Programs and Participants

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Program Schedules

Wednesday, June 10, 2015

09:30 ~ 10:00 Coffee  
10:00 ~ 11:00 Introduction to Complex Materials (slide) Carme Calderer (University of Minnesota)
11:15 ~ 12:15 The Landau-de Gennes Model for Nematic Liquid Crystalline Films (slide) Dmitry Golovaty (University of Akron)
12:15 ~ 13:15 Lunch  
13:15 ~ 14:15 Defects in Landau-de Gennes Theory (slide) Jonathan Robbins (University of Bristol)
14:30 ~ 15:30 Active Matter Models and Their Applications in Life Science Qi Wang (University of South Carolina / Beijing Computational Science Research Center)

Thursday, June 11, 2015

09:00 ~ 10:00 Numerical Methods for Elasticity (slide) Douglas Arnold (University of Minnesota)
10:15 ~ 11:15 Numerical Schemes for Materials with Fine Scale Structure (slide) Noel Walkington (Carnegie Mellon University)
11:30 ~ 12:30 A Finite Element Method for Nematic Liquid Crystals with Variable Degree of Orientation (slide) Shawn Walker (Louisiana State University)
12:30 ~ 13:30 Lunch  
13:30 ~ 14:30 Mechanical Metamaterials (slide) Graeme Milton (University of Utah)
14:45 ~ 15:45 Bilayer Plates: Model Reduction, Discrete Gradient Flow and Gamma-Convergent Finite Element Approximation (slide) Ricardo Nochetto (University of Maryland)
20:00 ~ Workshop Dinner (at Louise Restaurant) map

Friday, June 12, 2015

09:00 ~ 10:00 A Membrane Theory for Swelling Polymer Gels (slide, video1, video2) Alessandro Lucantonio (SISSA)
10:15 ~ 11:15 Cohesive Dynamics and Fracture (slide) Robert Lipton (Louisiana State University)
11:30 ~ 12:30 Modelling Colloidal Particles in a Liquid Crystal Matrix (slide) Paula Dassbach (University of Minnesota)
12:30 ~ 13:30 Lunch  
13:30 ~ 14:30 Mathematical Problems in Gels and Applications (slide) Carme Calderer (University of Minnesota)
14:45 ~ 15:45 Poroelasticity in Geoscience and Medicine (slide1, slide2) Jan Nordbotten & Kent-Andre Mardal (University of Bergen / University of Oslo)

 

Abstracts

 

The Landau-de Gennes Model for Nematic Liquid Crystalline Films

Speaker: Dmitry Golovaty

Abstract:
I will discuss the behaviour of the Landau-de Gennes model for a nematic liquid crystalline film in the limit of vanishing thickness. In this asymptotic regime, surface energy has a significant influence on the structure of the minimizers of the derived two-dimensional energy.

 

Defects in Landau-de Gennes theory

Speaker: Jonathan Robbins

Abstract:
We present some recent results concerning point defects in liquid crystals in two dimensions within the Landau-de Gennes model.  In the deep nematic regime, we establish the existence of global minimisers of the Landau-de Gennes energy for defects of arbitrary degree, and obtain explicit profiles in the limit of vanishing elastic constant.  The stability of index-1/2 defects under relaxed assumptions and the case of unequal elastic constants will be briefly discussed.

 

Active Matter Models and Their Applications in Life Science

Speaker: Qi Wang

Active matter is a type of materials that energy is converted at the molecular level from chemical form to mechanical form providing energy input to the molecular motion. Active matter is abundant in nature and man-made materials. Faithfully modelling active matter is cutting-edge research. I will present a modelling framework for developing mathematical models for multiphase complex fluids that involve active matter. Then, I will discuss some basic properties of fluid flows of active matter and numerical schemes that can be used to simulate the complex fluid flows. Numerical simulations of several biological systems will be presented including cell motion, mitosis, and vesicles filled with the active matter.

 

Numerical Schemes for Materials with Fine Scale Structure

Speaker: Noel J. Walkington

Abstract:
Many material models consist of the momentum equation coupled to an equation modelling the structure of the material.  Examples include liquid crystals, polymers, and crystalline solids undergoing plastic deformation.  These systems posses a Hamiltonian structure which reveals the subtle structure of the terms coupling of the equations, and a delicate balance between inertia, transport, and dissipation. This talk will focus on the development and analysis of numerical schemes which inherit the Hamiltonian structure, and hence stability, of the continuous problem. Examples of schemes to approximate the Ericksen Leslie equations, Oldroyd-B fluids, and problems in plasticity, will be presented it illustrate the mathematical and numerical properties of this class of materials.

 

A Finite Element Method for Nematic Liquid Crystals with Variable Degree of Orientation

Speaker: Shawn W. Walker

Abstract:
We present a finite element method (FEM) for computing equilibrium configurations of liquid crystals with variable degree of orientation. The model consists of a Frank-like energy with an additional "s" parameter that allows for line defects with finite energy, but leads to a degenerate elliptic equation for the director field.  Our FEM uses a special discrete form of the energy that does not require regularization, and allows us to obtain a stable (gradient flow) scheme for computing minimizers of the energy.  Simulations in 2-D and 3-D are presented to illustrate the method.

