Comparison of Computer Algebra Systems for Closed Form Stiffness Matrix Generation

2012 ◽  
Author(s):  
Panos S. Shiakolas ◽  
David C. Wilhite ◽  
Sara E. McCaslin
Keyword(s):  
Closed Form ◽  
Computer Algebra ◽  
Stiffness Matrix ◽  
Source Code ◽  
Computer Algebra Systems ◽  
Symbolic Form ◽  
Source Codes ◽  
Stiffness Matrices ◽  
Time Savings ◽  
Common Subexpressions

Computer algebra systems (CAS) have been advantageously employed to generate closed form expressions for finite elements. The advantages relate to the time improvements or savings realized by employing closed form generated expressions as compared to numerical integration. However as the element order increases, the size of the closed form generated expressions become unmanageable causing the source code files to possibly become unusable due to their size. One approach to reducing the size of the source files is to take advantage of the utilities found in CAS to identify common expressions or sub-expressions. In this manuscript we present on-going research by comparing two widely used CAS, Mathematica and Maple, as they relate to identifying common expressions in low order tetrahedral finite element stiffness matrices generated in symbolic form, associated time savings and possible issues. The results indicate that the use of CAS could be advantageously employed to identify common subexpressions through pattern matching to further reduce the size of the generated source files and realize time improvements during execution of the source codes. In addition, the developed procedures could be easily applied to higher order elements with much larger number of entries of closed form expressions where even more savings could be realized.

A New Approach to Obtaining Closed-Form Solutions for Higher Order Tetrahedral Finite Elements Using Modern Computer Algebra Systems

2011 ◽  
Author(s):  
Sara E. McCaslin ◽  
Panos S. Shiakolas ◽  
Brian H. Dennis ◽  
Kent L. Lawrence
Keyword(s):  
Parallel Processing ◽  
Closed Form ◽  
Computer Algebra ◽  
High Order ◽  
Computer Algebra Systems ◽  
Element Analysis ◽  
New Approach ◽  
Closed Form Solutions ◽  
Modern Computer ◽  
Stiffness Matrices

Closed-form solutions for straight-sided tetrahedral element stiffness matrices used in finite element analysis have been proven more efficient than numerically integrated solutions. These closed-form solutions are symbolically integrated using computer algebra systems such as Mathematica or Maple. However, even with memory and processing speed available on desktop computers today, major hindrances exist when attempting to symbolically evaluate the stiffness matrices for high order elements. This research proposes a new approach to obtaining closed-form solutions. Results are presented that demonstrate the feasibility of obtaining the stiffness matrices for high order tetrahedral elements through p-level 9 by use of parallel processing tools in Mathematica 7. Comparisons are made between serial and parallel approaches based on memory required to generate a solution. The serial approach requires more memory and can only generate closed-form solutions up to 7th order. The parallel processing approach presented requires less memory and can generate solutions up to 9th order.

scholarly journals Online Judging Platform Utilizing Dynamic Plagiarism Detection Facilities

Computers ◽  
2021 ◽  
Vol 10 (4) ◽  
pp. 47
Author(s):  
Fariha Iffath ◽  
A. S. M. Kayes ◽  
Md. Tahsin Rahman ◽  
Jannatul Ferdows ◽  
Mohammad Shamsul Arefin ◽  
...  
Keyword(s):  
Source Code ◽  
Large Data ◽  
Large Data Sets ◽  
Detection Technique ◽  
Data Sets ◽  
Plagiarism Detection ◽  
Source Codes ◽  
Efficient Detection ◽  
Mathematical Problems ◽  
Automatic Scoring

A programming contest generally involves the host presenting a set of logical and mathematical problems to the contestants. The contestants are required to write computer programs that are capable of solving these problems. An online judge system is used to automate the judging procedure of the programs that are submitted by the users. Online judges are systems designed for the reliable evaluation of the source codes submitted by the users. Traditional online judging platforms are not ideally suitable for programming labs, as they do not support partial scoring and efficient detection of plagiarized codes. When considering this fact, in this paper, we present an online judging framework that is capable of automatic scoring of codes by detecting plagiarized contents and the level of accuracy of codes efficiently. Our system performs the detection of plagiarism by detecting fingerprints of programs and using the fingerprints to compare them instead of using the whole file. We used winnowing to select fingerprints among k-gram hash values of a source code, which was generated by the Rabin–Karp Algorithm. The proposed system is compared with the existing online judging platforms to show the superiority in terms of time efficiency, correctness, and feature availability. In addition, we evaluated our system by using large data sets and comparing the run time with MOSS, which is the widely used plagiarism detection technique.

scholarly journals New ideas for handling of loop and angular integrals in D-dimensions in QCD

2021 ◽  
Vol 2021 (6) ◽  
Author(s):  
Valery E. Lyubovitskij ◽  
Fabian Wunder ◽  
Alexey S. Zhevlakov
Keyword(s):  
Computer Algebra ◽  
Cross Sections ◽  
Orthogonal Basis ◽  
Computer Algebra Systems ◽  
Covariant Formalism ◽  
Linear Combinations ◽  
Recursion Relations ◽  
New Ideas ◽  
Rotational Invariant ◽  
Geometric Picture

Abstract We discuss new ideas for consideration of loop diagrams and angular integrals in D-dimensions in QCD. In case of loop diagrams, we propose the covariant formalism of expansion of tensorial loop integrals into the orthogonal basis of linear combinations of external momenta. It gives a very simple representation for the final results and is more convenient for calculations on computer algebra systems. In case of angular integrals we demonstrate how to simplify the integration of differential cross sections over polar angles. Also we derive the recursion relations, which allow to reduce all occurring angular integrals to a short set of basic scalar integrals. All order ε-expansion is given for all angular integrals with up to two denominators based on the expansion of the basic integrals and using recursion relations. A geometric picture for partial fractioning is developed which provides a new rotational invariant algorithm to reduce the number of denominators.

The Fischer-Clifford Matrices and Character Table of a Maximal Subgroup of Fi24

2010 ◽  
Vol 17 (03) ◽  
pp. 389-414 ◽  
Author(s):  
Faryad Ali ◽  
Jamshid Moori
Keyword(s):  
Automorphism Group ◽  
Conjugacy Class ◽  
Maximal Subgroup ◽  
Computer Algebra ◽  
Character Table ◽  
Computer Algebra Systems ◽  
Split Extension ◽  
Sporadic Group ◽  
Local Subgroup ◽  
Fischer Group

The Fischer group [Formula: see text] is the largest 3-transposition sporadic group of order 2510411418381323442585600 = 222.316.52.73.11.13.17.23.29. It is generated by a conjugacy class of 306936 transpositions. Wilson [15] completely determined all the maximal 3-local subgroups of Fi24. In the present paper, we determine the Fischer-Clifford matrices and hence compute the character table of the non-split extension 37· (O7(3):2), which is a maximal 3-local subgroup of the automorphism group Fi24 of index 125168046080 using the technique of Fischer-Clifford matrices. Most of the calculations are carried out using the computer algebra systems GAP and MAGMA.

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