Multiscale Modeling, Simulation, Analysis and Applications
(1 Nov 2011 - 20 Jan 2012)
## ~ Abstracts ~
Rohan Abeyaratne, National University of Singapore"Multiscale analysis" is an important and active field of research in Mechanics. For example, one class of problems being studied involves deducing the macroscopic response of a body from knowing its microscopic response. Another involves using a macroscale analysis over part of a body, a microscale analysis over the remainder of the body, and then having these two analyses mesh together. Such problems are often considered in settings that involve quite complicated physics. In contrast, in this talk I will discuss a simple toy problem to elucidate some questions involved in multiscale analyses. The microscopic model considered will be a one-dimensional chain of particles, each attached to its nearest neighbors by nonlinear elastic springs. Emphasis will be placed on the fact that the relevant macroscopic model depends on the underlying class of motions of the microscopic model -- in particular on the length and time scales, sometimes in subtle ways. For example, for certain classes of microscopic motions one can derive the classical equations of nonlinear elasticity. For others, one can derive the classical equations of reversible thermoelasticity. For yet others, the macroscopic model comprises of Schroedinger's equation. And so on.Some attention will also be paid to the fact that the microscopic model here is always conservative, while on the other hand, if the motion of the macroscopic model involves a shock wave, it is dissipative.
Weizhu Bao, National University of SingaporeIn this talk, we study asymptotically and numerically the nonlinear eigenvalue problems under constraints arising from Bose-Einstein condensation, nonlinear optics, quantum physics and chemistry. We begin with the time-independent nonlinear Schrodinger equation also known as Gross-Pitaevskii equation and reformulated it into singular perturbed nonlinear eigenvalue problems under constraints. Matched asymptotic approximations for the eigenfunctions an eigenvalues as well as the corresponding energy are presented in strongly interaction regimes. Boundary and/or interior layers and their width are presented.An efficient and accurate numerical method based on the normalized gradient flow is proposed for computing the first and/or other eigenfunctions of the nonlinear eigenvalue problems. Numerical results in one dimension (1D), 2D and 3D are reported. Finally, the analysis results and numerical method are extended to nonlinear eigenvalue problems of system of nonlinear Schrodinger equations.
Weizhu Bao, National University of SingaporeIn this talk, I will review our recent works on numerical methods and analysis for solving the nonlinear Klein-Gordon (KG) equation in the nonrelativistic limit regime, involving a small dimensionless parameter which is inversely proportional to the speed of light. In this regime, the solution is highly oscillating in time and the energy becomes unbounded, which bring significant difficulty in analysis and heavy burden in numerical computation. We begin with four frequently used finite difference time domain (FDTD) methods and obtain their rigorous error estimates in the nonrelativistic limit regime by paying particularly attention to how error bounds depend explicitly on mesh size and time step as well as the small parameter. The we consider a numerical method by using spectral method for spatial derivatives combined with an exponential wave integrator (EWI) in the Gautschi-type for temporal derivatives to discretize the KG equation. Rigorious error estimates show that the EWI spectral method show much better temporal resolution than the FDTD methods for the KG equation in the nonrelativistic limit regime. In order to design a multiscale method for the KG equation, we establish error estimates of FDTD and EWI spectral methods for the nonlinear Schrodinger equation perturbed with a wave operator. Finally, a multiscale method is presented for discretizing the nonlinear KG equation in the nonrelativistic limit regime based on large-small amplitude wave decompostion. This multiscale method converges uniformly in spatial/temporal discretization with respect to the small parameter for the nonlinear KG equation in the nonrelativistic limite regime. Finally, applications to several high oscillatory dispersive partial differential equations will be discussed.
Weizhu Bao, National University of SingaporeIn this talk, I will review our recent work on quantized vortex stability and dynamics in Ginzburg-Landau-Schrodinger and nonlinear wave equations for modeling superfluidity and superconductivity as well as nonlinear optics. The reduced dynamic laws for quantized vortex interaction are reviewed and solved analytically in several cases. Direct numerical simulation results for Ginzburg-Landau-Schrodinger and nonlinear wave equations are reported for quantized vortex dynamics and they are compared with those from the reduced dynamics laws. References: [1] Y. Zhang, W. Bao and Q. Du, The Dynamics and Interaction of Quantized Vortices in Ginzburg-Landau-Schroedinger equations, SIAM J. Appl. Math., Vol. 67, No. 6, pp. 1740-1775, 2007 [2] Y. Zhang, W. Bao and Q. Du, Numerical simulation of vortex dynamics in Ginzburg-Landau-Schrodinger equation, Eur. J. Appl. Math., Vol. 18, pp. 607-630, 2007. [3] W. Bao, Q. Du and Y. Zhang, Dynamics of rotating Bose-Einstein condensates and their efficient and accurate numericalcomputation, SIAM J. Appl. Math., Vol. 66 , No. 3, pp. 758-786, 2006.
Weizhu Bao, National University of SingaporeIn this tutorial, efficient and accurate numerical methods and their error estimates are presented for nonlinear dispersive equations with applications in quantum and plasma physics. I will begin with the derivation of the nonlinear Schrodinger equation (NLS) or Gross-Pitaevskii equation (GPE) for modeling Bose-Einstein condensation (BEC) and nonlinear optics, then review its main properties, present numerical methods for computing ground & excited states and dynamics of NLS, and extend these numerical methods for dampled NLS, GPE with angular momentum rotation, coupled GPEs, Schrodinger-Poisson equations, etc. Then I will present and compare different numerical methods for nonlinear wave-type equation, and extend them for nonlinear dispersive coupled systems including Zakharov system for plasma physics, Klein-Gordon-Schrodinger equations, Maxwell-Dirac system, etc. I will also carry out rigorous error estimates for some numerical methods based on the energy method. Finally, emerging applications of these methods for problems arising in quantum and plasma physics will be presented.
Pavel Bochev, Sandia National Laboratories, USAWe present a new, optimization-based strategy for modeling and simulation of complex multiphysics problems whose constituent components are governed by physical laws with disparate mathematical structures. Our approach relies on optimization and control ideas to (i) assemble and decompose multiphysics operators in a way that enables parallelism, and (ii) preserve their fundamental physical properties in the discretization process. In so doing we are able to synthesize -> compatible, feature-preserving discretizations of multiphysics problems, from high-resolution compatible discretizations of their constituent physics components, and -> efficient, scalable multiphysics solvers from fast solvers for their constituent physics components. The optimization-based modeling and simulation framework departs substantially from the dominnant computational strategies for multiphysics problems. Established approaches treat discretization and solution as separate tasks, and the former is burdened with the preservation of physical properties such as maximum principles, positivity, or monotonicity. Because our framework assembles discrete multiphysics models from high-resolution discretizations of their constituent physics components, the operator decomposition required for fast, scalable solvers is embedded in the problem formulation from the onset. Our approach also relieves discretization from tasks that impose severe geometric constraints on the mesh, or tangle accuracy and resolution with the preservation of physical properties. Two examples will illustrate the scope of our approach approach: an optimization-based framework for multi-physics coupling, and an optimization-based algorithm for constrained interpolation and transport. This is joint work with D. Ridzal, (Sandia).
Yongyong Cai, National University of SingaporeThe error estimates of finite difference methods for nonlinear Schrödinger equations are only available for conservative schemes and one dimensional case in literature. In this work, we presented a unified approach for the error estimates both for conservative and non-conservative schemes and one, two, three-dimensions. Especially, we establish uniform error estimates of finite difference methods for the nonlinear Schrödinger (NLS) equation perturbed by the wave operator (NLSW) with a perturbation strength described by a small dimensionless parameter e, where the NLSW collapses to the standard NLS as the perturbation vanishes. In the small perturbation parameter regime, the solution of NLSW is perturbed from that of NLS with a function oscillating in time with e^2 wavelength att e^2 and e^4 amplitudes for ill-prepared and well-prepared initial data, respectively. This high oscillation of the solution in time brings significant difficulties in establishing error estimates uniformly in e of the standard finite difference methods for NLSW, such as the conservative Crank-Nicolson finite difference (CNFD) method, semi-implicit finite difference (SIFD) method. We obtain error bounds uniformly in e at the order of O(h^2+k^{2/3}) and O(h^2+k)$ with time step k and mesh size h for ill-prepared and well-prepared initial data, respectively, for both CNFD and SIFD in the l^2-norm and discrete semi-H^1 norm. Our error bounds are valid for general nonlinearity in NLSW and for one, two and three dimensions. Key techniques in the analysis include the energy method, cut-off of the nonlinearity.
Yongyong Cai, National University of SingaporeBose-Einstein condensate (BEC) with dipole-dipole interaction has received considerable research interests recently. At zero temperature, the dipolar BEC is well-described by a three dimensional (3D) Gross-Pitaevskii equation (GPE) with a nonlocal dipole-dipole interaction term. With strongly anisotropic confining potentials, the three dimensional dipolar GPE will result in effective two-dimensional (2D) equation for disk-shaped BEC or effective one-dimensional (1D) equation for cigar-shaped BEC . Upon a new formulation of the 3D dipolar GPE, we obtain the corresponding effective lower dimensional equations. Ground state and dynamics for the 3D and lower dimensional equations are discussed. Extensions to multi-layered dipolar condensate will be also discussed.
