Abstract : The mini-symposium will focus on mathematical aspects of multiscale phenomena in materials and complex fluids. New scientific problems along with novel mathematical techniques and computational tools have emerged from the study of multiscale phenomena, for example, in polycrystalline materials, biomaterials, flow through porous media, as well as liquid crystals, to name a few. The mini-symposium will bring together experts in the area of mathematical aspects of materials and complex fluids and will feature talks on the latest advances in the field that range from mathematical modeling and analysis of partial differential equations to algorithm design, simulation and data analysis.
Organizer(s) : Yekaterina Epshteyn, Chun Liu, Masashi Mizuno
[00142] New perspectives on modeling and analysis of grain growth in polycrystals
Format : Talk at Waseda University
Author(s) :
Yekaterina Epshteyn (University of Utah)
Katayun Barmak (Columbia University)
Chun Liu (IIT)
Masashi Mizuno (Nihon University)
Abstract : Grain growth in polycrystals is a very complex multiscale process. It can be regarded as the anisotropic evolution of a large cellular network, and can be described by a set of deterministic local evolution laws for the growth of an individual grain combined with stochastic models for the interaction between them. In this talk, we will present new perspectives on modeling, simulation and analysis of the evolution of the grain boundary network in polycrystalline materials.
[00183] Entropy dissipation methods for Nonlinear inhomogeneous Fokker-Planck models
Format : Talk at Waseda University
Author(s) :
Masashi Mizuno (Nihon University)
Yekaterina Epshteyn (University of Utah)
Chun Liu (Illinois Institute of Technology)
Chang Liu (University of Utah)
Abstract : This talk presents long-time asymptotic behavior for the nonlinear Fokker-Planck model: First, for the linear Fokker-Planck equation, we reformulate the entropy dissipation methods with the help of the velocity vector in the continuity equation. Next, we derive the evolution equation for the velocity vector for the nonlinear Fokker-Planck model. Finally, we give a sufficient condition to extend the entropy dissipation method to the nonlinear Fokker-Planck model.
[00198] Structure-preserving variational discretizations to generalized gradient flows
Format : Online Talk on Zoom
Author(s) :
Chun Liu (Illinois Institute of Technology)
Yiwei Wang (University of California, Riverside)
Abstract : We'll present a numerical framework for developing structure-preserving variational schemes for various complex fluids models built by the energetic variational approach. The numerical approach starts with the energy-dissipation law of the underlying system and can combine different spatial discretizations, including Eulerian, Lagrangian, particle, and neural-network-based methods. The numerical procedure guarantees the developed schemes are energy stable and can preserve the intrinsic physical constraints. Several applications of this numerical approach will be discussed.
[00248] Towards upscaling and simulation of coupled [THM] systems with applications to permafrost modeling
Format : Online Talk on Zoom
Author(s) :
Malgorzata Peszynska (Oregon State UniversityOregon State University)
Naren Vohra (Oregon State University)
Abstract : In the talk we discuss our progress towards multiscale computational modeling of coupled processes in permafrost, frozen ground which is ubiquitous in the Arctic. We focus on its active layer close to the surface whose depth is changing due to changing climate conditions, and we model its thermal state (temperature, and phase status: frozen or thawed), hydrological conditions, and mechanical response to hydrological and thermal controls. We present convergence studies and upscaling from porescale.
00137 (2/3) : 3D @E802 [Chair: Yekaterina Epshteyn]
[00202] Diffuse-interface approach to competition between viscous flow and diffusion in pinch-off dynamics
Format : Talk at Waseda University
Author(s) :
Weizhu Bao (National University of Singapore)
Fukeng Huang (National University of Singapore)
Tiezheng Qian (Hong Kong University of Science and Technology)
Abstract : In this talk, we present numerical simulations for the pinch-off dynamics in the Stokes regime and the
diffusion-dominated regime by adopting the Cahn-Hilliard-Navier-Stokes model derived by applying
Onsager's variational principle. The Cahn-Hilliard-Navier-Stokes model is solved by using an accurate
and efficient spectral method in a cylindrical domain with axisymmetry. Ample numerical examples are
presented to show the pinch-off processes in the Stokes regime and the diffusion-dominated regime,
respectively. In particular, the crossover between these two regimes is investigated numerically and
analytically to reveal how the scaling behaviors of similarity solutions are to be qualitatively changed as
the characteristic length scale is inevitably accessed by the pinching neck of the interface. Discussions
are also provided for numerical examples that are performed for the breakup of long liquid filaments
and show qualitatively different phenomena in different scaling regimes. This is a joint work with
Fukeng Huang and Tiezheng Qian.
