  We develop efficient and accurate numerical methods to solve a class of
shallow shell problems of the von Karman type. The governing equations form a
fourth-order coupled system of nonlinear biharnomic equations for the
transverse deflection and Airy's stress function. A second-order finite
difference discretization with three iterative methods (Picard, Newton and
Trust-Region Dogleg) are proposed for the numerical solution of the nonlinear
PDE system. Three simple boundary conditions and two application-motivated
mixed boundary conditions are considered. Along with the nonlinearity of the
system, boundary singularities that appear when mixed boundary conditions are
specified are the main numerical challenges. Two approaches that use either a
transition function or local corrections are developed to deal with these
boundary singularities. All the proposed numerical methods are validated using
carefully designed numerical tests, where expected orders of accuracy and rates
of convergence are observed. A rough run-time performance comparison is also
conducted to illustrate the efficiency of our methods. As an application of the
methods, a snap-through thermal buckling problem is considered. The critical
thermal loads of shell buckling with various boundary conditions are
numerically calculated, and snap-through bifurcation curves are also obtained
using our numerical methods together with a pseudo-arclength continuation
method. Our results are consistent with previous studies.

Numerical Analysis