C1‐continuous space‐time discretization based on Hamilton's law of varying action

Abstract

We develop a class of C1‐continuous time integration methods that are applicable to conservative problems in elastodynamics. These methods are based on Hamilton's law of varying action. From the action of the continuous system we derive a spatially and temporally weak form of the governing equilibrium equations. This expression is first discretized in space, considering standard finite elements. The resulting system is then discretized in time, approximating the displacement by piecewise cubic Hermite shape functions. Within the time domain we thus achieve C1‐continuity for the displacement field and C0‐continuity for the velocity field. From the discrete virtual action we finally construct a class of one‐step schemes. These methods are examined both analytically and numerically. Here, we study both linear and nonlinear systems as well as inherently continuous and discrete structures. In the numerical examples we focus on one‐dimensional applications. The provided theory, however, is general and valid also for problems in 2D or 3D. We show that the most favorable candidate — denoted as p2‐scheme — converges with order four. Thus, especially if high accuracy of the numerical solution is required, this scheme can be more efficient than methods of lower order. It further exhibits, for linear simple problems, properties similar to variational integrators, such as symplecticity. While it remains to be investigated whether symplecticity holds for arbitrary systems, all our numerical results show an excellent long‐term energy behavior.