J. Shanghai Jiaotong Univ. (Sci.), 2014, 19(3): 346-353 DOI: 10.1007/s12204-014-1508-4 Modified Numerical Model for Simulating Fluid-Filled Structure Response to Underwater Explosion with Cavitation XIAO Wei1 (肖 巍), ZHANG A-man1∗ (张阿漫), WANG Yu2 (汪 玉) (1. College of Shipbuilding Engineering, Harbin Engineering University, Harbin 150001, China; 2. Unit 92857 of PLA, Beijing 100161, China) © Shanghai Jiaotong University and Springer-Verlag Berlin Heidelberg 2014 Abstract: In this paper, a numerical method is established to analyze the response of fluid-filled structure to underwater explosion with cavitation and the validation of the method is illustrated. In the present implementa- tion, the second-order doubly asymptotic approximation (DAA2) other than curved wave approximation (CWA) is used to simulate non-reflecting boundary. Based on the method, the difference between DAA2 non-reflecting boundary and CWA non-reflecting boundary is investigated; then, the influence of internal fluid volume and the influence of cavitation on dynamic response of spherical shell are analyzed. Compared with CWA non-reflecting boundary, DAA2 non-reflecting boundary treats added mass effects better. When the internal fluid is full, the displacement and velocity of spherical shell decrease, but, when the internal fluid is half, the displacement and velocity of spherical shell increase. The effect of cavitation is more obvious at the trailing point than at the leading point of spherical shell. Key words: spectral element, finite element, doubly asymptotic approximation (DAA), fluid structure interac- tion, fluid filled structure CLC number: U 661, O 383 Document code: A 0 Introduction Fluid filled structure widely exists in ship and ocean engineering field, such as double cylindrical shell of submarine, and double bottom of vehicle. A large number of scholars have investigated the dynamic re- sponse of structure without internal fluid to under- water explosion[1-5], but there are few researches on the dynamic response of fluid filled structure. Geers and Zhang[6-7] developed doubly asymptotic approxi- mation (DAA) for submerged structure with internal fluid through the method of operator matching pre- viously used for external acoustic domains. Sprague and Geers[8] solved analytically the dynamic response of submerged spherical shell to planar and spherical exponentially-decaying wave. In addition, when the underwater shock wave propagates to the free sur- face or the structure surface, it gets reflected. The reflected rarefaction wave causes the pressure in the water to fall below its vapor pressure, and then cav- itation occurs. The generation of cavitation makes fluid-filled structure separated from fluid; the collapse Received date: 2013-06-06 Foundation item: the National Security Major Basic Re- search Program of China (No. 613157), the Outstanding Youth Fund of China (No. 51222904), and the National Natural Science Foundation of China (No. 51279038) ∗E-mail:
[email protected] of cavitation results in fluid striking fluid-filled struc- ture again, and then reloading phenomenon occurs[9]. So it is necessary to consider the effect of cavita- tion. Boundary element method and finite element method are the main tools to analyze the dynamic response of structure to underwater explosion. DAA developed by Geers[10-11] is the most widely used boundary ele- ment method. The advantage of DAA lies in saving the computation time through transferring the volume integration to the surface integration, but the DAA cannot simulate the cavitation occurring in the fluid domain. To simulate the cavitation effect, Felippa and Deruntz[12] developed a cavitating acoustic finite ele- ment (CAFE), in which trilinear, isoparametric, eight- node brick elements are used to discretize the fluid do- main. However, CAFE exhibits high numerical disper- sion during the propagation of wave, due