Abstract
We consider thermal fluid structure interaction with a partitioned approach, where typically, a finite volume and a finite element code would be coupled. As a model problem, we consider two coupled Poisson problems with heat conductivities \(\lambda _1\), \(\lambda _2\) in one dimension on intervals of length \(l_1\) and \(l_2\). Hereby, we consider linear discretizations on arbitrary meshes, such as finite volumes, finite differences, finite elements. For these, we prove that the convergence rate of the Dirichlet-Neumann iteration is given by \(\lambda _1l_2/\lambda _2l_1\) and is thus independent of discretization and mesh.
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References
Badia, S., Nobile, F., Vergara, C.: Fluid-structure partitioned procedures based on Robin transmission conditions. J. Comp. Phys. 227, 7027–7051 (2008)
Birken, P., Gleim, T., Kuhl, D., Meister, A.: Fast solvers for unsteady thermal fluid structure interaction. Int. J. Num. Meth. Fluids 79, 16–29 (2015)
Causin, P., Gerbeau, J.F., Nobile, F.: Added-mass effect in the design of partitioned algorithms for fluid-structure problems. Comput. Methods Appl. Mech. Eng. 194, 4506–4527 (2005)
Farhat, C.: CFD-based nonlinear computational aeroelasticity. In: Stein, E., de Borst, R., Hughes, T.J.R. (eds.) Encyclopedia of Computational Mechanics, vol. 3: Fluids, ch. 13, pp. 459–480. Wiley (2004)
Gander, M.J., Kwok, F., Mandal, B.C.: Dirichlet-Neumann and Neumann-Neumann waveform relaxation algorithms for parabolic problems. ETNA 45, 424–456 (2016)
Görtz, M.: Convergence rate of the Dirichlet-Neumann algorithm for coupled Poisson equations. Master thesis, Lund University (2019)
Görtz, M.: Analysis-of-the-1D-Dirichlet-Neumann-algorithm, GitHub (2020). https://github.com/morgan-gortz/Analysis-of-the-1D-Dirichlet-Neumann-Algorithm
Heck, U., Fritsching, U., Bauckhage, K.: Fluid flow and heat transfer in gas jet quenching of a cylinder. Int. J. Numer. Methods Heat Fluid Flow 11, 36–49 (2001)
Henshaw, W.D., Chand, K.K.: A composite grid solver for conjugate heat transfer in fluid-structure systems. J. Comp. Phys. 228, 3708–3741 (2009)
Kowollik, D., Tini, V., Reese, S., Haupt, M.: 3D fluid-structure interaction analysis of a typical liquid rocket engine cycle based on a novel viscoplastic damage model. Int. J. Num. Meth. Eng. 94, 1165–1190 (2013)
Kowollik, D.S.C., Horst, P., Haupt, M.C.: Fluid-structure interaction analysis applied to thermal barrier coated cooled rocket thrust chambers with subsequent local investigation of delamination phenomena. Prog. Propuls. Phys. 4, 617–636 (2013)
Monge, A.: Partitioned methods for time-dependent thermal fluid-structure interaction. Ph.D. thesis, Lund University, 2018
Monge, A., Birken, P.: On the convergence rate of the Dirichlet-Neumann iteration for unsteady thermal fluid-structure interaction. Comp. Mech. 62, 525–541 (2018)
van Brummelen, E.H.: Added mass effects of compressible and incompressible flows in fluid-structure interaction. J. Appl. Mech. 76, 021206 (2009)
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Görtz, M., Birken, P. (2020). On the Convergence Rate of the Dirichlet-Neumann Iteration for Coupled Poisson Problems on Unstructured Grids. In: Klöfkorn, R., Keilegavlen, E., Radu, F.A., Fuhrmann, J. (eds) Finite Volumes for Complex Applications IX - Methods, Theoretical Aspects, Examples. FVCA 2020. Springer Proceedings in Mathematics & Statistics, vol 323. Springer, Cham. https://doi.org/10.1007/978-3-030-43651-3_32
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DOI: https://doi.org/10.1007/978-3-030-43651-3_32
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