逆向工程中基于NURBS的曲面重构

曲面重构是逆向工程中的关键技术.根据非均匀有理B样条曲面矩阵表达式,对于造型曲面上的(2m+1)×(2n+1)个型值数据点,推导了一个仅利用型值点数据反算二次非均匀有理B样条曲面控制顶点的算法.数值算例表明了该算法的有效性.     

NURBS曲面的形状修改的一种方法

NURBS曲面是计算机辅助几何设计和计算机图形中最常用的参数曲面。本文采用NURBS曲面的齐次坐标表示，给出了通过控制顶点和权因子同时改变来修改NURBS曲面形状的一种方法。     

Characterization of Trimmed NURBS Surface Boundaries Using Topological Criteria

This paper presents an approach based on topology for the determination of characteristics and properties of curves used in the trimming of NURBS surfaces. Through discrete subdivision and topological criteria, a method is presented to determine characteristics of the boundary; such as whether the set of trimming curves forms a set of closed loops, whether trimming curves contain singularities or self intersections, and whether the boundary is simply connected. A surface mesh partitionning the parameter space is used, formed of isoparametric lines in both parametric directions. Topological properties of the cells of this mesh and their intersections with the trimming curves allow to localize the boundary. Topological treatment of this localization allows to define the interior and exterior of the face, and to refine the boundary localization. Singularities and self intersections of the boundary as well as voids in the face are investigated through the study of topological properties of neighbors. As an application, an algorithm for point localization is presented that very rapidly allows to determine whether a given point in parameter space lies inside, on the boundary or outside of the trimmed surface.     

Adaptive FEM and a posteriori error control for stationary coupled bulk-surface PDEs

We consider a system of two coupled elliptic equations, one defined on a bulk domain and the other one on the boundary surface. The numerical error of the finite element solution can be controlled by a residual a posteriori error estimator which takes into account the approximation errors due to the discretisation in space as well as the polyhedral approximation of the surface. The estimators naturally lead to refinement indicators for an adaptive algorithm to control the overall error. Numerical experiments illustrate the performance of the a posteriori error estimator and the adaptive algorithm. (© 2016 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Adaptive FEM with Stabilized Elements for Parameter Identification of Incompressible Hyperelastic Materials

This work is concerned with the identification of material parameters for isotropic, incompressible hyperelastic material models. Besides the principal stretch-based strain-energy function by Ogden an invariant-based strain-energy function by Rivlin/Saunders is considered for which parameter sensitivities are derived. The identification is formulated as a least-squares minimization problem based on the finite element method to account for inhomogeneous states of stresses and strains in the experimental data which is obtained by optical measurements. For the finite element method low-order tetrahedral elements in a mixed displacement-pressure formulation with stabilization are considered. Special attention is payed to an adaptive mesh-refinement based on a goal-oriented a posteriori error indicator to gain reliable material parameters. To approximate error terms an element-wise recovery technique based on enhanced gradients is introduced. (© 2012 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Curvature-Based Real-Time NURBS Surface Interpolator with Look-Ahead ACC/DEC Control

Real-time parametric surface interpolation is very useful in high-performance machining. A curvature-based NURBS surface interpolator with look-ahead acceleration/deceleration (ACC/DEC) control has been developed. The cutter contact (CC) paths are planned through iso-parametric line method. The CC feedrate profile is optimized using global flexible control strategy and local adaptive ACC/DEC optimization strategy according to the look-ahead algorithm. The real-time surface interpolation algorithm was programmed on Windows XP platform. The stability and efficiency of the proposed interpolation method were verified by a NURBS curve and a NURBS surface. It is shown that the proposed parametric interpolation algorithm can satisfy the high speed and high precision requirements of high-speed CNC systems.     

非线性时滞系统自适应模糊动态面控制

针对未知非线性系统控制器设计过程中引入逼近器过多的问题,提出一种简化的自适应模糊动态面控制器设计方案.在控制器设计过程中,仅采用一个模糊逻辑系统作为逼近器,使得所有的未知项得到补偿,同时采用自适应技术在线辨识未知参数和逼近误差上界.文中的控制方案克服了传统backstepping控制器中"复杂性膨胀"的问题.通过构造合适的Lyapunov函数,证明闭环系统的所有信号为半全局最终一致有界.仿真实例验证所提出的控制方案的有效性.     

离散系统运动方程的Galerkin有限元EEP法自适应求解

对于结构动力分析中的离散系统运动方程,现有算法的计算精度和效率均依赖于时间步长的选取,这是时间域问题求解的难点.基于EEP(element energy projection)超收敛计算的自适应有限元法,以EEP超收敛解代替未知真解,估计常规有限元解的误差,并自动细分网格,目前已对诸类以空间坐标为自变量的边值问题取得成功.对离散系统运动方程建立弱型Galerkin有限元解,引入基于EEP法的自适应求解策略,在时间域上自动划分网格,最终得到所求时域内任一时刻均满足给定误差限的动位移解,进而建立了一种时间域上的新型自适应求解算法.     

杆件轴向受迫振动的Galerkin有限元EEP法自适应求解

基于单元能量投影(element energy projection,EEP)法自适应分析在杆件静力问题以及离散系统运动方程组中所取得的成果,以直杆轴向受迫振动为例,研究并建立了一种在时间域和一维空间域同时实现自适应分析的方法.该方法在时间和空间两个维度都采用连续的Galerkin有限元法(finite element method,FEM)进行求解,根据半离散的思想,由空间有限元离散将模型问题的偏微分控制方程转化为离散系统运动方程组,对该方程组进行时域有限元自适应求解;然后再基于空间域超收敛计算的EEP解对空间域进行自适应,直至最终的时空网格下动位移解答的精度逐点均满足给定误差限要求.文中对其基本思想、关键技术和实施策略进行了阐述,并给出了包括地震波输入下的典型算例以展示该法有效可靠.     

