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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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Bi-penalty stabilized explicit finite element algorithm for one-dimensional contact-impact problems
Kolman, Radek ; Kopačka, Ján ; Tkachuk, A. ; Gabriel, Dušan ; Gonzáles, J.A.
In this contribution, a stabilization technique for finite element modelling of contact-impact problems based on the bipenalty method and the explicit predictor-corrector time integration is presented. The penalty method is a standard method for enforced contact constrains in dynamic problems. This method is easily implemented but the solution depends on numerical value of the stiffness penalty parameter and also the stability limit for explicit time integration is effected by a choice of this parameter. The bipenalty method is based on penalized not only stiffness term but also mass term concurrently. By this technique with a special ratio of mass and stiffness penalty parameters, the stability limit of contact-free problem is preserved. In this contribution, we also present a modification of the explicit time scheme based on predictor-corrector form. By meaning of this approach, spurious contact oscillations are eliminated and the results do not depend on numerical parameters.
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Recent progress in numerical methods for explicit finite element analysis
Kolman, Radek ; Kopačka, Ján ; Gonzalez, J. ; Gabriel, Dušan ; Cho, S.S. ; Plešek, Jiří ; Park, K.C.
In this paper, a recent progress in explicit finite element analysis is discussed. Properties and behaviour of classical explicit time integration in finite element analysis of elastic wave propagation and contact-impact problems based on penalty method in contact-impact problems are summarized. Further, stability properties of explicit time scheme and the penalty method as well as existence of spurious oscillations in transient dynamics are mentioned. The novel and recent improving and progress in explicit analysis based on a local time integration with pullback interpolation for different local stable time step sizes, bipenalty stabilization for enforcing of contact constrains with preserving of stability limit for contact-free problems and using a direct inversion of mass matrix are presented. Properties of the employed methods are shown for one-dimensional cases of wave propagation and contact-impact problems.
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An explicit time scheme with local time stepping for one-dimensional wave propagation in a bimaterial bar
Kolman, Radek ; Cho, S.S. ; Gonzalez, J.G. ; Park, K.C. ; Berezovski, A.
In this paper, we test a two-time step explicit scheme with local time stepping. The standard explicit time scheme in finite element analysis is not able to keep accuracy of stress distribution through meshes with different local Courant numbers for each finite element. The used two-time step scheme with the diagonal mass matrix is based on the modification of the central difference method with pullback interpolation. We present a numerical example of one-dimensional wave propagation in a bimaterial elastic bar. Based on numerical tests, the employed time scheme with pullback interpolation and local stepping technique is able to eliminate spurious oscillations in stress distribution in numerical modelling of shock wave propagation in heterogeneous materials.
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Dispersion properties of finite element method: review
Kolman, Radek ; Okrouhlík, Miloslav ; Plešek, Jiří ; Gabriel, Dušan
Review of the dispersion properties of plane square bilinear finite element used in plane elastic wave propagation problems is presented. It is assumed the grid (spatial) dispersion analysis and, further, the temporal-spatial dispersion analysis for explicit direct time integration based on the central difference method. In this contribution, the dispersion surfaces, polar diagrams and error dispersion graphs for bilinear finite element are depicted for different Courant numbers in explicit time integration. Finally, recommendation for setting the mesh size and the time step size for the explicit time integration of discretized equations of motion by the bilinear finite element method is provided.
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