GF Library Workshop Summary

October 15-16, 1998

Contents

  1. Introduction
  2. Theory
  3. Applications
    1. Acoustics
    2. Solid Mechanics
    3. Fracture
  4. Feedback
  5. Next Steps
  6. Attendance

Introduction

A workshop on "Library of Green's Functions and its Industrial Applications" was held on Thursday and Friday, October 15-16, 1998, at the National Institute of Standards and Technology in Gaithersburg, Maryland. The goal of the workshop, sponsored by the National Institute of Standards and Technology, was to demonstrate the idea and application of a library of discretized Green's functions to industrial problems.  The Green's Function Library working group has developed software which solves classes of problems by pre-computing Green's functions for industrially relevant geometries, and included these GFs in easy-to-use software for modeling the behavior of a variety of physical phenomena.  In this workshop, much of this software was presented to industrial working group participants for evaluation and discussion of the future direction of the working group.  The workshop was directed by Dr. Vinod Tewary of NIST (Boulder), and Prof. Frank Rizzo of Iowa State University.

The workshop was held at the CTCMS computer room. A server was available for two participants. Each participant was given a temporary computer account. The workshop consisted of brief tutorials on the methods, a demonstration of the library, and hands-on experience with the library for the participants. Included were some new boundary element strategies associated with the library, for problems in stress analysis, fracture mechanics, ultrasonic wave propagation, and composite materials.  Applications to other areas, such as acoustics, electromagnetic and thermal phenomena, were discussed. An open forum was held on potential uses of the library on specific industrial problems and its strengths and weaknesses. The plan is to make the library web based.

The inaugural talk was given by Dr. Sharon Glotzer, Director of the CTCMS. Dr. Les Smith, Director of the MSEL, visited the workshop on Oct 15 and met each participant. Dr. Hans Engler, Program director Mathematical and Computational Sciences Division, NSF, visited the workshop on Oct. 16.

Theory

Prof. Frank Rizzo began with a discussion of the theory of the Boundary Element Method (BEM).  For a linear partial differential equation (PDE) for the field u on a body P, if one knows the solution on its boundary dP and would like the solution at a point x inside P, one need only integrate a function of u over dP.  Therefore, calculation of the field throughout P reduces to its calculation on dP.  Note that the body need not be finite; the boundary may represent a finite interruption of an infinite body, such as an obstacle to a propagating wave (see Acoustics below).

The BEM assumes that either either the field value u or its normal derivative t=t.n (where t is the gradient of u) is known everywhere on that surface, and the goal is to calculate u where t is known and vice versa.  The method calculates an approximation to this solution by discretizing dP and calculating the unknowns at individual points xj along dP.  This turns the continuum problem into a linear system of equations for u(xj) and t(xj), of the form

Aij (tj, uj) = Bij (uj, tj)
where (uj, tj) on the right is the vector of known field values and derivatives at the points xj, (tj, uj) on the left is that of the unknowns, and Aij and Bij are dense matrices containing integrals of the Green's functions over the interpolant functions at xi and xj.  The solution is simply
(tj, uj) = Aij-1Bij (uj, tj).
Most of the computational work goes into calculating the inverse matrix Aij-1 (or its functional equivalent, the LU decomposition of Aij), because the computational work for this step is proportional to the number of unknowns cubed.

The concept of the Green's Function Library introduced by Prof. Frank Rizzo enters when we consider the boundary of P to have two components, which we call S1 and S2.  If S1 is more intricate than S2, and we would like to know the solutions for several different configurations of S2, we can accelerate the computation as follows: we divide (tj, uj) and (uj, tj) into their S1 and S2 components, and Aij into A11ij, A12ij, A21ij and A22ij, with similar partition for Bij.  The system of equations then becomes

