Use Of Symbolic Manipulation Research Articles

SYMBOLIC-NUMERICAL IMPLEMENTATION OF THE GALERKIN METHOD FOR APPROXIMATE SOLUTION OF THE WAVEGUIDE DIFFRACTION PROBLEM

In this paper, we construct a symbolic-numerical implementation of the Galerkin method for approximate solution of the waveguide diffraction problem at the junction of two open planar three-layer waveguides. The Gelerkin method is implemented in the Maple computer algebra system by symbolic manipulations; its software implementation is based on the scprod symbolic-numerical procedure, which enables the numerical calculation of scalar products for the Galerkin method based on symbolic expressions. The use of symbolic manipulations makes it possible to speed up the calculation of integrals in the Galerkin method owing to single-run symbolic calculation of integrals typical for the problem, rather than multiple numerical integration.

PyFR: An open source framework for solving advection–diffusion type problems on streaming architectures using the flux reconstruction approach

High-order numerical methods for unstructured grids combine the superior accuracy of high-order spectral or finite difference methods with the geometric flexibility of low-order finite volume or finite element schemes. The Flux Reconstruction (FR) approach unifies various high-order schemes for unstructured grids within a single framework. Additionally, the FR approach exhibits a significant degree of element locality, and is thus able to run efficiently on modern streaming architectures, such as Graphical Processing Units (GPUs). The aforementioned properties of FR mean it offers a promising route to performing affordable, and hence industrially relevant, scale-resolving simulations of hitherto intractable unsteady flows within the vicinity of real-world engineering geometries. In this paper we present PyFR, an open-source Python based framework for solving advection–diffusion type problems on streaming architectures using the FR approach. The framework is designed to solve a range of governing systems on mixed unstructured grids containing various element types. It is also designed to target a range of hardware platforms via use of an in-built domain specific language based on the Mako templating engine. The current release of PyFR is able to solve the compressible Euler and Navier–Stokes equations on grids of quadrilateral and triangular elements in two dimensions, and hexahedral elements in three dimensions, targeting clusters of CPUs, and NVIDIA GPUs. Results are presented for various benchmark flow problems, single-node performance is discussed, and scalability of the code is demonstrated on up to 104 NVIDIA M2090 GPUs. The software is freely available under a 3-Clause New Style BSD license (see www.pyfr.org). Program summaryProgram title: PyFR v0.1.0Catalogue identifier: AETY_v1_0Program summary URL:http://cpc.cs.qub.ac.uk/summaries/AETY_v1_0.htmlProgram obtainable from: CPC Program Library, Queen’s University, Belfast, N. IrelandLicensing provisions: New style BSD licenseNo. of lines in distributed program, including test data, etc.: 12733No. of bytes in distributed program, including test data, etc.: 214183Distribution format: tar.gzProgramming language: Python, CUDA and C.Computer: Variable, up to and including GPU clusters.Operating system: Recent version of Linux/UNIX.RAM: Variable, from hundreds of megabytes to gigabytes.Classification: 6.5, 12.External routines: Python 2.7, numpy, PyCUDA, mpi4py, SymPy, MakoNature of problem:Compressible Euler and Navier–Stokes equations of fluid dynamics; potential for any advection–diffusion type problem.Solution method:High-order flux reconstruction approach suitable for curved, mixed, unstructured grids.Unusual features:Code makes extensive use of symbolic manipulation and runtime code generation through a domain specific language.Running time:Many small problems can be solved on a recent workstation in minutes to hours.

MATHEMATICS IN ECONOMICS: REDUCIBILITY AND/OR APPLICABILITY?

The use of mathematics in physics is not restricted to non-formal use with emphasis on approximation and computability. There is also a domain of creative use of symbolic manipulation as well as of metaphoric imagination that makes applicability in the sciences so effective. The challenge to mathematical use in economics is to find ways to use it in these creative ways, including finding new discursive and writing strategies. For this to be possible, the nature of realism in economics has to be delineated since the way in which mathematics is used in economics seems to be in the "reductive" mode instead of the "applied" mode.

