Schedule for: 24w5504 - Structured Mesh Methods for Moving Interface and Free Boundary Problems and Applications

A set dinner is served daily between 5:30pm and 7:30pm in the Xianghu Lake National Tourist Resort.

Breakfast is served daily between 7 and 9am in the Xianghu Lake National Tourist Resort

We present a novel Front-Tracking method, the Edge-Based Interface Tracking (EBIT) method, for multiphase-flows simulation. In the EBIT method, the marker points are located on the grid edges and the interface can be reconstructed without storing the connectivity of the markers explicitly, which makes addition or removal of the markers easier than in the traditional Front-Tracking method. The EBIT method also allows almost automatic parallelization due to the localized feature of markers. In our previous paper about the EBIT method[1], we presented the kinematic part of the EBIT method, which includes the interface advection and linear interface reconstruction algorithms. In the current work, we extend the EBIT method by combining the interface advection algorithm with a full Navier-Stokes solver. To identify the reference phase and distinguish ambiguous topological configurations, we introduce a crucial concept, the “color vertex”, into the EBIT method. We calculate the volume fraction basing on the position of markers and the color vertex, then the viscosity and the density based on the reconstructed volume fractions and the surface tension based on the Height-Function method. In addition, we introduce an automatic topology change algorithm into the current EBIT method, which enables the simulation of more complex topologies. A planar version of the new EBIT method is implemented on the free platform Basilisk[1], and is validated by five benchmark test cases: (1) capillary waves, (2) shape oscillations of an inviscid droplet, (3) the Rayleigh-Taylor instability, (4) rising bubble and (5) the Plateau-Rayleigh instability. The results are compared with those obtained previously in a similar code by Volume-Of-Fluid tracking.

Capillary folding is the process of folding planar objects into three-dimensional (3D) structures using capillary force. We propose a 3D model for the capillary folding of thin elastic sheets with pinned contact lines. The energy of the system consists of interfacial energies between the different phases and the elastic energy of the sheet. The later is given by the nonlinear Koiter’s model which allows large deformation of the sheet. From the energy, we derive the governing equations for the static system using a variational approach. We then discuss numerical methods to find equilibrium solutions via a relaxation dynamics. Finally, we present simulation results which are in good agreement with physical experiments and exhibit rich and fully 3D behaviors not captured by previous 2D models. If time allows, I will also discuss moving contact lines on elastic sheets.

Lunch is served daily between 11:30am and 1:30pm in the Xianghu Lake National Tourist Resort

This study presents the development of a multiple-distribution-function lattice Boltzmann model (MDF-LBM) for the accurate simulation of multi-component and multi-phase flow. The model is based on the diffuse interface theory and free energy model, which enable the derivation of hydrodynamic equations for the system. These equations comprise a Cahn-Hilliard (CH) type mass balance equation, which accounts for cross diffusion terms for each species, and a momentum balance equation. By establishing a relationship between the total chemical potential and the general pressure, the momentum balance equation is reformulated in a potential form. This potential form, together with the CH type mass balance equation, is then utilized to construct the MDF-LBM as a coupled convection–diffusion system. Numerical simulations demonstrate that the proposed MDF-LBM accurately captures phase behavior and ensures mass conservation. Additionally, the calculated interface tension exhibits good agreement with experimental data obtained from laboratory studies.

Studying effects of moving particles on fluids is of fundamental importance for understanding particle dynamics and binding kinetics.  We compute the fluid dynamics using an accurate boundary integral method with 3rd order accuracy in space. A unique feature of our method is that we can calculate the stress on the particle surface for a prescribed particle velocity profile. It is well known that a boundary layer develops along an infinite plate under oscillatory motion in a Newtonian fluid. When the flow becomes viscoelastic, however, the boundary layers are fundamentally different than those observed in Newtonian fluids.

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 ≥ 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 break-down 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.

When developing numerical algorithms, computational mathematicians often assume that the problem domain has simple topology and/or geometry. However, a real-world application often has complex topology and/or geometry. Currently, most state-of-the-art methods such as level-set methods, phase-field methods, and VOF methods avoid problems of complex topology and geometry by converting them into numerical PDEs. In contrast, we tackle these problems by tools from topology and geometry. Specifically, we propose Yin sets as a topological model for 2D and 3D continua, give them a complete classification up to homeomorphisms, describe topological changes as sudden switching from one set of Betti numbers to another, develop a Boolean algebra for continua both theoretically and algorithmically, equip the space of Yin sets with a metric to analyze explicit interface tracking methods, and propose fourth- and higher-order methods for interface tracking and curvature estimation. These techniques are collected under the framework of MARS (mapping and adjusting regular semianalytic sets). We further couple MARS to a fourth-order and energy-stable projection Method for solving the incompressible Navier-Stokes equations on moving domains. Results of numerical experiments confirm our analysis and the fourth-order accuracy of the entire solver.

