Symbolic Derivation Research Articles

Symbolic computation of high order compact difference schemes for three dimensional linear elliptic partial differential equations with variable coefficients

We present a symbolic computation procedure for deriving various high order compact difference approximation schemes for certain three dimensional linear elliptic partial differential equations with variable coefficients. Based on the Maple software package, we approximate the leading terms in the truncation error of the Taylor series expansion of the governing equation and obtain a 19 point fourth order compact difference scheme for a general linear elliptic partial differential equation. A test problem is solved numerically to validate the derived fourth order compact difference scheme. This symbolic derivation method is simple and can be easily used to derive high order difference approximation schemes for other similar linear elliptic partial differential equations.

A modeling tool for biomedical systems

This paper describes the tool P ansym for biomedical system modeling which has been designed primarily for representation of kinetics, transport and metabolism of biological substances. New modeling tools are needed in this research field because available simulation environments are often limited when applied to multidisciplinary studies in biomedicine. The aim of the new software is to allow a flexible specification of system structures using concurrently different formalisms that are conventionally used in different modeling domains. Starting from an intuitive model specification, mathematical system equations are first derived in symbolic form and subsequently coded as source code for different target applications. This multidomain modeling environment was built using bond graphs as common basis for automated analysis of system structures and symbolic derivation of system equations. Preliminary experience with the software showed that the adopted design strategy comes up to expectations, especially as regards the definition of complex hierarchical models that exhibit interactions between heterogeneous subsystems, e.g. cardiovascular and metabolic functions. Moreover, the specification of model structures is close to graphical object representation which helps defining a model of a system more in terms of its structure rather than its equations.

ANALYTICAL DESCRIPTION AND OPTIMIZATION OF THE DYNAMIC BEHAVIOUR OF PASSIVELY SUSPENDED ROAD VEHICLES

A simple two-degree-of-freedom linear model is used to derive a number of analytical formulae describing the dynamic behaviour of passively suspended vehicles running on randomly profiled roads. Two different power spectral densities are considered for modelling the road irregularity. The derived analytical formulae can be used either during preliminary design or for other special purposes, especially when approximated results are acceptable.An optimization method, based on Multi-Objective Programming and Monotonicity analysis, is introduced and applied for the symbolic derivation of analytical formulae featuring the best compromise among conflicting performance indices pertaining to the vehicle suspension system, i.e., discomfort, road holding and working space. The optimal settings of the relevant vehicle suspension parameters (i.e., tyre radial stiffness, spring stiffness and damping) are derived either symbolically and/or numerically.

Nonlinear joint fractional transform correlator

An important tool in optical pattern recognition, the joint fractional transform correlator (JFTC), was introduced recently. We analyze the peak properties of fractional correlation (FC) by symbolic derivation and computer simulation. We show that the FC has a maximum correlation peak when the second fractional Fourier transform is reduced to the conventional Fourier transform. We introduce nonlinear operations in a joint fractional transform power spectrum and propose a differential JFTC and a binary differential JFTC. Numerical simulations show that such nonlinear JFTCs exhibit remarkable improvement in correlation peak intensity, discrimination capability, and signal-to-noise ratio. An optoelectronic setup that can implement such nonlinear JFTCs is also proposed.

Structural parameter sensitivity analysis of cantilever- and bridge-type accelerometers

The paper deals with the analytical estimation of bandwidth and device sensitivity of cantilever- and bridge-type monolithic piezoresistive and capacitive accelerometers using different beam models and of their design sensitivity due to the change of geometrical parameters. Using a simplified two-beams model, closed-form formulae have been derived to give the relationship between the rates of length, width and thickness of the beams and the lowest eigenfrequency characterizing the bandwidth as well as the physical sensitivity of the accelerometers. By means of symbolic derivation, a structural design sensitivity analysis has been carried out to obtain information for the optimal selection of the geometrical parameters.

Object-oriented finite elements. IV. Symbolic derivations and automatic programming of nonlinear formulations

In previous papers [Comput. Methods Appl. Mech. Eng. 132 (1996) 277–304, 259–276; 154 (1998) 41–68] , the authors have presented the main ideas, which have led to the creation of an environment for symbolic derivations of variational problems and automatic programming of finite elements codes. In these papers, only linear formulations were treated, e.g., Galerkin formulation for elastodynamics [Adv. Eng. Soft. 27 (1996) 3–10] or Galerkin least-squares (GLS) formulation of Stokes flow problem. In this paper, we wish to show a natural extension of symbolic derivation concepts to nonlinear formulations, using the general notion of directional derivative. The application to the numerical solution of Navier–Stokes equations is given.

