Approximate time integration of the finite element semi-discrete solution in structural dynamics

Explicit time integration is the gold standard for simulating the deformation of complex bodies, sometimes made up of hundreds or more trimmed patches. Its efficiency stems from mass lumping techniques that substitute the consistent mass with a (usually diagonal) approximation. Although the impact of mass lumping has been well studied for classical finite element methods based on the Lagrange basis, it remains poorly understood when combined with newer discretization techniques employing more general non-interpolatory bases, such as isogeometric analysis. This thesis partly addresses this shortcoming. Firstly, some special and practically relevant properties of lumped mass matrices are proved and later extended to (block-)banded and Kronecker product matrices for isogeometric discretizations on single-patch, multi-patch, and trimmed geometries. Secondly, we explore the possibility of constructing interpolatory spline bases for improving the accuracy of mass lumping techniques in isogeometric analysis. Although reminiscent of the spectral element method, this strategy comes with its lot of surprises and challenges, which are critically assessed.

Apart from speeding up the solution process, mass lumping techniques are also praised for increasing the critical time step of explicit time integration methods. In classical finite element analysis, the critical time step is constrained by the so-called "optical branches", representing the inaccurate high-frequency part of the spectrum. In contrast, smooth isogeometric analysis features far fewer inaccurate frequencies, which are mostly limited to a few so-called "outliers". Removing those outliers is paramount in explicit dynamics and solution techniques are often discretization-dependent. Rigorous outlier removal techniques for structured finite element discretizations give way to heuristic mass scaling techniques for unstructured ones. While the field has been flourishing over the years, it still lacks a strong theoretical basis and mostly relies on numerical experiments as the only means of assessment. This thesis thoroughly reviews existing mass scaling methods and connects them to established linear algebra results to derive rigorous eigenvalue bounds and condition number estimates. Our results cover some of the most successful instances of mass scaling, unraveling for the first time well-known numerical observations.

Outlier frequencies, however, generally have multiple origins. The high-order nature of certain partial differential equations and the slenderness of structural elements all impose a stringent constraint on the critical time step, and, unfortunately, badly cut elements in immersed finite element discretizations further aggravate the issue. For smooth isogeometric discretizations, Leidinger [1] first showed that mass lumping removed the dependency of the critical time step on the size of trimmed elements. This finding has attracted considerable attention but has unfortunately overshad