The geodesic dynamic relaxation method for problems of equilibrium with equality constraint conditions

This paper presents an extension to the existing dynamic relaxation method to include equality constraint conditions in the process. The existing dynamic relaxation method is presented as a general, gradient‐based, minimization technique. This representation allows for the introduction of the projected gradient, discrete parallel transportation and pull back operators that enable the formulation of the geodesic dynamic relaxation method, a method that accounts for equality constraint conditions. The characteristics of both the existing and geodesic dynamic relaxation methods are discussed in terms of the system's conservation of energy, damping (viscous, kinetic, and drift), and geometry generation. Particular attention is drawn to the introduction of a novel damping approach named drift damping. This technique is essentially a combination of viscous and kinetic damping. It allows for a smooth and fast convergence rate in both the existing and geodesic dynamic relaxation processes. The case study was performed on the form‐finding of an iconic, ridge‐and‐valley, pre‐stressed membrane system, which is supported by masts. The study shows the potential of the proposed method to account for specified (total) length requirements. The geodesic dynamic relaxation technique is widely applicable to the form‐finding of force‐modeled systems (including mechanically and pressurized pre‐stressed membranes) where equality constraint control is desired. Copyright © 2014 John Wiley & Sons, Ltd.     

Acoustic emission source location on a cylindrical surface

Acoustic emission (AE) is a stress wave, generated in a structural material when it is stimulated by perturbations. One of the important features of this technique is to detect the location of the AE source in the structure from the differences in the times of arrival of stress waves at several sensors placed on it. A generalized mathematical formalism for the evaluation of the path travelled by an acoustic wave on the surface of a structure, on the concept of geodesics, is presented. Based on this concept, a mathematical method is developed for the calculation of the coordinates of an AE source on a cylindrical surface by three AE sensors. This concept has not been attempted analytically so far, to our knowledge.     

基于黎曼流形的图像投影配准算法

提出了基于黎曼流形的图像投影配准优化方法, 根据投影变换的特点, 用SL(3)表征目标的图像投影变换, 研究SL(3)的几何结构, 通过变分的方法求出了SL(3)上的测地线, 给出相应的黎曼指数映射, 设计了一种新的基于SL(3)群上黎曼分析的平面投影配准算法, 分析了算法的优点, 并对其收敛性做出了证明. 模拟图像数据和真实图像序列测试的对比实验结果表明, 本文算法在效率和精度上较现有文献中基于欧氏空间的图像投影配准算法有显著提高, 优于基于李群的图像配准算法.     

圆环面上纤维轨迹的计算机辅助设计

曲面上的曲线造型是计算机图形领域的一个新的研究热点,而且它们在纤维织物编织,三维服装裁剪以及复合材料的纤维缠绕轨迹设计等领域有十分广泛的应用,为了解决圆环面上纤维轨迹的计算机辅助设计问题,研究了圆环面上测地线的解析解以及拟测地线数值求解的具体算法并给出了其表达式,测地线是曲面上两点之间最短距离的曲线段,在一般曲面上没有解析解,但是在圆环面上却可求出其精确的解析解,但在曲面的边沿部分,测地线因不能实现自然的折返过渡,于是拟测地线就被引进到曲面上的曲线造型设计之中,在拟测地线分析研究基础上,给出了圆环面上拟测地线的方程及数值解法,通过其在一个实例中的应用结果证明,该方法可获得织物的纺织条纹以及缠绕物体的纤维轨迹。     

基于MMP三角曲面测地线算法研究

测地线的计算在计算机图形处理等方面有着广泛的应用。采用基于MMP（Mitchell, Mount,Papadimitrious）方法,实现了三角曲面上测地线的计算,修正了Vitaly Surazhsky等采用的测地线算法中的误差。该方法首先在窗口传播上摒弃了原有的近似结束条件,采用光源射线法。特别在窗口相交处理过程中采用多种情况的分层枚举,补充了Vitaly Surazhsky讲述的单一情况,窗函数多交点时的测地线偏差情况,并且提供简洁的回溯方法。实验结果表明,该方法所需时间相当于Vitaly Surazhsky算法,可以代替Vitaly Surazhsky采用的算法。     

Woven Surface and Form

The Advanced Geometry Unit (AGU) at Arup, founded by Cecil Balmond and Charles Walker, has become synonymous with a highly mathematical, topological approach to architecture. It has, however, collaborated on some of the most exciting experimental fabric structures of recent years, including Anish Kapoor's Marsyas at Tate Modern and Rem Koolhaas's Cosmic Egg at the Serpentine Gallery. Here, the unit's Tristan Simmonds, Martin Self and Daniel Bosia describe how the AGU has progressed research into textile techniques that encompass tailored biomorphic forms alongside knot diagrams. Copyright © 2006 John Wiley & Sons, Ltd.     

