/MONVOL/FVMBAG1

• Gas materials are specified in separate /MAT/GAS cards.
• Composition of injected gas mixture and injector properties are specified in separate /PROP/INJECT1 or /PROP/INJECT2 cards.
• Automatic Finite Volume meshing in specified coordinate system, given by a frame.

Format

Definitions

Comments

1. The airbag external surface should be built only from 4- and 3-noded shell elements. The airbag external surface cannot be defined with option /SURF/SEG, nor with /SURF/SURF, if a sub-surface is defined in /SURF/SEG.
2. External surfaces shall compose a closed volume with normals must oriented outwards.
3. Abscissa scale factors are used to transform abscissa units in airbag functions, for example:(1)
   $$\text{F}(t\prime )={\text{f}}_t\left(\frac{t}{{\mathit{Ascale}}_t}\right)$$
   Where,

   $t$

   Time

   ${\mathrm{f}}_t$

   Function of fct_ID_t

   (2)

   $$\text{F}(P\prime )={\text{f}}_P\left(\frac{P}{{\mathit{Ascale}}_P}\right)$$
   Where,

   $P$

   Pressure

   ${\mathrm{f}}_P$

   Function of fct_ID_P

   The options are obsolete. Normally, the curve scaling parameters are used instead.

4. Pressure and temperature of external air and the initial pressure and temperature of air inside of airbag is set to P_ext and T₀.
5. The gas flow in FVMBAG1 is solved using finite volumes.

   Some of these finite volumes can be entered by you through a group of solids, located inside the airbag and filling a part or the total internal volume. If there still exists a part of the internal volume which is not discretized by user-defined solids, an automatic meshing procedure produces the remaining volumes. This can be used for example to model a canister.

   A finite volume consists in a set of triangular facets. Their vertices do not necessarily coincide with the nodes of the airbag. The airbag envelope can be modeled with 4-node or 3-node membranes; however, 3 nodes are recommended.

   
   
   Figure 1.

   
   
   Figure 2.

6. Venting through vent holes:

   If I_form = 1, venting velocity is computed from Bernoulli equation using local pressure in the airbag.

   The exit velocity is given by:(3)

   $$u^2=\frac{2\gamma }{\gamma -1}\frac{P}{\rho }\left(1-{\left(\frac{P_{\mathit{ext}}}{P}\right)}^{\frac{\gamma -1}{\gamma }}\right)$$

   The mass out flow rate is given by:

   If I_form = 2, venting velocity is computed from the Chemkin equation:(4)

   $$v=Fscal{e}_v\cdot f_v(P-P_{ext})$$

   Where, $f_v$ is defined by fct_ID_v.

   If I_form = 3, venting velocity is equal to the component of the local fluid velocity normal to vent hole surface. Local density and energy are used to compute outgoing mass and energy through the hole.

7. When there is no sensor which activates gas injection, the vent holes and porosity becomes active, if time T becomes greater than the T_start, or if the pressure P exceeds P_def value longer than the time given in $\text{Δ}t{P}_{def}$ .
8. When at least one of the injectors is activated by the sensor, then activation of venting and porosity options is controlled by I_ttf.

   T_inj is the time of the first injector to be activated by the sensor.

   I_ttf = 0

   	Venting, Porosity
   Activation	When $P>\text{Δ}{P}_{def}$ longer than the time $\text{Δ}t{P}_{def}$ , or $T>T_{start}$
   Deactivation	Tstop
   Time dependent functions	No shift

   I_ttf = 3

   	Venting, Porosity
   Activation	When $T>T_{inj}$ and $P>\text{Δ}{P}_{def}$ longer than the time $\text{Δ}t{P}_{def}$ , or $T>T_{inj}+T_{start}$
   Deactivation	$T_{inj}+T_{stop}$
   Time dependent functions	Shifted by $T_{inj}+T_{start}$

   All other related curves are active when the corresponding venting, porosity or communication option is active.

   The variety of I_ttf values comes from historical reasons. Values I_ttf=1 and 2 are obsolete and should not be used. Usual values are I_ttf=0 (no shift) or I_ttf=3 (all relative options are shifted by T_inj).

9. If surf_ID_v ≠ 0 (surf_ID_v is defined) the vent hole area is computed as:(5)
   $$vent\_holes\_area=A_{vent}\cdot {\mathrm{f}}_A\left(\frac{A}{A_0}\right)\cdot {\mathrm{f}}_t\left(t\right)\cdot {\mathrm{f}}_P\left(P-P_{ext}\right)$$
   Where,

