Numerical study on the water impact of 3D bodies by an explicit finite element method

Ocean Engineering 78 (2014) 7 3 - 8 8

.ELSEVIER

Contents lists available at S c i e n c e D i r e c t

Ocean Engineering

j o u r n a l h o m e p a g e : w w w . e l s e v l e r . c o m / l o c a t e / o c e a n e n g

Numerical study on the water impact of 3D bodies by an explicit finite

element method

Shan Wang, C. Guedes Scares *

Centre for Marine Tedmology and Engineering (CENTEC), Instituto Superior Técnico, University of Lisbon, Lisboa, Portugal

A R T I C L E I N F O A B S T R A C T

T h e h y d r o d y n a m i c p r o b l e m o f t h e w a t e r i m p a c t o f t h r e e - d i m e n s i o n a l b u o y s is i n v e s t i g a t e d b y t h e e x p l i c i t finite e l e m e n t m e t h o d w i t h a n A r b i t r a r y - L a g r a n g i a n E u l e r i a n (ALE) s o l v e r . T h e fluid is s a l v e d b y u s i n g a n E u l e r i a n f o r m u l a t i o n , w h i l e t h e s t r u c t u r e is d i s c r e t i z e d b y a L a g r a n g i a n a p p r o a c h , a n d a p e n a l t y c o u p l i n g a l g o r i t h m e n a b l e s t h e i n t e r a c t i o n b e t w e e n t h e b o d y a n d t h e fluids. T h e r e m a p s t e p i n t h e ALE a l g o r i t h m a p p l i e s a d o n o r c e l l + H I S ( H a l f - I n d e x - S h i f t ) a d v e c t i o n a l g o r i t h m t o u p d a t e f l u i d v e l o c i t y a n d h i s t o r y v a r i a b l e s . T h e i n t e r f a c e b e t w e e n t h e s o l i d s t r u c t u r e a n d t h e fluids is c a p t u r e d b y V o l u m e o f F l u i d m e t h o d . C o n v e r g e n c e s t u d i e s are c a r r i e d o u t f o r t h r e e d i m e n s i o n a l h e m i s p h e r e a n d c o n e s w i t h d i f f e r e n t d e a d r i s e angles. I t is f o u n d t h a t t h e m e s h d e n s i t y o f t h e i m p a c t d o m a i n is v e r y i m p o r t a n t t o t h e q u a l i t y o f t h e s i m u l a t i o n r e s u l t s . T h e c o n t a c t s t i f f n e s s b e t w e e n t h e c o u p l i n g n o d e s a f f e c t s t h e l o c a l p e a k p r e s s u r e v a l u e s . T h e n u m e r i c a l c a l c u l a t i o n s a r e v a l i d a t e d b y c o m p a r i n g w i t h o t h e r a v a i l a b l e r e s u l t s , f o r b o t h t h e d r o p cases a n d t h e o n e s w i t h c o n s t a n t i m p a c t v e l o c i t y . © 2013 E l s e v i e r L t d . A l l r i g h t s r e s e r v e d . Article history: Received 26 April 2013 Accepted 14 December 2013 Available online 31 January 2014 Keywords:

Numerical modeling Water impact

Explicit finite element method Impact coefficient

Slamming Wave converter

1. I n t r o d u c t i o n

Ocean waves are a s i g n i f i c a n t resource o f i n e x h a u s t i b l e , n o n -p o l l u t i n g energy. Waves are caused b y t h e w i n d b l o w i n g over the surface o f t h e ocean. I n m a n y areas o f t h e w o r l d , t h e w i n d b l o w s w i t h e n o u g h consistency a n d f o r c e t o p r o v i d e c o n t i n u o u s waves. A v a r i e t y o f technologies have b e e n p r o p o s e d t o capture t h e energy f r o m waves, and t h e y d i f f e r i n t h e i r o r i e n t a t i o n t o t h e waves w i t h w h i c h t h e y are i n t e r a c t i n g a n d i n t h e m a n n e r i n w h i c h t h e y c o n v e r t t h e energy o f t h e w a v e s i n t o o t h e r energy f o r m s . W a v e e n e r g y converters p r o v i d e a m e a n s o f t r a n s f o r m i n g w a v e energy i n t o usable electrical energy.

Point absorbers are one t y p e o f w a v e e n e r g y converters t h a t have s m a l l d i m e n s i o n s r e l a t i v e t o t h e i n c i d e n t w a v e l e n g t h . T h e y can capture wave energy f r o m a w a v e f r o n t t h a t is larger t h a n the d i m e n s i o n s o f the absorber. Several types o f w a v e absorbers have b e e n p r o p o s e d based o n d i f f e r e n t m e c h a n i s m s o f o b t a i n i n g r e l a t i v e m o t i o n s b e t w e e n t w o bodies. Due t o t h e i r r e l a t i v e l y s m a l l size, t h e a m o u n t o f energy t h a t t h e y c a n capture is r e l a t i v e l y s m a l l as c o m p a r e d w i t h devices based o n o t h e r p r i n c i p l e s i n Guedes Soares et al. (2012) and Silva et a l . (2013). To o v e r c o m e t h i s l i m i t a t i o n a p o s s i b i l i t y is h a v i n g a large p l a t f o r m f i x e d or floating a r o u n d w h i c h several s m a l l floaters have h e a v i n g t y p e o f m o t i o n s , w h i c h can t h e n be c o n v e r t e d i n p o w e r b y t h e p o w e r take o f f

* Corresponding author. Tel.: + 3 5 1 218417957: fax: + 3 5 1 218474015. E-mail address: guedess@mar.ist.utl.pt (C. Guedes Soares).

0029-8018/$-see f r o n t matter o 2013 Elsevier Ltd. A l l rights reserved. http://dx.doi.Org/10.1016/j.oceaneng.2013.12.008

m e c h a n i s m i n V a n t o r r e et al. ( 2 0 0 4 ) , L e n d e n m a n n et a l . ( 2 0 0 7 ) , Estefen e t al. ( 2 0 0 8 ) a n d M a r q u i s et al. (2010). H o w e v e r , i n t h i s process i t m a y h a p p e n t h a t the floaters w h e n at r e s o n a n c e have t o o h i g h v e r d c a l displacements a n d w i l l m o v e o u t o f t h e water, i m p a c r i n g i t a t t h e entrance. This p r o b l e m has b e e n d e t e c t e d by De Backer e t a l . ( 2 0 0 8 ) , w h o gives a b r i e f i n t r o d u c t i o n o n h o w t h e p o w e r a b s o r p t i o n is calculated, h o w t h e s l a m m i n g r e s t r i c t i o n is f o r m u l a t e d a n d f u l f i l l e d , a n d t h e y f o u n d t h a t t h e r e is a s i g n i f i c a n t r e d u c t i o n i n p o w e r a b s o r p t i o n due to t h e s l a m m i n g r e s t r i c t i o n . Since, i n any case, t h e p e n a l t y t o o v e r c o m e s l a m m i n g o f t h e p o i n t absorbers c o m p l e t e l y w i l l be t o o h i g h a n d a c e r t a i n level o f s l a m m i n g w i l l u s u a l l y be a l l o w e d , i t is i m p o r t a n t t o k n o w t h e m a g n i t u d e o f t h e s l a m m i n g load o n t h e floating objects w i t h d i f f e r e n t shapes.

De Backer e t al. ( 2 0 0 9 ) , c o n d u c t e d an e x p e r i m e n t a l s t u d y o f t h e i m p a c t o f 3 D bodies d u r i n g w a t e r e n t r y , i n o r d e r t o assess the s l a m m i n g loads i n these buoys a p p r o p r i a t e to t h e w a v e energy devices u n d e r c o n s i d e r a t i o n . This paper uses t h e s e e x p e r i m e n t a l results as references t o v a l i d a t e 3 D n u m e r i c a l studies, w h i c h f o l l o w eariier w o r k i n 2 D .

Early studies o n t h e local s l a m m i n g p r o b l e m f o c u s e d o n t h e analysis o f t w o - d i m e n s i o n a l structures, since s l a m m i n g o n ships has b e e n a m a j o r c o n c e r n a n d t h e 2 D s t r i p t h e o r y has b e e n w i d e l y used i n s h i p m o t i o n s research. The i m p o r t a n t p i o n e e r i n g s t u d y o n t h i s s u b j e c t can be a t t r i b u t e d to v o n K a r m a n ( 1 9 2 9 ) w h o p r o p o s e d t h e f i r s t t h e o r e t i c a l m e t h o d o n t h e analysis o f seaplane l a n d i n g . T h e n , W a g n e r ( 1 9 3 2 ) d e v e l o p e d an a s y m p t o t i c s o l u t i o n f o r w a t e r e n t r y o f t w o - d i m e n s i o n a l bodies w i t h s m a l l local deadrise angles

74 , Wang. C. Guedes Soares/Ocean Engineering 78 (20U) 73-88

b y a p p r o x i m a t i n g t h e m w i t h a flat plate, w h i c h c o n s i d e r e d t h e local w a t e r surface e l e v a t i o n . For t h e i d e a l i z e d case o f a w e d g e e n t e r i n g t h e c a l m w a t e r , D o b r o v o l ' s k a y a , 1969 d e r i v e d an a n a l y -t i c a l s o l u -t i o n by -t r a n s f e r r i n g -t h e p o -t e n -t i a l flow p r o b l e m f o r -t h e c o n s t a n t w a t e r e n t r y i n t o a self s i m i l a r flow p r o b l e m i n c o m p l e x plane, w h i c h t o o k advantage o f t h e s i m p l i c i t y o f t h e b o d y g e o m e t r y a n d is v a l i d f o r a n y deadrise angle.

