(September 9) Prof. Peter Smereka : Modeling and Simulation of Heteroeptixial Growth - In this talk I will describe a discrete kinetic Monte Carlo (KMC) model used for the simulation of heteroeptixial growth. Our KMC model contains all the physics accounted for in most continuum descriptions. However, it naturally captures physical effects not easily modeled in continuum formulations such as stochastic effects and nucleation. The KMC model is computationally challenging due the long range nature of elastic interactions. This talk will discuss the Fourier-Multigrid method for fast computation of the elastic displacement field. A technique for obtaining inexpensive upper bounds on transition rates will be presented. Finally, the principle of energy localization is explained which combined with the expanding box method allows one to accurately compute changes elastic energy using local calculations, resulting in a ten to one hundredfold increase in computation speed. These ideas are combined to allow one simulate heteroeptitaxy using KMC in physically interesting regimes. This is joint work with A. Baskaran, J. Devita, T. Schulze, and G. Russo.
(September 16) Dr. Bin Zhang: New Finite Element Methods for Fourth Order Partial Differential Equations - Developing accurate and efficient numerical approximations of solutions of high order PDEs is a challenging research topic. There are several difficulties that may be encountered in the finite element approximations of high order PDEs. In this talk, we will introduce two new finite element methods that directly discretize the fourth order curl equations (involving curl^4) in three dimensions arising from magnetohydrodynamics models. These elements provide nonconforming approximations for which the number of degrees of freedom is very small. The inter-element continuity is imposed weakly along the tangential directions which is appropriate for the approximation of the magnetic field. We will also show the detailed construction of basis functions and optimal error estimates for model problems containing both second order and fourth order terms.
(September 23) Prof. Chen Greif: Block Preconditioners for Saddle Point Linear Systems - Saddle point linear systems arise in a variety of constrained PDE and optimization problems. When these systems are very large and sparse, iterative methods must be used to compute a solution. A challenge here is to derive and apply preconditioners that exploit the properties and the structure of the given discrete operators, and yield fast convergence while imposing reasonable storage requirements. In this talk I will provide an overview of block preconditioners and discuss their spectral properties, bounds on convergence, and computational qualities. We will look at a variety of such techniques: Schur complement based approaches, constraint preconditioners, and augmentation methods.
(September 30) Prof. Sergei Sukharev: Tension-activated membrane channels - I will present a physical picture of ion channel activation with tension propagating through the surrounding lipid bilayer. The description includes the structures of the two bacterial mechanosensitive channels MscL and MscS, their positioning in the bilayer, the profile of forces acting on them, models of different states and the predicted pathways for the conformational transitions. Ample experimental data suggests several different energetic contributions, including solvation, that may stabilize the two end states (closed and open) for the channels as well as define the pathway between them. The discussion will also include the degree of structural symmetry that can be maintained along the transition and the benefits of possible asymmetric states. In the end I will illustrate the functional roles of moderately stable non-covalent associations between the structural helical elements in the regulation of channel response, i.e. in choosing between opening and inactivation.
(October 7) Elias Balaras: Embedded-boundary methods for fluid-structure interactions in biological flows - In this seminar we will discuss computational algorithms applicable to fluid-structure interaction problems from medicine and biology. Particular emphasis will be given on an embedded boundary formulation we have recently developed, where the fluid flow equations are solved on a fixed grid that does not conform to the structure, and boundary conditions are imposed using a local reconstructions. The structure that undergoes both linear-elastic and large-angle/large-displacement rigid body motions is strongly coupled to the fluid using a predictor-corrector approach. Results for both laminar and turbulent flow problems will be discussed covering applications from the cardiovascular circulation and locomotion in nature.
(October 21) Prof. Jacques Rappaz: A new algorithm to simulate the dynamics of a glacier - In this talk we propose a new Eulerian algorithm to compute the changes of a glacier geometry for given mass balances. The surface of a glacier is obtained by solving a transport equation for the Volume of Fluid (VOF). The surface mass balance is taken into account by adding an interfacial term in the transport equation. An unstructured mesh with standard stabilized finite elements is used to solve the nonlinear Stokes problem. The VOF function is computed on a structured grid with high resolution. The algorithm is stable for Courant numbers larger than unity and conserves mass to high accuracy. To demonstrate the potential of the algorithm, we applied it to reconstruct the past glacial states of a small valley glacier, Vadret Muragl, in the Swiss Alps, as well as the disappearance of the Rhone’s Glacier. This work is joint with G. Jouvet, M. Picasso, and H. Blatter.
(November 13) Prof. Gregoire Allaire: Diffractive behavior of the wave equation in periodic media - We study the homogenization and singular perturbation of the wave equation in a periodic media for long times of the order of the inverse of the period. We consider initial data that are Bloch wave packets, i.e., that are the product of a fast oscillating Bloch wave and of a smooth envelope function. We prove that the solution is approximately equal to two waves propagating in opposite directions at a high group velocity with envelope functions which obey a Schrodinger type equation. Our analysis extends the usual WKB approximation by adding a dispersive, or diffractive, effect due to the non uniformity of the group velocity which yields the dispersion tensor of the homogenized Schrodinger equation. This is a joint work with M. Palombaro and J. Rauch.
(November 14) Prof. Gregoire Allaire: Topology optimization of structures - The typical problem of structural optimization is to find the "best" structure which is, at the same time, of minimal weight and of maximum strength or which performs a desired deformation. In this context I will present the combination of the classical shape derivative and of the level-set methods for front propagation. This approach has been implemented in two and three space dimensions for models of linear or non-linear elasticity and for various objective functions and constraints on the perimeter. It has also been coupled with the bubble or topological gradient method which is designed for introducing new holes in the optimization process. Since the level set method is known to easily handle boundary propagation with topological changes, the resulting numerical algorithm is very efficient for topology optimization. It can escape from local minima in a given topological class of shapes and the resulting optimal design is largely independent of the initial guess.
(November 18) Dr. Guy Baruch:
Numerical solution of the nonlinear Helmholtz equation -
The nonlinear Helmholtz equation models the propagation of intense laser beams in Kerr media such as water, silica and air. It is a semilinear elliptic equation which requires non-selfadjoint radiation boundary-conditions, and remains unsolved in many configurations. Its commonly-used parabolic approximation, the nonlinear Schrodinger equation (NLS), is known to possess singular solutions. We therefore consider the question, which has been open since the 1960s: do nonlinear Helmholtz solutions exist, under conditions for which the NLS solution becomes singular? In other words, is the singularity removed in the elliptic model?
In this work we develop a numerical method which produces such solutions in some cases, thereby showing that the singularity is indeed removed in the elliptic equation. We also consider the subcritical case, wherein the NLS has stable solitons. For beams whose width is comparable to the optimal wavelength, the NLS model becomes invalid, and so the existence of such "nonparaxial solitons" requires solutions of the Helmholtz model. Numerically we consider the case of grated material, that has material discontinuities in the direction of propagation. We develop a high-order discretization which is "semi-compact", i.e., compact only in the direction of propagation, that is optimal for this case. Joint work with Gadi Fibich and Semyon Tsynkov.