total Mass(kg)

1 2 3

1.00 1.00 1.00

0.78 0.78 0.78

π/4 π/4 π/4

1.00 1.50 1.00

1.50 1.50 3.00

1.50 1.50 3.00

0.1, 0.2, 0.4, 0.8, 1.2 0.1, 0.2, 0.4, 0.8, 1.2 0.1, 0.2, 0.4, 0.8, 1.2

404 kg 456 kg 487 kg

3

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Fig. 5. The installation of pressure sensors and accelerometers.

Fig. 4. The distribution of measuring points: (a) pressure points; (b) accelera­ tion points.

Table 2 Specification of equipment and sensors. Items Data acquisition system Sampling freq. Maximum sampling freq. Voltage meas. range Resolution of A/D conversion Date storage Operating temp. range Weight Pressure sensor Range Safe temperature Safe overload rating Excitation voltage Mass Non-linearity Dedicated adhesive Accelerometers Range Safe temperature Safe overload rating

Unit

Value

kHz kHz Volt bits GB o C kg

100 100 �20 16 1 0–50 0.5

MPa C % V kg % –

10 20–70 120 1–2 0.009 �1 RC-19

g C %

0–1000 10–80 120

o

o

Fig. 6. The repeatability of the experiments: (a) pressure; (b) acceleration.

3. Experimental setup A series of drop tests for three specimens are carried out in water impact tank (see Fig. 3). A model was attached to the electrical magnet of a carriage. The carriage is released from a predetermined height, and the specimen is dropped into freefall at lowest point of the guide rail by cutting the electrical current of the magnet. To make experimental tests

repeatable, the water is drained out from the specimen prior to each drop. Furthermore, to make sure there is no water inside during impact process, the top and bottom face sheets of test samples are sealed before tests. Some small open holes in closure plate are used to prevent water inside. 4

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Table 3 Physical and mechanical properties of steel. Type

Density ρ(kg/m3)

E (GPa)

σ (MPa)

σu (MPa)

Poisson Ratio μ

Tangent hardening Modulus Et (GPa)

A36 (Q235) S304

7850.0 8015.2

216.33 211.60

239.29 325.01

312.3 1156.8

0.26 0.29

2.5 2.1

acceleration can reach 500 g (1 g ¼ 9.8 m/s2). In addition, the duration of water impact loading time 2τs is approximately 1–100 ms. Therefore, to capture the peak acceleration value, the sampling frequency must be greater than 10 kHz. Thus, in experimental tests, the accelerator sam­ pling frequency is 100 kHz. To avoid noise, a Butterworth digital lowpass filter is used for experimental results and numerical simulations. To extract fundamental pressure pulse, the lowest filtering frequency should be above the fundamental frequency [43]. In our pressure-time history analysis, a Butterworth digital low-pass filter is adopted. The fundamental frequency is approximately 110 Hz [44]. To avoid the debonding of pressure transducers and accelerometers, 7103 adhesive is used to protect the box from water seep. In the study, the repeatability of experiments is certified firstly. Here, if the variation of maximum pressure (acceleration) of one test at a certain drop height is greater than 5%, this test is treated as invalid. The result of final experimental curve is an average of five curve tests. To demonstrate their repeatability, five pressure (acceleration) timehistory curves for the centre point of specimen No.1 at a drop-height h ¼ 0.2 mare proposed as an example. As plotted in Fig. 6, the five water-impact tests with drop heights of h ¼ 0.2 m for specimen No.1 are given. The shapes of water impact duration of these five tests are similar, and the black bold curve in Fig. 6(a) is the average of these five pressurehistory curves.

Table 4 Material properties (the fresh water and the air). Material

LS-DYNA model

Input parameter and its Magnitude

Water

Polynomial Equation of State

Air

Gama Law

a1 ¼ 2.314 � 109Pa, a2 ¼ 6.561 � 109Pa, a3 ¼ 1.126 � 1010Pa, b0 ¼ 0.4934, b1 ¼ 1.3937, b2 ¼ 0.0000, ρ0 ¼ 1025 kg/m3,ew ¼ 3750.4J/kg γ ¼ 1.40,ρa ¼ 1.185 kg/m3, ea ¼ 287J/kg

4. Numerical model and verification

Fig. 7. The finite element model.

4.1. Materials model for steel, water and air

Photographs and the detailed coordinates of the pressure transducers and accelerometers are shown in Fig. 4. The specifications for these sensors are listed in Table 2. And the detailed installations of the pres­ sure sensors are plotted in Fig. 5. These installation steps can be divided into two stages. The first step is the installation of the aluminium block because the face sheet is too thin (tb ¼ 1.0 mm) to install. Here, the threaded holes must have been previously manufactured in the aluminium block. In the second step, the pressure sensors are installed on aluminium block. To model the clamped boundary condition, the bolts are used to connect the specimen and rigid supporting structures. The strain measurements are not considered in the present experi­ ments. During the manufacturing process of LASCOR sandwich panels, we find that it is very difficult to install a strain gauge due to the special structures of LASCOR sandwich panels. Here, if we want to measure the dynamic strain characteristics of water-contacting face, the strain gauge must be installed at the bottom face sheet. Thus, the electrical cables connecting the strain gauge and sampling device must pass though top face sheet. In addition, numerous open holes on top face sheet are also required. If multiple strain gauges are installed, numerous open holes on top face sheet will break the continuity of the LASCOR sandwich panels (Fig. 5). Second, if we want to measure the dynamic strain character­ istics of water-contacting face, the strain gauge must be installed before laser welding. This is mainly because numerous open holes on top face sheet are required if the installation of strain gauge follows laser weld­ ing. Furthermore, the precise directions (0, π /4, π/2) for strain gauges are very hard to satisfy in small open holes (the installation of the pressure sensor and the acceleration sensor do not have challenging directions). In addition, if the strain gauges are installed before laser welding process, the thermal energy generated by laser welding would damage the strain gauges. According to previous numerical investigations, the peak pressure value may be over 500 kPa at a velocity ofv0 ¼ 3–5 m/s, and the peak

