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PROCEEDINGS OF THE EIGHTH INTERNATIONAL CONFERENCE ON COMPUTATIONAL STRUCTURES TECHNOLOGY
Edited by: B.H.V. Topping, G. Montero and R. Montenegro
A High Performance Algorithm to solve the Static Stiffness Problem of a Catenary
A. Alberto1, E. Arias2, D. Cebrian1, T. Rojo2, F. Cuartero2 and J. Benet2
1Albacete Computer Science Research Institute, Spain
A. Alberto, E. Arias, D. Cebrian, T. Rojo, F. Cuartero, J. Benet, "A High Performance Algorithm to solve the Static Stiffness Problem of a Catenary", in B.H.V. Topping, G. Montero, R. Montenegro, (Editors), "Proceedings of the Eighth International Conference on Computational Structures Technology", Civil-Comp Press, Stirlingshire, UK, Paper 174, 2006. doi:10.4203/ccp.83.174
Keywords: structure catenary, static stiffness problem, interaction pantograph-catenary, equilibrium equation, high performance computing, software tool.
In this paper, we will present a high performance implementation that has been carried out on various algorithms to solve the equilibrium equation of the system in a catenary model, using spans with different numbers of droppers and different positions. The implementations have been performed both for the continued and stitched structure of the catenary.
In the static problem the static position of the system is determined when we apply a vertical force over the contact wire. Thus, it is necessary to configurate the stiffness matrix and the independent term of the system and then solving the lineal system for the position of the nodes of the wire, and can thus obtain the balance position.
Once the equation that determines the position of every node forming the structure of the catenary has been solved, it is necessary to tackle another study that will produce an iterative process. By taking a point inside the first span we exercise a vertical force on the contact wire, so that we must solve the equilibrium equation again, to obtain the new position of every node inside the structure of the catenary. This process is iterated for several points inside every span. Once the study has been performed inside the first span, the same process has to be repeated for each of the remainig spans.
Most of the execution time is spent in solving this problem. Thus, it is important to bring about a reduction related to the storage requirements, such as in the execution time.
We have used iterative methods with preconditioning for the resolution of this system . It has been chosen both from the precision and from the time of calculation. A second advantage of these implementations can be taken into account, which is the considerable saving of storage memory requirements obtained.
The stiffness matrix , has a high sparsity degree. In the scientific literature we can found a lot of storage schemes and methods to deal with sparse matrices. SPARKIT  is a software packet witch allows us to work with different storage schemes (COO, CSR, CSC), in particular we have used the COO format because it is simple and ideal for this algorithm, and CSR formats because it allows us to operate easily with sparse matrices (matrix*vector, methods for solving sparse systems of equations). In addition to the SPARTKIT library, some routines of the linear algebra library BLAS  have been used to achieved the matrix-vector products. In order to calculate the computational improvement we have realized a set of study cases.
The original algorithms stored the stiffness matrix in dense format, thus, they had large memory storage requirements. The high performance algorithms store the same matrix but in sparse format, and this reduces the memory requirements, and benefits the efficiency of the algorithms. The execution time has also been drastically reduced compared to the original implementation. All these algorithms have been used as part of a user-friendly, interactive and graphically oriented software tool, wich is currently being used in the industry.
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