Analyzing and Synthesizing Images by Evolving Curves with the Osher-Sethian Method

Numerical analysis of conservation laws plays an important role in the implementation of curve evolution equations. This paper reviews the relevant concepts in numerical analysis and the relation between curve evolution, Hamilton-Jacobi partial differential equations, and differential conservation laws. This close relation enables us to introduce finite difference approximations, based on the theory of conservation laws, into curve evolution. It is shown how curve evolution serves as a powerful tool for image analysis, and how these mathematical relations enable us to construct efficient and accurate numerical schemes. Some examples demonstrate the importance of the CFL condition as a necessary condition for the stability of the numerical schemes.     

对短程线微分方程精确解的初步分析

得出了短程线微分方程的一个精确解，其结论显示不能完全用来描述行星的轨道运动，是短程线微分方程的一种局域精确解．     

Motion4D-library extended

The new version of the Motion4D-library now also includes the integration of a Sachs basis and the Jacobi equation to determine gravitational lensing of pointlike sources for arbitrary spacetimes.

New version program summary

Program title: Motion4D-libraryCatalogue identifier: AEEX_v3_0Program summary URL:http://cpc.cs.qub.ac.uk/summaries/AEEX_v3_0.htmlProgram obtainable from: CPC Program Library, Queen?s University, Belfast, N. IrelandLicensing provisions: Standard CPC licence, http://cpc.cs.qub.ac.uk/licence/licence.htmlNo. of lines in distributed program, including test data, etc.: 219 441No. of bytes in distributed program, including test data, etc.: 6 968 223Distribution format: tar.gzProgramming language: C++Computer: All platforms with a C++ compilerOperating system: Linux, WindowsRAM: 61 MbytesClassification: 1.5External routines: Gnu Scientic Library (GSL) (http://www.gnu.org/software/gsl/)Catalogue identifier of previous version: AEEX_v2_0Journal reference of previous version: Comput. Phys. Comm. 181 (2010) 703Does the new version supersede the previous version?: YesNature of problem: Solve geodesic equation, parallel and Fermi-Walker transport in four-dimensional Lorentzian spacetimes. Determine gravitational lensing by integration of Jacobi equation and parallel transport of Sachs basis.Solution method: Integration of ordinary differential equations.Reasons for new version: The main novelty of the current version is the extension to integrate the Jacobi equation and the parallel transport of the Sachs basis along null geodesics. In combination, the change of the cross section of a light bundle and thus the gravitational lensing effect of a spacetime can be determined. Furthermore, we have implemented several new metrics.Summary of revisions: The main novelty of the current version is the integration of the Jacobi equation and the parallel transport of the Sachs basis along null geodesics. The corresponding set of equations read(1)(2)(3) where (1) is the geodesic equation, (2) represents the parallel transport of the two Sachs basis vectors s1,2, and (3) is the Jacobi equation for the two Jacobi fields Y1,2.The initial directions of the Sachs basis vectors are defined perpendicular to the initial direction of the light ray, see also Fig. 1,(4a)(4b)A congruence of null geodesics with central null geodesic γ which starts at the observer O with an infinitesimal circular cross section is defined by the above mentioned two Jacobi fields with initial conditions and . The cross section of this congruence along γ is described by the Jacobian . However, to determine the gravitational lensing of a pointlike source S that is connected to the observer via γ, we need the reverse Jacobian JS→O. Fortunately, the reverse Jacobian is just the negative transpose of the original Jacobian JO→S,(5)J:=JS→O=−T(JO→S). The Jacobian J transforms the circular shape of the congruence into an ellipse whose shape parameters (M_±: major/minor axis, ψ: angle of major axis, ε: ellipticity) read(6a)(6b)ψ=arctan2(J₂₁cosζ₊+J₂₂sinζ₊,J₁₁cosζ₊+J₁₂sinζ₊),(6c) with(7) and the parameters α=J₁₁J₁₂+J₂₁J₂₂, . The magnification factor is given by(8) These shape parameters can be easily visualized in the new version of the GeodesicViewer, see Ref. [1]. A detailed discussion of gravitational lensing can be found, for example, in Schneider et al. [2].In the following, a list of newly implemented metrics is given:

• • 

  BertottiKasner: see Rindler [3].

• • 

  BesselGravWaveCart: gravitational Bessel wave from Kramer [4].

• • 

  DeSitterUniv, DeSitterUnivConf: de Sitter universe in Cartesian and conformal coordinates.

• • 

  Ernst: Black hole in a magnetic universe by Ernst [5].

• • 

  ExtremeReissnerNordstromDihole: see Chandrasekhar [6].

• • 

  HalilsoyWave: see Ref. [7].

• • 

  JaNeWi: Janis–Newman–Winicour metric, see Ref. [8].

• • 

  MinkowskiConformal: Minkowski metric in conformally rescaled coordinates.

• • 

  PTD_AI, PTD_AII, PTD_AIII, PTD_BI, PTD_BII, PTD_BIII, PTD_C Petrov-Type D – Levi-Civita spacetimes, see Ref. [7].

• • 

  PainleveGullstrand: Schwarzschild metric in Painlevé–Gullstrand coordinates, see Ref. [9].

• • 

  PlaneGravWave: Plane gravitational wave, see Ref. [10].

• • 

  SchwarzschildIsotropic: Schwarzschild metric in isotropic coordinates, see Ref. [11].

