The basic equation of motion solved by an implicit transient dynamic analysis is

(10–1)

Where:

m = mass matrix

c = damping matrix

k = stiffness matrix

F(t) = load vector

At any given time, t, these equations can be thought of as a set of "static" equilibrium equations that also take into account inertia forces and damping forces. The Newmark time integration method (or an improved method called HHT) is used to solve these equations at discrete time points. The time increment between successive time points is called the integration time step.

The partial differential equations to be solved in an Explicit Dynamics analysis express the conservation of mass, momentum, and energy in Lagrangian coordinates. These, together with a material model and a set of initial and boundary conditions, define the complete solution of the problem.

For the Lagrangian formulations currently available in the Explicit Dynamics system, the mesh moves and distorts with the material it models and conservation of mass is automatically satisfied. The density at any time can be determined from the current volume of the zone and its initial mass

(10–2)

The partial differential equations that express the conservation of momentum relate the acceleration to the stress tensor σ_ij .

(10–3)

Conservation of energy is expressed via:

(10–4)

These equations are solved explicitly for each element in the model, based on input values at the end of the previous time step. Small time increments are used to ensure stability and accuracy of the solution. Note that in Explicit Dynamics we do not seek any form of equilibrium; we simply take results from the previous time point to predict results at the next time point. There is no requirement for iteration.

In a well-posed Explicit Dynamics simulation, mass, momentum, and energy should be conserved. Only mass and momentum conservation is enforced. Energy is accumulated over time and conservation is monitored during the solution. Feedback on the quality of the solution is provided via summaries of momentum and energy conservation (as opposed to convergent tolerances in implicit transient dynamics).