Zhao and Faltinsen ( 1 9 9 3 , 1 9 9 6 ) proposed a n o n l i n e a r b o u n d a r y e l e m e n t m e t h o d to s t u d y the w a t e r e n t r y o f a t w o - d i m e n s i o n a l b o d y o f a r b i t r a r y cross-section a n d generalized the W a g n e r (1932)'s t h e o r y to presented a s i m p l e a s y m p t o t i c s o l u t i o n f o r s m a l l deadrise angles. As a f u r t h e r d e v e l o p m e n t w o r k , a f u l l y n o n l i n e a r n u m e r i c a l s i m u l a t i o n m e t h o d w h i c h includes flow separation f r o m k n u c k l e s o f a b o d y w a s p r e s e n t e d by Zhao et al., 1996. S u n and Faltinsen ( 2 0 0 6 ) d e v e l o p e d a t w o - d i m e n s i o n a l b o u n d a r y e l e m e n t m e t h o d to s i m u l a t e t h e w a t e r flow d u r i n g the w a t e r i m p a c t o f a r i g i d h o r i z o n t a l c i r c u l a r a n d a n elastic c y l i n d r i c a l shell. Exact free surface c o n d i t i o n s w e r e satisfactory.

Ramos e t al. ( 2 0 0 0 ) c o n d u c t e d an e x p e r i m e n t a l p r o g r a m assessing t h e s l a m i n d u c e d loads o n a s e g m e n t e d s h i p m o d e l t h a t w i t h several i n t e r c o n n e c t e d l o n g w e d g e s w h i l e the p r e v i o u s studies d e a l t w i t h i n d i v i d u a l 2 D wedges, w h i c h was a n a l y z e d w i t h the m e t h o d u s e d b y Ramos a n d Guedes Soares (1998).

M o s t investigations o f w a t e r e n t r y problems, i n c l u d i n g the researches m e n t i o n e d above have been focused o n the t w o -d i m e n s i o n a l impact, w h i l e f e w e r stu-dy have been con-ducte-d o n t h e three d i m e n s i o n a l cases w h i c h is m o r e consistent w i t h the real i m p a c t i n engineering. I n this field, early studies have been p u b l i s h e d by some researchers. S h i f t m a n and Spencer (1951) investigated the vertical s l a m m i n g o n spheres a n d cones based o n the analytical s o l u t i o n . T h e y are a m o n g the first to notice t h a t the l i q u i d m a y separate f r o m the sphere, l e a d i n g to cavity f o r m a t i o n , however, the stage o f the i m p a c t u n d e r consideration i n this study is b e f o r e separation w h i c h means the penetration d e p t h is less t h a n h a l f o f radius. S h i f t m a n a n d Spencer (1951) also give an explicit relationship o f i m p a c t c o e f f i c i e n t w i t h j{p)=\.6 f o r a cone w i t h deadrise angle 30". E l M a l k i A l a o u i e t al. (2012) recently f o u n d the e x p e r i m e n t a l l y d e t e r m i n e d equivalent as j(/3)=1.58 and the n o n d i m e n s i o n a l s l a m -m i n g c o e f f i c i e n t ]{/}) depends o n l y o n the deadrise angle p. By -means o f high-speed shock machine, t h e y studied the s l a m m i n g c o e f f i c i e n t o n a x i s y m m e t r i c bodies, and f o u n d t h a t Cs f o r hemisphere, u n l i k e t h e cones, depends o n the d e p t h o f immersions.

Based o n the W a g n e r ' s t h e o r y , Chuang ( 1 9 6 7 ) d e v e l o p e d a n a n a l y t i c a l f o r m u l a t i o n f o r t h e pressure d i s t r i b u t i o n o n a cone w i t h s m a l l deadrise angle, a n d Faltinsen a n d Zhao ( 1 9 9 7 ) p r o p o s e d a t h e o r e t i c a l m e t h o d f o r w a t e r e n t r y o f h e m i s p h e r e s a n d cones w i t h s m a l l deadrise angles. B a t t i s t i n and l a f r a t i ( 2 0 0 3 ) s t u d i e d t h e i m p a c t loads a n d pressure d i s t r i b u t i o n o n a x i s y m m e t r i c b o d i e s b y n u m e r i c a l s o l u t i o n . I n t h e field o f e x p e r i m e n t a l i n v e s t i g a t i o n , C h u a n g a n d M i l n e ( 1 9 7 1 ) p e r f o r m e d d r o p tests o n t h e c o n i c a l bodies, a n d r e c e n t l y Peseux et a l . ( 2 0 0 5 ) carried o u t t h e d r o p tests f o r cones w i t h s m a l l deadrise angles w h i c h i n c l u d e 6", 10" a n d 14".

M o t i v a t e d by t h e w o r k o f Stenius et al. (2006), w h o c o n d u c t e d the m o d e l i n g o f h y d r o elasticity i n w a t e r impacts o f ship bott:om-panels b y using LS-DYNA, Luo et al. (2011) and W a n g et al. (2012) investigated the s y m m e t r i c w a t e r i m p a c t o f t w o - d i m e n s i o n a l r i g i d w e d g e sections a n d ship sections, the predictions f r o m w h i c h had v e r y good a g r e e m e n t w i t h comparable measured values a n d o t h e r n u m e r i c a l results by a p p l y i n g t h e explicit finite e l e m e n t m e t h o d , a n d t h e n the effects o f t h e deadrise angle o n the s l a m m i n g l o a d w e r e presented i n W a n g a n d Guedes Soares (2012) a n d W a n g et al. ( S u b m i t t e d f o r p u b l i c a t i o n ) . They extended the research t o the a s y m m e t r i c w a t e r i m p a c t o f a b o w - f l a r e d section w i t h various r o l l angles i n W a n g a n d Guedes Soares (2013). I n the present w o r k , the explicit finite e l e m e n t m e t h o d is extended to study the h y d r o d y -n a m i c p r o b l e m o f t h r e e - d i m e -n s i o -n a l bodies, i -n c l u d i -n g h e m i s p h e r e

a n d cones w i t h d i f f e r e n t deadrise angles. The predictions are compared w i t h the e x p e r i m e n t a l results f r o m t h e d r o p tests o f De Backer et al. (2009) and theoretical calculations based o n Wagner (1932)'s m e t h o d , i n terms o f i m p a c t velocity, acceleration, penetra-t i o n d e p penetra-t h i n penetra-the w a penetra-t e r and penetra-t h e pressure hispenetra-tories o n penetra-the pressure sensors. The comparisons b e t w e e n t h e m are satisfactory i n t h e initial stage o f the w a t e r entry. Then, the v e r i f i e d m e t h o d is applied to estimate the impact coefficients o n a f a l l i n g hemisphere and a cone w i t h a deadrise angle 30", w h i c h s h o w g o o d consistency w i t h some analytical and theoretical predictions.

2 . M a t h e m a t i c a l f o r m u l a t i o n s

I n t h i s section, t h e e q u a t i o n s t h a t g o v e r n t h e fluid m o t i o n a n d t h e i n t e r a c t i o n b e t w e e n t h e fluid a n d s t r u c t u r e s i n this e x p l i c i t finite e l e m e n t m e t h o d are r e c a l l e d .

2.1. ALE description of Navier-Stokes equations

The g o v e r n i n g equation f o r incompressible a n d unsteady N a v i e r -Stokes fluid is described as:

— + u V u - 2 u ' ' V t ( u ) + Vp = b (2-1) dt

VU = 0 (2.2)

w h e r e u is the flow velocity, p is the pressure o f fluid, b means body force acting o n the fluid and e{u) represents the deviatoric stress tensor. The b o u n d a r y c o n d i t i o n a n d i n i t i a l c o n d i t i o n are ff=-pl + 2v''e{u) (2.3) e(u) = ^ ( V u + (Vu)^) (2.4) I n ALE f o r m u l a t i o n , a r e f e r e n c e c o o r d i n a t e w h i c h is n o t t h e Lagrangian c o o r d i n a t e a n d E u l e r i a n c o o r d i n a t e is i n d u c e d . The d i f f e r e n t i a l q u o t i e n t f o r m a t e r i a l w i t h respect t o the r e f e r e n c e c o o r d i n a t e is described as f o l l o w i n g e q u a t i o n . dt dt (2.5) w h e r e , X, is t h e Lagrangian c o o r d i n a t e , x, is t h e E u l e r i a n c o o r d i -nate, a n d W; is t h e r e l a t i v e v e l o c i t y . T h e r e f o r e , the ALE f o r m u l a t i o n c a n be d e r i v e d f r o m t h e r e l a t i o n b e t w e e n the time d e r i v a t i v e o f m a t e r i a l a n d t h a t o f reference g e o m e t r y c o n f i g u r a t i o n .

A s s u m e t h a t v represents t h e v e l o c i t y o f t h e m a t e r i a l , a n d u means t h e v e l o c i t y o f the mesh. I n o r d e r t o s i m p l i f y t h e above e q u a t i o n , relative v e l o c i t y w is i n d u c e d , w h i c h is g i v e n by w = v - u. T h e r e f o r e , ALE f o r m u l a t i o n can be o b t a i n e d f r o m f o l l o w i n g c o n -s e r v a t i o n equation-s: ( 1 ) T h e mass c o n s e r v a t i o n e q u a t i o n : a t dXi dXi (2.6) ( 2 ) T h e m o m e n t u m c o n s e r v a t i o n e q u a t i o n T h e g o v e r n i n g e q u a t i o n o f fluid is Navier-Stokes e q u a t i o n w h i c h is described b y t h e ALE m e t h o d : P _dt d_Vi dX;

T h e stress tensor is expressed b y : ö-,j = -p<5,j+/((Vij + Vj,,)

(2.7)

S. Wang. C. Guedes Soares / Ocean Engineering 78 (2014) 73-88 75

The i n i t i a l and b o u n d a r y c o n d i t i o n s are:

Vj = U° o n r , d o m a i n (2.9)

(Tjjnj = 0 o n r2 d o m a i n (2.10)

w h i l e

r , u r2 = r r , n r2 = o (2.11) w h e r e , r represents the w h o l e b o u n d a r y o f c o m p u t e d f i e l d ,

w h i l e T l and r2 means t h e parts o f r. n,- represents t h e u n i t v e c t o r o f b o u n d a r y i n o u t w a r d n o r m a l d i r e c t i o n , is K r o -necker ö f u n c t i o n . Assume t h a t the v e l o c i t y f i e l d at t i m e t = 0 i n t h e w h o l e c o m p u t e d d o m a i n is k n o w n as: v,(Xi,0) = 0 ( 3 ) The e n e r g y c o n s e r v a t i o n e q u a t i o n dE , dE (2.12) (2.13)

T h e Euler e q u a t i o n is d e r i v e d based o n the a s s u m p t i o n s t h a t t h e v e l o c i t y o f reference c o n f i g u r a t i o n is zero, a n d t h e r e l a t i v e v e l o c i t y b e t w e e n t h e m a t e r i a l a n d t h e reference c o n f i g u r a t i o n is t h e v e l o c i t y o f t h e m a t e r i a l . The t e r m s o f v e l o c i t y i n the E q u a t i o n ( 2 . 7 ) a n d t h e e q u a t i o n (2.9) are k n o w n as c o n v e c t i v e t e r m s w h i c h are used t o calculate t h e t r a n s p o r t a t i o n v o l u m e t h a t t h e m a t e r i a l f l o w s t h r o u g h the m e s h . The a d d i t i o n a l i t e m s are t h e reason t h a t the n u m e r i c a l s o l u t i o n o f t h e ALE e q u a t i o n is m u c h m o r e d i f f i c u l t t h a n t h a t o f a Lagrange e q u a t i o n i n w h i c h t h e r e l a t i v e v e l o c i t y is zero.