to using of low-order basis functions. According to the disadvan- tage of CAFE, Sprague and Geers[13] developed spectral element method (SEM) which can simulate the prop- agation of wave well. SEM uses the high-order basis function based on Legendre polynomial to discretize the fluid domain. Compared with the traditional displace- ment vector description, SEM uses scalar displacement potential to describe the fluid node variable, greatly reduces the number of fluid node and improves the computational efficiency. J. Shanghai Jiaotong Univ. (Sci.), 2014, 19(3): 346-353 347 In finite element analysis, an infinite fluid domain is simulated as a finite fluid domain with non-reflecting boundary. Curved wave approximation (CWA) is the mainly non-reflecting boundary used in finite element method[14-15], but CWA does not consider the added mass effect. The added mass effect can only be simu- lated by the fluid near fluid structure interaction (FSI) surface. In Ref. [12], the first-order doubly asymp- totic approximation (DAA1) is adopted as the non- reflecting boundary in CAFE, but the accuracy of DAA1 is not satisfactory, due to overestimating the fluid damping[11]. The second-order doubly asymptotic approximation (DAA2) can treat the added mass effect well and its accuracy is satisfactory. Consequently, in this paper, the advanced SEM ob- tained through adopting the DAA2 to replace the CWA as the non-reflecting boundary based on the SEM, is used to simulate the external and internal fluid do- mains. On the other hand, the fluid filled structure is discretized with the nonlinear finite element software ABAQUS. 1 Numerical Model The fluid structure interaction numerical model is es- tablished by developing the ABAQUS user subroutine interface to combine the fluid spectral element method with the structure finite element method, and then the validation of the model is illustrated. The model can be used to analyze the response of fluid-filled struc- ture to underwater explosion with cavitation. Based on the model, the difference between DAA2 non-reflecting boundary and CWA non-reflecting boundary is investi- gated, and then, the influence of internal fluid volume and the influence of cavitation on dynamic response of spherical shell are analyzed. Figure 1 shows the schematic diagram of the fluid structure interaction nu- merical model. Internal fluid volume Structure External fluid volume Non-reflecting boundary Displacement Pressure Displacement PressureDisplacement Pressure Spectral element method Spectral element method Finite element method Doubly asymptotic approximation Fig. 1 Schematic diagram of the fluid structure interaction numerical model 1.1 Fluid Discrete Equation In an underwater explosion problem, the total pres- sure field can be expressed as the sum of equilibrium field, incident field and scattered field[16]. The bilinear model is adopted to deal with the cavitation phenom- ena in this paper. The bilinear model of scattered field can be written as psc = { c2ssc, c 2ssc � −(peq + pinc) − (peq + pinc), c2ssc < −(peq + pinc) , (1) where, c is the sound speed in fluid; peq is the equilib- rium pressure; pinc is the incident pressure; psc is the scattered pressure; ssc is the scattered dynamic conden- sation. According to the continuity of equation, the motion equation and the bilinear equation, the fluid governing equation of scattered field represented by scalar dis- placement potential with cavitation can be expressed as [17] ssc = ∇2ψsc, (2) ψ̈sc = { c2ssc, c 2ssc � −(peq + pinc) − (peq + pinc), c2ssc < −(peq + pinc) , where ψsc is the scattered displacement potential. Discretizing the fluid governing Eq. (2) with the stan- dard Galerkin approach and using Green’s first formula, we can obtain the fluid discrete equation[18], Qscsc +Hψ c sc = b, (3) where, scsc is the column vector of the scattered dynamic condensation; ψcsc is the column