Show description

8) i=1 m ¯ eξ are weighting coefficients and N ˜ e is the number of nodes for where D αi defining the DQ discretization. Four transition conditions at an inter-element boundary of two adjacent elements are also necessary for constructing the overall discrete equation system. These four transition conditions are the continuities of displacements and displacement gradients, and the equilibriums of moments and transverse forces at the inter-element boundary of two adjacent elements i and i + 1. 9) where N i is the right boundary node of element i.

0 0 1 ¯ξ D 22 0 . . 0 0 ⎤ 0 0 0 ... ¯ ξ ... D ¯ξ ¯ξ ⎥ D 23 2(N −1) D2N ⎥ ... 1 0 0 ⎥ ⎥ . . ⎥ ⎥ . . ⎥ ⎥ . . ⎥ ⎥ 1 0 ⎦ 0 ... 0 1 0 ... 117) Using [T ] in Eq. 113), the weighting coefficients for the C 1 −C 0 EDQ model ˆ =N ¯ − 1. For applying the C 1 − C 0 EDQ model to can be obtained with N the transient structural analysis, the physical first order derivative at the first EDQ element boundary node is a discrete EDQ parameter. Consequently, the elements of the second column of the related EDQ weighting coefficient matrix ξm need to be multiplied by l.

E) Sinc functions Sinc functions are suitable for dealing problems with singularity property or infinite domain. Assume that the range of ξ is 0 ≤ ξ ≤ 1 and that the discrete points are equally spaced. 89) sα = (ξ − αh) h with h = N 1−1 . 90) Consider that only one DOF representing the variable function at the discrete point is used to define the DQ. Then sinc functions can be interpolation functions used to define the DQ weighting coefficients. Using Eqs. 90) in Eq. 1) weighting coefficients in explicit form can be obtained.