基于密切多项式近似的多项式插值算法框架

多项式插值技术是近似理论中一种常见的近似方法,被广泛用于数值分析、信号处理等领域。但传统的多项式插值技术大多是基于数值分析与实验结果相结合得到的,没有统一的理论描述和规律性的解决方案。为此,根据密切多项式近似理论为图像的多项式插值算法提出一个统一的理论框架。密切多项式近似的理论框架包括采样点数目、密切阶数和导数近似规则三个部分,它既可以用于分析现有的多项式插值算法,也可以用于开发新的多项式插值算法。分析了主流多项式插值技术在密切多项式近似理论框架下的表现形式,并以四点二阶密切多项式插值算法为例详细描述了利用密切多项式插值的理论框架开发新的多项式插值算法的一般流程。理论分析和数值实验表明大多数主流插值算法都属于密切多项式插值算法,它们的处理效果与采样点数目、密切阶数和导数近似规则有紧密的关系。     

计算机辅助几何设计中的保形插值理论及应用

本文主要研究计算机辅助几何设计中的分段多项式保形插值理论与算法 ,分段参数多项式保形插值方法及GHI问题 ,参数曲线弧长参数化的混合数值算法与近似方法 ,与给定任意切线多边形相切的保形逼近样条曲线 ,Bézier曲线和 NURBS曲线的等距线生成以及一般参数曲线等距线的保形逼近曲线。本文首先系统地研究了分段多项式的保形插值 ,建立了分段多项式的保形插值理论框架 ,导出了分段三次Hermite插值保形的充要条件 ,构造了一个 C1 连续的分段三次多项式保形插值算法 ,导出了 2 k+1次或 2 k次多项式保凸的充要条件 ,给出了插入内结点的区域…     

三次Hermite插值的扩展及其应用

在总结分析三次Hermite插值多项式的基础上，对三次Hermite插值公式进行了推广和扩展。通过改变三次Hermite插值的初始条件．得到了扩展的插值多项式计算公式。给出了扩展的三次Hermite插值格式的有理函数的近似表示方法以及有理函数的数值积分算法。     

Hamilton系统的保辛 守恒积分算法

给出了Hamilton系统基于辛矩阵乘法的显式时不变正则变换和时变正则变换．引入含参变量的近似Hamilton系统,并以近似Hamilton系统为基础进行辛矩阵乘法的正则变换．正则变换保证了数值积分的保辛性质,而通过调整引入的参变量可保证能量在积分格点上守恒．实现了Hamilton系统即保辛又保能量的算法．     

嵌套式稀疏网格随机配置法及其在随机门延时建模中的应用

为了提高随机工艺偏差下门延时建模的计算精度和效率,提出一种基于扩展Gauss积分理论及嵌套式稀疏网格技术的随机配置门延时建模方法.首先采用参数空间中具有指数收敛特性的随机正交多项式对随机门延时进行逼近;然后针对现有的基于传统Gauss积分理论的稀疏网格随机配置法所用的配置点不具有嵌套特性的问题,利用单变量扩展Gauss积分理论及稀疏网格技术构造了一组嵌套式多变量Gauss积分点,将其作为随机门延时建模的配置点.这组配置点既具有Gauss积分点的高精度,又满足嵌套性质,且在低阶积分配置点上已经得到的门延时可以在高阶积分时重复使用.与现有的基于非嵌套式配置点的随机配置法相比,该方法的计算精度和效率可以得到很大的提升,数值实验结果也验证了该方法在计算精度和效率上的优势.     

单变量均匀静态细分格式的连续性分析和构造

利用单变量均匀稳定细分格式C^k连续的充要条件,分析了已有的插值曲线格式各阶连续时参数的取值范围.首次指出了六点二重插值格式可以达到C³连续,并构造了一种新的C³连续的六点三重插值细分格式.     

Duffing方程的辛精细积分方法研究

辛精细积分方法汲取了辛几何算法保持动力学系统辛结构的优点和精细积分方法高精度的数值优点,其实现过程中涉及到大量矩阵求逆运算.为减小辛精细积分方法的运算量,本文在辛精细积分算法之前先将非齐次方程近似齐次化,使得矩阵求逆部分不显含时间,降低矩阵求逆计算量,并将这一方法应用于无阻尼Duffing方程的数值分析.通过与经典四阶Runge-Kutta格式及精细积分方法对比,发现辛精细积分方法在数值精度、计算耗时、保持系统能量等方面明显优于Runge-Kutta格式.此外,与精细积分方法相比,辛精细积分方法在保持系统能量方面存在明显优势.     

