Closed-Form Solutions of Higher Order Duration Measures

1988 ◽  
Author(s):  
Sanjay K. Nawalkha ◽  
Nelson J. Lacey
Keyword(s):  
Closed Form ◽  
Higher Order ◽  
Closed Form Solutions

An Algorithm for Reducing the Size of Finite Element Closed-Form Source Code Files

2009 ◽  
Author(s):  
Sara McCaslin ◽  
Kent Lawrence
Keyword(s):  
Finite Element Analysis ◽  
Finite Element ◽  
Closed Form ◽  
Source Code ◽  
Simple Algorithm ◽  
Higher Order ◽  
Error Estimator ◽  
Element Analysis ◽  
Closed Form Solutions ◽  
Large Source

Closed-form solutions, as opposed to numerically integrated solutions, can now be obtained for many problems in engineering. In the area of finite element analysis, researchers have been able to demonstrate the efficiency of closed-form solutions when compared to numerical integration for elements such as straight-sided triangular [1] and tetrahedral elements [2, 3]. With higher order elements, however, the length of the resulting expressions is excessive. When these expressions are to be implemented in finite element applications as source code files, large source code files can be generated, resulting in line length/ line continuation limit issues with the compiler. This paper discusses a simple algorithm for the reduction of large source code files in which duplicate terms are replaced through the use of an adaptive dictionary. The importance of this algorithm lies in its ability to produce manageable source code files that can be used to improve efficiency in the element generation step of higher order finite element analysis. The algorithm is applied to Fortran files developed for the implementation of closed-form element stiffness and error estimator expressions for straight-sided tetrahedral finite elements through the fourth order. Reductions in individual source code file size by as much as 83% are demonstrated.

Closed-Form Solutions of Higher-Order Duration Measures

1988 ◽  
Vol 44 (6) ◽  
pp. 82-84 ◽  
Author(s):  
Sanjay K. Nawalkha ◽  
Nelson J. Lacey
Keyword(s):  
Closed Form ◽  
Higher Order ◽  
Closed Form Solutions

Higher-order Closed-form Solutions for Thick Laminated Sandwich Shells

2005 ◽  
Vol 7 (4) ◽  
pp. 335-358 ◽  
Author(s):  
Rakesh Kumar Khare ◽  
Vijay Rode ◽  
Ajay Kumar Garg ◽  
Sam Philip John
Keyword(s):  
Closed Form ◽  
Higher Order ◽  
Closed Form Solutions ◽  
Sandwich Shells

Kinematic Synthesis of Adjustable Mechanisms—Part 2: Path Generation

1973 ◽  
Vol 95 (2) ◽  
pp. 423-429 ◽  
Author(s):  
Joseph F. McGovern ◽  
George N. Sandor
Keyword(s):  
Closed Form ◽  
Higher Order ◽  
Complex Numbers ◽  
Path Generation ◽  
Function Generator ◽  
Kinematic Synthesis ◽  
Closed Form Solutions ◽  
Straight Line ◽  
Adjustable Mechanisms

A method utilizing complex numbers similar to that used in Part 1 for adjustable function generator synthesis is applied to the synthesis of adjustable path generators. Finitely separated path points with prescribed timing as well as higher order approximations (infinitesimally separated path points) are treated, by way of analytical and closed form solutions. Adjustment of the path generator mechanism is accomplished by moving a fixed pivot. Mechanisms adjustable for different approximate straight line motions, for various path curvatures, and path tangents as well as several arbitrary paths can be synthesized. Four-bar and geared five-bar mechanisms are considered. Examples are included describing synthesized mechanisms.

Closed Form Solutions of Joint Water-Filling for Coordinated Transmission

2010 ◽  
Vol E93-B (12) ◽  
pp. 3461-3468 ◽  
Author(s):  
Bing LUO ◽  
Qimei CUI ◽  
Hui WANG ◽  
Xiaofeng TAO ◽  
Ping ZHANG
Keyword(s):  
Closed Form ◽  
Closed Form Solutions ◽  
Water Filling
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