T h e r e are t w o approaches to solve t h e ALE e q u a t i o n , w h i c h are s i m i l a r t o t h e m e t h o d s a p p l i e d t o Euler's v i e w p o i n t i n h y d r o d y -n a m i c s . The f i r s t m e t h o d is s o l v i -n g f u l l y c o u p l e d e q u a t i o -n s u s i -n g c o m p u t a t i o n a l f l u i d mechanics, w h i c h can o n l y g o v e r n s i n g u l a r m a t e r i a l i n s i n g u l a r e l e m e n t . The second o n e was called d e t a c h e d o p e r a t o r m e t h o d , o f w h i c h t h e c a l c u l a t i o n i n each time step is separated i n t o t w o parts. First, t h e Lagrange a p p r o a c h is executed, w h e n t h e m e s h m o v e s w i t h m a t e r i a l . D u r i n g t h i s process, t h e e q u i l i b r i u m e q u a t i o n s are:

p'^=ffijj+pbi (2.14)

P^ = OijVij+pbiVi (2.15) I n t h e Lagrange process, t h e r e is n o m a t e r i a l f l o w i n g t h r o u g h

e l e m e n t b o u n d a r y , so t h e c a l c u l a t i o n satisfies the mass conserva-tion. T h e n t h e t r a n s p o r t a t i o n v o l u m e , i n t e r n a l e n e r g y a n d k i n e t i c e n e r g y o f m a t e r i a l s t h a t f l o w t h r o u g h t h e b o u n d a r i e s o f e l e m e n t are c a l c u l a t e d i n t h e second stage. It can be c o n s i d e r e d as r e m a p p i n g t h e meshes back t o t h e i r i n i t i a l o r a r b i t r a r y p o s i t i o n s .

As t o each node, t h e v e l o c i t y a n d d i s p l a c e m e n t are u p d a t e d a c c o r d i n g t o f o l l o w i n g e q u a t i o n : (2.16)

l+Atu"+l

/2

(2.17) w h e r e , F^, is vector o f i n t e r n a l f o r c e , a n d F"^^, is v e c t o r o f e x t e r n a l f o r c e . T h e y are i n r e l a t i o n w i t h b o d y f o r c e a n d b o u n d a r y c o n d i -tions. M is d i a g o n a l m a t r i x o f mass.

2.2. Fluid-structure coupling algoritlim

I n a n e x p l i c i t time i n t e g r a t i o n p r o b l e m , a f t e r c o m p u t a t i o n o f f l u i d a n d s t r u c t u r e n o d a l forces, t h e c o u p l i n g forces o f t h e nodes o n t h e f l u i d s t r u c t u r e i n t e r f a c e are c o m p u t e d i n the time step. For

each s t r u c t u r e node, a d e p t h p e n e t r a t i o n d is i n c r e m e n t a l l y u p d a t e d at each time step, u s i n g t h e r e l a t i v e v e l o c i t y ( v T - v ^ ) at t h e s t r u c t u r e node, w h i c h is c o n s i d e r e d as a slave node, and t h e master n o d e w i t h i n the E u l e r i a n e l e m e n t . The l o c a t i o n o f the master n o d e is c o m p u t e d u s i n g t h e i s o p a r a m e t r i c coordinates o f

^11 t h e f l u i d e l e m e n t . A t time t = f", t h e d e p t h p e n e t r a t i o n d is u p d a t e d b y :

-(v.. n +

1/2

Vf

)Af

(2.18)

w h e r e A t is the i n c r e m e n t o f time, is t h e v e l o c i t y o f t h e slave node, v} is the f l u i d v e l o c i t y at t h e m a s t e r n o d e l o c a t i o n , i n t e r -p o l a t e d f r o m t h e nodes o f t h e f l u i d e l e m e n t at t h e c u r r e n t time, a n d t h e v e c t o r ~d means t h e p e n e t r a t i o n d e p t h o f t h e s t r u c t u r e inside the f l u i d d u r i n g t h e time step. T h e c o u p l i n g f o r c e acts o n l y i f p e n e t r a t i o n occurs.

Penalty c o u p l i n g behaves like a s p r i n g s y s t e m a n d p e n a l t y forces are calculated p r o p o r t i o n a l l y to t h e p e n e t r a t i o n d e p t h a n d s p r i n g stiffness. The h e a d o f t h e s p r i n g is a t t a c h e d t o t h e s t r u c t u r e o r slave n o d e and t h e t a i l o f t h e s p r i n g is a t t a c h e d t o t h e m a s t e r n o d e w i t h i n a f l u i d e l e m e n t t h a t is i n t e r c e p t e d b y t h e s t r u c t u r e . S i m i l a r l y t o p e n a l t y c o n t a c t a l g o r i t h m , t h e c o u p l i n g force is described by: F = kd (2.19) w h e r e k represents t h e s p r i n g s t i f f n e s s , a n d d means t h e p e n e t r a -tion. The c o u p l i n g f o r c e F is a p p l i e d t o b o t h m a s t e r n o d e a n d slave n o d e i n o p p o s i t e d i r e c t i o n at t h e c o u p l i n g i n t e r f a c e . The m a i n d i f f i c u l t y i n t h e c o u p l i n g p r o b l e m is t h e e v a l u a t i o n o f t h e stiffness k.

I n t h i s paper, t h e stiffness o f t h e s p r i n g is based o n t h e e x p l i c i t p e n a l t y c o n t a c t a l g o r i t h m i n LS-DYNA, a n d t h e n u m e r i c a l s t i f f n e s s b y u n i t area is g i v e n i n t e r m o f t h e b u l k m o d u l u s K o f t h e f l u i d e l e m e n t i n t h e c o u p l i n g c o n t a i n i n g t h e slave s t r u c t u r e node, t h e v o l u m e V o f t h e f l u i d e l e m e n t t h a t c o n t a i n s t h e m a s t e r f l u i d node, a n d t h e average area A o f t h e s t r u c t u r e elements c o n n e c t e d to the s t r u c t u r e node.

k=pf I(A

V (2.20)

H o w e v e r , to a v o i d n u m e r i c a l i n s t a b i l i t i e s , a p e n a l t y f a c t o r pj- is i n t r o d u c e d f o r scaling t h e e s t i m a t e d s t i f f n e s s o f t h e i n t e r a c t i n g ( c o u p l i n g ) system. For i m p a c t p r o b l e m s , i t is a l w a y s necessary to e x a m i n e t h e i n f l u e n c e o f t h i s p a r a m e t e r o n t h e s o l u t i o n ( A q t i e l e t e t al. ( 2 0 0 6 ) ) . For t h e p r o b l e m o f t w o - d i m e n s i o n a l w e d g e , Luo e t al. (2011) c o n d u c t e d a p a r a m e t r i c study, i n c l u d i n g t h e p e n a l t y factor, t i m e step factor, m e s h size a n d t h e n u m b e r o f t h e c o n t a c t p o i n t s , a n d v a l i d a t e d t h i s m e t h o d b y c o m p a r i n g t h e p r e d i c t i o n s w i t h t h e e x p e r i m e n t a l results f r o m Zhao et al., 1996. The results s h o w t h a t m e s h size is o f g r e a t i m p o r t a n c e f o r t h e s i m u l a t i o n s , w h i l e o t h e r aspects a f f e c t l i t t l e .

3 . N u m e r i c a l m o d e l i n g

3.1. Description of the 3D structures

I n t h i s w o r k , d i f f e r e n t k i n d s o f t h r e e - d i m e n s i o n a l s t r u c t u r e s , i n c l u d i n g a h e m i s p h e r e a n d cones w i t h d i f f e r e n t d e a d r i s e angles are s t u d i e d . To v a l i d a t e t h e m e t h o d u s e d i n p r e s e n t w o r k , t h e p r e d i c t i o n s f r o m a h e m i s p h e r e , a c o n e 2 0 ' a n d a cone45" e n t e r i n g c a l m w a t e r w i t h d r o p v e l o c i t y , are c o m p a r e d w i t h t h e m e a s u r e d values f r o m De Backer et al. ( 2 0 0 9 ) . T h e m a i n p a r a m e t e r s o f t h e tested bodies w h i c h are a p p l i e d i n t h e n u m e r i c a l l y m o d e l i n g , are l i s t e d i n Table 1. As seen, t h e d i a m e t e r s o f t h e bodies are 3 0 c m w h i c h is c o n s i d e r e d s u f f i c i e n t t o reduce surface t e n s i o n effects.

76 S. Wang, C. Guedes Soares / Ocean Engineering 78 (2014) 73-88

Since t l i e s t r u c t u r e s are m a d e f r o m p o l y u r e t h a n e a n d ttie m a t e r i a l thiclmesses are large, t h e d e f o r m a t i o n s o f t h e m d u r i n g the w a t e r i m p a c t are c o n s i d e r e d l i m i t . It m u s t be n o t e d t h a t t h e m e a s u r e d i n i t i a l v e l o c i t i e s l i s t e d i n Table 1 are l o w e r t h a n t h e t h e o r e t i c a l c a l c u l a t i o n s based o n the d r o p h e i g h t s due t o the f r i c t i o n i n t h e g u i d i n g s y s t e m o f t h e test.