Yongyong Cai, National University of SingaporeIn the first part of the talk, I will present the error estimates of finite difference methods for the nonlinear Schrodinger equations. I will focus on the conservative Crank-Nicolson finite difference method and semi-implicit finite difference method, with unified approach for one, two and three dimensions, as well as general nonlinearity. In the second part of the talk, I will talk on the recent progress towards the error analysis of time-splitting methods, due to C. Lubich (2008).
Shuyang Dai, Hong Kong University of Science and Technology, Hong KongWe present a continuum model for the coherent-incoherent twin boundary junctions based on the Peierls-Nabarro framework, incorporating both the long-range strain field and the local atomic structure. The obtained results agree well with the atomistic and dislocation dynamics simulations and the experimental results. We also present a continuum model to simulate the glide force acts on the other dislocation array due to the perturbed grain boundary and its effects on the formation of different dislocation array pattern.
Xuan Chun Dong, National University of SingaporeI begin by the computation for the Schrödinger-Poisson-Slater (SPS) system, which serves as a local single particle approximation of the Hartree-Fock equations for a quantum system of N electrons interacting via Coulomb potential. The focus lies on the comparisons of different numerical methods for computing the ground state and dynamics, and the emphasis is put on the various approaches for the Hartree potential, with a conclusion that the spectral approach based on sine bases is the best choice in 3D. Also, simplified spectral methods for spherically symmetric SPS system are presented, which reduce the memory and computational load significantly. Next, an application of the results obtained for SPS system to compute the relativistic Hartree equation, which models a quantum system of N bosons with relativistic dispersion interacting through Coulomb potential, is presented. Some intriguing phenomena observed in the numerical experiments are shown.
Xuan Chun Dong, National University of SingaporeIn the first part of this talk, I begin by the computation for the Schrödinger-Poisson-Slater (SPS) system, which serves as a local single particle approximation of the Hartree-Fock equations for a quantum system of N electrons interacting via Coulomb potential. The focus lies on the comparisons of different numerical methods for computing the ground state and dynamics, and the emphasis is put on the various approaches for the Hartree potential, with a conclusion that the spectral approach based on sine bases is the best choice in 3D. Also, simplified spectral methods for spherically symmetric SPS system are presented, which reduce the memory and computational load significantly. Next, an application of the results obtained for SPS system to compute the relativistic Hartree equation, which models a quantum system of N bosons with relativistic dispersion interacting through Coulomb potential, is presented. Some intriguing phenomena observed in the numerical experiments are shown. In the second part of this talk, numerical comparisons between sine-Gordon and perturbed NLS equations for light bullets are made, with emphasis put on the regime beyond critical collapse of cubic NLS. The results validate that perturbed NLS is an effective model for the light bullets, and much cheaper to simulate than sine-Gordon due to disparate scales involved. References: [1]. Y. Zhang, X. Dong, On the computation of ground state and dynamics of Schrödinger-Poisson-Slater system, J. Comput. Phys., 230 (7), pp. 2660-2676, 2011 [2]. X. Dong, A short note on simplified pseudospectral methods for computing ground state and dynamics of spherically symmetric Schrödinger-Poisson-Slater system, J. Comput. Phys., 230 (22), pp. 7917-7922, 2011 [3]. W. Bao, X. Dong, Numerical methods for computing ground states and dynamics of nonlinear relativistic Hartree equation for boson stars, J. Comput. Phys., 230 (10), pp. 5449-5469, 2011 [4]. W. Bao, X. Dong, J. Xin, Comparisons between sine-Gordon equation and perturbed nonlinear Schrödinger equations for modeling light bullets beyond critical collapse, Physica D, 239 (13), pp. 1120-1134, 2010
Qiang Du, Pennsylvania State University, USAExploring complex energy landscape is a challenging issue in many applications. Besides locating equilibrium states, it is often also important to identify transition states given by saddle points. In this talk, we will discuss existing and new algorithms for the computation of such transition states with a focus on the newly developed Shrinking Dimer Dynamics and present some related mathematical theory on stability and convergence. We will consider a number of applications including the study of frustrations of interacting particles and nucleation in solid state phase transformations.
Weinan E, Princeton University, USAIn this series of lectures, I will give an elementary introduction to the analytical tools in multiscale analysis, including matched asymptotics, averaging methods, geometric optics, homogenization methods, scaling and renormalization group methods.
Yalchin Efendiev, Texas A&M University, USAThe development of numerical algorithms for simulations of diffusion processes in highly heterogeneous media is challenging because of high variability and complex spatial length scales. It is usually necessary to resolve a wide range of length and time scales, which can be prohibitively expensive, in order to obtain accurate predictions. In practice, some types of coarsening (or upscaling) of the detailed model are usually performed. Many approaches have been developed and applied successfully when a scale separation adequately describes the spatial variability of the media that have bounded variations. The quality of these approaches deteriorates for complex heterogeneities without scale separation and high contrast. In this talk, I will describe multiscale model reduction techniques that can be used to systematically reduce the degrees of freedoms of fine-scale simulations and discuss applications to preconditioners and coupling to global model reduction tools. Numerical results will be presented that show that one can improve the accuracy of multiscale methods by systematically adding new coarse basis functions, obtain contrast-independent preconditioners for complex heterogeneities, and get reduced order models at low cost.
Klemens Fellner, University of Cambridge, UKWe present some recent results and ongoing work on drift-diffusion-reaction systems modelling organic photovoltaic devices. While classical semiconductors show recombination typically throughout the whole device feature organic photovoltaic devices significant charge generation only in the very proximity of an interface between two different organic polymers. We discuss basic questions of modelling, existence and stationary states. Moreover, we present some interesting asymptotic approximations and discuss the use of entropy. Joint work with Dan Brinkman, Peter Markowich, Marie-Therese Wolfram (DAMTP, University of Cambridge)
Yuan Ping Feng, National University of SingaporeTo support the continued down-scaling of electronic devices, new materials and new concepts are in demand. It is highly possible that current electronic devices be replaced by spintronic devices which make use of both charge and spin, two fundamental properties of electron. A major challenge of spintronics is in generating, controlling and detecting spin-polarized current. Materials design based on computational methods can play a very important role in exploration of spintronic materials. Among various methods, the first-principles electronic structure method based on density functional theory is ideal for designing new materials because such methods do not require experimental inputs and prior knowledge on the materials. We have been using first-principles method to study properties of advanced materials and to design new materials for spintronic applications. Some of our recent work will be discussed. Using first-principles methods, we have successfully predicted a number of potentially useful dilute magnetic semiconductors such as ZnO doped by 2p light elements (carbon, nitrogen, etc.) and ZnO doped by Li, etc. Some of our predictions have been successfully confirmed experimentally. Recently, we explored linear carbon chains and graphene nanoribbons for spin logic gates. The unique spin-dependent electron transport property of zigzag graphene nanoribbons (ZGNRs) allows control of spin current through the ZGNRs by a bias voltage and/or the magnetic configurations of the electrodes and makes it possible to design a complete set of spintronic devices.
Chee Kwan Gan, Institute for High Performance ComputingRealistic materials simulations usually require the ability to describe the interactions between the constituent atoms accurately. Density functional theory (DFT) based methods (such as local density approximation (LDA) or generalized gradient approximation (GGA)) are very accurate and powerful in this respect. However, they are often limited by the size of the system DFT can treat. On the other hand, interatomic potentials often used in the classical molecular dynamics (MD) are extremely efficient compared to the DFT methods, where simulations up to thousands of atoms can be done routinely, even on a personal computer. However, the validity of the classical MD results depends critically on how well the interatomic potentials describe the region where bond making and bond breaking are made. In this talk, I shall outline a method that sits between DFT and MD based methods. This method, called a distance-dependent tight-binding method, is essentially inter-atomic based, but with ingredients from the density functional theory. I shall briefly outline the theory and implementation of this tight-binding method. Some details on the use of the code will be presented. The application of tight-binding method to heat transport properties of graphene nanoribbons will be illustrated.
Max Gunzburger, Florida State University, USABased on notions for nonlocal fluxes between volumes and nonlocal balance laws and a nonlocal vector calculus we have developed, we introduce nonlocal models for diffusion and the nonlocal peridynamics continuum model for mechanics. A feature of the nonlocal problems that has important practical consequences are that constraints, e.g., of Dirichlet type, are applied over volumes and not along bounding surfaces. A brief review of the nonlocal calculus is given, including definitions of nonlocal divergence, gradient, and curl operators and derivations of a nonlocal Gauss theorem and Green's identities. Through appropriate limiting processes, relations between the nonlocal operators and their differential counterparts are established. The nonlocal calculus is used to define weak formulations of the nonlocal diffusion and mechanics problems which are then analyzed, showing, for example, that unlike elliptic partial differential equations, these problems do not necessary result in the smoothing of data. We briefly consider connections to fractional Laplacian problems (which are special cases of our models) and finite element methods for nonlocal problems, focusing on solutions containing jump discontinuities; in this setting, discontinuous Galerkin methods are conforming and nonlocal problems can lead to optimally accurate approximations. We also show how our models can be implemented so that they are multi-scale mono-models, i.e., they are single models that are valid and tractable over a wide range of scales.
