WATER ENTRY HYDROELASTICITY ANALYSIS OF LATTICE SANDWICH PANEL WITH IMPERFECTION : SIMULATION AND ENGINEERING MODEL

In the present work, the three dimensional (3D) hydroelasticity characteristics of imperfect lattice sandwich panel (ILSP) subjected to water entry via analytical prediction and numerical simulations are proposed. Firstly, numerical investigations are performed on water entry characteristics based on Arbitrary Eulerian-Lagrange (ALE) coupling method for modeling fluid-structure interaction (FSI) at an impact velocity of 5.0m/s. The results show the impact pressure on total FSI surface of ILSP is generally lower than that of the perfect lattice sandwich panel. Then a novel semi-analytical method to calculate the elastic constants of ILSP is introduced. Based on this approach, an engineering computational model is developed to predict the deformation of ILSP, in which the total deformation is separated into two parts; local field deformation and global field deformation. Good agreement between the numerical and analytical results is achieved. And the effects of geometric parameters such as the thickness of face sheet, height of ILSP and relative density of core are discussed.


Introduction
It has been revealed that slamming on ship bow section may cause serious damages.Thus, in the conceptual design of ship and offshore structures, the prediction of hydrodynamic pressure acting on an impacting body is very important.Early in 1929, Von Karman [1] firstly introduced a significant contribution on this subject.Following Karman's classic works on hull-water impact, a series of studies on the theoretical analysis of water entry problem have been reported (Wagner [2], Kapsenberg [3], Faltinsen [4][5] Morabito [6], Korobkin [7][8] and Abrate [9] et al.).Due to the complexities of this problem (such as multi-physics interaction phenomenon, free surface tracking et al), only a few special cases of wedges entry water vertically at a constant velocity can be solved analytically.So, water entry problem has been solved by some different numerical methods (e.g.Boundary Element Method (BEM) [10][11], Arbitrary Lagrange-Euler Method (ALE) [12][13], Volume of Fluid Method (VOF) [14][15][16], Smoothed Particle Hydrodynamics (SPH) [17][18], et al).
Most of previous investigations concern on the traditional ship and offshore structures such as flat plate [19], stiffened panel [20][21], V-type wedges [22][23][24], spherical ball [25][26], cylindrical projectile [26] and propeller blade wedge section [27] et al.In these studies, the common perspective of water entry phenomenon is a typical FSI problem.Obviously, the elasticity of the structures is expected to influence the results of this problem (here, which calls "hydroelasticity").Some impressive investigations concern on the role of hydroelasticity when this effect is significant.Jones may be the first person to discuss this topic [28].After that, Berezniski [29] performed an impressive review of hydroelastic behaviour for Wagnertype impact by using explicit FEM code LS-DYNA.Then, Stenius [30][31] also illustrated the hydroelastic effects are typically characterized by a relation between loading period and natural period of vibration of the structure.Based on model experiments and numerical hydroelastic analysis results of the symmetric wedges (made by composite materials), Panciroli et al. [32][33][34] showed that the hydroelastic effects are negligible when the ratio between the wetting impact time and the first wet natural period is greater than 5.However, most of these hydroelastic interaction investigations are limited in simple steel structures.
More recently, the interest of sandwich structures with various lattice core topologies has grown rapidly for their significant advantages of light weight, high specific stiffness and strength [35][36][37].The static and dynamic mechanical behaviour of sandwich structures with lattice truss core is quite different from other traditional constructions, including those with honeycomb core or corrugated core.In the most previous studies, the static mechanism and energy absorption behaviour of sandwich structures with lattice truss core subjected to air explosion, underwater explosion and fragment penetration are widely concerned [38][39][40][41][42]. Up to date, water entry characteristics of sandwich structures with lattice truss core have rarely been studied.In our previous work [43], the dynamic responses of perfect lattice sandwich panel (PLSP) under water entry are investigated.But due to the manufacturing reasons, defects in the lattices may be either in the form of vacancies, owing to missing trusses, or topological imperfections owing to displaced nodes.Thus, we shall explore theoretically and numerically the hydroelastic mechanism of imperfect lattice sandwich panel (ILSP) based on some simple cases [44].
In the present study, the hydroelasticity characteristics of imperfect lattice sandwich panel (ILSP) are further discussed.A new computational method is utilized for determination of the vibration performance of ILSP (the random moved missing and partly moved missing cases are both considered here).And these two models are validated by comparing against vibration numerical analysis.Furthermore, in order to calculate the global deflection of partly moved missing ILSP, a novel computational method is derived by adopting the simple support boundary assumption for missing area [44].Results from this proposed model agree well with those from the 3D FSI analysis.Then, the geometric parameters effects of ILSP such as the thickness of face sheet, the relative density of core, the height of ILSP core et al. are also examined.The proposed model may be considered as initial point for modeling the design of the lattice sandwich structures subjected to water entry.