( A11ij A12ij ) ( t1j, u1j ) = ( B11ij B12ij ) ( u1j, t1j ).
A21ij A22ij t2j, u2j B21ij B22ij u2j, t2j
We first solve the top row for (t1j, u1j)
A11ij (t1j, u1j) + A12ij (t2j, u2j) = B11ij (u1j, t1j) + B12ij (u2j, t2j)
(t1j, u1j) = A11ij-1 [B11ij (u1j, t1j) + B12ij (u2j, t2j) - A12ij (t2j, u2j)].
Then we solve the bottom row for (t2j, u2j)
A21ij (t1j, u1j) + A22ij (t2j, u2j) = B21ij (u1j, t1j) + B22ij (u2j, t2j)
A22ij (t2j, u2j) = B21ij (u1j, t1j) + B22ij (u2j, t2j) - A21ij A11ij-1[B11ij (u1j, t1j) + B12ij (u2j, t2j) - A12ij (t2j, u2j)]
[A22ij - A21ijA11ij-1 A12ij] (t2j, u2j) = B21ij (u1j, t1j) + B22ij (u2j, t2j) - A21ijA11ij-1 [B11ij (u1j, t1j) + B12ij (u2j, t2j)]
(t2j, u2j) = [A22ij - A21ijA11ij-1 A12ij]-1[B21ij (u1j, t1j) + B22ij (u2j, t2j) - A21ij A11ij-1[B11ij (u1j, t1j) + B12ij (u2j, t2j)]].
The important thing to note here is that since S1 is more intricate (and therefore has more unknowns) than S2, the calculation of A11ij-1 will dominate the solution time of the system.  However, because only S2 changes from run to run, A11ij-1 need only be calculated once for a given S1!

Hence the concept of a Library.  For often-used geometries of S1, one stores the matrix A11ij-1 for subsequent retrieval.  This library can take several forms with varying degrees of distribution and permanence, from a national library which might have simple and often-used geometries, to a local library at a corporation containing A11ij-1 for selected parts, to a A11ij-1 calculated just for a single session in which a team of engineers is testing the effect of design changes on surface hoop stresses.

This working group is built around developing and commercializing this Library concept.

Applications

Acoustics

The first set of demonstrations involved calculating the scattering of a plane wave traveling through an infinite medium by a void or rigid inclusion.  This problem involved a single surface, and the demonstrated tool calculated the amplitude, phase shift and polarization of scattered dilational and shear waves in all directions.  Matrices A11ij-1 were available for several wave numbers and simple geometries, so solution time was very short.  The visualization tool for the results was a very impressive OpenGL-based 3-D color surface manipulator.

Solid (Linear Elastic) Mechanics

Here there were two problems with demonstrated solutions: a 3-D cantilevered beam with a hole through the thickness, and a 2-D beam with an interior hole.  In the latter case, the hole could be dragged with the mouse and the displacement field updated very quickly with each new hole position.  Available hole shapes included a circle or ellipse, a polygon, and a straight-line crack, which is the basis for the next section.

Fracture

In the 2-D elastic mechanics program mentioned above, the crack hole geometry allowed calculation of the stress intensity factors at both ends of the crack for the two fracture modes in 2-D.  Under cyclic load, a crack will grow by fatigue according to Paris' law for fatigue crack growth, da/dn = R (delta K)c, where da/dn is the change in crack length per fatigue cycle, delta K is the difference between maximum and minimum stress intensity factor at the tip throughout a fatigue cycle, and R and c are properties of the material.

A program has been written which steps through time, calculating the displacement field on the surface and stress intensity factors at crack tips.  This in turn gives the direction and length of the crack extension at each time step according to Paris' law.  The crack is extended by adding a segment to S2 at each end.  The program terminates when the critical stress intensity factor is reached, resulting in failure.  The result is a complete description of fatigue crack development from start to finish.

[Question: does this model neglect crack bifurcation?  I've forgotten just enough fracture mechanics to not remember under what circumstances this happens, i.e., whether it requires anisotropy on nonuniformity.]

Feedback

An open forum was held on the second day in two parts. In the first part, all the industry participants were asked to comment upon various aspects of the workshop, in particular, about the potential usefulness of the GF library in their actual problems.  A summary of the discussions and comments is given below.

A questionnaire (MS Word format) was given to participants on the first day with a request for them to fill out and return it on the second day.  A summary of questionnaire responses is below.

