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Results for “Symbolic and Algebraic Manipulation”

Search indexed NASA NTRS and DOE OSTI research on propulsion, heat transfer, battery materials and energy systems. Follow report and document links to the original sources.

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At least 19 records
All sourcesNASA NTRSDOE OSTI
NASA NTRS1993

Application of symbolic and algebraic manipulation software in solving applied mechanics problems

As its name implies, symbolic and algebraic manipulation is an operational tool which not only can retain symbols throughout computations but also can express results in terms of symbols. This report starts with a history of symbolic and algebraic manipulators and a review of the literatures. With the help of selected examples, the capabilities of symbolic and algebraic manipulators are demonstrated. These applications to problems of applied mechanics are then presented. They are the application of automatic formulation to applied mechanics problems, application to a materially nonlinear problem (rigid-plastic ring compression) by finite element method (FEM) and application to plate problems by FEM. The advantages and difficulties, contributions, education, and perspectives of symbolic and algebraic manipulation are discussed. It is well known that there exist some fundamental difficulties in symbolic and algebraic manipulation, such as internal swelling and mathematical limitation. A remedy for these difficulties is proposed, and the three applications mentioned are solved successfully. For example, the closed from solution of stiffness matrix of four-node isoparametrical quadrilateral element for 2-D elasticity problem was not available before. Due to the work presented, the automatic construction of it becomes feasible. In addition, a new advantage of the application of symbolic and algebraic manipulation found is believed to be crucial in improving the efficiency of program execution in the future. This will substantially shorten the response time of a system. It is very significant for certain systems, such as missile and high speed aircraft systems, in which time plays an important role.

Tsai, Wen-Lang↗
NASA NTRS1986

On the symbolic manipulation and code generation for elasto-plastic material matrices

A computerized procedure for symbolic manipulations and FORTRAN code generation of elastoplastic material matrix for finite element applications is presented. Special emphasis is placed on expression simplifications during intermediate derivations, optimal code generation, and interface with the main program. A systematic procedure is outlined to avoid redundant algebraic manipulations. Symbolic expressions of the derived material stiffness matrix are automatically converted to RATFOR code which is then translated into FORTRAN statements through a preprocessor. To minimize the interface problem with the main program, a template file is prepared so that the translated FORTRAN statements can be merged into the file to form a subroutine (or a submodule). Three constitutive models; namely, von Mises plasticity, the Drucker-Prager model, and a concrete plasticity model, are used as illustrative examples.

Chang, T. Y.↗
NASA NTRS1991

On the symbolic manipulation and code generation for elasto-plastic material matrices

A computerized procedure for symbolic manipulations and FORTRAN code generation of an elasto-plastic material matrix for finite element applications is presented. Special emphasis is placed on expression simplifications during intermediate derivations, optimal code generation, and interface with the main program. A systematic procedure is outlined to avoid redundant algebraic manipulations. Symbolic expressions of the derived material stiffness matrix are automatically converted to RATFOR code which is then translated into FORTRAN statements through a preprocessor. To minimize the interface problem with the main program, a template file is prepared so that the translated FORTRAN statements can be merged into the file to form a subroutine (or a submodule). Three constitutive models; namely, von Mises plasticity, Drucker-Prager model, and a concrete plasticity model, are used as illustrative examples.

Chang, T. Y.↗
NASA NTRS1974

Nonlinear finite-element analysis of laminated composite shells

Two aspects of the geometrically nonlinear analysis of laminated composite shells are considered in this paper. The first pertains to the simplifications and computational advantages resulting from the use of group-theoretic methods in conjunction with computerized symbolic (algebraic) manipulation in the element development. The second aspect pertains to the combined effects of shear deformation and nonlinearity on the accuracy and convergence of two-dimensional, shear-flexible stiffness finite element models.

