To do this, we form two groups: either time-dependent or time-independent problems.
To solve these problems, we commonly use ‘implicit’ and/or ‘explicit’ methods.
Explicit analyses aim to solve for acceleration (or otherwise ).
In most cases, the mass matrix is considered as “lumped” and thus a diagonal matrix.
For more information, this Wikipedia page provides great examples with illustrations of how both methodologies give numerical approximations to solutions of time-dependent and PDE equations.
Explicit analysis offers a faster solution in events where there is a dynamic equilibrium or otherwise: The explicit method should be used when the strain rates/velocity is over 10 units/second or 10 m/s respectively.
These events can be best exampled by extreme scenarios such as an automotive crash, ballistic event, or even meteor impact.
In these cases, the material models do not only need to account for the variation of stress with strain but also the strain rate.
This method requires additional computation and can be harder to implement.
However, it will be used in lieu of explicit methodologies when problems are still and using alternative analysis methods is impractical.