Determination of radar coverage diagrams in complex environments using closed form ray tracing

Radio Science ◽  
1991 ◽  
Vol 26 (1) ◽  
pp. 45-50
Author(s):  
Shakir Abdulla ◽  
Samir Raouf ◽  
Abdulamir Al-anbari ◽  
Ashraf Kadri
Keyword(s):  
Closed Form ◽  
Ray Tracing ◽  
Complex Environments

scholarly journals Simple and Exact Closed-Form Expression for Determination of a Penetrated Ray Path in a Ray Tracing

2013 ◽  
Vol 2013 ◽  
pp. 1-7 ◽  
Author(s):  
Hyun Wook Moon ◽  
Woojoong Kim ◽  
Sewoong Kwon ◽  
Jaeheung Kim ◽  
Young Joong Yoon
Keyword(s):  
Closed Form ◽  
Approximate Method ◽  
Ray Tracing ◽  
Conventional Method ◽  
Closed Form Expression ◽  
Indoor Environments ◽  
Straight Line ◽  
Ray Path ◽  
Ray Tracing Simulation

A simple and exact closed-form equation to determine a penetrated ray path in a ray tracing is proposed for an accurate channel prediction in indoor environments. Whereas the penetrated ray path in a conventional ray tracing is treated as a straight line without refraction, the proposed method is able to consider refraction through the wall in the penetrated ray path. Hence, it improves the accuracy in ray tracing simulation. To verify the validation of the proposed method, the simulated results of conventional method, approximate method, and proposed method are compared with the measured results. The comparison shows that the proposed method is in better agreement with the measured results than the conventional method and approximate method, especially in high frequency bands.

Solutions to the Vector Tetrahedron Equation

1965 ◽  
Vol 87 (2) ◽  
pp. 228-234 ◽  
Author(s):  
Milton A. Chace
Keyword(s):  
Closed Form ◽  
Digital Computer ◽  
Three Dimensional ◽  
Dimensional Vector ◽  
Spherical Coordinates ◽  
Tetrahedron Equation ◽  
Position Determination ◽  
Closed Form Solutions

A set of nine closed-form solutions are presented to the single, three-dimensional vector tetrahedron equation, sum of vectors equals zero. The set represents all possible combinations of unknown spherical coordinates among the vectors, assuming the coordinates are functionally independent. Optimum use is made of symmetry. The solutions are interpretable and can be evaluated reliably by digital computer in milliseconds. They have been successfully applied to position determination of many three-dimensional mechanisms.

Validation of closed-form expressions for the atmospheric altimetry correction in ground-based GNSS reflectometry based on rigorous ray-tracing

2021 ◽  
Author(s):  
Thalia Nikolaidou ◽  
Marcelo Santos ◽  
Simon D. P. Williams ◽  
Felipe Geremia-Nievinski
Keyword(s):  
Closed Form ◽  
Ray Tracing ◽  
Specular Reflection ◽  
Elevation Angle ◽  
Rate Of Change ◽  
Linear Component ◽  
Total Delay ◽  
Atmospheric Effect ◽  
Refraction Effect ◽  
Gnss Reflectometry

<p>GNSS reflectometry (GNSS-R) ability to remote sense the Earth’s surface is affected by an atmospheric bias, as pointed out by several recent studies. In particular, sea level altimetry retrievals are biased in proportion to the reflector height, while by-products, such as tidal amplitudes, are underestimated. Previously, we developed an atmospheric ray-tracing procedure to solve rigorously the three-point boundary value problem of ground-based GNSS-R observations. We defined the reflection-minus-direct or interferometric delay in terms of vacuum distance and radio length. We clarified the roles of linear and angular refraction in splitting the total delay in two components, along-path and geometric. We introduced for the first time two subcomponents of the atmospheric geometric delay, the geometry shift and geometric excess. Finally, we defined atmospheric altimetry corrections necessary for unbiased altimetry retrievals based on half of the rate of change of the atmospheric delays with respect to sine of elevation angle. Later, for users without access to ray-tracing software, we developed closed-form expressions for the atmospheric delay and altimetry correction. The first expression accounts for the angular component of refraction (bending), leading to a displaced specular reflection point. The second one accounts for the linear component (speed retardation) in a homogeneous atmosphere. The expressions are parametrized in terms of refractivity and elevation bending, which can be obtained from empirical models, such as the GPT2 or Bennet’s, or fine-tuned based on in situ pressure and temperature. We also provide a correction for the satellite elevation angle such that the refraction effect is nullified. We validated these expressions against rigorous ray-tracing results and showed that the discrepancy is caused by assumptions in the derivation of the closed formulas. We found the corrections to be beneficial even for small reflector heights, as approximated half of the atmospheric effect originates above the receiving antenna at low satellite elevation angles.</p>

