Partitioned coupling strategies for multi-physically coupled radiative heat transfer problems
Introduction
As a result of the ever increasing computational capacity, physical processes can be modeled with increased accuracy, allowing more realistic predictions of real-life problems. Here, multi-physically coupled processes leading to complex physical models are of special interest for many industrial applications. Especially the transfer of thermal energy is an important issue – and in many cases an in-depth realistic description needs to take the thermal interactions between the body and its environment into consideration. Thereby, in addition to heat conduction and convection, a radiative heat transfer occurs. As thermal radiation is caused by electromagnetic waves, heat transfer will occur under a vacuum too, meaning that there is no specific medium necessary to transfer thermal energy. Thermal radiation is of particular significance for industrial applications involving highly heated tools, where it may be the dominating mechanism to transfer thermal energy. In this article, the influence of thermal radiation is studied on the electro–thermo-mechanical process of the field-assisted sintering technology [38], [39], among others, which is an innovative processing technique for the sintering of powder materials.
In order to simulate a multi-physically coupled process, there are two fundamental strategies to solve this problem. The first one contemplates the entire system simultaneously within one time increment and is denoted as a monolithic scheme. This scheme is unconditionally stable for implicit time-integration, but involves asymmetrical global system matrices for the full problem, which may lead to extremely large systems.
The partitioned approach as the second strategy divides the entire multi-physical system into an iteration of several sub-problems, so that the fields involved are solved individually. Here, it might be necessary to solve the sub-problems successively until the fields are balanced. The main drawback of the partitioned approach stems from its property of being only conditionally stable. To this end, methods to stabilize the algorithm are required. Such methods are known from strongly coupled fluid–structure interactions [6], [18], [30], [32], [47] and can also be applied to thermo-mechanical [13] and electro–thermo-mechanical [15] problems. The implementation of these methods into the overall algorithm is straightforward and they work independently of the field solvers. Furthermore, the partitioned approach is much more flexible than the monolithic scheme, so different solvers, software and discretizations techniques may be used for the individual fields. Due to the high flexibility of the partitioned approach, the electro–thermo-mechanical problem, for example, can easily be extended by a thermal radiation field, so that detailed radiative heat transfer effects may be taken into account too.
The responsible transport equation for thermal radiation can be approximated by different numerical methods. In the general case of thermal radiation in a participating medium, the most commonly used methods are the method of spherical harmonics [28] and the discrete ordinates method in its modern finite volume formulation [7] to name a few. An additional approximation method is the Monte Carlo method proposed by [24], which can model irregular radiation behavior. However, due to its statistical nature, it is difficult to combine this method with others.
In a transparent medium or a vacuum, the heat transfer reduces to a purely thermal radiation. The radiation arising from the participating surfaces can be determined by means of view factors. Due to their plain structure, we employ the method of spherical harmonics, the finite volume discrete ordinates method as well as the view factor method (VFM) in this article. The benefits of these methods are studied in detail in this article, especially with regard to the integration into the partitioned solution strategy, the accuracy and computational efficiency.
The structure of this paper is as follows: we start with a brief overview of the electro–thermo-mechanical modeling and the required linearization of each individual field in Section 2. Thereafter, the radiative heat transfer and the corresponding numerical methods for vacuum and participating medium are treated in Section 3, followed by the description of the partitioned coupling algorithm in Section 4. Here, we also consider the data transfer between different discretization schemes and methods leading to convergence acceleration. In Section 5, the proposed solution strategy is analyzed based on several numerical examples, followed by concluding remarks in Section 6.
Section snippets
Electro–thermo-mechanical modeling
This section provides a brief overview of the partitioned treatment of electro–thermo-mechanically coupled problems. For this purpose, the mechanical, the thermal and the electric fields are treated separately – allowing us to point out the interdependencies between the individual fields. A linearization is required too, as the mechanical and the thermal field are nonlinear. The electric field is considered to be linear in the electric potential, so there is no linearization required here.
Radiative heat transfer modeling
In this section, we focus on the radiative heat transfer between surfaces in a vacuum and between surfaces in a participating medium. The different models applicable to the computation of thermal radiation are described as well.
In order to simulate the physical behavior of the body as realistically as possible, the thermal interactions between the body and the surroundings have to be considered. In the simplest case, the radiative heat transfer in a vacuum can be treated as a nonlinear
Partitioned coupling algorithm
In this section, we present a global partitioned solution strategy for four coupled fields. In accordance with the formulation proposed in [15] for three coupled fields, we extend the process by the thermal radiation field. The formulation allows us to solve each field individually while all other fields are assumed to be constant. This approach was already successfully applied to several multi-physical problems like fluid–structure interactions [6], [30], [32], [36], among others,
Numerical examples
In this section, we present four different numerical examples to study the properties of the partitioned coupling approach for four coupled fields. In the first example, we present a benchmark for thermal radiation to verify the coupling of the radiation field with the thermal field by means of two different solvers. Thereafter, the second example serves to analyze the convergence behavior of the radiation field in combination with the thermal field and to figure out the properties of the
Conclusions
In this article, we extend the strategies of a partitioned coupled electro–thermo-mechanical problem to a thermal radiation field. Based on this multi-field consideration, we are able to take complex thermal interactions with the environment into account. This is of particular interest in many industrial applications where highly heated tools in an enclosure play a decisive role. An external solver was integrated into the partitioned solution process in order to examine this additional field.
Acknowledgements
We gratefully acknowledge the financial support provided by the German Research Foundation (DFG, grant No. DU 405/3-2). We further thank F. Frischmann and E. Rank, Chair for Computation in Engineering at the Technische Universität München, for helping us with the mesh generation.
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