Admissibility of simplified equations in the dynamics of gyroscopic systems

Based on the asymptotic approach of /1/, rigorous mathematical methods are used to single out some known simplified models from the theory of gyroscopic systems and to prove that they may legitimately be employed to solve problems in dynamics (including stability problems). The initial system is of the singularly perturbed type /2/. The use of methods from stability theory /3, 4/ yields conditions under which transition to a simplified (computational) model is permissible. Several papers have been devoted to the solution of such problems for singularly perturbed equations /5/ by methods of Lyapunov theory.     

Exact and numerical solutions of a micropolar fluid in a vertical annulus

In this article, we have analyzed the effects of heat transfer on a peristaltic flow of a micropolar fluid in a vertical annulus. The governing equations of two‐dimensional micropolar fluid are simplified by using the assumptions of long wavelength and neglecting the wave number. A close form solutions are obtained for velocity field υ_x and microrotation component υ_θ. Further, the numerical solutions of the simplified equation of υ_θ are computed and the results are compared with the exact solution. The influence of pertinent parameters are analyzed through graphs. Trapping phenomena is also discussed for different parameters. © 2009 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 2010     

Simplified method and synchronization for a class of complex chaotic systems

This paper is devoted to investigate a class of complex chaotic systems and a linear correlation between the real and imaginary component of complex variables in these systems is found. Based on this linear relationship, a simplified law is proposed. First, complex Lorenz system is given to show the linear correlation, then it is simplified. Second, a simplified law is proposed to determine whether the complex system can be simplified, and the complex Lü system and hyperchaotic complex Lü system are used to verify the simplified law. Finally, a new synchronization control is proposed to synchronize complex Lorenz system and real Lorenz system. The theoretical analysis and numerical simulation prove the feasibility and better performance of this method.     

Two-parameter bifurcation in a two-dimensional simplified Hodgkin–Huxley model

The dynamical behaviors of a two-dimensional simplified Hodgkin–Huxley model exposed to external electric fields are investigated based on the qualitative analysis and numerical simulation. A necessary and sufficient condition is given for the existence of the Hopf bifurcation. The stability of equilibrium points and limit cycles is also studied. Moreover, the canards and bifurcation are discussed in the simplified model and original model. The dynamical behaviors of the simplified model are consistent with the original model. It would be a great help to further investigations of the original model.     

炉温曲线的数学模型及求解

针对2020年全国大学生数学建模竞赛A题——炉温曲线,在一些简化假设下给出了电路板在回焊炉中进行自动焊接时的炉温曲线的数学模型,求出了给定温度设置下的最大过炉速度,讨论了两种最优曲线的温度和过炉速度设置.最后对学生竞赛论文中的一些情况进行了评述.     

Comparative study of 1D, 2D and 3D simplified gas kinetic schemes for simulation of inviscid compressible flows

With assumption that all the particles in the phase velocity space are concentrated on a circle and on a sphere, the circular function-based gas kinetic scheme and sphere function-based gas kinetic scheme have been developed by Shu and his coworkers [21], [22], [23]. These schemes are simpler than the Maxwellian function-based gas kinetic schemes. The simplicity is due to the fact that the integral domain of phase velocity of circular function and sphere function is a finite region while the integral domain of Maxwellian distribution function is infinite. In this work, the 1D delta function-based gas kinetic scheme is also developed to form a complete set of the simplified gas kinetic schemes. The 1D, 2D and 3D simplified gas kinetic schemes can be viewed as the truly 1D, 2D and 3D flux solvers since they are based on the multi-dimensional Boltzmann equation. On the other hand, to solve the 3D flow problem, the tangential velocities are needed to be approximated by some ways for the 1D and 2D simplified gas kinetic schemes, and to solve the 1D flow problem, the tangential velocities should be taken as zero for the 2D and 3D simplified gas kinetic schemes. The performances of these three schemes for simulation of inviscid compressible flows are investigated in this work by their application to solve the test problems from 1D to 3D cases. Numerical results showed that the efficiency of the delta function-based gas kinetic scheme is slightly superior to that of the circular function- and sphere function-based gas kinetic schemes, while its stability is inferior significantly to the latter. For simulation of the 3D hypersonic flows, the sphere function-based gas kinetic scheme could be the best choice.     