Next Steps

Industrial partners
Areas of industrial interest have been identified. The GF library will be further developed and adopted for application to these areas. One specific industry identified for this purpose is Ball Aerospace. We plan to start working on some problems of Ball Aerospace very soon. More industrial partners will be identified.
GF/ BEM consortium
Almost all the participants expressed a strong interest in using the GF library and joining the CTCMS working group as developers or users. A consortium has been formed with NIST in the coordinating role.  A web-based tool developed at Kent University for scientific collaboration is proposed to be used by the consortium to communicate and collaborate on the use of the GF library.
GF library- second generation
We will develop the second generation GF library that would include a more efficient mathematical algorithm, data structure, and a GUI.
The solder connection
A new application of the GF library has been identified by Adam Powell in electronic packaging problems the solder group. A new working group is formed with Vinod Tewary and Adam Powell as PIs for this work that would mark collaboration between two different working groups (GF and Solder) of the CTCMS.
Another meeting/workshop
We plan to have another meeting/workshop after some time. In this workshop, the participants will describe the industrial problems that they addressed using the GF library and their first hand experience with using the library. We will also discuss the second generation library.

Attendance

The attendance in the workshop was free and by invitation only. Invitations were sent to scientists and engineers in several
organizations, who we knew were interested in GF and BEM analysis. In addition, an announcement was placed on the CTCMS
web site. People interested in attending the workshop were invited to write to us giving a brief description of the kind of
problems in which they would apply the Green's function methods.

A total of 19 people attended the workshop, in addition to the PI (Vinod Tewary) and two principal collaborators (Frank
Rizzo and John Berger). Out of the 19 participants, 8 were from industry, 9 from universities, and 2 from NIST. A university
participant (Ravi Pandey) was sponsored by Dow Corning.
 
*Dr. Laocet Ayari Ball Aerospace [email protected]
*Prof. John R. Berger CSM [email protected]
Prof. Abhijit Chandra MTU [email protected]
Dr. Joseph Chao Allied Signal [email protected]
%Dr. Xinyu Dou Motorola [email protected]
Mr. Michael Edlin Digital Creators Boulder [email protected]
Dr. Sharon Glotzer NIST [email protected]
Dr. Ivan Gonsalves Carterpillar, Inc. [email protected]
Dr. Steve Langer NIST [email protected]
Prof. Yujin Liu Univ. of Cincinnati [email protected]
Dr. Shan Lu ISU [email protected]
Dr. Ambar Mitra ISU [email protected]
Dr. Raj Molan Battelle [email protected]
Mr. S. Mukundan ISU [email protected]
*Dr. Lingyung Pan Automated Analysis Corp. [email protected]
Prof. Ravi Pandey MTU [email protected]
Dr. Adam Powell NIST [email protected]
Dr. S. Raveendra Automated Analysis Corp. [email protected]
Mr. Moshen Rezayat SDRC [email protected]
*Prof. Frank Rizzo ISU [email protected]
Dr. Thomas Rudolphi ISU [email protected]
Mr. Bob Shepherd CSM [email protected]
Dr. Julia Slutsker Univ. of Maryland [email protected]
%Mr. Kevin Smith Pratt & Whitney [email protected]
#Dr. Vinod Tewary NIST [email protected]
*Member, Organizing Committee 
#Chairman, Organizing Committee 
%Unfortunately could not attend

Discussion Summary

Caterpillar (Ivan Gonsalves):
Primarily use BEM analysis for fuel injection problems. Encouraged by the potential of the GF library.
GF library will be useful for analyzing multiple geometries.
Will enable design and analysis done by same people.
Turn around time is extremely important. GF library will allow numerous parameters to be changed.
It may be a good idea to switch to NT platform.

GF lib would be a big help: (a) with problems requiring lots of iterations, (b) with turnaround time in general. (c) There are still modeling and CPU problems with large finite element discretizations - there would be advantages for the GF lib and BEM here. (d) Gf lib and BEM would definitely help with (speedup) problems involving small changes (perhaps probabalistic data) in big complicated structures.

Allied Signal (Joe Chao):
Deal with flight safety issues. Search for flaws. Stress analysis for lifetime estimate. Need reliable method for flaw detection and to estimate which part should be scrapped. They are studying damage tolerances by analyzing cracks. The FEM is time consuming. They need to play "what if" games, such as by moving a flaw around a part and perform stress analysis. This can be done efficiently by using the GF library.