vector of the scattered displacement potential; Q and H are coefficient matri- ces, and b is coefficient vector. These coefficient matri- ces can be written as Q = ∫ Ω φφTdΩ H = ∫ Ω ∇φ∇φTdΩ b = ∫ Γ (φ∇ψcsc)ndΓ ⎫⎪⎪⎪⎪⎪⎪⎬ ⎪⎪⎪⎪⎪⎪⎭ , (4) where, φ is the high order spectral element basis func- tion column vector based on Legendre polynomial; Ω is the fluid domain; Γ is the surface of fluid domain; n is the outward normal vector to Γ . Q, H and b can be solved by Gauss-Lobatto-Legendre (GLL) integration. The fluid boundaries include free surface boundary, non-reflecting boundary and fluid structure interaction boundary. The total pressure on free surface boundary 348 J. Shanghai Jiaotong Univ. (Sci.), 2014, 19(3): 346-353 is also equal to the atmospheric pressure, expressed as pinc|fs + psc|fs = 0. (5) Hence, the boundary integration can be rewritten as b = ∫ Γ (φ∇ψcsc)ndΓ =∫ Γnrb φφTucscdΓ + ∫ Γfsi φφTucscdΓ, (6) where, Γnrb is the non-reflecting boundary surface; Γfsi is the fluid structure interaction surface; ucsc is the col- umn vector of scattered normal displacement of fluid. The scattered normal displacement on non-reflecting boundary can be obtained by the non-reflecting bound- ary equation. Based on the continuity at the fluid structure inter- face, for submerged structure without internal fluid do- main, the scattered normal displacement on fluid struc- ture interaction boundary is given by ucsc = G Txn, (7) where, G is the transformation matrix relating the structural and fluid nodes; xn, the column vector of normal displacement of structure, is obtained from the structure discrete equation. For submerged structure with internal fluid domain, the scattered normal dis- placement on fluid structure interaction boundary is given by uesc = G Txn − ucinc uisc = −GTxn } , (8) where uesc is the column vector of scattered normal dis- placement at interface between structure and external fluid, uisc is the column vector of scattered normal dis- placement at interface between structure and internal fluid, and ucinc is the column vector of incident normal displacement at interface between structure and exter- nal fluid. For spherical wave, the incident normal displace- ment at interface between structure and external fluid is given by uinc = ( 1 ρ0c ∗ pinc + 1 ρ0l ∗∗ p inc ) ξ · n, (9) where, ρ0 is the density of saturated fluid; l is the dis- tance between non-reflecting boundary point and inci- dent wave center; ξ is the unit vector of incident wave propagation direction; an asterisk denotes a temporal integration. For planar wave, the incident normal dis- placement at interface between structure and external fluid is given by uinc = 1 ρ0c ∗ pincξ · n. (10) 1.2 Non-Reflecting Boundary Equation For scattered field, CWA non-reflecting boundary equation can be written as[18] usc = 1 ρ0c ∗ psc + 1 ρ0 κ ∗∗ p sc, (11) where usc is the scattered normal displacement of fluid, and κ is the curvature of non-reflecting boundary. For scattered field, DAA2 non-reflecting boundary equation can be written as[6] usc = 1 ρ0c ∗ psc+ χ+ κ ρ0 ∗∗ p sc+ cχβ−1γ ρ0 ∗∗∗ p sc− cχ∗usc, (12) where χ ≈ β−1γ − κ, β and γ are spatial integral oper- ators. These operators are defined as βf(q1) = ∫ Γnrb f(q2) |rq1q2 | dΓq2 γf(q1) = ∫ Γnrb rq1q2 · n |rq1q2 |3 f(q2)dΓq2 ⎫⎪⎪⎪⎪⎪⎬ ⎪⎪⎪⎪⎪⎭ , (13) where, q1 and q2 are the points on non-reflecting bound- ary; rq1q2 = rq1 − rq2 is the distance vector between point q1 and point q2; f(q1) and f(q2) denote spatial functions. 