Noor, A. K.↗
NASA NTRS1977

Proceedings of the 1977 MACSYMA users' conference (NASA)

The MACSYMA program for symbolic and algebraic manipulation enables exact, symbolic mathematical computations to be performed on a computer. This program is rather large, and various approaches to the hardware and software problems are examined.

Source record↗
NASA NTRS1981

Computerized symbolic manipulation in nonlinear finite element analysis

The potential of using computerized symbolic manipulation in the development of nonlinear finite elements is discussed. Three tasks which can be efficiently performed using computerized symbolic manipulation are identified: (1) generation of algebraic expressions for the stiffness coefficients of nonlinear finite elements, (2) generation of FORTRAN source code for numerical evaluation of stiffness coefficients, and (3) checking the correctness of the FORTRAN statements for the arrays of coefficients. The symbolic and algebraic manipulation system MACSYMA is used in the present study. Two sample MACSYMA programs are presented for the development of the nonlinear stiffness coefficients of two-dimensional, shear-flexible, doubly-curved deep shell elements. The first program is for displacement models and the second program is for mixed models with discontinuous stress-resultant fields at interelement boundaries.

Noor, A. K.↗
NASA NTRS1975

A computerized symbolic integration technique for development of triangular and quadrilateral composite shallow-shell finite elements

Computerized symbolic integration was used in conjunction with group-theoretic techniques to obtain analytic expressions for the stiffness, geometric stiffness, consistent mass, and consistent load matrices of composite shallow shell structural elements. The elements are shear flexible and have variable curvature. A stiffness (displacement) formulation was used with the fundamental unknowns consisting of both the displacement and rotation components of the reference surface of the shell. The triangular elements have six and ten nodes; the quadrilateral elements have four and eight nodes and can have internal degrees of freedom associated with displacement modes which vanish along the edges of the element (bubble modes). The stiffness, geometric stiffness, consistent mass, and consistent load coefficients are expressed as linear combinations of integrals (over the element domain) whose integrands are products of shape functions and their derivatives. The evaluation of the elemental matrices is divided into two separate problems - determination of the coefficients in the linear combination and evaluation of the integrals. The integrals are performed symbolically by using the symbolic-and-algebraic-manipulation language MACSYMA. The efficiency of using symbolic integration in the element development is demonstrated by comparing the number of floating-point arithmetic operations required in this approach with those required by a commonly used numerical quadrature technique.

Anderson, C. M.↗
NASA NTRS1976

A computer program for anisotropic shallow-shell finite elements using symbolic integration

A FORTRAN computer program for anisotropic shallow-shell finite elements with variable curvature is described. A listing of the program is presented together with printed output for a sample case. Computation times and central memory requirements are given for several different elements. The program is based on a stiffness (displacement) finite-element model in which the fundamental unknowns consist of both the displacement and the rotation components of the reference surface of the shell. Two triangular and four quadrilateral elements are implemented in the program. The triangular elements have 6 or 10 nodes, and the quadrilateral elements have 4 or 8 nodes. Two of the quadrilateral elements have internal degrees of freedom associated with displacement modes which vanish along the edges of the elements (bubble modes). The triangular elements and the remaining two quadrilateral elements do not have bubble modes. The output from the program consists of arrays corresponding to the stiffness, the geometric stiffness, the consistent mass, and the consistent load matrices for individual elements. The integrals required for the generation of these arrays are evaluated by using symbolic (or analytic) integration in conjunction with certain group-theoretic techniques. The analytic expressions for the integrals are exact and were developed using the symbolic and algebraic manipulation language.

Andersen, C. M.↗
DOE OSTI2024

On amplitudes and field redefinitions

We derive an off-shell recursion relation for correlators that holds at all loop orders. This allows us to prove how generalized amplitudes transform under generic field redefinitions, starting from an assumed behavior of the one-particle-irreducible effective action. The form of the recursion relation resembles the operation of raising the rank of a tensor by acting with a covariant derivative. This inspires a geometric interpretation, whose features and flaws we investigate.

