In this section, the Explicit Dynamics time integration scheme is described and compared with an implicit formulation.
For implicit time integration, Ansys solves the transient dynamic equilibrium equation using the Newmark approximation (or an improved method known as HHT). For more information, see Transient Analysis.
For linear problems, the implicit time integration is unconditionally stable for certain integration parameters. The time step size will vary to satisfy accuracy requirements.
For nonlinear problems:
The solution is obtained using a series of linear approximations (Newton-Raphson method), so each time step may have many equilibrium iterations.
The solution requires inversion of the nonlinear dynamic equivalent stiffness matrix.
Small, iterative time steps may be required to achieve convergence.
Convergence tools are provided, but convergence is not guaranteed for highly nonlinear problems.
The Explicit Dynamics solver uses a central difference time integration scheme (often referred to as the Leapfrog method).
After forces have been computed at the nodes of the mesh (resulting from internal stress, contact, or boundary conditions), the nodal accelerations are derived by equating acceleration to force divided by mass.
Therefore, the accelerations are
 | (10–5) |
Where:
are the components of nodal acceleration (i=1,2,3)
are the forces acting on the nodal points
are the components of body acceleration
m is the mass attributed to the node.
With the accelerations at time n determined, the velocities at time 
are found from
 | (10–6) |
and finally the positions are updated to time n+1 by integrating the velocities
 | (10–7) |
The advantages of using this method for time integration for nonlinear problems are:
The equations become uncoupled and can be solved directly (explicitly). There is no requirement for iteration during time integration.
No convergence checks are needed because the equations are uncoupled.
No inversion of the stiffness matrix is required. All nonlinearities (including contact) are included in the internal force vector.
To ensure stability and accuracy of the solution, the size of the timestep used in Explicit time integration is limited by the CFL (Courant-Friedrichs-Lewy [1]) condition. This condition implies that the timestep be limited such that a disturbance (stress wave) cannot travel farther than the smallest characteristic element dimension in the mesh, in a single timestep. Thus the timestep criteria for solution stability is
 | (10–8) |
Where
Δt is the time increment
f is the stability timestep factor (= 0.9 by default)
h is the characteristic dimension of an element
c is the local material soundspeed in an element
The element characteristic dimension, h is calculated as follows:
Table 10.1: Characteristic Element Dimensions
|
Hexahedral/Pentahedral |
The volume of the element divided by the square of the longest diagonal of the zone
and scaled by |
|
Tetrahedral |
The minimum distance of any element node to it’s opposing element face |
|
Quad Shell [a] |
The square root of the shell area |
|
Tri Shell [a] |
The minimum distance of any element node to it’s opposing element edge |
|
Beam |
The length of the element |
[a] Quad and Tri Shells: The characteristic element dimension is calculated based on
minimal time-step calculation given in Kennedy et. al.(Recent developments in explicit
finite element techniques and their application to reactor structures). The calculations
are based on computing the element shear, bending, and membrane eigenfrequencies, and
choosing the 
as the stable time-step. In most cases, the membrane time step will be
the critical one, except for very thick shells.
The time steps used in Explicit time integration will generally be smaller than those used in Implicit time integration.
For example, for a mesh with a characteristic dimension of 1mm and a material soundspeed of 5000m/s. The resulting stability time step would be 0.18µ seconds. To solve this simulation to a termination time of 0.1 seconds will require 555,556 time increments.
Note: The minimum value of h/c for all elements in the model is used to calculate the time step that will be used for all elements in the model. This implies that the number of time increments required to solve the simulation is dictated by the smallest element in the model. Care should therefore be taken when generating meshes for Explicit Dynamics simulations to ensure that one or two very small elements do not control the timestep. The patch-independent meshing methods available in Workbench will generally produce a more uniform mesh with a higher timestep than patch-dependent meshing methods.
In some instances, the time step per cycle is not controlled by the CFL condition in the smallest element. Proximity Based (Gap) contact, large rotations in rigid bodies, bending or transverse shear stability in shells, artificial viscosity, and Flanaghan-Belytschko hourglass control can also impose restrictions on the time step.
The maximum timestep that can be used in Explicit time integration is inversely proportional to the soundspeed of the material, hence directionally proportional to the square root of the mass of material in an element
 | (10–9) |
Where
Cii is the material stiffness (i=1,2,3)
ρ is the material density
m is the material mass
V is the element volume
By artificially increasing the mass of an element, one can increase the maximum allowable stability timestep, and reduce the number of time increments required to complete a solution. When mass scaling is applied in an Explicit Dynamics system, it is applied only to those elements which have a stability timestep less than a specified value. If the model contains a relatively small number of small elements, this can be a useful mechanism for reducing the number of time steps required to complete an Explicit simulation.
Note: Mass scaling changes the inertial properties of the portions of the mesh to which scaling is applied. The user is responsible for ensuring that the model remains representative for the physical problem being solved.
Note: Mass scaling is only effective in increasing the simulation time step when the time step is being controlled by the CFL condition in some elements. If the time step is being controlled by other factors (see Explicit Time Integration for a list of factors), the desired time step may not be achievable through mass scaling.