Mikhail A. Dudaev

Irkutsk state transport university

The article proposes a modification to the classical minimum degree algorithm, designed to optimize the structure of the global coefficient matrix in the system of linear algebraic equations used in the finite element method. The aim is to improve the efficiency of the computational algorithm when solving contact problems in elasticity theory, specifically when modeling the connection of parts using special contact elements that determine the iterative solution with a gradual refinement of the state. The proposed modification is implemented in conjunction with the direct Cholesky method for solving systems of linear algebraic equations. The computational efficiency of this method is demonstrated to the greatest extent when applied to systems with a symmetric, positivedefinite coefficient matrix, which often has a sparse structure when using the finite element method. A significant problem in the Cholesky decomposition method is the filling of the "triangular multiplier" relative to the original matrix during the decomposition process. The reduction of this filling is typically achieved using well-known heuristic algorithms, among which the minimum degree algorithm has become widely used. The modification of the algorithm proposed in this work, by symmetric permutation of rows and columns of the initial global matrix, reorders the structure of this matrix in a way that, during the iterative solution of contact problems of elasticity theory, allows for partial Cholesky factorization to be performed in repeated iterations, thereby saving calculation time compared to performing a full factorization. The paper demonstrates that overall the matrix structure ordered by the modified algorithm is less preferable than that produced by the classical minimum-degree algorithm, since it is undergoing more filling. However, the efficiency of the computational method is achieved through repeated iterations that gradually specified the state of the contact finite elements of the structures coupling. Using a test finite element model of a rotor, as well as a model of the real aviation gas turbine engine, portraits of the initial matrices and their triangular multipliers are determined and presented, estimates of the number of arithmetic operations and the estimated time required for their execution are calculated using both a classical and a modified minimum degree algorithm.

finite element, finite element method, contact problem, Cholesky method, minimum degree algorithm, scalar products, filling, symmetric permutation