General background and approach to multibody dynamics for space applications

Multibody dynamics for space applications is dictated by space environment such as space-varying gravity forces, orbital and attitude perturbations, control forces if any. Several methods and formulations devoted to the modeling of flexible bodies undergoing large overall motions were developed in recent years. Most of these different formulations were aimed to face one of the main problems concerning the analysis of spacecraft dynamics namely the reduction of computer simulation time. By virtue of this, the use of symbolic manipulation, recursive formulation and parallel processing algorithms were proposed. All these approaches fall into two categories, the one based on Newton/Euler methods and the one based on Lagrangian methods; both of them have their advantages and disadvantages although in general, Newtonian approaches lend to a better understanding of the physics of problems and in particular of the magnitude of the reactions and of the corresponding structural stresses. Another important issue which must be addressed carefully in multibody space dynamics is relevant to a correct choice of kinematics variables. In fact, when dealing with flexible multibody system the resulting equations include two different types of state variables, the ones associated with large (rigid) displacements and the ones associated with elastic deformations. These two sets of variables have generally two different time scales if we think of the attitude motion of a satellite whose period of oscillation, due to the gravity gradient effects, is of the same order of magnitude as the orbital period, which is much bigger than the one associated with the structural vibration of the satellite itself. Therefore, the numerical integration of the equations of the system represents a challenging problem. This was the abstract and some of the arguments that Professor Paolo Santini intended to present for the Breakwell Lecture; unfortunately a deadly disease attacked him and shortly took him to death, leaving his work unfinished. In agreement with Astrodynamics Committee it was decided to prepare a paper based on some research activities that Paolo Santini performed during almost 50 years in the aerospace field. His researches spanned many arguments, encompassing flexible space structures, to optimization, stability analysis, thermal analysis, smart structure, etc. just to mention the ones more related to the space field (Paolo Santini was also one the pioneers of the studies of composite wing structures, aeroelasticity and unsteady aerodynamics for aeronautical applications). Following notes have been prepared by Paolo Gasbarri who was one of Paolo Santini collaborators for almost 15 years, they will attempt to offer a sketch of Professor Santini's activity by focusing on three main topics: the stability of flexible spacecrafts, the dynamics of multibody systems and the use of the smart structure technology for the space applications.

Neuro‐expert approach to characteristics recognition for process monitoring and performance evaluation

A process behavior feature extraction and recognition method based on neuro-expert approach is proposed for process monitoring and control. This approach provides an effective way to the analysis and automatic extraction of different types of process behavior features during operation without precise description of the process model. By combining use of symbolic manipulation, fuzzy membership function, rule-based reasoning and neural computation, the characteristic patterns which can capture system behavior features are deduced from on-line data and subsequently used for process monitoring and control. A multi-mode information extraction and representation mechanism is provided to meet the needs of different tasks of system supervision. This leads a more general and effective way to make full advantage of on-line information. The basic methodology and a general application of this approach to process monitoring and performance evaluation are discussed in this article. © 1997 John Wiley & Sons, Inc.

Neuro-expert approach to characteristics recognition for process monitoring and performance evaluation

A process behavior feature extraction and recognition method based on neuro-expert approach is proposed for process monitoring and control. This approach provides an effective way to the analysis and automatic extraction of different types of process behavior features during operation without precise description of the process model. By combining use of symbolic manipulation, fuzzy membership function, rule-based reasoning and neural computation, the characteristic patterns which can capture system behavior features are deduced from on-line data and subsequently used for process monitoring and control. A multi-mode information extraction and representation mechanism is provided to meet the needs of different tasks of system supervision. This leads a more general and effective way to make full advantage of on-line information. The basic methodology and a general application of this approach to process monitoring and performance evaluation are discussed in this article. © 1997 John Wiley & Sons, Inc.

Characteristics recognition for the supervision of knowledge-based control systems

This paper presents a pattern-based information-processing approach, called characteristics recognition (CR), for on-line supervision of knowledge-based control systems. This approach provides a method for the analysis and automatic extraction of different types of process behaviour features during operation without a precise description of the process model. By combining the use of symbolic manipulation, graded fuzzy membership and numeric computation, the characteristics patterns that can capture system behaviour features are deduced from on-line data, and subsequently used for process monitoring and control. A multi-mode information-extraction and representation mechanism is provided to meet the needs of different tasks of system supervision. This leads to an effective and practical way of taking full advantage of on-line information. The basic methodology and a general application of CR to process monitoring and supervisory control are given.

Symbolic Computations in Operations Research

While symbolic computation is enjoying success in its applications towards engineering and physical sciences, its application in operations research is still in an infancy stage. The purpose of this article is to demonstrate the use of symbolic manipulations for the automatic generation of solutions necessary for theoretical and practical analyses of operations research problems. The advantage of symbolic computations is that it enables one to eliminate a good deal of the drudgery (and errors) of formal manipulations in establishing closed form solutions.

Designing without numbers

The author describes how computers are taking even more of the drudgery of analysis out of circuit design. This is being achieved by the use of symbolic manipulation. The author describes an example using the Nodal and Mathematical tool to show how the technique of symbolic manipulation is used in circuit design.

