/BEM/DAA

Block Format Keyword Doubly Asymptotic Approximation for Underwater Explosion, where the fluid mass matrix is computed by boundary element method.

Format

(1) (2) (3) (4) (5) (6) (7) (8) (9) (10)
/BEM/DAA/daa_ID/unit_ID
daa_title
surf_ID grav_ID            
ρ C Pmin        
Xs Ys Zs        
Iform Ipri Ipres     Kform Freesurf Afterflow Integr  
Insert if Ipres=1
(1) (2) (3) (4) (5) (6) (7) (8) (9) (10)
Pm θ a a θ MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqGqFfpeea0xe9vq=Jb9 vqpeea0xd9q8qiYRWxGi6xij=hbba9q8aq0=yq=He9q8qiLsFr0=vr 0=vr0db8meaabaqaciGacaGaaeqabaWaaeaaeaaakeaacaWGHbWaaS baaSqaaiabeI7aXbqabaaaaa@3A30@    
Insert if Ipres=2
(1) (2) (3) (4) (5) (6) (7) (8) (9) (10)
fct_IDP   FscaleP        
(1) (2) (3) (4) (5) (6) (7) (8) (9) (10)
Xc Yc Zc        
Insert if grav_ID > 0 or Freesurf=2
(1) (2) (3) (4) (5) (6) (7) (8) (9) (10)
XA YA ZA        
Dir-X Dir-Y Dir-Z        

Definitions

Field Contents SI Unit Example
daa_ID DAA block identifier

(Integer, maximum 10 digits)

 
unit_ID Unit Identifier

(Integer, maximum 10 digits)

 
daa_title DAA block title

(Character, maximum 100 characters)

 
surf_ID Wet surface identifier. 2 3

(Integer)

 
grav_ID /GRAV option identifier.

(Integer)

 
ρ Fluid density.

(Real)

[ kg m 3 ]
C Fluid sound speed.

(Real)

[ m s ] MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaamWaaeaada Wcaaqaaiaab2gaaeaacaqGZbaaaaGaay5waiaaw2faaaaa@39DE@
Pmin Pressure cutoff (< 0).

Default = -1030 (Real)

[ Pa ]
Xs X-coordinate of standoff point. 3

(Real)

[ m ]
Ys Y-coordinate of standoff point. 3

(Real)

[ m ]
Zs Z-coordinate of standoff point. 3

(Real)

[ m ]
Iform BEM solution flag.
=1 (Default)
Gauss Integration.
= 2
Analytical integration.

(Integer)

 
Ipri Printout flag level.
=1 (Default)
Reduced printout.
= 2
Full printout.

(Integer)

 
Ipres Pressure loading flag. 6
=1
Pressure computed as P i ( t ) = P m e t θ MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqGqFfpeea0xe9vq=Jb9 vqpeea0xd9q8qiYRWxGi6xij=hbba9q8aq0=yq=He9q8qiLsFr0=vr 0=vr0db8meaabaqaciGacaGaaeqabaWaaeaaeaaakeaaciGGqbWaaS baaSqaaiaacMgaaeqaaOWaaeWaaeaacaWG0baacaGLOaGaayzkaaGa eyypa0JaamiuamaaBaaaleaacaWGTbaabeaakiaadwgadaahaaWcbe qaaiabgkHiTmaalaaabaGaamiDaaqaaiabeI7aXbaaaaaaaa@43A9@ .
= 2
Input by function.

(Integer)

 
Kform Analysis flag.
=1 (Default)
DAA1 formulation
= 2
High frequency

(Integer)

 
Freesurf Free surface flag. 6
=1 (Default)
No
= 2
Yes

(Integer)

 
Afterflow Afterflow computation. 7
=1
No
= 2 (Default)
Yes

(Integer)

 
Integr Time integer flag.
=1
First order
= 2 (Default)
Second order

(Integer)

 
Pm Maximum pressure. 5

(Real)

[ Pa ]
θ Decay time.

(Real)

[ s ]
a Maximum pressure constant. 5

(Real)

 
a θ MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqGqFfpeea0xe9vq=Jb9 vqpeea0xd9q8qiYRWxGi6xij=hbba9q8aq0=yq=He9q8qiLsFr0=vr 0=vr0db8meaabaqaciGacaGaaeqabaWaaeaaeaaakeaacaWGHbWaaS baaSqaaiabeI7aXbqabaaaaa@3A30@ Pressure decay time constant. 5

(Real)

 
fct_IDP Incident pressure function identifier.

(Integer)

 
FscaleP Ordinate (pressure) scale factor for fct_IDP.

(Real)

[ Pa ]
XC X-coordinate of explosive charge.

(Real)

[ m ]
YC Y-coordinate of explosive charge.

(Real)

[ m ]
ZC Z-coordinate of explosive charge.

(Real)

[ m ]
XA X-coordinate of point A on the free surface.

