A Method for Treating Temporal and Spatial High Gradient Phenomena in the Dynamics of Solids

Abstract eng:
Recently the development of variational integrators (see, e.g., [1, 2]) has offered additional insight into implicit long-duration time integration with excellent energy and momentum conserving properties. When the lower modes dominate the transient responses, these and previously reported implicit integration schemes are deemed adequate for practical applications. There remains a challenge in the short duration responses characterized by discontinuous waves propagating through heterogeneous solids. For most large-scale problems the high-frequency (and discontinuous) zones are often interlaced with low- frequency zones, for which several explicit-implicit algorithms have been proposed (see, e.g., [3]). In the time integration of multi-scale and multiphysics problems, the spatial and scale- level interfaces often characterized with discontinuities. To address such interface heterogeneities in the dynamics of solids, a variational construction of time-discontinuous, space-discontinuous (TDSC) transient analysis algorithms was recently developed for the simulation of solid mechanics problems that involve high temporal and spatial gradients/discontinuities[4]. The TDSC algorithm was developed by employing Hamilton’s action integral with two parameters whose choices can be tailored to problem characteristics on hand. The addition of space-discontinuity is realized by ab initio embedding of spatial jumps in the form of stress jumps and/or strain jumps. Numerical experiments focused on one-dimensional stress wave propagations indicate that the TDSC-VI method offers moderate to substantial improvements over existing explicit algorithms, especially the central difference method when the Courant number deviates appreciably from unity. In the present paper, we bring in the desirable features of previous work on explicit integration methods [5, 6] and infuse them into the time-discontinuous, space-discontinuous (TDSC) variational integrator. In doing so, the paper presents several schemes to accommodate the stress and strain jumps. The performance of various spatial jumps is assessed for one-dimensional wave propagation of heterogeneous bars.

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