Symbolic derivation and numeric computation of dyadic Green's functions using Mathematica

This paper presents a mathematical-software functional package that is capable of performing symbolic derivation and numeric computation of dyadic Green's functions for certain multilayered structures: a planar stratified multilayered medium, a spherical multilayered medium, a cylindrical multilayered medium, and a conducting rectangular waveguide with a multilayered dielectric load. The algorithms of this software package are based on the eigenfunction-expansion method. Using Mathematica/sup TM/, two packages were written to fulfill the aforementioned objectives. Upon completion of the software development, dyadic Green's functions for three-layered media were generated. A comparison of these outputs with published results showed good agreement. This demonstrated the applicability of the symbolic package. For the numeric package, the Green's dyadics for a particular three-layered spherical isotropic multilayered medium were generated as an illustration. These packages have been successfully implemented, and future derivation of dyadic Green's functions for these media may be performed.

Sensitivity analysis of the human body mechanical model

AbstractThe mechanical model of human body is introduced. Model consists of rigid segments connected with joints. Articulated rigid‐multibody model is used as a tool for investigation of the injury mechanism during the car crush event. The sensitivity analysis of proposed mechanical model of human body is discussed. Modified direct approach to the sensitivity analysis is introduced, which enables calculation of the sensitivity coefficients in a straightforward way. The approach is semi‐analytical and is baaed on symbolic derivatives of the differential equations of motion. The results of a case study are presented. They illustrate the intrisinct effect of human body geometry and other influential parameters on head acceleration. It is shown that the sensitivity analysis allows for a deeper insight into human body injury criteria.

Automatic derivation of sensitivity terms for mixed Galerkin‐collocation problems

AbstractThe paper presents the use of the SMS system (Symbolic Mechanics System) for the automatic derivation and coding of sensitivity terms in finite element analysis. General symbolic derivation of analytical sensitivity terms for general mixed Galerkin‐collocation problems is difficult due to the complex (implicit) interactions between different levels of computation. The approach implemented in SMS avoids this problem by combining several techniques: symbolic capabilities of Mathematica, automatic differentiation technique, simultaneous optimisation of expressions and stochastic evaluation of the formulae.

Symbolic software tools in the development of finite elements

Symbolic software has been used in a number of projects concerned with the development of finite element procedures, primarily aiming at complex, i.e. interacting and higher order instabilities, where high accuracy in formulations is required.The symbolic tools improve the efficiency and documentation of the developed procedures, in order to facilitate comparisons between different element assumptions. Beam formulations for plane and space models were developed, in total displacement and co-rotational contexts, respectively. Symbolic derivation allowed analytical verification of equivalence between certain formulations within these two contexts. Treatment of finite space rotations, based on the rotational vector makes the history-less treatment of rotations easier, which is needed in the evaluation of critical equilibrium subsets in higher-dimensional parameter space. A co-rotational viewpoint, where local element displacements can be obtained from global variables in a systematic manner, allowed different element expressions in a common framework. Different simple, linear elements have been tested with respect to computational efficiency. A field consistence approach was used to develop highly accurate beam and plane stress elements. The common element formulations, based on the matrix multiplication B T DB, is often inefficient, due to the large number of operations needed in the matrix product. Other formulations, based on an analytical integration and differentiation of the strain energy, producing explicit expressions for the stiffness terms, were considerably more efficient for certain elements.

Object-Oriented Symbolic Derivation and Automatic Programming of Finite Elements in Mechanics

Symbolic approaches to assist in the development of finite element formulations have been used since the late 1970s. Today, symbolic mathematical software such as Mathematica, Maple, etc., has proved to be helpful when testing formulations. In earlier work, the authors introduced a new way of integrating naturally symbolic concepts in numerical finite element codes, taking advantage of an objectoriented code organization. In this paper, we wish to prove on practical examples that the proposed approach is very attractive and promising today, leading to an alternative way of conceiving finite element codes. After presenting a state-of- the-art of symbolic approaches for finite element developments, we first give a practical application of symbolic developments (for discontinuous space-time formulations), and then examples of Computer Aided Software Engineering tools that can be introduced into such a finite element environment.

Symbolic Derivation of Equations in Engineering Using Computer Algebra Systems

In this paper, the derivation of analytical solutions for a problem of a current investigation shall be demonstrated utilizing the program Maple V.

