**6. Explicit dynamics approach in FEM for the impact scenarios**

The finite element method has been used by many scientists and engineers for decades to advance systems in different engineering applications for the validations, while reducing the experimental investments and the expenditure of the tests. This widely known method developed via the contributions of the researchers all around the world. While FEM problem is thriving, we will break the problem domain into a collection of 1D (dimensional), 2D and 3D shapes to form the discretization. Starting with the simplest way, {*F*} = [*K*] {*D*} where *F, K*, and *D*, are the nodal force vector, global stiffness matrix and nodal displacement vector, respectively, as usually known. The displacements in this equation are the unknowns which should be found and lead us to figure out the strains and stresses, etc. To solve the equations, generally, the direct stiffness method, variational technique, or weighted residual approaches are used. However, this fundamental technique is now capable enough to handle the issues lately popular like the fracture mechanics with an extended finite element method, cohesive modeling, contact algorithms, smoothed-particle hydrodynamics (SPH) and Arbitrary Lagrangian–Eulerian Method (ALE), etc. Regarding recent a few decades, requisition of a nonlinear mechanics or engineering plasticity, for the delicate and slightly designs which should be robust in tough conditions, induced the durability tests including impacts and crashes. Explicit dynamics approach in FEM emerges with this idea to untangle the difficulties of the impacts, drop tests, explosions, and collision scenarios.

Vehicle technology under the perspective of the crash worthiness and safety during the collisions via crash tests to explicit dynamics simulations of the FEM is a trustworthy way to follow up. If we have a dekko to the literature some valuable studies can be determined [5, 6, 19] about the vehicle crash worthiness and the safety, while some books [6, 10, 19] explain the details of explicit dynamics one of the most challenging nonlinear problems in structural mechanics.

The implicit FE solution technique delivers accurate results as long at the stress levels in the beam do not exceed the yield-stress value, thus remaining in the linear (elastic) region of the material curve. Because the implicit solver usually obtains a result with a single step, according to the slope of the material curve, which is represented by Young's modulus, for the plastic region of the material there will be an inaccuracy whereat the slope is changed up the plastic characteristics. It means that the implicit solver is very powerful for linear (static) finite element analysis (FEA); thence, it provides an unquestioningly stable solution methodology.

Unfortunately, the implicit solution approximation may have difficulties with the immensely nonlinear finite element models. Additionally, inverting extremely large stiffness matrices with the implicit solver might consume too much time and memory. Although this method is not logical to use in highly nonlinear problems, it can able to calculate the equations with the help of the iteration cycle, which is applying the load in minor increments called as "timesteps" [19].

During the accident, the vehicle body faces with the impact loads propagate localized plastic hinges and buckling. So the large deformations and rotations with contact and heaping between the many parts can be determined. One of the most important ideas for the explicit dynamics is the wave effects, primarily involved in the deformation stage, associated with the higher stress levels. When the stress level exceeds the yield strength and/or the critical buckling load and the localized structural deformations comprise throughout a few wave transits in the frame of the vehicle. The effects of the inertia will chase this step and also dominate the ensuing transient reaction. The collision could be considered as a less dynamic incident if there is a comparison between the ballistic impact and the vehicle crash. Closed-form analytical problems for the impact scenarios in structural mechanics offer tough times for everyone. Therefore, the numerical methods arise with the practical options to decipher the unknowns. A set of nonlinear partial differential equations of motion in the space-time domain is solved numerically by the explicit dynamics approximation, related to the stress–strain properties of the material about the initial and boundary conditions. To obtain a set of second order differential equations in time, the equations in space are discretized by the solution in the course of formulating the problem in a weak variational form and supposing an acceptable displacement field. Subsequently, in the time domain, discretization calculates the system of equations. If the selected integration parameters deliver the equations coupled, the technique is defined as implicit, while the solution is unconditionally stable. Furthermore, if the solution will be defined as explicit, the equations should be decoupled by the selection of the integration parameters, and it is conditionally stable [6].