The tensile strength test of the smooth axisymmetric specimens on a WDW-100 universal testing machine was run at a quasi-static strain rate (0.2 � 10 3s 1) and room temperature (293 K). The tensile properties of these two steels are given in Table 3. To define the material–work piece of the high-impact deformation during the water impact process, the elastic-plastic material model with a kinematic–isotropic hardening is adopted. The effect of the strain rate is considered by scaling the static yield stress by a given factor, derived using the Cowper-Symonds [45] relation: � �ε_ �1=q σ YD ε_ pl pl ¼1 þ (1) σY D where σ YD is the dynamic yield stress, σ Y is the static yield stress, ε_ is the strain rate, and D and q are constants of the Cowper-Symonds relation. The constants D and q have values of 40.4 S 1 and 5, respectively, for typical steel during dynamic responses [45]. To model fresh water, a polynomial state equation is conducted. The state equation (EOS) of fresh water is given as � � a1 μ þ a2 μ2 þ a3 μ3 þ b0 þ b1 μ þ b2 μ2 ρ0 ew ðμ > 0Þ P¼ (2) a1 μ þ ðb0 þ b1 μÞρ0 ew ðμ < 0Þ where μ¼(ρ-ρ0)/ρ0, ρ0 is the initial density of the fresh water; ew is the specific internal energy per unit mass; and a1, a2, a3, b0, b1, and b2 are constants for water. The upper equation applies to water in a com­ pressed state, while the lower applies to a fluid in an expanded state. The constants for this equation for fresh water are provided in Table 4. The gamma law gas model is adopted for the EOS of air [45], P ¼ ðγ

5

1Þρa ea

(3)

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Fig. 8. Water impact phenomena (h ¼ 1.20 m for specimen No.3): (a) experimental; (b) simulation.

6

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analysis. And it is known that the results of ALE algorithm are sensitive to Euler mesh density. Therefore, the Euler mesh needs to be fine enough to capture the highest gradients of pressure fields, yet a coarser mesh is favourable in terms of computational cost. In addition, the selection of contact stiffness penalty is based on contact algorithms; the maximum pressures are required to have been approximately known before the algorithm is used. The non-physical contact penetration can be controlled. Some studies, such as Ref. [46–49], discussed the effects of penalty factors and mesh size in simulations. To verify the reliability of the developed FE model, a circular monolithic plate built for calibration purposes free falls on to water surface. In previous study, the numerical verification model for a monolithic plate water impact problem is illustrated [50]. A study of the convergence of penalty factors and mesh size for simulation model is included in Appendix A. 5. Results and discussion 5.1. Basic water impact characteristics As an illustration (h ¼ 1.20 m for specimen No.3), the water impact process captured by high-speed camera is plotted in Fig. 8(a), and the corresponding numerical results are given in Fig. 8(b) respectively. The impact durations 2τsfor both simulations and experimental results are also given. To clarify the physical mechanisms of water impact, the whole process is divided into three stages: Stage 1: Structure begins to drop → Air compression begins. Stage 2: Air compression begins → Structure begins to contact water. Stage 3: Structure begins to contact water → Structure immerses in water. In this case, the duration 2τs ¼ 9.2 ms predicted by numerical simu­ lation coincides with the estimations of experimental data (2τs ¼ 8.7 ms). In addition, for water jet shapes, the results from simu­ lation also agree well with those from experimental ones during t ¼ 0.51s–0.52s. When t > 0.52 s, the jet begins to split into two or three parts. Moreover, the time of water contact observed by the numerical simulation is t~0.514s, while the high-speed camera result is about t~0.515s. The water impact jet of high-speed camera results is formed at the timet�0.52s, while the result of numerical simulation is t � 0.518s. The shape of water impact jet calculated by ALE coupling algorithm cannot capture the spattering of droplets. With the development of water impact jet, it is distorted (Fig. 8). This phenomenon is probably caused by the instability of fluid flow. Some small perturbations may be from some uncontrollable factors, such as the minor asymmetries of experi­ mental tests. The velocity vector distribution of fluid particles is illustrated in Fig. 9. In first stage, the direction of the air flow field velocity vector is nearly vertical to horizontal plane. In second stage t ¼ 0.51 s–0.52s, when the specimen is closed to horizontal plane, the air between a sandwich panel and horizontal plane is compressed and overflowed. As shown in Fig. 9, the direction of air flow field velocity vector changes (the angle between air flow field velocity vector and horizontal plane is approximately ϕ ¼ π/4). When the LASCOR sandwich panel begins to contact water, the angle ϕ is nearly equal to π /3. In third stage, the velocity rapidly decreases due to the water resistance.

