Dynamic programming algorithms for the optimal cutting of equal rectangles

This paper presents dynamic programming algorithms for generating optimal guillotine-cutting patterns of equal rectangles. The algorithms are applicable for solving the unconstrained problem where the blank demand is unconstrained, the constrained problem where the demand is exact, the unconstrained problem with blade length constraint, and the constrained problem with blade length constraint. The algorithms are able to generate the simplest optimal patterns to simplify the cutting process. When the sheet length is longer than the blade length of the guillotine shear used, the dynamic programming algorithm is applied to generate optimal layouts on segments of lengths no longer than the blade length, and the knapsack algorithm is employed to find the optimal layout of the segments on the sheet. The computational results indicate that the algorithms presented are more efficient than the branch-and-bound algorithms, which are the best algorithms so far that can guarantee the simplest patterns.     

Dynamic Programming Algorithms for Generating Optimal Strip Layouts

This paper presents dynamic programming algorithms for generating optimal strip layouts of equal blanks processed by shearing and punching. The shearing and punching process includes two stages. The sheet is cut into strips using orthogonal guillotine cuts at the first stage. The blanks are punched from the strips at the second stage. The algorithms are applicable in solving the unconstrained problem where the blank demand is unconstrained, the constrained problem where the demand is exact, the unconstrained problem with blade length constraint, and the constrained problem with blade length constraint. When the sheet length is longer than the blade length of the guillotine shear used, the dynamic programming algorithm is applied to generate optimal layouts on segments of lengths not longer than the blade length, and the knapsack algorithm is employed to find the optimal layout of the segments on the sheet. Experimental computations show that the algorithms are efficient.     

三叶片垂直轴水轮机结构动力响应分析

提出采用改进离散涡和几何精确梁理论混合方法对三叶片垂直轴水轮机进行结构动力响应分析.相比传统的有限元方法,该方法具有求解速度快、建模简单、计算精确等优点.在模态分析中,计算了不同叶片高度下,水轮机叶片和整体的前五阶固有频率,分析了水轮机半径大小和叶片高度对固有频率的影响,结果显示:随着尺寸的增加,叶片和整体固有频率显著减小,整体固有频率更易受到半径大小的影响.在瞬态分析中,考虑了离心载荷和叶片的水动力载荷,得到在工作状况下,旋转一周过程中叶片的最大变形曲线;分析了在不同H/R（叶片高度和半径的比值）的情况下的叶片强度问题,结果显示:当H/R大于3.0时,叶片强度将会失效.     

On some regularities in dynamic response of cyclic periodic structures

The paper deals with cyclic periodic structures modelling bladed disk assemblies of blades with friction elements for vibration damping. These elements placed between adjacent blades reduce the vibration amplitudes by means of dry friction resulting from centrifugal forces acting on the elements and relative displacements of the blades. However, the application of these friction elements results in an additional dynamical coupling which together with mistuning of some system parameters (e.g., blade eigenfrequency or contact parameters) may cause localization of vibration. In the present paper a linear approximation of such a system is investigated. The structure composed of cyclic periodic cells modelled each as a clamped-free beam interacting with each other by means of viscoelastic elements of complex stiffness is applied for dynamic system analysis. In case of free vibrations as well as in case of steady-state dynamic response to a harmonic pressure field, a perfect periodic structure and the structures with periodically mistuned parameters (blade eigenfrequencies and contact parameters) are studied. Some regularities in the dynamic response of the systems with mistuning have been noticed. Despite the fact that only a linear approximation has been used, the results and conclusions can be applied for models which describe the blade interaction in a nonlinear way.     

