Abstract
The application of the finite element method invariably involves the solution of large sparse systems of linear algebraic equations, and the solution of these systems often represents a significant or even dominant component of the total cost of applying the method. The object of this paper is to describe and relate various sparse matrix techniques which have been developed to make the solution of these equations more efficient.
Work supported in part by Canadian National Research Council Grant A8111.
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References
P.O. Araldsen, “The application of the superelement method in the analysis and design of ship structures and machinery components’, National Symposium of Computerized Structural Analysis and Design, George Washington University, March 1971.
J.H. Argyris and O.E. Bronlund, “The natural factor formulation of the stiffness for the matrix displacement method”, Computer Methods in Applied Mechanics and Engineering 5 (1975), pp. 97–119.
E. Cuthill and J. McKee, “Reducing the bandwidth of sparse symmetric matrices”, Proc. 24th Nat. Conf., Assoc. Comput. Mach., ACM Publ. P-69, 1969.
Carlos A. Felippa, “Solution of linear equations with skyline-stored symmetric matrix”, Computers and Structures, 5 (1975), pp. 13–29.
C. Felippa and R.W. Clough, “The finite element method in solid mechanics” pp.210–252 of Numerical Solution of Field Problems in Continuum Mechanics, (G. Birkhoff and R.S. Varga, eds.), SIAM-AMS Proceedings, American Math. Society, Providence, Rhode Island, 1970.
G. von Fuchs, J.R. Roy, and E. Schrem, “Hypermatrix solution of large sets of symmetric positive-definite linear equations”, Computer Methods in Applied Mechanics and Engineering 1 (1972), pp. 197–216.
Alan George, “Nested dissection of a regular finite element mesh”, SIAM J. Humer. Anal., 10 (1973), pp. 345–363.
Alan George and Joseph W.H. Liu, “An automatic partitioning and solution scheme for solving sparse positive definite systems of linear algebraic systems”, Trans. on Math. Software, to appear.
Alan George, “On block elimination for sparse linear systems”, SIAM J. Numer. Anal., 11 (1974), pp. 585–603.
Alan George, “Numerical experiments using dissection methods to solve n by n grid problems”, Research Report CS-75–07, University of Waterloo, March 1975.
Alan George and Joseph W.H. Liu, “A note on fill for sparse matrices”, SIAM J. Humer. Anal., 12 (1975), pp. 452–455.
N.E. Gibbs, W.G. Poole, and P.K. Stockmeyer, “An algorithm for reducing the bandwidth and profile of a sparse matrix”, SIAM J. Numer. Anal., to appear.
G. Hachtel, “The sparse tableau approach to finite element assembly”, Sparse Matrix Computations, Plenum Press, N.Y., 1976.
M.J.L. Hussey, R.W. Thatcher, and M.J.M. Bernal. “Construction and use of finite elements”, JIMA, 6 (1970), pp. 262–283.
B. Irons, “A frontal solution program for finite element analysis”, Internat. J. Humer. Meth. Engrg., 2 (1970), pp. 5–32.
A. Jennings, “A compact storage scheme for the solution of symmetric simultaneous equations”, Comput. J., 9 (1966), pp. 281–285.
A. Jennings and A.D. Tuff, “A direct method for the solution of large sparse symmetric simultaneous equations” in Large Sparse Sets of Linear Equations (J.K. Reid, editor ), Academic Press, 1971.
I.P. King, “An automatic reordering scheme for simultaneous equations derived from network problems”, Internat. J. Numer. Meth. Engrg., 2 (1970), pp. 523–533.
R.J. Melosh and R.M. Bamford, “Efficient solution of load deflection equations”, J. Struct. Div., ASCE, Proc. Paper No. 6510, (1969), pp. 661–676.
Christian Meyer, “Solution of linear equations–State-of-the-art”, J. of the Struct. Div., ASCE, Proc. Paper 9861, July 1973, pp. 1507–1527.
B. Speelpenning, “The generalized element method”, unpublished manuscript.
J.H. Wilkinson, The Algebraic Eigenvalue Problem, Clarendon Press, Oxford, England, 1965.
J.H. Wilkinson and C. Reinsch, Handbook for Automatic Computation, vol.Il, Linear Algebra, Springer Verlag, 1971.
F.W. Williams, “Comparison between sparse stiffness matrix and sub-structure methods”, Internat. J. Numer. Meth. Engrg., 5 (1973), pp. 383–394.
E.L. Wilson, K.L. Bathe, and W.P. Doherty, “Direct solution of large systems of linear equations”, Computers and Structures, 4 (1974), pp. 363–372.
C. Zienkfewicz, The Finite Element Method in Engineering Science, McGraw-Hill, London, 1971.
M. Tibial, “A finite element procedure for the second order of accuracy”, Humer. Math., 14 (1970), pp. 394–402.
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George, A. (1976). Sparse Matrix Aspects of the Finite Element Method. In: Glowinski, R., Lions, J.L. (eds) Computing Methods in Applied Sciences and Engineering. Lecture Notes in Economics and Mathematical Systems, vol 134. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-85972-4_1
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DOI: https://doi.org/10.1007/978-3-642-85972-4_1
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