Symbolic Manipulation Program Research Articles

Three dimensional singular element

The finite element method has become a powerful tool for computation of stress intensity factors in fracture mechanics. The simulation of singular behavior in the stress field is accomplished using “quarter points,” following the methods of Barsoum[1] and Henshell-Shaw[2]. The analysis has also been extended to cubic elements [3] and transition elements [4]. However, these concepts cannot be easily extended to three dimensional cases without additional conditions. Progress has been hampered firstly due to a variety of possible shapes the element may possess near the singular edge of the crack, and secondly due to the complexity of algebraic expressions that have to be manipulated. In the present investigation we extensively used MACSYMA[5], a large symbolic manipulation program at MIT, thereby alleviating some of these difficulties. A simple condition between mid-side nodes has been derived which simulates the proper singular behavior along the crack. In the investigation we first study a simple collapsed brick element. This is then generalized to a curved crack front. A few results are derived which can be used to compute the stress intensity factors. The concept of the transitional element has also been outlined. The stability of singular element has been discussed. Some of these ideas have been applied to a specific problem with unusual crack geometry. The analysis was carried out using ADINA on VAX machine. ADINA was implemented on VAX by W. E. Lorensen.

Quadratic and cubic transition elements

AbstractQuadratic and cubic quadrilateral isoparametric elements are investigated for use as transition elements for use a crack tip elements. Computations for a single element are explicitly carried out using a symbolic algebraic manipulation program and in addition to the desired square root singularity, additional singularities are revealed. Cubic transitional elements were used for a double‐edge crack problem and only marginal improvement is found for a well‐laid out grid.

Symbolic evaluation of dimensionally regularized Feynman diagrams

A modification of the symbolic and algebraic manipulation program REDUCE is reported which allows the treatment of vector and gamma algebra in an arbitrary number of dimensions. The number of dimensions serves as a cut-off for divergences in the Feynman diagrams one has to evaluate in a theory like quantum gravity.

Analysis of the availability of computer systems using computer-aided algebra

Analytical results, related to the availability of a computer system constructed of unreliable processors, are presented in this paper. These results are obtained by using various computer-aided algebraic manipulation techniques. A major purpose of this paper is to demonstrate that the difficulties of obtaining analytical solutions to Markov processes can be considerably reduced by the application of symbol manipulation programs. Since many physical systems can be modeled by Markov and semi-Markov processes, the potential range of application of these techniques is much wider than the problem of availability analyzed here.

An application of symbolic computation to crystal physics

This paper describes a practical application of symbolic computation to a problem in chemical physics. We compute a series approximation to a fairly complicated integral using straightforward techniques, allowing the computer to take over when the problem becomes too messy. Even though Altran [1] has no direct way to handle non-rational functions, introducing a few auxiliary variables renders the problem tractable. We certainly hope that more users will attempt to solve standard problems of this sort using symbolic manipulation programs.

An algebraic simplification program in PL/I

One of the objectives of the project reported on was to explore the usability of PL/I for the purpose of writing a symbolic formula manipulation program in a higher level language. The first step along this line is writing an algebraic simplification program.