Erratum to “Rattleback: A model of how geometric singularity induces dynamic chirality” [Phys. Lett. A 381 (2017) 2772]

2018 ◽  
Vol 382 (44) ◽  
pp. 3230
Author(s):  
Z. Yoshida ◽  
T. Tokieda ◽  
P.J. Morrison
Keyword(s):  
Geometric Singularity

Effect of a geometric singularity on a surface on ductile fracture

2014 ◽  
Vol 2014 (1) ◽  
pp. 44-48
Author(s):  
S. E. Alexandrov ◽  
D. Vilotic ◽  
E. A. Lyamina
Keyword(s):  
Ductile Fracture ◽  
Geometric Singularity

scholarly journals Rattleback: A model of how geometric singularity induces dynamic chirality

2017 ◽  
Vol 381 (34) ◽  
pp. 2772-2777 ◽  
Author(s):  
Z. Yoshida ◽  
T. Tokieda ◽  
P.J. Morrison
Keyword(s):  
Geometric Singularity

scholarly journals Singularity Conditions of 3T1R Parallel Manipulators With Identical Limb Structures

2012 ◽  
Vol 4 (1) ◽  
Author(s):  
Semaan Amine ◽  
Mehdi Tale Masouleh ◽  
Stéphane Caro ◽  
Philippe Wenger ◽  
Clément Gosselin
Keyword(s):  
Parallel Manipulator ◽  
Jacobian Matrix ◽  
Parallel Manipulators ◽  
Singularity Analysis ◽  
Type Synthesis ◽  
Constraint Analysis ◽  
Fixed Direction ◽  
Geometric Singularity ◽  
Grassmann Geometry ◽  
Grassmann Cayley Algebra

This paper deals with the singularity analysis of parallel manipulators with identical limb structures performing Schönflies motions, namely, three independent translations and one rotation about an axis of fixed direction (3T1R). Eleven architectures obtained from a recent type synthesis of such manipulators are analyzed. The constraint analysis shows that these architectures are all overconstrained and share some common properties between the actuation and the constraint wrenches. The singularities of such manipulators are examined through the singularity analysis of the 4-RUU parallel manipulator. A wrench graph representing the constraint wrenches and the actuation forces of the manipulator is introduced to formulate its superbracket. Grassmann–Cayley Algebra is used to obtain geometric singularity conditions. Based on the concept of wrench graph, Grassmann geometry is used to show the rank deficiency of the Jacobian matrix for the singularity conditions. Finally, this paper shows the general aspect of the obtained singularity conditions and their validity for 3T1R parallel manipulators with identical limb structures.

Numerical Issues Associated With the Full-Zone Model of the Optically Heated Floating-Zone Used in Semiconductor Processing

2008 ◽  
Author(s):  
Han Li ◽  
Brent C. Houchens
Keyword(s):  
Boundary Conditions ◽  
Taylor Series Expansion ◽  
Internal Symmetry ◽  
Base Flow ◽  
Strong Form ◽  
Zone Model ◽  
Semiconductor Processing ◽  
Velocity Gradients ◽  
Redundant Conditions ◽  
Geometric Singularity

Recent models of the thermocapillary driven liquid bridge or full-zone (FZ) have highlighted numerical difficulties in the system, associated with the large velocity gradients near the free surface and the geometric singularity at r = 0. High resolution spectral solutions have been developed to account for these issues. These result in complex representations and highly specialized numerical procedures. After a brief review of these methods, a simplified formulation for the FZ model with strong form boundary conditions is proposed and discussed. Comparisons are made using base flows and stability analyses. Existing solutions have overcome the geometric singularity either by moving the grid away from the r = 0 axis, or by maintaining the correct Taylor series expansion in the representation of each dependent variable. The former has the weakness that an important constraint is not applied. The later formulation is rigorous, but results in complex expressions for the governing equations. To decrease the load associated with the mathematical manipulation and numerical implementation of this method, here a more general Chebyshev polynomial representation of the stream function is applied to the axisymmetric base flow. This removes the need to maintain the proper expansion and instead offers a set of equations in strong form by treating the axis as a boundary known from the spatial symmetry of the model. However, this does not guarantee that momentum is conserved at the internal symmetry boundaries. Various applications of the other boundary conditions are also studied. In the most accurate representation, all boundary conditions except the thermocapillary condition are cast in the strong form via orthogonality. These strong equations must be chosen carefully to avoid introducing redundant conditions. However, the result is a mathematically simpler representation that mimics the accuracy of previous methods.

