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My research interests have been concentrated on three subjects:
The main results obtained in each of the above areas are discussed below.
Some of the problems under study belong to more than one of the above domains.
For instance, the study of granular flows has led us to problems belonging to
all three domains.
Materials science and granular flows
In collaboration with
D. Schaeffer (Duke University) and
M. Shearer (NCSU), we work on the study of flow of granular material
in silos. Using a combination of analytical and numerical methods, the structure
of the flow, shock and rarefaction wave solutions, will be investigated. The
project is partially supported by ARO (P.A. Gremaud, PI), NSF (P.A.
Gremaud and M. Shearer, PIs). This research is, and will be, pursued in consultation
with engineers at Jenike and Johanson, Inc.
My former student, J.V.
Matthews, (Duke) has made fundamental contributions to the project.
In a joint project with K. Coffey, Ph.D. Student, NCSU (P.A. Gremaud adviser),
C.T. Kelley (NCSU) and A.T. Royal, Jenike
and Johanson, Inc. , we also study problems of powder consolidation
(settlement). Those give rise to nonlocal free boundary problems corresponding
to involved integrodifferential equations.
Some other and earlier projects in the broad area of materials science
have also been considered. In collaboration with
M. Luskin and his
coworkers (University of Minnesota), I got interested in the approximation
of equilibrium states of new materials such as shape memory alloys. After
modeling, the resulting mathematical problems are of a nonconvex variational
type.
Additional material
Click on thumbnails. For some reason (
bug), Netscape N, where N < 4.75, seems to mess up the following PNG
files. Alternative links (to much larger pdf or ps files) are provided if
needed.
granular flow poster p.1 (
pdf)

granular flow poster p.2 (
pdf)

powder consolidation poster p.1 (
ps)

powder consolidation poster p.2 (
ps)

detailed flow calculations


Achievements:
 use of a nonconforming finite element, and derivation of
the first error estimate in a multidimensional case for
nonconvex variational problems related to shape memory alloys
(i.e., convergence rate of the mixture of the deformation gradients)
[5]; before this work, only
onedimensional results were available;
 use of a simulated annealing method to test, in the two
dimensional case, the famous Morrey conjecture about the
nonequivalence of quasiconvexity and rank1 convexity (an
example has recently be given for the three dimensional case)
[6]; this work is one of the rare
applications of simulated annealing like methods to high dimensional
problems;
 construction and implementation of simple cellular automata
models for granular flows [10];
this project was initiated in consultation with R. Behringer
(Dept.of Physics, Duke University) and M. Shearer (NCSU);
 numerical study of simple asymptotic equations related to
various models of granular flow in a silo around an inverted
conical insert [15]; this had
led to a better understanding of the relationship between, and
limitations of, models used in the engineering literature;
 analytical and numerical study of the stability of similarity
solutions (Jenike's radial solutions) for hopper flows
[16];
 implementation of the first high accuracy method for the
resolution of the stress and velocity equations for granular flows
in hoppers [17],
[18]; the equations correspond to
systems of nonlinear hyperbolic conservation laws with several
nonstandard features such as algebraic constraints;
 analysis and numerical analysis of powder settlement problems,
using respectively Integral equation theory and numerical
methods for DAEs [19];
 study of alternate formulations of plasticity models for
granular flows (PDAEs);
 first calculations and analysis of selfsimilar flows in "general"
hoppers [21]. Before this
work, existing were essentially restricted to axisymmetric flows.
Hyperbolic equations and related problems
In collaboration with
B. Cockburn (University of Minnesota), we have undertaken a general study
of numerical methods for nonlinear conservation laws. Our original approach
has already led to several significant results. This project is supported
by an ARO research grant (P.A. Gremaud, PI), as well as an ARO Augmentation
Awards for Science and Engineering Research Training grant (AASERT, P.A. Gremaud,
PI).
A different, but connected, project consists
of numerical studies related to the concept of viscosity solutions for general
first order nonlinear PDEs. For such equations, the notion of solution is
often problematic in the vectorial case (non existence of entropy functions
for systems of conservation laws, collapse of the notion of viscosity solution
for HamiltonJacobi problems). Many applications call, however, for the use
of vectorial equations. We use systems of HamiltonJacobi equations
as a way to formulate various problems from Solid Mechanics, a fundamentally
new approach. Our numerical investigations take place in this context.
Achievements:
 derivation of the first and only error
estimate for finite element approximations of nonlinear multidimensional
conservation laws [7],
[8]; in particular, this has led to fully justified error estimators,
a unique feature in this field;
 the approach showed that, contrary to
widespread beliefs, a priori estimates,
can be obtained for nonlinear conservation
laws [11], [12]
, [13];
 derivation of the first and only optimal
a priori estimates for conservation laws in the truly multidimensional
case;
 first explanation of supraconvergence
for problems with low regularity; clarification of the proper role played
by the consistency of the methods;
 precise understanding of the effects
of the size, the (un)uniformity and the ``(non)Cartesianity'' of the grids
in the approximation process, for conservation laws;
 accurate calculations of viscoelasticity/capillarity
solutions to scalar and vectorial HamiltonJacobi equations in the multidimensional
case [14]; those calculations are delicate since
consistency with the selection criterion demands the use of high order methods
(here, third order); unlike ours, most of the published work is restricted
to onedimensional infinite domain and/or scalar problems.
Free boundary problems
Some older results, mostly about the Stefan
problem and related questions.
Achievements:
 construction and study of a complete
model of alloy solidification processes, including convective effects in
the liquid zone;
 development, study and implementation
of a new numerical scheme for the thermal problem based on the use of finite
elements in space and semiimplicit finite differences in time
[1];
 derivation of the first error estimate
for an algorithm based on explicit finite differences in time and finite
elements in space in the framework of the Stefan problem
[2];
 study and implementation of accelerated
explicit methods [9];
 study of methods for the numerical resolution
of the NavierStokes equations in time dependent domains (liquide zone)
[3].
