Numerical methods for thermally stressed shallow shell equations

TL;DR

This paper introduces efficient finite difference numerical methods for solving nonlinear shallow shell equations of von Karman type, addressing boundary singularities and applying them to thermal buckling problems with validated accuracy and performance.

Contribution

The paper develops novel numerical techniques, including iterative solvers and boundary singularity treatments, for nonlinear shallow shell equations with applications to thermal buckling analysis.

Findings

Validated numerical methods with expected accuracy and convergence.

Efficient performance demonstrated through run-time comparisons.

Accurate computation of critical thermal loads and bifurcation curves.

Abstract

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.