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Green's Functions Experts Meeting
Boulder, CO
March 25-26, 2K2
List of Participants,
Agenda, and Abstracts
Sponsored by
Center of Theoretical & Computational Materials Science
National Institute of Standards & Technology
Organizing Commitee
Prof. Laura Bartolo, Kent State University
Dr. Adam Powell, MIT
Dr. Vinod Tewary, NIST, Boulder (Chairman)
March 25: Technical Sessions
There were several sessions on a variety of topics, covering Green's
functions themselves, their use in boundary element analysis, and the
NSF Digital Library. The talks themselves are described quite well in
the abstracts; here are presented brief summaries of them, focusing information
not explicitly presented in the abstracts.
GF-1, chair: Lingyun Pan
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David Barnett: A Comparison of Methods for the Computation of Anisotropic
Elastic Green's Functions and Their Derivatives
Analytic Green's functions are known for isotropic and hexagonal
elasticity in 3-D, but are impossible to calculate for other anisotropic
systems. Three methods for computing these functions are Fourier transforms
with numerical integration, Lothe-Gundersen iteration based on Stroh's
6x6 [N] matrix, and a technique of Lothe, Malén and Lavagnino (sp?)
based on angular derivatives ofr Stroh's eigenvectors. Of these, Fourier
transforms are the most straightforward but also the most time consuming,
Lothe-Gunderson iteration is the fastest for the Green's function
itself, and the Lothe-Malén-Lavagnino method is fastest for its derivatives.
Important work lies ahead in the use of these functions in multipole
algorithms for efficient boundary element calculations.
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Tom Ting: Recent Advances in Green's Functions
A taxonomy of Green's functions organizes a long list of papers
on Green's functions which was uploaded two years ago to the GF website
at Kent/NIST. These references and the functions they present are
categorized in terms of the equations they satisfy, space dimensionality,
coordinate system, and geometry. Since the list was uploaded, major
advances have been made by K.-C. Wu on dynamic motion of force lines
and dislocations, by E. Pan on 3-D anisotropic systems, by Z. Q, Yue,
and by C.-C. Ma on dynamic loading around a crack.
In another area, a puzzling situation arises when one considers
the image forces for anisotropic elasticity in a half-space: in 2-D,
an image force at one point is satisfactory, but in 3-D, the required
image force is on a line extending from the mirror image point to
infinity along the normal to the plane. Vinod Tewary and Dave Barnett
offered explanations for this phenomenon based on the difference between
log(r) and 1/r behavior of the Green's functions in 2-D and 3-D, but
the details of the cancellation of the infinite integrals remain unresolved.
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Len Gray: Direct Evaluation of 3-D Hypersingular Integrals
Hypersingular integrals of Green's functions at their singularities
are known to diverge, leading to the practice of using Stokes' theorem
to transform the integral into a path integration around neighboring
elements, which is then evaluated numerically. A new geometric construction
based on two polar coordinate transformations results in the cancellation
of the divergent parts and integrability by reducing the order of
the singularity. This is illustrated for the Laplace equation; future
work will extend this to nonlinear elements (which cannot be integrated
analytically), elastic behavior near a crack tip, graded materials,
and anisotropic elasticity.
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Paul Martin: Green's Function for a Three-Dimensional Exponentially-Graded
Elastic Solid
The Green's function discussed solves the equation for anisotropic
elasticity with a varying stiffness tensor given by
cijkl = Cijkl exp (2bmxm).
The solution is obtained by transformation to spherical coordinates
with the b vector as the axis, and involves double integrals
of elementary functions (and single integrals of modified Bessel functions,
which can be expressed as double integrals of elementary functions).
The exponential form of the properties makes the solution possible,
and with complex b vector, permits periodic variation of the
properties. The solution discussed here by no means in its "final"
form.
GF-2, chair: Jay Gillis
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James Beck: Survey of Green's Function Research Related to Transient
Heat Conduction
In addition to describing a variety of transient heat conduction Green's
functions, this work suggests a taxonomy for organizing such Green's
functions for a particular set of problems involving transient conduction
on a parallelpiped. It is hoped that this work will inspire a more
general overall taxonomy of Green's functions. The functions presented
allow transient heat conduction calculations with outstanding accuracy,
whose results may be are important for verification of other numerical
methodologies such as finite elements.
