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Direct construction of reciprocal mass matrix and higher order fininite element method
Cimrman, Robert ; Kolman, Radek ; González, J. A. ; Park, K. C.
When solving dynamical problems of computational mechanics, such as contact-impact problems or cases involving complex structures under fast loading conditions, explicit time-stepping algorithms are usually preferred over implicit ones. The explicit schemes are normally combined with the lumped (diagonal) mass matrix so that the calculations are efficient and moreover dispersion errors in wave propagation are partially eliminated. As an alternative to lumping with advantageous properties, the reciprocal mass matrix is an inverse mass matrix that has the same sparsity structure as the original consistent mass matrix, preserves the total mass, captures well the desired frequency spectrum and leads thus to efficient and accurate calculations. In the contribution we comment on the usability of the reciprocal mass matrix in connection with higher order FEM.
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Localized formulation of bipenalty method in contact-impact problems
Kolman, Radek ; González, J. A. ; Dvořák, Radim ; Kopačka, Ján ; Park, K.C.
Often, the finite element method together with direct time integration is used for modelling of contact-impact problems of bodies. For direct time integration, the implicit or explicit time stepping are gen-\nerally employed. It is well known that the time step size in explicit time integration is limited by the stability limit. Further, the trouble comes with the task of impact of bodies with different critical time step sizes for each body in contact. In this case, this numerical strategy based on explicit time stepping with the same time step size for both bodies is not effective and is not accurate due to the dispersion behaviour and spurious stress oscillations. For that reason, a numerical methodology, which allows independent time stepping for each body with its time step size, is needed to develop. In this paper, we introduce the localized variant of the bipenalty method in contact-impact problems with the governing equations derived based on the Hamilton’s principle. The localized bipenalty method is applied into the impact problems of bars as an one-dimensional problem. The definition of localized gaps is presented and applied into the full concept of the localized bipenalty method.
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