A computational model of blast loading on the human eye

Abstract

Ocular injuries from blast have increased in recent wars, but the injury mechanism associated with the primary blast wave is unknown. We employ a three-dimensional fluid–structure interaction computational model to understand the stresses and deformations incurred by the globe due to blast overpressure. Our numerical results demonstrate that the blast wave reflections off the facial features around the eye increase the pressure loading on and around the eye. The blast wave produces asymmetric loading on the eye, which causes globe distortion. The deformation response of the globe under blast loading was evaluated, and regions of high stresses and strains inside the globe were identified. Our numerical results show that the blast loading results in globe distortion and large deviatoric stresses in the sclera. These large deviatoric stresses may be indicator for the risk of interfacial failure between the tissues of the sclera and the orbit.

Appendix

1.1 Flow solver

To resolve the propagation and scattering of a blast (shock) wave, we considered the full compressible Navier-Stokes equations for air. The equations are written in a conservative form as,

$$\begin{aligned}&\frac{\partial \rho }{\partial t}+\frac{\partial (\rho u_i )}{\partial x_i }=0,\end{aligned}$$

(4)

$$\begin{aligned}&\frac{\partial (\rho u_i )}{\partial t}+\frac{\partial (\rho u_i u_j )}{\partial x_j }+\frac{\partial p}{\partial x_i }-\frac{\partial \tau _{ij} }{\partial x_j }=0,\end{aligned}$$

(5)

$$\begin{aligned}&\frac{\partial e}{\partial t}+\frac{\partial (u_j (e+p))}{\partial x_j }-\frac{\partial (u_k \tau _{jk} +q_j )}{\partial x_j }=0,\end{aligned}$$

(6)

$$\begin{aligned}&e=\frac{p}{\gamma -1}+\frac{1}{2}\rho u_i u_i , \end{aligned}$$

(7)

where $\rho , u_{i}, p$, and $e$ are the density, velocity, pressure, and total energy, respectively, and $\tau _{ij}$ is the viscous stress, $q_{j}$ is the heat flux, and $\gamma$ is the specific heat ratio (1.4 for air). Equations (4–6) were spatially discretized by a sixth-order central compact finite difference scheme (Lele 1992) and integrated in time using a four-stage Runge-Kutta method. An eight-order implicit spatial filtering proposed by Gaitonde et al. (1999) was applied at the end of each time step to suppress high-frequency dispersion errors. In order to resolve the discontinuity in the flow variables caused by a shock wave with the current non-dissipative numerical scheme, the artificial diffusivity method proposed by Kawai and Lele (2008) had been applied. In this method, the viscous stress and heat flux are written as

$$\begin{aligned} \tau _{ij}&= \left( {\mu +\mu ^{*}} \right) \left( {\frac{\partial u_i }{\partial x_j }+\frac{\partial u_j }{\partial x_i }} \right) +\left( {\beta ^{*}-\frac{2}{3}\mu } \right) \frac{\partial u_k }{\partial x_k }\delta _{ij},\end{aligned}$$

(8)

$$\begin{aligned} q_j&= \left( {\kappa +\kappa ^{*}} \right) \frac{\partial T}{\partial x_j },\quad T=\frac{p}{\rho R}, \end{aligned}$$

(9)

where $\mu$ and $\kappa$ are the physical viscosity and thermal diffusivity, respectively, while $\mu ^{*}$ is the artificial shear viscosity, $\beta ^{*}$ is the artificial bulk viscosity, and $\kappa ^{*}$ is the artificial thermal diffusivity. On the non-uniform Cartesian grid, these artificial diffusivities were adaptively and dynamically evaluated by

$$\begin{aligned} \mu ^{*}&= C_\mu \rho \overline{\left| {\frac{\partial ^{4}S}{\partial x_k^4 }} \right| \Delta x_k^6 } ,\quad \beta ^{*}=C_\beta \rho \overline{\left| {\frac{\partial ^{4}S}{\partial x_k^4 }} \right| \Delta x_k^6 } ,\end{aligned}$$

(10)

$$\begin{aligned} \kappa ^{*}&= C_\kappa \frac{\rho c}{T}\overline{\left| {\frac{\partial ^{4}}{\partial x_k^4 }\left( {\frac{{\text{ RT }}}{\gamma -1}} \right) } \right| \Delta x_k^5 }, \end{aligned}$$

(11)

where $C_\mu , C_\beta$, and $C_\kappa$ are user-specified constants, $c$ is the speed of sound, $S$ is the magnitude of the strain rate tensor,

$$\begin{aligned} S_{ij} =\frac{1}{2}\left( {\frac{\partial u_i }{\partial x_j }+\frac{\partial u_j }{\partial x_i }} \right) , \end{aligned}$$

(12)

$\Delta x_k$ is the grid spacing, and over-bar denotes Gaussian filtering (Cook and Cabot 2005). We used $C_\mu =0.002,C_\beta =1.0$, and $C_\kappa =0.01$ as suggested in Kawai and Lele (2008), and the fourth derivatives were computed by a fourth-order central compact scheme (Lele 1992). From Eqs. (10) and (11), artificial diffusivities are significantly larger only in the region where the steep gradient of flow variables exists, and ensure numerical stability in that region.

