2007 DOE Summer School in Multiscale Mathematics and High Performance Computing
2007 Summer School Research Talks
Introduction to Finite Difference, Finite Element and Finite Volume Methods for the Numerical Solution of Partial Differential Equations
Vrushali Bokil - Assistant Professor, Department of Mathematics, Oregon State University
Nathan Gibson - Assistant Professor, Department of Mathematics, Oregon State University
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Abstract:
Finite difference (FD), finite element (FE) and finite volume (FV) methods are three of the classical methods used for the numerical solution of continuum models described by partial differential equations (PDEs). In this talk we will give an introduction to these three methods by applying them to different types of PDEs.
We will start by classifying PDEs into three categories: elliptic, hyperbolic and parabolic, along with a discussion of appropriate initial/boundary conditions in order to obtain well-posed problems. We will then introduce conservation laws and their differential and integral representations. The theory of the three numerical methods will be demonstrated by applying them to various PDEs. A comparison of the methods will be conducted in order to highlight similarities, relative advantages and disadvantages as well as the difficulties that are encountered in their implementation. Various theoretical issues such as consistency, stability and convergence of the methods will be discussed along with important concepts such as the Courant-Friedrich-Lewy (CFL) condition and the Lax Equivalence theorem.
Survey of New Continuum Numerical Multiscale Approaches and Limitations
Malgorzata Peszynska - Associate Professor, Department of Mathematics, Oregon State University
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Abstract:
Direct numerical discretization of stationary and/or transient PDEs with multiscale coefficients requires solution of huge linear systems or very stiff ODEs. The associated complexity can be handled in some cases by use of various multigrid and multilevel solvers.
On the other hand, in the last few years a plethora of multiscale numerical methods based on finite elements have been proposed for various applications. These include the heterogeneous multiscale method, the variational multiscale method, and the multiscale FE (with overlapping variants) as well as subgrid or mortar methods. Most are designed to compute the macroscale solution, while some are able to recover next order effects. Some methods work best for periodic coefficients, and some can be extended to handle any coefficients including random. Some assume scale separation, and some use special test functions in the classical or mixed variational formulation. Finally, some methods can be naturally applied to transient and nonlinear problems. In the talk we give an overview of main ideas and issues as well as discuss some open problems in various applications.
Atomistic Simulation Methods and Multiscale Modeling
Sergey Medyanik - Assistant Professor, School of Mechanical and Materials Engineering, Washington State University
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Abstract:
In the first part of the lecture, the most common atomistic methods, such as molecular dynamics (MD), molecular mechanics (MM), and Monte-Carlo (MC) methods will be reviewed. Brief introduction to these methods will be provided and typical applications will be discussed. Examples of the simulations will be presented, including those from the speaker’s own research.
In the second part of the lecture, known limitations of the atomistic methods will be outlined that motivate the development of the multi-scale modeling techniques. Novel multi-scale approaches will be discussed with the emphasis on spatial and temporal multi-scale methods. Some recent results, including those obtained by the speaker, will be presented.
An Introduction to Multifractal Measurements
Nicolas Hô, Ph.D - Research Scientist, National Security Directorate, Pacific Northwest National Laboratory
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Abstract:
Multifractal measurements have shown to be relevant tools in the study of irregular, multiscale data sets and have been applied to a variety of complex systems. A description of fractals and multifractal measurements will first be presented, followed by an overview of numerical techniques used to measure the multifractal spectrum from an experimental data set.
Coupling atomistic and continuum descriptions in multi-scale modeling
Mark O. Robbins - Department of Physics and Astronomy, Johns Hopkins University
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Abstract:
Many physical phenomena involve a wide range of length scales. For example friction, fracture and adhesion involve breaking of atomic bonds at interfaces, but the stresses on these bonds are determined by macroscopic deformations in the surrounding regions. The discreteness of matter is critical to bond breaking, but large scale deformations are most efficiently treated with approaches based on continuous strain fields. Coupling these qualitatively different descriptions of matter poses both fundamental and algorithmic challenges. The talk will describe and illustrate different strategies for bridging between descriptions. One strategy is hierarchical or sequential in nature. Calculations are performed with different methods at overlapping scales and forced to be consistent through parameterization of constitutive laws or boundary conditions. Simple applications to fluids and solids will be described. A second strategy is to treat different regions of space with different physical models and solve them simultaneously. The treatment of the interface between atomistic and continuum regions poses special challenges and different strategies will be reviewed.
Multiscale Modeling of Preferential Flow
Ralph Showalter - Professor and Chair of Department of Mathematics, Oregon State University
Abstract:
In this talk we focus on saturated preferential flow and coupled transport.
First we present a new multiscale model of fully saturated flow coupled to transport through a highly heterogeneous medium without the traditional assumptions of scale separation. The model includes the usual diffusion effects from various spatial scales and retains the effects of local advective transport and is mass conserving. Previous models captured the observed tailing effects of slow storage, but the advection in the cell problems was lost or was not accounted for in the global transport and flow equations.