Katarina Gustavsson, KTH Royal Institute of Technology, Sweden and National University of Singapore, SwedenGravity induced sedimentation of slender rigid fibers in a highly viscous fluid is investigated by large scale numerical simulations. The fiber suspension is considered at a microscopic level and the flow is described by the Stokes equations in a three dimensional periodic domain/box. For this specific problem, it is suitable to work with a boundary integral formulation. Since the fibers are slender, a non-local slender body approximation can be used to accurately capture the dynamics of the fibers. A great advantage of such formulation is that it leads to a reduction in dimensionality of the problem. Instead of solving the full three dimensional problem, a system of one dimensional boundary integrals is solved to determine the dynamical behavior of the fibers. Using numerical simulations we will demonstrate that a suspension with an initial homogeneous and random distribution of fibers will form large scale inhomogeneities in the fiber distribution during sedimentation. Elongated fiber-dense streamers will form, surrounded by regions of clear fluid. Within these streamers smaller clusters of fibers are created and dispersed in a repetitive fashion. We observe a strong correlation between the creation and dispersion of clusters and the fluctuations in the sedimentation velocity of the suspension. We have also studied the effect of the micro-structure of the suspension on averaged quantities such as the mean sedimentation velocity. Our results show that two simulations with the same macroscopic properties (fiber concentration and periodic box geometry) but with different random initial distribution of fibers, can exhibit very different dynamical behavior at the fiber level, causing large differences in e.g. the mean sedimentation speed. Hence, in order to obtain reliable data of such quantities, the results need to be averaged over a number of simulations. We will present ensemble averages of the sedimentation velocity and fiber orientation for different values of the effective concentration of fibers and the results are compared to existing experimental data. Finally, we will also present preliminary results from a recent study of sedimenting clouds of rigid fibers.
Feimin Huang, Academia Sinica, ChinaIn this lecture, I will present recent progress on the time asymptotic behavior of solutions of compressible Navier-Stokes equations toward to the Riemann solutions of compressible Euler equations.
Kerson Huang, Institute of Advanced Studies, NTU Massachusetts Institute of Technology, USA We propose a theory of the big bang in which the universe was born as a superfluid with quantum turbulence. All matter present today was created in the turbulence, in an era that lasted about 10 ^{26}s, during which the universe expanded by a factor of 10^{27} from a radius of Planck length, about 10^{-35 }cm. After this era, the standard "hot big bang" theory takes over to describe CMB, nucleosynthesis, and galaxy formation, with one important difference: the universe remains a superfluid. Superfluidity and quantized vortices offer understanding of many unexplained observations. These include dark energy, dark matter, galactic voids, cosmic jets, and "non-thermal filaments". Our model spans different fields: general relativity, quantum field theory, and condensed matter physics. My aim in these lectures is to introduce the background physics and mathematics, describe the model, and suggest directions of future research. 1. GR (General relativity): the way of the very large 2. QFT (Quantum field theory): the way of the very small 3. RG (Renormalization group): navigating the space of all possible theories 4. Superfluidity and quantum turbulence 5. The big bang 6. Topics for future research
Zhongyi Huang, Tsinghua University, ChinaIn this talk, we consider the propagation of (non)linear high frequency waves in heterogeneous media with periodic microstructures. We are interested in the case where the typical wavelength is comparable to the period of the medium, and both of which are assumed to be small on the length-scale of the considered physical domain. Therefore, we consider this system as a two-scale asymptotic problem with different scalings of the nonlinearity, and we solve it by Bloch decomposition based method. In particular we discuss (nonlinear) mass transfer between different Bloch bands and also present three-dimensional simulations for lattice Bose-Einstein condensates in the superfluid regime.
Ahmad A. Hujeirat, University of Heidelberg, GermanyDark energy stars, black holes and a certain type of neutron stars belong to the family of ultra-compact relativistic objects. Such objects have been observed in our Galaxy and in the nearby galaxies, whereas the centers of almost all known galaxies are considered to be occupied by supermassive black holes. In the case of ultra compact neutron stars, the internal and rotational energies stored in their cores are sufficient to retain the fluidity phase of matter, but subsequently may cool and turn into superfluids. Therefore, a large variety of fluid-flows are expected to govern the dynamics inside and outside these objects, ranging from weakly to strongly compressible, ideal, dissipative or even to superfluids. Magnetic fields and radiation are expected to play an important role in the dynamics of these flows . In this talk I will briefly describe the properties of such flows, present the hierarchical solution scenario in combination with the numerical solver "GRIPS_I: A General Relativistic Implicit Plasma Solver" and will finally discuss the results of some applications.
Shi Jin, University of Wisconsin, USAWe consider systems coupling the compressible or incompressible Navier-Stokes equations to the Vlasov-Fokker-Planck equation. Such a problem arises in the description of particulate flows. We design numerical schemes to simulate this system. These schemes are asymptotic-preserving, thus efficient in both the kinetic and hydrodynamic regimes. It has a numerical stability condition controlled by the non-stiff convection operator, with an implicit treatment of the stiff drag term and the Fokker-Planck operator. Yet, consistent to a standard asymptotic-preserving Fokker-Planck solver or an incompressible Navier-Stokes solver, only the conjugate-gradient method and fast Poisson and Helmholtz solvers are needed. Numerical experiments are presented to demonstrate the accuracy and asymptotic behavior of the schemes, with several interesting applications. This is a joint work with T. Goudon, J.-G. Liu and Bokai Yan.
Quansen Jiu, Capital Normal University, ChinaIn this talk, we will present some recent results on the asymptotic stability of rarefaction waves for the compressible isentropic Navier-Stokes equations with density-dependent viscosity. Thehese results hold for large-amplitudes rarefaction waves and arbitrary initial perturbations. This is joint with Yi Wang and Zhouping Xin.
Ansgar Juengel, Institute for Analysis and Scientific Computing, AustriaThe microelectronics industry is heavily based on the construction of faster, smaller, and more efficient semiconductor devices. Quantum semiconductor structures might fulfill these requirements also in the future. In this talk, some modeling and simulation approaches for quantum semiconductors will be presented. First, quantum Navier-Stokes equations will be introduced. This model has a surprising nonlinear structure and posseses two different energy functionals. It is shown how this structure can be used to derive global existence results, and a connection to "Noether symmetries" in a Lagrangian mechanics framework is highlighted. Second, transient numerical simulations of a 3-D quantum waveguide are presented, based on a time-splitting pseudo-spectral approximation of the Schroedinger equation. Furthermore, self-induced oscillations of a tunneling diode in a small electric circuit are shown.
Jesper Karlsson, King Abdullah University of Science and Technology (KAUST), Saudi ArabiaWe present novel weak error estimates for the tau-leap approximation of kinetic Monte Carlo models. These error estimates are utilized to control the global weak error by an adaptive time-stepping algorithm. Adaptive methods are relevant for efficient numerical simulation and here their impact is in applied fields such as simulation of complex chemical reactions. To our knowledge no such adaptive method for global error control has previously been developed for the tau-leap method. We also present numerical experiments that show the gain from using this adaptive tau-leap method in practical computations. This work is done in collaboration with Markos Katsoulakis (Umass Amherst), Anders Szepessy (KTH) and Raul Tempone (KAUST).
Pawel Keblinski, Rensselaer Polytechnic Institute, USAIn the first part of the presentation I will review major methods used to model thermal conduction of materials and discuss associated computational challenges. In particular I will discuss thermal transport characterization at atomic- and micro-scales via molecular dynamics simulations and Boltzmann transport equation and the connection between these methods and the continuum-level description of thermal transport. I will also discuss phonon-transport theory-based and computationally intensive calculations of thermal conductivity. In the second part of the presentation I review several key thermal transport problems involving nanoscopic and interfacial structures including nanocomposites, nanofluids, and nanoparticle based thermal therapies, and discuss computational challenges associated with modeling of the relevant materials systems.
Claude Le Bris, École Nationale Des Ponts Et Chaussées, FranceWe present a series of ongoing works with X. Blanc and PL. Lions where we introduce suitable algebras of functions to derive a theory of homogenization for elliptic PDEs with nonperiodic coeffients. The nonperiodicity typically encodes the presence of one or many defects in the material at the microscale. The perspective is both theoretical and computational.
Claude Le Bris, École Nationale Des Ponts Et Chaussées, FranceThis set of lectures will survey recent developments in the field of multiscale modelling of materials with a focus on the stochastic context. It will be shown how some appropriately chosen "weakly random" generalizations of existing frameworks can lead to theories that are both practically relevant and computationally efficient. Some specific issues regarding fully random problems, such as variance reduction issues, will also be addressed. The material presented in the talk covers joint work with X. Blanc, PL. Lions, F. Legoll, A. Anantharaman, R. Costaouec, F. Thomines. A good introduction and written support for the lectures is the article: Introduction to Numerical Stochastic Homogenization and the Related Computational Challenges: Some Recent Developments (A Anantharaman, R Costaouec, C Le Bris, F Legoll and F Thomines), in Lecture Notes Series, Institute for Mathematical Sciences, National University of Singapore - Vol. 22 MULTISCALE MODELING AND ANALYSIS FOR MATERIALS SIMULATION edited by Weizhu Bao (National University of Singapore, Singapore) & Qiang Du (Pennsylvania State University, USA). Further references will be given during the lectures.
