title: Higher-order accurate and locking-free explicit dynamics in isogeometric structural analysis
Creators: name: Nguyen, Thi Hoa
Publisher: name: UNSPECIFIED
Year: 2023

description: Explicit structural dynamics codes simulating, for example, crash-tests and metal forming processes rely on the spectral properties of the chosen finite elements combined with locking-preventing mechanisms, such as reduced quadrature, to achieve higher-order spatial accuracy. To achieve highly efficient computations, these codes rely on three key ingredients: (1) low memory requirements; (2) an efficient solver; and (3) relatively large critical time step values. These three ingredients are present in contemporary linear finite element codes based on mass lumping, which, however, generally limits the spatial accuracy to second order. Overcoming this limitation to obtain a higher-order accurate and locking-free explicit scheme is the main objective of this work. We focus on isogeometric discretizations which are particularly attractive for higher-order accuracy due to their well-behaved spectral properties.

To this end, this thesis accomplishes the following tasks: (i) We propose to “measure”
locking by assessing the spectral accuracy of different finite element discretizations. (ii) We introduce a variational approach based on perturbed eigenvalue analysis for improving spectral properties of isogeometric multipatch discretizations. (iii) We develop an isogeometric Petrov-Galerkin formulation that enables higher-order spatial accuracy in explicit dynamics when the mass matrix is lumped, and (iv) we extend this approach to a Hellinger-Reissner mixed formulation, attempting to eliminate membrane locking for Kirchhoff-Love shells.

In the first task, we use eigenvalue and mode errors to assess five finite element formulations in terms of their locking-related efficiency: the displacement-based formulation with full and reduced integration, the B-bar, discrete strain gap, and Hellinger-Reissner methods. In the second task, we demonstrate that our approach allows for a much larger critical time step size in explicit dynamics calculations, which does not depend on the polynomial degree of spline basis functions. In the third task, we discretize the test functions using the so-called “approximate” dual functions that are smooth, have local support, and satisfy approximate bi-orthogonality with respect to a trial space of B-splines. This enables higher-order accurate mass lumping using the standard row-sum technique. In the last task, to increase efficiency, we integrate a boundary treatment with built-in Dirichlet boundary constraints, a strong outlier removal approach to increase the critical time step size, and a reduced quadrature rule with a minimal number of quadrature points. We numerically demonstrate, via spectral analysis and convergence studies of beam, plate, and shell models, that our Petrov-Galerkin approach leads to higher-order accurate and locking-free computations in explicit dynamics. In addition, we extend the horizon of this work by exploring the application of isogeometric analysis, together with the outlier removal approach, to nonlinear dynamics of shear- and torsion-free rods, combining with a robust implicit time integration scheme.