Adaptive FEM for PDE Constrained Optimization with Pointwise Constraints on the Gradient of the State

We are concerned with an interior penalty method for an optimal control problem with an elliptic PDE constraint and additional pointwise constraints on the gradient of the state variable. We will derive an estimator that gives qualitative information about the error in the cost functional due to the interior penalty parameter and for the discretization of the PDE. These can be used to balance the error contributions of both types of errors. (© 2008 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

NURBS曲线的一个插值性质

本介绍了非均匀有理B样条曲线，并给出了非均匀有理B样条曲线的一个插值性质。     

Isogeometric Collocation: Cost Comparison with Galerkin Methods and Extension to Adaptive Hierarchical NURBS Discretizations

Collocation is based on the discretization of the strong form of the underlying partial differential equations, which requires basis functions of sufficient order and smoothness. Consequently, the use of isogeometric analysis (IGA) for collocation suggests itself, since splines can be readily adjusted to any order in polynomial degree and continuity required by the differential operators. In addition, they can be generated for domains of arbitrary geometric and topological complexity, directly linked to and fully supported by CAD technology. The major advantage of isogeometric collocation over Galerkin type IGA is the minimization of the computational effort for numerical quadrature. (© 2013 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Adaptive coupling of FEM and BEM: Simple error estimators and convergence

A posteriori error estimators and adaptive mesh-refinement have themselves proven to be important tools for scientific computing. For error control in finite element methods (FEM), there is a broad variety of a posteriori error estimators available, and convergence as well as optimality of adaptive FEM is well-studied in the literature. This is, however, in sharp contrast to the boundary element method (BEM) and to the coupling of FEM and BEM. In our contribution, we present an easy-to-implement error estimator for some FEM-BEM coupling which, to the best of our knowledge, has not been proposed in the literature before. The derived mesh-refining algorithm provides the first adaptive coupling procedure which is mathematically proven to converge. (© 2011 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Adaptive FEM with goal-oriented error estimation and an approximation of the dual problem for inelastic problems

An approximation of the dual problem for goal-oriented error estimation is derived, which preserves the numerical structure of the primal problem, usually found in standard finite element implementations for solid mechanics. For the error estimation we apply a higher-order recovery procedure. An application to elasto-plasticity is given. Numerical examples demonstrate the effectiveness of the procedure. (© 2015 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

非线性薄壳理论的研究

利用正交多项式系上的Fourier展开就容易由方程的解直接得到位移和应力的明确表示.从薄壳的虚功原理出发导出各阶平衡方程.作为数学分析的基础,证明了关于Legendre级数逐项求导的定理.从而明确了对函数的要求,分析便不再只是形式的了.具体给出了三阶近似的平衡方程,可供对低阶近似的精度分析作参考.分析说明直法线理论只能是一阶的近似,法线无伸长地倾斜的假设本质上也是一阶的近似.     

An Asymptotic Dynamic Theory for Cylindrical Shells

By expanding the displacement and stress components together with the axial length scale in terms of a small thin shell parameter, three asymptotic shell theories are obtained which incorporate thickness effects in a systematic way. The expansions are made in the equations of linear three-dimensional elasticity. These theories are used to examine the problem of longitudinal wave propagation in a shell of infinite length.     

Approximation Algorithms for Minimum-Width Annuli and Shells

Let S be a set of n points in \re ^d . The ``roundness' of S can be measured by computing the width ω ^* =ω ^* (S) of the thinnest spherical shell (or annulus in \re ² ) that contains S . This paper contains two main results related to computing an approximation of ω ^* : (i) For d=2 , we can compute in O(n log n) time an annulus containing S whose width is at most 2ω ^* (S) . We extend this algorithm, so that, for any given parameter ε >0 , an annulus containing S whose width is at most (1+ε )ω ^* is computed in time O(n log n + n/ε ² ) . (ii) For d \geq 3 , given a parameter ε > 0 , we can compute a shell containing S of width at most (1+ε)ω ^* either in time O ( n / ε ^d ) log ( \Delata / ω ^* ε ) or in time O ( n / ε ^d-2 ) log n + 1 / εlog \Delata / ω ^* ε , where Δ is the diameter of S . Received July 6, 1999, and in revised form April 17, 2000. Online publication August\/ 11, 2000.     

Adaptive Surface Reconstruction Based on Tensor Product Algebraic Splines

Surface reconstruction from unorganized data points is a challenging problem in Computer Aided Design and Geometric Modeling. In this paper, we extend the mathematical model proposed by Juttler and Felis (Adv. Comput. Math., 17 (2002), pp. 135-152) based on tensor product algebraic spline surfaces from fixed meshes to adaptive meshes. We start with a tensor product algebraic B-spline surface defined on an initial mesh to fit the given data based on an optimization approach. By measuring the fitting errors over each cell of the mesh, we recursively insert new knots in cells over which the errors are larger than some given threshold, and construct a new algebraic spline surface to better fit the given data locally. The algorithm terminates when the error over each cell is less than the threshold. We provide some examples to demonstrate our algorithm and compare it with Jiittler's method. Examples suggest that our method is effective and is able to produce reconstruction surfaces of high quality.AMS subject classifications: 65D17