   $A$

   Area of surface surf_ID_v

   $A_0$

   Initial area of surface surf_ID_v

   ${\mathrm{f}}_t$ , ${\mathrm{f}}_P$ and ${\mathrm{f}}_A$

   Functions of fct_ID_t, fct_ID_P and fct_ID_A

10. In the case of activated venting closure the vent holes surface is computed as:(6)
    $$vent\_holes\_area=A_{vent}\cdot A_{non\_impacted}\cdot {\mathrm{f}}_t\left(t\right)\cdot {\mathrm{f}}_P\left(P-P_{ext}\right)\cdot {\mathrm{f}}_A\left(\frac{A_{non\_impacted}}{A_0}\right)$$
    (7)
    $$+B_{\mathit{vent}}\cdot A_{\mathit{impacted}}\cdot {\text{f}}_{t^{\prime }}\left(t\right)\cdot {\text{f}}_{P^{\prime }}\left(P-P_{\mathit{ext}}\right)\cdot {\text{f}}_{A^{\prime }}\left(\frac{A_{\mathit{impacted}}}{A_0}\right)$$
    With impacted surface:(8)

    $$A_{\mathit{impacted}}=\sum _{e\in S_{\mathit{vent}}}\frac{n_c\left(e\right)}{n\left(e\right)}{A}_e$$
    and non-impacted surface:(9)

    $$A_{\mathit{non}\_\mathit{impacted}}=\sum _{e\in S_{\mathit{vent}}}\left(1-\frac{n_c\left(e\right)}{n\left(e\right)}\right)A_e$$

    
    
    Figure 3.

    Where for each element e of the vent holes surf_ID_v, $n_c\left(e\right)$ means the number of impacted nodes among the $n\left(e\right)$ nodes defining the element.

    A₀ is the initial area of surface surf_ID_v

    f_t, f_P and f_A are functions of fct_ID_t, fct_ID_P and fct_ID_A

    f_t', f_P' and f_A' are functions of fct_ID_t', fct_ID_P' and fct_ID_A'

11. Radioss ends with a Starter error, if surf_ID_v = 0 (surf_ID_v is not defined) (I_form=1 or 2).
12. Functions fct_ID_t and fct_ID_P are equal to 1, if they are not specified (null identifier).
13. Function fct_ID_A is assumed to be equal to 1, if it is not specified.
14. To account for contact blockage of vent holes and porous surface areas, flag I_BAG must be set to 1 in the correspondent interfaces (Line 3 of interface /INTER/TYPE7 or /INTER/TYPE23). If not, the nodes impacted into the interface are not considered as impacted nodes in the previous formula for A_impacted and A_{non_impacted}.
15. Leakage by porosity formulations, the mass flow rate flowing out is computed as:
    • Iform_ps = 1 ${\dot{m}}_{\mathit{out}}=A_{\mathit{eff}}\sqrt{2P\rho }{Q}^{\frac{1}{\gamma }}\sqrt{\frac{\gamma }{\gamma -1}\left[1-Q^{\frac{\gamma -1}{\gamma }}\right]}$ (Isentropic - Wang Nefske)
    • Iform_ps = 2 ${\dot{m}}_{\mathit{out}}=A_{\mathit{eff}}\rho v(P-P_{\mathit{ext}})$

      Where, v is the outflow gas velocity (Chemkin)

    • Iform_ps = 3 ${\dot{m}}_{\mathit{out}}=A_{\mathit{eff}}\sqrt{2\rho (P-P_{\mathit{ext}})}$ (Graefe)

    The effective venting area A_eff is computed according to the input in the /LEAK/MAT input for fabric materials of TYPE19 or TYPE58.

16. If leakage blockage is activated, Iblockage=1, the effective venting area is modified as:(10)
    $$A_{\mathit{eff}}=A_{\mathit{non}\_\mathit{impacted}}$$

    $A_{\mathit{non}\_\mathit{impacted}}$ is non-impact surface 10

    The blockage will be active only if flag I_BAG is set to 1 in the concerned contact interfaces (line 3 of interface TYPE7 and TYPE23).

17. Automatic finite volume meshing parameters.
18. The finite volumes are generated in two steps.
    • The first step generates vertices lying exclusively on the envelope of the airbag. You can update the finite volume along with the deformation of the envelope and correspond to the following procedure (displayed in 2D for purpose of clarity):

      
      
      Figure 4.

      This procedure requires the input of the direction $V_3$ , named cutting direction, and of the direction $V_1$ . A second direction $V_2$ in the plan normal to the cutting direction will be computed. In order to position the finite volumes and to determine the cutting width in both direction $V_1$ and $V_2$ , an origin O must be provided as well as a length L_i, counted both positively and negatively from the origin, and a number of steps N_i. The cutting width is then given by:(11)

      $$W_i=\frac{2L_i}{N_i}$$

      It is required that the box drawn in the horizontal plane (normal to $V_3$ ) by the origin O and the length L_i, counted both positively and negatively from O, includes the bounding-box of the envelope of the volume to mesh projected in this plane. This is necessary to ensure that this volume in entirely divided into finite volumes.