Frédéric Legoll, École Nationale Des Ponts Et Chaussées, FranceWe introduce and analyze a micro/macro parareal algorithm for the time-parallel integration of singularly perturbed ordinary differential equations. The system we consider includes some fast and some slow variables, the limiting dynamics of which (in the limit of infinite time scale separation) is known. The algorithm first computes a cheap but inaccurate macroscopic solution using a coarse propagator (by only evolving the slow variables according to their limiting dynamics). This solution is iteratively corrected by using a fine-scale propagator (simulating the full microscopic dynamics on both slow and fast variables), in the parareal algorithm spirit. The efficiency of the approach will be demonstrated on the basis of numerical analysis arguments and representative numerical experiments. Joint work with T. Lelievre and G. Samaey.
Tony Lelievre, École Nationale Des Ponts Et Chaussées, FranceIn this talk, I will present two recent works concerning multiscale models for polymeric fluids. First, I will present a numerical closure procedure that we recently proposed in [1] to get, from a microscopic model, a closed macroscopic model. This procedure is related to the so-called quasi-equilibrium approximation method, and can be seen as a justification of this approach. Second, I will discuss the longtime behaviour of some models for rigid polymers (liquid crystals models). Such models are interesting since their longtime behaviour may be quite complicated, including convergence to periodic in time solutions. I will explain how such convergence can be proven using entropy techniques, see [2]. References: [1] V. Legat, T. Lelièvre and G. Samaey, A numerical closure approach for kinetic models of polymeric fluids: exploring closure relations for FENE dumbbells, Computers and Fluids, 43, 119-133, (2011). [2] L. He, C. Le Bris and T. Lelièvre, Periodic long-time behaviour for an approximate model of nematic polymers, http://arxiv.org/abs/1107.3592.
Jia Li, National University of SingaporeX-ray computed tomography (CT) has been playing an important role in diagnostic of cancer and radiotherapy. However, high imaging dose added to healthy organs during CT scans is a serious clinical concern. Imaging dose in CT scans can be reduced by reducing the number of X-ray projections. In this presentation, we consider 2D CT reconstructions using very small number of projections. For some existed regularization based reconstruction methods, like the total variation (TV) based reconstruction and balanced approach with anisotropic wavelet frame based regularization, at least 40 projections is usually needed to get a satisfactory reconstruction. In order to keep radiation dose as minimal as possible, while increase the quality of the reconstructed images, one needs to enhance the resolution of the projected image in the Radon domain without increasing the total number of projections. This presentation is to propose a CT reconstruction model with wavelet frame based regularization and Radon domain inpainting. The proposed model simultaneously reconstructs a high quality image and its corresponding high resolution measurements in Radon domain. In addition, we discovered that using the isotropic wavelet frame regularization which is superior than using its anisotropic counterpart. Our proposed model can be solved rather efficiently by split Bregman algorithm.
Wenlei Li, Jilin University, ChinaThe Galoisian approach to study the integrability of classical Hamiltonian systems, so called Morales-Ramis theory, has been proved to be useful and powerful by many applications. Here, an analogous form of the Morales-Ramis theory for general dynamical systems will be given. Galois group of the corresponding variational equation will be studied, and some necessary conditions of the system to possess a certain number of integrals are presented. An application will be given at last to illustrate our result.
Xiantao Li, Pennsylvania State University, USAWe will present a reduction method, motivated by boundary integral method for continuum elastostatics models, to reduce the dimension of a molecular statics model. In the reduced model, the new degrees of freedom are associated with the atoms at the remote boundary and the atoms at the interface with local defect. We will discuss the application for fracture initiations.
Tiejun Li, Peking University, ChinaWe consider the nucleation of stochastic Cahn-Hilliard dynamics with the standard double well potential. We design the string method for computing the most probable transition path in the zero temperature limit based on large deviation theory. We derive the nucleation rate formula for the stochastic Cahn-Hilliard dynamics through finite dimensional discretization. We also discuss the algorithmic issues for calculating the nucleation rate, especially the high dimensional sampling for computing the determinant ratios.
Liren Lin, National Taiwan University, TaiwanA redistribution of masses between different components of a BEC system will decrease the kinetic energy. In this talk I shall introduce this simple principle, and show that some properties of ground states of spin-1 BECs are direct consequences of special cases of this principle. If time allows, further possible usage of this principle will also be discussed.
Jie Liu, National University of SingaporeWhen solving the equations that describe the motion of incompressible viscous flow, one can use a Poisson equation of the pressure to replace the incompressibility constraint. But how to properly formulate the boundary conditions for the pressure on the no-slip and traction boundaries is a longstanding problem. We will present our solutions and discuss their influence on the stability and accuracy of the resulting schemes. For Stokes equations, we will present some 3rd order unconditionally stable schemes with no-slip boundary condition. And then a 1st order unconditionally stable scheme when part of the boundary is of traction type. The first part of the talk is based on joint works with J.-G. Liu and R. L. Pego.
Jie Liu, National University of SingaporeFluid structure interaction studies the dynamics of a deformable structure and its surrounding or enclosed fluid. The interface between the fluid and the structure is a free boundary whose position is completely determined by the continuity of velocity and balance of force along the interface. We introduce a stable explicit interface advancing scheme for this problem and study the nice mathematical properties of this scheme. In particular, we show why Navier-Stokes flow is more stable than Stokes flow in this case. Some benchmark numerical tests will be presented to confirm our theoretical results.
Gang Lu, California State University Northridge, USAWe present a multiscale modeling approach that can simulate multimillion atoms effectively via density functional theory (DFT). The method is based on the framework of the quasicontinuum (QC) approach with DFT as its sole energetics formulation - there is no empirical input. The local QC part is formulated by the Cauchy-Born hypothesis with DFT calculations for strain energy and stress. The nonlocal QC part is treated by a self-consistent DFT-based embedding approach. The method-QCDFT-is applied to several materials problems, including a nanoindentation study of an Al thin film in the presence and absence of Mg impurities; the system contains over 60 million atoms. In addition, recent results on ductile fracture of Al alloys and the effects of H impurities on the crack plasticity will be presented. The work was supported by DOE, NSF and ONR.
Mitchell Luskin, University of Minnesota, USAMany materials problems require the accuracy of atomistic modeling in small regions, such as the neighborhood of a cracktip. However, these localized defects typically interact through long-range elastic fields with a much larger region that cannot be computed atomistically. Materials scientists have proposed many methods to compute solutions to these multiscale problems by coupling atomistic models near alocalized defect with continuum models where the deformation isnearly uniform on the atomistic scale. During the past several years, a mathematical structure has been given to the description and formulation of atomistic-to-continuum coupling methods, and corresponding numerical analysis has clarified the relation between the various methods and their sources of error. This lecture will present recent developments for the numerical analysis of atomistic-to-continuum coupling methods.
Jeremy Mason, Institute for Advanced Study, USAA persistent question with front-tracking simulations of grain growth is the correct formulation of the equations of motion for a triangulated surface. We propose a scheme based on the von Neumann-Mullins and MacPherson-Srolovitz relations in two and three dimensions, respectively, and implement this in front-tracking simulations of some of the most extensive microstructures in the literature. Our purpose with these simulations is to address an often overlooked issue, namely, the identification of the steady-state microstructure. Meaningful comparisons of grain growth simulations is nearly impossible without identifying at least this structure as a reference state.
Norbert Mauser, Universität Wien, AustriaWe show how to obtain "effective one-particle Nonlinear Schrodinger equations" as proximations of the "exact" linear N-particle Schrodinger equations with Coulomb interaction. The simplest "mean field" approximations are "Hartree" equations for boson condensates, like the simple cubic NLS, or Hartree-Fock equations for fermions that contain "exchange interactions" due the Pauli principle. The Multi Configuration Time Dependent Hartree Fock (MCTDHF) equations are a complicated system of coupled ODEs and PDEs that allows to include "correlation" and that yields an approximation hierarchy that, in principle, converges to the exact model. We present the equations and state-of-the-art mathematical results on local/global well-posedness.
Florian Méhats, Université de Rennes 1, FranceWe introduce a new micro-macro decomposition of collisional kinetic equations in the specific case of the diffusion limit, which naturally incorporates the incoming boundary conditions. The idea is to write the distribution fonction $f$ in all its domain as the sum of an equilibrium adapted to the boundary (which is not the usual equilibrium associated with $f$) and a remaining kinetic part. This equilibrium is defined such that its incoming velocity moments coincide with the incoming velocity moments of the distribution function. A consequence of this strategy is that no artificial boundary condition is needed in the micro-macro models and the exact boundary condition on $f$ is naturally transposed to the macro part of the model. This method provides an 'Asymptotic preserving' numerical scheme which generates a very good approximation of the space boundary values at the diffusive limit, without any mesh refinement in the boundary layers. Our numerical results are in very good agreement with the exact so-called Chandrasekhar value, which is known in some simple cases.
