Some New Technical Reports & Papers
I will put stuff here as it is made ready.
Already published and accepted items can be found in the CV.
Roots of measures and impulsive control
Abstract. The question of what a ``square root of a measure'' is
investigated here by means of measure differential inclusions. This can
be extended to general ``nth roots of measures''.
University
of Iowa AMCS Technical report #105,
1998.
Constrained optimization and Morse theory
Abstract. In classical Morse theory the number and type (index)
of critical points of a smooth function on a manifold are related to topological
invariant of that manifold through the Morse inequalities. There the index
of a critical point is the number of negative eigenvalues of the Hessian
matrix.
Here definitions of ``critical point'' and ``index'' are given that
are suitable for functions on x in M where gi(x)
>= 0 for i = 1, ..., m. The new definitions are based on
Kuhn-Tucker theory of constrained optimization. The Morse inequalities
are proven for this new situation. These results are extended to smooth
manifolds with corners.
University
of Iowa AMCS Technical report #107,
1998.
Time-stepping for three-dimensional rigid body dynamics
With M. Anitescu (Argonne National Labs) and F.A. Potra (University of
Maryland, Baltimore County).
Abstract. Traditional methods for simulating rigid body dynamics
involves determining the current contact arrangement (e.g., each contact
is either a ``rolling'' or ``sliding'' contact). The development of this
approach is most clearly see in the work of Haug, Wu and Yang and Pfeiffer
and Glocker. However, there has been a controversy about the status of
rigid body dynamics as a theory, due to simple problems in the area which
do not appear to have solutions; the most famous, if not the earliest is
due to Paul Painlevé (1895). Recently, a number of time-stepping
methods have been developed to overcome these difficulties. These time-stepping
methods use integrals of the forces over time-steps, rather the
actual forces. This allows impulsive forces without the need for a separate
formulation, or special procedures, to cover this case. The newest of these
methods are developed in terms of complementarity problems. The
complementarity problems that define the time-stepping procedure are solvable
unlike previous methods for simulating rigid body dynamics with friction.
Proof of the existence of solutions to the continuous problem can be shown
in the sense of measure differential inclusions in terms of these
methods. In this paper, a number of these variants will be discussed, and
their essential properties proven.
Accepted for publication in Computer Methods in Applied Mechanics
and Engineering, July, 1998.
Convergence of an implicit time-stepping scheme for rigid body dynamics
Abstract. This paper gives convergence theory for a new implicit
time-stepping scheme for general rigid body dynamics with Coulomb friction
and purely inelastic collisions and shocks. An important consequence of
this work is the proof of existence of solutions of rigid body problems
which include the famous counterexamples of Painlevé. The mathematical
basis for this work is the formulation of the rigid body problem in terms
of measure differential inclusions of J.J.Moreau and Monteiro-Marques.
The implicit time stepping method is based on complementarity problems,
and is essentially a particular case of the algorithm described in Anitescu
and Potra, which in turn is based on the formulation in Stewart and Trinkle.
Accepted for publication in Archive for Rational Mechanics and Analysis,
November, 1997.
A Unified Approach to Frictional Contact Problems
With Jong-Shi Pang (Johns Hopkins University).
Abstract. We present a unified treatment of discrete or discretized
contact problems with Coulomb friction that include quasistatic and dynamic
problems involving rigid or elastic bodies undergoing small or large displacements.
A general existence theorem is established under broad assumptions that
are easily satisfied by many special models. The proof is based on a homotopy
argument. This result extends many existence results known to date for
discrete contact problems.
Submitted March, 1997.
A new algorithm for the SVD of a long product of matrices and the stability
of products
Abstract. Lyapunov exponents can be estimated by accurately computing
the singular values of long products of matrices, with perhaps 1000 or
more factor matrices. These products have extremely large ratios between
the largest and smallest eigenvalues. A variant of Rutishauser's Cholesky
LR algorithm for computing eigenvalues of symmetric matrices is used to
obtain a new algorithm for computing the singular values and vectors of
long products of matrices with small backward error in the factor matrices.
