Abstract : Dynamics of the interface, like deformation and reaction, play an important role in biology like cell aggregation, and industry like water-proof material. Modeling and simulation of the dynamics of the interface are challenging since multiphase-flow and multiphysics fields are evolved. Recently, machine learning-based methods like Neural networks are introduced to solve the obtained nonlinear coupled system more efficiently. The purpose of this symposium is to bring together researchers working on modeling, theory, and numerics for interface problems, to share the latest advances in the field, and to provide a forum for joint collaborations.
[01225] Solving elliptic interface problems using neural networks
Format : Talk at Waseda University
Author(s) :
Ming-Chih Lai (National Yang Ming Chiao Tung University)
Abstract : In this talk, we shall introduce a series of neural network methodology for solving elliptic interface problems that comprise of variable-coefficient Poisson equation and Stokes equations with interfaces. There are three novel
features in the present network; namely, (i) jump discontinuities are accurately captured, (ii) it is completely
shallow, comprising only one hidden layer, (iii) it is completely mesh-free so the problems in irregular domains
with irregular interfaces can be handled easily. Numerical results show better accuracy than the traditional finite difference method such as the immersed interface method.
[01276] Role of Cohesive Fiber-Fiber Interactions in Fibrin Networks
Format : Online Talk on Zoom
Author(s) :
Zhiliang Xu (University of Notre Dame)
Abstract : A novel structural mechanism of fibrin clots' mechanical response to external tensile loads is tested using newly developed three-dimensional computational model. This mechanism, underlying local strain-stiffening of individual fibers as well as global stiffening of the entire network, is based on previously neglected nascent cohesive pairwise interactions between individual fibers (crisscrossing) in fibrin networks formed under tensile load. The computational model enabled us to study structural details and quantify mechanical effects of the fiber-fiber cohesive crisscrossing during stretching of fibrin gels at various spatial scales. The results show that the nascent cohesive crisscrossing of fibers in stretched fibrin networks comprise an underappreciated important structural mechanism underlying the mechanical response of fibrin to (patho)physiological stresses that determine the course and outcomes of thrombotic and hemostatic disorders.
[01217] Variational Lagrangian schemes for interface problems
Format : Online Talk on Zoom
Author(s) :
Yiwei Wang (University of California, Riverside)
Chun Liu (Illinois Institute of Technology)
Abstract : In this talk, we present a systematic framework for deriving variational numerical methods for generalized diffusions and gradient flows. The numerical framework is based on the energy-dissipation law, which describes all the physics and the assumptions in each system and can combine different types of spatial discretizations including Eulerian, Lagrangian, and particle-based approaches. The resulting semi-discrete equation inherits the variational structures from the continuous energy-dissipation law. We apply such an approach to construct variational Lagrangian schemes to several interface problems, including the Allen-Cahn type phase-field models and the porous medium equation. Numerical examples show the advantages of the variational Lagrangian schemes in capturing thin diffuse interfaces and free boundaries. This is joint work with Professor Chun Liu.
[01256] Helical organization of DNA-like liquid crystal filaments in cylindrical viral capsids
Format : Talk at Waseda University
Author(s) :
Pei Liu (Florida Institute of Technology)
Abstract : We study equilibrium configurations of ds-DNA in a cylindrical viral capsid. The state of the encapsidated DNA consists of a disordered inner core enclosed by an ordered outer region, next to the capsid wall. The DNA configuration is described by a unit helical vector field, tangent to an associated center curve, passing through properly selected locations. We postulate an expression for the energy of the encapsulated DNA based on that of columnar chromonic liquid crystals. A thorough analysis of the Euler--Lagrange equations yields multiple solutions. We demonstrate that there is a trivial, non-helical solution, together with two solutions with nonzero helicity of opposite sign. Using bifurcation analysis, we derive the conditions for local stability and determine when the preferred coiling state is helical. The bifurcation parameters are the ratio of the twist versus the bend moduli of DNA and the ratio between the sizes of the ordered and the disordered regions.
[01231] A phase-field model and an energy-law preserving method for vesicles
Format : Talk at Waseda University
Author(s) :
Ping Lin (University of Dundee)
Abstract : We will first show how to develop a thermodynamically consistent phase field model for the binary incompressible (quasi-incompressible) fluid. We then show how to apply the idea to model vesicle motions and deformations through a narrowed channel. We will also introduce a Lennard-Jones type of interaction potential for vesicle-vesicle and vesicle-channel wall interactions. An energy law preserving computational method is then developed for the model. A few computational examples including vesicle-wall and multi-vesicle interactions will be presented to demonstrate the model and the computational method.
