NEWTON METHOD FOR SOLVING A CLASS OF SMOOTH CONVEX PROGRAMMING

1ConvexProgrammingandExactPenaltyFunction Weconsiderthefollowingconvexprogramming:(P)minf(x)s.t.x∈S={x∈Rn:gi(x)≤0,i=1,…,m}.SupposethatSisacompactset.ThusthereexistsalargeboundedboxX,suchthatS={x∈Rn:gi(x)≤0,i=1,…,m}intX.Assumethatf(x),gi(x),i=1,…,m     

Dimensional synthesis of the optimal RSSR mechanism for a set of variable design parameters

This paper deals with the dimensional synthesis of the RSSR mechanism, also known as spatial four-bar linkage (R and S stand for revolute and spherical kinematic pairs respectively). To univocally describe the geometry of the RSSR mechanism a specific set of geometry parameters is necessary. Generally speaking, in a synthesis problem a subset of these parameters, defined as design parameters, is usually considered as assigned whereas the remaining ones, defined as design variables, have to be found by the synthesis procedure. That is, the goal of the synthesis procedure is to find the values of the design variables, while satisfying both functional requirements of the mechanism and constraints on the design parameters. In this paper each design parameter is assigned as variable within a given range rather than being assigned as a single value. In general, varying a design parameter means obtaining a different mechanism which has different functional performances as a consequence. This feature gives raise to a novel synthesis problem, which has not been treated in the literature yet. In particular, the RSSR mechanism synthesis presented in this paper takes the optimization of the force transmission as an objective function, while referring to a given range of values of each design parameter. In addition, prescribed constraints on given extreme angular positions for both the input and the output links, on the upper and lower bounds for the transmission ratio, and on the upper and lower bounds for the design variable values have to be satisfied. The synthesis problem, set as a constrained minimization problem, is solved numerically in two steps by means of a Genetic algorithm followed by a quasi-Newton algorithm. As an example of application, a dimensional synthesis of an RSSR mechanism used in a hand exoskeleton is reported.     

The Conley Index for Fast-Slow Systems I. One-Dimensional Slow Variable

We develop a qualitative theory for fast-slow systems with a one-dimensional slow variable. Using Conley index theory for singularity perturbed systems, conditions are given which imply that if one can construct heteroclinic connections and periodic orbits in systems with the derivative of the slow variable set to 0, these orbits persist when the derivative of the slow variable is small and nonzero.     

Hausdorff dimension of set generated by exceptional oscillations of a class of N-parameter Gaussian processes

A class of N-parameter Gaussian processes are introduced, which are more general than the N-parameter Wiener process. The definition of the set generated by exceptional oscillations of a class of these processes is given, and then the Hausdorff dimension of this set is defined. The Hausdorff dimensions of these processes are studied and an exact representative for them is given, which is similar to that for the two-parameter Wiener process by Zacharie （2001）. Moreover, the time set considered is a hyperrectangle which is more general than a hyper-scluare used by Zacharie （2001）. For this more general case, a Fernique-type inequality is established and then using this inequality and the Slepian lemma, a Levy＇s continuity modulus theorem is shown. Independence of increments is required for showing the representative of the Hausdorff dimension by Zacharie （2001）. This property is absent for the processes introduced here, so we have to find a different way.     

Hausdorff dimensnon of set generated by exceptional oscillations of a class of N-parameter Gaussian processes

A class of N-parameter Gaussian processes are introduced, which are more general than the N-parameter Wiener process. The definition of the set generated by exceptional oscillations of a class of these processes is given, and then the Hausdorff dimension of this set is defined. The Hausdorff dimensions of these processes are studied and an exact representative for them is given, which is similar to that for the two-parameter Wiener process by Zacharie (2001). Moreover, the time set considered is a hyperrectangle which is more general than a hyper-square used by Zacharie (2001). For this more general case, a Fernique-type inequality is established and then using this inequality and the ments is required for showing the representative of the Hausdorff dimension by Zacharie (2001). This property is absent for the processes introduced here, so we have to find a different way.     