非线性动力学系统最优控制问题的保辛求解方法

基于对偶变量变分原理提出了求解非线性动力学系统最优控制问题的一种保辛数值方法.以时间区段一端状态和另一端协态作为混合独立变量,在时间区段内采用拉格朗日插值近似状态变量与协态变量,然后利用对偶变量变分原理并将非线性最优控制问题转化为非线性方程组的求解,最终得到求解非线性动力学系统最优控制问题的保辛数值方法.数值实验验证了本文算法在求解精度与求解效率上的有效性.     

利用形状参数构造保凸插值的双曲多项式B样条曲线

把一个参数化的奇异多边形与双曲多项式B样务按某一个因子调配，可自动生成带形状参数且插值给定平面点列的C^2（或G^1）连续的双曲多项式B样条曲线．把这一曲线的曲率符号函数写为Bernstein多项式形式，并利用Bernstein多项式的非负性条件，得到形状参数的合适取值来保证样条曲线对插值点列的保凸性．此方法简单、方便，无需解方程组或迭代计算，生成的插值曲线具有较均匀的曲率．大量实例验证了算法的正确与有效．     

基于祖冲之类方法的多体动力学方程保能量保约束积分

针对一类多体动力学问题导出的微分 代数方程,提出一种保能量、保约束的算法.该算法基于祖冲之类方法和欧拉中点保辛差分,利用祖冲之类方法保证在时间格点上精确满足约束方程,避免约束违约问题;并进一步证明该算法在时间格点上可以精确保能量.数值算例进一步验证该算法的可靠性.     

Attitude manoeuvre of spacecraft with long cantilever beam appendage by momentum wheel

Attitude manoeuvre of spacecraft with a long cantilever beam appendage by momentum wheel is studied. The dynamics equation of the main body is derived by conservation of angular momentum. The dynamics equation of the appendage is derived by force equilibrium principle. Two feedback control strategies of the momentum wheel are applied for the attitude manoeuvre of the main body. The lateral vibration of the appendage is suppressed by active control in proportion to its lateral displacement and velocity. The variation of residual nutation angle of the spacecraft or the residual transversal angular velocity of the main body in the manoeuvre process is researched with changes of steady state time of the manoeuvre, appendage parameters, control parameters on the appendage and appendage location. In addition, spacecraft response is researched when there are no active controls on the appendage.     

Stabilized finite elements with equal order of interpolation for soil dynamics problems

Summary The accurate prediction of the behaviour of geostructures is based on the strong coupling between the pore fluid and the solid skeleton. If the relative acceleration of the fluid phase to the skeleton is neglected, the equations describing the problem can be written in terms of skeleton displacements (or velocities) and pore pressures. This mixed problem is similar to others found in solid and fluid dynamics. In the limit case of zero permeability and incompressibility of the fluid phase, the restrictions on the shape functions used to approximate displacements and pressures imposed by Babuska-Brezzi conditions or the Zienkiewicz-Taylor patch test hold. As a consequence, it is not possible to use directly elements with the same order of interpolation for the field variables. This paper proposes two alternative methods allowing us to circumvent the BB restrictions in the incompressibility limit, making it possible to use elements with the same order of interpolation. The first consists on introducing the divergence of the momentum equation of the mixture as an stabilization term, the second is a generalization of the two step-projection method introduced by Chorin for fluid dynamics problems.     

Complete determination of the dynamic coefficients of coupled journal and thrust bearings considering five degrees of freedom for a general rotor-bearing system

A complete method is presented for calculating the stiffness and damping coefficients of coupled journal and thrust bearings of a general rotor-bearing system considering five degrees of freedom. The Reynolds equations and their perturbation equations were derived by linearization of the bearing reaction with respect to the general five degrees of freedom, i.e., the tilting displacements and angular velocities as well as the translational displacements and velocities. The Reynolds equations and their perturbation equations were transformed into finite element equations by considering the continuity of pressure and flow at the interface between the journal and the thrust bearings. The Reynolds boundary condition was included in the numerical analysis so as to simulate the phenomenon of cavitation. The stiffness and damping coefficients of the proposed method were compared with those found from a numerical differentiation of the loads with respect to the finite displacements and velocities of the bearing center. It was shown that the proposed method may be used to calculate the dynamic coefficients of coupled journal and thrust bearings more accurately and efficiently than the differentiation method. The tilting motion was also been found to play an important role in the determination of force and moment coefficients.     

