Computer-aided modeling of wire ropes bent over a sheave

In the paper, two methods for building a 3-dimensional geometry of a wire rope bent over a sheave in Proe™ are presented. In order to obtain the centroidal axes of the wires, the first method is associated with deriving their coordinate equations based on the Serret–Frenet frame and the second method is characterized by using the ‘Variable Section Sweep’ tool in Proe™ to generate a helically twisting surface around the centroidal axis of the king wire or a strand with the two boundary curves being the centroidal axes of two double or triple helical wires. Finally, both methods use the ‘Variable Section Sweep’ tool to generate the geometries of the wires. The two methods can be applied to any helical-strand wire rope.     

Computer modelling of wire strands and ropes Part I: Theory and computer implementation

In this paper the mathematical geometric models of the single-lay wire strands and double-lay wire ropes with defined initial parameters are presented. The present geometric models fully consider the single-helix configuration of individual wires in the strand and the double-helix configuration of individual wires within the wound strands of the ropes. The mathematical representation of the single and double helixes is in form of parametric equations with variable input parameters which determine the centreline of an arbitrary circular wire of the right hand lay and left hand lay strands and ropes of the Lang lay and regular lay construction. The concrete forms of the parametric equations are derived and presented. The application of the derived geometric analytical model is illustrated by numerical examples. Techniques for the implementation of the derived mathematical models in CATIA V5 software and procedures for the generation of the rope model are briefly presented. Correctness of the derived parametric equations and a performance of the generated rope model are controlled by visualizations. The application of the derived mathematical model and the development of a finite element model for the numerical simulation of the multi-layered strand under tension tests are treated in the second part of the paper [1].     

Algorithms for finding the axis of a helix: Fast rotational and parametric least-squares methods

Several methods for finding the axis of a helix are presented and compared. The most accurate determines the helix axis as the axis of rotation necessary to map point i to point i + 1 of the helix. The fastest method calculates the helix axis as the best-fit line through the coordinates by a three-dimensional parametric linear least-squares algorithm, taking advantage of the sequential nature of the data.     

Constructing up to G 2 continuous curve on freeform surface

This paper presents new methods for G ¹ and G ² continuous interpolation of an arbitrary sequence of points on an implicit or parametric surface with prescribed tangent direction and both tangent direction and curvature vector, respectively, at every point. We design a G ¹ or G ² continuous curve in three-dimensional space, construct a so-called directrix vector field using the space curve and then project a special straight line segment onto the given surface along the directrix vector field. With the techniques in classical differential geometry, we derive a system of differential equations for the projection curve. The desired interpolation curve is just the projection curve, which can be obtained by numerically solving the initial-value problems for a system of first-order ordinary differential equations in the parametric domain associated to the surface representation for the parametric case or in three-dimensional space for the implicit case. Several shape parameters are introduced into the resulting curve, which can be used in subsequent interactive modification such that the shape of the resulting curve meets our demand. The presented method is independent of the geometry and parameterization of the base surface, and numerical experiments demonstrate that it is effective and potentially useful in patterns design on surface.     

Computer modelling and finite element analysis of spiral triangular strands

In this paper a new mathematical geometric model of spiral triangular wire strands with a construction of (3 + 9) and (3 + 9 + 15) wires is proposed and an accurate computational two-layered triangular strand 3D solid modelling, which is used for a finite element analysis, is presented. The present geometric model fully considers the spatial configuration of individual wires in the strand. The three dimensional curve geometry of wires axes in the individual layers of the triangular strand consists of straight linear and helical segments. The derived mathematical representation of this curve is in the form of parametric equations with variable input parameters which facilitate the determination of the centreline of an arbitrary circular wire of the right and left hand lay triangular one and two-layered strands. Derived geometric equations were used for the generation of accurate 3D geometric and computational strand models. The correctness of the derived parametric equations and performance of the generated strand model are controlled by visualizations. The 3D computational model was used for a finite element behaviour analysis of the two-layered triangular strand subjected to tension loadings. Illustrative examples are presented to highlight the benefits of the proposed geometric parametric equations and computational modelling procedures by using the finite element method.     

