Abstract : The phase field method and its relevant extensions have been widely used in various applications, including phase separations, crystal growth, and solid fracture dynamics. Meanwhile, it is still an active research field to develop thermodynamically consistent phase field models, design accurate, efficient, and stable numerical algorithms for these models, and apply them to various application problems. This mini-symposia brings together experts with diverse backgrounds in numerical analysis, PDE modeling and mathematical biology, machine learning, and data science, but with the same interest in phase field method and its relevant extension. Through this mini-symposia, we aim to foster active interdisciplinary discussions.
[05280] Approximating Structurally Unstable Over-determined Systems of PDEs
Format : Online Talk on Zoom
Author(s) :
Qi Wang (University of South Carolina)
Abstract : Models for nonequilibrium phenomena are normally consisted of differential equations, from which some physically important relations can be deduced. Together, they form an extended, over-determined system of equations which is normally structurally unstable. Structure-preserving approximation thus ensues, which are known numerically as structure-preserving algorithms/schemes. In this presentation, I will discuss some structure-preserving numerical strategies for developing numerical algorithms for thermodynamically consistent multiphase materials models, which are derived from thermodynamical principles.
[04941] Phase-field modelling of three-phase solidification with density variation
Format : Online Talk on Zoom
Author(s) :
Pengtao Yue (Virginia Tech)
Jiaqi Zhang (Beijing Normal University-Hong Kong Baptist University United International College)
Yichen Li (Virginia Tech)
Abstract : We present a non-isothermal quasi-incompressible phase-field model for three-phase solidification that involves water, ice, and air. Water-ice phase transition and water-air interface are handled by the Allen-Cahn and Cahn-Hilliard equations, respectively. Constitutive relations are derived based on non-negative entropy production. Our model automatically captures curvature and pressure effects as dictated by the Gibbs-Thomson and Clausius-Clapeyron equations, as well as volume expansion during solidification. Numerical results on tip formation of a freezing droplet will be presented.
[03012] Energy stability analysis and error estimate of a maximum bound principle preserving scheme for the dynamic Ginzburg--Landau equations under the temporal gauge
Author(s) :
Zhonghua Qiao (The Hong Kong Polytechnic University)
Abstract : This paper proposes a decoupled numerical scheme of the time-dependent Ginzburg--Landau equations under the temporal gauge. For the magnetic potential and the order parameter, the discrete scheme adopts the second type Ned${\rm \acute{e}}$lec element and the linear element for spatial discretization, respectively; and a linearized backward Euler method and the first order exponential time differencing method for time discretization, respectively. The maximum bound principle (MBP) of the order parameter and the energy dissipation law in the discrete sense are proved. The discrete energy stability and MBP-preservation can guarantee the stability and validity of the numerical simulations, and further facilitate the adoption of adaptive time-stepping strategy, which often plays an important role in long-time simulations of vortex dynamics, especially when the applied magnetic field is strong. An optimal error estimate of the proposed scheme is also given. Numerical examples verify the theoretical results of the proposed scheme and demonstrate the vortex motions of superconductors in an external magnetic field.
[04302] Multi-phase-field modeling of grain growth and multiphase flow in additive manufacturing
Format : Online Talk on Zoom
Author(s) :
Tomohiro Takaki (Kyoto Institute of Technology)
Abstract : A multi-phase-field model for predicting material microstructures formed during the powder bed fusion additive manufacturing of a metallic alloy is developed. Here, multi-phase-field models for solidification and multiphase flow problems are coupled to express polycrystalline solidification with melt flow in the melt pool.
[02938] Multiscale topology optimization method for lattice materials
Author(s) :
Yibao Li (Xi'an Jiaotong University)
Qing Xia (Xi'an Jiaotong University)
Xin Song (Xi'an Jiaotong University)
Qian Yu (Xi'an Jiaotong University)
Binhu Xia (Xijing University )
Abstract : In this talk, we will introduce an efficient multiscale topology optimization method for lattice materials. In macro-scale, we present a second-order unconditionally energy stable schemes for the topology optimization problem. Using porous media approach, our objective functional composes of five terms including mechanical property, Ginzburg-Landau energy, two penalized terms for solid and the volume constraint. A Crank-Nicolson method is proposed to discrete the coupling system. We prove that our proposed scheme is unconditionally energy stable. In macro-scale, we propose a simple volume merging method for triply periodic minimal structure. A modified Allen–Cahn type equation with a correction term is proposed. The mean curvature on the surface will be constant everywhere at the equilibrium state. Computational experiments are presented to demonstrate the efficiency of the proposed method.
