Author: Jungho Lee 


Title: A Hybrid Domain Decomposition Method and its Applications to Contact Problems

 Abstract: 
Our goal is to solve nonlinear contact problems. We consider bodies in contact
with each other divided into subdomains, which in turn are unions of elements.
The contact surface between the bodies is unknown a priori, and we have a
nonpen-etration condition between the bodies, which is essentially an
inequality constraint. We choose to use an active set method to solve such
problems, which has both outer iterations in which the active set is updated,
and inner iterations in which a (linear) minimization problem is solved on the
current active face. In the first part of this dissertation, we review the
basics of domain decomposition methods. In the second part, we consider how to
solve the inner minimization problems. Using an approach based purely on FETI
algorithms with only Lagrange multipliers as unknowns, as has been developed by
the engineering community, does not lead to a scalable algorithm with respect
to the number of subdomains in each body. We prove that such an algorithm has 
a condition number estimate which depends linearly on the number of subdomains 
across a body; numerical experiments suggest that this is the best possible
bound. We also consider a new method based on the saddle point formulation of
the FETI methods with both displacement vectors and Lagrange multipliers as 
unknowns. The resulting system is solved with a block-diagonal preconditioner 
which combines the one-level FETIand the BDDC methods. This approach allows 
the use of inexact solvers. We show that this new method is scalable with 
respect to the number of subdomains, and that its convergence rate depends 
only logarithmically on the number of degrees of freedom of the subdomains 
and bodies. In the last part of this dissertation, a model contact problem 
is solved by two approaches. The first one is a nonlinear algorithm which 
combines an active set method and the new method of Chapter 4. We also present
a novel way of finding an initial active set. The second one uses the SMALBE
algorithm, developed by Dostal et al. We show that the former approach has 
advantages over the latter.