Stress Intensity Factor Coefficients for Circumferential ID Surface Flaws in Cylinders for Appendix A of ASME Section XI

2013 ◽  
Author(s):  
Russell C. Cipolla ◽  
Darrell R. Lee
Keyword(s):  
Stress Intensity Factor ◽  
Stress Intensity ◽  
Intensity Factor ◽  
Flat Plate ◽  
Closed Form ◽  
Fourth Order ◽  
Surface Crack ◽  
Crack Size ◽  
Plate Geometry

The stress intensity factor (KI) equations for a surface crack in ASME Section XI, Appendix A are based on non-dimensional coefficients (Gi) that allow for the calculation of stress intensity factors for a cubic varying stress field. Currently, the coefficients are in tabular format for the case of a surface crack in a flat plate geometry. The tabular form makes the computation of KI tedious when determination of KI for various crack sizes is required and a flat plate geometry is conservative when applied to a cylindrical geometry. In this paper, closed-form equations are developed based on tabular data from API 579 (2007 Edition) [1] for circumferential cracks on the ID surface of cylinders. The equations presented, represent a complete set of Ri/t, a/t, and a/l ratios and include those presented in the 2012 PVP paper [8]. The closed-form equations provide G0 and G1 coefficients while G2 through G4 are obtained using a weight function representation for the KI solutions for a surface crack. These equations permit the calculation of the Gi coefficients without the need to perform tabular interpolation. The equations are complete up to a fourth order polynomial representation of applied stress, so that the procedures in Appendix A have been expanded. The fourth-order representation for stress will allow for more accurate fitting of highly non-linear stress distributions, such as those depicting high thermal gradients and weld residual stress fields. The equations developed in this paper will be added to the Appendix A procedures in the next major revision to ASME Section XI. With the inclusion of equations to represent Gi, the procedures of Appendix A for the determination of KI can be performed more efficiently without the conservatism of using flat plate solutions. This is especially useful when performing flaw growth evaluations where repetitive calculations are required in the computations of crack size versus time. The equations are relatively simple in format so that the KI computations can be performed by either spreadsheet analysis or by simple computer programming. The format of the equations is generic in that KI solutions for other geometries can be added to Appendix A relatively easily.

scholarly journals Closed-form rocking overturning conditions for a family of pulse ground motions

2016 ◽  
Vol 472 (2196) ◽  
pp. 20160662 ◽  
Author(s):  
Elias G. Dimitrakopoulos ◽  
Edwin Dat Win Fung
Keyword(s):  
Lower Bound ◽  
Finite Number ◽  
Closed Form ◽  
Physical Mechanism ◽  
Ground Motions ◽  
Critical Conditions ◽  
Rocking Block ◽  
Approximate Analytical Solutions ◽  
The Stability

This paper characterizes the stability of a rigid rocking block subjected to a family of multi-lobe pulse ground motions. It unveils a counter to intuition plurality of overturning (OT) modes despite the short duration and bounded energy of the examined ground motions. Accordingly, it describes with original closed-form expressions the critical conditions of all OT modes involving a finite number of impacts. It also proposes pertinent semi-analytical, exact analytical and approximate analytical solutions with respect to the determination of the (unknown) times of impact, as appropriate. The analysis reveals that the first , or lower bound , critical OT mode is in most cases toppling during free rocking after one impact specifically before the end of the pulse. For this case, it elucidates the physical mechanism behind the timing of impact that produces minimum amplitude and maximum amplitude critical OT, respectively, and proposes pertinent closed-form approximations. Finally, the study derives, in ‘universal’ terms, global ‘safety walls’ against rocking OT considering a large number of different pulse shapes.

Explicit formulation for the axial collapse of a pipe during drilling

2010 ◽  
Vol 224 (8) ◽  
pp. 1547-1549
Author(s):  
A Almasi
Keyword(s):  
Closed Form ◽  
Plastic Hinge ◽  
Variable Length ◽  
Experimental Results ◽  
Displacement Curve ◽  
Explicit Formulation ◽  
Theoretical Predictions ◽  
Hinge Zone ◽  
Good Agreement

New closed-form expressions are introduced for ax-symmetric progressive axial collapse of pipes that use a plastic folding mechanism based on variable length of an active plastic hinge zone. A procedure for determination of a load—displacement curve of axial pipe collapse is presented. Theoretical predictions give a good agreement with the experimental results owing to the influence of presented new refinements.

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