Optimal reduction of numerical dispersion for wave propagation problems. Part 2: Application to 2-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 2 p (the conventional elements) to order 4 p (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 2 p by the modification of the stiffness matrix. The numerical examples confirm the computational efficiency o f t he n ew h igh-order i sogeometric e lements w ith r educed d ispersion. We h ave a lso s howed t hat n umerical 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.

Explicit dynamics in impact simulation using a NURBS contact interface

“…This naturally provides a diagonal mass matrix. Using an isogeometric discretization along with explicit time integration, several techniques have been proposed like collocation Auricchio et al or lumping procedures . However, using higher‐order isogeometric spatial discretizations a lumping procedure for the mass matrix gives only a limited accuracy, thus a modification of both the stiffness and mass matrices is required.…”

“…Using an isogeometric discretization along with explicit time integration, several techniques have been proposed like collocation Auricchio et al 17 or lumping procedures. [18][19][20] However, using higher-order isogeometric spatial discretizations a lumping procedure for the mass matrix gives only a limited accuracy, thus a modification of both the stiffness and mass matrices is required. Finally, for the wave propagation problem, the interaction between the spatial and time discretizations needs to be considered.…”

Improvement of accuracy of the spectral element method for elastic wave computation using modified numerical integration operators

“…Ainsworth and Wajid [18] analytically found the optimal value of the blending ratio such that the numerical dispersion of the SEM is minimized. Note also that their ideas have been applied to the isogeometric analysis method, which is wellsuitable for structures with smooth curved surfaces [19][20][21][22][23]. While the above studies are for computation of the Helmholtz equation, an extension to the Maxwell equations is given by [24].…”

Optimal Reduction of Numerical Dispersion for Wave Propagation Problems. Application to Isogeometric Elements.

“…More numerical examples in the 1-D and 2-D cases solved by the new isogeometric elements can be found in our papers [12,13].…”

“…This order is optimal and cannot be further improved. More detailed derivations of the new technique for quadratic and cubic isogeometric elements in the 1-D and 2-D cases as well as the corresponding numerical examples can be found in our papers [12,13].…”