72 PHYSICS OF ELEMENTARY PARTICLES AND FIELDS↗
NASA NTRS1984

Analysis of lateral and torsional vibration characteristics of beams and shafts with end located rotational masses

Partial differential equations are derived for free lateral and torsional vibration of a uniform free-free beam with a rotational mass attached to each extremity. For appropriate boundary conditions, nonlinear algebraic equations are obtained using a symbolic manipulation computer program, the solutions of which enable the computation of the neutral frequencies and mode-shapes. The mode-shapes are linear combinations of trigonometric and hyperbolic sine and cosine functions. A computer program is written for the numerical solution of the algebraic equations mentioned above, which can compute the natural frequencies, mode-shapes, and node points for any given set of parameters, for any given number of modes.

Robertson, D. K.↗
NASA NTRS1975

Literal algebra for satellite dynamics

A description of the rather general class of operations available is given and the operations are related to problems in satellite dynamics. The implementation of an algebra processor is discussed. The four main categories of symbol processors are related to list processing, string manipulation, symbol manipulation, and formula manipulation. Fundamental required operations for an algebra processor are considered. It is pointed out that algebra programs have been used for a number of problems in celestial mechanics with great success. The advantage of computer algebra is its accuracy and speed.

Gaposchkin, E. M.↗
NASA NTRS1978

Structural Dynamics, Stability, and Control of Helicopters

The dynamic synthesis of gyroscopic structures consisting of point-connected substructures is investigated. The objective is to develop a mathematical model capable of an adequate simulation of the modal characteristics of a helicopter using a minimum number of degrees of freedom. The basic approach is to regard the helicopter structure as an assemblage of flexible substructures. The variational equations for the perturbed motion about certain equilibrium solutions are derived. The discretized variational equations can be conveniently exhibited in matrix form, and a great deal of information about the system modal characteristics can be extracted from the coefficient matrices. The derivation of the variational equations requires a monumental amount of algebraic operations. To automate this task a symbolic manipulation program on a digital computer is developed.

Meirovitch, L.↗
NASA NTRS2011

Engineering and Scientific Applications: Using MatLab(Registered Trademark) for Data Processing and Visualization

MatLab(R) (MATrix LABoratory) is a numerical computation and simulation tool that is used by thousands Scientists and Engineers in many cou ntries. MatLab does purely numerical calculations, which can be used as a glorified calculator or interpreter programming language; its re al strength is in matrix manipulations. Computer algebra functionalities are achieved within the MatLab environment using "symbolic" toolbo x. This feature is similar to computer algebra programs, provided by Maple or Mathematica to calculate with mathematical equations using s ymbolic operations. MatLab in its interpreter programming language fo rm (command interface) is similar with well known programming languag es such as C/C++, support data structures and cell arrays to define c lasses in object oriented programming. As such, MatLab is equipped with most ofthe essential constructs of a higher programming language. M atLab is packaged with an editor and debugging functionality useful t o perform analysis of large MatLab programs and find errors. We belie ve there are many ways to approach real-world problems; prescribed methods to ensure foregoing solutions are incorporated in design and ana lysis of data processing and visualization can benefit engineers and scientist in gaining wider insight in actual implementation of their perspective experiments. This presentation will focus on data processing and visualizations aspects of engineering and scientific applicati ons. Specifically, it will discuss methods and techniques to perform intermediate-level data processing covering engineering and scientifi c problems. MatLab programming techniques including reading various data files formats to produce customized publication-quality graphics, importing engineering and/or scientific data, organizing data in tabu lar format, exporting data to be used by other software programs such as Microsoft Excel, data presentation and visualization will be discussed. The presentation will emphasize creating practIcal scripts (pro grams) that extend the basic features of MatLab TOPICS mclude (1) Ma trix and vector analysis and manipulations (2) Mathematical functions (3) Symbolic calculations & functions (4) Import/export data files (5) Program lOgic and flow control (6) Writing function and passing parameters (7) Test application programs