[01394] Machine Learning of Self Organization from Observation
Format : Online Talk on Zoom
Author(s) :
Ming Zhong (Illinois Institute of Technology)
Abstract : Self organization (also known as collective behaviors) can be found in studying crystal formation, aggregation of cells/animals, social behaviors of insects and humans, etc. It is a challenging task to understand such behaviors from the mathematical point of view. We offer a statistical/machine learning approach to understand these behaviors quantitatively from observation data; moreover, our learning approach can aid in validating and improving the modeling of collective behaviors.
We develop a learning framework to derive physically meaningful models to explain self organization from observation. We also investigate the stead state properties of our learned models, and extend the learning framework to include more complicated structures. We extend the learning approach to infer dynamical models for agents constrained on Riemannian manifolds. We further improve our learning capability to infer interaction feature variables as well as interaction kernels. We even study the effectiveness of our learning method on the NASA Jet Propulsion Laboratory's modern Ephemerides. Upon careful inspection of our model, we discover that it even captures potion of the general relativity effects. A complete learning theory on second-order systems is presented, as well as two new models on emergence of social hierarchy and combination of flocking and synchronization.
[01214] Free boundary problems in cardiovascular diseases
Format : Talk at Waseda University
Author(s) :
Wenrui Hao (Penn State University)
Abstract : I will present several free boundary problems based on the pathophysiology of cardiovascular disease. As an example, a mathematical model of atherosclerosis, based on this modeling approach, provides a personalized cardiovascular risk by solving a free boundary problem. Some interesting mathematical problems are also introduced by this new model to help us understand cardiovascular risk.
[05636] Buckling on Erythrocyte Membranes in Narrow Capillary Flows
Format : Online Talk on Zoom
Author(s) :
Deyun Liu (Shanghai Jiao Tong University)
Kazuyasu Sugiyama (Osaka University)
Xiaobo Gong (Shanghai Jiao Tong University)
Abstract : Experiments and numerical simulations are conducted to understand the non-axisymmetric deformation of a single RBC in narrow tubes and the hydrodynamics associated. With decreasing capillary numbers, the stable deformation shapes of RBCs change from axisymmetric bullet shape to asymmetric deformation with buckling under the major effect of the negative pressure difference across cell membrane at the rear part of the deformed RBCs.
[01249] A phase field model for droplets suspended in electrolyte solution
Format : Talk at Waseda University
Author(s) :
Yuzhe Qin (Shanxi University)
Huaxiong Huang (Beijing Normal University, Zhuhai)
Shixin Xu (Duke Kunshan Univeristy)
Zilong Song (Utah State University)
Abstract : In this talk, we consider modeling the deformation of droplets suspended in electrolyte solution with the phase field method. Firstly, we derive the Poisson-Nernst-Planck-Navier-Stokes phase field model based on energy variational approach method. Secondly, we accomplish the asymptotic analysis for our model after nondimensionalizing the system and the sharp interface limits of our proposed model is consistent to the sharp interface model. We take a series of numerical experiments to validate the correctness and effectiveness about our model.
[01220] A deterministic particle simulation for micro-macro viscoelastic flows
Format : Talk at Waseda University
Author(s) :
Xuelian Bao (Beijing Normal University)
Chun Liu (Illinois Institute of Technology)
Yiwei Wang (University of California, Riverside)
Abstract : We propose a deterministic particle-FEM discretization to micro-macro models of dilute polymeric fluids, which combines a finite element discretization to the macroscopic fluid dynamic equation with a variational particle scheme to the microscopic Fokker-Planck equation. The discretization is constructed by a discrete energetic variational approach, and preserves the microscopic variational structure in the semi-discrete level. All numerical examples demonstrate the accuracy and robustness of the proposed deterministic particle-FEM approach.
[01227] A Bubble Model for the Gating of Kv Channels
Format : Online Talk on Zoom
Author(s) :
zilong song (Utah State University)
ROBERT EISENBERG (Illinois Institute of Technology)
Shixin Xu (Duke Kunshan Univeristy)
Huaxiong Huang (York University)
Abstract : Voltage-gated Kv channels play fundamental roles in many biological processes. In this talk, we propose a bubble model coupled with a Poisson-Nernst-Planck (PNP) system to capture the key characteristics, particularly the delay in the opening of channels. The coupled PNP system is solved numerically by a finite-difference method and the solution is compared with an analytical approximation. The predicted ensemble average of the currents and the Cole-Moore delay is consistent with experimental observations.