1.3 Structure Discrete Equation The discrete structural response equation can be rep- resented as[2-3] Mẍ+Cẋ+Kx = F , (14) where x is the column vector of structural displace- ment; M is the structural mass matrix; C is the struc- tural damping matrix; K is the structural stiffness ma- trix; F is the external force vector. According to the force equilibrium condition at the fluid structure interface, for submerged structure with- out internal fluid domain, the external force vector can be written as F = −GAf(pcinc + pcsc), (15) where, Af is the fluid area matrix; pcinc and p c sc are the column vectors of the incident pressure and the scat- tered pressure, respectively. For submerged structure with internal fluid domain, the external force vector can be written as F = −GAf(pcinc + pesc − pisc), (16) where, pesc is the column vector of scattered pressure at interface between structure and external fluid; pisc is the column vector of scattered pressure at interface between structure and internal fluid; pesc and p i sc are obtained from the fluid discrete equation. J. Shanghai Jiaotong Univ. (Sci.), 2014, 19(3): 346-353 349 2 Model Validation In order to illustrate the validation of numerical model, the interaction between a water-backed infinite rigid plate and a planar exponentially decaying wave is simulated by the model. The analytic solution of pressure and velocity of plate without cavitation can be written as[19] p(t) = pm 1− λ [ (2 − λ)e−λt/td − λe−t/td], (17) v(t) = λpm ρ0c(1− λ) (e −λt/td − e−t/td), (18) where, λ = 2ρ0ctd/m, m is unit area mass of plate; td is the decay time constant of incident wave; pm is the maximum pressure of incident wave; v is the velocity of plate; p is the pressure of plate; t denotes time. Figure 2 shows the interaction between a water- backed rigid plate and planar exponentially decaying shock wave with a maximum pressure pm = 1MPa and a decay time td = 1ms. The density of plate ρs is 7 800 kg/m3 and the thickness h is 0.02m. The density of saturated fluid ρ0 is 1 000 kg/m3, in which the speed of sound c is 1.5 km/s. Figure 3 shows the pressure and velocity histories of water-backed rigid plate. As shown in Fig. 3, the present solutions agree well with the analytical solu- tions, so the numerical model is effective. Incident wave Plate Fluidh Fig. 2 Interaction between a water-backed rigid plate and planar exponentially decaying shock wave 0 1 2 3 4 5 6 0.5 1.0 1.5 2.0 t/ms t/ms p/ M P a 0 1 2 3 4 5 6 0.1 0.2 0.3 0.4 0.5 0.6 v/ (m ·s − 1 ) Analytical solution, Present solution Fig. 3 The pressure and velocity histories of water-backed plate 3 Results and Discussion 3.1 Difference Between DAA2 and CWA Non- Reflecting Boundary The difference between DAA2 non-reflecting bound- ary and CWA non-reflecting boundary is illustrated through the dynamic response of spherical shell with- out internal fluid to the planar exponentially decaying wave. Figure 4 shows the planar exponentially decaying wave and the spherical shell without internal fluid. The included angle between the incident wave propagation and the surface normal of spherical shell is denoted as α. In Fig. 4, the spherical shell has radius r = 1m, thickness h = 0.02m, density ρs = 7 766 kg/m 3, elastic modulus E = 210GPa, Poisson’s ratio μ = 0.3, maxi- mum pressure of incident wave pm = 1MPa and decay time constant td = 0.685ms. The density of saturated fluid ρ0 is 997 kg/m3, in which the speed of sound c is 1 461m/s. In the process of simulation, the radius of outer fluid domain R is taken as 1.5, 2.0 and 3.0m, re- spectively. Both DAA2 and CWA are adopted to deal z y Incident wave Spherical shell AirFluid r O α h Fig. 4 The spherical shell without internal fluid and planar wave with non-reflecting boundary. Figure 5 shows the radial velocity histories of spher- ical shell at α = 0◦ and α = 90◦ for different outer fluid domains. Figures 5(a) and 5(c) indicate the results obtained by CWA non-reflecting boundary; Figs. 5(b) and 5(d) indicate the results obtained by DAA2 non- reflecting boundary. According to Figs. 5(a) and 5(c), 350 J. Shanghai Jiaotong Univ. (Sci.), 2014, 19(3): 346-353 −0.4 0 4 8 12 16 (a) CWA non-reflecting boundary, α=0° t/ms 20 24 28 0 0.4 0.8 0.9 R=1.5m, R=2.0m, R=3.0m v/ (m ·s − 1 ) v/ (m ·s − 1 ) −0.4 0 0.4 0.8 v/ (m ·s − 1 ) v/ (m ·s − 1 ) 0 4 8 12 16 (b) DAA2 