Abstract
In this contribution, we present a space-time formulation of the Newmark integration scheme for linear damped structures under both harmonic and transient excitations. The incremental set of equations of motion and the Newmark approximations are transformed into their corresponding space-time equivalents. The dynamic system is then represented by one algebraic space-time equation only. This equation is projected into a coupled pair of space-time equations, which is solved via the fixed point algorithm. The solution is iteratively assembled by enrichments, each of which is decomposed by a dyadic product of spatial and temporal enrichment vectors. The evolution of the spatial enrichment vectors is investigated during convergence and interpreted by comparing them to the set of linear modes of vibration. The new method is demonstrated by means of four numerical examples, presenting not only the excellent convergence behavior and the numerical efficiency but also the limits of the proposed approach.
Highlights
Evaluating the response time history of engineering structures subjected to transient excitation has become a key issue in structural dynamics
Space-time finite elements have been applied to contact mechanics [26,27,28], multiscale modeling [29,30], landslide dynamics [31], problems including fluid-structure interaction [32] and modeling of viscoelastic materials [33,34,35]
We present a new space-time solution strategy in structural dynamics
Summary
Evaluating the response time history of engineering structures subjected to transient excitation has become a key issue in structural dynamics. The wide range of conventional methods includes analytic strategies for linear systems [1] and step-by-step time integration schemes [2,3]. With emphasis on computational efficiency, several approximation schemes and model order reduction strategies have been proposed in the literature [4,5,6,7]. As a statistical pattern technique in incremental structural dynamics, the proper orthogonal decomposition is used to find the dominant motion patterns for an optimal low-order description of linear and nonlinear systems [8]. The transformation into low-order subspaces depends on the set of proper orthogonal modes that are obtained from parts of
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