scholarly journals Energy Stable Numerical Method for the TDGL Equation with the Reticular Free Energy in Hydrogel

2017 ◽  
Vol 35 (1) ◽  
pp. 37-51
Author(s):  
Dong Liao
Keyword(s):  
Free Energy ◽  
Numerical Method ◽  
Energy Stable ◽  
Tdgl Equation ◽  
Stable Numerical Method

scholarly journals A free–energy stable p–adaptive nodal discontinuous Galerkin for the Cahn–Hilliard equation

2021 ◽  
pp. 110409
Author(s):  
Gerasimos Ntoukas ◽  
Juan Manzanero ◽  
Gonzalo Rubio ◽  
Eusebio Valero ◽  
Esteban Ferrer
Keyword(s):  
Free Energy ◽  
Discontinuous Galerkin ◽  
Hilliard Equation ◽  
Energy Stable ◽  
Cahn Hilliard Equation

Efficient numerical algorithms for the phase field crystal equation

2021 ◽  
pp. 2150042
Author(s):  
Qingqu Zhuang ◽  
Shuying Zhai ◽  
Zhifeng Weng
Keyword(s):  
Free Energy ◽  
Lower Bound ◽  
Numerical Simulations ◽  
Lagrange Multiplier ◽  
Phase Field ◽  
Numerical Algorithms ◽  
Energy Potential ◽  
Energy Stable ◽  
2D And 3D ◽  
Phase Field Crystal

In this paper, based on the Lagrange Multiplier approach in time and the Fourier-spectral scheme for space, we propose efficient numerical algorithms to solve the phase field crystal equation. The numerical schemes are unconditionally energy stable based on the original energy and do not need the lower bound hypothesis of the nonlinear free energy potential. The unconditional energy stability of the three semi-discrete schemes is proven. Several numerical simulations in 2D and 3D are demonstrated to verify the accuracy and efficiency of our proposed schemes.

Phase-Dield Simulation of Rhombohedral and Tetragonal Phases in Ferroelectric Single Crystals

Aerospace ◽  
2006 ◽  
Author(s):  
T. Liu ◽  
C. S. Lynch
Keyword(s):  
Free Energy ◽  
Phase Transformations ◽  
Single Crystals ◽  
Electric Fields ◽  
Tetragonal Phase ◽  
Ferroelectric Materials ◽  
Integration Scheme ◽  
Ferroelectric Crystal ◽  
Domain Structures ◽  
Tdgl Equation

Ferroelectric materials exhibit spontaneous polarization and domain structures below the Curie temperature. In this work the phase field approach has been used to simulate phase transformations and the formation of ferroelectric domain structures. The evolution of phases and domain structures was simulated in ferroelectric single crystals by solving the time dependent Ginzburg-Landau (TDGL) equation with polarization as the order parameter. In the TDGL equation the free energy of a ferroelectric crystal is written as a function of polarization and applied fields. Change of temperature as well as application of stress and electric fields leads to change of the free energy and evolution of phase states and domain structures. In this work the finite difference method was implemented for the spatial description of the polarization and the temporal evolution of polarization field was computed by solving the TDGL equation with an explicit time integration scheme. Cubic to tetragonal, cubic to rhombohedral and rhombohedral to tetragonal phase transformations were modeled, and the formation of domain structures was simulated. Field induced polarization switching and rhombohedral to tetragonal phase transition were simulated.

scholarly journals Numerical method for computing the free energy of glasses

2020 ◽  
Vol 102 (6) ◽  
Author(s):  
H. A. Vinutha ◽  
Daan Frenkel
Keyword(s):  
Free Energy ◽  
Numerical Method

scholarly journals Stability in the gaming equation

1980 ◽  
Vol 21 (2) ◽  
pp. 207-210
Author(s):  
Jason Gait
Keyword(s):  
Numerical Method ◽  
Approximate Solutions ◽  
Linear Programs ◽  
Stable Numerical Method

We discuss a computationally stable numerical method for the solution of linear programs and games. The method is useful in obtaining approximate solutions to large numerically unstable linear programs.

Sign in / Sign up

Export Citation Format

Share Document