[04437] An efficient nonsmooth global optimization-based bound-preserving approach for the Cahn-Hilliard equation
Author(s) :
Xiangxiong Zhang (Purdue University)
Abstract : It is quite difficult to construct bound-preserving schemes for many high order time-dependent PDEs. For instance, it is difficult to prove that the Cahn-Hilliard equation with polynomial potential admits a bound-preserving solution, yet a bound-preserving numerical solution is often preferred. Instead of directly constructing a bound-preserving scheme, we consider a global optimization based post processing in each time. Due to the nonsmooth terms in the cost function, such an optimization based approach has often been regarded as inefficient. However, it is possible to obtain an efficient solver by using optimal parameters obtained from asymptotic convergence rate formula. We demonstrate that the selection of optimization algorithm parameters from combing such an asymptotic convergence rate formula with time continuation in a time-dependent problem can give an efficient high order accurate bound-preserving post-processing solver, which costs O(n) per time step.
[04079] New unconditionally stable higher-order consistent splitting schemes for the Navier-Stokes equations
Author(s) :
JIE SHEN (Purdue University)
Fukeng Huang (National University of Singapore)
Abstract : The consistent splitting schemes for the Navier-Stokes equations decouple the computation of pressure and velocity, and do not suffer from the splitting error. However, only the first-order version of the consistent splitting schemes is proven to be unconditionally stable for the time dependent Stokes equations.
We construct a new class of consistent splitting schemes of orders two to four for Navier-Stokes equations based on Taylor expansions at time $t_{n+k}$ where $k\ge 1$ is a tunable parameter. We show that, for some suitable choices of $k$, they are unconditionally stable for the time dependent Stokes equations, and by combining them with the generalized scalar auxiliary variable (GSAV) approach, we construct, for the very first time, unconditionally stable (in H^1 norm) and totally decoupled schemes of orders two to four for the velocity and pressure, and provide rigorous optimal error estimates. We shall also present some numerical results to show the computational advantages of these schemes.
[05375] Efficient decoupling energy stable approach for coupled type gradient flow systems with anisotropy for alloys
Author(s) :
Xiaofeng Yang (University of South Carolina)
Abstract : The multi-component alloy phase-field model is a highly complex coupled gradient-like flow model that involves the coupling of multiple Allen-Cahn/Cahn-Hilliard equations with different types of flow field equations. Our ultimate goal is to develop efficient numerical algorithms of the decoupling type for this model. In our initial attempt, we focus on designing a second-order linear unconditional energy stable scheme for the solidification of pure metal, which is further coupled with free flow incorporating Darcy's force. The numerical method is primarily based on the derivative class of the IEQ (Invariant Energy Quadratization) method, which has gained prominence in recent years. Specifically, we combine the so-called EIEQ method with the modified projection method and employ the novel ZEC (Zero-Energy-Contribution) decoupling method to obtain the desired numerical scheme. This approach is versatile and can be applied to a wide range of models involving flow, magnetic, and electric field coupling. To validate the effectiveness of the scheme, we conduct extensive 2D and 3D numerical simulations as well.
[04765] Mathematical modeling and numerical approximation of bulk-surface model
Format : Online Talk on Zoom
Author(s) :
Xueping Zhao (University of Nottingham Ningbo China)
Abstract : In biological systems, many molecules, such as proteins and RNAs can phase separate and form liquid condensates in living cells. Many of those biological molecules can bind to the biological surfaces with some domains. How the kinetic processes are affected by surface binding is still unknown. Here, we derive the governing equations of the bulk-membrane coupled system with membrane-binding using irreversible thermodynamic theory. A three-dimensional numerical solver for kinetic equations is developed to study the effects of membrane binding on the various kinetic process. Our results suggest that membrane binding play crucial roles in the underlying physical principles(e.g. time scales) of kinetic processes of the bulk-membrane system.
[03552] High-order exponential integrators for semilinear parabolic equations with nonsmooth data
Author(s) :
Shu MA (City University of Hong Kong)
Buyang Li (The Hong Kong Polytechnic University)
Abstract : A multistep exponential integrator is proposed for the semilinear parabolic equation with nonsmooth initial data, using variable stepsizes and contour integral approximations to address the initial singularity. The approach is extended to the semilinear subdiffusion equation with nonsmooth initial data. We propose an exponential convolution quadrature that combines contour integral representation of the solution, quadrature approximation of contour integrals, multistep exponential integrators for ordinary differential equations, and locally refined stepsizes to resolve the initial singularity. The proposed k-step exponential integrator and exponential convolution quadrature can have kth-order convergence for bounded measurable solutions of the semilinear parabolic and subdiffusion equations, respectively, based on the natural regularity of the solutions corresponding to the bounded measurable initial data.
[02279] Discovery of Governing Equations with Recursive Deep Neural Networks
Author(s) :
Jia Zhao (Utah State University)
Abstract : In this talk, I will focus on the model discovery problem when the data is not efficiently sampled. This is common due to limited experimental accessibility and labor/resource constraints. Specifically, we introduce a recursive deep neural network (RDNN) for data-driven model discovery. By embedding the known physics knowledge, this recursive approach can retrieve the governing equation in a simple and efficient manner, and it can significantly improve the approximation accuracy by increasing the recursive stages.
