Reduced basis methods are particularly attractive to use in order to diminish the number of degrees of freedom associated with the approximation of a set of partial differential equations. The main idea is to construct ad hoc basis functions with a large information content. In this note, we propose to develop and analyze reduced basis methods for simulating hierarchical flow systems, which is of relevance for studying flows in a network of pipes, an example being a set of arteries or veins. We propose to decompose the geometry into generic parts (e.g., pipes and bifurcations), and to contruct a reduced basis for these generic parts by considering representative geometric snapshots. The global system is constructed by gluing the individual basis solutions together via Lagrange multipliers.
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- 1.Berkooz, G., Holmes, P., and Lumley, J. L. (1993). The proper orthogonal decomposition in the analysis of turbulent flows. Ann. Rev. Fluid Mech. 25, 539–575.Google Scholar
- 2.Bernardi, C., Maday, Y., and Patera, A. T. (1990). A new nonconforming approach to domain decomposition: The mortar element method. In Brezis, H., and Lions, J. L. (eds.), Nonlinear Partial Differential Equations and Their Applications, College de France seminar, Pitman.Google Scholar
- 3.Christensen, E. A., Brøns, M., and Sørensen, J. N. (2000). Evaluation of proper orthogonal decomposition-based techniques applied to parameter-dependent nonturbulent flows. SIAM J. Sci. Comput. 21(4), 1419–1434.Google Scholar
- 4.Machiels, L., Maday, Y., Oliveira, Y., Patera, A. T., and Rovas, D. V. (2000). Output bounds for reduced-basis approximations of symmetric positive definite eigenvalue problems. C. R. Acad. Sci. Paris, Série I 331(2), 153–158.Google Scholar
- 5.Noor, A. K., and Peters, J. M. (1980). Reduced basis technique for nonlinear analysis of structures. AIAA J. 18(4), 455–462.Google Scholar
- 6.Prud'homme, C., Rovas, D. V., Veroy, K., Machiels, L., Maday, Y., Patera, A. T., and Turinici, G. Reliable real-time solution of parametrized partial differential equations: Reduced-basis output bound methods. J. Fluids Engineering, to appear.Google Scholar