[00195] Multiscale analysis of nonlinear material models with carrier kinetics
Format : Talk at Waseda University
Author(s) :
Qing Xia (KTH Royal Institute of Technology)
Ludmila Prokopeva (Purdue University)
William Henshaw (RPI)
Alexander Kildishev (Purdue University)
Gregor Kovacic (RPI)
Jeffrey Banks (RPI)
Donald Schwendeman (RPI)
Abstract : In this talk, we introduce Maxwell-Bloch equations for modeling interactions between light and nonlinear optics. The model is based on real-valued rate equations, which describe kinetics of electrons between the ground state and excited states in the multi-level atomic system. We will show the rate equation approach is connected to the complex-valued density matrix approach via the Schrödinger's equation. Different multi-level atomic systems will be shown and multi-scale analysis is performed.
[00209] Energetic-variational particle-based method for Fokker-Planck Models.
Format : Talk at Waseda University
Author(s) :
Kaitlin O'Dell (University of Utah)
Yekaterina Epshteyn (University of Utah)
Chun Liu (Illinois Institute of Technology)
Abstract : Fokker-Planck models with energy-dissipation structures arise in many scientific and engineering applications. We present a novel energetic-variational particle-based approach for simulation of such nonlinear high-dimensional Fokker-Planck systems which cannot be solved using traditional numerical methods. First, we compare the performance of the proposed particle-based scheme with the finite-volume structure-preserving method on low-dimensional Fokker-Planck systems. Then, we apply new method for the analysis of the high-dimensional Fokker-Planck equations that describe grain boundaries dynamics in polycrystalline materials.
[00214] Phase transitions in near-liquid solids
Format : Talk at Waseda University
Author(s) :
Yury Grabovsky (Temple University)
Lev Truskinovsky (ESPCI)
Abstract : We consider a class of two-dimensional compressible Hadamard materials with very small shear modulus and a double-well potential for the energy as a function of specific volume. Such energy is not rank one convex and is in need of relaxation. While computing energy relaxation seems hopeless, identifying its binodal - the boundary separating stable and unstable homogeneously deformed configurations seems more tractable. I will describe several necessary and sufficient conditions for stability and show how they allow us to bound hydrostatic strains on the binodal. Then, in a surprising twist, I will show how the optimality of our bound would generate an excellent approximation to the entire binodal. This is a joint work with Lev Truskinovsky.
[00243] A finer singular limit of the Kobayashi-Warren-Carter type functional and its gradient flow
Format : Talk at Waseda University
Author(s) :
Masaaki Uesaka (Arithmer, Inc.)
Yoshikazu Giga (the University of Tokyo)
Koya Sakakibara (Okayama University of Science)
Jun Okamoto (Kyoto University)
Abstract : We consider the singular limit of the Kobayashi-Warren-Carter type energy, which is derived from the physical model of grain boundary motion in polycrystals. The KWC-type energy contains the weighted total variation, and hence we must employ a topology finer than $L1$ to capture the singular limit's behavior in detail. We shall explain the key result of the singular limit of KWC-type energy and the behavior of its gradient flow.
[00249] Variational modeling of fluid in poroelastic medium
Format : Talk at Waseda University
Author(s) :
Arkadz Kirshtein (Tufts University)
James Haley Adler (Tufts University)
Xiaozhe Hu (Tufts University)
Abstract : In this talk I will discuss modeling fluid flow through a deformable porous medium. I will start from introducing a variational approach for fluids and elasticity in Lagrangian coordinates. Next I will discuss an existing approach based on Biot's consolidation model. Ultimately I will introduce a system derived using energetic variational approach and discuss numerical methods and simulations based on it.
[00226] Phase field model for volume-preserving mean curvature flow
Format : Talk at Waseda University
Author(s) :
Keisuke Takasao (Kyoto University)
Abstract : In this talk, we show a global existence of the weak solution for the volume-preserving mean curvature flow.
To construct the weak solution, we use the Allen-Cahn equation with non-local term given by the penalty method.
We prove the $L^2$-estimate of the non-local term and the monotonicity formula for the equation, which are the keys of the proof of the existence theorem.
[00257] A unified continuum model for grain boundary dynamics incorporating microscopic structure
Format : Talk at Waseda University
Author(s) :
Yang Xiang (Hong Kong University of Science and Technology)
Abstract : We develop a unified continuum framework to account for the underlying line defect mechanisms. Conditions on the continuum level are imposed to account for the underlying microscopic mechanisms, which makes the continuum model more efficient to describe the collective behaviors of grain boundary networks at larger length scales.