and *D*, are the nodal force vector, global stiffness matrix and nodal displacement vector, respectively, as usually known. The displacements in this equation are the unknowns which should be found and lead us to figure out the strains and stresses, etc. To solve the equations, generally, the direct stiffness method, variational technique, or weighted residual approaches are used. However, this fundamental technique is now capable enough to handle the issues lately popular like the fracture mechanics with an extended finite element method, cohesive modeling, contact algorithms, smoothed-particle hydrodynamics (SPH) and Arbitrary Lagrangian–Eulerian Method (ALE), etc. Regarding recent a few decades, requisition of a nonlinear mechanics or engineering plasticity, for the delicate and slightly designs which should be robust in tough conditions, induced the durability tests including impacts and crashes. Explicit dynamics approach in FEM emerges with this idea to untangle the difficulties

Vehicle technology under the perspective of the crash worthiness and safety during the collisions via crash tests to explicit dynamics simulations of the FEM is a trustworthy way to follow up. If we have a dekko to the literature some valuable studies can be determined [5, 6, 19] about the vehicle crash worthiness and the safety, while some books [6, 10, 19] explain the details of explicit dynamics one of the most challenging nonlinear problems in structural

The implicit FE solution technique delivers accurate results as long at the stress levels in the beam do not exceed the yield-stress value, thus remaining in the linear (elastic) region of the material curve. Because the implicit solver usually obtains a result with a single step, according to the slope of the material curve, which is represented by Young's modulus, for the plastic region of the material there will be an inaccuracy whereat the slope is changed up the plastic characteristics. It means that the implicit solver is very powerful for linear (static) finite element

Unfortunately, the implicit solution approximation may have difficulties with the immensely nonlinear finite element models. Additionally, inverting extremely large stiffness matrices with the implicit solver might consume too much time and memory. Although this method is not logical to use in highly nonlinear problems, it can able to calculate the equations with the help of the iteration cycle, which is applying the load in minor increments called as "timesteps"

During the accident, the vehicle body faces with the impact loads propagate localized plastic hinges and buckling. So the large deformations and rotations with contact and heaping between the many parts can be determined. One of the most important ideas for the explicit dynamics is the wave effects, primarily involved in the deformation stage, associated with the higher stress levels. When the stress level exceeds the yield strength and/or the critical buckling load and the localized structural deformations comprise throughout a few wave transits in the frame of the vehicle. The effects of the inertia will chase this step and also dominate the ensuing transient reaction. The collision could be considered as a less dynamic incident if there is a comparison between the ballistic impact and the vehicle crash. Closed-form analytical problems for the impact scenarios in structural mechanics offer tough times for everyone. Therefore, the numerical methods arise with the practical options to decipher the unknowns.

analysis (FEA); thence, it provides an unquestioningly stable solution methodology.

of the impacts, drop tests, explosions, and collision scenarios.

mechanics.

70 Autonomous Vehicle

[19].

As Du Bois et al. [6] described, a set of hyperbolic wave equations in the region of effectiveness of the wave front is figured out via the explicit dynamics, even then it does not require coupling of large numbers of equations. Yet, the unconditionally stable implicit solvers requiring assembly of a global stiffness matrix provide a solution for all coupled equations of motion. Especially considering the impact and vehicle crash simulations, including the utilization of contact, several material models and withal a combination of non-conventional elements, the explicit solvers show up more potent and computationally more productive than the implicit solvers, while the timestep is about two to three orders of magnitude of the explicit timestep. "Explicit" defines an equilibrium at a moment in time that the displacements of the whole spatial points are known in advance. With the help of the equilibrium, we can determine the accelerations; moreover, the use of central differencing technique assures to establish the displacements at the next timestep and reiterate the process. And also, the only inversion of the mass matrix, M, is enough for this procedure.

There should be stated that if the lumped-mass approximation is utilized the mass matrix will be diagonal, so there is no obligation of the matrix inversion. When the uncoupled equation system is figured out, the results will be in a very fast algorithm. For the minimum memory exigency, the assembly of a global stiffness matrix should be avoided by using an element-byelement approach for the calculation of the internal forces. The explicit integration will be second order accurate when it is applied attentively. Utilization of the related nodal displace‐ ments and/or velocities, the stresses are calculated discretely for each element, causing the effect of the loads on one part of the element onto the opposite part which is simulated by each timestep. Accordingly, the stress wave propagation is demonstrated through the element clearly. The deficiencies of the explicit solution technique are the conditional stability and the weak approximation of treating the static problems. The meaning of the conditional stability of the explicit integration algorithm is the integration timestep must be smaller than or equal

