Dirichlet-boundary stabilization in space–time computational analysis: Derivation in 1D elastodynamics

In the space–time (ST) computational analysis, the discretization methods, such as those with finite elements, have appeared in a number of variations over the past years. Most of the current methods employ discontinuous functions in time. The motivation for that is to avoid a cost associated with fully 4D computations. Consequently, the increase in the number of unknowns in time remains under control despite using ST elements. Such methods have been applied over the past decades to a very large number of fluid dynamics and advection–diffusion problems, which are governed by equations with first-order time derivatives, and to a far lesser extent to elastodynamics and other solid mechanics problems, which are governed by equations with second-order time derivatives. In both contexts, especially when the functions are continuous in space, the ST methods are often formulated in the framework of stabilized methods, such as the Streamline-Upwind/Petrov-Galerkin, Galerkin/Least-Squares, Pressure-Stabilizing/Petrov-Galerkin, and variational multiscale methods. These are sometimes supplemented with discontinuity-capturing techniques. [Formula: see text]With the widespread adoption of isogeometric analysis (IGA), the ST methods have also been synthesized with IGA and applied to a large number of problems, mostly in fluid dynamics. The synthesis, ST-IGA, enables and encourages the use of higher-order functions in time. Within the ST domain, increasing the polynomial order in time leads to the expected improvement in solution accuracy. At the lateral boundaries with Dirichlet condition, however, the fluxes do not gain increased accuracy with higher-order polynomials in time. Motivated by this concern, we focus here on 1D elastodynamics and how to treat the Dirichlet boundaries. We introduce two new stabilization methods and show, with test computations, how the stabilized equations used in computing the fluxes perform. The methods are consistent in the way the fluxes are computed and can be extended to 2D and 3D problems.