The basic product SVD algorithm can also be accelerated using hyperbolic
Givens' rotations. The method is competitive with Jacobi-based methods
for certain problems as numerical results indicate. Some properties of
the product SVD factorization are also discussed, including uniqueness
and stability. The concept of a stable product is introduced; for
such products, all singular values can be computed to high relative accuracy.
Published in Electronic Transactions
in Numerical Analysis, vol. 5, pp. 29-47, 1997.
Impulsive controls via measure differential inclusions
Abstract. Certain control problems which are most simply formulated
with control functions lying in L1(0,T) should be relaxed
to allow control functions to belong to the space of Borel vector measures
supported on [0,T]. The dynamics of suitable problems can then be
interpreted in a measure differential inclusion (MDI) sense, and combined
with a weak* closure property for solutions of MDI's, natural existence
results for the relaxed problem follow. Examples are given showing cases
both of solutions belong to L1, and impulsive solutions.
Submitted March, 1997.
Convergence of a time-stepping scheme for rigid body dynamics and resolution
of Painlevé's paradoxes
Abstract. This paper gives convergence theory for a new implicit
time-stepping scheme for general rigid body dynamics with Coulomb friction
and purely inelastic collisions and shocks. An important consequence of
this work is the proof of existence of solutions of rigid body problems
which include the famous counterexamples of Painlevé. The mathematical
basis for this work is the formulation of the rigid body problem in terms
of measure differential inclusions of J.-J. Moreau and Monteiro-Marques.
The implicit time stepping method is based on complementarity problems,
and is essentially a particular case of the algorithm described in Anitescu
and Potra, which in turn is based on the formulation in Stewart and Trinkle.
Now published in Comptes Rendus Acad. Sci. Paris, Ser. I, vol.
325, pp. 689-693, 1997.
Asymptotic cones and measure differential inclusions
Abstract. Asymptotic cones are used to give a new solution concept
for measure differential inclusions, ``dx/dt in K(t)
subset Rn'' where x(.) is a function of
bounded variation and K is a set-valued map with closed convex values
and has closed graph. Measure differential inclusions were first defined
by J.-J. Moreau, for studying rigid body with impacts, shocks and Coulomb
friction and assumed that K(t) is always a cone. The definition
given here extends that of Moreau in allowing for arbitrary closed convex
K(t),
and so also incorporates ordinary differential inclusions, provided the
asymptotic cone K(t)inf is always pointed.
Dynamics, Friction and Complementarity Problems
With J.C. Trinkle (Computer Science, Texas A&M University).
Abstract. An overview is presented here of recent work and approaches
to solving dynamic problems in rigid body mechanics with friction. It begins
with the differential inclusion approach to Coulomb friction where
the normal contact force is known. Numerical methods for these differential
inclusions commonly lead to LCP's or NCP's. Then the measure differential
inclusion formulation of rigid body dynamics due to Moreau and others
is presented. A novel algorithm for time-stepping for rigid body dynamics
is given which avoids many of the ``non-existence'' issues that arise in
many other formulations. The time-stepping algorithm requires the solution
of an NCP at each step. Results on the convergence of the numerical solutions
to solutions in the sense of measure differential inclusions are given,
and numerical results for an implementation of the method are shown.
In Complementarity and Variational Problems: State of the Art,
the proceedings of the International Conference on Complementarity Problems,
Baltimore, MD, November 1-4, 1995. Publ. SIAM, 1997.
Stick slip friction
Slip stick friction is a form of friction where the "static" and "dynamic"
coefficients of friction are different (with the static coefficient greater
than the dynamic coefficient). I hope to post a paper that I have recently
written with some ideas I have about the mathematical modelling of such
systems. The point of it is that really these models should be limits of
plain smooth ordinary differential equation models, but the right hand
side (i.e. the forces) approach a discontinuous function. If this is interpreted
as a differential inclusion, the trajectories don't "stick". More heuristic
models don't have the right sort of topological structure which ODE's (and
differential inclusions) have for solving two-point boundary value problems.