Bin Min, Peking University, ChinaWe numerically study the long time dynamics of Bose-Einstein condensates (BEC) expanding in a speckle potential in the parameter regime of the experiment [Nature 453, 891(2008)] by using a highly efficient method. We demonstrate that the cubic interaction will destroy the Anderson localization and both rms and participation length would grow with time proportionally to t^{\alpha}(\alpha <1/2) up to 15000 optical confinment periods. The reason why the rms measured after t=0.5s in the experiment [Nature 453, 891 (2008)] ceased increasing is explained and participation length is suggested to characterize the subdiffusion behavior in experiments. The nature of this subdiffusion phenomenon is validated as interaction-assisted hopping between different localized modes by a heuristic model.
Bin Min, Peking University, ChinaWe numerically study the long time dynamics of Bose-Einstein condensates (BEC) expanding in a speckle potential in the parameter regime of the experiment [Nature 453, 891(2008)] by using a highly efficient method. We demonstrate that the cubic interaction will destroy the Anderson localization and both rms and participation length would grow with time proportionally to t ^{a}(a< 1/2) up to 15000 optical confinement periods. The reason why the rms measured after t =0.5s in the experiment [Nature 453, 891 (2008)] ceased increasing is explained and participation length is suggested to characterize the subdiffusion behavior in experiments. The nature of this subdiffusion phenomenon is validated as interaction-assisted hopping between different localized modes by a heuristic model.
Lorenzo Pareschi, University of Ferrara, ItalyImplicit-Explicit Runge-Kutta schemes represent a powerful tool for the numerical treatment of stiff terms in partial differential equations. When necessary they can be designed in order to achieve suitable asymptotic preserving (AP) properties of the underlying systems of differential equations, like in the case of hyperbolic systems with relaxation [1]. Recently they have been extended to the case of stiff diffusive limits [2]. Similar techniques can be adopted when dealing with kinetic equation of Boltzmann-type. Here, however, the major challenge is represented by the complicated nonlinear structure of the collisional operator which makes prohibitively expensive the use of fully implicit solvers. Additional difficulties are given by the need to preserve some relevant physical properties like nonnegativity of the solution and entropy inequality. This problem in the context of splitting methods using exponential techniques has been successfully studied recently in [4]. In this talk, using a penalization approach introduced in [3], we will present some recent advances in this directions and construct IMEX schemes up to third order which are asymptotic preserving for the Boltzmann equation in the fluid-limit without having to invert expensive nonlinear algebraic systems. General conditions on the coefficients of the IMEX schemes in order to satisfy the AP-property are derived and requirements for nonnegativity and entropicity analyzed. Examples of nonnegative and entropic schemes are also discussed. Finally numerical evidence of the effectiveness of the new IMEX schemes for space homogeneous and non homogeneous problems is given. We refer to [5] for further details. References [1] L.Pareschi, G.Russo, "Implicit-Explicit Runge-Kutta methods and applications to hyperbolic systems with relaxation", J. Sci. Comp. (2005) [2] S.Boscarino, L.Pareschi, G.Russo, "Implicit-Explicit Runge-Kutta schemes for hyperbolic systems and kinetic equations in the diffusion limit", preprint (2011) [3] S. Jin, F. Filbet, "A class of asymptotic-preserving schemes for kinetic equations and related problems with stiff sources", J. Comp. Phys. (2010) [4] G. Dimarco, L.Pareschi, "Exponential Runge"Kutta methods for stiff kinetic equations", SIAM J. Num. Anal. (2011) [5] G. Dimarco, L.Pareschi, "Implicit-Explicit Runge-Kutta methods for stiff kinetic equations and related hyperbolic relaxation problems", preprint (2011)
Danny Perez, Los Alamos National Laboratory, USAA significant problem in the atomistic simulation of materials is that molecular dynamics simulations are limited to microseconds, while important reactions and diffusive events often occur on much longer time scales. Although rate constants for infrequent events can be computed directly, this requires first knowing the transition state. Often, however, we cannot even guess what events will occur. In this talk, I will discuss accelerated molecular dynamics approaches, which have been developed over the last decade, for treating these complex infrequent-event systems. The idea is to directly accelerate the dynamics to achieve longer times without prior knowledge of the available reaction paths. In some cases, we can achieve time scales with these methods that are many orders of magnitude beyond what is accessible to molecular dynamics. I will briefly introduce the methods and discuss an application of one of these methods, namely the Parallel Replica Dynamics method, to the plastic deformation of metallic nanowires under very low strain rates.
An Phan Thanh, Instituto Superior Tecnico, Lisbon, PortugalTo date, solving shortest path problems inside simple polygons or on terrains has usually relied on triangulation and graph theory. The question: "Can one devise a simple O(n) time algorithm for computing the shortest path between two points in a simple polygon (with n vertices), without resorting to a (complicated) linear-time triangulation algorithm?" raised by J. S.B. Mitchell in Handbook of Computational Geometry (J. Sack and J. Urrutia, eds., Elsevier Science B.V., 2000), is still open. The aim of this paper is to use the ideas of two known numerical optimization methods, namely, method of orienting curves (introduced by H. X. Phu in 1987) and multiple shooting method (introduced by H. G. Bock in 1981) for some constrained optimal control problems, for solving some shortest path problems inside simple polygons without triangulation and graph theory. We will present efficient algorithms based on first method (second method, respectively) for finding the convex hull of a finite points in 2D and 3D, a version of shortest path problems (for finding the shortest path between two points inside a simple polygon, respectively). In the case of monotone polygons, it is implemented by a C code and a numerical example shows that our algorithm significantly reduces running time and memory usage of the system. We also discuss the use of second method for solving approximate shortest descending paths in terrains. P.T. An (CEMAT, Instituto Superior Técnico, Lisbon), co-author with N.N. Hai (International University, HCM City), T.V. Hoai (HCM University of Technology, HCM City), L.H. Trang, and D.T. Giang (CEMAT, Instituto Superior Técnico, Lisbon)
Olivier Pironneau, University of Paris VI, FranceMany practical problems of importance require simulations of flow through porous media : underground water resource, oil reservoir assessment, pollution control etc. One particular example, nuclear waste underground disposal and the security assessment of the site requires careful and accurate numerical simulations of the flow around the site. I will take this problem as a test case to learn by example about multiscale problems and present some solutions. The scale differences are enormous because the site has the size of a football ground but it influences a region as large as 50 km over a period of hundred of thousands of years while the leaks happens over a few years only. There are multiple layers of soils with porosity which differs by 12 orders of magnitudes. There are also faults and cracks of different size possibly requiring double porosity models and homogenisation or MSFEM. In the 2 lectures we intend to cover the mathematical modelling and numerical methods on the following material: Lecture 1: 1. Description of the problem with two simplified 2 and 2D test cases. 2. Darcy's law and the convection diffusion equations for pollutants 3. Taking advantage of the large variations of porosities by layer decomposition 4. Competing transport and diffusion 5. Appropriate Finite Element Methods, conservation and positivity. Lecture 2 1. Multi-Scale FEM, the different approaches. 2. Homogenisation, basics 3. Perspectives: extension to higher Reynolds number flows.
Olivier Pironneau, University of Paris VI, FranceHPC is the equivalent of formula 1 racing compared to usual car driving. It is expensive, difficult and some dare to say, useless since a good rack of PC boards or even an allocation on Amazon's cloud can host almost all scientific simulations. Examples in this talk will be given to prove that some of the leading research problems need extreme computing. Moreover almost all scientific fields are concerned: astrophysics, chemistry, geology, medicine, applied mathematics, data-mining, social networks etc.
Tiezheng Qian, Hong Kong University of Science and Technology, Hong KongI will report our recent works on diffuse-interface modeling of contact line motion at solid surfaces, from binary mixtures to one-component liquid-gas systems. For binary mixtures, we derive our continuum hydrodynamic model through a variational approach based on the Onsager principle of minimum energy dissipation, with the slip boundary condition derived along with equations of motion in the bulk. Numerical predictions of our model show remarkable quantitative agreement with molecular dynamics simulation results. For one-component liquid-gas systems, the dynamic van der Waals theory has been presented for one-component fluids, capable of describing the two-phase hydrodynamics involving the liquid-gas transition [A. Onuki, Phys. Rev. E 75, 036304 (2007)]. We use its hydrodynamic equations to describe the continuum hydrodynamics in the bulk region and derive the boundary conditions for dissipative processes at the fluid-solid interface. The positive definiteness of entropy production rate is the guiding principle of our derivation. Numerical simulations have been carried out to investigate the dynamic effects of the newly derived boundary conditions, showing that the contact line can move through both phase transition and slip, with their relative contributions determined by a competition between the two coexisting mechanisms in terms of entropy production. The observed competition can be interpreted by the Onsager principle of minimum entropy production.
