stationary points
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2022 ◽  
Author(s):  
Hameed Ashraf ◽  
Abida Parveen ◽  
Hamood Ur Rehman ◽  
Muhammad Imran Asjad ◽  
Bander N. Almutairi ◽  
...  

Abstract This article addresses the analysis of the uniform film thickness and stationary points forthe Carreau thin fluid film flow. The flow of fluid on a vertically upward moving cylinder takesplace in the presence of a surface tension gradient. The resulting non-linear and inhomogeneousordinary differential equation is solved for the series form solution using Adomian decompositionmethods (ADM). Stokes number St, inverse capillary number C, Weissenberg number W e andfluid behavior index n emerged as flow control parameters. The analysis showed that thepositions of stationary points transferred towards the surface of the cylinder by the increase ofSt and C while towards the fluid-air interface by the increase of n. W e delineated vice versaeffects on positions of stationary points for the shear thickening fluid film and shear thinningfluid film. The width of uniform film thickness reduces by an increment in the St and Cwhereas it increases by an increment in the n. The width of shear thickening uniform filmthickness increases whilst shear thinning uniform film thickness decreases as the W e increases. A comparison between the linearly viscous fluid and Carreau fluid is also made.


Author(s):  
Zhaosong Lu ◽  
Zhe Sun ◽  
Zirui Zhou

In this paper, we consider a class of structured nonsmooth difference-of-convex (DC) constrained DC programs in which the first convex component of the objective and constraints is the sum of a smooth and a nonsmooth function, and their second convex component is the supremum of finitely many convex smooth functions. The existing methods for this problem usually have a weak convergence guarantee or require a feasible initial point. Inspired by the recent work by Pang et al. [Pang J-S, Razaviyayn M, Alvarado A (2017) Computing B-stationary points of nonsmooth DC programs. Math. Oper. Res. 42(1):95–118.], in this paper, we propose two infeasible methods with a strong convergence guarantee for the considered problem. The first one is a penalty method that consists of finding an approximate D-stationary point of a sequence of penalty subproblems. We show that any feasible accumulation point of the solution sequence generated by such a penalty method is a B-stationary point of the problem under a weakest possible assumption that it satisfies a pointwise Slater constraint qualification (PSCQ). The second one is an augmented Lagrangian (AL) method that consists of finding an approximate D-stationary point of a sequence of AL subproblems. Under the same PSCQ condition as for the penalty method, we show that any feasible accumulation point of the solution sequence generated by such an AL method is a B-stationary point of the problem, and moreover, it satisfies a Karush–Kuhn–Tucker type of optimality condition for the problem, together with any accumulation point of the sequence of a set of auxiliary Lagrangian multipliers. We also propose an efficient successive convex approximation method for computing an approximate D-stationary point of the penalty and AL subproblems. Finally, some numerical experiments are conducted to demonstrate the efficiency of our proposed methods.


Molecules ◽  
2022 ◽  
Vol 27 (1) ◽  
pp. 307
Author(s):  
Fan Wang ◽  
Xiaoyan Ji ◽  
Fei Ying ◽  
Jiatao Zhang ◽  
Chongyang Zhao ◽  
...  

We characterized the stationary points along the nucleophilic substitution (SN2), oxidative insertion (OI), halogen abstraction (XA), and proton transfer (PT) product channels of M− + CH3X (M = Cu, Ag, Au; X = F, Cl, Br, I) reactions using the CCSD(T)/aug-cc-pVTZ level of theory. In general, the reaction energies follow the order of PT > XA > SN2 > OI. The OI channel that results in oxidative insertion complex [CH3–M–X]− is most exothermic, and can be formed through a front-side attack of M on the C-X bond via a high transition state OxTS or through a SN2-mediated halogen rearrangement path via a much lower transition state invTS. The order of OxTS > invTS is inverted when changing M− to Pd, a d10 metal, because the symmetry of their HOMO orbital is different. The back-side attack SN2 pathway proceeds via typical Walden-inversion transition state that connects to pre- and post-reaction complexes. For X = Cl/Br/I, the invSN2-TS’s are, in general, submerged. The shape of this M− + CH3X SN2 PES is flatter as compared to that of a main-group base like F− + CH3X, whose PES has a double-well shape. When X = Br/I, a linear halogen-bonded complex [CH3−X∙··M]− can be formed as an intermediate upon the front-side attachment of M on the halogen atom X, and it either dissociates to CH3 + MX− through halogen abstraction or bends the C-X-M angle to continue the back-side SN2 path. Natural bond orbital analysis shows a polar covalent M−X bond is formed within oxidative insertion complex [CH3–M–X]−, whereas a noncovalent M–X halogen-bond interaction exists for the [CH3–X∙··M]− complex. This work explores competing channels of the M− + CH3X reaction in the gas phase and the potential energy surface is useful in understanding the dynamic behavior of the title and analogous reactions.


Author(s):  
Veronika Grimm ◽  
Daniel Nowak ◽  
Lars Schewe ◽  
Martin Schmidt ◽  
Alexandra Schwartz ◽  
...  