 

Mechanical Metamaterials

Speaker: Graeme Milton

Abstract:
Composite materials can have properties unlike any found in nature, and in this case they are known as metamaterials. Materials with negative Poisson's ratio or negative refractive index are now classic examples. The effective mass density, which governs the propagation of elastic waves in a metamaterial can be anisotropic, negative, or even complex. Even the eigenvectors of the effective mass density tensor can vary with frequency. We show that metamaterials can exhibit a "Willis type behaviour" which generalizes continuum elastodynamics. Non-linear metamaterials are also interesting and a basic question is what non-linear behaviours can one get in periodic materials constructed from rigid bars and pivots? It turns out that the range is enormous. Materials for which the only easy mode of macroscopic deformation is an affine deformation, can be classed as unimode, bimode, trimode, ..., hexamode, according to the number of easy modes of deformation. We give a complete characterization of possible behaviours of nonlinear unimode materials.

 

Bilayer Plates: Model Reduction, Discrete Gradient Flow and Gamma-Convergent Finite Element Approximation

Speaker: Ricardo H. Nochetto

Abstract:
The bending of bilayer plates is a mechanism which allows for large deformations via small externally induced lattice mismatches of the underlying materials. We discuss its mathematical modelling, which consists of a nonlinear fourth order problem with a pointwise isometry constraint. We devise a finite element discretization based on Kirchhoff quadrilaterals and prove its Gamma-convergence. We propose an iterative method that decreases the energy and study its convergence to stationary configurations. We explore its performance, as well as reduced model capabilities, via several insightful numerical experiments involving large (geometrically nonlinear) deformations. This work is in joint with S. Bartels (Freiburg) and A. Bonito (Texas A&M).

 

A Membrane Theory for Swelling Polymer Gels

Speaker : Alessandro Lucantonio

Abstract:
Stimuli-responsive materials deform in response to non-mechanical stimuli, such as temperature, pH, or humidity changes. These materials are employed, for instance, in shape-morphing applications, where the material is programmed to achieve a target shape upon activation by an external trigger, and as coating layers to alter surface properties of bulk materials, such as the characteristics of spreading and absorption of liquids. In these applications, stimuli-responsive materials are often in the form of membranes. In particular, polymer gel membranes experience swelling or shrinking when their solvent content changes and the non-homogeneous swelling field may be exploited to control their shape. Here, we develop a theory of swelling material surfaces to model polymer gel membranes and demonstrate its features by studying numerically applications in the contexts of biomedicine and micro-motility. We also specialize the theory to thermo-responsive gels, which are made of polymers that change their affinity with solvent as a function of temperature.

 

Cohesive Dynamics and Fracture

Speaker: Robert P. Lipton

Abstract:
Dynamic brittle fracture is a multiscale phenomena operating across a wide range of length and time scales. Apply enough stress or strain to a sample of brittle material and one eventually snaps bonds at the atomistic scale leading to fracture of the macroscopic specimen. At present there is a growing demand for new fracture models capable of predicting complex fracture patterns inside materials used in modern infrastructure. The peridynamic formulation introduced in the work of Silling 2000 is a promising method for modeling free crack propagation. Here we work with the peridynamic formulation and introduce a new type of nonlocal, nonlinear, cohesive continuum model for assessing the deformation state inside a cracking body. In this model short-range forces between material points are initially elastic and then become unstable and soften beyond a critical relative displacement. The dynamics inside the deforming body selects whether a material point lies inside or outside the ``process zone'' associated with nonlinear behavior corresponding to softening. This is in contrast to a classic cohesive zone fracture model that collapses the process zone onto predetermined surfaces and assumes linear elastic deformation away from these surfaces. Here the natural length scale that controls the size of the process zone is the radius of nonlocal interaction between material points. An explicit inequality is identified showing how the length scale of nonlocal interaction controls the volume of the process zone. The volume of the process zone is shown to vanish as the nonlocal interaction distance is decreased to zero. We apply Gamma convergence arguments coming from the theory of image processing to find that the limiting dynamics has an energy density associated with a process zone confined to a surface. Distinguished limits of cohesive evolutions are identified and are found to have both bounded linear elastic energy and Griffith surface energy. The limit dynamics corresponds to the simultaneous evolution of linear elastic displacement described by the classic wave equation together with a fracture set across which the displacement is discontinuous.

 

Modelling Colloidal Particles in a Liquid Crystal Matrix

Speaker: Paula Dassbach

Experiments show that a colloidal particle placed in a liquid crystal matrix will create a defect in the surrounding medium. This defect is dependent on factors such as particle size, anchoring strength, and the size of the container relative to the particle size. Our goal is to find a minimizer for the Landau-de Gennes model in order to replicate these defects numerically. For this, we assume strong anchoring conditions are prescribed on the walls of the container as well as our particle and allow the size of the container and colloidal particle to vary. The incompatibility of the liquid crystal anchoring on the container and the surface of the particle produce defects which in the numerical simulations which match the observed experimental data.