An order n formulation for the motion simulation of general multi-rigid-body tree systems

Multibody programs tend to be plagued by two basic inefficiencies. The first, and most severe, arises from computational requirements associated with the determination of state derivative values. If n is the number of degrees of freedom of a system, the number of computational operations required to calculate state derivative values tends to be a cubic function in n, and thus can become prohibitively large for even modest values of n. The second inefficiency manifests itself when equations are derived for the most general system, as they frequently are. Such implicit programs generally are replete with unnecessary computations or logical statements. This paper presents an approach which does not suffer from either of these deficiencies. State derivative values are calculated in a highly efficient manner, the number of computational operations being a linear function in n. Problem specific equations are generated through the use of symbolic manipulation, which yields explicit equations devoid of needless operations. Finally, the equations are cast in a highly concurrent form, allowing the production of a simulation code that can be used on conventional sequential computers, but is well suited for parallel computers with distributed architecture.

Computational attributes of the integral form of the equation of transfer

Difficulties can arise in radiative and neutron transport calculations when a highly anisotropic scattering phase function is present. In the presence of anisotropy, currently used numerical solutions are based on the integro-differential form of the linearized Boltzmann transport equation. In this paper, we depart from classical thought and present an alternative numerical approach based on application of the integral form of the transport equation. Use of the integral formalism facilitates the following steps: a reduction in dimensionality of the system prior to discretization, the use of symbolic manipulation to augment the computational procedure, and the direct determination of key physical quantities which are derivable through the various Legendre moments of the intensity. Our approach is developed in the context of radiative heat transfer in a plane-parallel geometry, and results are presented and compared with existing benchmark solutions. Encouraging results are presented to illustrate the potential of the integral formalism for computation. The integral formalism appears to possess several computational attributes which are well-suited to radiative and neutron transport calculations.

Use of symbolic manipulation in physical acoustics

The development of a mathematical theory in acoustics (or in any other field) depends on the insight of the theoretician. However, the formulation of an insight into an equation derived from first principles involves much time spent in activities that fall within the category of a symbolic manipulation. To demonstrate the role symbolic manipulation software can play in the solution process, classical problems in acoustics will be selected. It is known that the solution to the linearized governing equations of an acoustic field can be considered as a superposition of vorticity, thermal, and acoustic modes. However, these modes are coupled. The separation of these modes will be demonstrated. Symbolic manipulation will be used to explore how the higher‐order terms in the perturbation parameters, eμ, and eκ, affect the modal coupling. The issue of asymptotic degeneracy will also be addressed (the effects of varying the rates at which these parameters approach zero).

Nonlinear rotorcraft analysis using symbolic manipulation

Analyzing a system of nonlinear equations for the presence of limit cycles is often a tedious and time-consuming proposition, even for relatively simple systems. The large amount of algebraic manipulation required is extremely difficult to perform accurately, and can quickly degenerate into an intractable problem. This paper presents the development of a mathematical model for a helicopter lwith nonlinear damping in its landing gear. The use of symbolic manipulation in the derivation of the complex set of equations is stressed. The resulting equations represent an important extension to the type of nonlinear equations that have previously been analyzed, and allow a more accurate calculation of the system's response. the techniques used are described in detail and are applicable to the study of generic rotating systems.

Equilibrium considerations of the updated Lagrangian formulation of beam‐columns with natural concepts

AbstractBathe and co‐workers'1,2 updated Lagrangian formulation is reviewed for a two‐dimensional beam‐column element. From this review a physical interpretation and properties are derived for the matrices of the method. A variant of Argyris and co‐workers'4,5 natural concept is introduced in the updated Lagrangian formulation, and a new method to derive element matrices arises. A detailed derivation of the matrices of the twodimensional beam‐column element is presented to clarify further the UL formulations. Explicit expressions for these matrices, computed with the use of symbolic manipulation,17 are also presented to illustrate the concepts discussed. Finally, numerical results are shown to compare and evaluate different methods and matrices.

Three-dimensional linear dynamic analysis of buildings with 32-bit virtual-memory minicomputers

This paper focuses on computer-aided analysis of three-dimensional buildings with 32-bit, virtual-memory minicomputers. Various floor, inertial, geometry, stiffness, and displacement models of three-dimensional buildings are discussed with respect to implications for computational efficiency. From this discussion a model for three-dimensional buildings is selected for use with virtual-memory minicomputers. This paper also treats the use of symbolic manipulation, dynamic allocation of memory, and matrix storage and manipulation to achieve computational efficiency. Examples with computational statistics illustrate the ideas presented.

Signal-to-Symbol Transformation: HASP/SIAP Case Study

Artificial intelligence is that part of computer science that concerns itself with concepts and methods of symbolic inference and symbolic representation of knowledge. Its point of departure -- it's most fundamental concept -- is what Newell and Simon called (in their Turing Award Lecture) the physical symbol system. But within last fifteen years, it has concerned itself also with signals -- with interpretation or understanding of signal data. AI researchers have discussed signal-to symbol transformations, and their programs have shown how appropriate use of symbolic manipulations can be of great use in making signal processing more effective and efficient. Indeed, programs for signal understanding have been fruitful, powerful, and among most widely recognized of AI's achievements.