AbstractWhile single-level Nash equilibrium problems are quite well understood nowadays, less is known about multi-leader multi-follower games. However, these have important applications, e.g., in the analysis of electricity and gas markets, where often a limited number of firms interacts on various subsequent markets. In this paper, we consider a special class of two-level multi-leader multi-follower games that can be applied, e.g., to model strategic booking decisions in the European entry-exit gas market. For this nontrivial class of games, we develop a solution algorithm that is able to compute the complete set of Nash equilibria instead of just individual solutions or a bigger set of stationary points. Additionally, we prove that for this class of games, the solution set is finite and provide examples for instances without any Nash equilibria in pure strategies. We apply the algorithm to a case study in which we compute strategic booking and nomination decisions in a model of the European entry-exit gas market system. Finally, we use our algorithm to provide a publicly available test library for the considered class of multi-leader multi-follower games. This library contains problem instances with different economic and mathematical properties so that other researchers in the field can test and benchmark newly developed methods for this challenging class of problems.


2021 ◽  
Vol 932 ◽  
Author(s):  
Andrew J. Mitchell ◽  
Brian R. Duffy ◽  
Stephen K. Wilson

A detailed analysis of steady coating flow of a thin film of a viscous fluid on the outside of a uniformly rotating horizontal circular cylinder in the absence of surface-tension effects but in the presence of a non-uniform pressure distribution due to an irrotational airflow with circulation shows that the presence of the airflow can result in qualitatively different behaviour of the fluid film from that in classical coating flow. Full-film solutions corresponding to a continuous film of fluid covering the entire cylinder are possible only when the flux and mass of fluid do not exceed critical values, which are determined in terms of the non-dimensional parameters $F$ and $K$ representing the speed of the far-field airflow and the circulation of the airflow, respectively. The qualitative changes in the behaviour of the film thickness as $F$ and $K$ are varied are described. In particular, the film thickness can have as many as four stationary points and, in general, has neither top-to-bottom nor right-to-left symmetry. In addition, when the circulation of the airflow is in the same direction as the rotation of the cylinder the maximum mass of fluid that can be supported on the cylinder is always less than that in classical coating flow, whereas when the circulation is in the opposite direction the maximum mass of fluid can be greater than that in classical coating flow.


Author(s):  
Radu Ioan Boţ ◽  
Minh N. Dao ◽  
Guoyin Li

In this paper, we consider a broad class of nonsmooth and nonconvex fractional programs, which encompass many important modern optimization problems arising from diverse areas such as the recently proposed scale-invariant sparse signal reconstruction problem in signal processing. We propose a proximal subgradient algorithm with extrapolations for solving this optimization model and show that the iterated sequence generated by the algorithm is bounded and that any one of its limit points is a stationary point of the model problem. The choice of our extrapolation parameter is flexible and includes the popular extrapolation parameter adopted in the restarted fast iterative shrinking-threshold algorithm (FISTA). By providing a unified analysis framework of descent methods, we establish the convergence of the full sequence under the assumption that a suitable merit function satisfies the Kurdyka–Łojasiewicz property. Our algorithm exhibits linear convergence for the scale-invariant sparse signal reconstruction problem and the Rayleigh quotient problem over spherical constraint. When the denominator is the maximum of finitely many continuously differentiable weakly convex functions, we also propose another extrapolated proximal subgradient algorithm with guaranteed convergence to a stronger notion of stationary points of the model problem. Finally, we illustrate the proposed methods by both analytical and simulated numerical examples.


2021 ◽  
Vol 9 (4B) ◽  
Author(s):  
Bassam A. Albassam ◽  

This paper deals with designing a control force to create nodal point(s) having zero displacement and/or zero slope at selected locations in a vibrating beam structure excited by multiple harmonic forces. It is shown that the steady state vibrations at desired points can be eliminated using applied control forces. The control forces design method is implemented using dynamic Green’s functions that transform the equations of motion from differential to algebraic equations, in which the resulting solution is analytic and exact. The control problem is greatly simplified by utilizing the superposition principle that leads to calculating the control forces to create node(s) for each excitation frequency independently. The calculated control forces can be realized using passive elements such as masses and springs connected to the beam having reaction forces equal to the calculated control forces. The effectiveness of the proposed method is demonstrated on various cases using numerical examples. Through examples, it was shown that creating node(s) with zero deflection, as well as zero slope, not only results in isolated stationary points, but also suppresses the vibrations along a wide region of the beam.


2021 ◽  
Author(s):  
Xin-long Luo ◽  
Hang Xiao

Abstract The global minimum point of an optimization problem is of interest in engineering fields and it is difficult to be solved, especially for a nonconvex large-scale optimization problem. In this article, we consider the continuation Newton method with the deflation technique and the quasi-genetic evolution for this problem. Firstly, we use the continuation Newton method with the deflation technique to find the stationary points from several determined initial points as many as possible. Then, we use those found stationary points as the initial evolutionary seeds of the quasi-genetic algorithm. After it evolves into several generations, we obtain a suboptimal point of the optimization problem. Finally, we use the continuation Newton method with this suboptimal point as the initial point to obtain the stationary point, and output the minimizer between this final stationary point and the found suboptimal point of the quasi-genetic algorithm. Finally, we compare it with the multi-start method (the built-in subroutine GlobalSearch.m of the MATLAB R2020a environment) and the differential evolution algorithm (the DE method, the subroutine de.m of the MATLAB Central File Exchange 2021), respectively. Numerical results show that the proposed method performs well for the large-scale global optimization problems, especially the problems of which are difficult to be solved by the known global optimization methods.


Ultrasonics ◽  
2021 ◽  
pp. 106662
Author(s):  
Wei Bai ◽  
Kai Wang ◽  
Dongxing Du ◽  
Jianguo Zhang ◽  
Wen Huang ◽  
...  

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