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A Review on Fast Quasi-Newton and Accelerated Fixed-Point Iterations for Partitioned Fluid–Structure Interaction Simulation

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Advances in Computational Fluid-Structure Interaction and Flow Simulation

Abstract

The partitioned simulation of fluid–structure interactions offers great flexibility in terms of exchanging flow and structure solver and using existing established codes. However, it often suffers from slow convergence and limited parallel scalability. Quasi-Newton or accelerated fixed-point iterations are a very efficient way to solve the convergence issue. At the same time, they stabilize and speed up not only the standard staggered fluid–structure coupling iterations, but also the variant with simultaneous execution of flow and structure solver that is fairly inefficient if no acceleration methods for the underlying fixed-point iteration are used. In this chapter, we present a review on combinations of iteration patterns (parallel and staggered) and of quasi-Newton methods and compare their suitability in terms of convergence speed, robustness, and parallel scalability. Some of these variants use the so-called manifold mapping that yields an additional speedup by using an approach that can be interpreted as a generalization of the multi-level idea.

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Notes

  1. 1.

    Note that the update formula for x k+1 also shows that skipping the fixed-point iteration step (computing \(\tilde{x}^{k} = H(x^{k})\)) before using a quasi-Newton step would have led to linearly dependent columns in W k : We then would always correct x k to x k+1 by adding multiples of differences x k − x i from previous iterations as we would have to use W k  = (Δ x 0 k, Δ x 1 k, …, Δ x k−1 k) with Δ x i k = x k − x i in this case. Using induction over the iterations, we see that all columns of W k would be in the space spanned by x 0 and x 1 − x 0.

  2. 2.

    For the computation of C k F k †, we can either use a QR decomposition of F k as we do in our quasi-Newton approaches to compute V k † or singular value decompositions of C k and F k as described in [5]

  3. 3.

    http://www5.in.tum.de/wiki/index.php/PreCICE_Webpage

  4. 4.

    http://www.openfoam.org/

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Authors and Affiliations

  1. Faculty of Aerospace Engineering, Delft University of Technology, 5058, 2600 GB, Delft, The Netherlands

    David Blom & Alexander van Zuijlen

  2. Institute for Parallel and Distributed Systems, Universität Stuttgart, Universitätsstraße 38, Stuttgart, Germany

    Florian Lindner, Miriam Mehl & Klaudius Scheufele

  3. Department of Computer Science, Technische Universität München, Boltzmannstraße 3, Garching, Germany

    Benjamin Uekermann

Authors
  1. David Blom
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  2. Florian Lindner
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  3. Miriam Mehl
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  4. Klaudius Scheufele
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  5. Benjamin Uekermann
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  6. Alexander van Zuijlen
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Correspondence to Miriam Mehl .

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Editors and Affiliations

  1. Department of Structural Engineering, University of California, San Diego, La Jolla, California, USA

    Yuri Bazilevs

  2. Department of Modern Mechanical Engineering, Waseda University, Tokyo, Japan

    Kenji Takizawa

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Blom, D., Lindner, F., Mehl, M., Scheufele, K., Uekermann, B., van Zuijlen, A. (2016). A Review on Fast Quasi-Newton and Accelerated Fixed-Point Iterations for Partitioned Fluid–Structure Interaction Simulation. In: Bazilevs, Y., Takizawa, K. (eds) Advances in Computational Fluid-Structure Interaction and Flow Simulation. Modeling and Simulation in Science, Engineering and Technology. Birkhäuser, Cham. https://doi.org/10.1007/978-3-319-40827-9_20

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