×

zbMATH — the first resource for mathematics

Numerical stability of the finite element immersed boundary method. (English) Zbl 1186.76661
The authors address the stability of immersed boundary computations taking advantage of the natural energy estimates that arise from the use of a variational approach to the immersed boundary method. A two-dimensional incompressible fluid and a boundary in the form of a simple closed curve are considered. A linearization of the Navier-Stokes equations and a linear elasticity model are considered for proving the unconditional stability of the fully implicit discretization, achieved with the use of a backward Euler method for both the fluid and the structure evolution, and a CFL condition for the semi-implicit method, where the fluid terms are treated implicitly while the structure is treated explicitly. The last sections are devoted to numerical validation, conclusions and plans for the future works.

MSC:
76M10 Finite element methods applied to problems in fluid mechanics
76D05 Navier-Stokes equations for incompressible viscous fluids
74F10 Fluid-solid interactions (including aero- and hydro-elasticity, porosity, etc.)
65N12 Stability and convergence of numerical methods for boundary value problems involving PDEs
65N30 Finite element, Rayleigh-Ritz and Galerkin methods for boundary value problems involving PDEs
Software:
deal.ii
PDF BibTeX XML Cite
Full Text: DOI
References:
[1] DOI: 10.1137/0729022 · Zbl 0762.65052
[2] DOI: 10.1016/S0045-7949(02)00404-2
[3] DOI: 10.4203/csets.12.12
[4] DOI: 10.1007/978-1-4612-3172-1 · Zbl 0788.73002
[5] DOI: 10.1137/0719018 · Zbl 0487.76035
[6] DOI: 10.1016/S0893-9659(00)00127-0 · Zbl 1013.76021
[7] DOI: 10.1002/fld.1650200824 · Zbl 0881.76072
[8] Peskin C. S., Acta Numerica, 2002 (2002)
[9] DOI: 10.1016/0021-9991(89)90213-1 · Zbl 0668.76159
[10] DOI: 10.1006/jcph.1993.1051 · Zbl 0762.92011
[11] DOI: 10.1016/0021-9991(77)90100-0 · Zbl 0403.76100
[12] Rosar M. E., New York J. Math. 7 pp 281–
[13] DOI: 10.1006/jcph.1999.6297 · Zbl 0953.76070
[14] DOI: 10.1137/S0036139994267018 · Zbl 0839.35105
[15] DOI: 10.1137/0913077 · Zbl 0760.76067
[16] DOI: 10.1016/j.cma.2003.12.024 · Zbl 1060.74676
[17] DOI: 10.1016/j.cma.2003.12.044 · Zbl 1067.76576
[18] DOI: 10.1006/jcph.2002.7066 · Zbl 1130.76406
[19] DOI: 10.1063/1.1582476 · Zbl 1186.76611
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.