A finite element method for free surface flows of incompressible fluids in three dimensions. I: Boundary fitted mesh motion.

*(English)*Zbl 0989.76043From the summary: This paper presents a new finite element formulation for handling three-dimensional free surface problems with a boundary-fitted mesh and full Newton iteration, which solves for velocity, pressure, and mesh variables simultaneously. A boundary-fitted, pseudo-solid approach is used for moving the mesh, which treats the interior of the mesh as a fictitious elastic solid that deforms in response to boundary motion. To minimize mesh distortion near free boundary under large deformations, the mesh motion equations are rotated into normal and tangential components prior to applying boundary conditions. The Navier-Stokes equations are discretized using a Galerkin-least square/pressure stabilization formulation, which provides good convergence properties with iterative solvers. The result is a method that can track large deformations and rotations of free surface boundaries in three dimensions. The method is applied to two sample problems: solid body rotation of a fluid and extrusion from a nozzle with a rectangular cross-section. The extrusion example exhibits a variety of free surface shapes that arise from changing processing conditions.

##### MSC:

76M10 | Finite element methods applied to problems in fluid mechanics |

76D27 | Other free boundary flows; Hele-Shaw flows |

76D45 | Capillarity (surface tension) for incompressible viscous fluids |

##### Keywords:

capillarity; finite element method; pseudo-solid mesh motion; three-dimensional free surface problems; boundary-fitted mesh; Newton iteration; Navier-Stokes equations; Galerkin-least square/pressure stabilization formulation; extrusion
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\textit{R. A. Cairncross} et al., Int. J. Numer. Methods Fluids 33, No. 3, 375--403 (2000; Zbl 0989.76043)

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