To use explicit time-stepping for a model that includes any physics interface for explicit dynamics, add an Explicit Dynamics study step to a new or existing study.

The Explicit Dynamics study step creates a solver sequence with the Time-Dependent Solver configured to use an explicit solver type for time stepping. By default, the Verlet method is used, but other methods are also available. The Verlet method is similar to as so-called central difference scheme, often used for explicit dynamics. Two important properties of this method is that it updates velocities at the half-step of each increment, and that it only requires one evaluation of the internal forces per step. The latter is a key to the efficiency of the method when compared to alternative methods.

Also important for an efficient solution is the use of a lumped, or diagonal, mass matrix. When a lumped mass matrix is used, accelerations are trivial to compute with no need to invert a large mass matrix, since the system equations are effectively decoupled. A lumped mass matrix is used by default when adding the Explicit Dynamics study step. An alternative is to instead use a consistent mass matrix, that is, directly use the mass matrix assembled form the weak equations. In this case the solver will assemble the mass matrix and invert it by a linear solver every time step in order to compute accelerations.

A fundamental property of any explicit time-stepping method is its conditional stability; the allowable time-step size is limited. A too large time step will cause significant numerical errors to develop and the solution will grow unbounded to an eventually failure. The time step by which the solution remains bounded is referred to as the stable time step, or the critical time step.

The stability of any explicit method is given by the highest natural frequency, ωmax, of the discretized model equations solved. The stable time step is then

where the factor β depends to the time-stepping method. For the Verlet method, β = 2.

To find the true and optimal value for Δtstb, would require an eigenfrequency analysis of the entire model to find ωmax. Given that the number of modes is equal to the number of degrees of freedom of the model, the cost of such an operation is large. For nonlinear models, ωmax also changes with time and deformation of the structure, making such an approach for finding a stable time step unfeasible.

A better strategy is to approximate ωmax as the maximum of the highest natural frequency, ωmax,e, of all mesh elements in the model. Such approximation will always overestimate ωmax, that is

where tcell,e is called the cell time. It can be interpreted as the shortest time it takes for a wave to pass for a computational cell, that is, a mesh element. We can thus define

where he is the characteristic size of the mesh element and cmax,e is the maximum wave speed of the material.

All material models in the explicit dynamics interfaces provide a value for cmax,e computed from the material properties. If geometric nonlinearity is included, cmax,e is defined from the current (deformed) wave speed with respect to the material frame

The explicit dynamics interfaces provide the time-dependent solver with expressions to evaluate tcell,e over all domain mesh elements while time-stepping to update the time step Δt used. The actual time step used is defined as

where ftime is global time step safety factor used scale the time step. It can be used to, for example, reduce the time step in case there are stiffness and mass contributions added to the model not accounted for in the definition of tcell,e.

The fact that the velocity for the Verlet method lags by one half-step when evaluating the internal forces leads to damping having a negative impact on its stability. It can be shown that the stable time step is reduced by a factor proportional to the critical damping ratio ξ. When damping is added to a physics interface, the cell time estimate in Equation 2-33 is therefore modified to

where ξe is the critical damping ratio of the mesh element.

It can be observed that the time step estimate for each mesh element provided by Equation 2-33 decreases with the size of the mesh elements. Hence it is likely that the smallest mesh element will determine the time step used by the solver. For this reason, small mesh elements should be avoided, as emphasized in Geometry and Meshing for Explicit Dynamics. However, for realistic geometries it may not be possible to avoid small elements completely. Considering Equation 2-33, an alternative to increase the times step is then instead to decrease the wave speed cmax,e.

To decrease c, we can either decrease the stiffness E or increase the density ρ. Changing the stiffness of the material is likely to have a significant impact structural and material behavior and is therefore seldom recommended. Adding more mass by increasing ρ is likely to have less impact, and is therefore a viable option for limiting the negative impact of small mesh elements on the solution time. This technique is referred to as mass scaling.

To include mass scaling in an explicit dynamic analysis, add a Mass scaling node and select the domains where to add mass. The amount of added mass is specified by entering a target value for the cell time at zero deformation, tcell,0. An artificial density ρa is then defined by scaling the physical density ρ0 assigned to the material,

where ttarget is the target value entered. The mass added by Mass scaling only contributes with inertial forces and will not affect volume forces due to frame accelerations added by, for example, gravity.

Storing output too frequently will damage the computational efficiency of the time-explicit method. To manage this, be careful when selecting output times in Explicit Dynamics study step, and plan in advance when output is of interest.

Explicit dynamics is suitable for solving large-model structural mechanics problems since it requires less memory when compared to implicit methods. Storing the solution for large models with a desired time resolution can result in huge data files. To circumvent this, an alternative is to use the Store in Output option of the Explicit Dynamics study step. Using the Selection option in Output allows you to store data only for predefined selections. The list of sections can, for example, include a subset of domains of interest, boundaries of a symmetry plane, and a number points. When postprocessing the data, the best results are obtained by adding the same selections to the dataset used.

For true dynamic problems, the output sampling rate may be dictated by the required accuracy to resolve a signal in time to avoid aliasing. Recall the Nyquist theorem that specifies the minimum data sampling rate required to resolve a signal is two times the frequency of interest. If this leads to too may output times, using Probes for selected points of interest stored at times taken by the solver is an alternative. However, using too many probes can be expensive, especially when adding domain or boundary probes.

When using Results While Solving with explicit time-stepping, updating plot groups is often more costly then advancing the solution one time step. Therefore, only update plots at Time stored in output unless required for debugging the behavior of a model.