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Special Section: Computational Fluid Mechanics and Fluid–Structure Interaction

Flux Evaluation in Primal and Dual Boundary-Coupled Problems

[+] Author and Article Information
E. H. van Brummelen, K. G. van der Zee

 Eindhoven University of Technology, Faculty of Mechanical Engineering, Multiscale Engineering Fluid Dynamics Institute, P.O. Box 513, 5600 MB, Eindhoven, The Netherlands

V. V. Garg, S. Prudhomme

 Institute for Computational Engineering and Sciences, C0200, The University of Texas at Austin, 1 University Station, Austin, TX 78712

J. Appl. Mech 79(1), 010904 (Dec 13, 2011) (8 pages) doi:10.1115/1.4005187 History: Received May 01, 2011; Revised August 24, 2011; Published December 13, 2011; Online December 13, 2011

A crucial aspect in boundary-coupled problems, such as fluid-structure interaction, pertains to the evaluation of fluxes. In boundary-coupled problems, the flux evaluation appears implicitly in the formulation and consequently, improper flux evaluation can lead to instability. Finite-element approximations of primal and dual problems corresponding to improper formulations can therefore be nonconvergent or display suboptimal convergence rates. In this paper, we consider the main aspects of flux evaluation in finite-element approximations of boundary-coupled problems. Based on a model problem, we consider various formulations and illustrate the implications for corresponding primal and dual problems. In addition, we discuss the extension to free-boundary problems, fluid-structure interaction, and electro-osmosis applications.

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Copyright © 2012 by American Society of Mechanical Engineers
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Figures

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Figure 2

The dual solution wh computed using the unstable formulation (10) (left) and the stable formulation (13) (right) on a 1024-element mesh (h = 2−4 )

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Figure 3

Dual electric potential (µ) of the adjoint electro-osmosis problem corresponding to the formulation (28) based on direct flux evaluation (top) and its trace on the bottom boundary (bottom)

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Figure 4

Dual electric potential (µ) of the adjoint electro-osmosis problem corresponding to the regularized formulation (31) (top) and its trace on the bottom boundary (bottom)

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Figure 1

Convergence of the flux functionals jd (uh ) (dashed) and je (uh ) (solid) versus the mesh parameter h of the finite-element approximation, for model problem (1,1,1,1) (left) and model problem (2) (right)

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