RD-V: 0560 Air Burst

In Radioss, only the Mach Front region is implemented along the vertical axis (shown in the animation below):

Here the scaled distance $Z$ (cm/g^1/3) is introduced:

The scaled distance suggests that two charges with the same geometry, explosive composition, and ambient conditions will generate self-similar blast waves when their scaled distances match.

This allows the pressure to be computed from experimental data for a given charge ( $W$ ) and a given location ( $R$ ).

The equation used to compute the pressure in /LOAD/PBLAST is the modified Friedlander equation:

The profile of the modified Friedlander equation at a given location is:

In Radioss, the negative phase is ignored and $P^{-}$ is replaced by a $P_{min}$ value (defaults to 10^-20). In this case and until the end of this study, the pressure is only described by:

• $P_{max}$ which replaces $P^{+}$ .
• $d{t}_0$ which replaces $t^{+}$ .
• $P_{min}$ which replaces $P^{-}$ .
• $t^{-}$ and $I_{-}$ vanish.
• $t_a$ is still the same

From $H_c$ , $R_g$ , $Z_c$ and $Z_g$ can be computed and used to obtain $Z_t$ and $Z_g$ in the first chart:

Since there is only a limited number of curves for values of $Z_c$ an interpolation is required.

In general, the interpolation looks like this:

• Identify the two curves bounding ( $f_{k+1}, f_{k-1}$ )
• On each bounding curve, identify the closest value above and under on the abscissa ( ${\left.x_{k+1}\right|}_{f_{k-1}},{\left.x_{k-1}\right|}_{f_{k-1}},{\left.x_{k+1}\right|}_{f_{k+1}},{\left.x_{k-1}\right|}_{f_{k+1}}$ )
• For every value ${\left.x_{k\pm 1}\right|}_{f_{k\pm 1}}$ found previously, identity the corresponding value on the ordinate ( ${\left.y_{k+1}\right|}_{f_{k-1}},{\left.y_{k-1}\right|}_{f_{k-1}},{\left.y_{k+1}\right|}_{f_{k+1}},{\left.y_{k-1}\right|}_{f_{k+1}}$ )
• Calculate the interpolation coefficient on the abscissa on both curves to calculate the interpolate $y_{f_{k\pm 1}}$ on the two bounding curves:
  $$a_{{\left.x\right|}_{f_{k+1}}}=\frac{x_k-{\left.x_{k-1}\right|}_{f_{k+1}}}{{\left.x_{k+1}\right|}_{f_{k+1}}-{\left.x_{k-1}\right|}_{f_{k+1}}}$$
  $$a_{{\left.x\right|}_{f_{k-1}}}=\frac{x_k-{\left.x_{k-1}\right|}_{f_{k-1}}}{{\left.x_{k+1}\right|}_{f_{k-1}}-{\left.x_{k-1}\right|}_{f_{k-1}}}$$
  $$y_{f_{k+1}}={10}^{\left[\left(1-a_{{\left.x\right|}_{f_{k+1}}}\right).\log \left({\left.y_{k-1}\right|}_{f_{k+1}}\right)+a_{{\left.x\right|}_{f_{k+1}}}.\text{log}\left({\left.y_{k+1}\right|}_{f_{k+1}}\right)\right]}$$
  $$y_{f_{k-1}} ={10}^{\left[\left(1-a_{{\left.x\right|}_{f_{k-1}}}\right).\log \left({\left.y_{k-1}\right|}_{f_{k-1}}\right)+a_{{\left.x\right|}_{f_{k-1}}}.\text{log}\left({\left.y_{k+1}\right|}_{f_{k-1}}\right)\right]}$$
• Calculate the interpolation coefficient on $f$ to calculate the final interpolate $y$
  $$a_f=\frac{y_k-y_{f_k-1}}{y_{f_k+1}-y_{f_k-1}}$$
  $$y_k={10}^{\left[\left(1-a_f\right).\log \left(y_{f_{k-1}}\right)+a_f.\log \left(y_{f_{k+1}}\right)\right]}$$
• Now, adapt $f$ , $x$ and $y$ to interpolate any quantities. This interpolation is done for $Z_t$ , $P_{r_{\alpha }}$ and $I_{r_{\alpha }}$ .

After interpolation and determination of $Z_t$ , $H_t$ can be calculated.

If $H_s>H_t$ , the target is above the triple point and therefore should be charged with Exp_data = 1 (Free Air burst).

The quantities are obtained in the following charts after calculating $\alpha =\text{tan}\left(\frac{R_G}{H_c}\right)$ .

