The value of numerical amplification matrices in time integration methods

• A numerical derivation of the amplification matrices for time integration methods is considered and compared to the analytical derivation
• Proper minimum dimension of the amplification matrices depending on the nature of the algorithm is discussed.
• Proper initialization of the time integration algorithms which do not establish equilibrium at the ends of the time step is treated.
• Observed overshooting phenomena are related to the difference between analytical and numerical amplification matrices.
• The overshooting phenomena observed in Wilson and Hilber–Hughes–Taylor schemes are studied as examples.

Several frequently overlooked concepts in linear elastodynamics are reviewed. First that the amplification matrix may be obtained numerically and if obtained this way, it may give some extra information on the programmed algorithm. Second that the adequate dimension of that matrix depends on the algorithm at hand. Such dimension equals the number of independent initial conditions that must be prescribed. Third that those initial conditions should be consistent with the problem at hand and with the algorithm. Overshooting phenomena present in some time integration algorithms may be a consequence of overlooking such issues.