(Real)

[ m ]
YA Y-coordinate of point A on the free surface.

(Real)

[ m ]
ZA Z-coordinate of point A on the free surface.

(Real)

[ m ]
Dir-X X-component of the normal to the free surface plane.

(Real)

 
Dir-Y Y-component of the normal to the free surface plane.

(Real)

 
Dir-Z Z-component of the normal to the free surface plane.

(Real)

 

Comments

  1. The entire structure must be modeled. Symmetric analysis is not supported.
  2. The surface normal n MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqGqFfpeea0xe9vq=Jb9 vqpeea0xd9q8qiYRWxGi6xij=hbba9q8aq0=yq=He9q8qiLsFr0=vr 0=vr0db8meaabaqaciGacaGaaeqabaWaaeaaeaaakeaacaWHUbaaaa@385F@ should be pointed into the fluid.
  3. Standoff point defined with (Xs, Ys, Zs) is the location where the incident pressure wave is given at time t=0:


    Figure 1.
  4. A plane wave can be simulated using a spherical wave and putting the explosive charge far enough away.
  5. Pressure at the standoff point as a function of time is:(1)
    P i ( t ) = P m e t θ MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqGqFfpeea0xe9vq=Jb9 vqpeea0xd9q8qiYRWxGi6xij=hbba9q8aq0=yq=He9q8qiLsFr0=vr 0=vr0db8meaabaqaciGacaGaaeqabaWaaeaaeaaakeaaciGGqbWaaS baaSqaaiaacMgaaeqaaOWaaeWaaeaacaWG0baacaGLOaGaayzkaaGa eyypa0JaamiuamaaBaaaleaacaWGTbaabeaakiaadwgadaahaaWcbe qaaiabgkHiTmaalaaabaGaamiDaaqaaiabeI7aXbaaaaaaaa@43A9@
    Where,
    P m MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqGqFfpeea0xe9vq=Jb9 vqpeea0xd9q8qiYRWxGi6xij=hbba9q8aq0=yq=He9q8qiLsFr0=vr 0=vr0db8meaabaqaciGacaGaaeqabaWaaeaaeaaakeaacaWGqbWaaS baaSqaaiaad2gaaeqaaaaa@395B@
    Maximum pressure
    t MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqGqFfpeea0xe9vq=Jb9 vqpeea0xd9q8qiYRWxGi6xij=hbba9q8aq0=yq=He9q8qiLsFr0=vr 0=vr0db8meaabaqaciGacaGaaeqabaWaaeaaeaaakeaacaWG0baaaa@3861@
    Time
    θ
    Decay time
    The maximum pressure and decay time can be calculated using:(2)
    P m = K [ W 1 3 R ] a MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqGqFfpeea0xe9vq=Jb9 vqpeea0xd9q8qiYRWxGi6xij=hbba9q8aq0=yq=He9q8qiLsFr0=vr 0=vr0db8meaabaqaciGacaGaaeqabaWaaeaaeaaakeaacaWGqbWaaS baaSqaaiaad2gaaeqaaOGaeyypa0Jaam4samaadmaabaWaaSaaaeaa caWGxbWaaWbaaSqabeaadaWccaqaaiaaigdaaeaacaaIZaaaaaaaaO qaaiaadkfaaaaacaGLBbGaayzxaaWaaWbaaSqabeaacaWGHbaaaaaa @41C4@
    (3)
    θ = K θ W 1 3 [ W 1 3 R ] a θ MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqGqFfpeea0xe9vq=Jb9 vqpeea0xd9q8qiYRWxGi6xij=hbba9q8aq0=yq=He9q8qiLsFr0=vr 0=vr0db8meaabaqaciGacaGaaeqabaWaaeaaeaaakeaacqaH4oqCcq GH9aqpcaWGlbWaaSbaaSqaaiabeI7aXbqabaGccaWGxbWaaWbaaSqa beaadaWccaqaaiaaigdaaeaacaaIZaaaaaaakmaadmaabaWaaSaaae aacaWGxbWaaWbaaSqabeaadaWccaqaaiaaigdaaeaacaaIZaaaaaaa aOqaaiaadkfaaaaacaGLBbGaayzxaaWaaWbaaSqabeaacaWGHbWaaS baaWqaaiabeI7aXbqabaaaaaaa@47E9@
    W MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqGqFfpeea0xe9vq=Jb9 vqpeea0xd9q8qiYRWxGi6xij=hbba9q8aq0=yq=He9q8qiLsFr0=vr 0=vr0db8meaabaqaciGacaGaaeqabaWaaeaaeaaakeaacaWGxbaaaa@3844@
    Explosive mass.
