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Computer Algebra for Applied Physics
Schedule of the special session at the 23rd Conference on Applications of Computer Algebra (ACA 2017), July 17-21 2017
|AP-01||Edgardo S. Cheb-Terrab||Physics, Differential Equations and Mathematical Functions Maplesoft, Canada|
Editor, Computer Physics Communications
|Computer Algebra in Theoretical Physics|
Generally speaking, physicists still experience that computing with paper and pencil is in most cases simpler than computing on a Computer Algebra worksheet. On the other hand, recent developments in the Maple system have implemented most of the mathematical objects and mathematics in theoretical physics computations, and have dramatically approximated the notation used in the computer to the one used with paper and pencil, diminishing the learning gap and computer-syntax distraction to a strict minimum. In this talk, the Physics project at Maplesoft is presented and the resulting Physics package is illustrated by tackling problems in classical and quantum mechanics, using tensor and Dirac’s Bra-Ket notation, general relativity, including the equivalence problem, and classical field theory, deriving field equations using variational principles.
 L.D. Landau and E.M, Lifshitz, Course of Theoretical Physics, Elsevier (1975).
 E.S. Cheb-Terrab, Mini-Course: Computer Algebra for Physicists. Mapleprimes
|AP-02||Alexander Prokopenya||Warshaw University||Sliding of a Block on the Plane with Variable Friction Coefficient: Simulation with Mathematica|
Dry friction of solids is often encountered both in engineering practice and in
our everyday life. Its study has a long history and many different models were
proposed to explain its physical properties (see [1, 2]). In spite of a complexity of
the dry friction as a physical phenomenon, its basic laws are known since the works of Amontons and Coulomb (see [4, 3]), and they give simple quantitative estimates of the friction forces which are widely used in engineering applications. Remind that a body sliding on a rough surface is acted on by a friction force that is parallel to the surface and is directed opposite to the velocity of the body. According to the Amontons-Coulomb law, the friction force does not depend on the area of contact of the body and the surface and is proportional to the normal reaction force NFf r = mN : (1)
The coefficient of friction m is usually assumed to be constant during motion of the body. In the case when the body contacts the surface in one or two points one can easily obtain the equations of motion of the system because the points of application of the friction forces and the normal forces are known and the law (1) can be used directly. But in case of a finite dimension of the contact area the normal force N is inevitably a distributed force. It does not essential matter if the body slides on the surface with constant coefficient of friction but may become very important when the body crosses a boundary of two domains with different coefficients of friction. Then the normal force and the friction force become dependent of position of the body at the surface and this complicates the equations of motion considerably. To analyze motion of the body in such a case one has to choose a model for the normal force distribution and to combine symbolic and numerical calculations for solving the equations of motion. However, the problem of calculations can be efficiently solved with modern computer algebra systems, for example, the system Mathematica (see ) that is used in the present talk.
|AP-03||Jose Antonio Valleijo|
University of Mexico
|Symbolic computation of normal forms for Hamiltonian perturbed systems||Two of the main goals in the theory of dynamical systems are the determination of possible closed, stable orbits, and the computation of adiabatic invariants (of course, taking for granted the impossibility of solving key equations explicitly). Of particular interest is the case in which the Hamiltonian H is a perturbation of an integrable one. A widely used procedure to study it, consists in writing the Hamiltonian in the so-called normal form, that is, as a formal series. In this talk we will describe our use of symbolic computation for these normal forms.|
Lev Academic Center,
|Singular Perturbed Vector Fields (SPVF) Applied to Combustion of Spray of Diesel Droplets||In our research we present the concept of singular perturbed vector field method and its application to thermal explosion of diesel spray combustion. Given a system of governing equations, which consist of hidden Multiscale variables, the SPVF method transfer and decompose such system to fast and slow singularly perturbed systems (SPS). The resulting sub-system enable us to understand better the complex system, and simplify the calculations. Later powerful analytical, numerical and asymptotic methods can be applied to each subsystem. In this paper we compare the results obtained by the methods of integral invariant manifold and SPVFM applied to spray (polydisperse) droplets combustion model.|
|AP-05||Avi Karsenty and Yaakov Mandelbaum|
Lev Academic Center,
|Computer algebra in nanotechnology: Modelling of Nano Electro-Optic Devices using Finite Element Method (FEM)|
We will discuss the simulation of Silicon-based light-emitting and photodetectors nano-devices using computer algebra. These devices couple the hyperbolic equations of Electromagnetic Radiation, the parabolic equations of Heat Conduction, the elliptic equations describing electric potential, and the eigenvalue equations of Quantum Mechanics – with the nonlinear drift-diffusion equations of the semiconductor physics. These must be solved subject to generally mixed Dirichlet-Neumann boundary conditions in three-dimensional geometries.
