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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 non-convex 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 non-convex variational problems related to shape memory alloys (i.e., convergence rate of the mixture of the deformation gradients) [5]; before this work, only one-dimensional results were available;
  • use of a simulated annealing method to test, in the two dimensional case, the famous Morrey conjecture about the non-equivalence of quasi-convexity and rank-1 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 self-similar 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 Hamilton-Jacobi problems). Many applications call, however, for the use of vectorial equations. We use systems of Hamilton-Jacobi 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 Hamilton-Jacobi 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 one-dimensional 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 semi-implicit 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 Navier-Stokes equations in time dependent domains (liquide zone) [3].