A method for predicting unsteady potential flow about an aerofoil

A new model is presented for the calculation of the incompressible, inviscid flow around an arbitrary aerofoil undergoing unsteady motion. The technique was developed from the steady flow algorithm of Leishman and Galbraith¹ in which use was made of a linear distribution of panel vorticity. The procedure is in the same class as that of Basu and Hancock² but, because of the particular approach to the manner of specifying the shed vorticity, only a set of linear simultaneous equations needs be solved, unlike the method of Reference 2, complicated by the necessary solution of a quadratic. A brief history of unsteady flow modelling is given in the introduction, followed by the mathematical details of the current method. Results are presented and discussed for a number of cases which clearly illustrate relevant characteristics of unsteady flow.     

BV Estimates in Optimal Transportation and Applications

In this paper we study the BV regularity for solutions of certain variational problems in Optimal Transportation. We prove that the Wasserstein projection of a measure with BV density on the set of measures with density bounded by a given BV function f is of bounded variation as well and we also provide a precise estimate of its BV norm. Of particular interest is the case f = 1, corresponding to a projection onto a set of densities with an L ^∞ bound, where we prove that the total variation decreases by projection. This estimate and, in particular, its iterations have a natural application to some evolutionary PDEs as, for example, the ones describing a crowd motion. In fact, as an application of our results, we obtain BV estimates for solutions of some non-linear parabolic PDE by means of optimal transportation techniques. We also establish some properties of the Wasserstein projection which are interesting in their own right, and allow, for instance, for the proof of the uniqueness of such a projection in a very general framework.     

Universal Singular Sets in the Calculus of Variations

For regular one-dimensional variational problems, Ball and Nadirashvilli introduced the notion of the universal singular set of a Lagrangian L and established its topological negligibility. This set is defined to be the set of all points in the plane through which the graph of some absolutely continuous L-minimizer passes with infinite derivative. Motivated by Tonelli’s partial regularity results, the question of the size of the universal singular set in measure naturally arises. Here we show that universal singular sets are characterized by being essentially purely unrectifiable—that is, they intersect most Lipschitz curves in sets of zero length and any compact purely unrectifiable set is contained within the universal singular set of some smooth Lagrangian with given superlinear growth. This gives examples of universal singular sets of Hausdorff dimension two, filling the gap between previously known one-dimensional examples and Sychëv’s result that universal singular sets are Lebesgue null. We show that some smoothness of the Lagrangian is necessary for the topological size estimate, and investigate the relationship between growth of the Lagrangian and the existence of (pathological) rectifiable pieces in the universal singular set. We also show that Tonelli’s partial regularity result is stable in that the energy of a “near” minimizer u over the set where it has large derivative is controlled by how far u is from being a minimizer.     

An iterative method for the inverse elasto-static problem

This brief communication describes the use of an iterative algorithm to tackle the inverse elasto-static problem which has two important applications: identifying the required initial body shape which deforms into a desired one when subject to a given set of loads; identifying the unloaded shape of a structure from a given deformed shape and a prescribed set of loads. The iterative algorithm stems from the rather intuitive observation that a better estimate of the required initial shape can be obtained by adding a correction to the previous shape estimate equal to the computed mismatch in the deformed shapes. A detailed convergence analysis explains the reason behind the success of the proposed algorithm. Compared to other techniques described in the literature to solve such inverse problems, this iterative scheme offers the advantage of being easy to implement and readily compatible with the use of commercial solver packages. Two benchmark problems are treated to illustrate the success of the algorithm: that of a punctured membrane and that of a cantilever beam in cross flow. The latter illustrates that the concepts described here readily applies to steady fluid–structure interaction problems leading to an interesting range of new applications.     

