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Adaptive time-integration for goal-oriented and coupled problems

Author

Summary, in English

We consider efficient methods for the partitioned time-integration of multiphysics problems, which commonly exhibit a multiscale behavior, requiring independent time-grids. Examples are fluid structure interaction in e.g., the simulation of blood-flow or cooling of rocket engines, or ocean-atmosphere-vegetation interaction. The ideal method for solving these problems allows independent and adaptive time-grids, higher order time-discretizations, is fast and robust, and allows the coupling of existing subsolvers, executed in parallel.

We consider Waveform relaxation (WR) methods, which can have all of these properties. WR methods iterate on continuous-in-time interface functions, obtained via suitable interpolation. The difficulty is to find suitable convergence acceleration, which is required for the iteration converge quickly.

We develop a fast and highly robust, second order in time, adaptive WR method for unsteady thermal fluid structure interaction (FSI), modelled by heterogeneous coupled linear heat equations. We use a Dirichlet-Neumann coupling at the interface and an analytical optimal relaxation parameter derived for the fully-discrete scheme. While this method is sequential, it is notably faster and more robust than similar parallel methods.

We further develop a novel, parallel WR method, using asynchronous communication techniques during time-integration to accelerate convergence. Instead of exchanging interpolated time-dependent functions at the end of each time-window or iteration, we exchange time-point data immediately after each timestep. The analytical description and convergence results of this method generalize existing WR theory.

Since WR methods allow coupling of problems in a relative black-box manner, we developed adapters to PDE-subsolvers implemented using DUNE and FEniCS. We demonstrate this coupling in a thermal FSI test case.

Lastly, we consider adaptive time-integration for goal-oriented problems, where one is interested in a quantity of interest (QoI), which is a functional of the solution. The state-of-the-art method is the dual-weighted residual (DWR) method, which is extremely costly in both computation and implementation. We develop a goal oriented adaptive method based on local error estimates, which is considerably cheaper in computation. We prove convergence of the error in the QoI for tolerance to zero under a controllability assumption. By analyzing global error propagation with respect to the QoI, we can identify possible issues and make performance predictions. Numerical results verify these results and show our method to be more efficient than the DWR method.

Publishing year

2021-06-03

Language

English

Document type

Dissertation

Topic

  • Computational Mathematics
  • Mathematical Analysis

Keywords

  • time-adaptivity
  • goal-oriented problems
  • error estimation
  • coupled problems
  • waveform relaxation
  • asynchronous methods
  • conjugate heat transfer

Status

Published

Project

  • eSSENCE@LU 4:4 - 3M: Multiphysics, multicore, multirate solution of coupled dynamic problems

Research group

  • Numerical Analysis

ISBN/ISSN/Other

  • ISBN: 978-91-7895-930-3
  • ISBN: 978-91-7895-929-7

Defence date

2 September 2021

Defence time

14:00

Defence place

MH:H, Lund. Join via zoom: https://lu-se.zoom.us/j/65642304248

Opponent

  • Andreas Dedner (Associate professor (Reader))