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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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Transient response of layered orthotropic strip to transverse load
Adámek, V. ; Valeš, František ; Červ, Jan
This work concerns the transient response of an infinite two-layered strip subjected to a transverse load of impact character. The material of each layer is assumed to be specially orthotropic, i.e. the material and geometric axes coincide. Moreover, the material is modelled as linear viscoelastic using the model of standard linear viscoelastic solid such that the damping behaviour of the strip for long wavelengths and long times can be addressed. The non-stationary wave phenomena in the strip are studied using analytical approach. The system of equations of motion for the case of 2D plane-stress problem is solved using the classical method of integral transform. Once the formulas for the Laplace transforms of fundamental mechanical quantities are derived, the numerical inverse Laplace transform is used to obtain the response in time domain for a strip with free-fixed boundaries. The results for a strip composed of two orthotropic layers of specific material properties are presented in this work. Finally, this solution is confronted with the results of numerical simulations reached by a professional FE code.
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Využití numerické zpětné laplaceovy transformace při řešení úloh elastodynamiky
Adámek, V. ; Valeš, František ; Červ, Jan
Laplaceova transformace představuje jednu z nejvíce užívaných transformací v časové oblasti. Existují dva přístupy při hledání inversní Laplaceovy transformace, analytický a numerický. Analytická metoda je založena na exaktním vyjádření inversního integrálu pomocí Cauchyovy residuové věty. Podstata druhé metody spočívá v numerickém řešení inversního integrálu. Ukazuje se, že numerický přístup je rychlejší nežli analytické řešení. V neposlední řadě může být tento přístup využit ve složitějších případech, kde např. existence bodů rozvětvení činí inversní proces, založený na analytickém přístupu, mnohem komplikovanější.
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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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