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Kevin Cole: Steady Heat Conduction in Cartesian Coordinates and a
Library of Green's Functions
Expressing general Green's functions for direct solution of the
steady heat equation in finite geometries present interesting challenges
discussed here. In particular, the many combinations of possible boundary
conditions leads to hundreds of different functions. Like the transient
heat transfer Green's functions described by Beck, these provably
accurate functions serve an important role in the verification of
software using other numerical methods -- and furthermore, these steady-state
functions can be used to verify the independently-developed transient
Green's functions. They are organized into a systematic taxonomy based
on the types of boundary conditions on the faces of the parallelpiped,
and made available for easy retrieval at this
website.
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John Berger: Green's Functions and Applications for Steady-State
Heat Transfer in Functionally Graded Materials
Green's functions for heat conduction with graded conductivity .
In one application, an indirect boundary method is used with reference
points outside the boundary, eliminating the singularity, but requiring
careful choice of quadrature points and weights in order to produce
accurate results. In an inverse problem, temperatures at a set of
points in a steady conduction situation are used to estimate the exponential
conductivity function parameters, and give an excellent fit, but the
temperature data used are produced analytically, so the behavior of
the method in the presence of noise is not yet known. The Method of
Fundamental Solutions is particularly useful for such inverse problems,
and can also be used for problems in which the boundary position is
unknown.
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Ambar Mitra: Application of BEM in Dislocation Dynamics and Other
Problems in Micro- and Nano-Scale
In this 2-D model of dislocation interactions, surface motion and
dislocations are used to calculate the elastic displacement field.
Within the domain, dislocations are nucleated at a predetermined set
of point forces based on the local stress state, and moved according
to the local stress state, Burger's vector, and dislocation drag.
The simulation is run to a strain of 0.38%, and for a variety of grain
sizes, producing as many as 104 dislocations and running
for up to 105 timesteps. Calculated yield stress for several
yield criteria are plotted in terms of grain size, and shown to closely
follow the Hall-Petch relationship.
Software Development and Industrial Applications, chair: Len Gray
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Laocet Ayari: Meeting Fracture-Based Requirements in the Aerospace
Industry
Current safe-life analysis practice in the aerospace industry involves
using the finite element method to calculate the maximum stress, or
stress intensity factor, at the edge of a flaw of non-detectable size,
and showing that this level of stress will not result in failure.
Unfortunately, the difficulties involved in creating multiple finite
element meshes to handle various flaw geometries, locations and orientations
forces analysts to consider only a relatively small subspace of the
possible flaw arrangements, and intelligent estimation of important
arrangements to analyze is necessary to produce meaningful resultsin
a reasonable amount of time. Unfortunately, on several occasions,
those estimates have been incorrect, leading to failure of significant
components and even entire spacecraft. It is hoped that tools based
on boundary element methods can much more efficiently analyze various
defect configurations without remeshing, making this process considerably
more rigorous and helping to prevent failures of the types described.
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Lingyun Pan: Boundary Element Analysis Usage in Caterpillar
Caterpillar has made extensive use of boundary element software
for decades, due to its ability to solve problems in which finite
element meshing is very difficult, such as engine geometries with
intersecting holes. However, a few significant difficulties have arisen
which are hindering progress on certain types of problems. Modeling
of thin plates is very challenging, and as CAD software produces surface
geometries of increasing complexity and resolution, the solution time
grows very quickly. Progress has been made in increasing the performance
of direct solvers, but iterative solvers which might run more quickly
have not yet been successfully realized. It is hoped that extension
of the software to use multipole methods will make boundary element
analysis continue to be practical as the complexity of the problems
continues to increase.
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Jay Gillis: Possible Applications for Green's Functions in Simulation
Software
Highly complex modeling efforts such as those describing electromagnetic
interactions in very elaborate integrated circuits require considerable
time to run and are increasingly incompatible with short design cycles.
In particular, the requirement that simulations produce results "by
morning" limits the ability of these models to keep up with the physical
parts which they reproduce. It is hoped that efficient Green's function
techniques, such as those developed by Pan, Yang and Yuan for multi-layered
materials, can make this process more efficient.