The pressure t i m e h i s t o r y , t h e p o s i t i o n a n d d e c e l e r a t i o n o f t h e b o d y w e r e r e c o r d e d i n t h i s test w o r k . Pressure sensors w e r e used t o o b t a i n t h e pressure t i m e h i s t o r y o n t h e b o d y . The sensors w e r e l o c a t e d at a h o r i z o n t a l distance o f 0 . 0 4 - 0 . 0 9 m o n t h e bodies f r o m t h e s y m m e t r i c axis respectively, as p l o t t e d i n Fig. 1, i n w h i c h P l a n d P2 r e p r e s e n t t h e pressure sensors a n d is t h e deadrise angle o f the cones.

Besides, n o n - d i m e n s i o n a l i m p a c t c o e f f i c i e n t s o n h e m i s p h e r e a n d cones w i t h d i f f e r e n t deadrise angles are c o m p u t e d based o n t h e a s s u m p t i o n o f c o n s t a n t i m p a c t velocity. T h e y are c o m p a r e d w i t h available m e a s u r e d a n d n u m e r i c a l values as w e l l .

3.2. Description of tlie modeling

The e x p l i c i t f i n i t e e l e m e n t analysis is based o n a m u l t i - m a t e r i a l E u l e r i a n f o r m u l a t i o n a n d a p e n a l t y c o u p l i n g m e t h o d . The f l u i d is s o l v e d by u s i n g a E u l e r i a n f o r m u l a t i o n , w h i l e t h e w e d g e is d i s c r e d i t e d b y a Lagrangian a p p r o a c h . The f l u i d s ( w a t e r a n d a i r ) are d e f i n e d as t h e m u l t i - m a t e r i a l g r o u p , w h i c h means t h a t t h e effects o f t h e w a t e r a n d t h e a i r are a l l c o n s i d e r e d . The p e n a l t y c o u p l i n g a l g o r i t h m is a p p l i e d to activate t h e i n t e r a c t i o n b e t w e e n t h e f l u i d s a n d the s t r u c t u r e . I t behaves like a s p r i n g system, t h u s g e n e r a t i n g h i g h o s c i l l a t i o n s t o t h e c o u p l i n g f o r c e . The p e n a l t y forces are c a l c u l a t e d p r o p o r t i o n a l l y t o t h e p e n e t r a t i o n d e p t h a n d s p r i n g s t i f f n e s s . T h o u g h s o m e noise w i l l be g e n e r a t e d t o t h e pressure values o n t h e Lagrangean e l e m e n t s , t h e total force o n t h e s t r u c t u r e w i l l n o t be sensitive to t h e c o u p l i n g factor, since i t is a n average value. The c o m m e r c i a l code LS-DYNA is used as a t o o l t o solve t h e d i f f e r e n t i a l e q u a t i o n s t h a t g o v e r n the p h e n o m e n o n w i t h f o l l o w i n g hypotheses:

o The g r a v i t y effects are n e g l e c t e d .

o The surface t e n s i o n effects w i l l n o t be m o d e l e d . o The s t r u c t u r e s have no d e f o r m a t i o n a n d r o t a t e m o t i o n .

Table 1

Characteristics for the measured bodies.

Item Radius (m) Total mass (kg) Material thickness (m) Initial Velocity (m/s) Hemisphere 0.15 11.5 0.05 4.0 Cone 20' 0.15 9.8 0.03 3.85 Cone 45' 0.15 10.2 0.03 4.05 Based o n these a s s u m p t i o n s , t h e n u m e r i c a l l y m o d e l i n g is as f o l l o w s : • T h e c o o r d i n a t e s y s t e m o f t h e p r o b l e m

As illustrated i n Fig. 2, a Caitesian coordinate system (x, y, z) is introduced, and the (x, y)-plane is placed i n the u n d i s t u r b e d w a t e r surface, w h i l e the z-axis is located i n the axis o f the body. The b o d y enters the calm w a t e r w i t h a vertical velocity w h i c h is denoted as dz/dt, and t = 0 means the t i m e instance w h e n the body touches the water. The boundaries o f the w a t e r are denoted as S | , , S r and SB .

Fig. 2. Coordinate system of the problem.

Fig. 3. Mesh style of the fluids i n the x-y plane.

150mm

[

S. Wang, C. Cuedes Soares / Ocean Engineering 78 (2014) 73-88 77

• T h e m a t e r i a l a n d e l e m e n t types

T h e f l u i d , w a t e r a n d air, are m o d e l e d w i t h Solid164 e l e m e n t w h i c h is an 8-nodes b r i c k e l e m e n t , and t h e y are d e f i n e d as v o i d m a t e r i a l s w h i c h a l l o w s e q u a t i o n s o f state t o be c o n s i d e r e d w i t h o u t c o m p u t i n g d e v i a t o r i c stresses. The G r u n e i s e n e q u a t i o n o f state is used t o t h e w a t e r d o m a i n a n d the l i n e a r p o l y n o m i a l e q u a t i o n o f state is a p p l i e d f o r the a i r d o m a i n . The w e d g e is m o d e l e d w i t h S h e l l l 6 3 e l e m e n t w h i c h is a 4-nodes e l e m e n t a n d can o n l y be used i n e x p l i c i t d y n a m i c analysis, a n d r i g i d b o d y m a t e r i a l .

• B o u n d a r y c o n d i t i o n s

O n l y a q u a r t e r o f t h e m o d e l is established w i t h s y m m e t r i c b o u n d a r i e s o n ( y - z ) a n d ( x - z ) planes. T h e b o u n d a r i e s o f t h e f l u i d s are d e f i n e d as n o n - r e f l e c t i n g , except t h a t , o t h e r f l u i d s nodes are f r e e . For t h e b o d y , o n l y v e r t i c a l m o v e m e n t d o w n -w a r d s is released.

• N u m e r i c a l m o d e l

As k n o w n , t h e ALE c a l c u l a t i o n is time-consuming, so d i f f e r e n t m e s h types are a p p l i e d o n d i f f e r e n t regions to reduce m e m o r y a n d CPU r e q u i r e m e n t . Luo e t al. (2011) f o u n d t h a t t h e m e s h size i n t h e r e g i o n n e a r t h e c o n t a c t area b e t w e e n t h e s t r u c t u r e a n d t h e f l u i d s are o f g r e a t i m p o r t a n c e t o t h e s i m u l a t i o n . As t o t h e r e g i o n t h a t is f a r f r o m t h e i m p a c t , t h e m a p p e d area m e s h w h i c h contains o n l y q u a d r i l a t e r a l e l e m e n t s is e m p l o y e d , a n d the m e s h size i n t h i s d o m a i n is m o d e r a t e l y e x p a n d i n g t o w a r d s t h e b o u n d a r i e s . Fig. 3 a n d Fig. 4 s h o w the m e s h style o f t h e

Fig. 4. Mesh style of the fluids in the y-z plane.

a

f l u i d s i n x-y a n d y-z planes. F u r t h e r m o r e , t h e s t r u c t u r e is m e s h e d w i t h q u a d r i l a t e r a l e l e m e n t s as p l o t t e d i n Fig. 5. C o n -s i d e r i n g t h e c o m p u t a t i o n a l e f f o r t -s , t h e f l u i d -s d o m a i n i-s l i m i t e d t o 0.5 m X 0.5 m X 0.6 m , w h i c h m e a n s t h e d i m e n s i o n i n x-y p l a n e is L4 x L3 (0.5 m x 0.5 m ) , a n d t h e d i m e n s i o n s o f air d o m a i n and w a t e r d o m a i n i n z - d i r e c t i o n are L7-I-L8 (0.2 m + 0 . 4 m ) . The d i m e n s i o n o f i m p a c t d o m a i n is d e n o t e d as L l X L2 X ( L 5 + L 6 ) w h i c h is 0.18 m x 0.18 m x (0.05 m x 0.08 m ) . It is f o u n d t h a t t h e size o f t h e i m p a c t d o m a i n ' is o f g r e a t i m p o r t a n c e t o t h e n u m e r i c a l results. The selection o f t h e size o f t h e m o d e l i n p r e s e n t w o r k is based o n lots o f calcula-tions and the experience u s i n g t h e code. The d i s c u s s i o n o n t h e m o d e l s w i t h d i f f e r e n t size is n o t p r e s e n t e d here, because t h e convergence s t u d y focuses o n t h e m e s h d e n s i t y a n d t h e c o n t a c t s t i f f n e s s . For m o d e l i n g accurately o f t h e w a t e r i m p a c t p r o b l e m , a c a r e f u l s e l e c t i o n o f m e s h d e n s i t y a n d c o n t a c t s t i f f n e s s is r e q u i r e d . As m e n t i o n e d i n Section 2.2, t h e c o n t a c t s t i f f n e s s is r e l a t e d to t h e p e n a l t y f a c t o r a n d t h e v o l u m e V o f t h e f l u i d e l e m e n t t h a t contains t h e m a s t e r f l u i d node, so i t is a f f e c t e d b y t h e m e s h d e n s i t y o f t h e f l u i d s . I n t h e f o l l o w i n g section, a c o n v e r g e n c e s t u d y is c o n d u c t e d to o b t a i n a p r o p e r n u m e r i c a l m o d e l . 4. C o n v e r g e n c e s t u d y 4 . 7 . Mesh density

Three m e s h sizes,10 m m , 5 m m a n d 2.5 m m are selected f o r t h e f l u i d s o f the i m p a c t d o m a i n (L3 x L4 x {L5+L6)). T h e m e s h sizes are d e n o t e d b y 0.067R, 0.033R a n d 0.0167R, w h e r e R means t h e r a d i u s o f t h e h e m i s p h e r e o r t h e cones. Unless o t h e r w i s e s p e c i f i e d , t h e m e s h size o f t h e s t r u c t u r e is as same as t h a t o f t h e f l u i d s , a n d t h e v a l u e o f Pf is set as 0.1. I n p r e s e n t w o r k , t h e n u m e r i c a l c o n t a c t s t i f f n e s s l< is c o m p u t e d by e q u a t i o n ( 2 . 2 0 ) . For t h e t h r e e models, t h e v a l u e is 22.5 G p a / m , 4 5 G p a / m a n d 90 G p a / m , respectively.