Experimental study of a momentum-based method for identifying the inertia barycentric parameters of a human body

Physical modeling, simulation and analysis of an individual human body require inertia properties of the body segments of the human. Such subject-specific inertia data can be obtained only by measuring the individual human body as opposed to be derived from statistically generated anthropometric database. This paper presents experimental validation of a momentum-based approach for identifying the barycentric parameters of an individual human body which fully describes the inertia properties of the human. The identification algorithm is derived from the impulse–momentum equations of the human body which is assumed to be a multibody system with tree-type topology. Since the impulse–momentum equations are linear in terms of the unknown barycentric parameters, these parameters can be solved from the equations using a least-squares method. The approach does not require measuring or estimating accelerations and joint forces/torques because they do not appear in the impulse–momentum equations, and thus, the resulting identification procedure is less demanding on measurement data than the methods derived from the equations of motion. In this paper the test results of the identification method are validated by comparing the identified inertia parameters against the statistically established anthropometric data. Additionally, the identification results are also confirmed by comparing the contact forces using inverse dynamics to those obtained by forces plates.     

The control of angular momentum for asymmetric rigid bodies

A linear feedback law is obtained in closed form for the regulation of the angular momentum of an arbitrary rotating rigid body, without linearizing hypotheses of symmetry, yet optimal for a suitable quadratic energy cost. The evolution of the amplitude of the controlled angular momentum is thereby obtained in closed form, whence asymptotic closed-loop stability and exact open-loop terminal rate nulling are derived. Earlier steady-state results are recovered as particular cases.     

Exact solutions for dynamic analysis of composite plates with distributed piezoelectric layers

Exact solutions are presented for analyzing dynamics of composite plates with piezoelectric layers bonded at the top and the bottom surfaces. The expressions for mechanical displacements, stresses, electric displacements and potential are derived from constitutive relations and field equations for the piezoelectric medium under applied surface traction and electric potential. The procedure is illustrated with a simply supported symmetric cross-ply (0°/90°/0°) graphite–epoxy composite plate covered with piezoelectric material polyvinylidene fluoride (PVDF). Results are in good agreement with those obtained from finite element model.     

Comparative analysis of hybrid-trefftz stress and displacement elements

Summary The alternative stress and displacement models of the hybrid-Trefftz finite element formulation for the analysis of linear boundary value problems are derived in parallel form to emphasise the complementary nature of the fundamental concepts they develop from. In the stress model the stresses in the structural domain and the boundary displacements are independently approximated and inter-element stress continuity is enforced explicitly. Conversely, in the displacement model the displacements in the structural domain and the boundary tractions are independently approximated and inter-element linkage is enforced in the form of displacement continuity. In both models the approximation in the domain is constrained to satisfy locally all field equations, a feature typical of the Trefftz method. Duality is used to interpret physically the finite element equations, which are derived from the fundamental relations of elastostatics. Numerical tests are presented to compare the relative performance of the alternative stress and displacement models.     

Experiment and hydro-mechanical coupling simulation study on the human periodontal ligament

In this paper, a new method involving an experiment in vivo and hydro-mechanical coupling simulations was proposed to investigate the biomechanical property of human periodontal ligament (PDL). Teeth were loaded and their displacements were measured in vivo. The finite element model of the experiment was built and hydro-mechanical coupling simulations were conducted to test some PDL's constitutive models. In the simulations, the linear elastic model, the hyperfoam model, and the Ogden model were assumed for the solid phase of the PDL coupled with a model of the fluid phase of the PDL. The displacements of the teeth derived from the simulations were compared with the experimental data to validate these constitutive models. The study shows that a proposed constitutive model of the PDL can be reliably tested by this method. Furthermore, the influence of species, areas, and the fluid volume ratio on PDL's mechanical property should be considered in the modeling and simulation of the mechanical property of the PDL.     

Discontinuous and coupled continuous/discontinuous Galerkin methods for the shallow water equations

We consider the approximation of a simplified model of the depth-averaged two-dimensional shallow water equations by two approaches. In both approaches, a discontinuous Galerkin (DG) method is used to approximate the continuity equation. In the first approach, a continuous Galerkin method is used for the momentum equations. In the second approach a particular DG method, the nonsymmetric interior penalty Galerkin method, is used to approximate momentum. A priori error estimates are derived and numerical results are presented for both approaches.     

Finite element dynamic analysis of geometrically exact planar beams

We present a new strain-based finite element formulation for the dynamic analysis of highly flexible elastic planar beams. The formulation employs the geometrically exact Reissner planar beam theory which accounts for finite displacements and rotations, and finite membrane, shear and bending strains. The system of semi-discrete dynamic equations of motion is derived from the modified Hamilton principle in which only the strain variables are interpolated. Such a choice of the interpolated variables is an advantage over approaches, in which the displacements and rotations are interpolated, since the field consistency problem and related locking phenomena do not arise. The performance and accuracy of the formulation are illustrated by several numerical examples.