David Stewart, Professor
Department of Mathematics
University of Iowa


Picture of David Stewart
Professor David Stewart
Office: McLean Hall 325B
Email: dstewart (at) math.uiowa.edu
Paper Mail
Department of Mathematics
Room 14, McLean Hall
The University of Iowa
Iowa City, IA 52242-1419 USA
Phone:  voice: 319-335-3832
Fax: 319-335-0627


RESEARCH

Areas: Numerical analysis, Computational models of mechanics, friction etc, Scientific computing, Optimization & optimal control and Software Development.
Books: Meschach: Matrix Computations in C, Writing Scientific Sotware and Dynamics with Inequalities: impacts and hard constraints (forthcoming from SIAM).  Other books and Publications here.

Former PhD students

In reverse chronological order:

Summary: Differential Variational Inequalities

Differential Variational Inequalities (DVIs) are a means of modeling dynamical systems which have hard constraints or limits.  These extend the idea of differential equation, are closely related to differential inclusions, and are useful for modeling a wide variety of systems arising in mechanics, biology, economics, and engineering.

Summary: Rigid body dynamics & measure differential inclusions

I have done work on mathematical and computational models of rigid body mechanics (including friction and collisions). This involves pure mathematics (measure differential inclusions, which were invented bu J.J. Moreau in the 1980's) as well as numerical analysis and more classical applied mathematics. This relates to previous work on discontinuous ODE's and differential inclusions.

One of the major outstanding issues in the area is the resolution of Painlevé's paradoxes. I have recently been able to prove rigorously in terms of measure differential inclusions & equations, that these problems do indeed have solutions, provided the maximal dissipation form of Coulomb's law is used with the post impact velocity - at least for 1-dimensional frictional forces and one contact. This includes Painlevé's examples, and is the first general result on rigid body dynamics to do so.
This work has led to some new investigations and results about measure differential inclusions and equations.

More about my research below

My Erdös number is less or equal to 3.  Here is the proof.

Schedule for Spring 2011.

Courses. 

Places I have worked

1998-present      Mathematics Dept., University of Iowa
1996-1997          Mathematics Dept., Virginia Polytechnic Institute and State University
1995-1996          Mathematics Dept., Texas A&M University 

1991-1994          Australian National University, School of Mathematical Sciences and  advanced computation group
1990-1991          Mathematics Department, University of Queensland,Australia
More about Previous Research

I was part of the DaVinci (Differential Algebraic and Variational Inequalities in Control and sImulation) project.  It is about how to simulate and control non-smooth dynamical systems systems of different kinds.  These arise in the context of rigid-body dynamics (see below), electrical circuits with switching elements such as diodes and transistors, and hybrid control systems, for example.  Many of these systems can be modeled using complementarity theory ; complementarity conditions have the form

f(x,y) = 0,
0 <= x  orthogonal to  y >= 0

where x and y can be vectors (in which case "x >= 0" means "xi >= 0 for all i").  If f represents something like a differential equation, then this is a Dynamic Complementarity Problem (DCP).  A special case is the class of Linear Complementarity Systems (LCS), which have the form

dx/dt = Ax + Bu,
y = Cx + Du,
0 <= u(t)  orthogonal to  y(t) >= 0.

Recently, I have been working on convolution complementarity problems which have the form: Find u(t) satisfying

0 <= u(t)  orthogonal to  (k*u)(t) + q(t) >= 0

for all t >= 0 given the functions k(t) and q(t).  This has applications in elastic body impact problems.

Linear (and Nonlinear) Complementarity Problems

This work also relates to Linear Complementarity Problems (LCP's). LCP's are problems where given a square matrix M and a compatible matrix q, the task is to find vectors z and w such that

Mz+q = w >= 0, z >= 0, z Tw = 0

where the inequalities are understood componentwise. LCP's (in spite of their name) are truly nonlinear, and tools of nonlinear analysis such as degree and index theory can be successfully applied to LCP's (and NCP's). They can be represented in various ways in terms of nonlinear (but nonsmooth!) systems of equations.

Some recent work (with Jong-Shi Pang) has been on developing a unified complementarity formulation of contact problems with friction. (See the previous paragraph on rigid body dynamics as well.)

Optimization and Optimal Control

Other interests include optimization and optimal control. This includes some work on solving optimal control problems with discrete control values (e.g., {on, off}, or {-1, 0, +1}) with switching costs. (Without switching costs, the optimal solutions typically ``chatter'' rapidly between the allowed control values, which ``convexifies'' the problems and makes into standard optimal control problems.) These problems can be NP-hard, but there are ways of developing good, efficient, suboptimal algorithms.  Recently I have worked on optimal control problems where the dynamics are discontinuous.  In these systems the adjoint variables (essentially Lagrange multipliers) satisfy the usual differential equations, except that at times they have jumps.  

Dynamical Systems

I am interested in good computational/numerical methods for dealing with dynamical systems, fractals and related objects. One area of interest is the (numerically stable) calculation of Lyapunov exponents, which has led me to investigate singular value decompositions of products of matrices and the notion of stable products: small perturbations to the factor matrices should not lead to large relative changes in the singular values.

I have also worked on algorithms that can distinguish between the fractal and Hausdorff-Besicovitch dimensions.

Computational geometry

I have also done some work on using quadtrees and octrees to improve the asymptotic behavior of some algorithms for meshfree methods.  Meshfree methods are Galerkin methods for solving PDE's which don't rely on a mesh like standard Finite Element Methods do.  (This is particularly crucial since current meshing software has a great deal of difficulty avoiding triangles with small angles and similar pathologies with other elements in two and three dimensions.)  However, this means that there are more geometric tasks that have to be performed in the basic meshfree methods.
 

A long, long time ago, I can still remember...

Here is a picture from the January 1993 SCADE meeting in Auckland, New Zealand.  If you can identify anybody not already identified, let me know...  Or better yet, if you know xfig, then update the xfig file in the zip here: auckland93.zip.  Here is the list of the attendees.

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David Stewart

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