Fig. 9. Fluid flow field velocity vectors from numerical simulation.

where ρa is the density of air, γ is the heat capacity of the gas and ea is the specific internal energy of air. The initial pressure is equal to 1.0 � 105 Pa. 4.2. 2FE model mesh and coupling algorithm of FSI In numerical model, the top (bottom) face sheet and core are modelled as a planar plate using 52000 quadrilateral shell elements (specifically, shell 163 with a KEYHOFF formulation, hourglass control, five degrees of freedom per node (ux, uy, uz, θx, and θy), finite membrane strains elements, and five integration points). The FE model of the LASCOR sandwich panel can be seen in Fig. 7. The mass of pressure (accelerator) sensors are also included, and they are modelled as mass elements. To model the fluid inside and outside of the LASCOR sandwich panel, two Euler domains are used. The outer domain includes the LASCOR sandwich panel surface (including the top face sheet, bottom face sheet and out-off rigid wall) as part of the fluid boundary; the Euler material is outside the LASCOR sandwich panel surface, and there is no material inside. In addition, the contents inside are modelled in the inner domain, and this domain is also enclosed by the outer surface of the LASCOR sandwich panel. Therefore, both the Euler domains use the LASCOR sandwich panel surface as part of their ranges (Fig. 7). The hydrostatic pressure is also considered using the*INITIAL_HYDROSTATIC_ALE keyword card. The Euler mesh contains water and air. The outer boundary of the outer domain is given as a sufficiently large box. The fluid mesh used for this problem that consists of a block of elements and with the dimensions 2.5 m*2.5 m*4.0 m. The finest grid size of the Euler domain is 0.005 m in this simulation. These fluid blocks of water and air were meshed with 100*100*60 hexahedron elements, and the total number of fluid elements is about 900000. All boundary conditions for fluid mesh are given a “flow” boundary condition by adopting a nonreflecting boundary keyword card. The arbitrary Lagrange-Euler (ALE) coupling algorithm is used in

5.2. Water impact pressure characteristics 5.2.1. Time domain characteristics of pressure As shown in Fig. 10(a), the peak pressure amplitude on entire FSI surface increases rapidly. It is notable that an oscillation exists in the pressure-time curve. This is probably caused by local structural defor­ mation ofspecimen No.1 (tb ¼ tf ¼ 1.5 mm). When LASCOR sandwich panels contacts the water at timet2 ¼ 1.186s, the maximum pressure reaches Pmax ¼ 304.04 kPa (h ¼ 0.2 m), while the numerical result is 7

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Fig. 10. Water impact pressure-time characteristics: (a) specimen No.1, P3 at h ¼ 0.2 m; (b) specimen No.1, P3 at h ¼ 1.2 m; (c) specimen No.2, P3 at h ¼ 0.2 m; (d) specimen No.2, P3 at h ¼ 1.2 m; (e) specimen No.3, P3 at h ¼ 0.2 m; (f) specimen No.3, P3 at h ¼ 1.2 m.

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Fig. 11. Comparison of water impact duration time.

Table 5 Maximum pressures for P1~P5. (a) Specimen No.1 NO.

v (m/s)

P1(KPa)

P2(KPa)

P3(KPa)

P4(KPa)

P5(KPa)

0.1 m 0.2 m 0.4 m 0.8 m 1.2 m

1.401 1.981 2.801 3.962 4.852

225.96 286.81 335.58 408.37 482.84

226.67 293.63 404.58 465.79 502.45

228.99 304.04 382.55 485.79 568.97

255.87 347.42 402.22 555.70 601.81

96.53 159.52 200.34 206.11 261.07

(b) Specimen No.2 NO.

v (m/s)

P1(KPa)

P2(KPa)

P3(KPa)

P4(KPa)

P5(KPa)

0.1 m 0.2 m 0.4 m 0.8 m 1.2 m

1.401 1.981 2.801 3.962 4.852

229.05 297.86 340.62 426.34 502.31

228.49 301.39 411.17 491.12 573.45

232.17 311.11 395.71 517.16 589.77

257.79 357.43 426.22 563.22 644.58

122.30 193.11 239.89 252.02 313.19

Fig. 12. Water impact pressure-time characteristics for different pressure sen­ sors P1–P5: (a) specimen No.1 at h ¼ 0.2 m; (b) specimen No.3 at h ¼ 1.2 m.

(c)Specimen No.3

2τs ffi

NO.

v (m/s)

P1(KPa)

P2(KPa)

P3(KPa)

P4(KPa)

P5(KPa)

0.1 m 0.2 m 0.4 m 0.8 m 1.2 m

1.401 1.981 2.801 3.962 4.852

231.22 330.93 382.47 433.42 526.95

231.57 331.94 427.27 536.73 644.53

247.09 332.24 432.01 542.56 641.70

279.11 376.90 438.16 580.87 675.51

151.31 205.50 283.61 317.21 449.26

where Pmax

τ

2 s

; K1 ¼

2Pmax

τs

*

; Pmax ¼ k* vα0 ; v ¼

pffiffiffiffiffiffiffi 2gh

pffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2:6Þ a2 þ b2 C0

(6)

