The numerical results in format postscript: numer.ps.gz

Some numerical experiments

Applications of the fictitious domain method to elastodynamic waves

The case of a heterogeneous anisotropic elastic medium with a crack

We present here some numerical results on the simulation of a non destructive testing experiment. The domain of propagation is a 2D heterogeneous linear elastic medium which might represent part of a nuclear reactor. As we can see in figure 1, it is composed of three different metals. One is isotropic. The other two are anisotropic. The domain of computation is a rectangle, three of its boundaries are absorbing boundaries, the remaining boundary represents a free surface.

Figure 1: The geometry of the problem

The characteristics of the elastic medium we consider are the following

c₁₁¹ =2.324 , c₁₂¹= 1.208, c₂₂¹ = 2.324, c₃₃¹= 0.558

c₁₁² =2.324 , c₁₂²= 0.704, c₂₂² = 0.873, c₃₃²= 0.558

c₁₁³ =3.450 , c₁₂³= 1.160, c₂₂³ = 4.020, c₃₃³= 0.850

The coefficients c₁₃, c₂₃ are zero for the three materials and the density is taken equal to one. The size of the grid is 400 × 200, h=0.05m, N_L=10 and the source is located at the point (10m,0m). For the absorbing layer model we taken d = 15 h, d(x) = d₀ ( x/ d )² with d₀ = | log(1/ R) | 3 V_p/ 2 d and R=0.01. To visualize the propagation of an elastic wave in such a medium we first compute its response to a point source term located at the point S. The source emits pressure waves and the quantity represented is the norm of the velocity.

First experiment : without crack. Point source S.

t=0.997 s t=2.993 s t=4.987 s

t=6.983 s t=8.998 s t= 9.975 s

Figure 2: First experiment: A point load source. The norm of the velocity is represented at different times.

Second experiment : with a crack. Pressure field imposed on the free boundary.

Figure 3: The geometry of the problem with the crack

In the second experiment the source is a pressure field imposed on a small portion of the free boundary. More precisely we impose the following condition on the free boundary

t_xy(x) = 0

on the Free Surface

t_yy(x) =

ì
í
î

P₀(x)

(

1-cosg(x)

)

cos2 g(x)

if 0 £ g(x) £ 2 p

if not

with g(x) defined by

g(x) = 2 p f₀

æ
ç
ç
è

t +

x-x_q₁

2 V_p

ö
÷
÷
ø

P₀(x) =

ì
ï
ï
ï
í
ï
ï
ï
î

P₀

æ
ç
ç
è

1- cos

x-x_p₁

x_p-x_p₁

ö
÷
÷
ø

for x_p₁ £ x £ x_p

P₀

for x_p £ x £ x_q

P₀

æ
ç
ç
è

1- cos

x-x_q₁

x_q-x_q₁

ö
÷
÷
ø

for x_q £ x £ x_q₁

and where P₀ = 14.451 Pa, x_p₁ =6.45m, x_p =6.7m, x_q =7.7m, x_p₁ =7.95m and f₀ = V_s/ h N_L is the central frequency computed for the smallest V_s. This source will generate a truncated shear wave. The purpose of the experiment is to detect the possible presence of a crack. We consider here that the material has a crack, the new geometry of the problem is given in figure 3.

A free boundary condition is imposed on the two lips of the crack and the incident wave is directed toward the crack. We present in figure 4 the norm of the velocity at different times.

t = 0.997 sec t = 2.993 sec t = 4.987 sec

t = 6.983 sec t = 8.998 sec t = 9.975 sec

Figure 4: Second experiment: diffraction by a crack of a truncated Shear wave