Breakfast is served daily between 7 and 9am in the Xianghu Lake National Tourist Resort

In this talk we report on recent advances about our fictitious domain approach for fluid structure interactions. Our results include existence and uniqueness for the continuous problem, unconditional stability in time and stability in space for the discrete model. We discuss some implementation aspects, including quadrature error estimates for the integration of the coupling term and rigorous estimate of the condition number.

In this talk, I will present sharp interface models with anisotropic surface energy for simulating solid-state dewetting and the morphological evolution of patterned islands on a substrate. We will show how to derive the sharp interface model via thermovariation dynamics, i.e. variation of the interfacial energy via an open curve with two triple points moving along a fixed substrate. The sharp interface model tracks the moving interface explicitly and it is very easy to be handled in two dimensions via arc-length parametrization. An efficient and accurate parameteric finite element method (PFEM) was proposed for the sharp interface models. It is applied to study numerically different setups of solid-state dewetting including short and long island films, pinch-off, hole dynamics, semi-infinite film, tiny particle migration, etc. Our results agree with experimental results very well. In addition, extension to curved substrate and three dimensions will be discussed. Finally, we also present a reduced variatonal model via the Onsager's principle for small particle migration in solid-state dewetting. This is joint works with Wei Jiang, Yifei Li, David J. Srolovitz, Carl V. Thompson, Yan Wang and Quan Zhao.

Consider the following optimal minimization problem in the cylindrical domain $\Omega=D\times(0,1)$: $$\min_{\bar{\mathcal{R}}_\beta^D} \Phi(f)$$ where $\bar{\mathcal{R}}^D_\beta=\left\{f(x)\in L^\infty(\Omega)\colon f(x',x_n)=f(x'),\,\, 0\leq f \leq 1,\,\,\int_D fdx=\beta \right\},$ \noindent $u_f\in W^{1,2}_0(\Omega)$ is the unique solution of $\Delta u_f=0$, and $\Phi(f)=\int_\Omega |\nabla u_f|^2dx$. We show the existence of the unique minimizer. Moreover, we show that for a particular $\alpha>0$ the function $U=\alpha-u_f$ minimizes the functional with with nonlocal obstacle acting on function $V(x')=\int_0^1 U(x', t) dt$ $$\int_\Omega \frac{1}{2}|\nabla U(x)|^2dx +\int_D V(x')^+\,dx',$$ and solves the equation $$\Delta U(x',x_n) = \chi_{\{V>0\}}(x') + \chi_{\{V=0\}}(x') [\partial_\nu U (x',0) + \partial_\nu U (x',1)],$$ where $\partial_\nu U$ is the exterior normal derivative of $U$. Several further regularity results are proven. It is shown that the comparison principle does not hold for minimizers, which makes numerical approximation we developed in \cite{LM} somewhat challenging.

This talk proposes a nonconforming extended virtual element method for solving elliptic interface problems with interface-unftted meshes. The discrete approximation form is presented by adding some special terms along the edges of interface elements and several stabilization terms in the discrete bilinear form. The well-posedness of the discrete scheme is obtained and the optimal convergence is proven under the energy norm. It is shown that all results are independent of the position of the interface relative to the mesh and the contrast between the diffusion coefcients. Furthermore, short edges are allowed to appear in the mesh by modifying the stabilization term of the nonconforming virtual element method. Some numerical experiments are performed to verify the theoretical results.

In diverse natural and industrial applications, multi-component solid-liquid-gas phase change processes, encompassing binary solidification and evaporation, exert a pivotal influence. The intricacies of flow patterns and interface morphology arise from intricate interactions among flow dynamics, temperature variations, and concentration fields (both solutal and vapor). This paper introduces a novel 3D adaptive octree-based sharp interface method \cite{article1, article2} designed to model such phenomena. The method ensures strong coupling among flow, temperature, and concentration fields at the interfaces. The volume-of-fluid (VOF) method is implemented to depict sharp interfaces, with geometric reconstruction employed to delineate distinct phases. Building upon the finite-volume method, the embedded boundary method (EBM) discretizes different phases sharply, enforcing precise jump conditions and conserving heat and mass transfer properties across the interfaces. Rigorous validation of the method's accuracy and robustness is undertaken through a comprehensive set of benchmark test cases. Notably, the efficacy of this sharp interface method is underscored through the exploration of challenging numerical scenarios, including Leidenfrost drop impact, directional solidification of an alloy, and double-diffusive melting of an ice sphere in a salted solution under forced convection

Level-set methods and VOF methods are often said to have the advantage of automatically handling topological changes of the tracked fluid. However, with these methods it appears to be difficult to pinpoint where and when topological changes occur along the interface. It is even not clear what a topological change is. Under the MARS framework, we model 2D and 3D continua as Yin sets, classify Yin sets up to homeomorphisms, equip the space of Yin sets with a Boolean algebra, and view a topological change as the sudden switch from one homeomorphic class to another under the action of the flow map. Then we develop an algorithm that determines the locations and instants of topological changes with third-order accuracy in the max-norm. Numerical results are presented in both two and three dimensions.

