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1

Multi-core and many-core SPMD parallel algorithms for construction of basins of attraction

Silveira Marcos, Gonçalves Paulo J.P., Balthazar José M.

Journal of Theoretical and Applied Mechanics

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2019

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Vol. 57 nr 4

1067--1079

EN

Construction of basins of attraction, used for the analysis of nonlinear dynamical systems which present multistability, are computationaly very expensive. Because of the long runtime needed, in many cases, the construction of basins does not have any practical use. Numerical time integration is currently the bottleneck of algorithms used for the construction of such basins. The integrations related to each set of initial conditions are independent of each other. The assignment of each integration to a separate thread seems very attractive, and parallel algorithms which use this approach to construct the basins are presented here. Two versions are considered, one for multi-core and another for many-core architectures, both based on a SPMD approach. The algorithm is tested on three systems, the classic nonlinear Duffing system, a non-ideal system exhibiting the Sommerfeld effect and an immunodynamic system. The results for all examples demonstrate the versatility of the proposed parallel algorithm, showing that the multi-core parallel algorithm using MPI has nearly an ideal speedup and efficiency.

2

Application of higher order Hamiltonian approach to nonlinear vibrating systems

Askari H., Nia Z. S., Yildirim A., Yazdi M. K., Khan Y.

Journal of Theoretical and Applied Mechanics

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2013

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Vol. 51 nr 2

287--296

EN

The higher order Hamiltonian approach is utilized to elicit approximate solutions for two nonlinear oscillation systems. Frequency-amplitude relationships and the model of buc kling of a column and mass-spring system are scrutinized in this paper. First, second and third approximate solutions of examples are achieved, and the frequency responses of the systems are verified by exact numerical solutions. According to the numerical results, we can conclude that the Hamiltonian approach is an applicable method for solving the nonlinear equations, and the accuracy of this method in the second and third approximates is very high and reliable. The achieved results of this paper demonstrate that this method is powerful and uncomplicated for solving of sophisticated nonlinear problems.

PL

W pracy przedstawiono zastosowanie metody Hamiltona wyższego rzędu do wyznaczania przybliżonych rozwiązań analitycznych dla dwóch nieliniowych układów drgających. Szczegółowej analizie poddano charakterystyki amplitudowo-częstościowe modelu ściskanej belki oraz dyskretnego układu sprężysto-inercyjnego. Otrzymano przybliżone rozwiązania pierwszego, drugiego i trzeciego rzędu, a odpowiedzi częstościowe układów porównano z dokładnymi rezultatami symulacji numerycznych. Na ich podstawie oceniono, że metoda Hamiltona jest stosowalna dla układów nieliniowych, a przybliżenia drugiego i trzeciego rzędu stanowią rozwiązania analityczne o wysokiej dokładności. Uzyskane w pracy wyniki przekonują, że zaproponowana metoda jest prostym i jednocześnie bardzo skutecznym narzędziem rozwiązywania nieliniowych problemów układów mechanicznych o dużym stopniu złożoności.

3

On the solution set of a vector Duffing equation

Papalini F., Papageorgiu N. S.

Demonstratio Mathematica

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2002

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Vol. 35, nr 4

763-772

EN

In this paper we consider a vector Duffing equation with periodic boundary conditions. First we prove an existence result assuming on f(t,x) Caratheodory type conditions. Then by imposing also a monotonicity assumption we show that the solution set is acyclic.