Descriptions of Imperfection Lattice Sandwich (ILSP)
In the practical cases, the imperfection of manufacture widely exists.Generally, there are three types imperfection in manufacture process including the core truss missing, initial wave less buckling of core truss, jointing invalidation between trusses and face sheet.And the core truss missing may be the most important.Thus, this type imperfection is considered here.Based on our previous studies [43][44], when the non-dimensional criteria 2τsfs>4 (here, 2τs is the duration of water entry loading, fs is the natural vibration frequency of structure including added mass effect) the hydroelastic effect of lattice sandwich panel is not significant.In the present paper, both 2τsfs>4 and 2τsfs<4 are considered.Some other researchers also give similar criteria to analysis the hydroelasticity of simple structures.Considering the added mass of the first vibration modal of structure for a v-shaped ship section, a similar definition given by Stenius et al. [30][31] is the ratio TLP /TNP.Where TLP is the loading period based on the Wagner theory and TNP is the first wet natural period of vibration.It is concluded that the hydroelastic effects are negligible and the response can be set as quasi-static for (TLP /TNP)> 4.
Here, by referring a practical monolithic flat panel of cruiser [43], the ILSP with an exposed area 595.1 mm × 270.5 mm is considered with 2τsfs~5.Besides, the ILSP with an exposed area 1200.0 mm×1000.0cm referred a stiffened plate of bulk carrier bottom structures is proposed with 2τsfs~1.ILSP consists of two thin face sheets attached to both sides of a light weight lattice core with the total mass of perfect lattice core sandwich plate Ms=67.6kg,see [43][44]), and the geometric description of ILSP is plotted in detail in Fig. 1.Here, the geometric parameters of referenced case C1 are φ=π/4, dc1=dc2=dc=54.100mm,tf=4.219mm,tb=4.219mm,tc=6.000mm,Hc=38.250mm(for case C1, the lattice sandwich panel is perfect, please see Table 1).Two missing imperfection types are considered in this study; the first type is randomly moved missing imperfection, and the second type is partly moved imperfection.And the missing core cell (here it means the reduction in density) percentage varies from 0.45% to 16.36% (the cases C2~C11, please also see Table 1).The boundary condition of panel can be taken as clamped.Therefore, a sealed off round structure made by rigid materials (rigid out wall, see Fig. 1) is used to model clamped boundary condition.Particularly, for each random moved missing cases C2(R) ~ C11(R), the five random types are considered here.So, there are totally 50 random moved missing cases.For partly moved missing cases C2(P) ~ C11(P), to simplify the analysis, only the central partly moved missing is considered (see Fig. 1).
2.1.22τsfs~1.0(the hydroelastic effect is significant) In this case, the parameters of C1' are φ=49.88○ , dc1=dc2=dc=100.00mm,tf=1.000mm,tb=1.000mm,tc=6.500mm,Hc=90.000mm(for case C1', the lattice sandwich panel is perfect, please see Table 2).Two missing imperfection types are also considered in this study; the first type is randomly moved missing imperfection, and the second type is partly moved imperfection.The missing core cell (here it means the reduction in density) percentage varies from 0.45% to 16.36% (the cases C2'~C11', please see Table 2).The other conditions are same as 2τsfs~5.As the complexity of PM-ILSP, only the elastic constants of RM-ILSP are considered here.Due to the discrete stochastic characteristics, the homogenization method is adopted.According to the concept of continue damage mechanics (CDM), the elastic constants of RM-ILSP (E*, G*) are given as follows.

