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A time adaptive multirate Quasi-Newton waveform iteration for coupled problems

Niklas Kotarsky, Philipp Birken

TL;DR

This work targets efficient partitioned simulations of time-dependent multiphysics problems coupled through an interface. It extends waveform relaxation with time-adaptive Quasi-Newton acceleration by introducing a fixed auxiliary time grid ${\cal T}_{QN}$ to keep the iteration dimension constant, enabling standard QN updates despite changing sub-solver grids. A linear analysis establishes finite convergence properties and characterizes interpolation error, which vanishes when ${\cal T}_{QN}={\cal T}_2$; the method is implemented in the open-source library preCICE. Numerical experiments on a heat transfer test case and a fluid-structure interaction scenario demonstrate faster convergence and robustness compared to relaxation and fixed-grid QNWR, while maintaining ease of integration with existing sub-solvers.

Abstract

We consider waveform iterations for dynamical coupled problems, or more specifically, PDEs that interact through a lower dimensional interface. We want to allow for the reuse of existing codes for the subproblems, called a partitioned approach. To improve computational efficiency, different and adaptive time steps in the subsolvers are advisable. Using so called waveform iterations in combination with relaxation, this has been achieved for heat transfer problems earlier. Alternatively, one can use a black box method like Quasi-Newton to improve the convergence behaviour. These methods have recently been combined with waveform iterations for fixed time steps. Here, we suggest an extension of the Quasi-Newton method to the time adaptive setting and analyze its properties. We compare the proposed Quasi-Newton method with state of the art solvers on a heat transfer test case, and a complex mechanical Fluid-Structure interaction case, demonstrating the methods efficiency.

A time adaptive multirate Quasi-Newton waveform iteration for coupled problems

TL;DR

This work targets efficient partitioned simulations of time-dependent multiphysics problems coupled through an interface. It extends waveform relaxation with time-adaptive Quasi-Newton acceleration by introducing a fixed auxiliary time grid to keep the iteration dimension constant, enabling standard QN updates despite changing sub-solver grids. A linear analysis establishes finite convergence properties and characterizes interpolation error, which vanishes when ; the method is implemented in the open-source library preCICE. Numerical experiments on a heat transfer test case and a fluid-structure interaction scenario demonstrate faster convergence and robustness compared to relaxation and fixed-grid QNWR, while maintaining ease of integration with existing sub-solvers.

Abstract

We consider waveform iterations for dynamical coupled problems, or more specifically, PDEs that interact through a lower dimensional interface. We want to allow for the reuse of existing codes for the subproblems, called a partitioned approach. To improve computational efficiency, different and adaptive time steps in the subsolvers are advisable. Using so called waveform iterations in combination with relaxation, this has been achieved for heat transfer problems earlier. Alternatively, one can use a black box method like Quasi-Newton to improve the convergence behaviour. These methods have recently been combined with waveform iterations for fixed time steps. Here, we suggest an extension of the Quasi-Newton method to the time adaptive setting and analyze its properties. We compare the proposed Quasi-Newton method with state of the art solvers on a heat transfer test case, and a complex mechanical Fluid-Structure interaction case, demonstrating the methods efficiency.

Paper Structure

This paper contains 14 sections, 3 theorems, 60 equations, 5 figures, 2 tables.

Table of Contents

  1. Introduction
  2. Waveform iterations
  3. Waveform iterations for interface coupled PDEs
  4. Quasi-Newton method
  5. Generalization to time adaptive sub-solvers
  6. Implementation details of time adaptive QNWR in the open source coupling library preCICE
  7. Linear analysis of the new method
  8. Heat test case
  9. Preliminary experiments
  10. Interpolation error and termination
  11. Comparison of different auxiliary time grids for the time adaptive case
  12. Efficiency of the time adaptive QNWR method
  13. Fluid-Structure interaction
  14. Conclusion

Key Result

Lemma 1

If the coupled problem is linear and constant time grids are employed, then the interpolation error of the time adaptive QNWR timeAdaptiveQN is given by

Figures (5)

  • Figure 1: Number of iteration compared to the number of base time steps for different number of QN time steps.
  • Figure 2: Time discretization error of the last time step compared to the number of base time steps for different number of QN time steps.
  • Figure 3: The computational efficiency of the QNWR algorithm for the three different time grid selection strategies inside the time adaptive QNWR.
  • Figure 4: The error of the last time step compared to different tolerances.
  • Figure 5: Simulation results for the coarse test case with $T_f = 0.2$.

Theorems & Definitions (7)

  • Definition 1
  • Definition 2
  • Lemma 1
  • proof
  • Theorem 1
  • Theorem 2
  • proof