Fourier Spectral Moving Mesh Method for Willmore Equation of ...

where a 1 represents the surface tension, H = κ 1+κ 22is the mean curvature of the membranesurface, with κ 1 and κ 2 as the principle curvatures, and G = κ 1 κ 2 is the Gaussian curvature.a 2 is the bending rigidity and a 3 the stretching rigidity. c 0 is the spontaneous curvaturethat describes the asymmetry effect of the membrane or its environment. The equilibriummembrane configurations are the minimizers of the energy subject to given surface area andvolume constraints. The simplified case ofis like the Willmore Problem.∫E = (H − c 0 ) 2 ds,ΓPhase field method has been extensively applied to modeling microstructure evolutionfor various materials process such as the mesoscale pattern formation and interface motion.A set of spatially dependent field variables are described by the phase function. Particularly,the membrane surface Γ is defined as the zero level set of the phase field function φ . Thecorrespondent phase field model is given by [?]∫E(φ) =Γ12ξ (ξ∆φ + (1 ξ φ + c √0 2)(1 − φ 2 )) 2 dxThe surface area and volume constrains can be specified as∫B(φ) =Ω∫A(φ) =Ωφdx = α[ ξ 2 |∇φ|2 + 1 4ξ (φ2 − 1) 2 ]dx = β.Semi-implicit Fourier-spectral methods have been employed as an efficient numericalalgorithms for phase field simulations. For the reason that this kind of method can bothincrease the time step and improve the stability, while achieving high accuracy in space.[?]However, as the width of interface ξ in the phase-field model goes to zero, in order tomaintain high resolution, more grid points should be used.On the other hand, uniformgrids are required in order to use FFT algorithm; which results in the high cost of numericalcomputation.The studies on adaptive meshing techniques have led to significant improvement of thecomputational efficiency of traditional fixed grid methods in many applications, includingphase field modeling. It is clear that adaptive mesh methods are useful for microstructureswith a very small interfacial width compared to the domain size. .2