CNCS Center for Nonlinear and Complex Systems
   Search Help Login pdf version printable version

David G Schaeffer, James B. Duke Professor of Mathematics

David G Schaeffer
Contact Info:
Office Location:  132B Physics
Office Phone:  (919) 660-2814
Email Address: send me a message
Web Page:  http://www.math.duke.edu/~dgs

Teaching (Spring 2012):

  • MATH 108.03, ORD & PRTL DIFF EQUATIONS Synopsis
    Physics 047, TuTh 08:30 AM-09:45 AM
  • MATH 132S.01, NONLIN ORD DIFF EQUA Synopsis
    Physics 259, TuTh 01:15 PM-02:30 PM
Education:

B.S., Physics, University of Illinois, 1963
Ph.D., Mathematics, MIT, 1968
Specialties:

Applied Math
Research Interests: Applied Mathematics, especially Partial Differential Equations

Granular flow is the major focus of my research, and this is what I describe in the first part of this page. In the second part, I discuss a course I began teaching in 1996 that has expanded my research horizons.

Part I: Granular flow

Besides finding granular-flow problems fascinating, I am interested in them because of their practical importance. Manufacturing industries must handle heaps (literally) of raw materials in granular form, and difficulties in handling them are very expensive. There are also important geophysical applications involving granular flow: e.g., avalanches, earthquakes, beach erosion, etc. (Incidentally, [B97] and [JNB96] in the references contain surveys of experiments in granular flow.)

Ill-posed partial differential equations are one of the mathematical challenges which arise in granular flow. (Although it is more than a decade old, I still have occasion to refer to my original demonstration of this behavior, item [S87] in the references.) Physically, ill-posedness appears at the point when sheared material cracks and then breaks into separate pieces which move independently: in technical terms, when a shear band forms. A model problem exhibiting this behavior was developed in [S92] in the references.

Currently I am struggling to understand what mathematical sense can be made of these ill-posed problems. The equations are ill-posed in a manner analogous to

utt = uxx - uyy ,

the wave equation with a space-like direction appearing as the time axis. Specifically, there is a wedge of directions with the property that any plane wave exp{i(k1x+k2y)} whose wave vector k belongs to this wedge suffers catastrophic amplification. (As with this example, the steady-state equations of granular flow are hyperbolic.) With a linear ill-posed problem such as above, no solution is possible unless the initial data are unrealistically smooth (analytic, to be precise). However, in the nonlinear equations describing granular flow, the wedge of ill-posed directions rotates as the solution varies, and this may control growth. Working with the model problem of [S92] in the references, we have found two tentative conclusions:
(i) Although transients are unpredictable, nevertheless at large times the solution tends to a well defined steady-state. (This is an appealing result for industry since they study only steady-state equations.)
(ii) Under some circumstances the long-term, steady-state solution appears to "ignore" some of the boundary conditions. So far we have been unable to predict this behavior reliably. In applications, it is vital to understand such issues in order to know what boundary conditions should be imposed for the (hyperbolic) steady-state equations of granular flow. I hope that the model problem will shed light on this question.

Below are annotated references to work in various other directions in granular flow. The first five items, like the work described above, relate to a continuum description of granular material; the latter two relate to discrete models for granular materials. Industry uses continuum models because they are simpler. However, a granular medium consists of many individual particles, and the separation between microscales and macroscales is incomparably smaller than in a fluid. Thus, a fundamental investigation of granular flow must include its discrete nature.

(a) In [GM99a,b] of the references, an initial-value problem is solved for the (hyperbolic) steady-state equations for granular flow in a conical hopper, with Cauchy data specified at the top of the hopper. (As hinted above, it is not known whether this is the most physical boundary problem.)

(b) In [GSS99] of the references we developed an approximation for the analogue of Jenike's radial solution in a conical hopper with an inverted conical insert. In the near future we plan to study numerically solutions of the steady-state equations in this geometry.

(c) Describing the unloading that occurs near a shear band leads to an interesting free-boundary problem for the wave equation. This problem is analyzed in [SS93,94] in the references.

(d) The dramatic role of imperfections near the onset of shear banding is demostrated in [SS97] in the references.

(e) Models for liquefication of soils are analyzed in [HS98,99] in the references. In one of these problems the solution of an initial/boundary value problem depends on the boundary data in a discontinuous, fractal, manner.

(f) Although continuum models for granular materials deal exclusively with the average behavior of materials, recent experiments by Behringer and collaborators have emphasized that fluctuations from the average may be substantial. Two simple probabilistic models are studied in [SS98,SSS99] in the references; these are preliminary attempts to understand fluctuations theoretically.

