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

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.Comment: slightly condensed the manuscript, added references, numerical results unchange

Implementation of Space-Time Finite Element Formulation in Elastodynamics

Elastodynamics is an academic field that is involved in solving problems related to the field of wave propagation in continuous solid medium. Finite element methods have long been an accepted way of solving elastodynamics problems in the spatial dimension. Considerable thought has been given to ways of implementing finite element discretization in the temporal dimension as well. A particular method of finite element solving called space-time finite element formulation is explored in this thesis, which is a relatively recent technique for discretization in spatial and temporal dimensions. The present thesis explores the implementation of the Space-Time finite element formulation in solving classical elastodynamics examples, such as the mass-on-spring for a single degree of freedom and for an axially vibrating bar with multiple degrees of freedom. The space-time formulation is compared with existing finite difference techniques, such as the central difference method, for computational expenditure and accuracy. In the mass-on-spring case, the central difference method and linear time finite elements yield relatively similar results, whereas quadratic time finite elements are more accurate but take more time computationally. In the axially vibrating bar case, central difference is computationally more efficient than the Space- Time finite element method. The final section concludes our findings and critiques the numerical effectiveness of the space-time finite element formulation

Structure-preserving space-time discretization in a mixed framework for multi-field problems in large strain elasticity

The present work deals with the design of structure-preserving numerical methods in the field of nonlinear elastodynamics with an extension to multi-field problems. A new approach to the design of energy-momentum (EM) consistent time-stepping schemes for nonlinear elastodynamics is proposed. Moreover, we extend the formalism to multi-field problems