 

Mathematics of Liquid Crystal Electrokinetics

Speaker: Maria-Carme Calderer

Abstract:
The presence of electric ions in liquid crystals give rise to a host of new electrokinetic phenomena not encountered in isotropic fluids. The electric charge separation in the latter occurs by pre-existing electrostatic double-layers surrounding dielectric or metal colloidal particles in the liquid; fluid motion that follows by application of an electric field. In liquid crystals, charge separation over large regions occurs by gradients of the molecular alignment, either due to defects present in the system or by patterned substrate. The defects emerge as a result of incompatible boundary conditions. Liquid crystal electrokinetics, electrophoresis and electro-osmosis, is showing remarkable new non-linear features not found in classical isotropic linear electrokinetics. It shows the capability to control and manipulate microfluidic phenomena, such as three-dimensional particle path, and separation and sorting, at a very fine level. In particular, it may allow for AC-driven electrophoresis. From a different point of view, analogies between ionic liquid crystals and biological systems, with scale gaps of the order of 10−6 allow for model sharing. Certain types of liquid crystals are found to form packing structures analogous to those of viral DNA confinement, such as bacteriophages. We present a survey of relevant experiments and their models, and analyze well-posedness of time-dependent flow problems and as well as free boundary problem of DNA confinement. The experiments discussed in this presentation were performed at the laboratory of Dr. Oleg Lavrentovich at Kent State University, Ohio, USA.

 

Stable finite volume methods for coupled Darcy flow and deformation in geological media

Speaker: Jan Nordbotten

Abstract:
The rise of advanced geoengineering applications such as enhanced geothermal systems together with energy and CO2 storage, leads to settings where coupled poro-mechanical systems need to be considered. The geological complexity requires numerical methods adapting to complex grids, discontinuous material coefficients, and possibly also fractures. Herein, we explore the possibilities of using finite volume methods as a unifying framework for solving both flow and mechanical deformation. We illustrate the applicability using both synthetic examples as well as a preliminary comparison to field data.

 

Poroelastic Modelling of the Central Nervous System

Speaker: Kent-Andre Mardal

Abstract:
In this talk we will discuss simulations of the central nervous system using poroelastic modelling. We will consider the remodelling that occurs in the spinal cord under abnormal flow and pressure. Furthermore, we will discuss sensitivity with respect to modelling choices and aspects related to simulations.  Finally, we will give a short overview of the processes that governs fluid flow within the brain and the challenges faced. 

 

List of Participants

NAME DEPARTMENT AFFILIATION
Arnold, Douglas N. School of Mathematics University of Minnesota, USA
Ausrand, Peder Department of Mathematical Sciences Norwegian University of Science and Technology, Norway
Calderer, Maria-Carme School of Mathematics University of Minnesota, USA
Christiansen, Snorre H. Department of Mathematics University of Oslo, Norway
Dassbach, Paula School of Mathematics University of Minnesota, USA
Falk, Richard S. Department of Mathematics Rutgers University, USA
Golovaty, Dmitry Department of Mathematics and Computer Science University of Akron, USA
Hansen, Erik Department of Mathematics University of Bergen, Norway
Johansson, August Biomedical Computing Simula Research Laboratory, Norway
Lee, Jeonghun J. Department of Mathematics University of Oslo, Norway
Licht, Martin W. Department of Mathematics University of Oslo, Norway
Lipton, Robert P. Department of Mathematics Louisiana State University, USA
Lucantonio, Alessandro Structural Mechanics SISSA, Italy
Mardal, Kent-Andre Department of Mathematics University of Oslo, Norway
Milton, Graeme Department of Mathematics University of Utah, USA
Nochetto, Ricardo H. Department of Mathematics University of Maryland, USA
Nordbotten, Jan M. Department of Mathematics University of Bergen, Norway
Ridder,Johanna Department of Mathematics University of Oslo, Norway
Risebro, Nils H. Department of Mathematics University of Oslo, Norway
Robbins, Jonathan School of Mathematics University of Bristol, United Kingdom
Rognes, Marie E. Biomedical Computing Simula Research Laboratory, Norway
Stenberg, Rolf Department of Mathematics and Systems Analysis Aalto University, Finland
Walker, Shawn W. Department of Mathematics Louisiana State University, USA
Walkington, Noel J. Department of Mathematical Sciences Carnegie Mellon University, USA
Wang, Qi Department of Mathematics University of South Carolina, USA / CSRC, China
Weber, Franziska Department of Mathematics University of Oslo, Norway
Winther, Ragnar Department of Mathematics University of Oslo, Norway
Wyller, John A. Department of Mathematical Sciences and Technology Norwegian University of Life Sciences, Norway

 

This workshop is part of ERC project n° 339643 which has received funding from the European Research Council under the European Union's Seventh Framework Programme (FP7/2007-2013)

Published Feb. 27, 2015 6:59 PM - Last modified Oct. 10, 2023 10:57 AM