1.2 Structural solver

The displacement vector $\mathbf{d}(\mathbf{x, }t)$ describes the motion of each point in the deformed solid as a function of space x and time $t$. The deformation gradient tensor $F_{ik}$ can be defined in terms of the displacement gradient tensor $\frac{\partial d_i }{\partial x_k }$ as follows:

$$\begin{aligned} F_{ik} =\delta _{ik} +\frac{\partial d_i }{\partial x_k }, \end{aligned}$$

(13)

where $\delta _{ik}$ is the Kronecker delta symbol, defined as follows:

$$\begin{aligned} \delta _{ik} =\left\{ {{\begin{array}{ll} {1,}&{} {i=k} \\ {0,}&{} {i\ne k} \\ \end{array} }} \right. , \end{aligned}$$

(14)

The right Cauchy green tensor is defined in terms of the deformation gradient tensor as follows:

$$\begin{aligned} C_{ij} =F_{ki} F_{kj} , \end{aligned}$$

(15)

The invariants of the right Cauchy green tensor are defined as follows:

$$\begin{aligned} {\begin{array}{l} {I_1 =\lambda _1 +\lambda _2 +\lambda _3 } \\ {I_2 =\lambda _1 \lambda _2 +\lambda _2 \lambda _3 +\lambda _3 \lambda _1 } \\ {I_3 =\lambda _1 \lambda _2 \lambda _3 } \\ \end{array} }, \end{aligned}$$

(16)

where $\lambda _{i}$ are eigenvalues of the right Cauchy green tensor. The strain–energy density function $\psi$ of a neo-Hookean, quasi-incompressible solid is written as (Holzapfel 2006):

$$\begin{aligned} \psi (C)=\frac{G}{2}\left( {\overline{I_1 } -3} \right) +\frac{K}{4}\left( {I_3 -\ln I_3 -1} \right) , \end{aligned}$$

(17)

where $G$ and $K$ are shear and bulk moduli and $\overline{I} _1 =I_3^{-1/3} I_1$. The Cauchy stress is given in terms of strain energy function $\psi$ as follows:

$$\begin{aligned} \sigma _{ij} =\frac{1}{J}\frac{\partial \psi }{\partial F_{ik} }F_{jk} , \end{aligned}$$

(18)

where $J = {\text{ det }}(\mathbf{F})$ denotes the volume change ratio. The governing equations for the structure are the Navier equations (momentum balance equation in Lagrangian form) and are written as follows:

$$\begin{aligned} \rho _s{\frac{\partial ^{2}d_{i}}{\partial t^2}}={\frac{\partial \sigma _{ij}}{\partial x_{j}}}+\rho _{s}f_{i}, \end{aligned}$$

(19)

where $i$ and $j$ range from 1 to 3, $\rho _{s}$ is the density of the structure, $d_{i }$ is the displacement component in the $i$ direction, $t$ is the time, $\sigma$ is the Cauchy stress tensor, and $f_{i}$ is the body force component in the $i$ direction. The momentum balance equation was solved by finite elements using the Galerkin method for spatial discretization, which yielded the following system of ordinary differential equations for the nodal displacement vector d:

$$\begin{aligned} M{{\ddot{d}}_{n+1}} +Kd_{n+1} =F_{n+1} , \end{aligned}$$

(20)

where $M$ is the lumped mass matrix and $K$ is the stiffness matrix. The Galerkin method was implemented in Tahoe$^{\copyright }$,⁴ an open-source, Lagrangian, three-dimensional, finite element solver. The central-difference method was used for the time integration, which resulted in an explicitly and conditionally stable second-order scheme (Hughes 1987):

$$\begin{aligned} d_{n+1}&= d_n +\Delta t{\dot{d}}_n +\frac{\Delta t^{2}}{2}{\ddot{d}}_n \nonumber \\ {\dot{d}}_{n+1}&= {\dot{d}}_n +\frac{\Delta t}{2}\left( {{\ddot{d}}_n +{\ddot{d}}_{n+1} } \right) , \end{aligned}$$

(21)

The constraints of the time step of the governing equations for the fluid and solid are different and are governed by the wave speed in the respective domain. In the present case, the wave speed inside the eye is larger than that in the ambient air (see Sect. 2.2) because of the higher wave speed in the water-like fluids (aqueous and vitreous humor) inside the eye. Thus, the time step was chosen to resolve the longitudinal wave speed within the structure. All simulations were performed on quad core 2.83 GHz Intel$^{\circledR }$ Xeon$^{\circledR }$ processors in a parallel computing Linux environment.

