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Finite element crash codes are based on updated Lagrangian mechanics. The equations of motion are obtained from stating the balance of linear momentum in an integral form and introducing spatial discretization by linear isoparametric elements. The semidiscretized second-order set of equations of motion can be written as

Ma = P(x, t) - Q(x, t)

where M is the diagonal mass matrix, a is the acceleration vector, P is the external force vector, Q is the nodal internal force vector, x is a spatial coordinate, and t is time. The solution of the previous set of equations in time is accomplished by the explicit central difference technique. The integration scheme, though conditionally stable, has the advantage of avoiding implicit integration and iterative solution of the stiffness matrix.

Reference:

Khalil, T. B. and Vander Lugt, D. A. 1989, Identification of vehicle front structure crashworthiness by experiments and finite element analysis.

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