where C0 is the speed of sound in air (C0 ¼ 340 m/s). It is observed that the impulse duration 2τs for all cases of LASCOR sandwich panels is .83 ms–9.70 ms from Eq. (4), which coincides well with experimental and numerical simulation results. According to the analyses of Wagner [51], Faltinsen [23], and Kor­ obkin [25,26], the load of the water impact is complex. The load is related to the impact velocity, structural flexibility, and acceleration of the impacting body. In the experimental studies proposed by Okada and Sumi [52], a detailed clarification of three types of water impacts, including the trapped-air region (0� <β < 1.0� , here, β is the impact angle, which means the angle between the body surface and the still water surface at impact), the transitional region (1.0� <β < 3.0� ) and the Wagner-type impact region (3.0� <β), was given. The typical loading pressure shape curves of a trapped-air impact and Wagner-type impact are also given in Ref. [52]. And an empirical formula for the duration of water impact is adopted as follows [52]: � �λ b 2τs ffi α (7) v0

approximately 260.79 kPa. The impulse duration is nearly 10.5 ms. When the height h ¼ 1.2 m, the maximum pressure value Pmax is approximately 568.97 kPa (the numerical result Pmax ¼ 520.14 kPa). And the impulse duration is2τs~9.0 ms, which is shorter than that of h ¼ 0.2 m (see Fig. 10 (b)). The shape of time-pressure curves can be described by a two-parameter parabolic mathematical expression [50], � K1 t2 þ K2 tð0 � t � 2τs Þ pðtÞ ¼ (4) 0 ð2τs � tÞ

K1 ¼

ð2:1

(5)

where Pmax is the maximum value of water impact pressureimpulse,2τs is the duration of impulse, and α*,k* are the parameters-based on geo­ metric dimensions and material properties of structures. In addition, the expression for 2τs is defined as [50],

α ¼ 6:94 � 10 8 ; λ ¼ 0:673

(8)

where b is the breadth of the plate and v0 is the initial velocity at impact. A comparison of the estimation of 2τs based on Eq. (7) and our for­ mula is plotted in Fig. 11. The comparison shows that our formula es­ timations agree well the experimental results within the range 9

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Fig. 13. Water impact acceleration-time characteristics: (a) specimen No.1, P3 at h ¼ 0.2 m; (b) specimen No.1, P3 at h ¼ 1.2 m; (c) specimen No.2, P3 at h ¼ 0.2 m; (d) specimen No.2, P3 at h ¼ 1.2 m; (e) specimen No.3, P3 at h ¼ 0.2 m; (f) specimen No.3, P3 at h ¼ 1.2 m.

v0 ¼ 1.981–4.852 m/s (h ¼ 0.2–1.2 m). The error of estimation is larger when the structure is flexible. Thus, the proposed formula is not suitable if the structure is extremely flexible. However, it can be modified by the coefficient C0. For test specimen No.2, as shown in Fig. 10(c), the oscillations also exist in pressure-time curves. The duration 2τs ¼ 11.0mswhenh ¼ 0.2 m, which the duration 2τs ¼ 9.0mswhenh ¼ 1.2 m. In addition, Fig. 10(c)– (d) indicates that Pmax ¼ 311.11 kPa when h ¼ 0.2 m (the numerical result Pmax ¼ 268.98 kPa), while the pressure Pmax ¼ 588.77 kPa when h ¼ 1.2 m (the numerical result Pmax ¼ 540.78 kPa). For test specimen No. 3, as plotted in Fig. 10(e)–(f), when h ¼ 0.2 m, Pmax ¼ 332.24 kPa (the numerical result Pmax ¼ 301.01 kPa). When h ¼ 1.2 m, the Pmax ¼ 644.53 kPa (the numerical result Pmax ¼ 590.32 kPa). In addi­ tion, the duration 2τs ¼ 11.0 ms when h ¼ 0.2 m (2τs ¼ 9.0 ms when h ¼ 1.2 m). The pressure-time curves can also be described by a twoparameter parabolic mathematical expression. Compared with the pressure-time characteristics of these three specimens, the main difference is the severity of oscillation. As shown in Fig. 10(a)–(f), the oscillation phenomenon of water-impact pressure history for specimen No. 1 with h ¼ 0.2 m is much more significant than that of specimen No. 3. We know that the local stiffness of specimen No. 1 is 1/8 that of specimen No. 3, thus the local hydroelastic effect is more significant. This phenomenon can also be found in Chuang’s classic experimental tests [16].

Furthermore, the time-history curves of pressure sensors P1–P5 for the two different drop heights, h ¼ 0.2 m (for specimenNo. 1) and 1.2 m (for specimen No. 3), are given in Fig. 12. In most cases, the pressure values of P2 are lower than those of P3 (some cases differ, such as when the drop height is h ¼ 0.4 for test specimen 1 and 2). The time-history curves also show that the impulses can be described using two-parameter parabolic mathematical expres­ sions for most cases. Some special cases, such as the water impact duration curves for the P1 and P5 gauges seem more complex. Moreover, all the water impact pressure values for P3 are lower than those of P4 (see Table 5). According to most previous investigations, the pressure values at the central point are usually the highest. But some in­ vestigations also noted that the coordinates of the maximum water impact pressure also occur at some other positions [15,53]. A series of circular plate water-impact experimental investigations also revealed that the distance between the centre of the plate point and the maximum impact pressure position is approximately r/3 (r is the radius of circular plates) when the initial imperfections are considered [53]. In our study, this phenomenon also exists (for example, the maximum values of pressure for P2 and P4 are greater than those for P3). There may be two reasons for this. First, the initial imperfection is inevitable during the laser welding. Second, the initial deflection of the face sheet of water impact stage 1 plays as an imperfection due to the special structural configurations of the LASCOR sandwich panel, which can change the spatial distribution of impact pressure. Furthermore, the experimental results of maximum impact pressure are usually larger than those from the numerical simulations. There are two probable reasons. First, the welding line may have a local

5.2.2. Spatial domain characteristics of pressure Compared with the maximum pressure value distribution of P1~P5, listed in Table 5, the impact pressure rises in proportion tov1.0~1.1. 10

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Fig. 14. Impact hydrodynamic forces: (a) specimen No.1 (b) specimen No.2, (c) specimen No.3.

