scholarly journals Coordinate singularities in harmonically sliced cosmologies

2000 ◽  
Vol 62 (4) ◽  
Author(s):  
Simon D. Hern
Keyword(s):  
Coordinate Singularities

scholarly journals SYMMETRY WITHOUT SYMMETRY: NUMERICAL SIMULATION OF AXISYMMETRIC SYSTEMS USING CARTESIAN GRIDS

2001 ◽  
Vol 10 (03) ◽  
pp. 273-289 ◽  
Author(s):  
MIGUEL ALCUBIERRE ◽  
BERND BRÜGMANN ◽  
DANIEL HOLZ ◽  
RYOJI TAKAHASHI ◽  
STEVEN BRANDT ◽  
...  
Keyword(s):  
Numerical Simulation ◽  
Black Holes ◽  
General Relativity ◽  
Finite Difference ◽  
Three Dimensional ◽  
Central Plane ◽  
Spherical Coordinate ◽  
Finite Differencing ◽  
A New Technique ◽  
Coordinate Singularities

We present a new technique for the numerical simulation of axisymmetric systems. This technique avoids the coordinate singularities which often arise when cylindrical or polar-spherical coordinate finite difference grids are used, particularly in simulating tensor partial differential equations like those of 3+1 numerical relativity. For a system axisymmetric about the z axis, the basic idea is to use a three-dimensional Cartesian(x,y,z) coordinate grid which covers (say) the y=0 plane, but is only one finite-difference-molecule–width thick in the y direction. The field variables in the central y=0 grid plane can be updated using normal (x,y,z)-coordinate finite differencing, while those in the y≠ 0 grid planes can be computed from those in the central plane by using the axisymmetry assumption and interpolation. We demonstrate the effectiveness of the approach on a set of fully nonlinear test computations in 3+1 numerical general relativity, involving both black holes and collapsing gravitational waves.

Kinematic and dynamic modeling of spherical joints using exponential coordinates

2013 ◽  
Vol 228 (10) ◽  
pp. 1777-1785
Author(s):  
Jinwook Kim ◽  
Sung-Hee Lee ◽  
Frank C Park
Keyword(s):  
Dynamic Modeling ◽  
Angular Acceleration ◽  
Euler Angle ◽  
Range Limits ◽  
Inverse Dynamic ◽  
Inverse Dynamic Analysis ◽  
Computational Procedures ◽  
Joint Range ◽  
Robust Procedures ◽  
Coordinate Singularities

Traditional Euler angle-based methods for the kinematic and dynamic modeling of spherical joints involve highly complicated formulas that are numerically sensitive, with complex bookkeeping near local coordinate singularities. In this regard, exponential coordinates are known to possess several advantages over Euler angle representations. This paper presents several new exponential coordinate-based formulas and computational procedures that are particularly useful in the modeling of mechanisms containing spherical joints. Computationally robust procedures are derived for evaluating the forward and inverse formulas for the angular velocity and angular acceleration in terms of exponential coordinates. We show that these formulas simplify the parametrization of joint range limits for spherical joints, and lead to more compact equations in the forward and inverse dynamic analysis of mechanisms containing spherical joints.

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