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Inverse mass matrix for higher-order finite element method in linear free-vibration problems
Kolman, Radek ; González, J.G. ; Cimrman, Robert ; Kopačka, Ján ; Cho, S.S. ; Park, B.G.
In the paper, we present adirect inverse mass matrix in the higher-orderfinite element method forsolid mechanics. The direct inverse mass matrix is sparse, has the same structure as the consistent mass matrixand preserves the total mass. The core of derivation of the semi-discrete mixed form is based on the Hamilton’s principle of leastaction. The cardinal issue is finding the relationship between discretized velocities and discretized linear momentum. Finally, the simple formula for the direct inversemass matrix is presented as well as thechoice of density-weighted dual shape functions for linear momentum with respect to the displacement shape functionwith achoice of the lumping mass method for obtaining the correct and positive definitive velocity-linear momentum operator. The application of Dirichlet boundaryconditions into the direct inversemass matrix forafloating system is achieved usingthe projection operator. The suggested methodology is tested on a free-vibration problem of heterogeneous bar for different ordersof shape functions.
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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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Component-wise partitioned finite element method in linear wave propagation problems: benchmark tests
Kolman, Radek ; Cho, S.S. ; Červ, Jan ; Park, K.C.
A novel explicit time scheme for nite element computations of wave propagation problems in solids is presented. The presented algorithm, with the component-wise partition of equations of motion to the longitudinal and shear parts, is designed to more precisely integrate wave propagation in accordance with their dierent propagation wave speeds. The suggested three-time step integrator is fully explicit with the diagonal mass matrix, of second-order accuracy, conditionally stable and it exhibits minimal sensitivity behavior on time step size satisfying the stability limit. We present two numerical tests of wave propagation phenomena to show accuracy and performance of the proposed method.
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