Weiqing Ren, Institute for High Performance Computing and National University of SingaporeIn many areas of science and engineering, we face the problem that we are interested in analyzing the macroscale behavior of a given system, but we do not have a compete and accurate model for the macroscale quantities that we are interested in. On the other hand, we often do have a microscale model with satisfactory accuracy - the difficulty being that solving the full microscale model is far too inefficient, due to the disparate spatial and temporal scales that have to be resolved in such simulations. Therefore, it is desirable to develop multiscale models and algorithms that are based on a combination of the two formulations, in order to take the advantage of both the efficiency of the macroscale model and the accuracy of the microscale model. In this tutorial, I will first give an overview of the physical models at different scales, followed by some classical examples of multiscale methods including the Car-Parribello Molecular dynamics and the quasi-continuum methods. Then I will talk about three general frameworks for designing multiscale methods: the equation-free approach, the heterogeneous multiscale method, and the seamless algorithm.
Weiqing Ren, Institute for High Performance Computing and National University of SingaporeI will present a seamless multiscale method for the study of multiscale problems. The multiscale method aims at capturing the macroscale behavior of the system with the help of a microscale model. The macro model provides the necessary constraint for the micro model and the micro model supplies the missing data (e.g. the constitutive relation or the boundary conditions) for the macro model. In the seamless algorithm, the macro and micro models evolve simultaneously using different time steps, and they exchange data at every step. The micro model uses its own appropriate (micro) time step. The macro model uses a macro time step but runs at a slower pace than required by accuracy and stability considerations in order for the micro dynamics to have sufficient time to adapt to the environment provided by the macro state. The method has the advantage that it does not require the reinitialization of the micro model at each macro time step or each macro iteration step. The data computed from the micro model is implicitly averaged over time. In this talk, I will discuss the algorithm of the multiscale method, the error analysis, and its application to complex fluids.
Matthias Rosenkranz, National University of SingaporeDipolar Bose-Einstein condensates serve as a versatile experimental tool for studying magnetic materials used, for example, in hard disks and magnetic sensors. The underlying magnetic dipole-dipole interaction between atoms is nonlocal and acts over long distances. These properties give rise to new phenomena in quantum matter, such as supersolids or topological order. We introduce models for the experimentally relevant cases of 1D and 2D dipolar Bose-Einstein condensates. We also present extensions of these models for a stack of parallel 2D layers of dipolar Bose-Einstein condensates formed by a standing wave of laser light. All models are derived through a dimension reduction of the Gross-Pitaevskii equation. For a multilayer stack we find that a simple 2D potential acting across different layers describes well interactions that are induced by the long-range nature of the dipole-dipole interaction. We computed numerically the ground states of our models using the normalized gradient flow method. We investigate their validity by comparing them to ground states of the original 3D problem. Furthermore, we show that the dipole-dipole interaction in all models changes the aspect ratio of the ground state, which is observable with current experimental techniques.
Nicolas Rougerie, Université Grenoble 1 and CNRS, FranceA rotating trapped Bose-Einstein condensate is usually described using the Gross-Pitaevskii (GP) theory. Of particular interest are the quantized vortices of the condensate, unveiling its superfluid nature, and it is a major issue to understand these within the framework of GP theory. When increasing the rotation speed, three critical values are crossed at which the distribution of the vortices in the ground state of the condensate changes drastically. From a vortex free state, the condensate evolves to a vortex lattice state, then to a vortex lattice surrounding a giant vortex and finally to a pure giant vortex state. In this talk we will focus on the third critical speed, marking the transition to a giant vortex state. We will provide a rigorous estimate of this speed in the so-called Thomas-Fermi regime and investigate the nature of the phase transition, which turns out to depend on the type of trapping potential considered in the model. This is joint work with Michele Correggi, Florian Pinsker and Jakob Yngvason.
Christian Schmeiser, University of Vienna, AustriaThe 2D parabolic-elliptic Keller-Segel system features finite time concentration effects and global existence of suitably defined measure solutions. A stochastic particle approximation can be used as basis for numerical approximations. Its convergence as the number of particles tends to infinity will be discussed. This is joint work with Jan Haskovec.
Jie Shen, Purdue University, USAI shall present some recent work on phase-field model for multiphase complex fluids. Particular attention will be paid to develop models which are valid for problems with large density ratios and which obey an energy law. I shall present efficient and accurate numerical schemes for solving the coupled nonlinear system for the multiphase complex fluid, in many case prove that they are energy stable, and show ample numerical results which not only demonstrate the effectiveness of the numerical schemes, but also validate the flexibility and robustness of the phase-field model.
Christof Sparber, University of Illinois at Chicago, USAWe consider semiclassically scaled Schroedinger equations with an external potential and a highly oscillatory periodic potential. We construct asymptotic solutions in the form of semiclassical wave packets, i.e. coherent states propagating within a given Bloch energy band. These solutions are concentrated (both, in space and in frequency) around the effective semiclassical phase-space flow and involve a slowly varying envelope whose dynamics is governed by a homogenized Schrödinger equation with time-dependent effective mass. In the linear case, the corresponding adiabatic decoupling of the slow and fast degrees of freedom is shown to be valid up to Ehrenfest time scales.
David Srolovitz, Institute for High Performance Computing and National University of SingaporeMechanical forces play a key role in the shaping of versatile morphologies of thin structures in natural and synthetic systems. Understanding large deformations and instabilities of thin objects is central to ongoing efforts to developing programmable microfabrication techniques and novel functional devices including artificial muscles, stretchable electronics and bio-inspired robots. In this presentation, we discuss the morphology and deformation of thin ribbons, plates and rods and their instabilities through both theoretical modeling and table-top experiments. We'll first review some basic aspects of elasticity, differential geometry and stationarity principles as applied to the spontaneous bending and twisting of ribbons with tunable geometries. We show that helicity arises from mechanical anisotropy and the misorientation between the principal axes of the effective surface stresses and geometric axes of the ribbon. We report closed form analytic results (with no adjustable parameters) and validate these with simple, table-top experiments. For large deformation of ribbons and plates, a more general theory is developed to account for mechanical instability induced by geometric nonlinearity, due to the competition between inhomogeneous bending and mid-plane stretching energy. This model leads to unique predictions that are validated with additonal series of table-top experiments. Moreover, we show how edge effects alter the energy landscape of the two locally stable states when the in-plane dimensions are asymmetric (i.e., when the length does not equal the width). If time permits, we'll also discuss the buckling of rods embedded in an elastic medium and in the presence of external torques applied at the two ends. Such a study is relevant for instability-related phenomena in a broad spectrum of natural and synthetic systems, including helical growth of plant roots, buckling of cytoskeletal tissues, and helical buckling of oil pipes in wellbores. Authors: Zi Chen, Carmel Majidi, Mikko Haataja, and David J. Srolovitz
Qinglin Tang, National University of SingaporeIn this talk, I will present efficient and accurate numerical methods for studying the quantized vortex dynamics and their interaction of the two-dimensional (2D) Ginzburg-Landau equation (GLE) and Nonlinear Schrodinger equation(NLSE) with a dimensionless parameter $\varepsilon>0$ in bounded domains under either Dirichlet or homogeneous Neumann boundary condition. I will begin with a review of various reduced dynamical laws for time evolution of quantized vortex centers and show how to solve these nonlinear ordinary differential equations numerically. Then, I will present some results on the quantized vortex interaction under various different initial setup, and identify the cases where the reduced dynamical laws agree qualitatively and/or quantitatively as well as fail to agree with those from GLE or NLSE. Some other interesting phenomena will also be presented.
Tao Tang, Hong Kong Baptist University, Hong KongNumerical simulations for many physical problems require large time integration; as a result large time-stepping methods become necessary. In this talk, we will concentrate on the adaptive time stepping methods for physical problems with energy stable properties. The physical problems involving complex fluids, phase separations, epitaial growth of thin films, etc. By using the energy stable schemes, we are able to propose some time adaptivity strategies to resolve both the solution dynamics and the steady state solutions. Numerical simulation results will be reported and discussed.
Mechthild Thalhammer, University of Innsbruck, AustriaIn this talk, I shall primarily address the issue of efficient numerical methods for the space and time discretisation of nonlinear Schrödinger equations such as systems of coupled time-dependent Gross-Pitaevskii equations arising in quantum physics for the description of multi-component Bose-Einstein condensates. For the considered class of problems, a variety of contributions confirms the favourable behaviour of pseudo-spectral and exponential operator splitting methods regarding efficiency and accuracy. However, due to the fact that in the absence of an adaptive local error control in space and time, the reliability of the numerical solution and the performance of the space and time discretisation strongly depends on the experienced scientist selecting the space and time grid in advance, I will exemplify different approaches for the reliable time integration of Gross-Pitaevskii systems on the basis of a local error control for splitting methods.
