TOMAR SUS PRECAUCIONES

All numerical modelling contained within this present study has been carried out using LS-DYNA version 971[122][123]. LS-DYNA is a general-purpose finite element program capable of simulating highly non-linear transient dynamic events using explicit time integration. In a transient dynamic event, such as an explosion it is vital to account for the non-linear material behaviour (plasticity and fracture), contact non-linearity (varying boundary conditions) and geometric non-linearity (large deformations).

6.2.1

Finite Element techniques

Several different finite element techniques can be used to describe the response of structures due to blast loads. Some of these are: (1) pure Lagrangian formulation, (2) an initial Eulerian simulation (to determine the load) followed by a Lagrangian simulation (for the structural response) and (3) a hybrid technique that combines the advantages of the Eulerian and Lagrangian methods [124], known as Arbitrary La- grangian Eulerian (ALE). These are also known as uncoupled and coupled approaches as described in Chapter 2 Section 2.6.3. These solvers are briefly described in the sections below.

6.2.2

Lagrangian solver

In the Lagrangian code the computational grid is fixed in the material and as such it moves and deforms with the material, as shown in Figure 6.1. This means that the quadrature points will coincide with the material points, ensuring that the boundary nodes remain on the boundary throughout the evolution of the problem (Belytschko

et al.[125]). This helps to simplify imposition of the boundary conditions in the Lagrangian mesh.

However there are limitations associated with this method. Since Lagrangian meshes deform with the material, they become distorted in simulations with severe deformation. This distortion can lead to inaccuracies in the output and potential termination of the analysis when an element becomes so distorted it effectively folds in on itself, resulting in a negative element volume. This is normally overcome using adaptive meshing techniques.

6.2. THE FINITE ELEMENT CODE 113

Figure 6.1: Illustration of a two-dimensional shearing of a block showing Lagrangian mesh[13].

6.2.3

Eulerian solver

In the Eulerian code the computational grid is fixed in space, whilst the material passes through it, as illustrated in Figure 6.2. This means that the nodal grid has to be large enough so that the body does not leave the domain whilst deforming, resulting in larger run times.

Figure 6.2: Illustration of a two-dimensional shearing of a block showing Eulerian mesh[13].

On the plus side, large deformations in the material do not cause distortion in the meshing, allowing for unlimited material deformation without causing degradation of the elements. Eulerian formulations are best suited for fluid mechanics problems.

6.2.4

Arbitrary Lagrangian Eulerian (ALE)

The Arbitrary Lagrangian Eulerian method has been developed to combine the advantages of both Eulerian and Lagrangian finite elements, whilst minimising the disadvantages. The ALE technique allows the mesh to deform, somewhat independently of the material, allowing the mesh to adapt to the deformation of the material, whilst avoiding overly-distorted elements, depicted in Figure 6.3.

(a) ALE (b) Lagrangian

Figure 6.3: ALE vs Lagrangian mesh[14].

By allowing the mesh to adhere to the deformation of the material, boundaries are more easily treated with ALE than with Eulerian meshing. However because the mesh and material are not bound together, the evaluation of the constitutive equations in the material points are not as convenient as when using a Lagrangian mesh. This problem can be addressed by using an operator split method and further details can be found in various references by Belytschoet al.[125] and by Olovosson[126][127].

In ALE formulations the nodes do not follow the material flow, as shown in Figure 6.2. There is a flux of material between the elements and this complicates the governing equations. Using an operator split technique, LS-DYNA first computes the Lagrangian time derivative and updates the history variables. Subsequently the relative motion between mesh and material is computed and the history variables are updated once more[126].

The numerical modelling in the present study was carried out primarily within the Lagrangian domain. The Eulerian and ALE methods are also briefly discussed. Ideally all blast simulations should be carried out using a fully coupled Eulerian- Lagrangian approach, however this is often not practicable as the computational time increases considerably when going from a pure Lagrangian simulation to a fully coupled Eulerian-Lagrangian simulation.

6.2. THE FINITE ELEMENT CODE 115

6.2.5

Explicit time integration

Implicit and explicit are two types of time integration methods used to perform dynamic simulations. For practical purposes the differences between the two methods are related to stability and cost. The explicit solution is the most common for blast and short impact events. The implicit solution is used for static and long duration events.

In the explicit approach, internal and external forces are summed at each node point, and a nodal acceleration is computed by dividing by the nodal mass. Integrating this acceleration in time advances the solution. The maximum time step size is limited by the Courant condition, producing an algorithm which typically requires many relatively inexpensive time steps [123]. Therefore explicit methods are conditionally stable. This method is well suited for dynamic simulations, however it can become prohibitively expensive to conduct long durations, such as static events.

In the implicit method, a global stiffness matrix is computed, inverted, and applied to the nodal out-of-balance force to obtain a displacement increment[123]. The advantage to this is that the user can select the time step size, making this method unconditionally stable. The disadvantage with this method is it requires a large numerical effort to form, store, and factorize the stiffness matrix. In general implicit simulations involve a relatively small number of expensive time steps.

All the numerical modelling described in this study was carried out within the explicit time integration domain, unless otherwise stated.

Explicit time integration procedure

An overview of the procedure of explicit time integration is available from LS- DYNA[122]. Dynamic equilibrium is solved from the following equation:

Ma+Cv+Kx−F_E =0 (6.1)

whereM is the mass matrix,C is the damping matrix,K is the stiffness matrix and FE represents the applied or external forces acting on the system. The nodal

displacements are represented by x while v and a represent the nodal velocities and accelerations, respectively. If damping is considered negligible this can be re-written as:

Internal forces are defined asFI =Kx, thus:

Ma=F_E−F_I (6.3)

This set of equations are solved at the beginning of each time step. When building a model the mass matrix is known, as it is formulated from the geometry and the material properties. The applied forces are also known, as the user applies these. The unknown parameter is the internal forces, which must be calculated in order to solve the nodal accelerations. These are derived from the displacement at the beginning of each time step and the accelerations are solved from:

a=M−1(FE−FI) (6.4)

Once the accelerations are calculated, the central difference method is used to determine the velocities and nodal displacements at the end of each time step. The stresses and strains are calculated from the nodal displacements. This process is demonstrated below for a single element:

FE =kxt =mat (6.5) FE =kxt+dt =mat+dt (6.6) x_t+dt=xtvt+dt/2dt (6.7) v_t+dt/2=v_t−dt/2dt+atdt (6.8) ma_t+dt =FE−k(xt =v_t−dt 2 dt+atdt2 (6.9)

The central difference method does not have any iteration and thus no conver- gence checks are required. This advantage comes at a price, and this price is that this method is conditionally stable. This means that in order to maintain stability, the time step must be kept within a certain limit (critical time step) to capture meaningful results.

Critical time step

The critical time step∆tcr must not be exceeded when performing explicit integration.

For damped materials the size∆tcr is determined by the largest natural frequencyωmax

6.3. MATERIAL MODELLING 117