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Table 6 Peak accelerations for A1~A6. (a) Specimen No.1 NO

v (m/s)

A1(m/s2)

A2(m/s2)

A3(m/s2)

A4(m/s2)

A5(m/s2)

A6(m/s2)

0.1 m 0.2 m 0.4 m 0.8 m 1.2 m

1.401 1.981 2.801 3.962 4.852

570.45 693.70 1354.60 2749.42 3669.06

660.07 1036.19 1287.79 1055.80 1086.39

600.35 988.27 1492.08 1185.77 1274.03

955.37 1254.19 2187.23 2215.29 2944.76

778.37 977.94 1652.93 2393.15 3049.35

753.64 813.17 1320.09 3031.12 4538.40

NO

v (m/s)

A1(m/s2)

A2(m/s2)

A3(m/s2)

A4(m/s2)

A5(m/s2)

A6(m/s2)

0.1 m 0.2 m 0.4 m 0.8 m 1.2 m

1.401 1.981 2.801 3.962 4.852

623.06 862.50 1657.53 2554.43 4387.90

1126.59 1277.79 1343.59 1654.59 2770.89

672.59 741.54 926.51 1928.58 3913.09

995.42 1129.73 2696.73 3222.04 4241.34

981.25 1235.55 3018.39 3079.20 4780.50

1561.90 1676.97 2002.24 4601.31 5181.67

NO

v (m/s)

A1(m/s2)

A2(m/s2)

A3(m/s2)

A4(m/s2)

A5(m/s2)

A6(m/s2)

0.1 m 0.2 m 0.4 m 0.8 m 1.2 m

1.401 1.981 2.801 3.962 4.852

911.45 1101.94 1859.93 3087.54 5661.81

1275.66 1694.30 1822.26 2025.22 5372.65

1042.89 1231.83 1401.10 1848.48 5231.15

1330.22 1983.38 1708.77 2613.15 5836.69

1292.80 2128.93 2910.59 4898.04 5937.16

1617.12 2111.17 2282.96 4506.18 6522.05

(b) Specimen No.2

(c)Specimen No.3

result Amax ¼ 5278.12 m/s2). The relationship between Amax and v1.1~1.2 seems to be linear. The results of experiments and numerical simulations for test specimen No. 3 (when the drop heights are h ¼ 0.2 m & 1.2 m, respectively) are also plotted in Fig. 14(e)–(f). As given in Table 6, along with the increase of face sheet bending stiffness, Amax increases from 600.35 m/s2 to 1042.89 m/s2(A3 accel­ eration gauge, h ¼ 0.1 m). And this trend coincides with the pressure response, but is more significant. Compared with the acceleration-time curves, the oscillation phenomenon is more significant if the local stiffness value is lower (i.e., the face sheet is thinner). In addition, the decay time of specimenNo. 1 is longer than those of specimens No. 2 and No. 3 for h ¼ 0.2 m. This means the local hydroelastic response is more significant for specimenNo. 1.

reinforcement design. Second, the instalment of the pressure sensors and shim (or spacer) may have a local reinforcement design. The time of experimental maximum pressure value does not coincide with that of the numerical simulation results (this phenomenon also exists for the test specimens No. 2 and No. 3). This is mainly because of the minor asymmetry of the water impact process. 5.2.3. Hydrodynamic forces of LASCOR sandwich panels The hydrodynamic forces, which can be calculated by ∬ PHF ds, are also considered in our scope, where PHF is the global FSI impact pressure. Here, the corresponding case of three specimens at entry height h ¼ 0.1 m and 1.2 m are also considered here. All the results are given in Fig. 13. All of the time-hydrodynamic force curves are also can be described by a two-parameter parabolic mathematical expression, which are similar with that of water impact pressure. The maximum hydrodynamic force of the test specimen No. 1, when entry height h ¼ 0.1 m, is 18.78 KN (Fig. 13 (a)). With the same water entry height, the hydro­ dynamic forces of test specimen No. 2 and No. 3 are 16.35 KN and 13.65 KN (Fig. 13 (b) and (c)). It seems that the thickness of sandwich face sheet and sandwich core has little effects on the hydrodynamic forces. Unlikely, while water entry height h ¼ 1.2 m, the maximum hydro­ dynamic force of test specimen No. 1, is 172.6 KN. And the maximum hydrodynamic force of the test specimen No. 2 and No. 3, are 174.1 KN, 309.1 KN (Fig. 13 (b)–(c)). It means the contacted water face sheet thickness has significant effects on the hydrodynamics forces.