Arthur Voter, Los Alamos National Laboratory, USAMany important materials processes take place on time scales that vastly exceed the nanoseconds accessible to molecular dynamics simulation. Typically, this long-time dynamical evolution is characterized by a succession of thermally activated infrequent events involving defects in the material. Accelerated molecular dynamics methods (hyperdynamics, parallel replica dynamics and temperature accelerated dynamics) and adaptive kinetic Monte Carlo methods, developed over the past 10-15 years, have made significant progress on this problem. These methods evolve the system from state to state in a dynamically accurate way, sometimes reaching time scales orders of magnitude longer than MD can. A significant remaining challenge arises from the fact that the more realistic the system under study, the more likely that many of the processes have low activation barriers, and this characteristic can severely limit the computational boost factor. After an introduction to these long-time atomistic methods, I will discuss some of our latest thinking on ways to mitigate this problem, including the use of the accelerated superbasin kinetic Monte Carlo method [A. Chatterjee and A.F. Voter, J. Chem. Phys., 132, 194101 (2010)] in the context of the accelerated molecular dynamics methods.
Arthur Voter, Los Alamos National Laboratory, USAThe molecular dynamics method, although extremely powerful for materials simulations, is limited to time scales of roughly one microsecond or less. On longer time scales, dynamical evolution typically consists of infrequent events, which are usually activated processes. This six-hour course is focused on understanding infrequent-event dynamics, on methods for characterizing infrequent-event mechanisms and rate constants, and on methods for simulating long time scales in infrequent-event systems, emphasizing the recently developed accelerated molecular dynamics methods (hyperdynamics, parallel-replica dynamics, and temperature accelerated dynamics). Some familiarity with basic statistical mechanics and molecular dynamics methods will be assumed.
Jian-Sheng Wang, National University of SingaporeWhen a disconnected junction is suddenly connected, what is the time-dependent energy current, and how is the steady-state current predicted by Landauer formula reached? In this talk, we'll try to answer these questions using the nonequilibrium Green's function (NEGF) method. In addition to the current, we also discuss the problem of calculating high order moments of energy transferred in a given time, which is known as full counting statistics.
Shouhong Wang, Indiana University, USAGas-liquid transition is one of the most basic problems in equilibrium phase transitions. In the pressure-temperature phase diagram, the gas-liquid coexistence curve terminates at a critical point C, also called the Andrews critical point. It is, however, still an open question why the Andrews critical point exists and what is the order of transition going beyond this critical point. To answer this basic question, using the Landau's mean field theory and the Le Chatelier principle, a dynamic model for the gas-liquid phase transitions is established, and the model is consistent with the van der Waals equation in steady state level. With this dynamic model, we are able to derive a theory on the Andrews critical point C: 1) the critical point is a switching point where the phase transition changes from the first order with latent heat to the third order, and 2) the liquid-gas phase transition going beyond Andrews point is of the third order. This clearly explains why it is hard to observe the liquid-gas phase transition near the critical point. In addition, the study suggests an asymmetry principle of fluctuations, which we also discover in phase transitions for ferromagnetic systems. The analysis is based on the dynamic transition theory we have developed recently with the philosophy to search the complete set of transition states. The theory has been applied to a wide range of nonlinear problems. This is joint with Tian Ma.
Heyu Wang, Zhejiang University, ChinaThis paper deals with the application of a moving mesh method for kinetic/hydrodynamic coupling model in two dimensions. With some criteria, the domain is dynamically decomposed into three parts: kinetic regions where fluids are far from equilibrium, hydrodynamic regions where fluids are near thermodynamical equilibrium, and buffer regions which are used as a smooth transition. Boltzmann-BGK equation is solved in kinetic regions, while Euler equations in hydrodynamic regions and both equations in buffer regions. By a well defined monitor function, our moving mesh method smoothly concentrate the mesh grids to the regions containing rapid variation of the solutions. In each step of the mesh moving, the solutions are conservatively updated to the new mesh, and the cut-off function is rebuilt first to consist with the region decomposition after the mesh moving. In such a framework, the evolution of the hybrid model and the moving mesh procedure can be implemented independently, therefore keep the advantages of both. Numerical examples are presented to demonstrate validation and efficiency of the method.
David T. Wu, Institute for High Performance ComputingThe properties of a polycrystalline material may be strongly influenced by the nature of its grain boundary network. We focus on the nucleation and growth mechanism for generating polycrystals and analyze how processing conditions affect the geometric and topological properties of the resulting grain boundary network. Modeling nucleation and growth is challenging computationally because nucleation is a stochastic process that requires molecular resolution, whereas the final mean size of grains formed by the impingement of growth fronts may be on the order of a micron or larger. We show how molecular simulations can be used to estimate input for continuum models of microstructure evolution, the result of which can be used by simulations that determine material properties.
Haijun Wu, Nanjing University, ChinaEver since the pioneer work of Ihlenburg and Babu\v{s}ka [{\em Finite element solution of the Helmholtz equation with high wave number. Part I: The $h$-version of the FEM}, Comput. Math. Appl., 30 (1995), pp.~9--37], a lot of pre-asymptotic analyses have been carried out for finite element methods (FEM) for the Helmholtz equation, but the pre-asymptotic error analysis of the FEM for the Helmholtz equation in two and three dimensions has not been done yet. This paper addresses pre-asymptotic stability and error estimates of some continuous interior penalty finite element method (CIP-FEM) and the FEM using piecewise linear polynomials for the Helmholtz equation in two and three dimensions. The CIP-FEM, which was first proposed by Douglas and Dupont for elliptic and parabolic problems in 1970's and then successfully applied to advection dominated problems as a stabilization technique, modifies the bilinear form of the FEM by adding a least squares term penalizing the jump of the gradient of the discrete solution at mesh interfaces, while the penalty parameters are chosen as complex numbers here instead of the real numbers as usual. It is proved that, if the penalty parameter is a pure imaginary number $\i\ga$ with $0<\ga\le C$, then the CIP-FEM is stable (hence well-posed) without any mesh constraint. Moreover the CIP-FEM satisfies the error estimates $C_1kh+C_2k^3h^2$ in the $H^1$-norm when $k^3h^2\le C_0$ and $C_1kh+\frac{C_2}{\ga}$ when $k^3h^2> C_0$ and $kh$ is bounded, where $k$ is the wave number, $h$ is the mesh size, and the $C$'s are positive constants independent of $k$, $h$, and $\ga$. By taking $\ga\to 0+$ in the CIP-FEM, it is shown that, if $k^3h^2$ is small enough, then the FEM attains a unique solution and satisfies the error estimate $C_1kh+C_2k^3h^2$ in the $H^1$-norm. Previous analytical results had been shown with the assumption that $k^2h$ is small, but it is too strict for medium or high wave number. Optimal order $L^2$ error estimates are also derived. Numerical results are provided to verify the theoretical findings. It is shown that the penalty parameters may be tuned to greatly reduce the pollution errors. The theoretical and numerical results reveal that the CIP-FEM have advantages over the FEM and is worth to be investigated further.
Congmin Wu, Xiamen University, ChinaThe spreading dynamics of a liquid on a solid surface plays a key role in many practical processes and has attracted considerable research interest. We perform molecular dynamics simulations to investigate the dynamics of droplet spreading driven by a long-range van der Waals force. We observe a transition from partial wetting to complete wetting by gradually increasing the coupling constant in the attractive van der Waals interaction. There exists a critical value of the coupling constant, above which the spreading is pioneered by a precursor film. The dynamically determined critical value quantitatively agrees with that determined by the energy criterion that the spreading coefficient equals zero. In the regime of complete wetting, the radius of the spreading droplet varies with square root of time. The behavior is also found in molecular dynamics simulations where the wetting dynamics is driven by the short-range Lennard-Jones interaction between liquid and solid. Joint work with Tiezheng Qian and Ping Sheng (Hong Kong University of Science and Technology).
Yang Xiang, The Hong Kong University of Science and Technology, Hong KongDislocations are line defects in crystals. The collective motion and interaction of dislocations determine the plastic properties of crystalline materials. Such study is very challenging due to the multiscale nature of dislocation modeling. In this series of lectures, I will first give an introduction to the dislocation theory, and then present some of our recent work on modeling and simulation of dislocations at multiple length scales, including the Peierls-Nabarro models, dislocation dynamics simulations, and continuum models based on dislocation densities.
Yang Xiang, The Hong Kong University of Science and Technology, Hong KongIn the first part, we present a continuum model for the core relaxation of incoherent twin boundaries based on the Peierls-Nabarro framework, incorporating both the long-range strain field and the local atomic structure. The continuum model is applied to the finite size effect of twin boundaries and interactions of dislocations with twin boundaries. The obtained results agree well with the experimental and atomistic simulation results. In the second part, we study the glide force due to stress on the constituent dislocations of slightly perturbed low-angle grain boundaries. We show that the stabilizing force comes from both the long-range interaction of the constituent dislocations and their local line tension effect. We also present a continuum model for this glide force.
Zhouping Xin, Chinese University of Hong Kong, Hong KongStudy of boundary layers for large Reynolds numbers is one of the central topics in fluid dynamics, and rigorous mathematical theory of the boundary layers remains a challenging area despite great past progress and efforts. There have been many fundamental questions to be addressed in the near future. In this short course, I would like to discuss the following topics: 1. Unsteady Navier-Stokes system with various physical boundary conditions, such no-slip, Navier-Slip, and in-flow and out-flow conditions. 2. Formal theories and matched asymptotic analysis. 3. Some well-posedness results for Prandtl?s boundary layer system. 4. Some recent works on convergence of viscous flows as the viscosity tends to zero. 5. Open problems.