Comparison of the bifurcation curves of a two-variable and a three-variable circadian rhythm model

Two dynamical systems describing the circadian fluctuation of two proteins (PER and TIM) in cells are compared. A simplified model with two variables has already been investigated. Detailed study of the possible bifurcation has been carried out. Periodic solutions of the differential equations with 24-h period have been obtained numerically. Here the general, more realistic model having three variables is investigated. The possible phase portraits and local bifurcations are studied in detail. The saddle-node and Hopf-bifurcation curves are determined in the plane of two parameters by using the parametric representation method. Using these curves the number and the type of the stationary points can be determined. The relation of the two bifurcation curves and the Takens–Bogdanov bifurcation points are also studied. The bifurcation curves are compared to those obtained for the simplified two-variable system.     

Group decision making with simplified neutrosophic ordered weighted distance operator

Simplified neutrosophic set is a convenient tool proposed for dealing with complex problems; it is effective in providing more data for decision‐making process. In this study, we develop a simplified neutrosophic ordered weighted distance operator which combines the neutrosophic distance measures and the ordered weighted average distance in the same formulation. It is a new handy aggregation operator that considers the situations where the input data are represented in simplified neutrosophic numbers, and it also contains diverse distance aggregation operators. Parameterized families of simplified neutrosophic ordered weighted distance operator are handled. Moreover, we establish a new neutrosophic group decision‐making method based on the simplified neutrosophic ordered weighted distance operator, which has 2 extended approaches for determining the weights of decision makers and decision attributes in decision‐making process, respectively. Finally, an illustrative example demonstrates the application of the proposed method. The effectiveness and advantages of the proposed method are shown by the comparative analysis with existing relative methods.     

An improved approximate Newton method for implicit Runge-Kutta formulas

Implicit Runge-Kutta (IRK) methods (such as the s-stage Radau IIA method with s=3,5, or 7) for solving stiff ordinary differential equation systems have excellent stability properties and high solution accuracy orders, but their high computing costs in solving their nonlinear stage equations have seriously limited their applications to large scale problems. To reduce such a cost, several approximate Newton algorithms were developed, including a commonly used one called the simplified Newton method. In this paper, a new approximate Jacobian matrix and two new test rules for controlling the updating of approximate Jacobian matrices are proposed, yielding an improved approximate Newton method. Theoretical and numerical analysis show that the improved approximate Newton method can significantly improve the convergence and performance of the simplified Newton method.     

An effective and efficient approach for nonlinear seismic analysis of bridges

A simplified and practical approach for nonlinear seismic analysis of highway bridges is proposed. Efficiency and effectiveness of the proposed model is discussed in detail. The effects of soil-structure interaction, analysis methods, and bridge skews on bridge responses are also carefully examined. The results obtained from the simplified model are compared to those of refined models. The paper concludes with a number of real bridge examples and recommendations.     

Fracture and Fragmentation of Silicon Dies

Different failure modes of silicon are observed. Experimental results are not able to explain these variations clearly and therefore numerical simulations have been performed. In order to reduce high computational costs a simplified method to introduce weakened areas on silicon chips is presented. (© 2010 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Theoretical and numerical analysis for the quasi-continuum approximation of a material particle model

In many applications materials are modeled by a large number of particles (or atoms) where any one of particles interacts with all others. Near or nearest neighbor interaction is expected to be a good simplification of the full interaction in the engineering community. In this paper we shall analyze the approximate error between the solution of the simplified problem and that of the full-interaction problem so as to answer the question mathematically for a one-dimensional model. A few numerical methods have been designed in the engineering literature for the simplified model. Recently much attention has been paid to a finite-element-like quasicontinuum (QC) method which utilizes a mixed atomistic/continuum approximation model. No numerical analysis has been done yet. In the paper we shall estimate the error of the QC method for this one-dimensional model. Possible ill-posedness of the method and its modification are discussed as well.

