A quasi-elliptic transformation for moving boundary problems with large anisotropic deformations


Dimakopoulos Y., Tsamopoulos J.


We have developed a quasi-elliptic set of equations for generating a discretization mesh that optimally conforms to an entire domain that undergoes large deformations in primarily one direction. We have applied this method to the axisymmetric problem of the transient displacement of a viscous liquid by a high-pressure gas. The liquid initially fills completely a tube the diameter of which may be constant or change smoothly, suddenly or periodically along its finite length. Key ingredients for the success of the proposed transformation are limiting the orthogonality requirements on the mesh and employing an improved node distribution function along the deforming boundary. The retained orthogonal term along with the penalty method for the imposition of the boundary conditions overcome the inherent restrictions of a conformal transformation, producing meshes of high quality. This term also eliminates the discontinuous slopes of the coordinate lines that are normal to the free surface. These usually arise due to the harmonic transformation around highly deforming surfaces. The mathematical anisotropy in the mesh generating equations directly corresponds to the physical one, when the initially straight and normal to the axis of symmetry air/fluid interface, at the tube entrance, develops mostly along the axial direction, generating a long, open bubble. Moreover, the generalized node distribution imposes an optimal discretization of the bubble surface itself. Combining this scheme with the mixed finite element method produces a powerful tool, with extended robustness and accuracy. In order to reduce the computational cost, the resulting set of equations is solved with a non-linear, Gauss-Seidel technique and a variable time step. Specific applications of the proposed scheme are presented. © 2003 Elsevier B.V. All rights reserved.


Adaptive time-stepping, Finite element methods, Liquid displacement by gas, Mesh generation, Moving boundary problems

DOI: 10.1016/j.jcp.2003.07.027