Constructing G 1 continuous curve on a free-form surface with normal projection

In this paper, we develop a new method for G ¹ continuous interpolation of an arbitrary sequence of points on an implicit or parametric surface with a specified tangent direction at every point. Based on the normal projection method, we design a G ¹ continuous curve in three-dimensional space and then project orthogonally the curves onto the given surface. With the techniques in classical differential geometry, we derive a system of differential equations characterizing the projection curve. The resulting interpolation curve is obtained by numerically solving the initial-value problems for a system of first-order ordinary differential equations in the parametric domain associated to the surface representation for a parametric case or in three-dimensional space for an implicit case. Several shape parameters are introduced into the resulting curve, which can be used in subsequent interactive modification such that the shape of the resulting curve meets our demand. The presented method is independent of the geometry and parameterization of the base surface, and numerical experiments demonstrate that it is effective and potentially useful in surface trim, robot, patterns design on surface and other industrial and research fields.     

Computing general geometric structures on surfaces using Ricci flow

Systematically generalizing planar geometric algorithms to manifold domains is of fundamental importance in computer aided design field. This paper proposes a novel theoretic framework, geometric structure, to conquer this problem. In order to discover the intrinsic geometric structures of general surfaces, we developed a theoretic rigorous and practical efficient method, Discrete Variational Ricci flow.Different geometries study the invariants under the corresponding transformation groups. The same geometry can be defined on various manifolds, whereas the same manifold allows different geometries. Geometric structures allow different geometries to be defined on various manifolds, therefore algorithms based on the corresponding geometric invariants can be applied on the manifold domains directly.Surfaces have natural geometric structures, such as spherical structure, affine structure, projective structure, hyperbolic structure and conformal structure. Therefore planar algorithms based on these geometries can be defined on surfaces straightforwardly.Computing the general geometric structures on surfaces has been a long lasting open problem. We solve the problem by introducing a novel method based on discrete variational Ricci flow.We thoroughly explain both theoretical and practical aspects of the computational methodology for geometric structures based on Ricci flow, and demonstrate several important applications of geometric structures: generalizing Voronoi diagram algorithms to surfaces via Euclidean structure, cross global parametrization between high genus surfaces via hyperbolic structure, generalizing planar splines to manifolds via affine structure. The experimental results show that our method is rigorous and efficient and the framework of geometric structures is general and powerful.     

Robust uniform triangulation algorithm for computer aided design

This paper presents a new robust uniform triangulation algorithm that can be used in CAD/CAM systems to generate and visualize geometry of 3D models. Typically, in CAD/CAM systems 3D geometry consists of 3D surfaces presented by the parametric equations (e.g. surface of revolution, NURBS surfaces) which are defined on a two dimensional domain. Conventional triangulation algorithms (e.g. ear clipping, Voronoi-Delaunay triangulation) do not provide desired quality and high level of accuracy (challenging tasks) for 3D geometry. The approach developed in this paper combines lattice tessellation and conventional triangulation techniques and allows CAD/CAM systems to obtain the required surface quality and accuracy. The algorithm uses a Cartesian lattice to divide the parametric domain into adjacent rectangular cells. These cells are used to generate polygons that are further triangulated to obtain accurate surface representation. The algorithm allows users to control the triangle distribution intensity by adjusting the lattice density. Once triangulated, the 3D model can be used not only for rendering but also in various manufacturing and design applications. The approach presented in this paper can be used to triangulate any parametric surface given in S(u,v) form, e.g. NURBS surfaces, surfaces of revolution, and produces good quality triangulation which can be used in CAD/CAM and computer graphics applications.     

A pseudo-spectral method with volume penalisation for magnetohydrodynamic turbulence in confined domains

We present a Fourier pseudo-spectral method for solving the resistive magnetohydrodynamic equations of incompressible flow in confined domains. A volume penalisation method allows to take into account boundary conditions and the geometry of the domain. A code validation is presented for the z-pinch test case. Numerical simulations of decaying MHD turbulence in two space dimensions show spontaneous spin-up of the flow in non-axisymmetric geometries, which is reflected by the generation of angular momentum. First results of decaying MHD turbulence in a cylinder illustrate the potential of the new method for three-dimensional simulations.     