non-reflecting boundary, α=0° t/ms 20 24 28 0.9 −0.2 −0.4 0 4 8 12 16 (c) CWA non-reflecting boundary, α=90° t/ms 20 24 28 0 0.2 0.4 −0.2 −0.4 0 0.2 0.4 0 4 8 12 16 (d) DAA2 non-reflecting boundary, α=90° t/ms 20 24 28 Fig. 5 The radius velocity histories of spherical shell at α = 90◦ for different outer fluid domains as the outer radius of fluid domain R increases, the early time velocity response of spherical shell is basi- cally invariant, but the late time velocity response of spherical shell increases and is driven to stable stage. Because CWA does not consider the added mass effect, the added mass effect is simulated by the fluid near fluid structure interaction surface. As shown in Figs. 5(b) and 5(d), as the outer radius of fluid R increases, the velocity response of spherical shell is basically invariant. DAA2 can treat added mass effect well. 3.2 Influence of Internal Fluid Volume on Spherical Shell Response We simulate the interaction between the spherical shells with different fluid volume coefficients and the planar exponentially decaying wave to illustrate the in- fluence of internal fluid volume on structure response. The properties of spherical shell, fluid and planar wave are the same as those in Subsection 3.1. The fluid vol- ume coefficient (Cf) is defined as the ratio of the in- ternal fluid volume (Vf) to the volume surrounded by spherical shell (Vs), that is Cf = Vf/Vs. In this sec- tion, three Cf values of 0.0, 0.5 and 1.0 are carried out and correspond to empty, half and full internal fluid, respectively. Figure 6 shows the schematic dia- gram of the spherical shell with internal fluid and planar wave. z zO O y y Incident wave (a) Cf =0.5 (b) Cf =1.0 Incident wave Spherical shell FluidFluid Air Spherical shell FluidFluid O Fig. 6 The spherical shell with internal fluid and planar wave J. Shanghai Jiaotong Univ. (Sci.), 2014, 19(3): 346-353 351 The radial displacement responses of spherical shell with different fluid volume coefficients are shown in Fig. 7. Figures 7(a) and 7(b) are the radial dis- placements of spherical shell at α = 0◦ and α = 180◦, respectively. According to Fig. 7, as the vol- ume of internal fluid increases, the maximum of dis- placement decreases and the period increases. How- ever, the displacement oscillates severely when Cf = 0.5. Figure 8 shows the radial velocity of spherical shell with different fluid volume coefficients. Figures 8(a) and 8(b) are the radial velocities of spherical shell at α = 0◦ and α = 180◦, respectively. As shown in Fig. 8, the amplitudes of radial velocity of spherical shell cor- responding to Cf = 0.5, Cf = 0 and Cf = 1.0 increase in order. 0 0.2 0.4 0.6 0.8 1.0 u/ m m 0.2 0.4 0.6 0.8 1.0 1.2 u/ m m 4 8 12 16 t/ms 20 24 28 Cf =0, Cf =0.5, Cf =1.0 (a) α=0° 0 4 8 12 16 t/ms 20 24 28 (b) α=180° Fig. 7 The radial displacement of spherical shell with different fluid volume coefficients −0.7 −0.3 0.1 0.5 0.9 −0.9 −0.5 −0.1 0.3 0.7 1.1 0 4 8 12 16 t/ms 20 24 28 (a) α=0° 0 4 8 12 16 t/ms 20 24 28 (b) α=180° Cf =0, Cf =0.5, Cf =1.0 v/ (m ·s − 1 ) v/ (m ·s − 1 ) Fig. 8 The radial velocity of spherical shell with different fluid volume coefficients As discussion above, the internal fluid volume has a great influence on the dynamic response of spherical shell. When the internal fluid is full (Cf = 1.0), the displacement and velocity of spherical shell decrease as the added mass increases. When the internal fluid is half (Cf = 0.5), the displacement oscillates severely and the velocity increases. This results from the rarefied reflection of shock wave on free surface of internal fluid. 