Symbolic Manipulation Software Research Articles

HIGH ORDER SECOND DERIVATIVE DIAGONALLY IMPLICIT MULTISTAGE INTEGRATION METHODS FOR ODES

Construction of second derivative diagonally implicit multistage integration methods (SDIMSIMs) as a subclass of second derivative general linear methods with Runge–Kutta stability property requires to generate the corresponding conditions depending of the parameters of the methods. These conditions which are a system of polynomial equations can not be produced by symbolic manipulation packages for the methods of order p ≥ 5. In this paper, we describe an approach to construct SDIMSIMs with Runge–Kutta stability property by using some variant of the Fourier series method which has been already used for the construction of high order general linear methods. Examples of explicit and implicit SDIMSIMs of order five and six are given which respectively are appropriate for both non-stiff and stiff differential systems in a sequential computing environment. Finally, the efficiency of the constructed methods is verified by providing some numerical experiments.

The sky invariant: A new conformal invariant for Schwarzschild spacetime

A new class of conformal invariants for a given spacetime M is introduced exploiting the conformal geometry of any light ray [Formula: see text]. Each congruence of light rays passing through a given point p defines the sky S(p) of such point. The new conformal invariants are defined on the bundle of skies of the spacetime M, being called sky invariants accordingly. The natural conformal covariant derivative defined on a light ray and its associated covariant calculus allows us to show the existence of a natural conformal invariant differential of arc that, together with the restriction of the curvature of the conformal covariant derivative, can be used to construct a sky invariant that will be called the sky curvature. An algorithm, that can be implemented on any symbolic manipulation software system, to compute the sky curvature will be discussed and the main ideas and the explicit computation of the sky curvature are illustrated in Schwarzschild spacetime.

Symbolic-numeric integration of univariate expressions based on sparse regression

The majority of computer algebra systems (CAS) support symbolic integration using a combination of heuristic algebraic and rule-based (integration table) methods. In this paper, we present a hybrid (symbolic-numeric) method to calculate the indefinite integrals of univariate expressions. Our method is broadly similar to the Risch-Norman algorithm. The primary motivation for this work is to add symbolic integration functionality to a modern CAS (the symbolic manipulation packages of SciML, the Scientific Machine Learning ecosystem of the Julia programming language), which is designed for numerical and machine learning applications. The symbolic part of our method is based on the combination of candidate terms generation (ansatz generation using a methodology borrowed from the Homotopy operators theory) combined with rule-based expression transformations provided by the underlying CAS. The numeric part uses sparse regression, a component of the Sparse Identification of Nonlinear Dynamics (SINDy) technique, to find the coefficients of the candidate terms. We show that this system can solve a large variety of common integration problems using only a few dozen basic integration rules.

Exact Solutions and Conservation Laws of a Generalized (1 + 1) Dimensional System of Equations via Symbolic Computation

The aim of this paper is to compute the exact solutions and conservation of a generalized (1 + 1) dimensional system. This can be achieved by employing symbolic manipulation software such as Maple, Mathematica, or MATLAB. In theoretical physics and in many scientific applications, the mentioned system naturally arises. Time, space, and scaling transformation symmetries lead to novel similarity reductions and new exact solutions. The solutions obtained include solitary waves and cnoidal and snoidal waves. The familiarity of closed-form solutions of nonlinear ordinary and partial differential equations enables numerical solvers and supports stability analysis. Although many efforts have been dedicated to solving nonlinear evolution equations, there is no unified method. To the best of our knowledge, this is the first time that Lie point symmetry analysis in conjunction with an ansatz method has been applied on this underlying equation. It should also be noted that the methods applied in this paper give a unique solution set that differs from the newly reported solutions. In addition, we derive the conservation laws of the underlying system. It is also worth mentioning that this is the first time that the conservation laws for the equation under study are derived.