Symbolic derivation of finite difference approximations for the three-dimensional Poisson equation

Symbolic derivation of finite difference approximations for the three-dimensional Poisson equation

Details of the integral equation method applied to the analysis of an adhesive layer crack

The mathematical model for a crack in an elastic adhesive layer sandwiched between two adherends proposed by Fleck, Hutchinson and Suo ((1991) Crack path selection in a brittle adhesive layer. International Journal of Solids and Structures27(13), 1683–1703) was considered. The elastic mismatch between the adhesive and adherend materials modifies the far-field values of the stress intensity factors and of the T-stress in a manner that depends on the position of the crack inside the layer and on the Dundurs parameters. A complex-potential stress-function formulation, using dislocation distributions represented by truncated Chebyshev series, yielded an integral equation that was solved numerically by the method of collocations. The symbolic derivation of the integral equations was checked, and a few differences from Fleck, Hutchinson and Suo's expressions were identified and reconciled. The computational aspects of the solution were studied in detail using two programming languages, MATHCAD and C++, run on standard PC hardware. A palette of numerical techniques were utilized to study and control the consistency and accuracy of the solution. Symmetry and anti-symmetry arguments were used to identify numerically sensitive regions. Fast Fourier Transform calculation of sine and cosine Fourier integrals was used to increase speed. Convergence of intermediate and final results was examined. It was found that the method is very sensitive to the details of numerical computation, especially for combinations of parameters that lead to nearly singular matrices.

Object-oriented finite elements III. Theory and application of automatic programming

In a first paper [1], the authors have developed the key features of an object-oriented environment for the symbolic derivation of linear initial-boundary-value problems. The approach was illustrated on a simple example of an elastic bar in dynamics. In a companion paper [2], the structure of the proposed environment was described at length. In this article, the key features of the automatic generation of finite elements in a symbolic object-oriented environment are given. A few examples of derivation and automatic programming are developed and numerically tested. First, a classical Galerkin formulation of Stokes flow is compared to a Galerkin least-squares stabilization formulation. Then, various stabilized formulations are tested on a simple scalar diffusion model equation.

Fonctionnalité d’un environnement orienté objet pour le développement de code éléments finis

Abstract.Our purpose in this paper is to introduce aspects of useris and developer’s graphical interface associated with operational aspects in an object-oriented environment for symbolic derivation and automatic programming of finite elements. The approach is illustrated first on the example of two formulations for the linear elasticity and second on the example of a stabilized formulation for the Navier-Stokes problem.

Computer Algebra and Interpolation: a Lesson Plan

Polynomial interpolation and related numerical analysis questions are used to take users of computer algebra systems from issuing one-liners to writing more complex programs. An alternate use of the outlined lessons is to introduce students to interpolation and acquaint them with splines and numerical differentiation. Constructive symbolic derivations of formulae for polynomial interpolants are presented. Numerical differentiation formulae are derived using them. Necessary background material is introduced. The examples and assignments can be done using any general purpose symbolic manipulator; complete codes using the computer algebra system AXIOM are given.

Automatic derivation of thermodynamic property functions using computer algebra

The commercial computer algebra system known as Maple has been extended in order to make possible the automatic symbolic derivation and differentiation of thermodynamic property functions of mixtures with arbitrary numbers of components. The tools described here could be useful in the rapid development of new models for thermodynamic properties (and their derivatives).

Object-oriented finite element programming: an interactive environment for symbolic derivations, application to an initial boundary value problem

Object-oriented finite element programming has been receiving increasing attention recently. In recent papers the authors' research group has systematically developed a methodology of implementing finite elements within the context of an object-oriented paradigm, using several languages successively: Smalltalk, Ctalk and finally C ++. The key features of the object-oriented approach (data encapsulation, message passing, hierarchical code organization, inheritance and polymorphism) have been shown to produce codes with enhanced modularity and improved reusability, even among different programmers, and to lead to faster prototyping and easier debugging of new codes. In this article an attempt to go beyond this stage is illustrated. The principles of object-oriented programming are applied directly to the problem statement in differential form. The new approach is conducted as follows: • User input: strong form • Symbolic derivation: weak form • Symbolic derivation: Galerkin form • Symbolic derivation: matrix form • Numerical derivation: numerical matrix form All derivations are carried out in symbolic form except for that of the final numerical matrix form. The barriers between symbolic and numerical programming are broken and the code is generated automatically. The potential of the proposed approach is illustrated by its application to linear elastodynamics.

Object-oriented finite elements I. Principles of symbolic derivations and automatic programming

The development of a numerical finite element model for a given initial-boundary-value problem goes essentially through the derivation of a weak form, its discretization and finally the derivation of the matrix forms. These steps can, to a large extent, be automated, except for the ones which require intelligence or some form of decision making from the developer. The article outlines the essential steps of an interactive quasi-automatic approach and illustrates it in the example of a linear uniaxial bar formulation for dynamics. The implementation is done in SmalltalkV. A detailed description of the proposed environment is described in a companion article.