Partitioned methods for time-dependent thermal fluid-structure interaction

Abstract

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.

Abstract (Swedish)

The invention of computers revolutionized the way of doing science and in particular the field of mathematics. Computers were faster than any human mind in doing calculations and they did not make mistakes opening a wide range of possibilities that earlier where in practice uncomputable. Scientists observe the behavior of nature and find mathematical equations that model the different phenomena. In other words, they translate the multiplicity of phenomena observed in nature into mathematical language. Sometimes one is
able to find a solution of an equation in the classical way, by pen and paper. This is called an analytical solution. However, most of the times this is not possible because the equations are too complicated and either the analytical solution is yet unknown or it has been proved to not exist. In those cases, it is still possible to find a numerical solution which is a discrete approximation to the unknown continuous analytical solution. Numerical analysis is the discipline that builds and analyzes new numerical methods to approximate solutions to all kinds of equations; from climate models to rocket engines.
The work in this thesis is motivated by the simulation of thermal fluid-structure interaction (FSI). The thermal interaction between fluids and structures, also called conjugate heat transfer, occurs when a deformable or moving structure transfers or receives heat from a surrounding or internal fluid flow. Examples of this are cooling of gas-turbine blades, thermal anti-icing systems of airplanes, supersonic reentry of vehicles from space or gas quenching, which is an industrial heat treatment of metal workpieces. These problems are usually too complex to solve them analytically, and therefore, numerical simulations
of the conjugate heat transfer are essential.
There are three different aspects that one needs to take into account for the simulation of thermal FSI. Firstly, we need a fluid solver that models the behavior of the gas in the quenching process of metal workpieces or in the liquid chemicals of the anti-icing systems of airplanes. Secondly, a structure solver is needed to model the temperature distribution over the metal workpiece or the airplane. Thirdly, the temperature interaction in the
places where the fluid and the structure meet needs to be taken into consideration as well. There are basically two approaches for the numerical simulation of thermal FSI. On one hand, one can set a numerical method that includes the fluid model, the structure model and the corresponding interaction building a new holistic model for each specific application. This is called monolithic approach. As an alternative, one can reuse existing
models for the simulation of the fluid and the structure and set a coupled numerical method to handle the interaction between fields in an iterative manner. This is known as partitioned approach and even though the advantages with respect to the monolithic are clear because only the coupling needs to be taken into account, it depends on an iterative procedure that does not guarantee in general to achieve a solution.
My contribution is focused on providing efficient partitioned numerical methods for the simulation of thermal FSI. The efficiency of a partitioned method is measured through the speed of the iterative solver to achieve an accurate numerical solution. Three scenarios are possible; the iteration does not converge to any solution, the method approaches to the solution but very slowly, meaning that needs many iterates to find it or the method is very fast and achieves the solution in few iterates. The first scenario is uninteresting and the speed of the iteration to determine if the method is fast and efficient or slow and inefficient is measured through its rate of convergence.
In this thesis, we have measured the rate of convergence of the Dirichlet-Neumann iteration which is one of the classical coupled partitioned methods for FSI simulation. We are interested in time-dependent problems. This means that we find the numerical solution over a certain time grid corresponding to a time interval. Then, for each of the values of the time grid, one performs the Dirichlet-Neumann iteration to coordinate the solution of the fluid and the structure models. The rate of convergence of the Dirichlet-Neumann method is highly dependent on the materials that one couples. In particular, the rate will be very small and consequently the coupled method will be very fast when
there exist strong jumps in the material properties. This means that when for instance one couples air with steel where their densities and heat conductivities are strongly different, one gets a very fast method. Conversely, the rate will be larger and consequently the numerical method will be very slow or even divergent when the properties of the coupled materials are very similar to each other. In conclusion, the Dirichlet-Neumann iteration is
a very good choice when coupling fields with strongly different material properties. This is exactly the situation we have in the air cooling of metal workpieces.
In spite of the efficient behavior of the Dirichlet-Neumann iteration in the thermal FSI framework, it has a main disadvantage. The subsolvers for the fluid and the structure wait for each other, and therefore, they perform the iterative procedure sequentially. In order to increase time parallelization we use the Neumann-Neumann waveform relaxation (NNWR) algorithm as an alternative to the Dirichlet-Neumann method. Using the NNWR algorithm we were able to construct a new method that allows at each iteration to find the solution of the two subproblems in parallel over the whole time interval before
performing the coupling across the interfaces. In addition, this method also allows each of the subsolvers to freely construct their own time grid independently of each other. Therefore, our proposal increases parallelism and it is more efficient than the classical Dirichlet-Neumann method in most cases.

• author: Monge, Azahar
• supervisor: Philipp Birken; Claus Führer; Gustaf Söderlind
• opponent: Prof. Dr. Rheinbach, Oliver, Technische Universität Bergakademie Freiberg, Germany
• organization: Numerical Analysis (research group); Mathematics (Faculty of Sciences); eSSENCE: The e-Science Collaboration
• publishing date: 2018-09
• type: Thesis
• publication status: published
• subject: Natural Sciences; Engineering and Technology
• keywords: thermal FSI, Conjugate heat transfer, Domain decomposition, Time adaptivity, Coupled problems, multirate
• pages: 196 pages
• defense location: Hörmander lecture hall (MH), Matematikcentrum, Sölvegatan 18A, Lund
• defense date: 2018-09-10 12:15:00
• ISBN: 978-91-7753-776-2
• ISBN: 978-91-7753-775-5
• language: English
• LU publication?: yes
• id: a39adf53-8501-4ee1-9a2e-c9a86387be16
• date added to LUP: 2018-08-15 14:16:12
• date last changed: 2024-02-13 15:11:36