Simplifying the Reinsch algorithm for the Baker–Campbell–Hausdorff series

The Goldberg version of the Baker–Campbell–Hausdorff series computes the quantity Z(X,Y)=lneXeY=∑wg(w)w(X,Y), where X and Y are not necessarily commuting in terms of “words” constructed from the {X, Y} “alphabet.” The so-called Goldberg coefficients g(w) are the central topic of this article. This Baker–Campbell–Hausdorff series is a general purpose tool of very wide applicability in mathematical physics, quantum physics, and many other fields. The Reinsch algorithm for the truncated series permits one to calculate the Goldberg coefficients up to some fixed word length |w| by using nilpotent (|w| + 1) × (|w| + 1) matrices. We shall show how to further simplify the Reinsch algorithm, making its implementation (in principle) utterly straightforward using “off the shelf” symbolic manipulation software. Specific computations provide examples which help to provide a deeper understanding of the Goldberg coefficients and their properties. For instance, we shall establish some strict bounds (and some equalities) on the number of non-zero Goldberg coefficients. Unfortunately, we shall see that the number of nonzero Goldberg coefficients often grows very rapidly (in fact exponentially) with the word length |w|. Furthermore, the simplified Reinsch algorithm readily generalizes to many closely related but still quite distinct problems—we shall also present closely related results for the symmetric product S(X,Y)=lneX/2eYeX/2=∑wgS(w)w(X,Y). Variations on such themes are straightforward. For instance, one can just as easily consider the “loop” product L(X,Y)=lneXeYe−Xe−Y=∑wgL(w)w(X,Y). This “loop” type of series is of interest, for instance, when considering either differential geometric parallel transport around a closed curve, non-Abelian versions of Stokes’ theorem, or even Wigner rotation/Thomas precession in special relativity. Several other closely related series are also briefly investigated.

Energy-momentum tensors in linearized Einstein’s theory and massive gravity: The question of uniqueness

The question of the uniqueness of energy-momentum tensors in the linearized general relativity and in the linear massive gravity is analyzed without using variational techniques. We start from a natural ansatz for the form of the tensor (for example, that it is a linear combination of the terms quadratic in the first derivatives), and require it to be conserved as a consequence of field equations. In the case of the linear gravity in a general gauge we find a four-parametric system of conserved second-rank tensors which contains a unique symmetric tensor. This turns out to be the linearized Landau-Lifshitz pseudotensor employed often in full general relativity. We elucidate the relation of the four-parametric system to the expression proposed recently by Butcher et al. "on physical grounds" in harmonic gauge, and we show that the results coincide in the case of high-frequency waves in vacuum after a suitable averaging. In the massive gravity we show how one can arrive at the expression which coincides with the "generalized linear symmetric Landau-Lifshitz" tensor. However, there exists another uniquely given simpler symmetric tensor which can be obtained by adding the divergence of a suitable superpotential to the canonical energy-momentum tensor following from the Fierz-Pauli action. In contrast to the symmetric tensor derived by the Belinfante procedure which involves the second derivatives of the field variables, this expression contains only the field and its first derivatives. It is simpler than the generalized Landau-Lifshitz tensor but both yield the same total quantities since they differ by the divergence of a superpotential. We also discuss the role of the gauge conditions in the proofs of the uniqueness. In the Appendix, the use of the symbolic tensor manipulation software Cadabra is briefly described.

Robust Topology Optimization: A New Algorithm for Volume-constrained Expected Compliance Minimization with Probabilistic Loading Directions using Exact Analytical Objective and Gradient

This paper presents a new algorithm for volume-constrainedexpected compliance minimization of continuum structureswith probabilistic loading directions using analytically determinedexact objective and gradient functions. The algorithmis based upon the finding that for a particular set of statisticalparameters the integration in the expected compliance functioncan be done symbolically and automatically using symbolicmanipulation software. In this study, Mathematica wasused to integrate and simplify the highly nonlinear expectedcompliance function. It will be demonstrated by examples thatthe result of the symbolic pre-processing step is a simple linearfunction defined on a particular subset of the inverse stiffnessentries which is needed in the compliance computation. Thecoefficient set of this linear function forms the base of the exactanalytical gradient computation used in the optimal solutionsearching optimality criteria (OC) method to define the steepestdescent direction. Naturally, the applied OC method canbe replaced by any other appropriate nonlinear solver. Matlabcodes of the algorithm for 2D and 3D structures with exactanalytical sensitivities have been developed using the topologyoptimization codes presented by Andreassen et al. [1] for2D and Liu and Tovar [2] for 3D structures as starting points.Illustrative examples with Mathematica and Matlab codes arepresented to demonstrate the essence and viability of the proposedapproach and highlight the potential of the automaticsymbolic computation in structure optimization.