3.3 Influence of Cavitation on Spherical Shell Response In order to illustrate the influence of cavitation on fluid filled structure, we investigate the responses of fluid filled spherical shell (Cf = 1.0) with cavitation and without cavitation to the planar exponentially decaying wave. The spherical shell is located at 10 m depth. The properties of spherical shell, fluid and planar wave are the same as those in Subsection 3.1. The fluid pressure histories on spherical shell at α = 0◦ and α = 180◦ are shown in Fig. 9. The solid line denotes pressure histories with cavitation; the dashed line denotes pressure histories without cavitation. As shown in Figs. 9(a) and 9(b), there is almost no cav- itation in external and internal fluid near the leading point of spherical shell. According to Figs. 9(c) and 9(d), the cavitation occurs in external fluid near the trailing point of spherical shell during 2—6 ms, and the cavitation occurs in internal fluid near the trailing point of spherical shell during 1—9 ms. The reloading phe- nomenon due to the collapse of cavitation can be also observed in Figs. 9(c) and 9(d). The radial acceleration (denoted as a) and velocity histories of spherical shell are shown in Fig. 10, where 352 J. Shanghai Jiaotong Univ. (Sci.), 2014, 19(3): 346-353 −0.2 0.2 0.6 1.0 1.4 1.8 −0.3 0 0.3 0.6 0.9 1.2 −0.5 0 0.5 1.0 1.5 2.0 p/ M P a p/ M P a p/ M P a −0.5 0 0.5 1.0 1.5 2.0 p/ M P a Cavitation, No cavitation 0 4 8 12 16 (a) External fluid, α=0° t/ms 20 24 28 0 4 8 12 16 (b) Internal fluid, α=0° t/ms 20 24 28 0 4 8 12 16 (c) External fluid, α=180° t/ms 20 24 28 0 4 8 12 16 (d) Internal fluid, α=180° t/ms 20 24 28 Fig. 9 The pressure histories on spherical shell −200 0 200 400 600 800 a/ g a/ g Cavitation, No cavitation −800 −600 −400 −200 0 200 400 −0.4 −0.2 0 0.2 0.4 0.6 −0.5 −0.3 −0.1 0.1 0.3 0.5 0.7 0 4 8 12 16 (a) α=0° t/ms 20 24 28 0 4 8 12 16 (b) α=180° t/ms 20 24 28 0 4 8 12 16 (c) α=0° t/ms 20 24 28 0 4 8 12 16 (d) α=180° t/ms 20 24 28 v/ (m ·s − 1 ) v/ (m ·s − 1 ) Fig. 10 The radial and acceleration velocity histories of spherical shell J. Shanghai Jiaotong Univ. (Sci.), 2014, 19(3): 346-353 353 g is the acceleration of gravity (g = 9.8m/s2). Fig- ures 10 (a) and 10 (c) are the acceleration and velocity histories at α = 0◦. Figures 10(b) and 10(d) are the ac- celeration and velocity histories at α = 180◦. The solid line denotes the acceleration histories with cavitation; the dashed line denotes the acceleration histories with- out cavitation. According to Fig. 10, cavitation has a little effect on the radial acceleration and velocity of spherical shell at α = 0◦ and has a great influence on the radial acceleration and velocity of spherical shell at α = 180◦ during 2—10ms. 4 Conclusion In this paper, a numerical method is established to analyze the response of fluid-filled structure to under- water explosion with cavitation and the validation of the method is illustrated. Based on the model, the difference between DAA2 non-reflecting boundary and CWA non-reflecting boundary is investigated, and then, the influence of internal fluid volume and the influence of cavitation on dynamic response of spherical shell are analyzed. (1) The dynamic response of spherical shell with- out internal fluid to the planar exponentially decay- ing wave is investigated through DAA2 non-reflecting boundary and CWA non-reflecting boundary. Com- pared with CWA non-reflecting boundary, DAA2 non- reflecting boundary treats added mass effects better. (2) The dynamic responses of spherical shell with dif- ferent internal fluid volume coefficients are obtained by the established numerical model. When the internal fluid is full, the displacement and velocity of spherical shell decrease, but, when the internal fluid is half, the displacement and velocity of spherical shell increase. (3) The dynamic responses of spherical shell with cav- itation and without cavitation are simulated by the es- tablished numerical model. It is found that the effect of cavitation is more obvious at the trailing point than at the leading point of spherical shell. References [1] Hung C F, Lin B J, Hwang-Fuu J J, et al. 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