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Finite Element Analysis of Beams
Published in J. N. Reddy, Theories and Analyses of Beams and Axisymmetric Circular Plates, 2022
Figure 9.8.2 shows convergence of the Newton iteration procedure for a single degree-of-freedom system. Here T(U) denotes the slope of the tangent to the curve F=K(U)U at U. The Newton iteration converges for hardening as well as softening type nonlinearity. For the hardening type, convergence may be accelerated using the under-relaxation given in Eq. 9.8.7. The method may diverge for a saddle-point behavior. The steps involved in Newton's iteration solution approach are described in Box 9.8.2.
Dynamic response of elastic la ttices and discretised elastic membranes
Published in A. B. Movchan, N. V. Movchan, I. S. Jones, D. J. Colquitt, Mathematical Modelling of Waves in Multi-Scale Structured Media, 2017
A. B. Movchan, N. V. Movchan, I. S. Jones, D. J. Colquitt
Analysis of the signs of the Hessian determinant and second derivatives at the stationary points reveals that the first is a local maximum of f(ξ), the second is a saddle point, whilst the third is a local minimum. Indeed, since these are the only stationary points in the irreducible Brillouin zone of the reciprocal lattice, the local extrema are global extrema. Thus, the maximum value of ω which corresponds to the minima of f(ξ) is ω = 3. Hence, there exists a semi-infinite stop band for frequencies ω > 3, whilst propagating solutions are supported for 0 < ω ≤ 3. The saddle point frequency, corresponding to the saddle points of f(ξ), is ω=22 $ \omega = 2\sqrt 2 $ as stated earlier. Finally, as expected, the minimum value of ω, corresponding to the maxima of f(ξ) is ω = 0.
Systems of First Order Linear Differential Equations
Published in Vladimir A. Dobrushkin, Applied Differential Equations with Boundary Value Problems, 2017
10. A critical point x* of an autonomous vector equation X = f(x) is stable if all solutions that start sufficiently close to x* remain close to it. A critical point is said to be asymptotically stable if all solutions originated in a neighborhood of x* approach it. There are four kinds of stability:A center is stable, but not asymptotically stable.A sink is asymptotically stable.A source is unstable and all trajectories recede from the critical point.A saddle point is unstable, although some trajectories are drawn to the critical point and other trajectories recede.
On prediction of slope failure time with the inverse velocity method
Published in Georisk: Assessment and Management of Risk for Engineered Systems and Geohazards, 2023
Jie Zhang, Hong-zeng Yao, Zi-peng Wang, Ya-dong Xue, Lu-lu Zhang
The performances of the linear and non-linear INV methods are assessed using a landslide database with 55 case histories. The results obtained from this paper can be summarised as follows. There are two common pitfalls when applying the non-linear INV method, i.e. the saddle point and the ill-conditioned Hessian matrix. When these two pitfalls are encountered, the non-linear INV method cannot converge to the minimum point. For the landslides examined in this paper, the linear INV method is free from the two pitfalls.The saddle point can be assessed through the eigenvalues of the Hessian matrix. Whether the Hessian matrix is ill-conditioned can be examined by the condition number supplemented with an examination to see if the termination point is sensitive to the initial point. For the landslides studied in this paper, the Hessian matrix is ill-conditioned when the condition number is greater than 108.For the landslides examined in this paper, the linear INV method is not only more stable, but also more accurate than the non-linear INV method. For the above reasons, it is suggested that the linear INV method should be preferred over the non-linear INV method in future applications.
TDOA and RSSD Based Hybrid Passive Source Localization with Unknown Transmit Power
Published in IETE Journal of Research, 2020
Zengfeng Wang, Hao Zhang, Tingting Lu, Xing Liu, Zhaoqiang Wei, T. Aaron Gulliver
When the error statistics are known, the ML estimator is asymptotically optimal. The joint probability density function (PDF) of the hybrid TDOA and RSSD vector is given by and according to the assumption that and are independent Therefore, the ML estimate for hybrid TDOA and RSSD localization is This estimate has no closed-form solution, but it can be solved using iterative algorithms. However, they typically have high computational complexity and the performance depends on the initial solution used. Poor initialization may lead to converge to a local minimum or a saddle point causing large estimation error. Conversely, the proposed closed-form estimator is computationally efficient and does not require an initial solution.
Nash equilibrium computation of two-network zero-sum games with event-triggered communication
Published in Journal of Control and Decision, 2021
Hongyun Xiong, Jiangxiong Han, Xiaohong Nian, Shiling Li
The saddle point is an extreme point, which is neither a maximum nor a minimum. For a function , if there exists a pair such that then is called a saddle point of function f. In order to ensure that the saddle point set is nonempty, the constraint sets and usually are nonempty convex compact set, and is convex-concave.