An improved Vietoris sine inequality

We prove that if {ak} is a sequence of positive numbers, such that (2j−1)j+12jja2j+1≤a2j≤2j−12ja2j−1 for all j=1,2,…, then for all n=1,2,…, x∈[0,π], ∑k=1naksin(kx)≥0. An example is {ak}={1,12,12,342,13,563,…}={1,0.5,0.707,0.530,0.577,0.481,…} where ak=1/(k+1)/2 for odd k, and (k−1)/(kk/2), for even k. This improves the well-known Vietoris sine inequality, by relaxing the requirement that {an} has to be a nonincreasing sequence.The proof is based on a Lukács-type inequality and a result on positive trigonometric sums with “convex” coefficients (both established recently by the authors), the classical Sturm Theorem on the number of real roots of a polynomial, and a well-known comparison principle. The symbolic manipulation software MAPLE is used amply for various computations.

Enumeration of substitutional isomers with restrictive mutual positions of ligands. II. Counts with restrictions on (sub)symmetry

The present paper extends our previous discussion of paper I on “Overall Counts”, still focusing on enumerations of substitutional isomers with restrictive positioning of ligands. But now, we address the counts of such isomers with a specified subsymmetry of the symmetry of the parent skeleton. Constrained analogs of Polya’s cycle index still appear, but now we introduce more powerful technical tools to include subsymmetry-specified generalizations of the cycle index. This involves differential-operator approach for analytically treating newly derived hybrids of the the generalized cycle index and suitable F-polynomials. As a simple illustration of the general mathematical exposition, a specific problems are solved and some tasks for possible further consideration are also stated, where again the Maple symbolic manipulation package proves useful.

Enumeration of Rosenberg-type hypercompositional structures defined by binary relations

Every binary relation ρ on a set H,(card(H)>1) can define a hypercomposition and thus endow H with a hypercompositional structure. In this paper, binary relations are represented by Boolean matrices. With their help, the hypercompositional structures (hypergroupoids, hypergroups, join hypergroups) that emerge with the use of the Rosenberg’s hyperoperation are characterized, constructed and enumerated using symbolic manipulation packages. Moreover, the hyperoperation given by x∘x={z∈H|(z,x)∈ρ} and x∘y=x∘x∪y∘y is introduced and connected to Rosenberg’s hyperoperation, which assigns to every (x,y) the set of all z such that either (x,z)∈ρ or (y,z)∈ρ.

Closed form ultimate strength of multi-rectangle reinforced concrete sections under axial load and biaxial bending

The analysis of prismatic members made of reinforced concrete under inclined bending, especially the computation of ultimate loads, is a pronounced non-linear problem which is frequently solved by discretizing the stress distribution in the cross-section using interpolation functions. In the approach described in the present contribution the exact analytical stress distribution is used instead. The obtained expressions are integrated by means of a symbolic manipulation package and automatically converted to optimized Fortran code. The direct problem-computation of ultimate internal forces given the position of the neutral axis-is first described. Subsequently, two kinds of inverse problem are treated: the computation of rupture envelops and the dimensioning of reinforcement, given design internal forces. An iterative Newton-Raphson procedure is used. Examples are presented.

Similarity: generalizations, applications and open problems

ThisSpecialIssueoftheJournalofEngineeringMathematicsfocusesonrecenttheoreticalandapplieddevelopmentsinSimilarity,includingasymptotic/numericalsimilarityandassociatedsoftware.Inparticular,itisdevotedtopapersbyleadingresearcherswhoapplysimilarityanalysistoengineeringandbiologicalproblems(asymptoticsinvolvingsimilaritysolutions),developsignificantextensionsofsimilaritymethodstofindandusesymmetriesandconserva-tion laws of partial differential equations (PDEs), develop and use numerical methods for PDEs based on admittedsymmetries and/or conservation laws, and develop symbolic manipulation software to implement similarity analy-ses.Asaconsequence,itishopedthatthesefourdistinctgroupsofsimilarityresearcherswillmorereadilybecomeawareofrecentdevelopments andneedsinrelatedfields.Inturn,thisshouldleadtoimportantandfruitfulresearchdirections. These papers arose from presentations at a conference on Similarity: Generalizations, Applications andOpen Problems in Vancouver, British Columbia, August 11–15, 2009, sponsored by PIMS (Pacific Institute for theMathematicalSciences)andAMSI(AustralianMathematicalSciencesInstitute).Thisintroductoryarticleincludesa brief summary of the papers as well as a reference to part of the discussion on open problems during a session ofthe conference.

