Space-time variational material modeling

a new paradigm demonstrated for thermo-mechanically coupled wave propagation, visco-elasticity, elasto-plasticity with hardening, and gradient-enhanced damage

authored by: Philipp Junker, Thomas Wick
Abstract: We formulate variational material modeling in a space-time context. The starting point is the description of the space-time cylinder and the definition of a thermodynamically consistent Hamilton functional which accounts for all boundary conditions on the cylinder surface. From the mechanical perspective, the Hamilton principle then yields thermo-mechanically coupled models by evaluation of the stationarity conditions for all thermodynamic state variables which are displacements, internal variables, and temperature. Exemplary, we investigate in this contribution elastic wave propagation, visco-elasticity, elasto-plasticity with hardening, and gradient-enhanced damage. Therein, one key novel aspect are initial and end time velocity conditions for the wave equation, replacing classical initial conditions for the displacements and the velocities. The motivation is intensively discussed and illustrated with the help of a prototype numerical simulation. From the mathematical perspective, the space-time formulations are formulated within suitable function spaces and convex sets. The unified presentation merges engineering and applied mathematics due to their mutual interactions. Specifically, the chosen models are of high interest in many state-of-the art developments in modeling and we show the impact of this holistic physical description on space-time Galerkin finite element discretization schemes. Finally, we study a specific discrete realization and show that the resulting system using initial and end time conditions is well-posed.
Organisation(s): PhoenixD: Photonics, Optics, and Engineering - Innovation Across Disciplines
Institute of Continuum Mechanics
Institute of Applied Mathematics
Type: Article
Journal: Computational mechanics
Volume: 73
Pages: 365-402
No. of pages: 38
ISSN: 0178-7675
Publication date: 02.2024
Publication status: Published
Peer reviewed: Yes
ASJC Scopus subject areas: Computational Mechanics, Ocean Engineering, Mechanical Engineering, Computational Theory and Mathematics, Computational Mathematics, Applied Mathematics
Electronic version(s): https://doi.org/10.1007/s00466-023-02371-2 (Access: Open)
: Details in the research portal "Research@Leibniz University"

BibTeX

@article{590c610e5ea04cb7b07e5ff056a34e69,
title = "Space-time variational material modeling: a new paradigm demonstrated for thermo-mechanically coupled wave propagation, visco-elasticity, elasto-plasticity with hardening, and gradient-enhanced damage",
abstract = "We formulate variational material modeling in a space-time context. The starting point is the description of the space-time cylinder and the definition of a thermodynamically consistent Hamilton functional which accounts for all boundary conditions on the cylinder surface. From the mechanical perspective, the Hamilton principle then yields thermo-mechanically coupled models by evaluation of the stationarity conditions for all thermodynamic state variables which are displacements, internal variables, and temperature. Exemplary, we investigate in this contribution elastic wave propagation, visco-elasticity, elasto-plasticity with hardening, and gradient-enhanced damage. Therein, one key novel aspect are initial and end time velocity conditions for the wave equation, replacing classical initial conditions for the displacements and the velocities. The motivation is intensively discussed and illustrated with the help of a prototype numerical simulation. From the mathematical perspective, the space-time formulations are formulated within suitable function spaces and convex sets. The unified presentation merges engineering and applied mathematics due to their mutual interactions. Specifically, the chosen models are of high interest in many state-of-the art developments in modeling and we show the impact of this holistic physical description on space-time Galerkin finite element discretization schemes. Finally, we study a specific discrete realization and show that the resulting system using initial and end time conditions is well-posed.",
keywords = "Hamilton principle, Space-time formulation, Variational material modeling",
author = "Philipp Junker and Thomas Wick",
note = "Funding Information: This work is supported by the Deutsche Forschungsgemeinschaft (DFG) under Germany{\textquoteright}s Excellence Strategy within the cluster of Excellence PhoenixD (EXC 2122, Project ID 390833453). Moreover, we acknowledge the International Research Training Group 2657 (Deutsche Forschungsgemeinschaft (DFG) - Projektnummer 433082294) in which both authors and their groups investigate modern modeling and discretization space-time methods for high-dimensional problems. The authors furthermore thank V. Meine and D.R. Jantos for their support in the creation of the plots. We thank both reviewers for their numerous questions that helped to significantly improve the manuscript.",
year = "2024",
month = feb,
doi = "10.1007/s00466-023-02371-2",
language = "English",
volume = "73",
pages = "365--402",
journal = "Computational mechanics",
issn = "0178-7675",
publisher = "Springer Verlag",
number = "2",
}