    R MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqGqFfpeea0xe9vq=Jb9 vqpeea0xd9q8qiYRWxGi6xij=hbba9q8aq0=yq=He9q8qiLsFr0=vr 0=vr0db8meaabaqaciGacaGaaeqabaWaaeaaeaaakeaacaWGsbaaaa@383F@
    Distance to the explosion.
    K MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqGqFfpeea0xe9vq=Jb9 vqpeea0xd9q8qiYRWxGi6xij=hbba9q8aq0=yq=He9q8qiLsFr0=vr 0=vr0db8meaabaqaciGacaGaaeqabaWaaeaaeaaakeaacaWGlbaaaa@3838@ , α, K θ MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqGqFfpeea0xe9vq=Jb9 vqpeea0xd9q8qiYRWxGi6xij=hbba9q8aq0=yq=He9q8qiLsFr0=vr 0=vr0db8meaabaqaciGacaGaaeqabaWaaeaaeaaakeaacaWGlbWaaS baaSqaaiabeI7aXbqabaaaaa@3A1A@ and a θ MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqGqFfpeea0xe9vq=Jb9 vqpeea0xd9q8qiYRWxGi6xij=hbba9q8aq0=yq=He9q8qiLsFr0=vr 0=vr0db8meaabaqaciGacaGaaeqabaWaaeaaeaaakeaacaWGHbWaaS baaSqaaiabeI7aXbqabaaaaa@3A30@
    Constants depending on the explosive.
    If W MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqGqFfpeea0xe9vq=Jb9 vqpeea0xd9q8qiYRWxGi6xij=hbba9q8aq0=yq=He9q8qiLsFr0=vr 0=vr0db8meaabaqaciGacaGaaeqabaWaaeaaeaaakeaacaWGxbaaaa@3844@ in kg, R MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqGqFfpeea0xe9vq=Jb9 vqpeea0xd9q8qiYRWxGi6xij=hbba9q8aq0=yq=He9q8qiLsFr0=vr 0=vr0db8meaabaqaciGacaGaaeqabaWaaeaaeaaakeaacaWGsbaaaa@383F@ in meter, P m MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqGqFfpeea0xe9vq=Jb9 vqpeea0xd9q8qiYRWxGi6xij=hbba9q8aq0=yq=He9q8qiLsFr0=vr 0=vr0db8meaabaqaciGacaGaaeqabaWaaeaaeaaakeaacaWGqbWaaS baaSqaaiaad2gaaeqaaaaa@395B@ in MPa and in ms.
    K MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqGqFfpeea0xe9vq=Jb9 vqpeea0xd9q8qiYRWxGi6xij=hbba9q8aq0=yq=He9q8qiLsFr0=vr 0=vr0db8meaabaqaciGacaGaaeqabaWaaeaaeaaakeaacaWGlbaaaa@3838@ α K θ MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqGqFfpeea0xe9vq=Jb9 vqpeea0xd9q8qiYRWxGi6xij=hbba9q8aq0=yq=He9q8qiLsFr0=vr 0=vr0db8meaabaqaciGacaGaaeqabaWaaeaaeaaakeaacaWGlbWaaS baaSqaaiabeI7aXbqabaaaaa@3A1A@ a θ MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqGqFfpeea0xe9vq=Jb9 vqpeea0xd9q8qiYRWxGi6xij=hbba9q8aq0=yq=He9q8qiLsFr0=vr 0=vr0db8meaabaqaciGacaGaaeqabaWaaeaaeaaakeaacaWGHbWaaS baaSqaaiabeI7aXbqabaaaaa@3A30@
    TNT 52.12 1.180 0.0895 -0.185
    PETN 56.21 1.194 0.0860 -0.257
    HBX 53.51 1.144 0.0920 -0.247
  6. A free surface is a plane defined by a point and its normal vector.
  7. The afterflow normal velocity is calculated as:(4)
    v a f t e r f l o w = cos γ ρ R 0 t P ( t ) d t MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqGqFfpeea0xe9vq=Jb9 vqpeea0xd9q8qiYRWxGi6xij=hbba9q8aq0=yq=He9q8qiLsFr0=vr 0=vr0db8meaabaqaciGacaGaaeqabaWaaeaaeaaakeaacaWG2bWaaS baaSqaaiaadggacaWGMbGaamiDaiaadwgacaWGYbGaamOzaiaadYga caWGVbGaam4DaaqabaGccqGH9aqpdaWcaaqaaiGacogacaGGVbGaai 4Caiabeo7aNbqaaiabeg8aYjaadkfaaaWaa8qmaeaacaWGqbWaaeWa aeaacaWG0baacaGLOaGaayzkaaGaamizaiaadshaaSqaaiaaicdaae aacaWG0baaniabgUIiYdaaaa@524C@
    P
    Fluid point.
    C
    Explosive charge point.
    S
    Standoff point.


    Figure 2.
1 Littlewood, J. de Runtz T. 2004. "USA Code". Mecalog Workshop, Sophia Antipolis, France, 2004
2 Cole, Robert H. Underwater Explosion. Princeton University Press, 1948