1.3 Fluid–structure interaction coupling

A partitioned approach was used to couple the flow and the structure solvers (Bhardwaj and Mittal 2012). In this approach, flow and structure solvers are coupled such that they exchange data at each time step (Fig. 18a). In general, there are two coupling methods used in fluid–structure interaction algorithms—explicit (or weak, one-way) coupling and implicit (or strong, two-way) coupling. As the name suggests, explicit and implicit coupling integrate the governing equations of the flow and the structure domain explicitly and implicitly in time. Explicit coupling is computationally inexpensive and may be subject to stability constraints, which depends on the structure-fluid density ratio $(\rho _{s} /\rho _{f} )$ (Zheng et al. 2010). On the other hand, implicit coupling is robust, computationally expensive and does not introduce stability constraints. Explicit coupling is a good candidate in cases where $\rho _{s} /\rho _{f}$ is large, for example air–tissue interaction during phonation of vocal folds in the larynx, while an implicit scheme is needed for low values of $\rho _{s} /\rho _{f}$, for example blood–tissue interaction in cardiovascular flows. In the latter case, the structure will respond strongly even with small perturbations from the fluid and vice versa. In the present paper, $\rho _{s} /\rho _{f} \sim 800$ and explicit coupling is used for the simulations.

Fig. 18

a Partitioned approach, b data exchange between flow and solid solver, c algorithm of FSI solver

In explicit coupling, the flow solution is marched by one time step with the current deformed shape of the structure and the velocities of the fluid-structure interface act as the boundary conditions in the flow solver (Fig. 18b). This boundary condition represents continuity of the velocity at the interface (no slip on the solid surface):

$$\begin{aligned} u_{i,f} ={\dot{d}}_{i,s} \end{aligned}$$

(22)

where subscripts $f$ and $s$ denote fluid and structure, respectively. The pressure loading on the structure surface exposed to the fluid domain is calculated at the current location on the structure using the interpolated normal fluid pressure at the boundary intercept points via a tri-linear interpolation (bilinear interpolation for 2D) as described by Mittal et al. (2008). This boundary condition represents continuity of the traction at the solid–fluid interface:

$$\begin{aligned} \sigma _{ij,f} n_j =\sigma _{ij,s} n_j , \end{aligned}$$

(23)

where $n_{j}$ is the local surface normal pointing outward from the surface. The structure solver is marched by one time step with the updated fluid dynamic forces (Fig. 18c).

1.3.1 Immersed boundary method

The compressible Navier-Stokes equations for the fluid flow with complex structure boundaries inside the fluid domain were solved using the sharp-interface immersed boundary method of Mittal et al. (2008). In this method, the surface of the structure and the fluid domain are represented by an unstructured surface mesh and a Cartesian grid, respectively. The surface mesh of the structure and Cartesian grid of the fluid domain consists of triangular elements and cells (cube or cuboids), respectively. The surface mesh is “immersed” inside the fluid grid. At the pre-processing stage before integrating governing equations, the cells of the fluid domain were marked according to their location with respect to the surface mesh. The cells whose centers were located inside the surface mesh were identified and tagged as “body” cells, and the other points outside the surface mesh were “fluid” cells as shown in Fig. 19. Note that only cell centers are shown in Fig. 19. Any body cell which has at least one fluid cell neighbor was tagged as a “ghost cell” (Fig. 19). The center of this ghost cell is referred to as a “ghost point.” The boundary condition on the surface mesh was imposed by specifying an appropriate value at the ghost point. A “normal probe” was extended from the ghost point to intersect the surface mesh. This intersection point is referred to as the “body intercept.” The probe was further extended into the fluid to the “image point” such that the body intercept lies midway between the image and ghost points. A linear interpolation was used along the normal probe to compute the value at the ghost point based on the boundary intercept value and the value estimated at the image point. The value at the image point itself was computed through a tri-linear interpolation from the surrounding fluid nodes. This procedure leads to a nominally second-order accurate specification of the boundary condition on the surface mesh (Mittal et al. 2008).

Fig. 19

Schematic of the immersed boundary method

1.4 Validation of the model

In this section, we present the validation of our FSI model against the experiments reported for deformation of the globe by the blast wave. To the best of our knowledge, experimental data for the deformation of the globe at high-pressure blasts are not currently available and we have used data reported by Alphonse et al. (2012b) at low-pressure blasts for the validation of the model. They tested 10 g Pyrodex$^{\circledR }$ gunpowder and commercial firecrackers and created low-energy blasts to measure overpressures inside and outside the eye. The charge was offset 2.0 cm from the front of the cornea to minimize the amount of material projected toward the eye. We simulated the experiments conditions for blast of 10 g Pyrodex$^{\circledR }$ gunpowder at a distance of 7 cm from the cornea to the center of the charge (Fig. 20). Four numerical probes were used in the simulation at same respective positions of sensors mounted in the experimental setup (Fig. 20). The initial condition for the pressure and the computational domain are shown in Fig. 20. The initial pressure was calculated using energy of 10 g Pyrodex$^{\circledR }$ gunpowder (Alphonse 2012a). The measured static pressures outside the eye, IOP, and wave velocity outside the eye by Alphonse et al. (2012b) and respective calculated values are compared in Table 7. The maximum percentage error in numerical values as compared to mean experimental ones is around 7 % and it verifies the validity of our model at low-pressure blasts.

Fig. 20

Initial conditions and computational domain for simulation done for validation against experiments

Table 7 Comparison between measured values by Alphonse et al. (2012b) and values obtained by simulation