5.3.2. Spatial-domain characteristics of acceleration Compared with the pressure distribution, the discreteness of the datasets is more significant. According to test datainTable6, the peak accelerations of A1, A4, A5 and A6 are higher than those of A2 and A3. This is because the positions of A2 and A3 are close to the boundary. When h < 0.2 m, the peak acceleration value is usually lower than 100 g (1 g ¼ 9.81 m/s2). And when h > 0.8 m, the peak acceleration value is greater than 300 g. It is notable that the numerical results agree well with experimental data for the initial water impact stage, while the difference is relatively significant in later decay stages (this phenomenon also exists for test specimens No. 2 and 3). Based on the analyses of differences, the reasons for this may be as follows. First, a bracket is installed below the water level in order to avoid testing equipments falling into water. In addition, the testing equipment will be stopped when the test specimen hits the bracket. However, in simulation, the testing equipment continues dropping due to gravitational effects. Second, due to the effects of the water viscosity damping and flooding water mass, the responses of the LASCOR sandwich panel decay considerably in a short time. In simu­ lations, the water viscosity damping and flooding water masses are not included. Third, the frictional effects of experimental facilities are not considered.

5.3. Water impact acceleration characteristics 5.3.1. Time domain characteristics of acceleration The acceleration-time curves of all specimens for five different dropping heights are shown in Fig. 14(a)-(f). Here, for test specimen No.1, the peak acceleration value Amax ¼ 693.70 m/s2 from experimental data, and the simulation result Amax ¼ 770.89 m/s2. For test specimen No. 3, when h ¼ 1.2 m, the experimental maximum value Amax ¼ 3669.06 m/s2 (the numerical result Amax ¼ 3212.67 m/s2). For test specimen No. 2, Amax ¼ 862.50 m/s2 when h ¼ 0.2 m (for numerical simulation result, Amax ¼ 972.12 m/s2). When h ¼ 1.2 m, as illustrated in Fig. 14(d), Amax ¼ 4387.90 m/s2 (numerical simulation

5.4. Deformation of LASCOR sandwich panels The dynamic deformations of three points are considered. The first point is the central point of water contacted bottom face sheet No. DP1, 12

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Fig. 15. Deformation of LASCOR sandwich panel: (a) specimen No.1, at h ¼ 0.2 m; (b) specimen No.1 at h ¼ 1.2 m; (c) specimen No.2, at h ¼ 0.2 m; (d) specimen No.2, at h ¼ 1.2 m; (e) specimen No.3, at h ¼ 0.2 m; (f) specimen No.3, at h ¼ 1.2 m.

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Fig. 16. The detailed experimental work of Chuang [16,17] and Zhang [18]: (a) facilities of Chuang; (b) gage locations of Chuang; (c) gage locations of Zhang.

the second point is the central point of top face sheet No. DP2, and the third point is the corresponding single cell boundary point near the No. DP1, which is called No. DP3. The numerical simulation results of three specimens at h ¼ 0.1 m and 1.2 m are given in Fig. 15. For test specimen No. 1, when the h ¼ 0.1 m, the maximum defor­ mation of three points are 1.382 mm, 1.492 mm and 1.392 mm respec­ tively. As shown in Fig. 15(a), the local hydroelastic dynamic response components are included in the whole responses. And in later stages, the structural response amplitude decayed as the energy dissipation due to damping effects. The deformation difference between the top face sheet point No. DP2 and the bottom face sheet point No. DP1 is 0.11 mm, which also proves the existence of local hydroelastic effect. The defor­ mation difference between the No. DP1 and the No. DP3 is very minor (about 0.01 mm). When the h ¼ 1.2 m, the maximum deformation of three points are 20.42 mm, 21.50 mm and 20.51mmrespectively (see Fig. 15(b)). The structural response decayed speed seems faster compared with that of h ¼ 0.1 m. Moreover, as the deformation is large, the hydroelasticity is in the nonlinear elastic region. For test specimen No. 2, when the h ¼ 0.1 m, the maximum deformation of three points are 0.891 mm, 0.894 mm and 1.001mmrespectivelyas plotted in Fig. 15(c). And the local hydroelastic response components can also be found in the deformation-time curves. And the response decaying period is longer than that of the specimen No. 1. When the h ¼ 1.2 m, the nonlinear hydroelasticity effect is significant as shown in Fig. 15(d). As shown in Fig. 15(e), for specimen No. 3, when h ¼ 0.1 m, the

maximum deformation of three points are 0.481 mm, 0.483 mm and 0.520 mm respectively. It seems that the local frequency response is more significant than those of specimen No. 1 and No.2. This is due to the fact that the bending rigidity of water contacted face sheet of spec­ imen No. 3 is greater. When h ¼ 1.2 m, it is seen clearly from Fig. 15(f) that the response stage can be divided into two periods. The first period is the initial high frequency response stage. Thus, the maximum defor­ mation is 9.110 mm. In the second stage, the response oscillation of LASCOR sandwich panel is the main characteristics. 5.5. Further study of water impact behaviour According to previous studies, the water impact experimental results for traditional ship structures, such as rectangular panels or stiffened panels, are limited. A series of experimental results can be obtained from the work of Nagai [12–14], Maclean [15], Chuang [16,17], Huang and Zhang [18]. To enrich experimental results, the water impact pressure characteristics are compared discussed with Chuang [16,17], Huang and Zhang [18]. The experimental facilities of Chuang’s work are shown in Fig. 16(a). An identical 1/4-scale model KG-3, is fabricated and scaled geometri­ cally and structurally to represent a portion of the bottom of a seagoing vessel. The geometric parameters of structures are given in Refs. [16, 17]. And the gauge locations are shown in Fig. 16(b). In the study of Huang and Zhang [18], a flat-bottom body water 14

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Fig. 17. The pressure experimental results of Chuang [16,17]: (a) PE-3, PE-4; (b) PE-17, PE-18; (c) PE-09, PE-10, PE-20.