   
where f (φ) is a dimensionless function of φ (the meaning of notation φ sees Fig. 1), ρ0 is the relative density of RM-ILSP and k is a dimensionless constant (if η=0.00% means the perfect case).
Here, the dimensionless function f (φ) and the relative density ρ0 can be referred in literature, where the meanings of tc and dc can be seen in Fig. 1, l is the length of core truss, and ρs is of base solid material for ILSP.By combining Eqs. ( 1)-( 4) and neglecting the high order terms in the Taylor expansions, the expression of E* and G* can be written as follows, In the previous study, Wallach and Gibson [45] has given an observation that Young' modulus of lattice materials decreases linearly with increasing fraction of missing trusses, with a roughly 16.6% decrease in modulus for every 10% reduction in relative density.Thus, we assume that the value of k is set to be 0.66 here.If the equivalent elastic constants E* and G* are calculated, the response of RM-ILSP is easy to obtained based on our previous study [43][44] which is a classic structural dynamic problem in linear region.In this section, the boundary condition of partly moved ILSP (PM-ILSP) is analyzed.If the missing percentage of core truss is relative large, the above method will not be suitable.Thus, the local boundary effect must be considered.Based on the previous study [43][44], it shows that the core truss can be set as supported by rows of equidistant trusses or simply supported edges.To simplify the analysis, the shape of partly moved area is set to be rectangular, making the calculation of the vibration and static behavior of local partly moved area much easier.

Vibration Verification Analysis of ILSP
If the equivalent elastic properties of random moved ILSP are obtained from Eqs. ( 1)-( 5), the fundamental bending vibration frequencies of RM-ILSP are calculated by three steps [43] as follows.The first and second step is to calculate the equivalent shear (bending) rigidity of ILSP lattice core.And the last step is to calculate the natural frequency of RM-ILSP.In this key step, an approximate method [43][44][46][47] is adopted to analyze the fundamental vibration frequency.And the detailed expressions of the formulations are shown as follows.
Step 1: calculate the equivalent shear rigidity of RM-ILSP lattice core [43]   / 2 where C * G is the transverse shear rigidity matrix of ILSP core (the detailed expressions of the homogenized module of lattice core Cαβ *H (α, β=4, 5) can be found in literature [43]).
Step 2: calculate the bending rigidity ILSP lattice core [43]  Here, for the thin face sheets of ILSP, only bending deformation is considered (the bending stiffness Df of face sheets is included).
Step 3: calculate the natural frequency of ILSP [43].The third step is to calculate the natural frequency of ILSP.In the present investigation, an approximate method is adopted to analyze the fundamental vibration frequency of the ILSP.And the detailed steps of solving approximate equations can be found in our previous study [43] and classic literature [47].The vibration results for these two imperfection types are listed in Table 3.Compared with the FEM results in Table 3.a, the approximate analytical method can give a good prediction when the missing ratio is between 0.45%~16.36%,where the maximum error is less than 9.0%.For partly moved missing type, the vibration characteristics are complex (Table 3).The FEM results illustrated in Table .3.b show that the prediction method (Eqs.( 1)-( 7)) cannot reflect the vibration characteristics when the missing ratio is greater than 1.8%.And this is mainly caused by the local vibration mode due to the partly moved area.The whole vibration mode seems a mixture of local-global vibration interaction (please see Fig. 2).In practical, due to the complexity of boundary condition, the precision of analytical model is very difficult to achieve.As an initial estimation, the local vibration mode of missing area can be firstly calculated based on the model of section 3.1.2.In this section, a novel engineering model is performed to investigate the deformation characteristics of ILSP when the structural deformation is in linear region as shown in Fig. 3.The total deformation field can be divided into two components:

Engineering Prediction Model of Structural Deflection and Discussion
(1) The global bending deformation field of ILSP (wG).
(2) The local bending deformation field of each cell (wL).
where K1 and K2 are dynamic coefficient for total and local deformation respectively, when the dynamic effect is considered.And the calculation of dynamic deflection in linear elastic range under uniform impact pressure is given by following equation, where K (t) is the dynamic inertial effect factor function varied with time, Substitute Eq. ( 9) into Eq.( 8), it has, where ωG is the first circular frequency of global ILSP, ωL is the first circular frequency of ILSP local cell.ωG can be calculated from Eqs. ( 1)- (7).The method of calculation for ωG can be referred in [43].
In the water entry problem, generally, the compressibility effects can be divided as two parts.The first part of compressibility effects is the compressibility of water, while the second compressibility effect is the compressibility of air respectively.Some previous authors have included the effects of compressibility on the flow during impact [48][49][50].The water entry bodies can be divided as three types, which are plate body, blunt body and pointed body.For flat or nearly flat bodies, the hydroelasticity of plate and the compressibility of air cushion should be included.For blunt body, the compressibility of liquid should be included in the first instants as the flow is accelerated from rest.After the fluid particles have reached the velocities associated with the subsonic stage of water entry compressibility can be ignored.As pointed out by , there are many difficulties to describe and solve the water entry mathematical model, such as the multiphase, the fluid-solid interaction and the nonlinearity et al.One of these key difficulties is the compressibility of air cushion.To avoid this difficulty, based on the assumption the water impact pressure pmax is the peak impulsive pressure for FSI surface which can be calculated in the following, where p∞ is atmosphere pressure, A and B are also the empirical parameters.If the logarithm is used to the Eq. ( 10) left and right, we can obtain, We see that the logarithm value of (pmax-p∞) is linear with the logarithm value of water entry velocity v.To calculate the water impact pmax, the two unknown parameters in the Eq. ( 12) are A and B. And the parameters A and B can be determined from a least square linear fit based on a few numerical simulations or experiments.The values of A and B are constants though the range of water entry pressure varies widely.It must be pointed out that the coefficients A, B in Eq. ( 12) are different for the average (whole) FSI pressure and the central FSI pressure, respectively.For example, from our previous study [43], the performance of central FSI pressure for perfect lattice sandwich panels follows a linear trend, as determined from a least square linear fit,   max, ln 4.05 1.44 ln By means of the least square linear fit, the average (whole) FSI pressure for perfect lattice sandwich panels is also determined, The dynamic coefficient K is only associated with the fundamental circular frequency ω and the impact duration time 2τs.For the average (whole) FSI pressure, a simplified expression to define the impulsive shape pG(τ) in Eq. (10) as following [43], where C0 is the sound speed of air (C0= 340 m/s).
Similarly, for local FSI pressure, it has, The last two unknown parameters wG and wL can be calculated from following,   1 where C * G is the equivalent shear stiffness of lattice core, D * 1 are the bending stiffness of ILSP, wL is the local maximum deformation, pmax,L is the maximum value of water entry pressure loading on central cells, dc is the width of cell (see Fig. 1) and Df is the flexural rigidity of cell face sheet.And the other nomenclatures have the same meanings as mentioned in the previous study.
If the detailed expression of ωG, ωL, wG, wL, pmax, p(t) and 2τs are obtained from Eqs. ( 1)-( 25), the value of dynamic coefficient K can be easily obtained.And the maximum value of K(t) at t = t0 is commonly considered, which satisfies the following conditions [43],  The top (bottom) face sheet and corrugated core are modeled as a plane plate using 52000 quadrilateral shell elements(shell 163, KEYHOFF formulation, hourglass control, five degrees of freedom per node ux, uy, uz, θx, θy, finite membrane strains elements, with 5 integration points).The detailed Lagrange FE model of ILSP can be seen in Fig. 4. In the simulation, the Lagrange material of the ILSP is modeled to be S304 steel.In order to consider the strain rate effect, Cowper-Symonds model is adopted.And the material property constants can be referred in [43][44].
To model the fluid inside and outside the ILSP, two Euler domains are used.The outer domain has the ILSP surface (including top face sheet, bottom face sheet and out-off rigid wall) as part of the fluid boundary, Euler material is outside the ILSP surface and there is no material inside the ILSP surface.The contents inside the ILSP are modeled in the inner domain and this domain is also enclosed by the ILSP out surface.Therefore both Euler domains use the ILSP surface as part of their enclosure.Here, the outer boundary of the outer domain is given by a sufficiently large fixed box.Here, pressure at the water Euler domain is set to the hydrostatic pressure by using the *INITIAL_HYDROSTATIC_ALE keyword cards.The Euler mesh contains the water and the air on the top of the water.