GF lib would be very good in NDE, for detecting flaws (inverse problem), and for playing the many what-if games which are important in NDE.

Dr. Chao sees advantages in stress analysis as done in the company.  He is already using BEM/ GF library idea.

Ball Aerospace (Laoucet Ayari):
Build hardware for aerospace agencies such as NASA. Do stress analysis on components. Perform static and dynamic tests. They use FEM. Need a tool to assess stress field near cracks, predict flaw size and location.

GF library would provide basic solution for certain type of crack problems. He would like to use it. His colleagues don't know about GF. Some education would be needed. Suggested offering short courses on site.  BEM in general and GF lib in particular would be excellent for the many problems in fracture mechanics which seem to be of primary concern in the company. Education about the ideas seems to be an important first step. There was also interest in almost everything done at the workshop.

Automated Analysis/ Comet Acoustics (S. Raveendra):
This company is already committed to BEM in a big way for acoustics problems of particular types.

In cases such as noise reduction problems in cars, where an analysis is not repeated, GF library would not be of much use.  In cases such as architectural acoustics, concert-hall design, sound location and sound source identification, etc., the GF library would be very useful. For example, in concert hall acoustics, they need calculations like what happens if a certain instrument is moved from one place to other, or a certain absorber is introduced in the hall. In such cases where only "S2" changes, GF library would be useful.

GF library will be useful in acoustic holography. Methods for inverse problems will be useful to locate acoustic sources.

In general, GF library seems to be more suited to fracture, stress analysis, thermal problems etc. but may be tough for acoustic problems. However, he would consider using GF library for some acoustic problems; may be set up own GF libraries.

Digital Creators (Mike Edlin):
Very impressed by the software design. For placing the GF library on the Internet, JAVA would perhaps not work because it is not powerful enough for interactive calculations. CD ROM will be good. Suggested provision for downloading specific modules so that calculations can be done locally.
Automated analysis and Caterpillar (L. Pan):
Developing in-house BEM software for stress analysis. GF library will save time that is the chief advantage. It may have important applications in geomechanics (half-space calculations).
Dow Corning and MTU (Ravi Pandey):
Interested in interfacial failure. GF library will be useful in that kind of calculations. Dow Corning would be interested in exploring that.
Battelle (Raj Mohan):
Presently uses FEM. GF library would be useful in surface engineering. GF library should be particularly useful in diffusion calculations such as in electromigration problems.
SDRC (M. Rezayat):
The analysis component in the design of complicated structures is a real bottleneck.  There are some new advantages to BEM analysis as portrayed by the GF lib idea, and when developed to their full potential for 3-D problems, there could be some revolutionary changes in speed of analysis to allow the highest level managers take notice. There is a real need for this.  Nobody wants to do stress analysis anymore -especially if they have to wait at least 6 hours for it.  They simply want graphical results instantly and somewhat reliably, but some sacrifices in accuracy can be made for speed. Nobody really cares about meshes, elements, etc., anymore.

The GF lib has potential for all of this. Indeed the workshop showed what is possible for 2-D problems.  To be able to drag the circular hole around on the interior of a structure, and see results of the analysis almost immediately, was very impressive and new.  What we have is already valuable for some applications, certainly valuable for everyone as an educational tool for what is possible, and something of definite value in his organization if we can get comparable effects for 3-D problems.

GF library can be useful in doing real time calculations that would be useful in communication and collaboration in the analysis area.  Speed of calculation is most important.  GF library should be further developed to be applicable to materials of different shapes. With its present limitations, he does not think it would be useful to SDRC.

MTU (A. Chandra):
We should look carefully at the physics of the problem which will often dictate the choice of modeling scheme.
We should get a "shopping list" from the industry. We should define and target our initial goals.
ISU (Tom Rudolphi):
We are attempting to create a virtual handbook. The most important thing is that it should be easy to use.
CTCMS (Sharon Glotzer):
Form a working group/consortium of all the people involved. Promote direction; Problem formulation; Participation in providing ideas, sharing own experiences; Develop specific modules.