Fig. 6 presents the p r e d i c t e d n o n d i m e n s i o n a l i m p a c t c o e f f i -cients o f a r i g i d h e m i s p h e r e o f radius R e n t e r i n g v e r t i c a l l y i n t o i n i t i a l l y c a l m w a t e r w i t h a c o n s t a n t v e l o c i t y V, t o g e t h e r w i t h t h e available e x p e r i m e n t a l a n d n u m e r i c a l results. The n o n -d i m e n s i o n a l i m p a c t c o e f f i c i e n t is -d e f i n e -d as CS = 2F/PMR^V^, w h e r e F i s t h e t o t a l i m p a c t f o r c e a n d ^ = 1 0 0 k g / m ^ is the d e n s i t y o f t h e f l u i d , and t h e n o n - d i m e n s i o n a l time is d e n o t e d as c/(f)/R, w h e r e d ( t ) is the i n s t a n t a n e o u s p e n e t r a t i o n o f t h e sphere b e l o w t h e c a l m w a t e r . Here, t h e i m p a c t v e l o c i t y is 4 m/s, a n d the radius

b

78 Ü 4 3.5 3 2.5 2 1.5 1 0.5

S. Wang, C. Cnedes Soares / Ocean Engineering 78 (20U) 73-88

1.4 Miloh(1991)

O Nisewagnerexp.(1961) - - Battistin and lafrati (2003)

LS-DYNA Mestisize=0.067R - LS-DYNA Mesiisize=0.033R — r - L S - D Y N A Mesiisize=0.0167R : O 0.Ó5 0.1 0.15 d(t)/R 0.2 0.25 LS-DYNA Mestisize=0.0167R V=18m/s LS-DYNA Mesiisize=0.0167R V=4m/s 0.1 0.15 0.2 0.25 0.3 0.35 d(t)/R

Fig. 7. Tlie impact coefficient for rigid liemisptiere entering' calm water w i t h Fig. 6. The impact coefficient for a rigid hemisphere impacting w i t h calm water. d i f f e r e n t velocities.

Table 2

Three models w i t h d i f f e r e n t mesh densities.

Parameters Model 1 Model 2 Model 3 4

Mesh size 0.067R 0.033R 0.0167R

Number o f elements(Fluids) 43200 134400 510300 ₃

Number o f elements (Structures) 175 500 1600

CPU time^ 1 h 22 m i n 9 h 49 m i n 45 h 53 m i n Q.

O Note: It was run o n one PC w i t h 2.50 GHz processor and 3 Gigabytes of memory.

o f t h e s p h e r e is 0.15 m . Table 2 lists the m a i n p a r a m e t e r s f o r the t h r e e m o d e l s w i t h 0.015 s' s o l u t i o n t i m e .

As seen i n Fig. 6, w h e n t h e m e s h size is 0.0167R, t h e p r e d i c t e d i m p a c t c o e f f i c i e n t is i n g o o d a g r e e m e n t w i t h t h e e x p e r i m e n t a l m e a s u r e m e n t s f r o m N i s e w a n g e r ( 1 9 9 6 ) a n d t h e n u m e r i c a l calcu-lations f r o m B a t t i s t i n a n d l a f r a t i ( 2 0 0 3 ) , a f t e r the i n i t i a l stage o f the i m p a c t . A t t h e i n i t i a l stage, t h e i m p a c t c o e f f i c i e n t is h i g h e r t h a n t h e e x p e r i m e n t a l and n u m e r i c a l results. This is because, at t h i s stage, the i n t e r a c t i o n b e t w e e n t h e fluid a n d t h e s t r u c t u r e o n l y involves f e w elements, f r o m the b o t t o m o f the h e m i s p h e r e a n d the surface o f t h e w a t e r . The n u m e r i c a l i m p u l s e s o f pressure o n the e l e m e n t s are i n e v i t a b l e at t h e i n i t i a l i m p a c t , a n d t h e i m p a c t force is o b t a i n e d f r o m t h e i n t e g r a t i o n o f the pressures a l o n g t h e w e t t e d surface o f t h e s t r u c t u r e . For t h e a n a l y t i c a l calculations f r o m M i l o h (1991), t h e s i m p l i f i e d m e t h o d gives l o w e r p r e d i c t i o n s at t h e i n i t i a l stage a n d h i g h e r ones at t h e late stage.

W h e n t h e m e s h size is 0.033K a n d 0.067R, t h e p r e d i c t i o n s are n o t c o n s i s t e n t w i t h the e x p e r i m e n t a l m e a s u r e m e n t s . A t t h e m i d d l e a n d late stage o f t h e i m p a c t , as t h e m e s h size becomes large, t h e i m p a c t c o e f f i c i e n t is higher. It also s h o w s t h a t t h e n u m e r i c a l noises are a p p a r e n t f o r a larger m e s h size.

It is o b v i o u s t h a t t h e m o d e l w i t h 0.0167R m e s h size is m o r e a p p r o p r i a t e t o c a p t u r e the t i m e h i s t o r y o f i m p a c t force o n the h e m i s p h e r e e n t e r i n g c a l m water, a n d t h e c o m p u t a t i o n a l t i m e is acceptable. To v e r i f y the s t a b i l i t y o f t h e n u m e r i c a l results, d i f f e r e n t i m p a c t velocities are a p p l i e d t o t h e h e m i s p h e r e . The i m p a c t c o e f f i c i e n t s o n the h e m i s p h e r e w i t h l / = 4 m/s a n d V = 1 8 m/s are p l o t t e d i n Fig. 7, w h i c h shows v e r y g o o d consistency. T h e d i s -c r e p a n -c y at t h e i n i t i a l stage is s t i l l d u e t o t h e m e s h size.

To c a p t u r e the pressure d i s t r i b u t i o n o n the h e m i s p h e r e surface, v i r t u a l pressure sensors are l o c a t e d at the c e n t e r o f t h e shell e l e m e n t s o n l o c a t i o n y = 0 . The h e m i s p h e r e is m e s h e d w i t h 4799 shell e l e m e n t s , f r o m w h i c h 80 e l e m e n t s o n x - z plane are selected. Fig. 8 s h o w s the pressure d i s t r i b u t i o n s o n t h e w e t t e d h e m i s p h e r e

Fig. 8. The impact coefficient for a rigid hemisphere impacdng w i t h calm water at

d(t)/R=0.134.

surface o n x z plane at d{t)IR=0A34. T h e n o n d i m e n s i o n a l p r e s -sure Cp is d e f i n e d as 2plpV^, w h e r e p is t h e pres-sure v a l u e o b t a i n e d f r o m the pressure sensor. x/R denotes t h e p o s i t i o n o n t h e h e m i s p h e r e surface, w h e r e x is t h e c o o r d i n a t e o f t h e e l e m e n t a n d R is t h e radius. x / R = 0 m e a n s t h e l o w e s t p o i n t , a n d x / R = l is t h e highest p o i n t o n t h e h e m i s p h e r e . For d i f f e r e n t i m p a c t v e l o -cities, t h e pressure d i s t r i b u t i o n s have v e r y g o o d a g r e e m e n t . Some n u m e r i c a l noise is o b s e r v e d at t h e p o s i t i o n near t h e i n t e r s e c t i o n b e t w e e n t h e w a t e r surface a n d t h e s t r u c t u r e f o r b o t h cases. A t t h i s m o m e n t , t h e pressure is a l m o s t u n i f o r m l y d i s t r i b u t e d a l o n g t h e surface.

The p r e d i c t i o n s o f i m p a c t c o e f f i c i e n t and pressure d i s t r i b u t i o n f r o m the m o d e l 0.0167R w i t h d i f f e r e n t i m p a c t v e l o c i t i e s s h o w v e r y g o o d consistency. It is b e l i e v e d t h a t t h i s m o d e l is a p p r o p r i a t e f o r t h e h e m i s p h e r e . Fig. 9 p l o t s t h e p r e d i c t e d n o n - d i m e n s i o n a l i m p a c t c o e f f i c i e n t s o f a c o n e 2 0 ° o f radius R e n t e r i n g v e r t i c a l l y i n t o i n i t i a l l y c a l m w a t e r w i t h a c o n s t a n t v e l o c i t y V. Here, t h e m e s h size o f t h e s t r u c t u r e is 0.0167R f o r t h e t h r e e m o d e l s , so t h e n u m e r i c a l c o n t a c t s t i f f n e s s is respectively, 1.406 G p a / m , 11.25 G p a / m a n d 9 0 G p a / m . S i m i l a r t o t h e p r e d i c t i o n s o f the h e m i s p h e r e , t h e i m p a c t c o e f f i c i e n t is h i g h e r f o r a m o d e l w i t h larger m e s h size. W h e n the m e s h size is 0.0167R a n d 0.033R, t h e n u m e r i c a l calculations are close, especially f o r t h e values at t h e m i d d l e stage. A t t h e late stage, a h i g h i m p u l s e is o b s e r v e d i n t h e cui-ve o f t h e m o d e l w i t h 0.067R m e s h size. This is

S. Wang, C. Cuedes Soares / Ocean Engineering 78 (2014) 73-88 79 LS-DYNA M e s h s i z e = 0 . 0 6 7 R - LS-DYNA M e s h s i z e = 0 . 0 3 3 R LS-DYNA M e s h s i 2 e = 0 . 0 1 6 7 R / 1 I \ 0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 d(t)/R

Fig. 9. Tlie impact coefficient for a cone20 impacting w i t h calm water.

Ü

-LS-DYNA Meshsize=0.0167R V=6.15m/s -LS-DYNA Meshsize=0.0167R V=18m/s

0 0.05 0.1 0.16 0.2 0.25 0.3 0.35 d(t)/R

Fig. 11. The impact coefficient for a cone20 impacting w i t h calm water at different velocity. 15 10 • L S - D Y N A M e s h s i z e = 0 . 0 6 7 R L S - D Y N A M e s h s l z e = 0 . 0 3 3 R L S - D Y N A M e s h s i z e = 0 . 0 1 6 7 R i i M I I I I ' 1 I I¬ I 1 I I I I A. \

' ' ' !

