Optimal reduction of numerical dispersion for wave propagation problems. Part 1: Application to 1-D isogeometric elements

Abstract

A numerical technique with the optimal coefficients of the stencil equation has been suggested. Based on this approach, new high-order isogeometric elements with the reduced dispersion error have been developed for wave propagation problems in the 1-D case. By the modification of the mass and stiffness matrices, the order of the dispersion error is improved from order 2p (the conventional elements) to order 4p (the new elements) where p is the order of the polynomial approximations. It was shown that the new approach yields the maximum order of the dispersion error for the stencil equations related to the high-order isogeometric elements. The analysis of the dispersion error of the high-order isogeometric elements with the lumped mass matrix has also shown that independent of the procedures for the calculation of the lumped mass matrix, the second order of the dispersion error cannot be improved with the conventional stiffness matrix. However, the dispersion error for the lumped mass matrix can be improved from the second order to order 2p by the modification of the stiffness matrix. The numerical examples confirm the computational efficiency of the new high-order isogeometric elements with reduced dispersion. We have also showed that numerical results obtained by the new and conventional high-order isogeometric elements may include spurious oscillations due to the dispersion error. These oscillations can be quantified and filtered by the two-stage time-integration technique recently developed in our papers. The approach developed in the paper can be directly applied to other space-discretization techniques with similar stencil equations.