Application of non-smooth NES in vibration suppression of rotor-blade systems

The non-smooth nonlinear energy sink (NSNES) is used to suppress the vibration of the rotor-blade system. Firstly, the structure and working principle of the NSNES for rotor-blade system are introduced. Then, the dynamics model of the rotor-blade-NSNES system is established by Lagrangian method. And then, numerical simulations are applied to evaluate the vibration suppression ability of the NSNES on rotor and blade. The results show that the suppression rates of NSNES on the rotor and the blade can reach 81% and 74% in steady state resonance under given parameters, respectively; and for transient vibration of blade, a 1.85 times dissipating speed is obtained in rotor-blade system with NSNES than that without NSNES. In particular, NSNES has better vibration suppression capability than linear dynamic vibration absorber (LDVA) when both have the same vibration absorption mass.     

Optimal investment-reinsurance policy for an insurance company with VaR constraint

This paper investigates an investment-reinsurance problem for an insurance company that has a possibility to choose among different business activities, including reinsurance/new business and security investment. Our main objective is to find the optimal policy to minimize its probability of ruin. The main novelty of this paper is the introduction of a dynamic Value-at-Risk (VaR) constraint. This provides a way to control risk and to fulfill the requirement of regulators on market risk. This problem is formulated as an infinite horizontal stochastic control problem with a constrained control space. The dynamic programming technique is applied to derive the Hamilton-Jacobi-Bellman (HJB) equation and the Lagrange multiplier method is used to tackle the dynamic VaR constraint. Closed-form expressions for the minimal ruin probability as well as the optimal investment-reinsurance/new business policy are derived. It turns out that the risk exposure of the insurance company subject to the dynamic VaR constraint is always lower than otherwise. Finally, a numerical example is given to illustrate our results.     

基于动态VaR约束与随机波动率模型的最优投资策略

研究Stein-Stein随机波动率模型下带动态VaR约束的最优投资组合选择问题. 假设投资者的目标是最大化终端财富的期望幂效用,可投资于无风险资产和一种风险资产, 风险资产的价格过程由Stein-Stein随机波动率模型刻画. 同时, 投资者期望能在投资过程中利用动态VaR约束控制所面对的风险.运用Bellman动态规划方法和Lagrange乘子法, 得到了该约束问题最优策略的解析式及特殊情形下最优值函数的解析式; 并通过理论分析和数值算例, 阐述了动态VaR约束与随机波动率对最优投资策略的影响.     

用投影方法求耗散广义Hamilton约束系统的李群积分

针对耗散广义Hamilton约束系统,通过引入拉格朗日乘子和采用投影技术,给出了一种保持动力系统内在结构和约束不变性的李群积分法．首先将带约束条件的耗散Hamilton系统化为无约束广义Hamilton系统, 进而讨论了无约束广义Hamilton系统的李群积分法,最后给出了广义Hamilton约束系统李群积分的投影方法．采用投影技术保证了约束的不变性,引入拉格朗日乘子后,在向约束流形投影时不会破坏原动力系统的李群结构．讨论的内容仅限于完整约束系统, 通过数值例题说明了方法的有效性．     

Modelling and passive control of flexible guiding hoisting system with time-varying length

ABSTRACT

A coupled dynamic modelling of the flexible guiding hoisting system is established, which includes the transverse-longitudinal-coupled vibration and the rotational vibration. Substituting vibrational energy of the system into Hamilton principle and applying the dynamic constraint, a distributed parameter mathematical model of the multi-rope system is derived. It is governed by coupled partial differential equations and ordinary differential equations (PDEs-ODEs), where the dynamic constraint in the form of an unknown moving force is the only connection between the hoisting conveyance and the guiding ropes. Based on Galerkin method, the dynamic response of the system is validated by numerical calculation and ADAMS simulation. Besides, an absorber with artificial intelligence optimization is proposed to reduce system vibration. The simulation result has demonstrated that a hoisting conveyance resonance can be observed when the external disturbance frequency is close to the system natural frequencies. Moreover, a vibration absorber can effectively diminish the resonant peaks of the first three orders of the guiding rope.     

Aggregation of constraints in integer programming

Constraint aggregation provides a method of formulating equivalent integer programs with a smaller number of constraints. This approach was widely researched in the seventies but its use was discounted due to large coefficients in the equivalent problem. We provide a method that yields numerically smaller constraint coefficients. This method has enabled us to investigate other computational issues relating to the use of constraint aggregation in solving integer programming problems, more thoroughly than has previously been possible.     

