Keywords: high performance computing, interaction pantograph/catenary, static equilibrium equation, sparse linear algebra libraries.

During the recent last years, passengers transportation by railway has experimented a considerable increasing in some European countries (Germany, France, Spain, ...). For that reason, reaching higher velocities in railways has become a very important target. In that scenario, the pantograph/catenary system, with its dynamic behaviour, becomes a crucial component (see [1,3,4]), because at high speed it is very difficult to guarantee the permanent contact of pantograph head and contact wire. Moreover, it becames more difficult when noise and wear are considered.

In order to obtain an adequate behaviour in the pantograph/catenary system, it is necessary the existence of adequate conditions in the line, and this requires, among other aspects, a very precise mechanical calculus. Recent investigations have focused on dynamical behaviour by dynamical simulations in order to allow a better interaction of the pantograph and the catenary [4,2]; in this paper we will follow a more traditional approach, focusing in the catenary, modeled, as usual, by a set of coupled strings.

In fact, the best conditions in which the pantograph would obtain electric energy from the line are when the contact wire is parallel to the ground, Thus, an important problem is to determine the exact length of the droppers in order to allow the contact wire to acquire the correct shape. So, our objective is the development of a technique which allows us to implement a high precision calculation algorithm, and thus to develop a software tool to design high quality catenaries.

In this paper, a High Performance Computing Algorithm has been developed to obtain the solution of the static equilibrium equation of the pantograph/catenary system after an exhaustive study of the tradictionl mechanical approach based on a set of coupled strings. This approach is based on the folowing features:

• Making a mathematical model of the physical problem.
• Finding methods for solving the mathematical model. This point constitutes an study of the mathematical background in order to find what methods are available for the problem.
• Identifying the best method for solving the problem from a numerical point of view.
• Implementing the method of the previous step in a computer.

Sequential implementations of the resulting algorithm has been carried out by using standard libraries as BLAS [9] and SPARSKIT [10], in order to achieve good performance, portability, robutness and efficiency. A reduced execution time and low memory requirements allow to deal with more realistics (bigger) problems and, of course, to better solve the dynamic problem.

1

Poetsch G., J. Evans, R. Meisinger, W. Kortum, M. Baldauf, A. Veitl, & J. Wallaschek. "Pantograph/catenary dynamics and control", Vehicle System Dynamics, 28:159-195, 1997. doi:10.1080/00423119708969353

2

Carsten, N. J. "Nonlinear systems with discrete and continuous elements", PhD thesis, University of, 1997.

3

Poetsch G. and J. Wallaschek. "Symulating the dynamic behaviour of electrical lines for high-speed trains on parallel computers", International Symposium on Cable Dynamics, Liège, 1993.

4

Simeon, B. and Arnold M. "The simulation of pantograph and catenary: a PDAE approach", Technical Report 1990, Fachbereich Mathematik Technische Universitat Darmstadt, 1998.

5

Garfinkle, M. "Tracking pantograph for branchline electrification", Technical Report - School of Textiles and Materials Technology, University of Philadelphia, 1998.

6

K. J. Bathe. "Finite Element Procedures in Engineering Analysis", Ed. Prentice-Hall, 1996.

7

Zienkiewics, R. L. Taylor. "El Metodo de los Elementos Finitos", Ed. McGraw Hill, 1980.

8

Cook D. C., Malkus D. S., Plesha M. E. "Concepts and Applications of Finite Element Analysis", Ed. John Wiley and Sons, 1989.

9

J. J. Dongarra, J. Du Croz, I. S. Duff and S. Hammarling. "A set of Level 3 Basic Linear Algebra Subprograms", ACM Trans. Math. Soft, 1990.

10

Yousef Saad. "SPARSKIT: a basic tool kit for sparse matrix computations", University of Illinois and NASA Ames Research Center, 1994.

11

J. Benet, F. Cuartero and T. Rojo. "A tool to calculate catenaries in railways", Seventh International Conference on Computer in Railways, COMPRAIL-2000, 2000.

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