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Implementation of the FEM-FCT method for nonstationary convection-diffusion equations
Stará, Lenka ; Knobloch, Petr (advisor) ; Dolejší, Vít (referee)
The aim of this work is the implementation and the testing of the fi- nite element method flux corrected transport (FEM-FCT) for an evolutionary convection-diffusion-reaction equation with small diffusion parameter. The basic idea of this method lies in modification of algebraic equation which is obtained from the Galerkin finite element method in order to suppress spurious oscillations and not to smear the solution considerably. In the first section of this work we in- troduce the problem of solving a convection-diffusion-reaction equation. The next section is devoted to a short introduction of the finite element method and we pro- vide the Galerkin finite element formulation of the convection-diffusion-reaction problem. Afterward we derive formulae, which are necessary for implementation FEM-FCT method. In the last section we present numerical results, which are studied at body rotation problem. 1
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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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Implementation of the FEM-FCT method for nonstationary convection-diffusion equations
Stará, Lenka ; Knobloch, Petr (advisor) ; Dolejší, Vít (referee)
The aim of this work is the implementation and the testing of the fi- nite element method flux corrected transport (FEM-FCT) for an evolutionary convection-diffusion-reaction equation with small diffusion parameter. The basic idea of this method lies in modification of algebraic equation which is obtained from the Galerkin finite element method in order to suppress spurious oscillations and not to smear the solution considerably. In the first section of this work we in- troduce the problem of solving a convection-diffusion-reaction equation. The next section is devoted to a short introduction of the finite element method and we pro- vide the Galerkin finite element formulation of the convection-diffusion-reaction problem. Afterward we derive formulae, which are necessary for implementation FEM-FCT method. In the last section we present numerical results, which are studied at body rotation problem. 1
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Comparative study of finite element method, isogeometric analysis, and finite volume method in elastic wave propagation of stress discontinuities
Berezovski, A. ; Kolman, Radek ; Blažek, Jiří ; Kopačka, Ján ; Gabriel, Dušan ; Plešek, Jiří
A comparative study of Finite Element Method, Isogeometric Analysis, and Finite Volume Method in numerical simulation of one-dimensional wave propagation problems of stress discontinuities in elastic solids is presented. The special attention is paid to accuracy, convergence, and stability of tested numerical methods and the appearance of spurious oscillations and damping effects occurring close to theoretical sharp wavefronts.
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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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