Runge-kutta method, finite element method, and regular algorithms for hamiltonian system

The symplectic algorithm and the energy conservation algorithm are two important kinds of algorithms to solve Hamiltonian systems. The symplectic RungeKutta (RK) method is an important part of the former, and the continuous finite element method (CFEM) belongs to the later. We find and prove the equivalence of one kind of the implicit RK method and the CFEM, give the coefficient table of the CFEM to simplify its computation, propose a new standard to measure algorithms for Hamiltonian systems, and define another class of algorithms—the regular method. Finally, numerical experiments are given to verify the theoretical results.     

Lagrangean Structure and Propagation of Singularities in Multidimensional Compressible Flow

We study the propagation of singularities in solutions of the Navier–Stokes equations of compressible, barotropic fluid flow in two and three space dimensions. The solutions considered are in a fairly broad regularity class for which initial densities are nonnegative and essentially bounded, initial energies are small, and initial velocities are in certain fractional Sobolev spaces. We show that, if the initial density is bounded below away from zero in an open set V, then each point of V determines a unique integral curve of the velocity field and that this system of integral curves defines a locally bi-Hölder homeomorphism of V onto its image at each positive time. This “Lagrangean structure” is then applied to show that, if the initial density has a limit at a point of such a set V from a given side of a continuous hypersurface in V, then at each later time both the density and the divergence of the velocity have limits at the transported point from the corresponding side of the transported hypersurface, which is also a continuous manifold. If the limits from both sides exist, then the Rankine–Hugoniot conditions hold in a strict pointwise sense, showing that the jump in the divergence of the velocity is proportional to the jump in the pressure. This leads to a derivation of an explicit representation for the strength of the jump in the logarithm of the density, from which it follows that discontinuities persist for all time, convecting along fluid particle paths, and in the case that the pressure is strictly increasing in density, having strengths which decay exponentially in time.     

多体系统Euler-Lagrange方程组的一类新数值分析方法

针对受完整约束的多体系统,首先指出其动力学Euler-Lagrange方程组是高指标(index>2)的微分代数方程组;不同于传统的直接增广法和直接消去法,文中提出了一类将微分代数方程直接视为非线性代数方程组求解的新的数值分析方法;最后,以典型的单摆模型为例给出了新算法与其他方法的比较,结果表明新算法优于BDF方法及违约修正方法。     

Risk-sensitive reinforcement learning algorithms with generalized average criterion

A new algorithm is proposed, which immolates the optimality of control policies potentially to obtain the robusticity of solutions. The robusticity of solutions maybe becomes a very important property for a learning system when there exists non-matching between theory models and practical physical system, or the practical system is not static, or the availability of a control action changes along with the variety of time. The main contribution is that a set of approximation algorithms and their convergence results are given. A generalized average operator instead of the general optimal operator max (or min) is applied to study a class of important learning algorithms, dynamic programming algorithms, and discuss their convergences from theoretic point of view. The purpose for this research is to improve the robusticity of reinforcement learning algorithms theoretically.     

A continuous-discontinuous cellular automaton method for cracks growth and coalescence in brittle material简

A method of continuous-discontinuous cellular automaton for modeling the growth and coalescence of multiple cracks in brittle material is presented. The method uses the level set to track arbitrary discontinuities, and calculation grids are independent of the discontinuities and no remeshing are required with the crack growing. Based on Grif- fith fracture theory and Mohr-Coulumb criterion, a mixed fracture criterion for multiple cracks growth in brittle mate- rial is proposed. The method treats the junction and coales- cence of multiple cracks, and junction criterion and coales- cence criterion for brittle material are given, too. Besides, in order to overcome the tracking error in the level set ap- proximation for crack junction and coalescence, a dichotomy searching algorithm is proposed. Introduced the above the- ories into continuous-discontinuous cellular automaton, the present method can be applied to solving multiple crack growth in brittle material, and only cell stiffness is needed and no assembled global stiffness is needed. Some numerical examples are given to shown that the present method is efficient and accurate for crack junction, coalescence and percolation problems.     