experiment (with a drop height h ¼ 0.3 m), the impulse duration is approximately 19.4 ms–24.1 ms, which coincides well with the experi­ mental results of the pressure sensors PE-3, PE-4, PE-9, PE-10, PE-17, and PE-18 from Fig. 17. The impulse shape of pressure history curves for PE-3, PE-4, PE-9, PE-10, PE-17 and PE-18 (i.e., except for gauge PE20) can also be described using a two-parameter parabolic mathematical expression. Compared with the pressure-history curves of other gauges, as plotted in Fig. 17, the time of maximum value of gauge PE-20 shows some differences. The impulse duration of PE-20 is approximately 12.00 ms, which is much shorter. And the shape of PE-20 impact impulse may be simplified as an exponential decay type impulse. It is notable that the location of the PE-20 gauge is near the boundary of model, which has a significant effect on the water impact process. Furthermore, the oscillation phenomenon seen in PE-3 and PE-4 seems more signifi­ cant than those of PE-9, PE-10, PE-17 and PE-18. 5.5.2. Comparison of spatial-domain characteristics for pressure To discuss the differences of water impact responses between LAS­ COR sandwich panels and stiffened panels, our results are compared with the work proposed by Huang and Zhang [18]. First, the relationship between the peak water impact pressure and the impact velocity for LASCOR sandwich panel and stiffened plate are analysed. And the spatial distributions of water impact pressures for these two structures types are further discussed. The maximum impact pressure values for LASCOR sandwich panels and stiffened plates are written in non-dimensional styles. The non-dimensional variables for X-co­ ordinates and Y-coordinates are defined as follows:

Fig. 18. Water impact pressure compassion between the stiffened plate and the LASCOR sandwich panels.

impact problem is performed in the same tank of our study. The overall dimensions of model are 1000.0 mm long by 800.0 mm wide. The thickness of the model bottom is 20.00 mm. The dimensions of the bottom stiffener are 60 mm*40 mm. A pressure-time history curve with a drop height h ¼ 0.3 m is given in their study, which the pressure gauge locations are given in Fig. 16(c).

X¼

5.5.1. Comparison of time domain characteristics for pressure Using Eq. (6), the impact duration is estimated. For Chuang’s

x Pmax ;Y ¼ ða=2Þ ðPmax; center Þ

(9)

In the analysis of Ref. [18], the water impact height varies from 0.1 15

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Fig. 19. Water impact pressure distribution comparison between the stiffened plate and the LASCOR sandwich panels: (a) h ¼ 0.1 m; (b) h ¼ 0.2 m; (c) h ¼ 0.4 m.

m to 0.5 m (five cases). As shown in Fig. 18, the peak water impact pressure rises proportionally with v1.0~1.1 for LASCOR sandwich panels, while the peak water impact pressure rises proportionally with v1.9~2.0 for stiffened panels. In Fig. 18, if the maximum water impact pressure for a whole wet surface is selected as an indicator, the relationship between water impact velocity and pressure seems linear. But the relationship between water impact velocity and pressure for stiffened plates is nonlinear. This figure also explains that the position of maximum water impact pressure is not at original coordinate (centre of the wet surface), which can also be seen in Fig. 18. There may be two reasons for different proportionality with velocity. First, the 3D fluid flow distribution characteristics are different as the structural types are different. The 3D fluid flow distribution has direct connections with water impact pres­ sure. Second, for laser-welded lightweight corrugated sandwich panels, it is shown that both the local hydroelasticity and the global hydro­ elasticity phenomenon existed in our analysis. But the local hydro­ elasticity phenomenon is quite weak for a stiffened panel water entry problem. The local hydroelasticity also changes the pressure value. Fig. 19 illustrates the two-dimensional (2D) pressure characteristics of present study and Huang’s experimental results for X-coordinates (here, the results for Y-coordinates are similar). The water impact heights of present study vary from 0.1 m to 1.2 m, and the test cases of Huang’s study are the same. Three cases (h ¼ 0.1 m, 0.2 m and 0.4 m) are considered and compared. As plotted in Fig. 19, the symmetrical 2D spatial distribution of water impact pressure for a stiffened plate is very clear. This phenomenon is explained by experimental error [18]. Because the pressure sensor is not installed across the whole specimen, the corresponding results for LASCOR sandwich panels are strictly symmetrical, which an ideal case. Fig. 19(a) also shows that the position of the maximum pressure is not located in the centre of wet surface for either the stiffened plate or the LASCOR sandwich panel. However, the

values of water impact pressure at the central region of wet surface are often greater than those of other regions. The pressure fluctuation phenomenon is more significant for the case h ¼ 0.1min Fig. 19. This may be due to air cushion effects, which plays a more important role when the water impact velocity is relatively low. 6. Conclusions In present investigation, the experimental tests and numerical ana­ lyses are performed to study the water impact response of LASCOR sandwich panels. The experimental equipments are designed. A series of experiments are conducted by setting different drop heights (h ¼ 0.1–1.2 m) for three test specimens. Numerical simulations employing the LS-DYNA Explicit code are also carried out, in which the FSI effects are considered. The water impact phenomenon, the FSI pressure and the acceleration characteristics are both given. The following findings and conclusions are made: (1) The entire water impact process can be divided into three stages: Stage 1: structure begins to drop → air compression begins; Stage 2: air compression begins → structure begins to contact water; Stage 3: structure begins to contact water→ structure is immersed in water. From the high-speed cameras, it should be concluded that the numerical simulations can capture the basic physical processes of the water impacts, though some minor differences exist, especially the phenomenon of spattered droplets. (2) The shape of impact pressure can be described with a twoparameter parabolic mathematical expression. And the pre­ sented estimation formula of impulse duration 2τs can give a reasonable estimation. According to pressure distributions, the peak impact pressure rises in proportion with v1.0~1.1. The 16