11 1 0.2 0.4 0.6 X/R 0.8 1.1 1 0.9 0.8 0.7 0.6 0.5 0.4 pfac=0.01 pfac=0.1 pfac=0.5

1'

Fig. 10. Pressure distribution along the surface of cone20 at f„,

0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 d{t)/R

Fig. 12. The impact coefficient f o r a hemisphere impacting w i t h calm water w i t h

d i f f e r e n t value of pfac.

because t h e m e s h size o f w a t e r surface n e a r b y t h e s t r u c t u r e surface becomes l a r g e r as t h e w a t e r surface evolves d u r i n g t h e i m p a c t . For a cone20" w i t h R = 0 . 1 5 m , a n d v = 6 . 1 5 m / s , t h e t o t a l i m m e r s i o n occurs at t = 0 . 0 0 8 9 s w h i l e f = O s m e a n s t h e cone touches t h e c a l m w a t e r . As seen f o r t h e t h r e e m o d e l s , t h e m a x i m u m i m p a c t f o r c e occurs at a b o u t d ( f ) / R = 0 . 2 9 , a n d t h e c o r r e s p o n d i n g t i m e i n s t a n t t = 0 . 0 0 7 0 7 s w h i c h is i n g o o d agree-m e n t w i t h W a g n e r ' s t h e o r y p r e d i c t i n g f , „ o , ^ = ; r R t a n ^ / 4 l / = 0.007 s. Fig. 10 shows t h e pressure d i s t r i b u t i o n o n t h e w e t t e d surface o f cone20-' o n x - z plane at t^ax ( t h e t i m e i n s t a n t w h e n the peak value h a p p e n s ) . The pressure value is o b t a i n e d e v e n at t h e highest p o i n t . This m e a n s the w a t e r j e t is p r o d u c e d u n d e r t h e structure's surface a n d reaches the h i g h e s t p o s i t i o n . T h e p r e s e n t m e t h o d ' s p r e d i c -t i o n s s h o w -t h a -t -the m a x i m u m i m p a c -t f o r c e o n a cone 20" occurs a-t t h e t o t a l i m m e r s i o n o f t h e m o d e l .

As seen f r o m t h e t h r e e curves, t h e m a x i m u m pressures are located n e a r the r o o t o f t h e w a t e r j e t . This is c o n s i s t e n t w i t h t h e r e s u l t o b t a i n e d f r o m t h e 2 D w e d g e 2 0 " b e f o r e f l o w separation. W h e n t h e m e s h size is 0.067R, t h e pressure d i s t r i b u t i o n o b t a i n e d f r o m the sensors has m u c h noise a n d the pressure values are m u c h l o w e r t h a n the ones f r o m t h e m o d e l s w i t h 0.033R and 0.0167R m e s h size. W h e n t h e m e s h size is 0.033R o r 0.0167R, the pressure d i s t r i b u t i o n s are i n g o o d a g r e e m e n t , h o w e v e r , t h e one f r o m the m o d e l w i t h 0.0167R m e s h size is s m o o t h e r .

T h e i m p a c t c o e f f i c i e n t s o n t h e c o n e 2 0 " w i t h d i f f e r e n t i m p a c t v e l o c i t i e s are p l o t t e d i n Fig. 11. T h e p r e d i c t i o n s have g o o d agree-m e n t , w h i l e the curve f r o agree-m the agree-m o d e l vvith V = 1 8 agree-m/s is s agree-m o o t h e r .

4.2. Contact stiffness As m e n t i o n e d b e f o r e , a p e n a l t y f a c t o r p / ( p f a c ) is i n t r o d u c e d f o r s c a l i n g t h e e s t i m a t e d s t i f f n e s s o f the c o u p l i n g s y s t e m . To o b t a i n a p r o p e r value o f i t , t h e i n f l u e n c e o f this p a r a m e t e r o n t h e s o l u t i o n is e x a m i n e d . T h r o u g h t h e s e n s i t i v i t y s t u d y o f m e s h size, t h e m o d e l w i t h 0.0167R m e s h size is selected f o r w a t e r i m p a c t s o f t h e h e m i s p h e r e a n d cones, a n d f o r t h e m o d e l , t h e d e f a u l t v a l u e o f p f a c is 0.1. Here, t w o d i f f e r e n t values, 0.01 a n d 0.5 are a p p l i e d i n t h e s i m u l a t i o n s .

Fig. 12 p l o t s t h e i m p a c t c o e f f i c i e n t s o n t h e h e m i s p h e r e w i t h d i f f e r e n t p f a c values. Here, t h e m e s h size is 0.0167R, a n d t h e c o n s t a n t i m p a c t v e l o c i t y is 18 m/s. G e n e r a l l y s p e a k i n g , t h e t h r e e curves agree w e l l , t h o u g h some o s c i l l a t i o n s exist. O b v i o u s d i s t i n c -t i o n s are o b s e r v e d a-t -t h e i n i -t i a l m o m e n -t o f -t h e i m p a c -t a n d a-t -t h e m o m e n t t h a t t h e peak v a l u e occurs. As seen i n Fig. 6, t h e i m p a c t c o e f f i c i e n t f r o m t h e m o d e l w i t h 0.0167R m e s h size, t h e c o n t a c t s t i f f n e s s o f w h i c h is 90 G p a / m , agrees w e l l w i t h t h e e x p e r i m e n t a l m e a s u r e m e n t s , c o m p a r e d to the m o d e l s w i t h l o w e r c o n t a c t s t i f f n e s s . I t seems t h a t h i g h e r c o n t a c t s t i f f n e s s is b e t t e r f o r t h e i m p a c t m o d e l o f t h e h e m i s p h e r e . H o w e v e r , f o r t h e m o d e l w i t h 0.5 pfac, t h e c u r v e o f i m p a c t c o e f f i c i e n t does n o t b e c o m e better, a n d e v e n appears m o r e n u m e r i c a l noises.

The 80 pressure sensors, w h i c h are l o c a t e d at t h e c e n t e r o f t h e s h e l l e l e m e n t s o n x - z plane, are n u m b e r e d f r o m 1 t o 80, i n w h i c h sensor 1 m e a n s the l o w e s t o n e a n d sensor 8 0 d e n o t e s t h e h i g h e s t o n e . T h e pressure values c a p t u r e d b y t h r e e sensors are s h o w n i n Fig. 13. T h e results s h o w t h a t the m a x i m u m local p r e s s u r e v a l u e is

80 S, Wang, C. Cuedes Soares / Ocean Engineering 78 (2014) 73-88

0 10 20 0

"Time (s)

^

^g-^

- - pfac=0,01

•

_{--• pfac=0,1}

- pfac=0.5

Sensor 20

10

20

Fig. 13. Pressure histories on the b o t t o m surface of the hemisphere.

Ü

- - LS-DYNA Meshsize=0.0167R 90Gpa/m - - LS-DYNA Meshsize=0.0167R 22.5Gpa/m

10

0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 (d(t)/R

Fig. 14. The impact coefficient for a cone20 impacting w i t h calm water w i t h

d i f f e r e n t contact stiffness.

located at t h e l o w e s t p o i n t o f the h e m i s p h e r e . As t h e distance f r o m t h e p o i n t to t h e axis o f t h e h e m i s p h e r e becomes far, the local peak pressure decreases greatly.

It is f o u n d t h a t t h e local peak pressure o n t h e h e m i s p h e r e is sensitive t o t h e v a l u e o f pfac. The h i g h e r t h e c o n t a c t s t i f f n e s s is, the s m a l l e r t h e peak v a l u e is. These d i f f e r e n c e s b e c o m e s m a l l w h e n t h e p o s i t i o n o f t h e sensor is higher. FroiTi t h e pressure histories, i t is also possible t o f i n d t h a t the m a x i m u m pressure v a l u e occurs at t h e m o m e n t w h e n the h e m i s p h e r e t o u c h e s t h e c a l m w a t e r , d u r i n g t h e w a t e r i m p a c t .

The peak values are sensitive to t h e scale factor. This is c o n s i s t e n t w i t h t h e f a c t t h a t the c o u p l i n g force o n the m a s t e r a n d slave n o d e is c o m p u t e d b y m u l t i p l y i n g c o n t a c t s t i f f n e s s a n d p e n e t r a t i o n , w h i l e t h e c o n t a c t stiffness is scaled by t h e v a l u e o f pfac. The t o t a l i m p a c t forces o n the s t r u c t u r e are n o t sensitive t o t h e v a r i a t i o n o f t h e scale factor, since t h e y are average values o n the s t r u c t u r e .

For t h e m o d e l s o f t h e cone20", t h e o n e w i t h 0.0167R m e s h size is s t u d i e d here, f i r s t l y b y a l t e r i n g the m e s h size o f t h e L a g r a n g i a n e l e m e n t s , a n d s e c o n d l y b y a p p l y i n g d i f f e r e n t values o f pfac, t o i n v e s t i g a t e t h e i n f l u e n c e s o f t h e c o n t a c t s t i f f n e s s . Based o n CL O 1 1 9 0 G p a / m d ( t y R = 0 . 1 6 4 ™ — 90Gpa/m d(t)/R=0.303 - ^ - . 2 2 . 5 G p a / m d ( t ) / R = 0 . 1 6 4 22.5Gpa/m d(t)/R=0.303

il

i i i i 0.2 0.4 0.6 x / R

Fig. 15. Pressure d i s t r i b u t i o n along the surface o f cone20' at t w o time instants.

Eq. (2.20), the n u m e r i c a l c o n t a c t s t i f f n e s s is 22.5 G p a / m , w h e n the m e s h size o f t h e cone is 0.00835R. The p r e d i c t e d i m p a c t c o e f f i c i e n t and pressure d i s t r i b u t i o n s o n t h e w e t t e d surface o f cone 2 0 " are c o m p a r e d w i t h t h e calculations f r o m t h e m o d e l w i t h /<:=90 Gpa/m, as s h o w n i n Figs. 14 a n d 15.

