High Accuracy And Energy Stable Scheme For Thin Film Material Computation

In this thesis,two typical and important problems of wrinkling and solid-state dewetting in thin film/substrate systems are studied by constructing mathematical mod-els and their numerical schemes.The wrinkling phenomenon mainly occurs in a system in which the surface layer is relatively hard and the substrate is relatively soft,and the surface layer is subjected to compressive residual stress caused by the shrinkage effect of the substrate to form a wrinkle pattern on the interface;The solid-state dewetting phenomenon mainly occurs in a solid film on a rigid substrate,which is a spontaneous physical phenomenon similar to dewetting of liquid films when subjected to a temper-ature high enough and lower than the melting point of the film.The main work of the thesis includes the following three parts:For the wrinkle problem of the elastic film on the soft elastic substrate,the FvK theory of large deflection of thin plates and the linear elastic theory are applied to the thin film layer and the substrate,respectively.The total free energy of the system is expressed as the membrane energy and the bending energy of the thin film,and the elastic strain energy of the substrate.The evolution equation is obtained by first variation of the total energy and considering the L²gradient flow.Based on the SAV method and using the BDF and CN schemes discrete time derivatives,we obtain a class of uniquely solvable,high-order,unconditional energy stable numerical schemes.These SAV schemes only need to solve,twice,a decoupled linear,fourth-order differen-tial equations with constant coefficients at each time step,and these linear equations can be efficiently implemented by using Fourier spectral method to discretize spatial derivatives.Numerical results have demonstrated that these SAV schemes ensure en-ergy stability and accuracy in the time,and also has a better performance than the semi-implicit discrete scheme.Finally,we successfully numerical simulated four typical wrinkling modes,i.e.,stripe,checkerboard,labyrinth and herringbone patterns,which are commonly observed in experiments.For the isotropic solid-state dewetting problem,through the phase field model we obtain the evolution equation?i.e.,Cahn-Hilliard equation?and dynamic contact line boundary conditions.Based on the IEQ method,and using the BDF scheme and second-order central difference scheme to discretize temporal and spatial derivatives,respectively,we construct linearly,first-and second-order uniquely solvable schemes,these schemes ensure unconditional energy stability and total mass conservation.Differ-ent from the convex splitting and the SAV method,the method only needs to solve the linear differential equation with variable coefficient at each time step,thus facilitating the implementation of numerical calculation.By performing numerical simulations,we demonstrate that these IEQ schemes are accurate,energy stable and total mass conserv-ing.In addition,numerical simulations also investigated several interesting solid-state dewetting phenomena,such as hole dynamics,pinch-off and evolution under different Young contact angles.Finally,numerical calculations show that different mobilities may cause different numerical equilibrium states.For the solid-state dewetting problems with weakly anisotropic surface energy,we propose a new phase field model with uniform interface thickness.Using the H^-1gra-dient flow,the dynamic evolution equation?i.e.,anisotropic Cahn-Hilliard equation?and contact line boundary conditions are obtained.Based on the IEQ and stabilization method,a linear numerical scheme is given,and numerical calculations show that can overcome the non-physical oscillation caused by the anisotropic surface energy,and also ensure the stability of energy and the conservation of mass during the evolution.By performing numerical simulations,we investigated several interesting solid-state dewet-ting phenomena?such as different anisotropy degrees,contact angles and fold symmetry orders?,and it is shown that the complete dewetting and complete wetting phenomenon does not occur with weakly anisotropic surface energies.Finally,calculations show that different mobilities may also result in different numerical equilibrium states.The obtained results enhance the theory and numerical methods of wrinkling and solid-state dewetting of film/substrate systems,and provide some reliable bases for film material processing and manufacturing.