HPMultiscale: High Performance Simulation of Space-Time Multiscale Nonlinear Problems

• Applicant: Professor Dr. Daniel J. Rixen
• Subject Area: Mechanics ; Applied Mechanics, Statics and Dynamics
• Term: from 2017 to 2024
• Project identifier: Deutsche Forschungsgemeinschaft (DFG) - Project number 357361040

• Final Report Year: 2024

Final Report Abstract

In this project we have been investigating the efficiency of multi-rate and multiscale time integration techniques to solve problems where localized rapid and non-linear phenomenon occur. We mainly concentrate on the mutlirate aspect of the problem, since mulli-scale aspects can be handled, at least in a first stage, with methods like the Mortar approach, whereas multi-rate techniques were found to be much less advanced. So, we investigated how multi-rate methods found in literature perform in the context of domain decomposition (in particular when the FETI solver is used) and observed significant problems in terms of spurious waves and wave reflection, and in terms of convergence of the iteration solution techniques. Numerical damping approaches were developed to alleviate the problem of interface reflections, but they require tuning and do not handle the root-cause of the oscillations. Therefore, we extended to multi-rate domain decomposition approaches the idea of variational time-integrators, which inherently conserve energy and, at least in theory, allow to freely choose the time discretization on each side of the interface. On that principle, we developed several new techniques allowing for instance to reduce the number of evaluation of the local problems and enforcing the interface compatibility in a stable and accurate manner. Those approaches improved the problem of wave reflection, without eliminating them completely. Specific attention was also given to new variational formulations that allow to perform the time-simulation of multirate models in a time-stepping manner. This required special modification of the algorithms since, by nature, variational multi-rate approaches normally require to solve all time-steps at ones, which would lead to a high computational cost. Next to the multi-rate formulation itself, we investigate several techniques (preconditioners and recycling of search directions) to speed up the convergence of the parallel solver and to reduce its computational cost. The methods were tested on a simple academic problem (a 1D duffing oscillator) and on more engineering-like problems (like a 2D structure with non-linear material). The methods were implemented in an in-house opensource research code. Several additional capabilities (such as a simple modeling of fracture or the coupling on non-matching multi-scale grids) were also implemented as part of this project, but where not fully used due to lack of time and do to the fact that the development of the multi-rate formulations and solvers appeared to be more challenging than expected.

Publications

• Asynchronous time integration in structural mechanics. PAMM, 19(1).
  Seibold, Andreas S. & Rixen, Daniel J.
  
• A VARIATIONAL APPROACH TO ASYNCHRONOUS TIME-INTEGRATION OF STRUCTURAL DYNAMICS PROBLEMS IN THE CONTEXT OF FETI AND SPURIOUS OSCILLATIONS ON THE INTERFACES. Proceedings of the XI International Conference on Structural Dynamics, 26-43. EASD.
  Seibold, Andreas & Rixen, Daniel
  
• Localization of Nonlinearities and Recycling in Dual Domain Decomposition. Lecture Notes in Computational Science and Engineering, 474-482. Springer International Publishing.
  Seibold, Andreas S.; Leistner, Michael C. & Rixen, Daniel J.
  
• “A Variational-Based Multirate Time-Integrator for FETI and Structural Dynamics: Lagrange-Multiplier with Micro-Discretization”, in: 27th International Conference on Domain Decomposition Methods, ed. by al., Z. D. et, Prague, Czech Republic.
  Seibold, Andreas & Rixen, Daniel
  
• On Total Reuse of Krylov Subspaces for an iterative FETI-solver in multirate integration. Proceedings of the YIC 2021 - VI ECCOMAS Young Investigators Conference. Editorial Universitat Politècnica de València.
  Seibold, Andreas S.; Rixen, Daniel J. & Fresno, Zarza Javier Del
  
• Preconditioning of a FETI‐solver for a nonlinear asynchronous time‐integrator applied to structural dynamics. PAMM, 20(1).
  Seibold, Andreas S. & Rixen, Daniel J.
  
• A Variational-Based Multirate Time-Integrator for FETI and Structural Dynamics: Lagrange-Multiplier with Micro-Discretization. Lecture Notes in Computational Science and Engineering, 503-510. Springer Nature Switzerland.
  Seibold, Andreas S. & Rixen, Daniel J.