In the second part we show that preferential flow or channeling effects at the boundary of a porous medium can be effectively modeled by a Darcy-Brinkman interface system.
This is joint work with Malgozata Peszynska and Fernando Morales.
Homogenization, Microstructure and Optimal Design
Robert Lipton - Professor, Louisiana State University
Abstract:
We present multiscale methods for the design of microstructure. The approach given here is based upon the theory of homogenization for heterogeneous media. The first part of the presentation provides a brief overview of homogenization theory and its connection to the physical modeling of heterogeneous media. The second part introduces the notion of G-convergence of elliptic operators and the associated relaxation of of variational problems. We show how these ideas apply to the design of microstructure and how they motivate numerical methods for microstructure design. The last part describes numerical methods for optimal design of graded microstructure for thermal management.
Introduction to discrete particle methods I: Smoothed Particle Hydrodynamics
Alexandre Tartakovsky - Research Scientist, Computational & Information Sciences Directorate, Pacific Northwest National Laboratory
Abstract:
Smoothed Particle Hydrodynamics is a lagrangian particle method that uses particles to discretize a computational domain. Because of its Lagrangian particle nature SPH has several advantages for modeling non-linear processes involving large material deformations and complex physics: i) complex interface and free surface flows/deformations including material coalescence and splitting can be modeled without complex front tracking schemes; ii) in Lagrangian framework there is no non-linear term in momentum conservation equation, so that SPH allows accurate solution of momentum dominated flows; iii) complex physics can be implemented through simple particle interaction force. In addition, SPH method explicitly conserves mass, momentum and energy. This lecture gives an introduction to the SPH method and describes applications of SPH for the multi-scale simulations of multiphase flow and reactive transport.
Introduction to Parallel Programming
Kevin A. Glass, Ph.D. - Environmental Molecular Sciences Laboratory, Pacific Northwest National Laboratory
Doug Baxter - Environmental Molecular Sciences Laboratory, Pacific Northwest National Laboratory
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Abstract:
In this presentation, we will discuss fundamental issues related to the design and implementation of good parallel code. We will begin by establishing the goals for parallel programs then move into a discussion regarding the development of parallel code from “good” serial code. We will use a straightforward matrix multiplication algorithm as the basis for the discussion. After discussing the general problems associated with parallel coding, we will turn our focus to the design of shared memory code but our main focus will be on the design of distributed memory code. We will wrap-up the presentation with a discussion of debugging and analyzing parallel code.
Introduction to MPI
Kevin A. Glass, Ph.D. - Environmental Molecular Sciences Laboratory, Pacific Northwest National Laboratory
Doug Baxter - Environmental Molecular Sciences Laboratory, Pacific Northwest National Laboratory
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Abstract:
In this presentation, we will describe the design and implementation of distributed memory code using MPI. This talk will use ideas developed in the "Introduction to Parallel Programming" to demonstrate good programming and design style as it applies to MPI-based parallelism. Using the matrix multiplication algorithm, we will demonstrate several methods for parallelizing the code including an analysis of the effectiveness of our implementations. We will conclude this discussion with a demonstration of the use of "groups" in MPI code.
Using MPI Workshop
Kevin A. Glass, Ph.D. - Environmental Molecular Sciences Laboratory, Pacific Northwest National Laboratory
Doug Baxter - Environmental Molecular Sciences Laboratory, Pacific Northwest National Laboratory
Abstract:
In the "Introduction to MPI" section, we present several methods for parallelizing the matrix multiplication algorithm. Each of the implementations has flaws that inhibit the production of efficient parallel code. During this workshop, students are encouraged to use the information in previous two talks to refine these implementations. We will provide students with an introduction to the available computing resources and a set of code which from which to work.
An Introduction to the Quasicontinuum Method
Mitchell Luskin - Professor, University of Minnesota
Abstract:
Multiscale methods offer the potential to solve complex problems by utilizing a fine-scale model only in regions that require increased accuracy.
The quasicontinuum (QC) method is a multiscale computational method for crystals that uses an atomistic model only in regions with highly non-uniform deformations such as around dislocation defects and crack tips, but reduces the number of degrees of freedom by using a continuum model in regions where the deformation is nearly uniform.
The continuum model is based on a strain energy density for a deformed lattice that is computed from the atomistic potential.
Multigrid Robustification
Barry Smith - Computational Mathematician, Argonne National Laboratory
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Abstract:
Multigrid is an extremely efficient solver for many systems of linear equations arising from the the discretization of partial differential equations. It is, however, fragile, unless one is careful, making seemingly small changes to the equations can destroy multigrid's great convergence properties. I will present, through an interactive presentation, various techniques available to make multigrid more robust and thus expand its domain of applicability.
Homogenization II
Alexander Panchenko - Department of Mathematics, Washington State University
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Abstract:
This is the second lecture in the two-lecture series devoted to homogenization. Here we consider further applications of two-scale asymptotics. Two particular examples of interest are flow in porous media and acoustic wave propagation in solid-fluid composites.
