Farzin Shakib, Thomas J. R Hughes (auth.), James H. Kane,'s Solution of Superlarge Problems in Computational Mechanics PDF

By Farzin Shakib, Thomas J. R Hughes (auth.), James H. Kane, Arthur D. Carlson, Donald L. Cox (eds.)

ISBN-10: 1461278546

ISBN-13: 9781461278542

ISBN-10: 1461305357

ISBN-13: 9781461305354

There is a necessity to resolve difficulties in reliable and fluid mechanics that presently exceed the assets of present and foreseeable supercomputers. the problem revolves round the variety of levels of freedom of simultaneous equations that one must properly describe the matter, and the pc garage and pace boundaries which limit such options. The targets of tHis symposium have been to discover a number of the most modern paintings being performed in either and academia to resolve such super huge difficulties, and to supply a discussion board for the dialogue and prognostication of valuable destiny direc­ tions of either guy and computer. As evidenced during this complaints we think those targets have been met. Contained during this quantity are discussions of: iterative solvers, and their program to a number of difficulties, e.g. buildings, fluid dynamics, and structural acoustics; iterative dynamic substructuring and its use in structural acoustics; using the boundary point approach either by myself and along with the finite point approach; the appliance of finite distinction how to difficulties of incompressible, turbulent circulation; and algorithms amenable to concurrent computations and their purposes. in addition, discussions of present computational shortcomings from the large photograph perspective are awarded that come with innovations for destiny work.

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It is clear from equation (10) that for discretizations in Rd these products can be efficiently calculated in O(KNd+l) operations using sum-factorization methods [13]. Additional computational efficiency can be gained by exploiting the regular structure of the spectral (intra-element) operators and recognizing that the inner most kernel of the matrix-vector product, (Dip up;), is exactly equivalent to a matrix-matrix product which can be rapidly evaluated on many vector architectures. In addition to the 0 (K Nd+l) computational effort, the direct stiffness summation will require O(KN d - 1 ) operations, corresponding to the number of degrees-of-freedom lying on the element interfaces.

Numerical results clearly demonstrate the superiority of the iterative algorithm over implicit/direct and explicit strategies. The iterative algorithm, in its present form, is not applicable to completely general nonsymmetric systems, as we assume either the diagonal entries are positive, or the nodal block-diagonal submatrices admit Cholesky factorization. However, we conjecture these properties are possessed by many finite element discretizations of well-posed physical problems of interest. Current work is focused on reducing the data base of the unassembled element file.

The value of E that was used for checking convergence of eigenvectors as in Eq. (47) and following was 10- 9 . If the determinant of the 2 X 2 mass matrix was less than E, only a single vector was included in the reduced subspace for representing the eigenvector in question. If this determinant was less than two orders of magnitude smaller than EO, the included vector was a linear combination of the present and previous iterates, found by solving the associated 2 x 2 eigenvalue problem. If the determinant was more than two orders of magnitude smaller than EO, it was assumed that the present iterate was sufficiently accurate by itself.

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Solution of Superlarge Problems in Computational Mechanics by Farzin Shakib, Thomas J. R Hughes (auth.), James H. Kane, Arthur D. Carlson, Donald L. Cox (eds.)

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