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Fig. 21. The analysis of fluid mesh size: (a) velocity v ¼ 1.4 m/s; (b) velocity v ¼ 2.8 m/s. Fig. 20. The analysis of penalty factor fs: (a) velocity v ¼ 1.4 m/s; (b) velocity v ¼ 2.8 m/s.

finding is that the peak water impact pressure rises proportionally with v1.0~1.1 for the LASCOR sandwich panel, while the peak water impact pressure rises proportionally with v1.9~2.0 for a stiffened panel.

maximum pressure of all FSI surfaces (wet surfaces) may be not be located in centre. This may be caused by the initial deflection of face sheets in impact stage 1, which plays a role of imperfec­ tions due to the special structural configurations of a LASCOR sandwich panel. (3) The numerical simulation results agree well with experimental data in initial water impact stage, while the difference is rela­ tively significant in the later decay stage due to the effects of water flooding between the face sheets, the friction effect, etc. The relationship between the peak acceleration Amax and v1.1~1.2 is linear. (4) Compared with the water impact pressure responses of a stiffened panel, the pressure spatial distributions of a LASCOR sandwich panels are different. And the pressure fluctuation phenomenon is more significant for a LASCOR sandwich panel. Another key

Declaration of competing interest The authors declared that they have no conflicts of interest to this work. We declare that we do not have any commercial or associative interest that represents a conflict of interest in connection with the work submitted. Acknowledgement The authors are pleased to acknowledge the financial support of the National Natural Science Foundation of PR China (Contract No. 51079058, 51279065).

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Appendix A Several investigations, including those of Bereznitski [47], Stenius, Rose’n and Kuttenkeuler [48,49], discussed the parameter selection for the LS-DYNA ALE coupling algorithm. Stenius noted that the contacting damping and time scaling have limited effects, while the solution is highly dependent on the relation between the mesh density and contact stiffness. Thus, the discussions of convergence of two parameters are proposed here. As a comparison, the experimental results of pressure for a circular plate in free fall described in previous study [50]. A1) Penalty factor (fs) Ina FSI simulation, the fluid will pass through the structures without the contact stiffness. And a relationship is generated through an ‘elastic spring’ between the two bodies, where the contact force is equal to the product of the contact stiffness (k) and the penetration (δ). The amount of penetration (δ), or incompatibility, between the two bodies is therefore dependent on the stiffness k [48,49]. For an ideal case, there should be no penetration, but this implies that k ¼ ∞, which will lead to numerical instabilities. Thus, the value of k depends on the relative stiffnesses of the bodies in contact. In the LS-DYNA programme, the contact stiffness is determined using the following relationships [45,48,49]: k¼

fs � Area2 � K ; forsegementsonsolidelements Volume

(A.1)

k¼

fs � Area � K ; forsegementsonshellelements Minimum Diagonal

(A.2)

Area ¼ Areaofcontactsegment

(A.3)

K ¼ Thebulkmodulusofthefluidelement

(A.4)

fs ¼ Penaltyfactor

(A.5)

Generally, the contact stiffness parameters are automatically determined by LS-DYNA code. In addition, raising the value of fs above 0.1 will typically cause practical instabilities. If fs is too low, numerical fluid leakage will occur. Thus, the five different penalty factors (fs ¼ 0.001, 0.005, 0.01, 0.1, and 0.2) are selected. And two water impact velocities v ¼ 1.40 m/s, 2.80 m/s are considered. As shown in Fig. 20, the effects of different penalty factors have been plotted. Here, a higher penalty factor results in noise and an oscillating signal. For Fig. 20(a) (v ¼ 1.40 m/s) and Fig. 20(b) (v ¼ 2.80 m/s), the simulation is incorrect if fs ¼ 0.1, 0.2. When fs ¼ 0.001, the numerical fluid leakage is significant. Thus, the penalty factor fs is set as0.005 in simulation. A2) Mesh density To examine the mesh-density effect, five different mesh sizes are used in numerical analysis. The fluid mesh size is varied by increasing the number of elements (lmin,f ¼ 0.001 m, 0.005 m, 0.01 m, 0.02 m, 0.05 m) in the uniformly mesh region while the structural mesh density is fixed (lmin, s ¼ 0.0015 m). Thus, numerical computations are performed for five fluid/structure mesh-size combinations. In addition, the penalty factor fs is initially set to be 0.005. And two velocities v ¼ 1.40 m/s, 2.80 m/s are also considered. The maximum pressure at central point, with a mesh density varying from 0.001 m to 0.05 m, is shown in Fig. 21. And we can see that the initial investigation leads to a qualitatively correct pressure value for five different velocities and a mesh size of lmin,f ¼ 0.005 mm and has a computational time of approximately 56 h. In addition, when lmin,f ¼ 0.05 m, the computation process is terminated due to an LS-DYNA error. Supplementary video related to this article can be found at https://doi.org/10.1016/j.tws.2019.106452

References

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