On enumeration of hypergroups of order 3

In this paper we present a symbolic manipulation package that enumerates the hypergroups of order 3. It separates them into isomorphic classes and calculates their cardinality.

Hermite-Legendre-Gauss-Lobatto Direct Transcription in Trajectory Optimization

D IRECT methods have been widely applied for solving trajectory optimization problems [1–5]. Among the popular methods are the Hermite–Simpson method [6] and the Legendre pseudospectral (PS)method [7,8]. There has been some considerable interest in developing theory related to the Legendre PS method due to its high accuracy, although the Hermite–Simpson method continues to be applied to large-scale practical problems [9]. The main characteristic of the Hermite–Simpson method is the combination of reasonable accuracy with a highly sparse constraint Jacobian and Hessian matrix [10]. The PS method offers impressive accuracy (spectral accuracy) for smooth problems, but the constraint Jacobians are much denser than other methods. The sparsity of the constraint Jacobians can be increased in the PS method by using knots [11]. In addition to the Hermite–Simpson method, additional highorder methods have been proposed by Herman and Conway [2]. These methods are attractive from the point of view of accuracy, but in the framework proposed by Herman and Conway, they require detailed derivation when extended to arbitrary higher orders. For instance, in [2], the form of the constraints was derived via the symbolic manipulation software MAPLE. The purpose of this Note is to provide an alternative framework for arbitrary higher-order methods suitable for implementation on digital computers and in a reusable form. The optimal control problem is approximated by a discrete nonlinear programming problem (NLP) by expanding the state trajectories using local Hermite interpolating polynomials. For high accuracy, the collocation points are selected from the family of Gauss–Lobatto points. This also allows the integral performance index to be approximated viaGauss– Lobatto quadrature rules. For optimal control problems of the Bolza form, the natural choice of quadrature points are the Legendre– Gauss–Lobatto (LGL) points, because they are derived on the basis of a unity weight function, giving the highest accuracy for polynomial integrands. The generalization of the approach is referred to as the Hermite–Legendre–Gauss–Lobatto (HLGL) approach throughout the remainder of this Note.

On the improvement of a numerical method for solving high‐order non‐linear ordinary differential equations

AbstractThere have been many approaches to solve ordinary differential equations numerically. Even though many numerical methods can provide very good approximate solutions they need considerable calculation effort, often through iterations. The advance of symbolic manipulation packages such as Maple gives the opportunity for new approaches to this type of problems. This paper will discuss an improvement to one of these new approaches enabled by the availability of these packages, to obtain a numerical solution of an initial‐value problem governed by a high‐order non‐linear ordinary differential equation. The method we propose here is based on Zhu and Phan's earlier paper (Commun. Numer. Meth. Engng. 2003; 19:601–614) but we shall show an improved order of accuracy for the case of the highest‐order derivative being of an even order. In any case it provides the ability to further increase the order of accuracy under certain conditions of differentiability. Like Zhu and Phan's (Commun. Numer. Meth. Engng. 2003; 19:601–614) original method, an important feature of the new method is that no iterations are needed except in the preparatory phase for some cases. Copyright © 2006 John Wiley & Sons, Ltd.

