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Exploring islands of stability in the design space of cylindrical shell structures
Published in Wojciech Pietraszkiewicz, Wojciech Witkowski, Shell Structures: Theory and Applications Volume 4, 2017
In pure and applied mathematics, a rich literature on the so-called numerical continuation techniques exists that are used to explore the solution space of nonlinear ordinary or partial differential equations in terms of a set of arbitrary parameters that govern the intrinsic properties or external factors acting on the physical system. When coupled with the concepts developed in bifurcation theory, a numerical continuation algorithm is capable of tracing any nonlinear solution path, traverse and identify different instability points, and switch onto other solution branches if so required. Hence, such an algorithm, embedded within the finite element method, significantly enhances the engineer’s capability to design nonlinear structures.
Bifurcation Analysis of Spatial Xenon Oscillations in Large Pressurized Heavy Water Reactors Using Multipoint Reactor Kinetics with Thermal-Hydraulic Feedback
Published in Nuclear Science and Engineering, 2021
Abhishek Chakraborty, Suneet Singh, M. P. S. Fernando
MATCONT is a user-friendly software written in MATLAB® based on a method of numerical continuation that identifies the different types of bifurcations. The basic idea of numerical continuation is that “for any two values of the parameter close to each other, the equations to be solved are similar, and their solutions may be expected to be close too.”18 The MATCONT package is able to detect different types of bifurcations like Hopf, branch point, limit point bifurcation of cycles, Neimark-Sacker (torus) bifurcation, period doubling (flip) bifurcation, Bogdanov-Takens bifurcation, zero-Hopf bifurcation, double Hopf bifurcation, generalized Hopf (GH) (Bautin) bifurcation, etc.18
Bifurcation Analysis of Xenon Oscillations in Large Pressurized Heavy Water Reactors with Spatial Control
Published in Nuclear Science and Engineering, 2022
Abhishek Chakraborty, Suneet Singh, M. P. S. Fernando
The type of Hopf bifurcation can be determined using MATCONT (Ref. 20), which is a MATLAB-based program based on the numerical continuation method. This is very versatile software that can be used to determine different types of bifurcations. The nature of Hopf bifurcation can be easily identified using the first Lyapunov coefficient (FLC), which is estimated by the code using the Hessian matrix. If the FLC is positive, then it is a subcritical Hopf bifurcation, and if it is negative, then it’s a supercritical Hopf bifurcation. If the value is zero, then it is known as the generalized Hopf (GH) point, and it divides the parameter space into supercritical and subcritical Hopf regions.
A nonlinear optimization bifurcation tracking method for periodic solution of nonlinear systems
Published in Mechanics Based Design of Structures and Machines, 2023
It is often of interest to follow the evolution of the solution of nonlinear systems which can be tracked by using the numerical continuation method (Krauskopf, Osinga, and Galán-Vioque 2007; Allgower and Georg 2012) as the system parameters vary. A great deal of research about the continuation of periodic solutions is available and various numerical computation methods have been presented. Generally speaking, the methods can be mainly classified into two different categories. The first category includes the prediction–correction method and the second category includes the Asymptotic Numerical Method (ANM).