水 Water
The fluid mesh used for this problem are consisted of a block of elements, with the dimensions are 2.5m*2.5m*4.0m.And this fluid block of water and air are meshed with 100*100*90 hexahedron elements, for a total of 900000 fluid elements (Fig. 4).The finest gird size of Euler domain is 0.005 m in this simulation.All the boundary conditions for out fluid mesh are given a "flow" boundary condition by adopting non-reflecting boundary keyword card.In order to model fresh water, a polynomial equation of state was conducted.This state equation (EOS) of fresh water relates the pressure in the fluid to the acoustic condensation μ and the specific internal energy by: where μ= (ρ-ρw )/ρw , ρw is the initial density of fresh water, Ew is the specific internal energy per unit mass, and a1 , a2 , a3 , b0, b1 are constants for the fluid.And the upper equation applies to a fluid in a compressed state, while the lower applies to a fluid in an expanded state.The constants for this equation for fresh water are provided in literature [43][44].
The gamma law gas model is adopted for the EOS of air,   where ρa is the density of air, γ is the heat capacities of the gas and Ea is the specific internal energy of air.The initial pressure of air is set to 1.0×10 5 Pa.In this analysis, the Arbitrary Lagrange-Euler (ALE) Coupling algorithm is used.The Lagrangian and Eulerian meshes are geometrically independent, and interact via coupling surface.The coupling surface 'cuts across' Eulerian elements which contain multi-material air and water, changing their volume and surface areas.As the finite element mesh deforms under the action of the impact pressure from the Eulerian mesh, the resulting FEM deflection then influences subsequent material flow and pressure forces in the Eulerian mesh, resulting in automatic and precise coupling of FSI.In general, it is known that the results based on ALE algorithm are sensitive to the Euler mesh density.So the Euler mesh needs to be fine enough to capture the highest gradients in the pressure fields, yet a coarser mesh is favorable in terms of computational cost.In addition, the selection of the contact stiffness in penalty is based on contact algorithm, it is required the maximum pressures are approximately known ahead.The non-physical contact penetration can be controlled.To verify the reliability of the developed finite element model, compared with experimental results, the water entry analyses of a circular monolithic plate are illustrated in our previous study [19,43].

Basic Water Entry Mechanism
As an illustration, the case of C1' is analyzed here.The entire water entry process can be divided into three stages as follows.
First Stage; Structure begins to drop → Air compression begins; Second Stage; Air compression begins → Structure begins to contact water; Third Stage; Structure begins to contact water → Structure immerses in water.
In this case, the duration predicted by numerical simulation 2τs=11.5 ms coincides with the empirical estimations (2τs=10.7ms)from Eqs. ( 11)- (12).The contacting water time t observed by the numerical simulation is ~19.0ms.The result from the simulation shows that when t >25.0ms, the water jet begins to split into two parts as plotted Fig. 7. Furthermore, the shape of the water entry jet calculated by the ALE coupling algorithm cannot capture the spatter of the droplets.From the authors' view, this phenomenon is probably caused by the instability of fluid flow.Some small perturbations may be from some uncontrollable practical factors, such as the minor asymmetries in the simulation.From the simulation results, it is obtained that the global air cushion is significant from Fig. 5 (please see the black dash line).Due to the global elastic deformation, the shape of this air cushion is not rectangular.And the fluid materials volume distribution isosurface also shows the local air cushion exists during the initial water entry duration (Fig. 6).This local cushion is caused by lattice cell local deformation which has significant effects on the water entry pressure and dynamic responses of ILSP.Because of the existence of this local air cushion, which plays a role as a buffer device, the peak value of average FSI pressure of contacted-water wet face sheet is lower than that of monolithic plate which has been certificated in our previous study [43][44].
As the effects of local air cushion, the local hydroelasticity is important in the dynamic responses of ILSP.Due to the local loading, the high frequency responses of pressure are also induced as shown later in the pressure-time history curve (please see in Fig. 8).And this local high frequency response is more significant if the loss imperfection is more seriously (please see in Fig. 9).So, the whole response of ILSP is the combination of the multiple global deformations, the lattice cell local deformations, the air cushion and the free surface interaction.Furthermore, some other strong nonlinear phenomenon such as flow turbulence and multiphase flow mixture et al. phenomenon may be also exists which cannot captured by present analytical solution and numerical simulation.And it must be studied by future more finely experimental measurement of micro-scale flow structures in FSI surface.

Impact
Pressure Results for 2τsfs~5 (the hydroelastic effect is not significant) The relation between core truss missing ratio and water entry pressure is shown in Fig. 7 for RM-ILSP and PM-ILSP when water entry velocity is 5m/s.According to the results as plotted in Fig. 7, by means of the least square fit, the following observations can be drawn.

