Yana Di

Moving Mesh Finite Element Methods for the Incompressible Navier--Stokes Equations

This work presents the first effort in designing a moving mesh algorithm to solve the incompressible Navier--Stokes equations in the primitive variables formulation. The main difficulty in developing this moving mesh scheme is how to keep it divergence-free for the velocity field at each time level. The proposed numerical scheme extends a recent moving grid method based on harmonic mapping [R. Li, T. Tang, and P. W. Zhang, J. Comput. Phys., 170 (2001), pp. 562--588], which decouples the PDE solver and the mesh-moving algorithm. This approach requires interpolating the solution on the newly generated mesh. Designing a divergence-free-preserving interpolation algorithm is the first goal of this work. Selecting suitable monitor functions is important and is found challenging for the incompressible flow simulations, which is the second goal of this study. The performance of the moving mesh scheme is tested on the standard periodic double shear layer problem. No spurious vorticity patterns appear when even fairly coarse grids are used.

Precursor simulations in spreading using a multi-mesh adaptive finite element method

Using the phase-field model for immiscible two-phase flows, we have numerically investigated the wetting dynamics. The long-range van der Waals forces towards the solid, which drive the spreading of the wetting phase into the nonwetting phase, have been explicitly taken into account in the governing equations. Our continuum model uses the generalized Navier boundary condition (GNBC) to account for the fluid slipping at the solid surface. The accurate description of the molecular-scale contact-line hydrodynamics makes the numerical simulations cost too much to abide. In this work, we propose an efficient multi-mesh adaptive finite element method which approximates different components of the solution (velocity, pressure and phase variable) on different h-adaptive meshes because of their strongly different local behaviors. That allows us to study the early stage of spreading, wherein the precursor is initiated and developed if the van der Waals forces are strong enough. We find that there is indeed a transition in the spreading behavior across a critical value of the Hamaker constant. In particular, this critical value is noted to be the one that separates the partial wetting from complete wetting.

Moving Mesh Methods for Singular Problems on a Sphere Using Perturbed Harmonic Mappings

This work is concerned with developing moving mesh strategies for solving problems defined on a sphere. To construct mappings between the physical domain and the logical domain, it has been demonstrated that harmonic mapping approaches are useful for a general class of solution domains. However, it is known that the curvature of the sphere is positive, which makes the harmonic mapping on a sphere not unique. To fix the uniqueness issue, we follow Sacks and Uhlenbeck [Ann. of Math. (2), 113 (1981), pp. 1-24] to use a perturbed harmonic mapping in mesh generation. A detailed moving mesh strategy including mesh redistribution and solution updating on a sphere will be presented. The moving mesh scheme based on the perturbed harmonic mapping is then applied to the moving steep front problem and the Fokker-Planck equations with high potential intensities on a sphere. The numerical experiments show that with a moderate number of grid points our proposed moving mesh algorithm can accurately resolve detailed features of singular problems on a sphere.

Computation of Dendritic Growth with Level Set Model Using a Multi-Mesh Adaptive Finite Element Method

In this paper, we propose an efficient multi-mesh h-adaptive algorithm to solve the level set model of dendritic growth. Since the level set function is used to provide implicitly the location of the phase interface, it is resolved by an h-adaptive mesh with refinement only around the phase interface, while the thermal field is approximated on another h-adaptive mesh. The proposed method not only can enjoy the merits of the level set function to handle complex evolution of the free boundary, but also can achieve the similar accuracy as the front tracking method for the sharp interface model with about the same degrees of freedom. The algorithm is applied to the simulation of the dendritic crystallization in a pure undercooled melt. The accuracy is verified by comparing the computational dendrite tip velocity with solvability theory. Numerical simulations, both in 2D and 3D cases, are presented to demonstrate its capacity and efficiency.

Variational method for liquids moving on a substrate

A new variational method is proposed to calculate the evolution of liquid film and liquid droplet moving on a solid substrate. A simple time evolution equation is obtained for the contact angle of a liquid film that starts to move on a horizontal substrate. The equation indicates the dynamical transition at the receding side and the ridge formation at the advancing side. The same method is applied for the evolution of a droplet that starts to move on an inclined solid surface, and again the characteristic shape change of the droplet is obtained by solving a simple ordinary differential system. We will show that this method has a potential application to a wide class of problems of droplets moving on a substrate.

Theoretical analysis for meniscus rise of a liquid contained between a flexible film and a solid wall

We study the dynamics of the meniscus rise of a liquid contained in a narrow gap between a flexible film and a solid wall. We show that the meniscus rises indefinitely expelling liquid from the gap region, and that the height of the rising front h(t) increases with time as , while the gap distance e(t) decreases as . These results are consistent with the experiments of Cambau et al. (EPL, 96 (2011) 24001).

Sharp-interface limits of a phase-field model with a generalized Navier slip boundary condition for moving contact lines

The sharp-interface limits of a phase-field model with a generalized Navier slip boundary condition for binary fluids with moving contact lines are studied by asymptotic analysis and numerical simulations. The effects of the mobility number as well as a phenomenological relaxation parameter on the boundary condition are considered. In asymptotic analysis, we consider both the cases that the mobility number is proportional to the Cahn number and the square of the Cahn number, and derive the sharp-interface limits for several set-ups of the boundary relaxation parameter. It is shown that the sharp-interface limit of the phase-field model is the standard two-phase incompressible Navier–Stokes equations coupled with several different slip boundary conditions. Numerical results are consistent with the analysis results and also illustrate the different convergence rates of the sharp-interface limits for different scalings of the two parameters.

Thin film dynamics in coating problems using Onsager principle

A new variational method is proposed to investigate the dynamics of the thin film in a coating flow where a liquid is delivered through a fixed slot gap onto a moving substrate. A simplified ODE system has also been derived for the evolution of the thin film whose thickness is asymptotically constant behind the coating front. We calculate the phase diagram as well as the film profiles and approximate the film thickness theoretically, and agreement with the well-known scaling law as is found.

13-Moment System with Global Hyperbolicity for Quantum Gas

We point out that the quantum Grad’s 13-moment system (Yano in Physica A 416:231–241, 2014) is lack of global hyperbolicity, and even worse, the thermodynamic equilibrium is not an interior point of the hyperbolicity region of the system. To remedy this problem, by fully considering Grad’s expansion, we split the expansion into the equilibrium part and the non-equilibrium part, and propose a regularization for the system with the help of the new hyperbolic regularization theory developed in Cai et al. (SIAM J Appl Math 75(5):2001–2023, 2015) and Fan et al. (J Stat Phys 162(2):457–486, 2016). This provides us a new model which is hyperbolic for all admissible thermodynamic states, and meanwhile preserves the approximate accuracy of the original system. It should be noted that this procedure is not a trivial application of the hyperbolic regularization theory.

Anisotropic Meshes and Stabilization Parameter Design of Linear SUPG Method for 2D Convection-Dominated Convection–Diffusion Equations

We propose a numerical strategy to generate a sequence of anisotropic meshes and select appropriate stabilization parameters simultaneously for linear SUPG method solving two dimensional convection-dominated convection–diffusion equations. Since the discretization error in a suitable norm can be bounded by the sum of interpolation error and its variants in different norms, we replace them by some terms which contain the Hessian matrix of the true solution, convective field, and the geometric properties such as directed edges and the area of triangles. Based on this observation, the shape, size and equidistribution requirements are used to derive corresponding metric tensor and stabilization parameters. Numerical results are provided to validate the stability and efficiency of the proposed numerical strategy.