(This could also be crucial for optimal control theory.) Instead a "hybrid"
differential inclusion/heuristic method that is numerically implementable
but "sticks" properly and has the right topological properties is developed.
A talk on this material has been given at the WE Hereaus Conference on
"Analysis of Non-Smooth Dynamical Systems" at Bad Honnef, Germany, March
13th-17th, 1995.
An implicit time-stepping scheme for rigid body dynamics with inelastic
collisions and Coulomb friction
With J.C. Trinkle (Computer Science, Texas A&M University).
Abstract. In this paper a new time-stepping method for simulating
systems of rigid bodies is given which incorporates Coulomb friction and
inelastic impacts and shocks. Unlike other methods which take an instantaneous
point of view, this method does not need to explicitly identify impulsive
forces. Instead the treatment is similar to that of J.J.~Moreau and Monteiro-Marques,
except that the numerical formulation used here ensures that there is no
inter-penetration of rigid bodes, unlike their velocity-based formulation.
Numerical results are given for the method presented here for a spinning
rod impacting a table in two dimensions, and a system of four balls colliding
on a table in a fully three-dimensional way. These numerical results also
show the practicality of the method, and its convergence of the method
as the step size becomes small.
In International J. Numer. Methods Eng., vol. 39, pp. 2673-2691,
1996.
Note on the end game in homotopy zero curve tracking
With M. Sosonkina and L.T. Watson (Mathematics, Virginia Polytechnic Institute
and State University).
Abstract. Homotopy algorithms to solve a nonlinear system of
equations $f(x)=0$ involve tracking the zero curve of a homotopy map $\rho(a,\lambda,x)$
from $\lambda =0$ until $\lambda=1$. When the algorithm nears or crosses
the hyperplane $\lambda=1$, an ``end game'' phase is begun to compute the
solution $\bar x$ satisfying $\rho(a,\lambda,\bar x) = f(\bar x)=0 $. This
note compares several end game strategies, including the one implemented
in the normal flow code FIXPNF in the homotopy software package HOMPACK.
Published in ACM Trans. Mathematical Software, vol. 22, pp. 281-287,
1996.
An iterative aggregation/disaggregation procedure for modelling the long-term
behaviour of continuous-time evanescent random processes
With M. Bebbington (Dept. Statistics, Massey University, New Zealand).
Abstract. We describe an aggregation/disaggregation method for
finding the quasi-stationary distribution and decay parameters for continuous-time
Markov chains. Finding the quasi-stationary distribution is equivalent
to calculating the eigenvector corresponding to the smallest eigenvalue
of the $q$-matrix restricted to the non-absorbing class. The method presented
here is similar to an algebraic multigrid, with restriction operators that
depend on the current approximation to the solution. The smoothers are
short Arnoldi iterations or Gauss-Seidel iterations. Numerical results
are presented for a variety of models of differing character, including
simple epidemic, bivariate SIS, predator-prey, and the cubic auto-catalator.
These indicate that the number of cycles required grows only very slowly
with the size of the problem.
Published in J. Statistical Computation and Simulation, vol.
56, pp. 77-95, 1996.
A numerical method for friction problems with multiple contacts
Abstract. Friction problems involving ``dry'' or ``static'' friction
can be difficult to solve numerically due to the existence of discontinuities
in the differential equations appearing in the right-hand side. Conventional
methods only give first order accuracy at best; some methods based on stiff
solvers can obtain high order accuracy. The previous method of the author
[Numer. Math. 58, 299--328] is extended to deal with friction
problems involving multiple contact surfaces.
Published in J. Australian Math. Soc., Series B, vol. 37, pp.
68-79, 1996.