T h e i m p a c t c o e f f i c i e n t s have v e r y l i m i t e d d i f f e r e n c e s , w h i l e some noise is o b s e r v e d i n b o t h curves. I n Fig. 15, t h e pressure d i s t r i b u t i o n s at t w o t i m e i n s t a n t s d ( f ) / R = 0 . 1 6 4 a n d d ( f ) / R = 0 . 3 0 3 are presented. A t t h e f o r m e r t i m e i n s t a n t , t h e o n l y h a l f o f t h e s t r u c t u r e i m m e r s e s i n t o t h e w a t e r surface, w h i l e at t h e later t i m e instant, the s t r u c t u r e i m m e r s e i n t o t h e w a t e r c o m p l e t e l y , c o n -s i d e r i n g the w a t e r -surface e l e v a t i o n . It can be f o u n d that, b e f o r e f l o w separation, t h e m a x i m u m pressure is located a t the l o w e s t p o i n t o f the cone. W i t h d i f f e r e n t c o n t a c t stiffness, t h e general trends o f the pressure d i s t r i b u t i o n d o n o t change t o o m u c h . O n l y some differences are o b s e r v e d near t h e spray r o o t o f t h e w a t e r surface or at the l o w e s t l o w e r p a r t o n t h e cone surface. U n l i k e t h e s i m u l a t i o n s o f t h e h e m i s p h e r e , t h e pressure values are l a r g e r w h e n the contact s t i f f n e s s is h i g h e r f o r t h e cone20".

Figs. 16 and 17 s h o w t h e i m p a c t c o e f f i c i e n t s a n d pressure d i s t r i b u t i o n s o n t h e c o n e 2 0 " w i t h d i f f e r e n t value o f pfac. The i n f l u e n c e s o f t h e scale f a c t o r o n t h e results are s m a l l , t h o u g h s l i g h t

S. Wang. C. Cuedes Soares / Ocean Engineering 78 (2014) 73-88 81

d i f f e r e n c e s are f o u n d a r o u n d the m i d d l e stage o f t h e i m p a c t i n the curves.

4.3. Time step

A n ALE f o r m u l a t i o n consists o f a Lagrangian t i m e step f o l l o w e d by an a d v e c t i o n step, w h i c h updates t h e v e l o c i t y a n d d i s p l a c e m e n t o n each n o d e at o n e t i m e step. A stable t i m e step is o f great s i g n i f i c a n t to the n u m e r i c a l results. The t i m e step s h o u l d n o t be larger t h a n the c r i t i c a l one, o t h e r w i s e n e g a t i v e v o l u m e errors w i l l a p p e a r B u t i f the t i m e step is set t o one v a l u e t h a t is t o o s m a l l ,

o

0.15 0.2 0.25 0.3 0.35

d ( t ) / R

Fig. 16. Tlie impact coefficient for a cone20 impacting w i t h calm water w i t h p f

10 O pfac 0.5 pfac 0.1 • pfac 0.01 d(t)/R=0.164 0 I 0.1 0.2 0.3 0.4 0.5 X/R 0.6

Fig. 17. Pressure distributions on the surface of cone20 at d(t)//?=0.164 for

d i f f e r e n t

t h e n t h e c o m p u t a t i o n a l t i m e w i l l increase c o r r e s p o n d e n t l y . The c r i t i c a l t i m e step size is t h e m i n i m u m t i m e value t h a t t h e s o u n d travels t h r o u g h any e l e m e n t s i n t h e m o d e l . The c r i t i c a l t i m e step size can be a p p r o x i m a t e d f i r s t l y before t h e s i m u l a t i o n , i n o r d e r to set o n e scale f a c t o r t o o b t a i n one a p p r o p r i a t e t i m e step. In LS-DYNA T h e o r y m a n u a l , t i m e step calculations f o r d i f f e r e n t types o f e l e m e n t s are e x p l a i n e d t h r o u g h m a t h e m a t i c a l f o r m u l a t i o n s .

Obviously, t h e c r i t i c a l t i m e step is r e l a t e d t o t h e m i n i m u m size of t h e e l e m e n t , a n d t h e scale f a c t o r is b e t w e e n 0 and 1. For one n u m e r i c a l m o d e l , a p r o p e r t i m e step v a l u e can b e achieved b y a d j u s t i n g t h e scale factor. I n p r e s e n t w o r k , t h e t i m e step f o r t h e m o d e l s w i t h 0.067R m e s h size is 4.69E-07s, a n d t h e value is 2.28E-07s f o r t h e m o d e l s w i t h 0.033R m e s h size, 1.14E-2.28E-07s f o r t h e m o d e l s w i t h 0.0167R m e s h size. It is f o u n d t h a t the value is p r o p o r t i o n a l to t h e m e s h size. W h e n t h e i m p a c t v e l o c i t y is 18 m/s, t h e n u m e r i c a l s o l u t i o n t i m e is decreased greatly, so a v e r y s m a l l scale f a c t o r 0.05 is a p p l i e d t o m a k e the s o l u t i o n stable, w h i c h f o l l o w s a v e r y s m a l l t i m e step 5.71E-08s.

5. V a l i d a t i o n a n d results

A c c o r d i n g to t h e d r o p tests o f t h e t h r e e d i m e n s i o n a l bodies De Backer e t al. ( 2 0 0 9 ) , t h e acceleration, i m p a c t v e l o c i t y , p e n e t r a -t i o n d e p -t h and pressure d i s -t r i b u -t i o n s d u r i n g -t h e w a -t e r i m p a c -t are p r e d i c t e d a n d c o m p a r e d w i t h the m e a s u r e d values, as w e l l as t h e c a l c u l a t i o n s f r o m a s y m p t o t i c t h e o r y . In o r d e r t o reduce t h e i n f l u e n c e o f t h e a s s u m p t i o n s m e n t i o n e d above, o n l y the i n i t i a l stage o f t h e i m p a c t is i n v e s t i g a t e d . F u r t h e r m o r e , t h e i m p a c t c o e f f i c i e n t s o n t h e h e m i s p h e r e a n d cones are c o m p u t e d and c o m p a r e d to s o m e p u b l i s h e d results, w i t h t h e a s s u m p t i o n o f c o n s t a n t i m p a c t v e l o c i t y . To e x a m i n e t h e i n f l u e n c e s o f t h i s a s s u m p t i o n , t h e p r e d i c t i o n s f r o m t h e m o d e l w i t h d r o p v e l o c i t y a n d c o n s t a n t v e l o c i t y are c o m p a r e d f i r s t l y .

5 . 3 . Influence of impact velocity

Fig. 18 c o m p a r e s the i m p a c t c o e f f i c i e n t s o n t h e h e m i s p h e r e w i t h d i f f e r e n t types o f i m p a c t v e l o c i t y . T w o v e l o c i t i e s v = 4 m / s a n d v = 18 m/s are selected. As seen, at t h e i n i t i a l stage, t h e i m p a c t c o e f f i c i e n t s f r o m t w o m o d e l s agree w e l l , w h i l e t h e d i f f e r e n c e s b e t w e e n t h e m b e c o m e larger as t h e p e n e t r a t i o n d e p t h raises. For a h i g h e r i m p a c t v e l o c i t y , the i n f l u e n c e s are m o r e a p p a r e n t . Obviously, t h e i m p a c t f o r c e o n a h e m i s p h e r e e n t e r i n g w i t h a c o n s t a n t v e l o c i t y is higher, since t h a t t h e i m p a c t v e l o c i t y o f t h e d r o p case decays due t o t h e r e s u l t a n t f o r c e o n t h e s t r u c t u r e .

Fig. 19 c o m p a r e s the i m p a c t c o e f f i c i e n t s o n t h e cone20- w i t h d i f f e r e n t types o f i m p a c t v e l o c i t y . As seen, a t t h e i n i t i a l stage, the

1.2 1 0.8 Ö 0.6 0.4 0.2 0 ft') v> J - V=4m/s 0 0.05 0.1 0.15 0.2 0.25 d ( t y R

1

0.8

S

0-6 0.4 0.2 0 ( . ( . ' -V=18m/s 0.1 0.2 d(t)/R 0.4

82 S, Wang. C. Guedes Soares / Ocean Engineering 78 (2014) 73-88 Ü - V = 1 8 m / s — Drop VQ=18m/s 0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 d{t)/R

Fig. 19. Comparison o f impact coefficients on ttie cone20 entering water w i t l i

constant and drop velocity. V=18 m/s.

-zr 10 0- 0 S 10 0.5 1.5 Time (s) 2.5 3 x10"^ • V = 1 8 m / s - D r o p V p = 1 8 m / s

Fig. 20. Comparison of pressure histories on the cone20' entering water w i t h

constant and drop velocity. V=\8 mis.

i m p a c t c o e f f i c i e n t s f r o m t w o m o d e l s agree w e l l , w h i l e t h e d i f f e r -ences b e t w e e n t h e m b e c o m e larger as t h e p e n e t r a t i o n d e p t h raises. T h e peak v a l u e f r o m t h e m o d e l w i t h c o n s t a n t v e l o c i t y is m u c h larger, a l t h o u g h t h e n o n - d i m e n s i o n a l p e n e t r a t i o n d e p t h s o f the cone u n d e r w a t e r are v e r y close w h e n t h e peak values occur. T h e d i f f e r e n c e s b e t w e e n t h e i m p a c t forces are d u e to t h e pressures o n t h e w e t t e d s u r f a c e o f t h e s t r u c t u r e . Fig. 2 0 s h o w s t h e pressures o f t h r e e p o s i t i o n s , w h i c h are d e n o t e d b y x = 0 . 2 5 R , x = 0 . 5 R a n d x = 0 . 7 5 R , o n t h e cone i n x - z plane. For the m o d e l w i t h a d r o p t v e l o c i t y , t h e pressure decays m o r e q u i c k l y a f t e r the peak value occurs. T h u s , t h e d i f f e r e n c e s b e c o m e larger as the i m p a c t processes.