to an upper bound value known as the Courant condition,: Δ*t* ≤ *l c <sup>c</sup>* explaining that the timestep must not pass over the smallest of all element timesteps identified by dividing the element characteristic length through the acoustic wave speed through the material from which origin the element is made like Du Bois et al. [6] mentioned. And they also said that the numerical timestep of the analysis must be smaller than, or equal to, the time required for the physical stress wave to pass by the element. If we try to explain it in a straightforward way, c of the mild steel elements is about 5000 m/s with a characteristic length of 5 mm for the automotive field practices concluding with 1 μs for the analysis timestep. Consequently, incidents dealing with the short duration of time are appropriate for the explicit dynamics solution technique enclosed with superior nonlinear behaviors and loading velocities demanding small timesteps for more precision sequels.

The integration of the equations of motion defines the time integration:

$$\{M\}\{\ddot{\mathbf{x}}\} + \{C\}\{\dot{\mathbf{x}}\} + \{F\_{\text{int}}\} = \{F\_{\text{ext}}\} \tag{1}$$

or in a different representation,

$$M\ddot{\mathbf{x}}\_n + C\dot{\mathbf{x}}\_n + f\_{n\_{\text{in}}} = f\_{n\_{\text{out}}} \tag{2}$$

The basic problem is to determine the displacement, *xn*+1, at time *tn*+1. We can also say that [*K*] {*x*} = {*Fint*}. By the way, the explicit direct time integration can in a conceptual form be written as:

$$\mathbf{x}\_{n+1} = f\left(\mathbf{x}\_n, \dot{\mathbf{x}}\_n, \ddot{\mathbf{x}}\_n, \mathbf{x}\_{n-1}, \ldots\right) \tag{3}$$

The equation shows that the solution depends on nodal displacements, velocities, and accelerations at state *n*, quantities which are known. Therefore, the equation can be solved directly. And for the implicit:

$$\mathbf{x}\_{n+1} = f(\dot{\mathbf{x}}\_{n+1}, \ddot{\mathbf{x}}\_{n+1}, \mathbf{x}\_n, \dot{\mathbf{x}}\_n, \dots) \tag{4}$$

This equation represents the solution that depends on nodal velocities and accelerations at state *n* + 1, quantities which are unknown. This implies that an iterative procedure is needed to compute the equilibrium at time *tn*+1.

If we try to define the time increments:

$$
\Delta t\_{n+1\vert 2} = t\_{n+1} - t\_n \tag{5}
$$

$$\mathbf{t}\_{n+}\mathbf{y}\_{2}^{2} = \bigvee\_{\mathbf{y}\_{n+1}} (\mathbf{t}\_{n+1} + \mathbf{t}\_{n}) \tag{6}$$

$$\mathbf{V}\_{n} = \mathbf{t}\_{n+}\mathbf{Y}\_{n} - \mathbf{t}\_{n-}\mathbf{Y}\_{n} \tag{7}$$

The Reliability of Autonomous Vehicles Under Collisions http://dx.doi.org/10.5772/63974 73

$$\mathbf{t}\_{n+1} = \mathbf{t}\_n - \Delta \mathbf{t}\_{n+1} \mathbf{y} \tag{8}$$

Then, the central difference method is used for the velocity:

$$\dot{\mathbf{x}}\_{n+} \mathbf{y}\_{\overline{k}} = \frac{\mathbf{x}\_{n+1} - \mathbf{x}\_n}{\Delta t\_{n+} \mathbf{y}\_{\overline{k}}} \tag{9}$$

and

field practices concluding with 1 μs for the analysis timestep. Consequently, incidents dealing with the short duration of time are appropriate for the explicit dynamics solution technique enclosed with superior nonlinear behaviors and loading velocities demanding small timesteps

The basic problem is to determine the displacement, *xn*+1, at time *tn*+1. We can also say that [*K*] {*x*} = {*Fint*}. By the way, the explicit direct time integration can in a conceptual form be written

The equation shows that the solution depends on nodal displacements, velocities, and accelerations at state *n*, quantities which are known. Therefore, the equation can be solved