scholarly journals Preface by the Editors for the Special Issue "Geometric Singularity Theory''

2015 ◽  
Vol 48 (2) ◽  
Author(s):  
Wojciech Domitrz ◽  
Goo Ishikawa ◽  
Shyuichi Izumiya
Keyword(s):  
Singularity Theory ◽  
Special Issue ◽  
Geometric Singularity

CAT'S EYE RADIATION WITH BOUNDARY ELEMENTS: COMPARATIVE STUDY ON TREATMENT OF IRREGULAR FREQUENCIES

2005 ◽  
Vol 13 (01) ◽  
pp. 21-45 ◽  
Author(s):  
STEFFEN MARBURG ◽  
SIA AMINI
Keyword(s):  
Integral Equation ◽  
Large Range ◽  
Acoustic Radiation ◽  
Collocation Point ◽  
Normal Derivative ◽  
Boundary Integral ◽  
High Frequencies ◽  
Modified Method ◽  
Geometric Singularity ◽  
Helmholtz Integral

This paper reviews a number of techniques developed to overcome the well-known nonuniqueness problem in boundary integral formulations of acoustic radiation. Although the problem has received much attention, comparative studies are hardly known in this field. Furthermore, the problem has often been studied using an unsuitable example, namely a simple radiating sphere. In this case, often the addition of one collocation point in the centre of the sphere suffices to remove the nonuniqueness problem for a large range of wavenumbers. In contrast to the radiating sphere, the radiating cat's eye structure is considered in this paper. Solution of the discretized ordinary Kirchhoff–Helmholtz integral equation, also known as the Surface Helmholtz Equation, reveals a large number of so-called irregular frequencies, i.e. frequencies where the BEM fails. The paper compares the performance of different methods in alleviating this failure. The CHIEF method and its variation due to Rosen et al. are found to encounter difficulties at high frequencies. A much better performance is obtained by combining the Kirchhoff–Helmholtz integral equation with its normal derivative. In particular the method of Burton and Miller and a modification of it which avoids evaluating the hypersingular operator at nonsmooth points are tested. Both methods seem to provide reliable solutions. The modified method encounters minor failures in the frequency response function at a geometric singularity, although performing surprisingly well in many cases. More tests need to be carried out to assess fully the effectiveness of this method which allows easy use of continuous quadratic elements. However, it is the Burton and Miller formulation which appears to be the most reliable for acoustic radiation analysis. The use of CHIEF and its variations cannot be recommended.

A STABLE DIRECT SOLUTION OF PERSPECTIVE-THREE-POINT PROBLEM

2011 ◽  
Vol 25 (05) ◽  
pp. 627-642 ◽  
Author(s):  
SHIQI LI ◽  
CHI XU
Keyword(s):  
Classical Problem ◽  
Main Idea ◽  
Point Problem ◽  
Numerical Instability ◽  
Direct Solution ◽  
Unknown Parameters ◽  
Similar Triangle ◽  
Danger Cylinder ◽  
Geometric Singularity ◽  
Permutation Problem

The perspective-three-point problem (P3P) is a classical problem in computer vision. The existing direct solutions of P3P have at least three limitations: (1) the numerical instability when using different vertex permutations, (2) the degeneration in the geometric singularity case, and (3) the dependence on particular equation solvers. A new direct solution of P3P is presented to deal with these limitations. The main idea is to reduce the number of unknown parameters by using a geometric constraint we called "perspective similar triangle" (PST). The PST method achieves high stability in the permutation problem and in the presence of image noise, and does not rely on particular equation solvers. Furthermore, reliable results can be retrieved even in "danger cylinder", a typical kind of geometric singularity of P3P, where all existing direct solutions degenerate significantly.