5.2. Drop 3D structures

5.2.1. Hemisphere

Fig. 21 s h o w s t h e p r e d i c t e d , m e a s u r e d a n d t h e o r e t i c a l accel-e r a t i o n o f t h accel-e h accel-e m i s p h accel-e r accel-e d u r i n g 0.012s a f t accel-e r t h accel-e b o t t o m o f t h accel-e b o d y t o u c h e s w a t e r The t h e o r e t i c a l c a l c u l a t i o n s are based o n t h e

600 400 200 -200 -400 E X P ( A c c e l e r o m e t e t ) T h e o r e t i c a l ( A M ) T h e o r e t i c a l ( P l ) " — ' — L S - D Y N A 0 0.002 0.004 0.006 0.008 0.01 0.012 T i m e ( s )

Fig. 21. Predicted and measured acceleration on the hemisphere.

_o (D > • • EXP(Accelerometer) EXP{HSC) — Theoretical(AM) -Theoretical(PI) LS-DYNA • 0 0.002 0.004 0.006 0.008 0.01 0.012 T i m e ( s )

Fig. 22. Predicted and measured impact velocity on the hemisphere.

pressure i n t e g r a t i o n m e t h o d a n d a d d e d mass m e t h o d . A l t h o u g h t h e e x p e r i m e n t a l data has lots o f h i g h f r e q u e n c y noises, t h e p r e d i c t i o n o f LS-DYNA agree w e l l w i t h i t , w h i l e t h e t h e o r e t i c a l c a l c u l a t i o n s o v e r e s t i m a t e t h e a c c e l e r a t i o n o f t h e s t r u c t u r e .

As p l o t t e d i n Fig. 22, t h e i m p a c t v e l o c i t y o b t a i n e d b y LS-DYNA is l o w e r t h a n t h a t m e a s u r e d i n the test, a n d t h e d i f f e r e n c e b e t w e e n t h e m becomes larger as t i m e goes by, m a i n l y d u e t o t h e f r i c t i o n c r e a t e d f r o m t h e f r e e l y m o v e m e n t o f t h e s t r u c t u r e a l o n g the g u i d i n g system i n t h e test. As e x p e c t e d , t h e h e m i s p h e r e d r o p s m o r e q u i c k l y f r o m t h e p o i n t o f v i e w o f t h e o r e t i c a l s o l u t i o n s . C o r r e s p o n d i n g l y , t h e p e n e t r a t i o n d e p t h s b e l o w t h e c a l m w a t e r d u r i n g this t i m e s p a n o f t h e h e m i s p h e r e are c o m p a r e d i n Fig. 23, w h i c h also s h o w s t h a t the d e v i a t i o n b e t w e e n t h e m e a s u r e d value a n d t h e p r e d i c t e d ones f r o m p r e s e n t w o r k a n d t h e o r e t i c a l s o l u -t i o n s are o b s e r v e d m o r e o b v i o u s l y as -t i m e progresses. Fig. 24 p l o t s t h e pressure h i s t o r i e s o f t h e t w o p o i n t s a t r = 4 c m a n d r = 9 c m o n t h e h e m i s p h e r e as i l l u s t r a t e d i n Fig. 1. As m e n t i o n e d i n W a n g et a l . ( 2 0 1 2 ) , t h e a s y m p t o t i c t h e o r y o v e r -e s t i m a t -e s t h -e pr-essur-e o f s h i p - l i k -e s -e c t i o n s , i n p a r t i c u l a r f o r a s m a l l d e a d r i s e angle, the s i m i l a r b e h a v i o r is o b s e r v e d here f o r b o t h o f t h e p r e s s u r e p o i n t s . For t h e p r e s s u r e p o i n t at r = 9 c i n , t h e p r e d i c t i o n s f r o m LS-DYNA are i n g o o d a g r e e m e n t w i t h t h e m e a s u r e d ones, i n c l u d i n g t h e r i s i n g t i m e o f t h e p e a k v a l u e a n d as w e l l t h e m a x i m u m v a l u e o f the p r e s s u r e , t h o u g h s o m e n u m e r i c a l noises exist. As t o t h e p r e s s u r e p o i n t at r = 4 c m , t h e p r e d i c t e d peak v a l u e is s m a l l e r t h a n t h e m e a s u r e d one. T h i s is m a i n l y d u e t o t h e t h r e e - d i m e n s i o n a l i t y o f t h e s i m u l a t i o n , f o r w h i c h t h e p r e s s u r e c a p t u r e d b y t h e v i r t u a l s e n s o r is m o r e e a s i l y d i s t u r b e d b y f r e q u e n c y noises, a n d p r o b a b l y t h e pressure is a f f e c t e d b y t h e p o s i t i o n o f t h e sensor.

S. Wang, C. Guedes Soares / Ocean Engineering 78 (2014) 73-88 ₈₃ 0.05 0.04 0,03 E X P ( A c c e l e r o m e t e r ) E X P ( H S C ) - " n i e o r e t i c a l ( A M ) " n i e o r e t i c a l ( P I ) L S - D Y N A 0 0.002 0.004 0.006 0.008 0.01 T i m e ( s )

Fig. 23. Predicted and measured penetration on the hemisphere.

100 50 -100 -150 -200 E X P ( A c c e l e r o m e t e r ) - T t i e o r e t i c a l ( A M ) T h e o r e t i c a l ( P I ) • L S - D Y N A 0 0.002 0.004 0.006 0.008 0.01 0.012 T i m e ( s )

Fig. 25. Predicted and measured acceleration on the cone20 .

O - EXP t = 4 c m K30A • EXP r = 9 c m K 3 0 A A s y m p t o t i c theory r=4cm • A s y m p t o t i c theory r = 9 c m - L S - D Y N A r = 4 c m • L S - D Y N A r = 9 c m 0.002 0.006 T i m e ( s ) 0.008

Fig. 24. Predicted and measured pressure distribution at r = 4 c m on the

hemisphere.

5.2.2. Cone

20-Figs. 2 5 - 2 7 s h o w t h e p r e d i c t e d a n d m e a s u r e d acceleration, i m p a c t v e l o c i t y a n d p e n e t r a t i o n d e p t h f o r t h e r i g i d cone w i t h a deadrise angle o f 2 0 ' d u r i n g the i m p a c t . The c o m p a r i s o n s b e t w e e n t h e calculations i n t h i s w o r k w i t h t h e m e a s u r e d a n d t h e o r e t i c a l values are s i m i l a r t o t h a t o f the h e m i s p h e r e .

Fig. 28 plots t h e pressure histories o f t h e t w o p o i n t s at r = 4 c m a n d r = 9 c m o n t h e cone 20". The s i m u l a t e d r i s i n g t i m e o f t h e pressure p o i n t s at r = 4 c m is a l i t t l e b i t earlier t h a n those f r o m t h e tests. Probably i t is due to w a t e r j e t o f t h e f r e e surface i n t h e m o d e l i n g , w h i c h affects t h e pressure v a l u e earlier. For t h e peak values, t h e one at r = 4 c m is s m a l l e r t h a n t h a t f r o m t h e e x p e r i -m e n t a l a n d t h e o r e t i c a l s o l u t i o n , a n d t h i s d i f f e r e n c e w a s also observed i n t h e s t u d y o f 2 D w e d g e w i t h a deadrise angle 2 0 " b y W a n g a n d Guedes Soares ( S u b m i t t e d f o r P u b l i c a t i o n ) . S i m i l a r to t h e e x p e r i m e n t s , the p r e d i c t e d peak pressure at r = 9 c m is larger t h a n t h e one at r = 4 c m . 4.5 O 2.5 • EXP(Accelerometer) • EXP(HSC) Theoretical(AM) Theorefical(PI) LS-DYNA -0 -0.-0-02 -0.-0-04 -0.-0-06 -0.-0-08 -0.-01 -0.-012 T i m e ( s )

Fig. 26. Predicted and measured impact velocity on the cone20 .

0.05 -g- 0.04 sz & 0.03 c O I 0.02 (D C a> CL 0.01 EXP(Accelerometer) EXP(HSC) Theoretica!(AM) -Theoretical(PI) - LS-DYNA 5mm O L i i 0 0.002 0.004 0.006 0.008 0.01 0.012 T i m e ( s )

Fig. 27. Predicted and measured penetration on the cone20

Probably, i t is d u e to t h e w a t e r surface e l e v a t i o n d u r i n g t h e i m p a c t . H i g h i m p u l s e s are observed f o r these t w o pressures.

5.2.3. Cone 45'

Figs. 2 9 - 3 1 s h o w the p r e d i c t e d a n d m e a s u r e d acceleration, i m p a c t v e l o c i t y a n d p e n e t r a t i o n d e p t h f o r t h e rigid cone w i t h a d e a d r i s e angle o f 4 5 " d u r i n g t h e i m p a c t . Q u i t e g o o d agreements b e t w e e n t h e p r e d i c t i o n s a n d m e a s u r e d values are f o u n d i n t h e i n i t i a l stage, w h i l e t h e discrepancies increase as t i m e goes by.

Fig. 32 p l o t s the pressure histories o f t h e t w o p o i n t s at r = 4 c m a n d r = 9 c m o n t h e cone 45". Obviously, t h e p r e d i c t e d pressure peak at r = 4 c m is m u c h l o w e r t h a n t h e e x p e r i m e n t a l results. It can be n o t i c e d t h a t t h e r i s i n g t i m e o f the s i m u l a t e d pressure at r = 9 c m is earlier t h a n t h e m e a s u r e d a n d a s y m p t o t i c ones. 5.3. Impact coefficient The n o n - d i m e n s i o n a l i m p a c t c o e f f i c i e n t o n a h e m i s p h e r e is p l o t t e d i n Figure a n d c o m p a r e d w i t h e x p e r i m e n t a l m e a s u r e m e n t s a n d n u m e r i c a l calculations. It m u s t be n o t e d t h a t t h e i m p a c t v e l o c i t y o f t h e h e m i s p h e r e is c o n s t a n t here. As seen i n t h i s f i g u r e , t h e p r e d i c t i o n i n p r e s e n t w o r k has g o o d consistency w i t h o t h e r calculations at t h e l a t e stage. A t t h e early stage, the p r e s e n t m e t h o d u n d e r e s t i m a t e s t h e i m p a c t c o e f f i c i e n t . T h i s is c o n s i s t e n t w i t h the p r e d i c t i o n o f t h e pressure values at r = 4 c m w h i c h are p l o t t e d i n Fig. 2 4 . I t is d u e to t h e m e s h size o f