This equation represents the solution that depends on nodal velocities and accelerations at state *n* + 1, quantities which are unknown. This implies that an iterative procedure is needed

> 1 1 ( ) <sup>2</sup> 1

> > 1 1

int [ ]{ } [ ]{ } { } { } *Mx Cx F F* + += *ext* && & (1)

int *ext Mx Cx f f* && & *n nn n* ++= (2)

1 1 (,,, ,) *n nnnn x fx x x x* + - = ¼ & && (3)

1 11 , ,( ), , *n n n nn x fx x x x* + ++ = ¼ & && &(4)

*n nn* 12 1 *t tt* D =- + + (5)

<sup>2</sup> *<sup>n</sup> n n t tt* <sup>+</sup> = + <sup>+</sup> (6)

2 2 *<sup>n</sup> n n tt t* + - D= - (7)

The integration of the equations of motion defines the time integration:

for more precision sequels.

72 Autonomous Vehicle

or in a different representation,

directly. And for the implicit:

to compute the equilibrium at time *tn*+1.

If we try to define the time increments:

as:

$$\mathbf{x}\_{n+1} = \dot{\mathbf{x}}\_{n+} \mathbf{x}\_{n}^{2} \mathbf{x}\_{n+} \mathbf{x}\_{n} + \mathbf{x}\_{n} \tag{10}$$

When Δ*t n*+ 1 2 =*tn*+1 −*tn* is used, we get:

$$\dot{\mathbf{x}}\_{n+1} = \dot{\mathbf{x}}\_{n+} \underset{\mathbf{x}\_n}{\mathbf{x}}\_2 \left( t\_{n+1} - t\_n \right) + \mathbf{x}\_n \tag{11}$$

With the help of the acceleration, velocity can be stated as:

$$\dot{\mathbf{x}}\_{n} = \frac{\dot{\mathbf{x}}\_{n+} \mathbf{y}\_{2} - \dot{\mathbf{x}}\_{n-} \mathbf{y}\_{2}}{\mathbf{t}\_{n+} \mathbf{y}\_{2} - \mathbf{t}\_{n-} \mathbf{y}\_{2}} \tag{12}$$

With Δ*tn* =*t n*+ 1 2 −*t n*− 1 2 , the velocity is given below:

$$
\dot{\mathbf{x}}\_{n+} \mathbf{y}\_{1}^{2} = \dot{\mathbf{x}}\_{n-} \mathbf{y}\_{1}^{2} + \Delta t\_{n} a\_{n} \tag{13}
$$

From the Eq. (2), the acceleration at *a*n+1 can be found like:

$$\ddot{\mathbf{x}}\_{n+1} = m^{-1} \left( f\_{n+1\_{\text{av}}} - f\_{n+1\_{\text{av}}} - \mathbf{C} \dot{\mathbf{x}}\_{n+\frac{1}{2}} \mathbf{y}\_2 \right) \tag{14}$$

Remember, the damping force is taken at *x*˙ *n*+ 1 2 . The new velocity will be calculated with the old velocity and acceleration by Eq. (13). From this result, displacement will be updated at Eq. (10). Consequently, the new internal and external forces help for determination of the new acceleration [Eq. (14)].

Considering the LSDYNA, the widely known commercial explicit solver, the procedure for the solution steps is given below:

When *t* = 0, it will be the beginning of the solution loop or cycle. Whereupon, the displacement will be updated at Eq. (10), while the velocity is also updated at Eq. (13). Then, the internal forces (strain rate, strain, stress, and force magnitude) will be computed looping over the elements. Subsequently, external forces will be calculated to reach the computation of the accelerations at Eq. (14) which is finalizing the initial loop. Thereafter, following loops will go through the process again and again, until the convergence criteria will be satisfied.

For the robust design, stress wave propagation, timestep adjustment, critical element length and finally, the mesh size and quality are the factors, will be considered carefully to affect the simulation performance in a better way.

Due to the reliable side of the explicit dynamics solution methodology, we indicated the differences between the implicit and explicit solver for the approximation of the vehicle crash simulations and safety of the occupants and vehicles. We can possibly say that, for the collision of the regular or autonomous vehicles the accident results will be similar that should be taken care of precisely.