Stability analysis in spanwise-periodic double-sided lid-driven cavity flows with complex cross-sectional profiles

Three-dimensional linear instability analyses are presented of steady two-dimensional laminar flows in the lid-driven cavity defined by [15] and further analyzed in the present volume [1], as well as in a derivative of the same geometry. It is shown that in both of the geometries considered three-dimensional BiGlobal instability leads to deviation of the flow from the two-dimensional solution; the analysis results are used to define low- and high-Reynolds number solutions by reference to the flow physics. Critical conditions for linear global instability and neutral loops are presented in both geometries.     

High β and toroidal effects in three dimensions

The resistive MHD equations for toroidal plasma configurations are reduced by expanding to second order in ?, the inverse aspect ratio, allowing for high β = μ₀p/B² of order ?. The result is a closed system of nonlinear, three-dimensional equations where the fast magnetohydrodynamic time scale is eliminated. In particular, the equation for the toroidal velocity remains decoupled. These equations generalize reduced equations derived earlier. They are now solved numerically. As a first step, various types of axisymmetric equilibria are found by relaxing a given initial configuration. Then, three-dimensional perturbations are introduced and followed in time to investigate the linear and nonlinear properties of tearing modes, etc., for high-β plasmas in toroidal geometry.     

Transfer matrix analysis of asymmetric frame-shear wall systems

Frame-shear wall systems are common high-rise structural forms resisting efficiently lateral forces. High redundancy complicates their three-dimensional analysis by traditional methods. In this paper an efficient and reliable such analysis based on the transfer matrix technique is presented. It yields the coupled linear static as well as the free vibrational response of tall asymmetric buildings of quite general form. The application of the method to various frame-shear wall geometries produced results in good agreement with experimental data and other theoretical predictions. The developed analysis was also found to be applicable to a wider range of geometries than simplified methods would allow while condensing the overall problem to a size depending on the degrees of freedom of a single level of the structure.     

Modeling complex heterogeneous objects with non-manifold heterogeneous cells

This paper presents a new approach to model complex heterogeneous objects with simultaneous geometry intricacies as well as complex material distributions. Different from most of the existing approaches, which utilize manifold B-Rep and the assembly representations, the proposed scheme takes advantage of the non-manifoldcellular representations to model the geometries of the heterogeneous objects. With the aid of the cell adjacency information and attribute based reasoning, complex, smooth and versatile material distributions can be defined upon the intricate geometries. Compared with other similar approaches, the proposed scheme (1) generates heterogeneous object models with higher data consistencies and lower redundancies; (2) naturally avoids unnecessary/repetitive computations and thus improves computation efficiencies; (3) represents versatile material variations/distributions using different heterogeneous feature tree (HFT) structures. The detailed representation, associated algorithms and a prototype software package are presented. Example heterogeneous objects modeled with the proposed approach are provided.     

Dynamic instability of composite laminated rectangular plates and prismatic plate structures

Periodic dynamic loadings may cause dynamic instability of a structure through parametric resonance. In this paper, a B-spline finite strip method (FSM) is presented for the dynamic instability analysis of composite laminated rectangular plates and prismatic plate structures, based on the use of first-order shear deformation plate theory (SDPT). The equations of motion of a structure are established by using Lagrange's formulation and they are a set of coupled Mathieu equations. The boundary parametric resonance frequencies of the motion are determined by using the method suggested by Bolotin through a novel development which incorporates the Sturm sequence method and the multi-level substructuring technique to achieve reliability, efficiency and accuracy. Various loading patterns, arbitrary lamination and general boundary conditions are accommodated. A variety of numerical applications is presented to test the developed method and to study the dynamic instability behaviour of single plates and of complicated plate structures under various types of dynamic loading. A dynamic instability index (DII) is devised to measure the degree of instability against certain parameters which include the thickness-to-length ratio, the degree of orthotropy, the fibre orientation, the loading pattern and the boundary conditions.     

