Explicit Time Integration

There are two forms of time integration used in LS-DYNA. These are the explicit form and the implicit form. The explicit solution is the most common for impact and short duration events. The implicit solution is used for static and long duration events.

In the explicit solution, nodal accelerations are calculated using a force balance at every time step and time steps can become extremely small for increased accuracy. The implicit solution is used for static and long duration events in which a stiffness matrix is assembled and inverted at each time step resulting in a longer run time (more computationally expensive).

All numerical modelling described in this thesis was carried out within the explicit time integration domain.

3.5.1 Explicit Time Integration Procedure

The procedure of explicit time integration is outlined below (LSTC, 2007). Dynamic equilibrium is solved from the following equation:

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where FE represents the applied or external forces acting on the system, [K] is the stiffness matrix, [C] is the damping matrix and [M] is a mass matrix with x representing the nodal displacements, v the nodal velocities and a the nodal accelerations. If damping is considered to be negligible then the damping matrix is eliminated from the equilibrium equation and it can be re-written as:

0 (3.2)

Internal forces are defined as FI = [K]x, therefore:

(3.3) This set of equations is solved at the beginning of each time step. The mass matrix is

known, as defined by the user at the beginning i.e. formulated from the geometry and material properties of the model. The applied forces are also known, again applied to the model by the user. Therefore the internal forces need to be calculated in order to solve the nodal accelerations. These are derived from the displacements at the beginning of each time step and the accelerations are solved from:

(3.4)

Once the accelerations are known, 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 nodal displacements. The central difference method does not have any iteration and thus does not have to converge. It is only conditionally stable based on time step size which must be kept small enough to capture the solution. This process is demonstrated below for a single element.

(3.5)

(3.6)

/ (3.7)

/ / (3.8)

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(3.10)

3.5.1.1 Time Step Calculation

As highlighted by Zienkiewicz and Taylor (2000) the time step selection is governed by the Courant Condition (Courant et al, 1928). The Courant condition effectively limits the time step to be less than the time taken for a sound wave to traverse the smallest element in the model. In doing this, it is guaranteed that a stress wave is detected in an element. If the time step is too large stress waves can jump elements, thus leaving them un-detected in the analysis and therefore rendering the results incorrect.

Explicit time integration has a critical time step (∆dtcr). It is based on element size and stiffness. It is a measure of the natural frequency of each element. The critical time step must conform to the Courant Condition for stability which is:

∆ / (3.11)

where d is the characteristic length of the element and c is the speed of sound in the material. In LS-DYNA a time step factor is applied to this condition for safety so it is re-written as:

∆ / (3.12)

where tsf is the time step factor set at a default value of 0.9, however the user is able to vary the value if necessary pending on the element size chosen. In a study by Borvik et al (2009) on the perforation of high strength armour plates tsf was limited to 0.5 for greater accuracy. The speed of sound in a material is defined as:

⁄ (3.13)

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When time steps become very small the demand on CPU time is increased but this can be overcome if elements with fewer integration points are used.

3.6 Elements

The complicated geometry of the cables under consideration in the present study requires the use of 3D solid elements (Chapter 4 and 6). The most time-efficient solid element is an eight node element with a single integration point. However, this type of element is prone to zero energy deformation modes known as hourglassing. Any element motion that is not a rigid body motion and results in no straining of the element is a spurious singular deformation mode (Belytschko et al, 2000). Fig. 3.4 shows some examples of spurious deformation modes. The dashed line represents the outline of the element before spurious distortion occurs as indicted by the solid line.

Fig. 3.4 Spurious deformation modes (Belytschko et al, 2000)

A vertical pair of distorted elements in the first mode in Fig. 3.4 would look like an hourglass hence the term hourglassing. In problems which involve high velocity viscous flow, hourglass control is always recommended in conjunction with the use of fully integrated solid elements where necessary. This has been considered in the numerical simulations in this study. This is discussed further in Chapter 6.

3.7 Contact Modelling

In LS-DYNA, contact surfaces are defined from existing nodes and elements to prevent parts of the model passing through (or separating from) each other. Energy between impacting bodies is transferred via ‘contact energy’ in artificial springs located on the contacting surfaces. The contact stiffness is defined as (for solid elements):

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were Pf is a penalty factor, A is the area of the element face in contact, K is the bulk modulus of the material that the interface is attached to and Ve is volume of the element.

The surfaces that contact one another are designated as slave and master surfaces. The slave and master surfaces are user defined from a set of nodes, elements or parts within the model. A part could be a set of wires that make up the cable or a fragment. The specific contact algorithms used in this study are discussed in more detail in Chapters 4 and 6; however, the general principles are highlighted here.