     

Comparison of the simplified and full MHD models for laminar incompressible flow past a circular cylinder

In the majority of research on incompressible magnetohydrodynamic (MHD) flows, the simplified model with the low magnetic Reynolds number assumption has been adopted because it reduces the number of equations to be solved. However, because the effect of flow on magnetic field is also neglected, the solutions of the simplified model may be different from those of the full model. As an example, the flow of an electrically conducting fluid past a circular cylinder under a magnetic field is investigated numerically using the simplified and full models in this paper. To solve the problems, two second-order compact finite difference algorithms based on the streamfunction-velocity formulation of the simplified model and the quasi-streamfunction-velocity formulation of the full model are developed respectively.Numerical simulations are carried out over a wide range of Hartmann number for steady-state laminar problems with both models. For the full model, magnetic Reynolds number (Re_m) is chosen from 0.01 to 10. The computed results show that solutions of the simplified MHD model are not exactly the same as those of the full MHD model for this flow problem in most cases even if Re_m in the full model is very low. Only in the special case that a strong external magnetic field is exerted perpendicular to the dominant flow direction, can the simplified MHD model be regarded as an approximation of the full MHD model with low Re_m.     

Criterion of unitary similarity of minimal passive scattering systems with a given transfer function

We establish necessary and sufficient conditions under which all minimal passive scattering systems that have a given transfer operator function are unitarily equivalent. These conditions can be significantly simplified in special cases important for applications, in particular, in the case where a transfer function is rational and in a more general case where this function is pseudoextendable.     

柴油机供喷油过程的压力变化与控制

针对2019年全国大学生数学建模竞赛A题高压油管的压力控制问题,在一些简化的假设下给出了描述柴油机高压共轨管内燃油进入和喷出过程压力变化的数学模型,得到了几种情况下维持压力稳定的策略,并对学生竞赛答卷中的一些情况进行了评述.     

Flow topology in an L-shaped cavity with lids moving in the same directions

Flow development and eddy structure in an L-shaped cavity with lids moving in the same directions have been investigated using both tools from topological and numerical methods. In particular, structural bifurcation near a nonsimple degenerate point is investigated by making a local analysis of the velocity field based on a Taylor series expansion. The streamlines of a Hamiltonian vector field system are simplified by using the homotopy invariance of the index theory. A series of bifurcation curves are constructed to determine the sequence of flow structures by which eddies are generated in the L-shaped cavity.     

Decay mode solutions to cylindrical KP equation

A nonlinear transformation for the cylindrical KP(CKP) equation has been derived by using the simplified homogeneous balance method (SHB). The 1-decay mode and 2-decay mode solutions of the CKP equation have been obtained in terms of the nonlinear transformation derived here. The results obtained in the paper are different from those reported earlier.     

两端固定资源连续分配问题的一种算法

用动态规划可解一端固定,一端自由的资源连续分配问题,对两端固定的此类问题,动态规划解法过程复杂．针对目标函数及约柬条件均为线性函数的此类问题,给出一个简化的一般算法及相应算例,本算法极大简化了计算的复杂性．     

An effective solution for a real cutting stock problem in manufacturing plastic rolls

We confront a practical cutting stock problem from a production plant of plastic rolls. The problem is a variant of the well-known one dimensional cutting stock, with particular constraints and optimization criteria defined by the experts of the company. We start by giving a problem formulation in which optimization criteria have been considered in linear hierarchy according to expert preferences, and then propose a heuristic solution based on a GRASP algorithm. The generation phase of this algorithm solves a simplified version which is rather similar to the conventional one dimensional cutting stock. To do that, we propose a Sequential Heuristic Randomized Procedure (SHRP). Then in the repairing phase, the solution of the simplified problem is transformed into a solution to the real problem. For experimental study we have chosen a set of problem instances of com-mon use to compare SHRP with another recent approach. Also, we show by means of examples, how our approach works over instances taken from the real production process. All authors are supported by MEC-FEDER Grant TIN2007-67466-C02-01 and by contract CN-05-127 of the University of Oviedo and the company ERVISA, and by FICYT under grant BP04-021.     

平面弹性力学问题的混合元法的泡函数稳定性及其后验误差估计

该文讨论平面弹性力学问题的混合元法的泡函数稳定性,并导出基于简化的稳定化格式的一种先验误差估计和后验误差估计．这种简化的稳定化格式较通常的格式节省自由度．