(g) Several people (e.g., [A99] in the references) in the group at Duke are working on molecular-dynamics (MD) simulations of granular flow. Although MD calculations provide complete information about the flow, the calculations are too lengthy for practical situations. Ultimately we hope to merge continuum and MD methods, using MD at the finest level of an adaptive mesh refinement code and continuum equations at all higher levels; but this remains an elusive goal.

My collaborators in studying granular flow include Tom Witelski, Bill Allard, and Bob Behringer at Duke; Michael Shearer, and Pierre Gremaud at NC State; Brian Hayes at Stevens; and Tony Royal at Jenike & Johanson, Inc

Part II: Research growing out of teaching

Every spring semester since 1996 I have taught a course that has led to an expansion of my research. The process starts by my sending a memo to the science and engineering faculty at Duke, asking if they would like the assistance of a group of math graduate students working on mathematical problems arising in their (the faculty member's) research. I choose one area from the responses, and I teach a case-study course for math grad students focused on problems in that area. In broad terms, during the first half of the course I lecture on scientific and mathematical background for the area; and during the second half student teams do independent research, with my collaboration, on the problems isolated earlier in the semester. I also give supplementary lectures during the second half, and at the end of the semester each team lectures to the rest of the class on what it has discovered.

The topics and their proposers for each year have been:
1996 Lithotripsy L. Howle, P. Zhong (ME)
1997 Population models in ecology W. Wilson (Zoology)
1998 Electrophysiology of the heart I C. Henriquez (BME)
1999 Electrophysiology of the heart II D. Gauthier (Physics).

Lithotripsy is an alternative to surgery for treating kidney stones--focused ultrasound pulses are used to break the stones into smaller pieces that can be passed naturally. This course was written up in [HSSZ98] in the references; the research of one team was published in [MTHZ97] in the references.

Population models in ecology became the PhD-thesis topic for A. Ashih, working jointly with myself and Will Wilson. Some of his results will appear in [AW99] in the references, and two more publications are in preparation.

The 1999 heart course focused on the experiments of [HBG99] in the references, in which a small piece of cardiac tissue was subjected to periodic stimuli at various frequencies much above the normal pacing rate. Martin Hall included some of the ideas developed in the course in his Physics PhD thesis, [H99] in the references. Based on what I learned teaching this course, I have undertaken a joint research project with Dan Gauthier (Physics) and Wanda Krassowska (BME) studying mathematical models for the electrical response of the heart, in particular comparing their predictions with experiment.

Although there are many models for the electrical response of the heart, the simplest model involves a system of only two ODE, analogous to the ODE resulting from suppression of spatial extent in the FitzHugh-Nagumo equations. Such a model is well within the reach of undergraduates, and this fall I taught an undergraduate seminar on it. The course is culminating in research projects by three separate student teams. The problems are elementary by usual research standards, but their successful completion will further the research begun in [HBG99].

Curriculum Vitae
Current Ph.D. Students   (Former Students)

  • John W. Cain  
  • Aaron Ashish  
  • Shu Dai  
  • Kevin E. Gonzales  
  • Matthew M Bowen  
  • Michael Gordon  
  • Lianjun An  
  • Feng Wang  
  • Risto Lehtinen  
  • Maija Kuusela  
  • Joseph Fehribach  
  • E. Bruce Pitman  
  • John Goodrich  
Postdocs Mentored

  • Anne Catlla (2006 - 2008)  
  • Xiaopeng Zhao (2005 - 2007)  
  • Wenjun Ying (2005 - 2008)  
  • Elena Tolkacheva (2004 - 2006)  
  • J. Matthews (2000/09-2003/06)  
Recent Publications   (More Publications)

  1. K. Gonzales, Omur Kayikci, D.G. Schaeffer, and P. Magwene, Modeling mutant phenotypes and oscillatory dynamics in the \emph{Saccharomyces cerevisiae} cAMP-PKA pathway, PLoS Computational Biology (Submitted, Winter, 2010)
  2. S. Payne, B. Li, H. Song, D.G. Schaeffer, and L. You, Self-organized pattern formation by a pseudo-Turing mechanism (Submitted, Winter, 2010)
  3. S. Dai and D.G. Schaeffer, Bifurcation in a modulation equation for alternans in a cardiac fiber, ESAIM Mathematical modelling and numerical analysis, vol. 44 no. 6 (Winter, 2010)
  4. Y. Farjoun, D.G. Schaeffer, The hanging thin rod: a singularly perturbed eigenvalue problem, SIAM Sppl. Math. (Submitted, July, 2010)
  5. S. Dai and D.G. Schaeffer, Chaos in a one-dimensional model for cardiac dynamics, Chaos, vol. 20 no. 2 (June, 2010)