
 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 522421419 USA
 Phone: voice: 3193353832
 Fax: 3193350627

RESEARCH
Areas: Numerical analysis,
Computational models of mechanics, friction etc, Scientific
computing, Optimization & optimal control and Software
Development.
Books: Meschach:
Matrix
Computations in C (with Z. Leyk, 1992), Writing
Scientific Software (with S. Oliveira, 2006), Numerical
Solution of Ordinary Differential Equations (with K.E.
Atkinson and Weimin Han, 2009), Dynamics with Inequalities:
impacts and hard constraints (2011), and Building
Proof: a practical guide (with S. Oliveira, 2015). Other
books and Publications here.
Current PhD students
 Mario Barela, began in 2013.
 Benjamin Dill, began in 2014.
 Cole Stiegler, began in 2014.
Former PhD students
In reverse chronological order:
 Brian Gillispie;
obtained PhD in 2009, University of Iowa.
 Ted Wendt; obtained
PhD in 2008, University of Iowa; currently at Carroll College,
Montana.
 Ricardo Ortiz;
obtained PhD in 2007, University of Iowa; currently at KitWare,
Inc.
 KoungHee Leem;
coadvised with Suely Oliveira (Computer Science); obtained PhD
in 2003, University of Iowa; currently at Southern Illinois U,
Edwardsville.
 Jeongho Ahn; obtained
PhD in 2003, University of Iowa; currently at Arkansas State U.
 Christopher Cartright;
obtained PhD in 2002, University of Iowa; currently at Lawrence
Tech U.
 Teresa Leyk; advised
19921994 by myself, advised 19941997 by Steve Roberts, ANU;
obtained PhD 1997, Australian National University (ANU);
currently at Texas A&M U.
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 1dimensional 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.
Places I have worked
1998present Mathematics Dept., University of Iowa
 19961997 Mathematics Dept., Virginia Polytechnic Institute and
State University
19951996 Mathematics Dept., Texas
A&M University
19911994 Australian National University, School of Mathematical
Sciences and advanced
computation group
 19901991 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 nonsmooth
dynamical systems systems of different kinds. These arise in
the context of rigidbody 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 "x_{i} >= 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 ^{T}w = 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 JongShi 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 NPhard, 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 HausdorffBesicovitch 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.
More recently...
Here is a picture from the
DaVinci/SICONOS meeting at Grenoble in 2005 showing the DaVinci
group. And here is a
picture of some of us relaxing after the International Conference on
Complementarity Problems (ICCP) in Berlin in 2014. (Thanks to Uday
Shanbhag  not shown  for the picture.)
David
Stewart
To the Department of
Mathematics