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Earlin Lutz: What Online Mathematical Content Could Be Useful to
Commercial Software Development?
Considerable effort is expended in the translation of mathematical
content between various formats, such as journal articles and programming
implementations in specific languages, and error is inevitable in
this process. More generic machine-readable representations of these
data would permit automation of much of this process. However, presenting
machine-readable representations requires a change in the basic mindset
of the mathematician or programmer, toward one where the "programmer"
develops such representations and uses them in more general contexts
with greater flexibility and power than one's own code written to
solve a particular class of problems.
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Adam Powell: The Julian Boundary Element Code
Named for the late Julian Szekely, Julian is a generic boundary
element code designed to be as flexible as possible in solving a wide
variety of problems. It has a standardized Green's Function API which
permits scalar and vector equations, and can handle several element
types and distributions of nodes, arbitrary space dimensions, and
multiple surfaces with separate sub-matrices for efficient solution
of problems for various configurations of one or two of the surfaces.
It is also written to be portable across operating system platforms,
and internationalized such that translation of strings is a relatively
easy process which does not require recompilation. At this point,
calculation of scalar fields is working well, and vector fields and
graphical representation of solutions are nearing completion, so a
first developers' release is not far away.
GF-3, chair: Ambar Mitra
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Ravi Pandey: Calculation of Properties of Defects in Semiconductors
A hybrid approach involving electron density functional theory and
Green's functions is used to model the effect of point defects in
chalcopyrite semiconductors on their optical absorption properties.
This absorption coincides with the wavelength range of intended applications
of these materials, making their understanding crucial to such applications.
In addition, phonon scattering behavior of these defect centers can
be inferred from this model.
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Vinod Tewary: Elastic Green's Functions for Multiscale Modeling
A lattice static Green's function determines the effects of point
defects, such as vacancies, substitutions and interstitial atoms,
on the surrounding material. This model shows that such effects tend
to be limited to the region immediately aronud the defects themselves,
including the first- or second-nearest neighbor atoms. However, although
strain effects are highly localized, other properties are very sensitive
to defect concentrations. The various properties calculated by this
methodology can be used to develop a thermodynamic understanding of
defect free energy, which can be used to help predict their concentration
under certain circumstances. This implicitly multiscale approach to
these phenomena is many orders of magnitude more efficient than other
methodologies such as molecular dynamics.
Digital Library, chair: Laocet Ayari
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Dave Fulker: NSDL and its "Core Integration" Effort
The National S(cience) Digital Library is an "education layer" over
the Web, allowing intelligent use of available resources through organized
collections of targeted material in specific areas, and tools which
simplify discovery of material in these collections, all knit together
by a Core Integration team into a seamless library. In this context,
such resources as software can be seen in several ways: as a discoverable
resource in a collection, as a helper to use other resources, or as
a means to actually invoke NSDL services which support them.
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Greg Shreve: Integrating Domain Specidic Content and Document Description
Markup with Collection Metadata in a Green's Functions Digital Library
In the GREEN Digital Library, "metadata", or data describing data,
is used to organize available resources from books and literature
papers to software, in the same way as a card catalog organizes a
traditional library. However, all of these resources have internal
data, such as chapters within books and equations, which are not adequately
described in metadata but which would serve valuable research functions.
Part of the GREEN project therefore includes an effort to put data
in such a searchable format, and as the only engineering collection
funded to date, the methodologies developed to accomplish this task
will form important examples for other engineering disciplines to
follow. These methodologies begin with online forms which facilitate
the entry of data, and development of other useful data types and
tools will require input from practitioners in the field, such as
those represented here.
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Laura Bartolo: Building the Green's Functions Digital Library
In building the current Green's functions site into a useful digital
library, there are several important tasks ahead. In addition to the
web forms discussed by Greg Shreve, and the static content of the
site, we will need to define the roles of an advisory board, and address
issues of sustainability, in order for the collection to remain useful
into the indefinite future. With the help of this group, a preliminary
launch of the collection is planned for December, 2002, with the material
in final form and the library in a sustainable state by the October,
2003 end of the current grant.
March 26: Green's Function Digital Library open discussion
(To be added later.)
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