• • 

  SchwarzschildTortoise: Schwarzschild metric in tortoise coordinates, see Ref. [11].

• • 

  Sultana-Dyer: A black hole in the Einstein–de Sitter universe by Sultana and Dyer [12].

• • 

  TaubNUT: see Ref. [13].

The Christoffel symbols and the natural local tetrads of these new metrics are given in the Catalogue of Spacetimes, Ref. [14].To study the behavior of geodesics, it is often useful to determine an effective potential like in classical mechanics. For several metrics, we followed the Euler–Lagrangian approach as described by Rindler [10] and implemented an effective potential for a specific situation. As an example, consider the Lagrangian for timelike geodesics in the ?=π/2 hypersurface in the Schwarzschild spacetime with α=1−2m/r. The Euler–Lagrangian equations lead to the energy balance equation with the effective potential V(r)=(r−2m)(r²+h²)/r³ and the constants of motion and . The constants of motion for a timelike geodesic that starts at (r=10m,φ=0) with initial direction ξ=π/4 with respect to the black hole direction and with initial velocity β=0.7 read k≈1.252 and h≈6.931. Then, from the energy balance equation we immediately obtain the radius of closest approach rmin≈5.927.Beside a standard Runge–Kutta fourth-order integrator and the integrators of the Gnu Scientific Library (GSL), we also implemented a standard Bulirsch–Stoer integrator.Running time: The test runs provided with the distribution require only a few seconds to run.References:

• [1] 

  T. Müller, New version announcement to the GeodesicViewer, http://cpc.cs.qub.ac.uk/summaries/AEFP_v2_0.html.

• [2] 

  P. Schneider, J. Ehlers, E. E. Falco, Gravitational Lenses, Springer, 1992.

• [3] 

  W. Rindler, Phys. Lett. A 245 (1998) 363.

• [4] 

  D. Kramer, Ann. Phys. 9 (2000) 331.

• [5] 

  F.J. Ernst, J. Math. Phys. 17 (1976) 54.

• [6] 

  S. Chandrasekhar, Proc. R. Soc. Lond. A 421 (1989) 227.

• [7] 

  H. Stephani, D. Kramer, M. MacCallum, C. Hoenselaers, E. Herlt, Exact Solutions of the Einstein Field Equations, Cambridge University Press, 2009.

• [8] 

  A.I. Janis, E.T. Newman, J. Winicour, Phys. Rev. Lett. 20 (1968) 878.

• [9] 

  K. Martel, E. Poisson, Am. J. Phys. 69 (2001) 476.

• [10] 

  W. Rindler, Relativity – Special, General, and Cosmology, Oxford University Press, Oxford, 2007.

• [11] 

  C.W. Misner, K.S. Thorne, J.A. Wheeler, Gravitation, W.H. Freeman, 1973.

• [12] 

  J. Sultana, C.C. Dyer, Gen. Relativ. Gravit. 37 (2005) 1349.

• [13] 

  D. Bini, C. Cherubini, Robert T. Jantzen, Class. Quantum Grav. 19 (2002) 5481.

• [14] 

  T. Muller, F. Grave, arXiv:0904.4184 [gr-qc].

     

An updated version of the Motion4D-library

We present an updated version of the Motion4D-library that can be used for the newly developed GeodesicViewer application.

New version program summary

Program title: Motion4D-libraryCatalogue identifier: AEEX_v2_0Program summary URL:http://cpc.cs.qub.ac.uk/summaries/AEEX_v2_0.htmlProgram obtainable from: CPC Program Library, Queen's University, Belfast, N. IrelandLicensing provisions: Standard CPC licence, http://cpc.cs.qub.ac.uk/licence/licence.htmlNo. of lines in distributed program, including test data, etc.: 153 757No. of bytes in distributed program, including test data, etc.: 5 178 439Distribution format: tar.gzProgramming language: C++Computer: All platforms with a C++ compilerOperating system: Linux, Unix, WindowsRAM: 31 MBytesCatalogue identifier of previous version: AEEX_v1_0Journal reference of previous version: Comput. Phys. Comm. 180 (2009) 2355Classification: 1.5External routines: Gnu Scientific Library (GSL) (http://www.gnu.org/software/gsl/)Does the new version supersede the previous version?: YesNature of problem: Solve geodesic equation, parallel and Fermi-Walker transport in four-dimensional Lorentzian spacetimes.Solution method: Integration of ordinary differential equations.Reasons for new version: To be applicable for the GeodesicViewer (accepted for publication in Comput. Phys. Comm. (COMPHY) 3941, doi:10.1016/j.cpc.2009.10.010 [program AEFP_v1_0]), there were several minor adjustments to be done.Summary of revisions:

1.

Calculation of embedding diagrams are improved.

2.

Physical units can be used for some metrics.

3.

Tests for the constraint equation within the metric classes are slightly modified.

4.

New metrics: AlcubierreWarp, GoedelScaled, GoedelScaledCart, Kasner.

Running time: The test runs provided with the distribution require only a few seconds to run.