For RM-ILSP:
   The local FSI pressure at the center point increases slowly (<4%) when the missing ration varies from 0.45% to 16.36%.But the water entry pressure of total FSI surface for RM-ILSP decreases linearly with fraction of missing trusses, with a roughly 4.5% (from 212KPa to 195KPa) respectively in pressure for every 10% reduction in relative density.This is mainly because the core of RM-ILSP becomes softer when E*, G* are smaller.Compared with the Eq. ( 29) and Eq.(31), it shows that the changing trend of missing ratio is different between RM-ILSP and PM-ILSP.This phenomenon reveals that the random missing truss could change the space distribution of water entry pressure.And the local area of RM-ILSP may be becoming 'harder'.This conclusion also certificated from Fig. 7 (a) which shows the discrete pressure value of central point.And the maximum difference between the lower point and upper point value is about 80KPa (~15%).This gives a conclusion that the core topology structure is very important in the FSI.The least square fit data also illustrate that the decrease changing trend of PM-ILSP is faster than that of RM-ILSP.Compared with Fig. 8 (a) and Fig. 9 (a), the pressure-time curve shows that the oscillation amplitude of PM-ILSP is greater in the whole water entry process.The same conclusion of total FSI pressure also can be summarized from Fig. 8 For PM-ILSP: When the hydroelastic effect is significant, the numerical simulation results reveal that the random missing imperfection effect on the dynamic response of RM-ILSP is complex as shown in Fig. 10 (a).When the missing percentage η is less than 5%, the local FSI water entry pressure seems rise up.But when η>5%, the local FSI pressure is becoming lower while this oscillation amplitude is minor.The same phenomenon also can be found in the response of PM-ILSP.According to the numerical results in Fig. 10 (b), the changing trend is decreasing about 9.8% for whole FSI surface when η=16%.Compared with Figs. 10 (a) and (b) (here, also compared with Eqs. ( 34) and ( 36)), an approximate linear decreasing relation between missing rate and pressure value can be given.In Fig. 11 and Fig. 12, the water entry pressure time history curves are given for local coordinate and whole surface.Compared the water entry curves detailed, the random missing imperfection also has little effect on dynamic response.But the effect of local partly missing imperfection is more significant (Fig. 12).Compared with Fig. 9 and Fig. 12, both for local central point pressure and total FSI surface pressure, the smoothness of pressure curve is more significant when 2τsfs~1.0.This may the high frequency pressure response components would be excited by smaller core cell.

2τsfs~5 (the hydroelastic effect is not significant)
For RM-ILSP, compared with FSI simulation results, with water entry speed v=5m/s, the maximum deformations by using above FE analysis model are illustrated in Fig. 13.It is clear that the changing trend of simulation result is consistent well with theoretical predictions when the reduction of relative density is lower than 10~11%.But the error of this analytical method is greater than 15% when the missing percentage is greater than 12%.This is mainly because the precision of first homogenization approximation in Eq. ( 4) is not precise enough when the reduction of relative density becomes sufficiently high.Another reason is probably the local deflection of imperfection cell.For PM-ILSP, a detailed comparison analysis in Fig. 13 shows that the maximum deflection has a rapid increase when η=9-10%.Thus, this result indicates that there exists a critical missing ratio point.As shown in Fig. 14, the critical deformation point also exits when hydroelastic effect is significant both for RM-ILSP and PM-ILSP.For RM-ILSP, the critical point is nearly equal to 12%.But for PM-ILSP, the critical point is nearly equal to 2%.The computational results reveal that the maximum deformation is increasing very fast (~300%) for PM-ILSP when the missing percentage is about 1%.This means the maximum deformation is insensitive with imperfection ratio.Compared with Fig. 13 and Fig. 14, the different critical deformation point exists when hydroelasticity indicator 2τsfs is different.For RM-ILSP cases, the critical deformation point of 2τsfs~1.0 (about 12%) is very close to that of 2τsfs~5.0 (about 10~11%).But for PM-ILSP cases, the critical point between 2τsfs~1.0 (about 2%) and 2τsfs~5.0(about 10~11%) is very variable.This may the effects of different missing trusses region.

Geometric Parameters Effects Study
When the hydroelastic effect is not significant (2τsfs~5), the effects of parameters can be referred our previous investigation [44].Thus, only 2τsfs~1 is considered here.Both for RM-ILSP and PM-ILSP, three parameters including the face sheet thickness tb (tf), the height Hc and the section cross area tc*tc are considered here.

