Partitioned methods for time-dependent thermal fluid-structure interaction

Sammanfattning: The efficient simulation of thermal interaction between fluids and structures is crucial in the design of many industrial products, e.g. turbine blades or rocket nozzles. The main goal of this work is to present a high order time adaptive multirate parallel partitioned coupled method for the efficient numerical solution of two parabolic problems with strong jumps in the material coefficients that can be further extended to thermal fluid-structure interaction simulation. Our starting point was to analyze the convergence rate of the Dirichlet-Neumann iteration, which is one of the basic methods for simulating fluid-structure problems, for the fully discretized unsteady transmission problem. Specifically, we consider the coupling of two linear heat equations on two identical non overlapping domains with jumps in the material coefficients across these as a model for thermal fluid-structure interaction. We provide an exact formula for the spectral radius of the iteration matrix in 1D. We then show numerically that the 1D result estimates the convergence rates of 2D examples and even of nonlinear thermal fluid-structure interaction test cases with unstructured grids.However, an important challenge when coupling two different time-dependent problems is to increase parallelization in time. We suggest a multirate Neumann-Neumann waveform relaxation algorithm to solve two heterogeneous coupled heat equations as an alternative to the Dirichlet-Neumann method. In order to fix the mismatch produced by the multirate feature at the space-time interface a linear interpolation is constructed. Furthermore, we perform a one-dimensional convergence analysis for the nonmultirate fully discretized heat equations to find the optimal relaxation parameter in terms of the material coefficients, the step size and the mesh resolution. This gives a very efficient method which needs only two iterations. Numerical results confirm the analysis and show that the 1D nonmultirate optimal relaxation parameter is a very good estimator for the multirate 1D case and even for multirate and nonmultirate 2D examples. Finally, we also include in this work a time adaptive version of the multirate Neumann-Neumann waveform relaxation method mentioned above. Building a variable step size multirate scheme allows each of the subsolvers to freely construct its own time grid independently of each other. Therefore, the overall coupled method is more efficient than the previous multirate version.