求解非定常非完整约束多体系统动力学的一种方法

通过对约束矩阵进行奇异值分解,将系统的动力学方程沿与约束相容和不相容的两个方向上投影,求解受非定常非完整约束的多体系统动力学问题,并给出了求约束反力的公式和避免违约的一种方法;这种解法不仅不依赖于描述系统的坐标选择,而且计算效率高。最后举了一个说明性例子。     

旋翼/机身非线性气弹耦合配平及稳定性分析

根据Hamilton原理,采用中等变形梁理论,将桨叶离散为15自由度梁单元,用准定常气动模型建立旋翼/刚性机身耦合的有限元非线性方程,用时间有限元法进行气弹耦合配平计算,得到桨叶和机身运动的周期解．在此基础上,引入Peters动态入流模型分析耦合系统的稳定性．并研制相应的计算程序,可用于桨叶响应、桨叶和桨毂载荷、旋翼操纵等方面的分析计算．算例分析结果与相关文献吻合较好,且同时满足桨叶响应和配平方程的收敛性要求．     

Generating optimal cutting patterns for rectangular blanks of a single size

This paper deals with the problem of minimizing trim loss in cutting rectangular blanks of a single size from a rectangular sheet using orthogonal guillotine cuts. First we prove that we can obtain the unconstrained optimal layout by searching among normal multi-section layouts. Next we present an unconstrained algorithm to search for it. The unconstrained algorithm uses a branch-and-bound method with a tight upper bound. Later we discuss the algorithm for the constrained problem where the blank demand must be met exactly. Finally, the unconstrained algorithm is extended to cope with the blade length constraint. Experimental computations show that the algorithms are extremely efficient.     

A new component mode synthesis method: Quasi-static mode compensation

A new component mode synthesis method is presented in this paper that combines the computational efficiency of the well-known constraint mode approach with the dynamic compensation accuracy obtained by higher-order expansion methods. Instead of employing static constraint modes, quasi-static modes are used to capture inertial effects of the truncated modes. The method is ideally suited for mid-band frequency analysis in which both high-frequency and low-frequency modes may be omitted. A tuning parameter, designated as the centering frequency, controls the dynamic range of the quasi-static modes. Numerical examples are provided which demonstrate the improved accuracy of the proposed method.     

Regularity properties of optimal controls for problems with time-varying state and control constraints

In this paper we report new results on the regularity of optimal controls for dynamic optimization problems with functional inequality state constraints, a convex time-dependent control constraint and a coercive cost function. Recently, it has been shown that the linear independence condition on active state constraints, present in the earlier literature, can be replaced by a less restrictive, positive linear independence condition, that requires linear independence merely with respect to non-negative weighting parameters, provided the control constraint set is independent of the time variable. We show that, if the control constraint set, regarded as a time-dependent multifunction, is merely Lipschitz continuous with respect to the time variable, then optimal controls can fail to be Lipschitz continuous. In these circumstances, however, a weaker Hölder continuity-like regularity property can be established. On the other hand, Lipschitz continuity of optimal controls is guaranteed for time-varying control sets under a positive linear independence hypothesis, when the control constraint sets are described, at each time, by a finite collection of functional inequalities.     

Knowledge Extraction from Aerodynamic Design Data and its Application to 3D Turbine Blade Geometries

Applying numerical optimisation methods in the field of aerodynamic design optimisation normally leads to a huge amount of heterogeneous design data. While most often only the most promising results are investigated and used to drive further optimisations, general methods for investigating the entire design dataset are rare. We propose methods that allow the extraction of comprehensible knowledge from aerodynamic design data represented by discrete unstructured surface meshes. The knowledge is prepared in a way that is usable for guiding further computational as well as manual design and optimisation processes. A displacement measure is suggested in order to investigate local differences between designs. This measure provides information on the amount and direction of surface modifications. Using the displacement data in conjunction with statistical methods or data mining techniques provides meaningful knowledge from the dataset at hand. The theoretical concepts have been applied to a data set of 3D turbine stator blade geometries. The results have been verified by means of modifying the turbine blade geometry using direct manipulation of free form deformation (DMFFD) techniques. The performance of the deformed blade design has been calculated by running computational fluid dynamic (CFD) simulations. It is shown that the suggested framework provides reasonable results which can directly be transformed into design modifications in order to guide the design process.     