The face sheet thickness tb (tf)
For RM-ILSP, as plotted in Fig. 15, when the top face sheet thickness tf varies from 0.5mm to 5.5mm, the central FSI pressure Pc rises about 11% even η=12.50%.Furthermore, when tf is greater than 0.5mm, the Pc is nearly about 550KPa.The whole FSI surface pressure Pw rises about 12% 0.5 tf<1.5mm.But when tf>1.5mm, the Pw~370KPa.Moreover, when tf changes, the characteristic of maximum deformation is given in Fig. 16.When tf varies from 0.5mm to 5.5mm, the deformation value becomes lower dramatically as the bending stiffness increases fast.Particularly, this trend is not significant when tf >2.5mm.Compared with tf, the effect of tb is significant (Fig. 17).As plotted in Fig. 17, Pc increases fast when tf <2.5mm while this trend is becoming slow if tf >2.5mm.Thus, the maximum deformation decrease fast when tf <2.5m (Fig. 18).The effects of face sheet thickness tb (tf) for perfect lattice sandwich panel has been investigated by many previous scholars in other field such as bending, vibration and blast et al [38][39][40][41], some similar conclusions have been given.
For PM-ILSP, if η=12.50%, the central FSI pressure rises Pc about 3% when the top face sheet thickness tf varies from 0.5mm to 5.5mm, while the FSI pressure Pc increases 5%.That means the change of tf has little effect on the water entry pressure.And this trend also exits in the analysis of deformation.This is mainly because the majority of deformation is local deformation for water contacted surface.Though the bending stiffness increases fast, the global deformation has little effect on whole deformation.Compared with Fig. 16 and Fig. 19, the effect of tf is more significant for RM-ILSP cases.We think this may the average of random missing plays a key role in the response which makes ILSP core softer.While the central effect of local missing is more important for PM-ILSP cases.For RM-ILSP, Hc has significant effect on the dynamic behavior.As shown in Fig. 20, the water entry FSI pressure is decreasing about 40% when Hc increases from 70.00 to 110.00mm.And the changes for the deformation of ILSP is dramatically (Fig. 21) from 50.0 mm to 5.0 mm.But this trend becomes slow when Hc>90.0 mm.This can be interpreted as the local buckling phenomenon occurs when the Hc is relative high.Compared with Fig. 20(a) and Fig. 20(b), the decreasing ratio of central pressure value and total FSI pressure value is nearly the same (~30%).The minimum deflection critical point in Fig. 21 shows that the optimization is needed in the parameter design of ILSP.Moreover, it shows that the effects of Hc on the changing trend of water entry pressure is more dramatically than that of tb(tf).And the conclusion is also suited for the effects of defection characteristics.
For PM-ILSP, the changing trend of dynamic behavior curve is similar with RM-ILSP.When 20.0 mm<Hc<40.0mm, the water entry pressure decreasing trend is more dramatically (nearly ~50%) as shown in Fig. 22.Furthermore, the changing trend seems more complex when 40.0 mm<Hc<60.0mm.We think the basic reason for this phenomenon is may be the changing of pressure spatial distribution.For the characteristics deformation of PM-ILSP, we find it decreases dramatically when Hc<90.0mm (Fig. 23).This may be caused by two reasons.The first is the deceasing of water entry pressure; the second is the increasing of bending stiffness.Compared with Fig. 21 and Fig. 23, it seems minimum deflection critical point is not existed for PM-ILSP.For random moved ILSP, the last parameter is the section cross area tc*tc.According to the numerical simulation results, the changing characteristic is rather different when tc increases from 5.0mm to 7.0mm.As shown in Fig. 24, the central water entry pressure Pc is decreasing fast when tc<6.0mm.But the changing trend is becoming increasing when tc>6.0mm.This means the water entry pressure also has a critical point when tc changes.This may be caused by the non-uniform distribution of water entry pressure due to local stiffened truss.For global FSI pressure Pw, the changing trend of is increasing fast when tc>6.5mm.This increasing trend is slow when tc<4.5mm.Compared in Fig. 24, the changing trend between Pc and Pw is different.As we known, the pressure Pc is a local indicator of pressure distribution; while the pressure Pw is a global indicator of pressure distribution.The structural deformation of ILSP is also sensitive when tc is changing.As shown in Fig. 25, the maximum deformation is decreasing fast when tc changes from 2.5mm to 6.5mm.From the numerical simulation, the local truss buckling occurs when tc is about equal to 2.5mm.For PM-ILSP, compared with Fig. 24 and 26, the changing trend of water entry pressure for central point is different with that of RM-ILSP.For the central pressure Pc, the water entry pressure is decreasing slowly from 650KPa to 540KPa.But the Pw is increasing about 30KPa (Fig. 26).Compared with the effect of Hc, it shows that the effect of tc*tc is minor.Thus, in the parameter design of ILSP, the section area can be seen as a secondary consideration.As plotted in Fig. 27, the changing trend of deformation curve also concludes that the tc has effect on the dynamic response of PM-ILSP, which is similar with that of RM-ILSP.The central deformation of PM-ILSP is decreasing about 10% when tc is increasing from 5.0mm to 7.0mm.

