Variationally consistent inertia templates for B-spline- and NURBS-based FEM: Inertia scaling and customization

Abstract

In this contribution, variationally consistent inertia templates for B-spline and NURBS-based finite elements are proposed as a unified concept for two different purposes: Customization of the template allows construction of masses and reciprocal masses with desired properties like higher-order accuracy or improved dispersion behavior;Inertia scaling allows substantial speed-up for explicit dynamics by increased critical time steps.The derivation of the template is based on a three-field parametrized functional as in previous works of the authors’ group, but with modified primary variables, namely displacement, velocity and mass-specific linear momentum. The latter allows for mass-preservation for non-constant density throughout the domain and is therefore an enhancement to the formulation proposed in Tkachuk and Bischoff (2015).With the focus on B-spline and NURBS-based finite elements, the proposed template provides alternatives to the row-sum-lumped mass matrix, which is only 2nd order accurate independent of the polynomial order. Earlier proposed algebraically constructed higher order masses from the literature can be reconstructed in the variational setting described here as special instances. Furthermore, higher-order reciprocal masses can be constructed from the template. They are especially attractive for explicit dynamics as there is no extra expense per time step compared with lumped mass for linear problems. For non-linear problems only small overhead is expected, but this paper focuses on linear problems only and mainly undistorted meshes. Tuning the method towards inertia scaling, a reduction of the maximum eigenfrequency by 25%–40% is obtained in the examples herein, whereas the accuracy is higher than for lumped or consistent mass.