A new constraint test-case generator and the importance of hybrid optimizers

In the past decade, significant progress has been made in understanding problem complexity of discrete constraint problems. In contrast, little similar work has been done for constraint problems in the continuous domain. In this paper, we study the complexity of typical methods for non-linear constraint problems and present hybrid solvers with improved performance. To facilitate the empirical study, we propose a new test-case generator for generating non-linear constraint satisfaction problems (CSPs) and constrained optimization problems (COPs). The optimization methods tested include a sequential quadratic programming (SQP) method, a penalty method with a fixed penalty function, a penalty method with a sequence of penalty functions, and an augmented Lagrangian method. For hybrid solvers, we focus on the form that combines two or more optimization methods in sequence. In the experiments, we apply these methods to solve a series of continuous constraint problems with increasing constraint-to-variable ratios. The test problems include artificial benchmark problems from the test-case generator and problems derived from controlling a hyper-redundant modular manipulator. We obtain novel results on complexity phase transition phenomena of the various methods. Specifically, for constraint satisfaction problems, the SQP method is the best on weakly constrained problems, whereas the augmented Lagrangian method is the best on highly constrained ones. Although the static penalty method performs poorly by itself, by combining it with the SQP method, we show a hybrid solver that is significantly better than any of the individual methods on problems with moderate to large constraint-to-variable ratios. For constrained optimization problems, the hybrid solver obtains much better solutions than SQP, while spending comparable amount of time. In addition, the hybrid solver is flexible and can achieve good results on time-bounded applications by setting parameters according to the time limits.     

改进的NSGA-Ⅱ算法研究风力机叶片多目标优化

将一种采用精英控制策略和动态拥挤方法用于快速非支配排序遗传算法（NSGA-Ⅱ）,并应用到风力机叶片的优化研究中,获得了一种新颖的风力机叶片多目标优化设计方法．作为应用算例,以设计风速下的功率系数最大和叶片质量最小为优化目标,用该方法设计了5 MW大型风力机叶片．优化结果表明,此算法在处理风力机多目标优化问题取得了良好的效果,给出的是一个Pareto最优解集,而不是传统优化方法追求的单个最优解,为风力机多目标优化设计提供新的思路和通用的算法．     

Vibration and flutter of stiff-inplane elastically tailored composite rotor blades

Aeroelastic response, blade and hub loads, and shaft-fixed aeroelastic stability is investigated for a helicopter with elastically tailored stiff-inplane composite rotor blades. A free wake model for nonuniform rotor inflow is integrated with a recently developed finite-element-based aeroelastic analysis for helicopters with tailored composite blades. Pitch-flap and pitch-lag elastic couplings, introduced through the anisotropy of the plies in the blade spar, have a significant effect on the dynamic elastic torsion response. Positive and negative pitch-flap couplings reduce vertical hub shear forces approximately 20% in the high vibration transition flight regime, however, negative pitch-flap elastic coupling significantly increases inplane hub shear forces at all flight speeds. The influence of pitch-flap, pitch-lag, and extension-torsion elastic couplings on the rotating frame blade bending moments is small. Ply-induced composite couplings have a powerful effect on blade stability in both hover and forward flight. Positive pitch-flap, positive pitch-lag, and positive extension-torsion couplings each have a stabilizing effect on lag mode damping. Negative pitch-lag coupling has a strong destabilizing effect on blade lag stability, resulting in a mild instability at moderate flight speeds.