Optimal reduction of numerical dispersion for wave propagation problems. Part 2: Application to 2-D isogeometric elements

Abstract

Based on the optimal coefficients of the stencil equation, a numerical technique for the reduction of the numerical dispersion error has been suggested. New isogeometric elements with the reduced numerical dispersion error for wave propagation problems in the 2-D case have been developed with the suggested approach. By the minimization of the order of the dispersion error of the stencil equation, the order of the dispersion error is improved from order 2p (the conventional isogeometric elements) to order 4p (the isogeometric elements with reduced dispersion) where p is the order of the polynomial approximations. Because all coefficients of the stencil equation are obtained from the minimization procedure, the obtained accuracy is maximum possible. The corresponding elemental mass and stiffness matrices of the isogeometric elements with reduced dispersion are calculated with help of the optimal coefficients of the stencil equation. The analysis of the dispersion error of the 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 with 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 isogeometric elements with reduced dispersion. The numerical results obtained by the new and conventional 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 developed recently. The approach developed in the paper can be directly applied to other space-discretization techniques with similar stencil equations.