After interpolation and determination of $P_{r_{\alpha }}=P_{so},$ the scaled distance $Z$ is determined from the Exp_data = 1 (Free Air Burst) chart and used to find the corresponding values for $P_r$ and $t_a$ .

Repeat the same process with:

and find $i_r=\frac{I_r}{W^{\frac{1}{3}}}$ and $d{t}_0$ .

From these two quantities, the pressure applied to the surface also satisfies the following:

$$P_{blast}\left(t\right)=\left\{\begin{array}{c}{\cos }^2\left(\theta \right).{\text{P}}_{\text{r}}\left(\text{t}\right)+\left(1+\text{cos}\left(\theta \right)-2.{\cos }^2\left(\theta \right)\right).{\text{P}}_{\text{so}}\left(t\right)\\ P_{so}\left(t\right)\end{array}\right.\begin{array}{c}if\cos \left(\theta \right)>0\\ if\cos \left(\theta \right)\le 0\end{array}$$

Where, $\theta$ is the angle between the surface normal and the direction of the detonation point.

This corresponds to the measurement data obtained with the captors (UFC charts).

Previously, $I_{\pm }$ was introduced as the integral of the pressure with respect to time (positive or negative phase). The same process is applied to ${\text{P}}_{\text{r}}$ and $P_{so}$ , with the impulses denoted by ${\text{I}}_{\text{r}}$ and $I_{so}$ . In the charts, ${\text{i}}_{\text{r}}$ and $i_{so}$ are introduced as the impulse divided by $W^{1/3}$ ; the same applies to $d{t}_0$ and $t_a$ .

Model Description

The boundary conditions are represented below:

The two models are tested with 2D shell elements using the /PROP/TYPE1 (SHELL) property and the /MAT/LAW1 (ELASTIC) material, with a rigid body attached. The elements are placed at distance of $R$ from the detonation point ( $W$ is imposed).

The first goal is to compare ${\text{P}}_{\text{r}}$ , $P_{so}$ , ${\text{I}}_{\text{r}}$ , $I_{so}$ , and $d{t}_0$ from the Radioss outputs and the UFC charts. To achieve this, $W$ is imposed $\left(W={500.10}^{3}g\right)$ and only $H_c$ and $R_g$ vary. The second goal is to evaluate Equation 10 for every 10 degrees from 0 degrees to 180 degrees for different points of the list.

Points Considered

UFC Quantities Evaluation

For each point in the table, $P_{so}$ , ${\text{P}}_{\text{r}}$ , $I_{so}$ , ${\text{I}}_{\text{r}}$ and $d{t}_0$ are measured from the pressure profile on the 2D shell elements.

As shown in the graphs, there are a lot of errors in the impulses. These errors come from the construction of the blast wave using the modified Friedlander model. In this model, the decay parameter $b$ is derived from the desired impulse, the peak pressure value ( $P_{so}$ or $P_r$ ) and $d{t}_0$ , under the constraint that the resulting pressure profile cannot be concave. This constraint implies that the maximum achievable impulse corresponds to a triangular pressure shape. Therefore, if the impulse required by the UFC charts exceeds the area of such a triangle, it is impossible to reproduce the desired profile accurately. In this case, a warning is issued.

Without the errors from the impulses, the maximum error obtained is 0.67% and the average error across all data is 0.02%.

These errors are due to interpolation between charts data points and the discretization of the time pressure response.

Formula Evaluation

For point B, M and X, the formula is evaluated every 10 degrees from 0 degree to 180 degrees:

$$P_{blast}\left(t\right)=\left\{\begin{array}{c}{\cos }^2\left(\theta \right).{\text{P}}_{\text{r}}\left(\text{t}\right)+\left(1+\text{cos}\left(\theta \right)-2.{\cos }^2\left(\theta \right)\right).{\text{P}}_{\text{so}}\left(t\right)\\ P_{so}\left(t\right)\end{array}\right.\begin{array}{c}if\cos \left(\theta \right)>0\\ if\cos \left(\theta \right)\le 0\end{array}$$

The average error obtained is 0.02% with a maximum error of 0.04%.

Again, these errors are due to interpolation between charts data points.

I_z Option Evaluation

To evaluate I_z, a displacement is imposed from point L to point O, and the detonation is launched after the displacement.

For I_z = 1, since $R$ is constant and $Z$ is evaluated at time = 0, the pressure ( ${\text{P}}_{\text{r}}$ ) obtained should correspond to point L, even after the displacement.