Breakfast is served daily between 7 and 9am in the Xianghu Lake National Tourist Resort

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 extensive numerical simulation.

We address an analysis of an Eulerian finite element method used for solving the Navier–Stokes system for incompressible viscous fluid in a time-dependent domain. The domain’s evolution is assumed to be known and independent of the solution to the problem at hand. The numerical method employed in the study combines a standard backward differentiation formula-type time-stepping procedure with a geometrically unfitted finite element discretization technique on a structured mesh. We discuss available error analysis for several velocity–pressure elements that are inf-sup stable and touch on open problems.

Multi-phase flow problems with moving interface play essential roles in numerous daily life activities and industrial processes. In this talk, I will present immersed boundary methods for simulating some moving inteface problems, including: membrane motion inside the two-phase flow, and thermal migration of the surfactant-laden droplets. For membrane motion, the total energy (including kenetic energy, surface energy and bending energy) stable immersed boundary method is developed, tank-treading and tumbling phenomena are simulated. For themral migration of the droplet, both surfactant and non-isothermal condition are considered, besides, non-Newtonian (viscoelastic) effect is also taken into account, results show that both surfatant and elasticity delay the migration of the droplets. Finally, if time allows, I will also present the nonlinear stability study for a surfactant laden viscoelastic thread, where numerical results show that surface viscoelasticity of surfactant can eliminate the satellite droplets in the beads-on-a-string structure at large deformation.

Breakfast is served daily between 7 and 9am in the Xianghu Lake National Tourist Resort

In this talk I will discuss a variety of gradient flow models for multi-phase problems, derived from an energy variational formulation. The models include fractional differential equations, equations describing the interfacial dynamics of immiscible and incompressible two-phase fluids, dendritic crystal growth model, thermal phase change problems etc. The talk starts with a review of the models and numerical methods for these models. Then a new class of time-stepping schemes will be discussed.

In this talk, we will discuss port-Hamiltonian formulations and their numerical discretization for several classes of hyperbolic partial differential equations. Firstly, based on the classical formulations, we derive generalized Hamiltonian formulations of the incompressible Euler equations with a free surface using the language of differential forms. Three sets of variables, including velocity, solenoidal velocity, potential, vorticity, and free surface, are used to represent the incompressible Euler equations with a free surface. Additionally, we derive the corresponding Poisson bracket for these sets of variables and express the Hamiltonian systems using these Poisson brackets. Our main results are the construction and proof of Dirac structures in suitable Sobolev spaces of differential forms for each variable set, which provides the core of any port-Hamiltonian formulation. We obtain discontinuous Galerkin (DG) finite element discretizations of a class for linear hyperbolic port-Hamiltonian dynamical systems. The accuracy and capabilities of the methods developed in this chapter are demonstrated by presenting several numerical experiments.

The swimming dynamics of marine organisms has been the subject of many theoretical and computational studies, especially for microscopic organisms (such as bacteria, spermatozoa or plankton) that exist in the viscous-dominated Stokes flow regime where the Reynolds number (Re) is very small. This previous work has focused on individual microswimmers, although there has been a surge of recent interest in simulating suspensions of large numbers of such swimmers using algorithms that exploit approximate analytical solutions and fast solvers tailored for Stokesian dynamics. However, relatively little attention has been paid to creatures like nematodes that swim in an intermediate regime where inertial and viscous effects are comparable (corresponding roughly to 0.1 < Re < 100). Our objective in this study is to develop a better understanding of the hydrodynamic interactions and collective behaviour in dilute suspensions of these worm-like "mesoswimmers" in the inertial regime. We model the fluid-structure interaction problem using the immersed boundary method, in which the incompressible Navier-Stokes equations capture the fluid dynamics and simple spring-like forces are employed to mimic the elastic properties and muscle contractions of the swimming organisms. We investigate individual swimming dynamics as well as pairwise/collective interactions between multiple swimmers, and show that we can capture observed behaviours such as synchronization of swimming strokes and aggregation.