Conclusion
In this study, the hydroelastic behavior of lattice sandwich panels with imperfection (ILSP) subjected to water entry is investigated both analytically and numerically.A novel engineering estimation method is proposed to calculate the dynamic response characteristics of ILSP.The FSI numerical model is presented to obtain the detailed water entry behavior for ILSP.To identify the effect of missing ratio on the water entry characteristics, various imperfection scenarios are simulated.Finally, the effects of key parameters such as the face sheet thickness, core height and section of core truss are discussed.This study provides an engineering method for water entry problem of sandwich panels how hydroelastic pressure and maximum structural deformation are affected by the missing imperfection during the manufacture.And the following conclusions could be drawn from the semi-analytical method and numerical simulation results.
(1) When the hydroelastic effect is not significant (2τsfs~5), the random missing imperfection has little effect on water entry pressure of local central point if the missing percentage varies from 0.45%-16.36%;though a more significant effect on the whole water entry pressure.When the core of ILSP becomes softer, the water entry pressure of central point will decrease by about 10%.
(2) When the hydroelastic effect is significant (2τsfs~1), the random missing imperfection also has little effect on water entry pressure of local central point if the missing ration is varied from 0.45%-16.36%.With the increasing of missing percentages, the water entry pressure decreases.An oscillation phenomenon is identified in the water entry pressure curve.And the water entry pressure of central point decreases about 6.4% when the missing percentage is 16.36%.
(3) The results obtained by engineering estimation model agree well with that of numerical simulation when the imperfection type is random missing and the missing percentage is lower than 12%.But if the missing percentage is greater than 12%, the error becomes larger.And this is mainly caused by the first order approximation in the assumption.
(4) A very interesting phenomenon is that a critical missing percentage exists both for random missing and partly moved imperfection types.If the missing percentage is greater than the critical one, the maximum deformation increases dramatically.Thus, in the engineering practice, the missing percentage must be limited in the critical region which must be emphasized.
(5) The parametric study results show that the thickness of water contacted face sheet, the core height and the section of core truss has significant effect on the water entry response characteristics.If the core truss buckling occurs in the water entry process, this effect is more significant.On the contrary, the top face sheet has neglected effect on the response of ILSP.

Fig. 2
Fig. 2 Vibration of partly moved imperfect lattice sandwich plate

Fig. 3
Fig. 3 Deformation characteristics of lattice sandwich plate

Fig. 6
Fig.5The basic water entry process of ILSP Relation between missing percentage and water entry pressure for RM-ILSP (left: central, right: whole FSI surface) Relation between missing percentage and water entry pressure for PM-ILSP Fig.7Relation between missing percentage and water entry pressure (2τ s f s ~5)

Fig. 8 Fig. 9
Fig. 8 Comparison of perfect and imperfect water entry pressure curves for RM-ILSP (left: central, right: whole FSI surface) (2τ s f s ~5) (b) and Fig. 9 (b).5.2.3 Impact Pressure Results for 2τsfs~1.0(the hydroelastic effect is significant) Relation between missing percentage and water entry pressure for RM-ILSP (left: central, right: whole FSI surface) Missing(%) (b) Relation between missing percentage and water entry pressure for PM-ILSP (left: central, right: whole FSI surface) Fig. 10 Relation between missing percentage and water entry pressure (2τ s f s ~1)For RM-ILSP:

Fig. 15 Fig. 16 Fig. 17
Fig. 15 Effect of t f on water entry pressure (left: P c , right: P w ), RM-ILSP

Fig. 24
Fig. 24 Effect of t c on water entry pressure (left: P c , right: P w ) , RM-ILSP

Fig. 25 Fig. 26
Fig. 25 Effect of t c on maximum deformation-Random missing

Fig. 27
Fig. 27 Effect of t c on maximum deformation-Partly moved missing

Table 2
Computational cases when the hydroelastic effect is significant

Table 3
Vibration analysis of imperfect lattice sandwich plate