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ANSYS: Finite Element Analysis
Published in Paul W. Ross, The Handbook of Software for Engineers and Scientists, 2018
The ANSYS program uses an eigenvalue formulation to perform linear buckling analysis, determining the scaling factors (eigenvalues) for the stress stiffness matrix, which offset the structural stiffness matrix through the following equation: () ([K])−λ[S]{u}=0 where [K] is the structural stiffness matrix[S] is the stress stiffness matrixλ are eigenvalues representing the scale factors{u} is the eigenvector representing the buckled shape The point at which buckling occurs, where the two paths of the force-deflection curve intersect, is called the bifurcation point. Once the bifurcation point is exceeded, the structure will either buckle or continue to take on load in an unstable state.
Stability and bifurcation in geomechanics
Published in G. Swoboda, Numerical Methods in Geomechanics Innsbruck 1988, 2017
Experiments with ‘perfect’ boundary conditions and ‘perfectly’ homogeneous material do not generally secure homogeneous deformation, as various bifurcation modes of the deformation are possible and do actually develop. Bifurcation of the deformation process means that at some critical state the deformation process does not follow its ‘straight-ahead’ continuation but turns to an entirely different mode. Typical examples of bifurcation phenomena are buckling, barrelling, necking, shear banding and liquefaction phenomena that are observed in soil specimens with lubricated ends. Mathematically bifurcation means that the equations describing the equilibrium continuation do not provide a unique solution. Equilibrium bifurcation analyses in deformable solids are mostly based on the fundamental work of Hill (1958).
Bifurcations
Published in Arturo Buscarino, Luigi Fortuna, Mattia Frasca, ®and Laboratory Experiments, 2017
Arturo Buscarino, Luigi Fortuna, Mattia Frasca
Bifurcation analysis detects the critical conditions when a qualitative shift of the system behavior appears. An important point in the analysis of the behavior of a system is to understand how and when a bifurcation occurs. Often, based on the physical knowledge of the process, from a primary investigation of the system it is straightforward to understand if some bifurcation occurs. For instance, bifurcations occur in structurally ill-conditioned systems. Let us consider, for example, the buckling of a column. The column has length L and section area S. Let us consider the parameter P = L/S. If P is large and a small weight is placed on top of the column, it can support the load and the column shape remains unaltered. Let us imagine increasing at small steps the load. At first the column will still be in vertical position. Then, for further increase the vertical position of the column will become unstable and a catastrophic behavior (the breakage of the column) will occur. Therefore, a thin column is a system that bifurcates (Figure 3.1). On the contrary, if P is small, the catastrophic behavior previously discussed does not occur. The column with small P is in fact structurally stable (Figure 3.2).
A noise-robust Koopman spectral analysis of an intermittent dynamics method for complex systems: a case study in pathophysiological processes of obstructive sleep apnea
Published in IISE Transactions on Healthcare Systems Engineering, 2023
Phat K. Huynh, Arveity R. Setty, Trung B. Le, Trung Q. Le
Intermittency is defined as the erratic alternations between periodic (i.e., regular and laminar) dynamics and chaotic (i.e., irregular and turbulent), commonly characterized by short bursts in the signal (Kantz & Schreiber, 2004). Intermittency also exists in the other form of chaotic dynamics called crisis-induced intermittency (Grebogi et al., 1987). Two groups of explanations have been proposed to clarify the origins of the intermittency phenomena. First, intermittency may originate from Hamiltonian chaos (Zaslavsky & Zaslavskij, 2005) and hydrodynamical systems (Bessaih et al., 2015). Second, intermittent behaviors can also arise from small control parameter fluctuations around critical values (Contoyiannis et al., 2002; Hramov et al., 2014). To avoid confusion between the non-intermittent dynamics mode of non-stationarity and the dynamical intermittency for fixed parameters of dynamical systems, the statistics of intermittent phases and chaotic bursts need to be studied (Kantz & Schreiber, 2004). The study of intermittent dynamics is considerably challenging. Bifurcation analysis from the governing equations is the classical approach to studying the system behaviors as the parameters are perturbed. However, there are increasing numbers of complex systems for which we have abundant measurement data from sensors but do not have access to the underlying parameterized governing equations. Hence, an interpretable data-driven method that accurately models the intermittent dynamics of high-dimensional complex systems with unknown governing equations is needed.
Bifurcation, chaotic and multistability analysis of the $(2+1)$-dimensional elliptic nonlinear Schrödinger equation with external perturbation
Published in Waves in Random and Complex Media, 2022
Samina Samina, Adil Jhangeer, Zili Chen
Recently, applying bifurcation analysis to the study of differential equations is an interesting subject of research. In Ref. [27], by using bifurcation theory, the authors obtained a group of solutions to Klein–Gordon–Zakharov equations. More recently [28], exact solutions for resonant nonlinear Schrödingers equation are obtained with the Kerr law nonlinearity using bifurcation theory. Several authors have been studying the concepts of bifurcation of dynamic systems in the perturbed and unperturbed framework [4, 25, 26, 29–33]. However, the presence of external perturbation includes several dynamical structures which exhibit an external disturbance to a nonlinear integrable wave equation, generally quasi-periodic and chaotic behaviors [27, 33]. Furthermore, in [34] bifurcations, quasi-periodic motions and chaotic motions of the positron acoustic waves are reported.