Three dimensional eddy current computation using the surface impedance method considering geometric singularity

1995 ◽  
Vol 31 (3) ◽  
pp. 1400-1403 ◽  
Author(s):  
Yong-Gyu Park ◽  
Tae-Kyung Chung ◽  
Hyun-Kyo Jung ◽  
Song-Yop Hahn
Keyword(s):  
Eddy Current ◽  
Surface Impedance ◽  
Three Dimensional ◽  
Impedance Method ◽  
Geometric Singularity

Singularity Analysis of the 4-RUU Parallel Manipulator Based on Grassmann-Cayley Algebra and Grassmann Geometry

2011 ◽  
Author(s):  
Semaan Amine ◽  
Mehdi Tale Masouleh ◽  
Ste´phane Caro ◽  
Philippe Wenger ◽  
Cle´ment Gosselin
Keyword(s):  
Parallel Manipulator ◽  
Jacobian Matrix ◽  
Parallel Manipulators ◽  
Singularity Analysis ◽  
Graph Representation ◽  
Fixed Direction ◽  
Cayley Algebra ◽  
Geometric Singularity ◽  
Grassmann Geometry ◽  
Grassmann Cayley Algebra

This paper deals with the singularity analysis of parallel manipulators with identical limb structures performing Scho¨nflies motions, namely, three independent translations and one rotation about an axis of fixed direction. The study is developed through the singularity analysis of the 4-RUU parallel manipulator. The 6 × 6 Jacobian matrix of such manipulators contains two lines at infinity, namely, two constraint moments, among its six Plu¨cker lines. The Grassmann-Cayley Algebra is used to obtain geometric singularity conditions. However, due to the presence of lines at infinity, the rank deficiency of the Jacobian matrix for the singularity conditions is not easy to grasp. Therefore, a wrench graph representation for some singularity conditions emphasizes the linear dependence of the Plu¨cker lines of the Jacobian matrix and highlights the correspondence between Grassmann-Cayley algebra and Grassmann geometry.

scholarly journals Geometrical Classification of Self-Similar Motion of Two-Dimensional Three Point Vortex System by Deviation Curvature on Jacobi Field

2021 ◽  
Vol 2021 ◽  
pp. 1-14
Author(s):  
Yuma Hirakui ◽  
Takahiro Yajima
Keyword(s):  
Point Vortex ◽  
Point Vortices ◽  
The Self ◽  
Fourth Power ◽  
Jacobi Field ◽  
Self Similarity ◽  
Vortex System ◽  
Geometric Singularity ◽  
Self Similar

In this study, we geometrically analyze the relation between a point vortex system and deviation curvatures on the Jacobi field. First, eigenvalues of deviation curvatures are calculated from relative distances of point vortices in a three point vortex system. Afterward, based on the assumption of self-similarity, time evolutions of eigenvalues of deviation curvatures are shown. The self-similar motions of three point vortices are classified into two types, expansion and collapse, when the relative distances vary monotonously. Then, we find that the eigenvalues of self-similarity are proportional to the inverse fourth power of relative distances. The eigenvalues of the deviation curvatures monotonically convergent to zero for expansion, whereas they monotonically diverge for collapse, which indicates that the strengths of interactions between point vortices related to the time evolution of spatial geometric structure in terms of the deviation curvatures. In particular, for collapse, the collision point becomes a geometric singularity because the eigenvalues of the deviation curvature diverge. These results show that the self-similar motions of point vortices are classified by eigenvalues of the deviation curvature. Further, nonself-similar expansion is numerically analyzed. In this case, the eigenvalues of the deviation curvature are nonmonotonous but converge to zero, suggesting that the motion of the nonself-similar three point vortex system is also classified by eigenvalues of the deviation curvature.

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