Partitioned methods for timedependent thermal fluidstructure interaction
(2018) 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 fluidstructure interaction simulation.
Our starting point was to analyze the convergence rate of the DirichletNeumann iteration, which is one of the basic methods for simulating fluidstructure problems, for the fully discretized unsteady transmission problem. Specifically, we consider the coupling of two... (More)  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 fluidstructure interaction simulation.
Our starting point was to analyze the convergence rate of the DirichletNeumann iteration, which is one of the basic methods for simulating fluidstructure 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 fluidstructure 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 fluidstructure interaction test cases with unstructured grids.
However, an important challenge when coupling two different timedependent problems is to increase parallelization in time. We suggest a multirate NeumannNeumann waveform relaxation algorithm to solve two heterogeneous coupled heat equations as an alternative to the DirichletNeumann method. In order to fix the mismatch produced by the multirate feature at the spacetime interface a linear interpolation is constructed.
Furthermore, we perform a onedimensional 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 NeumannNeumann 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. (Less)  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... (More)  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 fluidstructure 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 gasturbine blades, thermal antiicing 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 antiicing 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 DirichletNeumann iteration which is one of the classical coupled partitioned methods for FSI simulation. We are interested in timedependent 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 DirichletNeumann iteration to coordinate the solution of the fluid and the structure models. The rate of convergence of the DirichletNeumann 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 DirichletNeumann 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 DirichletNeumann 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 NeumannNeumann waveform relaxation (NNWR) algorithm as an alternative to the DirichletNeumann 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 DirichletNeumann method in most cases. (Less)
Please use this url to cite or link to this publication:
https://lup.lub.lu.se/record/a39adf5385014ee19a2ec9a86387be16
 author
 Monge, Azahar ^{LU}
 supervisor

 Philipp Birken ^{LU}
 Claus Führer ^{LU}
 Gustaf Söderlind ^{LU}
 opponent

 Prof. Dr. Rheinbach, Oliver, Technische Universität Bergakademie Freiberg, Germany
 organization
 publishing date
 201809
 type
 Thesis
 publication status
 published
 subject
 keywords
 thermal FSI, Conjugate heat transfer, Domain decomposition, Time adaptivity, Coupled problems, multirate
 pages
 196 pages
 publisher
 Lund University, Faculty of Science, Centre for Mathematical Sciences
 defense location
 Hörmander lecture hall (MH), Matematikcentrum, Sölvegatan 18A, Lund
 defense date
 20180910 12:15:00
 ISBN
 9789177537762
 9789177537755
 language
 English
 LU publication?
 yes
 id
 a39adf5385014ee19a2ec9a86387be16
 date added to LUP
 20180815 14:16:12
 date last changed
 20190308 16:50:12
@phdthesis{a39adf5385014ee19a2ec9a86387be16, 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 fluidstructure interaction simulation. <br/>Our starting point was to analyze the convergence rate of the DirichletNeumann iteration, which is one of the basic methods for simulating fluidstructure 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 fluidstructure 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 fluidstructure interaction test cases with unstructured grids.<br/>However, an important challenge when coupling two different timedependent problems is to increase parallelization in time. We suggest a multirate NeumannNeumann waveform relaxation algorithm to solve two heterogeneous coupled heat equations as an alternative to the DirichletNeumann method. In order to fix the mismatch produced by the multirate feature at the spacetime interface a linear interpolation is constructed. <br/>Furthermore, we perform a onedimensional 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. <br/>Finally, we also include in this work a time adaptive version of the multirate NeumannNeumann 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.}, author = {Monge, Azahar}, isbn = {9789177537762}, language = {eng}, publisher = {Lund University, Faculty of Science, Centre for Mathematical Sciences}, school = {Lund University}, title = {Partitioned methods for timedependent thermal fluidstructure interaction}, url = {https://lup.lub.lu.se/search/files/49505455/Azahar_Monge_WEBB.pdf}, year = {2018}, }