Special Session 68: Recent advances on interfaces dynamics modeling, simulation and applications

Coupled Capillary-Perivascular Flow and Mass Transport: Modeling and Computation

Xuelian Bao South China University of Technology Peoples Rep of China Co-Author(s):

Abstract: In this talk, we will present a coupled model for the Capillary-Perivascular Flow and Mass Transport of oxygen.

Numerical Studies for Multicomponent Vesicles

Zhenlin Guo Beijing Computational Science Research Center Peoples Rep of China Co-Author(s):    Shuqi Tang, John Lowengrub

Abstract: In this presentation, we will introduce a thermodynamically consistent phase field model for multi-component vesicles in fluids. Our model encompasses dual two-phase fluid systems: one to depict the interaction dynamics between the vesicle and its ambient fluid, and the other to capture the dynamics of the multi-component on the vesicle membrane. These two systems are coupled within a diffuse domain framework, which provides a high-accuracy approach for solving partial differential equations on complex surfaces. We will illustrate a series of numerical examples showcasing classical scenarios encountered in multicomponent vesicles, including phase separation, tank treading, and phase treading. Additionally, we will validate the accuracy of our model by comparing these simulations with experimental observations.

Interfaced neural networks for solving (parametric) interface PDE problems

Benzhuo Lu Academy of Mathematics and Systems Science, Chinese Academy of Sciences Peoples Rep of China Co-Author(s):    Sidi Wu, Aiqing Zhu, Yifa Tang

Abstract: Machine learning is used as an accessible meshless approach for solving PDEs. For interface problem, the computational domain can be decomposed into several subdomains, and the solution on each subdomain can be accordingly represented by one network. In this talk, we will present, 1) a Physics-Informed Neural Network, interfaced neural network (INN) to solve interface problems with discontinuous coefficients as well as irregular interfaces, 2) an interfaced operator network (IONet) to solve parametric elliptic interface PDEs, where different coefficients, source terms, and boundary conditions are considered as input features. The convergence of the INN will be discussed as well.

A phase field description of droplet dynamics with ion transport

Yuzhe Qin Shanxi University Peoples Rep of China Co-Author(s):    Hoaxing Huang, Zilong Song, Shixin Xu

Abstract: We present a phase field model to describe the charged droplets suspended in a viscous fluid with ion transport across the membrane. Our model incorporates spatial variations in electric permittivity and diffusion constants, as well as interfacial capacitance. An ion pump is added to describe the ion active transport to maintain the ion difference between the inner and outer regions. We perform a detailed asymptotic analysis to demonstrate the convergence of the diffusive interface model to the sharp interface model as the interface thickness approaches zero. A series of numerical experiments are conducted to validate the asymptotic analysis and demonstrate the model`s effectiveness. Our numerical results show the pump with phase field method is effective in maintaining the ion difference across the interface. Besides, the motions of multiple droplets have been investigated in the numerical tests. The numerical solutions confirm our model is a reasonable model and easy to develop a series of models to consider the interface problems.

Convergent finite element approximations of surface evolution with relaxed minimal deformation

Rong Tang The Hong Kong Polytechnic University Hong Kong Co-Author(s):    Guangwei Gao, Buyang Li

Abstract: The finite element approximation of surface evolution under an external velocity field is studied. An artificial tangential motion is designed by using harmonic map heat flow from the initial surface onto the evolving surface. This makes the evolving surface have minimal deformation (up to certain relaxation) from the initial surface and therefore improves the mesh quality upon discretization. By exploiting and utilizing an intrinsic cancellation structure in this formulation and the role played by the relaxation term, convergence of the proposed method in approximating surface evolution in the three-dimensional space is proved for finite elements of degree $k\geq 4$. One advantage of the proposed method is that it allows us to prove convergence of numerical approximations by using the normal vector of the computed surface in the numerical scheme, instead of evolution equations of normal vector (as in the literature). Another advantage of the proposed method is that it leads to better mesh quality in some typical examples, and therefore prevents mesh distortion and breakdown of computation. Numerical examples are presented to illustrate the convergence of the proposed method and its advantages in improving the mesh quality of the computed surfaces.