Zhouping Xin, Chinese University of Hong Kong, Hong KongIn this lecture, I will discuss some problems with sready compressible flows past a curved nozzle. Some of the recent progress will be surveyed.
Zhouping Xin, Chinese University of Hong Kong, Hong KongIn this talk, I will discuss some of the recent developments on the global in time existence of classical solutions to the multi-dimensional compressible Navier-Stokes equations for data which may be of large oscillations and contain vacuums. Some new phenomena and analysis will be reported.
Chuanju Xu, Xiamen University, ChinaThe fractional calculus is almost as old as it's more familiar integer-order counterpart. The fractional partial differential equations are novel extensions of the traditional models, based on fractional calculus. They are now winning more and more scientific applications cross a variety of fields including control theory, biology, electrochemical processes, viscoelastic materials, polymer, finance, and etc. In this talk, we will explain a number of fractional models using the stochastic formulation of transport phenomena in terms of a random walk process. We will also present some efficient methods for the numerical solution of the time-space fractional diffusion equation. Particularly, we discuss the existence and uniqueness of the weak solution, and its spectral approximations based on the weak formulations. Finally, some interesting applications to viscoelastic materials, turbulence, and molecular biology will be addressed.
Zhi-Guo Xu, Beijing Computational Science Research Center, ChinaGinzburg-Landau equation (GLE) is well known for modeling superconductivity. In this talk, I will introduce the dynamical of quantized vortices of GLE based on particle interaction. I just consider the reduced dynamical laws for time evolution of quantized vortex centers in GLE and show some results of the nonlinear ordinary differential equation (ODE). First, two non-autonomous first integrals of ODE will be given. According to these first integrals, I discuss the collision of the vortices. Second, I show the dynamical of three vortices completely. Finally, I present several conjecture.
Boris Yakobson, Rice University, USATo transform a geometrical concept of a line or a plane into a model of material object, one must invoke some basic physical measures, like energy or force. We will discuss how such 'innocuous' model extension inevitably leads to the notion of discrete atom, chemical bond, and all the complexity of quantitative materials science. Illustrations will be 1D-nanowires, nanotubes, polymer chains, or 2D-graphene, h-BN, ribbons, edges and interfaces, mainly from recent research of my group at Rice.
Zhijian Yang, Wuhan University, ChinaA coupled atomistic and continuum model for numerical simulations of dynamics of crystalline solids will be presented. The method combines the continuum nonlinear elasto-dynamics model, which models the stress waves and physical loading conditions, and molecular dynamics model, which provides the nonlinear constitutive relation and resolves the atomic structures near local defects. The coupling of the two models is achieved based on a general framework for multiscale modeling - the heterogeneous multiscale method (HMM). An explicit coupling condition at the atomistic/continuum interface is derived. Application to the dynamics of brittle cracks will be presented. Results of the coupled model will be compared with the empirical continuum models. In particular, process zone, stress intensity factor, stress field, and loading curve will be discussed in details. Different types of loadings will be applied to study the interaction between elastic waves and crack tip behavior. The inertia effects of the crack tip will also be investigated.
Xingye Yue, Soochow University, ChinaIn this talk, we will give a short review on multiscale methods for elliptic homogenization problems. We will emphasize the intrinsic links between some popular methods such as generalized .nite element methods, residual-free bubble methods, variational multiscale methods, multiscale .nite element methods and heterogeneous multiscale methods.
Pingwen Zhang, Peking University, ChinaThe rich and fascinating phases, resulting from the subtle balance of two or more completing factors, as well as the phase transition in actual condensed physical systems, have been received tremendous attention in scientific community and industry. Some well-known illustrations come from block copolymer systems, liquid crystal systems, electric-carrying systems etc. How to efficiently discover new phases and study their phase translation presents a set of interesting and challenging problems. Mathematically, these problems can be translated into the study of the corresponding nonlinear variation (NV). In this talk, we focus on the numerical study of the NV, which can be divided into three main parts: the discrete methods, the iterative methods, and the reasonable initial values. The iterative methods are heavily dependent on the mathematical structures of the NV itself. The way of the system modeling will result in different mathematical features of the NV. And the initial values can be screened by the comprehension of physical mechanism. Therefore, solving the NV with profound physical background needs a comprehensive consideration. Lastly, the block copolymer systems are taken as an example to demonstrate the approach. We present a novel self-consistent field theory for general polymeric systems using the Gaussian functional integration technique. It reveals many favorable mathematical features, one of which is that it points out the descent and ascent directions of the effective Hamiltonian. With these properties, a series of gradient-typed algorithms are developed to accelerate the self-consistent field iteration procedure with a careful comparison with the traditional iterative methods. If time allows, we will show some more results of the discovery of new phases and the ordered nucleation phase transition in block copolymers.
Yanzhi Zhang, Missouri University of Science and Technology, USAIn this talk, a quadrature-rule type method is presented to approximate the quasi-continuum method for atomistic mechanics. For both the short-range and long-range interaction cases, the complexity of the quadrature-rule type method depends on the number of representative particles but not on the total number of particles. Simple analysis and numerical experiments are presented to illustrate the accuracy and performance of the method. It is shown that, for the same accuracy, our quadrature-rule type method is much less costly than the quasi-continuum method. References: [1] Y. Zhang and M. Gunzburger, Quadrature-rule type approximations to the quasi-continuum method for long-range interatomic interactions, Comput. Methods Appl. Mech. Engrg., 199 (2010), pp. 648-659. [2] M. Gunzburger and Y. Zhang, A quadrature-rule type approximation to the quasi-continuum method, SIAM Multiscale Model Simul., 8 (2010), pp. 571-590.
Yong Wei Zhang, Institute for High Performance Computing and National University of SingaporeUnderstanding the structural and mechanical properties of graphene is important for the development of graphene-based electronic and sensing devices. We report our recent research work on the mechanical properties of graphene using continuum mechanics and molecular dynamics/mechanics simulations. We have characterized edge mechanical properties graphene nanoribbons, and shown that edge stresses introduce intrinsic ripples in free-standing graphene sheets even in the absence of any thermal effects. Based on elastic plate theory, we identify scaling laws for the amplitude and penetration depth of edge ripples as a function of wavelength. We demonstrate that edge stresses can lead to twisting and scrolling of nanoribbons as seen in experiments. We also study how thermal fluctuation affects the elastic bending rigidity. It is found that the bending rigidity of single-layer graphene decreases exponentially. This is in stark contrast with recent atomistic Monte Carlo simulation result that the bending rigidity of a single-layer graphene increases with increasing temperature. We have investigated the mechanical properties of hydrogen functionalized graphene for H-coverages spanning the entire range from graphene (H-0%) to graphane (H-100%). It is found that the Young?s modulus, tensile strength, and ductility of the functionalized graphene deteriorate drastically with increasing H coverage up to about 30%. Beyond this limit the mechanical properties remain insensitive to H-coverage. The underlying reasons are discussed. We have also studied the epitaxial relation between graphene nanoflakes and Si-terminated SiC surface. It is found the rotation angle of the nanoflakes plays an important role in determining epitaxial relation.
Lei Zhang, University of Oxford, UKWe present a new variant of the geometry reconstruction approach for the formulation of atomistic/continuum coupling methods (a/c methods). For multi-body nearest-neighbour interactions on the 2D triangular lattice, we show that patch test consistent a/c methods can be constructed for arbitrary interface geometries. Moreover, we prove that all methods within this class are first-order consistent at the atomistic/continuum interface and second-order consistent in the interior of the continuum region.
Yong Zhang, Tsinghua University, ChinaIn this talk, we deal with the computation of ground state and dynamics of the Schrödinger-Poisson-Slater (SPS) system. To this end, backward Euler and time-splitting pseudopectral methods are proposed for the nonlinear Schrödinger equation with the nonlocal Hartree potential approximated by solving a Poisson equation. The approximation approaches for the Hartree potential include fast convolution algorithms, which are accelerated by using FFT in 1D and fast multipole method (FMM) in 2D and 3D, and sine/Fourier pseudospectral methods. The inconsistency in 0-mode in Fourier pseudospectral approach is pointed out, which results in a significant loss of high-order of accuracy as expected for spectral methods. Numerical comparisons show that in 1D the fast convolution and sine pseudospectral approaches are compatible. While, in 3D the fast convolution approach based on FMM is second-order accurate and the Fourier pseudospectral app roach is better than it from both efficiency and accuracy point of view. Among all these app roaches, the sine pseudospectral one is the best candidate in the numerics of the SPS system. Finally, we apply the backward Euler and time-splitting sine pseudospectral methods to study the ground state and dynamics of 3D SPS system in different setup.
Huijiang Zhao, Wuhan University, ChinaThis talk is concerned with the construction of global smooth solution to the Vlasov-Poisson-Boltzmann system near a given Maxwellian in the whole space for non-hard sphere interactions. It is based on a recent work joint with Renjun Duan and Tong Yang. |
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