Closed‐form impedance model for annular through‐silicon via pairs in three‐dimensional integration

2015 ◽  
Vol 9 (8) ◽  
pp. 808-813 ◽  
Author(s):  
Aobo Chen ◽  
Feng Liang ◽  
Gaofeng Wang ◽  
Bing‐Zhong Wang
Keyword(s):  
Closed Form ◽  
Three Dimensional ◽  
Through Silicon Via ◽  
Impedance Model

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.

Online three-dimensional SLAM by registration of large planar surface segments and closed-form pose-graph relaxation

2010 ◽  
Vol 27 (1) ◽  
pp. 52-84 ◽  
Author(s):  
Kaustubh Pathak ◽  
Andreas Birk ◽  
Narunas Vaskevicius ◽  
Max Pfingsthorn ◽  
Sören Schwertfeger ◽  
...  
Keyword(s):  
Closed Form ◽  
Three Dimensional ◽  
Planar Surface

Analysis of tunnels in shaly rock considering three-dimensional stress effects on swelling

2005 ◽  
Vol 42 (1) ◽  
pp. 1-12 ◽  
Author(s):  
B C Hawlader ◽  
K Y Lo ◽  
I D Moore
Keyword(s):  
Finite Element ◽  
Closed Form ◽  
Cold Water ◽  
Principal Direction ◽  
Three Dimensional ◽  
Closed Form Solution ◽  
Time Dependent ◽  
Stress Effect ◽  
Stress Effects ◽  
Laboratory Test Results

Underground structures in shales or shaly rocks endure time-dependent swelling effects. Laboratory test results show that the swelling of these shales is dependent on three-dimensional stresses; an external stress on a specimen in one principal direction reduces the swelling not only in that direction but also perpendicularly. The effectiveness of a time-dependent swelling model that considers the three-dimensional stress effect is presented in this paper. A finite element algorithm incorporating the new constitutive model is used for a numerical analysis. The finite element program is used to analyze two tunnels in southern Ontario: the Heart Lake storm sewer tunnel, and the Darlington cold-water intake tunnel. The predicted results agree well with the records of field performance of these tunnels. The comparison between present analyses and the existing closed-form solution shows that the existing solution overestimates the time-dependent swelling effects. The three-dimensional stress effects on swelling are not considered in the closed-form solutions and are the cause of this discrepancy. The pseudo-Poisson's effect is a key parameter for modelling the observed time-dependent swelling. The use of these solutions in design is discussed.Key words: time-dependent swelling, shale, modelling, three-dimensional stress effect, finite element method, tunnel.

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