Comsol Multiphysics modelling software is employed integrated with Matlab-Simulink and Zemax. The physical equations are discretized on a mesh using the Galerkin Finite Element Method (FEM), and to a lesser extent the method of Finite Volumes (FVM). The equations can be implemented in a variety of forms such as directly as a PDE, or as variational integral, the so called weak form. Boundary conditions may also be imposed directly or using variational constraint and reaction forces. Both choices have implication for convergence and physicality of the solution. The mesh is assembled from triangular or quadrilateral elements in two-dimensions, and hexahedral or prismatic elements in three dimensions, using a variety of algorithms. Solution is achieved using direct or iterative linear solvers and non-linear solvers. The former are based on conjugate gradients, the latter generally on Newton-Raphson iterations.
The general framework of FEM discretization, meshing and solver algorithms will be presented together with techniques for dealing with challenges such as multiple time scales, shocks and non-convergence; these include load-ramping, segregated iterations, and adaptive meshing.
O. Gleisner ,
Lev Academic Center
Ben Gurion University
Algebraic Processing of Sequential Fluoroscopy Images for Quantitative Evaluation of Partial Obstruction of the Upper Urinary Tract
To develop a novel method for the quantitative evaluation of partial obstruction of the upper urinary tract in
patients who have undergone percutaneous nephrolithotomy (PCNL). For this purpose, sequential fluoroscopic
images obtained during a postoperative nephrostogram were processed in order to calculate the residual amount of contrast material in the renal collecting system and evaluate the urine flow rate.
Lev Academic Center,
|Computer Algebra in Satellite Imaging||Computer algebra is ubiquitously used in satellite imaging and in particular in the exploitation of satellite images. A couple of examples developed in our Remote Sensing Laboratory are given, one related to the automatic atmospheric correction of images, the other related to the deconvolution of images to improve the image exploitation process.|
On the Applicability of
Method in Astronomy:
Influence of the Noise
Small number of objects poses often a problem in the analysis of large-scale structure of the Universe,
especially if one is interested in studying fractal structures – estimating the fractal dimension or similar
characteristics. So, pairwise separations method that uses not coordinates of objects (n sets of coordinates for n
objects) but pairwise distances (n(n-1)=2 distances) looks very attractive. We studied the applicability of the
pairwise separations method in astronomy. Description of the method and some applications of it in astronomy
can be found in  and .
This method may be used, in particular to analyze fractal sets: for a fractal set with Hausdorff-Bezicovich
dimension D, the distribution of pairwise distances f (l) behaves asymptotically as f (l) α lD-1 for small l. Since
large enough data set is needed to estimate the fractal dimension, using this method looks promising, especially
in the case when using a small sample of data - as pairwise separations method indicates, pairwise distances are
used rather than points; thus, dealing with n(n-1)=2 distances as compared to n original data points.
In , the authors made simulations to estimate applicability of the method, however, large noiseless data sets
for experiments were used. Here, we use more realistic data for simulations. Iterated function systems (IFS, see,
e.g., ) were used to generate model fractal sets, then noise was added to the data. Estimates of fractal
dimension using pairwise-separations method were conducted where results
were compared with the dimension of the attractor of the IFS and with estimates of the box-counting
dimension. In the simulations, classic 2D fractals – Sierpinsky carpet and Sierpinsky gasket as well as 3D fractals
of the Menger Sponge family were used. These simulations were executed using computer algebra system
Wolfram Mathematica 11 to generate fractal sets and estimate dimension of these sets
using pairwise separations method. To test applicability of the method in practice, noise to the data was added
in order to evaluate how it affects the results. A series of simulations were also done without noise to test the
influence of the sample size.