    • The second step performs horizontal cutting of the finite volumes, and may be useless in many cases of tightly folded airbags. It is required especially when injection is made in a canister filled by the injected gas before unfolding the airbag.
    This second step may generate vertices located inside the airbag. In order for them to be moved along with the inflation of the airbag, each is attached to a vertical segment (parallel to direction $V_3$ ) between two vertices lying on the envelope of the airbag (Figure 4). The local coordinates of the vertex within its reference segment remain constant throughout the inflation process.

    
    
    Figure 5.

    The horizontal cutting width is given by:(12)

    $$W_3=\frac{2L_3}{N_3}$$

    It is not necessary that the segment given in the $V_3$ direction by the origin O and length L₃, counted both positively and negatively, includes the bounding-box of the envelope of the volume to mesh projection on the $V_3$ direction, since at the second step only existing finite volumes are cut.

19. Actual vector $V_1$ used for automatic meshing is obtained after orthogonalization of the input vector with respect to vector $V_3$ .
20. When a finite volume fails during the inflation process of the airbag (volume becoming negative, internal mass or energy becoming negative), it is merged to one of its neighbors so that the calculation can continue. Two merging approaches are used:
    • Global merge: a finite volume is merged if its volume becomes less than a certain factor multiplying the mean volume of all the finite volumes. The flag I_gmerg determines if the mean volume to use is the current mean volume (I_gmerg =1) or the initial mean (I_gmerg =2). The factor giving the minimum volume from the mean volume is C_gmerg.
    • Neighborhood merge: a finite volume is merged if its volume becomes less than a certain factor multiplying the mean volume of its neighbors. The factor giving the minimum volume from the mean volume is C_nmerg.
21. In the case of both C_gmerg and C_nmerg are not equal to 0, means both merging approaches will be used simultaneously. In case of a strong shock, it is recommended to set q_a = 1.1 and q_b = 0.05.
22. When two layers of fabric are physically in contact, there should be no possible flow between finite volumes, which is numerically not the case because of interface gap. H_min represents a minimum height for the triangular facets below which the facet is impermeable. Its value should be close to the gap of the self-impacting interface of the airbag.
23. N_layer, N_facmax, and N_ppmax are memory parameters that help the finite volume creation process. Changing their value cannot cause the calculation to stop. Increasing the leads to a higher amount of memory and a smaller computation time for automatic meshing.
24. During the finite volume creation process, plane polygons are first created, which are then assembled into closed polyhedra and decomposed into triangular facets. N_ppmax is the maximum number of vertices of these polygons.
25. Automatic finite volume meshing based on reference geometry can be activated with flag I_ref=1. It only works with a reference geometry based on /REFSTA and /XREF. The flag is not supported when disjointed reference geometry /EREF is used. Note that for I_ref=1, the frame definition for automatic meshing should refer to non-folded reference geometry.
26. The option kmesh controls type of FVM meshing of internal airbag volume. The polyhedron meshing method, kmesh =1 was the default method used in 2017.2 and before. If grbric_ID ≠ 0, kmesh is ignored and the tetra FVM mesh is specified by the user created.
27. Surface surf_ID_in is used to take internal surfaces or baffles into account as obstacles to the gas flow inside the monitored volume. Internal surfaces are taken into account in FVM only if the monitored volume is meshed automatically with polyhedron or if it is filled with solid elements, like TETRA4 (possibly HEXA and PENTA) with nodes coinciding with the monitored volume external and internal surface nodes (these solids must be declared in grbrick_ID). A porosity ranging from 0: no porosity up to 1: full porosity (vent) can be applied to internal surface fabrics only if their material model is LAW19 or LAW58. Injector surface can also be defined on an internal surface in which case the gas flow direction is opposite to the internal surface normal orientation.
28. The lost heat flow is given by:(13)
    $$\dot{\text{Q}}\left(x,t\right)=H_{\mathit{conv}}\cdot \text{Area}\left(x,t\right)\cdot \left(\text{T}\left(x,t\right)-T_0\right)$$
29. If an element of a vent hole surface (surf_ID_v) belongs to an injector (surf_ID_inj) it will be ignored from the vent hole. A constant correction factor f computed at time t=0 is applied to the total vent hole surface:(14)
    $$f=\frac{S_{\mathit{vent}}}{S_{\mathit{vent}}-{\text{S}}_{\mathit{injector}}}$$
30. If an element of a porous surface also belongs to an injector (surf_ID_inj), it will be ignored from the porous surface.
31. The time to switch T_switch to Uniform Pressure is relative to the time to fire.
32. With option I_switch=2, the airbag is always computed with finite volume method, even when only 1 finite volume remains. The gas parameters are identical before and after switching to a single finite volume. Some variation of pressure or gas parameters may be seen with a switch to uniform pressure method (I_switch=1).
33. P_switch is the ratio of standard deviation of the Finite Volume pressures to the airbag average pressure.(15)
    $$P_{switch}=\frac{\text{SD(FV pressure)}}{\text{Average pressure}}$$

    This ratio can be output using the /TH/MONVOL variable UPCRIT. P_switch approaches zero as the pressure in each finite volume approaches the average pressure in the airbag.