This work is devoted to introducing new spline quasi-interpolants for the sharp approximation of data in one and several dimensions. The new construction aims to obtain accurate approximations close to singularities of the function from which the data is obtained. The technique relies in an accurate knowledge of the position of the singularity, which can be known or approximated, and that allows for the obtention of accurate approximations of the jumps of the function and its derivatives at this point. With this information, it is possible to compute correction terms for the B-spline bases of the spline that are affected by the presence of singularities. The result is a piecewise smooth reconstruction with full accuracy (meaning by full accuracy, the accuracy of the spline at smooth zones). The numerical experiments presented support the theoretical results obtained.

We develop a second-order Cartesian grid based numerical method to solve moving contact line problems, which are modeled by the incompressible Navier-Stokes equations with the Navier-slip condition and the contact angle condition (CAC). The resulting moving interface problem are solved using a hybrid immersed interface method and parametric finite element method (PFEM). With this method, we detect topological changes in the interface by the inconsistency of neighboring normal vectors, which are directly computed through the PFEM. Second-order accuracy of the proposed method in both the interface and the contact line positions before and after topological changes has numerically validated. Moreover, with the help of the numerical method, the merging and collision dynamics of droplets on the substrates are easily investigated.

Numerical simulation of kinetic equations poses significant challenges due to their inherently high-dimensional nature. This talk introduces a novel geometric approach to achieve model reduction while preserving essential structural properties of the equations under certain conditions. By employing projections onto tangent bundles of finite-dimensional approximate solution manifolds, our framework naturally yields first-order hyperbolic systems. We introduce criteria for selecting Riemannian metrics for kinetic equations, which act as analogues of symmetrizers for first-order PDEs, ensuring hyperbolicity and linear stability. Furthermore, we establish, for the first time, a rigorous connection between the H-theorem for kinetic equations and the linear stability conditions for the reduced models. Joint work with Ruo Li.

The Hiptmair-Xu preconditioner is a popular solver for computing discrete H(curl) and H(div) problems on Euclidean domains. In this talk, I will extend the HX preconditioning to hypersurfaces and present user-friendly preconditioners on unstructured triangulated surfaces. In addition, I will describe a new iterative solver for computing discrete harmonic vector fields on Euclidean domains as well as surfaces.

We propose a fourth-order cut-cell finite-volume method for solving the incompressible Navier-Stokes equations (INSE) with moving boundaries under the MARS framework. The moving boundary is embedded inside a static background Cartesian mesh and tracked by a fourth-order MARS method. Employing the Reynolds transport theorem，we integrate the GePUP formulation of INSE over each time-varying control volume that results from one or multiple fixed control volumes cut by the moving boundary. The integral of a spatial operator over a control volume is approximated by fitting a multivariate polynomial from cell averages over a set of nearby control volumes; this set is generated by coupling AI search algorithm, approximation theory, and group theory. Within each time step, we employ a geometric multigrid method to solve the linear systems with optimal complexity. We simulate the motion of incompressible fluids around an oscillating cylinder and propagation of internal waves in density-stratified flows; our numerical results confirm the fourth-order accuracy both in space and in time.

Breakfast is served daily between 7 and 9am in the Xianghu Lake National Tourist Resort

In this talk we will present a multi-domain model for water and ion transport in optic nerve. The main objective of our model is to investigate the roles of glial cells and perivascular space in potassium clearance under normal and pathological conditions.

An accurate and efficient numerical method has been proposed for degenerate interface problem with extreme conditions such as very big jump ratio, coefficient blow-up and geometric singularity interface. The schemes combine Puiseux series asymptotic technique with augmented and DNN technique to build semi-decoupling and decoupling high accuracy methods for the problem, respectively. Error estimates are obtained. Numerical examples confirm the theoretical analysis and efficiency of the method. We also apply this method for solving time dependent problems and 2D problems.

Clebsch maps offer a framework for representing velocity fields via scalar functions that encapsulate essential information about the flow. For example, closed integral curves of the vorticity field correspond to level sets of the vorticity Clebsch map, making these maps a valuable tool for visualisation and analysis in fluid dynamics. We propose a Clebsch-based method for the simulation of free-surface vortical flows, in which the fluid state is described by a quaternion-valued wave function evolving under the Euler equations with incompressibility constraints. To mitigate numerical instabilities near dynamic interfaces, we integrate a level-set method with a wave-function correction scheme and a wave-function extrapolation algorithm. This combination leverages the Clebsch wave function’s ability to represent complex vortical structures and the level-set method’s capacity for tracking interfacial dynamics, enabling the simulation of vortex-interface interactions with detailed free-surface features on a Cartesian grid. We demonstrate the robustness of the proposed method by simulating various free-surface flow phenomena, including horseshoe vortices, sink vortices, bubble rings, and free-surface wake vortices.