Modified integral-level set method for the constrained solving Global Optimization

The constrained global optimization problem being considered, a modified integral-level set method was illustrated based on Chew-Zheng‘s paper on Integral Global Optimizationand Wu‘s paper on Implementable Algorithm Convergence of Mottled Integral-Level SetMethod for Global Optimization Problem. It has two characters : 1 ) Each phase must constructa new function which has the same global optimal value as that of primitive objective function ;2) Comparing it with Zheng‘s method, solving level set procedure is avoided. Animplementable algorithm also is given and it is proved that this algorithm is convergent.     

Stable periodic solutions for the hypercycle system

We consider the hypercycle system of ODEs, which models the concentration of a set of polynucleotides in a flow reactor. Under general conditions, we prove the omega-limit set of any orbit is either an equilibrium or a periodic orbit. The existence of an orbitally asymptotic stable periodic orbit is shown for a broad class of such systems.     

The globalization of Durand-Kerner algorithm

I.Intr0ductionForflndingallzerosofacomplexpolynomialofdegree;Ip(z)=zu olz'-l ..' a.-1z a.(l.l)whenzerosareal1simple,manymethodsareavailable.AmongthemaretheDurand-Kerneralgorithm,theAberthalgorithm,thesquarerootingaIg0rithmandtheirvarietiesI3'[1'141.Thecom…     

基于物理统一存储大规模数字导航地图道路网拓扑自动生成算法

道路网络的拓扑信息是GIS进行空间分析,如最优路径、地图匹配等算法的数据基础。目前,获取和建立道路网络的拓扑信息非常繁琐,不仅费时、费力,并且容易出错。采用逻辑分幅物理统一的存储策略,在探讨了拓扑生成的一般算法的前提下,提出了大规模超大规模数字地图自动生成道路网络拓扑关系的步骤和算法。该算法采用网格索引检索每个子图的元素,用hash索引映射实体ID和实体对象信息,并将整图的拓扑信息生成转化为对每个子图的拓扑求取,并对跨子图道路拓扑求解特别讨论。然后,对算法复杂度进行了分析,并且通过建立不同道路数的多个虚拟道路网络子图对算法性能进行了测试和比较。最后用本算法跟踪处理了南京市道路网络（部分）,并给出了结果。本算法在保留地理数据完整性的前提下,解决了常规方法的内存限制,并且具有准线性的运算代价,并能够自动恢复数据处理。     

Numerical algorithm for rigid body position estimation using the quaternion approach

This paper deals with rigid body attitude estimation on the basis of the data obtained from an inertial measurement unit mounted on the body. The aim of this work is to present the numerical algorithm, which can be easily applied to the wide class of problems concerning rigid body positioning, arising in aerospace and marine engineering, or in increasingly popular robotic systems and unmanned aerial vehicles. Following the considerations of kinematics of rigid bodies, the relations between accelerations of different points of the body are given. A rotation matrix is formed using the quaternion approach to avoid singularities. We present numerical procedures for determination of the absolute accelerations of the center of mass and of an arbitrary point of the body expressed in the inertial reference frame, as well as its attitude. An application of the algorithm to the example of a heavy symmetrical gyroscope is presented, where input data for the numerical procedure are obtained from the solution of differential equations of motion, instead of using sensor measurements.     

Finite element analysis of flow in a gas-filled rotating annulus

Linearized multidimensional flow in a gas centrifuge can be described away from the ends by Onsager's pancake equation. However a rotating annulus results in a slightly different set of boundary conditions from the usual symmetry at the axis of rotation. The problem on an annulus becomes ill-posed and requires some special attention. Herein we treat axially linear inner and outer rotor temperature distributions and velocity slip. An existence condition for a class of non-trivial, one-dimensional solutions is given. New exact solutions in the infinite bowl approximation have been derived containing terms that are important at finite gap width and non-vanishing velocity slip. The usual one-dimensional, axially symmetric solution is obtained as a limit. Our previously reported finite element algorithm has been extended to treat this new class of problems. Effects of gap width, temperature and slip conditions are illustrated. Lastly, we report on the compressible, finite length, circular Couette flow for the first time.