Symbolic-numeric efficient solution of optimal control problems for multibody systems

This paper presents an efficient symbolic-numerical approach for generating and solving the boundary value problem-differential algebraic equation (BVP-DAE) originating from the variational form of the optimal control problem (OCP). This paper presents the method for the symbolic derivation, by means of symbolic manipulation software (Maple), of the equations of the OCP applied to a generic multibody system. The constrained problem is transformed into a nonconstrained problem, by means of the Lagrange multipliers and penalty functions. From the first variation of the nonconstrained problem a BVP-DAE is obtained, and the finite difference discretization yields a nonlinear systems. For the numerical solution of the nonlinear system a damped Newton scheme is used. The sparse and structured Jacobians is quickly inverted by exploiting the sparsity pattern in the solution strategy. The proposed method is implemented in an object oriented fashion, and coded in C++ language. Efficiency is ensured in core routines by using Lapack and Blas for linear algebra.

Complete Zonal Problem of the Artificial Satellite: Generic Compact Analytic First Order in Closed Form

This paper is a contribution to the Theory of the Artificial Satellite, within the frame of the Lie Transform as canonical perturbation technique (elimination of the short period terms). We consider the perturbation by any zonal harmonic J n (n ≥ 2) of the primary on the satellite, what we call here the complete zonal problem of the artificial satellite. This is quite useful for primaries with symmetry of revolution. We give an analytical formula to compute directly the first order averaged Hamiltonian. The computation is carried out in closed form for all terms, avoiding therefore tedious expansions in the eccentricity or in any anomaly; this feature makes the averaging process, not only valid for all kind of elliptic trajectories but at the same time it yields the averaged Hamiltonian in a very short and compact way. The formula allows us to now skip the averaging process, which means an asymptotic gain of a factor 3n/2 regarding the computational cost of the n th zonal. Our analytical formulae have been widely checked, by comparison on one hand with published works (Brouwer, 1959) (which contained results for particular zonal harmonics, let’s say typically from J 2 to J 8), and on the other hand with the results of 3 symbolic manipulation software, among which the MM (standing for ‘Moon’s series Manipulator’), which has already been used and described in (De Saedeleer B., 2004). Additionally, the first order generator associated with this transformation is given into the same closed form, and has also been validated.

Elastic–plastic response spectra for exponential blast loading

The design of structures subjected to loads due to explosions is often treated by means of elastic–plastic response spectra. Such spectra that are currently available in the literature were computed on the basis of triangular shape of blast pressure with respect to time. In the present paper, response spectra based on an exponential distribution of blast pressure, which is in better agreement to experimental data, are proposed. To that effect, analytical expressions of the solutions of the pertinent equations of motion have been obtained via symbolic manipulation software, and have been used to carry out an extensive parametric study. A comparison of the spectra obtained by the proposed approach to the existing ones, reveals that the commonly used assumption of triangular blast load evolution with time can sometimes be slightly unconservative, particularly for flexible structural systems, but can also be significantly overconservative for stiffer structures.

Study on the true and spurious eigensolutions of two-dimensional cavities using the dual multiple reciprocity method

In this paper, true and spurious eigensolutions for a circular cavity using the dual multiple reciprocity method (MRM) are analytically derived and numerically verified by the developed program. The roots of spurious eigenequation are found analytically by using symbolic manipulation software. A more efficient method is proposed by choosing a fewer number of equations from the dual MRM instead of all of the equations in the dual MRM. Numerical experiments are performed by using dual MRM program for comparison purposes. A circular cavity of radius 1 m with Neumann boundary conditions is considered, and the results match very well between the theoretical prediction and the numerical experiments for the first four true eigenvalues and the first two spurious eigenvalues. Also, a noncircular case of square cavity is numerically implemented. The true eigensolutions can be easily solved by the dual MRM program in conjunction with the singular value decomposition technique. At the same time, the boundary modes and the multiplicities of the true eigenvalues can also be determined.

On the uncontrollable damped triple inverted pendulum

In this paper, the controllability of the damped triple inverted pendulum is investigated. The work is concerned with the form of the cancelling pole and zero which appear in the transfer functions of an uncontrollable system, and follows on from earlier work on the damped double inverted pendulum. The investigation considers first the cases where only one of the three arm frictions is non-zero, and then explores the cases when two of the three arm frictions are non-zero. Due to the complexity of this problem, and the difficulties with the symbolic manipulation software, exploratory numerical investigations have been carried out to facilitate the symbolic investigations, all of which are reported here.