C1continuous spacetime discretization based on Hamilton's law of varying action
Abstract
We develop a class of C1continuous 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 C1continuity for the displacement field and C0continuity for the velocity field. From the discrete virtual action we finally construct a class of onestep 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 onedimensional 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 p2scheme  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 longterm energy behavior.
 Publication:

arXiv eprints
 Pub Date:
 October 2015
 arXiv:
 arXiv:1510.07863
 Bibcode:
 2015arXiv151007863M
 Keywords:

 Mathematics  Numerical Analysis;
 Computer Science  Computational Engineering;
 Finance;
 and Science
 EPrint:
 slightly condensed the manuscript, added references, numerical results unchanged