Sen, Syamal K.↗
NASA NTRS2011

Engineering and Scientific Applications: Using MatLab(Registered Trademark) for Data Processing and Visualization

MatLab(TradeMark)(MATrix LABoratory) is a numerical computation and simulation tool that is used by thousands Scientists and Engineers in many countries. MatLab does purely numerical calculations, which can be used as a glorified calculator or interpreter programming language; its real strength is in matrix manipulations. Computer algebra functionalities are achieved within the MatLab environment using "symbolic" toolbox. This feature is similar to computer algebra programs, provided by Maple or Mathematica to calculate with mathematical equations using symbolic operations. MatLab in its interpreter programming language form (command interface) is similar with well known programming languages such as C/C++, support data structures and cell arrays to define classes in object oriented programming. As such, MatLab is equipped with most of the essential constructs of a higher programming language. MatLab is packaged with an editor and debugging functionality useful to perform analysis of large MatLab programs and find errors. We believe there are many ways to approach real-world problems; prescribed methods to ensure foregoing solutions are incorporated in design and analysis of data processing and visualization can benefit engineers and scientist in gaining wider insight in actual implementation of their perspective experiments. This presentation will focus on data processing and visualizations aspects of engineering and scientific applications. Specifically, it will discuss methods and techniques to perform intermediate-level data processing covering engineering and scientific problems. MatLab programming techniques including reading various data files formats to produce customized publication-quality graphics, importing engineering and/or scientific data, organizing data in tabular format, exporting data to be used by other software programs such as Microsoft Excel, data presentation and visualization will be discussed.

Sen, Syamal K.↗
NASA NTRS1986

The role of computerized symbolic manipulation in rotorcraft dynamics analysis

The potential role of symbolic manipulation programs in development and solution of the governing equations for rotorcraft dynamics problems is discussed and illustrated. Nonlinear equations of motion for a helicopter rotor blade represented by a rotating beam are developed making use of the computerized symbolic manipulation program MACSYMA. The use of computerized symbolic manipulation allows the analyst to concentrate on more meaningful tasks, such as establishment of physical assumptions, without being sidetracked by the tedious and trivial details of the algebraic manipulations. Furthermore, the resulting equations can be produced, if necessary, in a format suitable for numerical solution. A perturbation-type solution for the resulting dynamical equations is shown to be possible with a combination of symbolic manipulation and standard numerical techniques. This should ultimately lead to a greater physical understanding of the behavior of the solution than is possible with purely numerical techniques. The perturbation analysis of the flapping motion of a rigid rotor blade in forward flight is presented, for illustrative purposes, via computerized symbolic manipulation with a method that bypasses Floquet theory.

Crespo Da Silva, Marcelo R. M.↗
NASA NTRS1969

COGENT programming manual

COGENT /COmpiler and GENeralized Translator/ programming system is a compiler whose input language enables a description of symbolic and linguistic manipulation algorithms. Primarily for use as a compiler-compiler, it is also applicable to algebraic manipulation, mechanical theorem proving, and heuristic programming.

Reynolds, J. C.↗
NASA NTRS1991

Transfer Functions Via Laplace- And Fourier-Borel Transforms

Approach to solution of nonlinear ordinary differential equations involves transfer functions based on recently-introduced Laplace-Borel and Fourier-Borel transforms. Main theorem gives transform of response of nonlinear system as Cauchy product of transfer function and transform of input function of system, together with memory effects. Used to determine responses of electrical circuits containing variable inductances or resistances. Also possibility of doing all noncommutative algebra on computers in such symbolic programming languages as Macsyma, Reduce, PL1, or Lisp. Process of solution organized and possibly simplified by algebraic manipulations reducing integrals in solutions to known or tabulated forms.

Can, Sumer↗