基于参数化设计的三维桥梁模型构建

参数化设计是几何体建模的重要手段,而桥梁的三维可视化是当今桥梁信 息化的发展趋势。论文针对桥梁外观参数化模型可视化的需要,对几何体参数化建模的原理 和方法进行了深入探讨。结合桥梁结构特点,分析了主要桥梁组件结构（如T 梁、盖梁柱 式墩、轻型桥台等）构件图的几何和拓扑关系,对组件进行了参数化设计,实现组件的三维 建模;并对组件中特殊图元（如圆弧倒角形墩柱、桥台近似锥面护坡）的绘制算法进行详细 研究;根据各组件间空间位置的拓扑关系,利用参数化变量驱动计算各组件空间位置坐标, 进行桥梁的快速拼接。完成的桥梁三维参数化建模以参数为驱动,用户可以对设计结果进行 可视化修改。所实现的三维参数桥梁可视化系统具有模型构建精致,参数化建模彻底、建模 速度快,参数分类清晰,可视化效果好的特点。     

OFF,Open source Finite volume Fluid dynamics code: A free,high-order solver based on parallel,modular, object-oriented Fortran API

OFF, an open source (free software) code for performing fluid dynamics simulations, is presented. The aim of OFF is to solve, numerically, the unsteady (and steady) compressible Navier–Stokes equations of fluid dynamics by means of finite volume techniques: the research background is mainly focused on high-order (WENO) schemes for multi-fluids, multi-phase flows over complex geometries. To this purpose a highly modular, object-oriented application program interface (API) has been developed. In particular, the concepts of data encapsulation and inheritance available within Fortran language (from standard 2003) have been stressed in order to represent each fluid dynamics “entity” (e.g. the conservative variables of a finite volume, its geometry, etc…) by a single object so that a large variety of computational libraries can be easily (and efficiently) developed upon these objects. The main features of OFF can be summarized as follows:     

Fictitious domain method and separated representations for the solution of boundary value problems on uncertain parameterized domains

A tensor-based method is proposed for the solution of partial differential equations defined on uncertain parameterized domains. It provides an accurate solution which is explicit with respect to parameters defining the shape of the domain, thus allowing efficient a posteriori probabilistic or parametric analyses. In the proposed method, a fictitious domain approach is first adopted for the reformulation of the parametric problem on a fixed domain, yielding a weak formulation in a tensor product space (product of space functions and parametric functions). The paper is limited to the case of Neumann conditions on uncertain parts of the boundary. The Proper Generalized Decomposition method is then introduced for the construction of a tensor product approximation (separated representation) of the solution. It can be seen as an a priori model reduction technique which automatically captures reduced bases of space functions and parametric functions which are optimal for the representation of the solution. This tensor-based method is made computationally tractable by introducing separated representations of variational forms, resulting from separated representations of the parameterized indicator function of the uncertain domain. For this purpose, a method is proposed for the construction of a constrained tensor product approximation which preserves positivity and therefore ensures well-posedness of problems associated with approximate indicator functions. Moreover, a regularization of the geometry is introduced to speed up the convergence of these tensor product approximations.     

A full-wave solver of the Maxwell's equations in 3D cold plasmas

A new solver for Maxwell's equations in three-dimensional (3D) plasma configurations is presented. The new code LEMan (Low-frequency ElectroMagnetic wave propagation) determines a global solution of the wave equation in a realistic stellarator geometry at low frequencies. The code is aimed at the applications with relatively small computational resources and is very efficient in the Alfvén frequency range. In the present work, the cold plasma model is implemented. Finite elements are applied for the radial discretization and the spectral representation is used for the poloidal and toroidal angles. Special care is taken to avoid the numerical pollution of the spectrum as well as to ensure the energy conservation. The numerical scheme and the convergence properties are discussed. Several benchmarks and results in different geometries are presented.