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Manfred Bischoff, "Variational methods for consistent singular and scaled mass matrices"
Mass matrices in explicit and implicit finite element analysis mostly come along in their consistent or lumped version. Consistency in this context refers to a derivation from a given spatial discretization along with a strong functional relationship in time of displacement, velocity and momentum. In some cases, however, more freedom and flexibility in designing mass matrices is desirable. One example is the technique of mass scaling with the aim of increasing the critical time step in explicit dynamics without significant loss of accuracy [2]. Moreover, alternative mass matrices based on mass redistribution techniques may help to reduce oscillating contact forces in impact problems [1].In general, these techniques lack a sound variational basis and they may be inconsistent, thus compromising convergence and accuracy. Therefore, in [3] a new variational method for formulation of consistent mass matrices has been proposed. It is based on a new penalized Hamilton’s principle, where relations between variables for displacement, velocity and momentum are imposed via a penalty method. Independent spatial discretization of the variables along with a local static condensation for velocity and momentum yields a parametric family of consistent mass matrices which allows for constructions which are more flexible than classical approaches based on a one-field Hamilton’s principle.
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