An efficient unconditional energy stable scheme for multiphase flow simulations

Xiaoping Wang The Chinese University of Hong Kong (Shenzhen) and the Shenzhen International Center for Industrial and Applied Mathematics Peoples Rep of China Co-Author(s):

Abstract: We design an efficient and unconditionally energy stable method for simulating the dynamics of the multi-phase flow based on the the Cahn-Hilliard-Navier-Stokes phase field model with variable density and viscosity. An improved SAV type scheme is developed. We introduce some nonlocal auxiliary variables and associated ordinary differential equations to decouple the nonlinear terms. The resulting scheme is completely decoupled and unconditionally energy stable. The accuracy and stability of the algorithm are verified by numerical simulation.

The Onsager variational principle and physics preserving numerical schemes

Xianmin XU Chinese Academy of Sciences Peoples Rep of China Co-Author(s):

Abstract: The Onsager variational principle is a fundamental law for irreversible processes in non-equilibrium statistical physics. It has been used to model many complicated phenomena in soft matter. By using the Onsager principle, one can formulate partial differential equation for diverse gradient flow systems. In this talk, we will show it also acts as a natural framework for constructing energy-stable time discretization schemes. It provides a robust basis for developing numerical schemes that uphold crucial physical properties. Within this framework, several widely used schemes emerge naturally, showing its versatility and applicability.

A Cartesian grid method for nonhomogeneous elliptic interface problems on unbounded domains

Wenjun Ying Shanghai Jiao Tong University Peoples Rep of China Co-Author(s):    Wenjun Ying

Abstract: We will present a Cartesian grid based fast and accurate method for indirectly evaluating boundary and volume integrals in a boundary-volume integral approach for nonhomogeneous elliptic interface problems on unbounded domains. The indirect calculation is done by solving equivalent but simple interface problems. We accelerate the computation by introducing an intermediate, transitional circle or sphere and taking advantages of super-convergent numerical quadrature or series expansion on circles/spheres. We first map the boundary or volume integral on the irregular boundary or domain to the intermediate circle/sphere; then evaluate the boundary integral on the intermediate circle/sphere to get boundary conditions for the simple interface problem. We will also show numerical results of examples in both two and three space dimensions.

Simulation of wetting/dewetting process on a permeable and inextensible elastic sheet

Zhen Zhang Southern University of Science and Technology Peoples Rep of China Co-Author(s):    Weidong Shi, Weiqing Ren, Xiaoping Wang, Zhen Zhang

Abstract: In this work, we derive a two-dimension continuum model for wetting/dewetting on a thin, permeable and inextensible elastic sheet in a thermodynamically consistent framework. The derived model satisfies an energy dissipation law. We present an Eulerian weak formulation for the former and an arbitrary Langrangian-Eulerian weak formulation for the latter. The two weak formulations are discretized by finite element method on the moving mesh. Numerical experiments show the convergence and effectiveness of the numerical method.

Thermodynamically consistent hydrodynamic phase-field computational modeling for fluid-structure interaction with moving contact lines

Jia Zhao Binghamton University USA Co-Author(s):

Abstract: In this talk, I will present a novel computational modeling approach for investigating fluid structure interactions with moving contact lines. By applying the generalized Onsager principle, we develop a coupled hydrodynamics and phase-field system in a thermodynamically consistent manner. The resulting partial differential equation (PDE) model consists of the Navier Stokes equations and a nonlinear Allen Cahn type equation, with volume conservation enforced through an additional penalty term. We propose a fully discrete, structure preserving numerical scheme that combines several techniques to solve this coupled PDE system effectively and accurately. Finally, various numerical simulations will be shown to verify the model`s capabilities and demonstrate the scheme`s effectiveness, accuracy, and stability.