The structure of weighting coefficient matrices of harmonic differential quadrature and its applications

Abstract

KEY WORDS: harmonic differential quadrature method, matrix, differential quadrature, numerical method, centrosymmetric matrix. SUMMARY The structure of weighting coefficient matrices of Harmonic Differential Quadrature (HDQ) is found to be either centrosymmetric or skew centrosymmetric depending on the order of the corresponding derivatives. The properties of both matrices are briefly discussed in this paper. It is noted that the computational effort of the harmonic quadrature for some problems can be further reduced up to 75 per cent by using the properties of the above-mentioned matrices.

An iterative algorithm for a least squares solution of a matrix equation

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A Note on the Dq Analysis of Anisotropic Plates

“…The weighting coefficient matrices in the DQ and HDQ methods were proven to be either centrosymmetric matrix for the derivatives of even order or skew centrosymmetric matrix for the derivatives of odd order if a grid spacing is symmetric [3,4]. This is often seen in many situations such as equally spaced grid points or zeros of the Chebyshev and the Legendre polynomials.…”

“…This is often seen in many situations such as equally spaced grid points or zeros of the Chebyshev and the Legendre polynomials. In the following we establish the notations on the centrosymmetric and skew centrosymmetric matrices first [3,4,6]. The centrosymmetric and skew centrosymmetric matrices can be factorized into two smaller sub matrices in the calculation of their determinant, inverse and eigenmodes.…”

“…Therefore, the respective computing effort and storage requirements can be reduced by 75 percent and 50 percent, respectively. The details on centrosymmetric and skew centrosymmetric matrices can be found in references [3,4,6].…”

“…According to Lemmas 2 and 3, the resulting coefficient matrix in the formulation equation (6) is a centrosymmetric matrix. Therefore, the reduction algorithm based on the factorization properties of centrosymmetric matrix [3,4,6] is applicable for the cases, namely, the computing effort and storage requirements are reduced to 75 percent and 50 percent as much as that in references [1,2].…”