G. R. Liu | Searcharxiv

arXiv2024

An Efficient Finite Difference-based Implicit Solver for Phase-Field Equations with Spatially and Temporally Varying Parameters

The phase field method is an effective tool for modeling microstructure evolution in materials. Many efficient implicit numerical solvers have been proposed for phase field simulations under uniform and time-invariant model parameters. We use Eyre's theorem to develop an unconditionally stable implicit solver for spatially non-uniform and time-varying model parameters. The accuracy, unconditional stability, and efficiency of the solver is validated against benchmarking examples. In its current form, the solver requires a uniform mesh and may only be applied to problems with periodic, Neumann, or mixed periodic and Neumann boundary conditions.

math.NA↗

arXiv2014

On stability, convergence and accuracy of bES-FEM and bFS-FEM for nearly incompressible elasticity

We present in this paper a rigorous theoretical framework to show stability, convergence and accuracy of improved edge-based and face-based smoothed finite element methods (bESFEM and bFS-FEM) for nearly-incompressible elasticity problems. The crucial idea is that the space of piecewise linear polynomials used for the displacements is enriched with bubble functions on each element, while the pressure is a piecewise constant function. The meshes of triangular or tetrahedral elements required by these methods can be generated automatically. The enrichment induces a softening in the bilinear form allowing the weakened weak (W2)procedure to produce a high-quality solution, free from locking and that does not oscillate. We prove theoretically that both methods confirm the uniform inf-sup and convergence conditions. Four numerical examples are given to validate the reliability of the bES-FEM and bFS-FEM.

math.NA↗