Thin shell structures are widely constructed and manufactured because a well-designed shell can sustain a large load with remarkably little material. While elastic shells can exhibit great strength, their behaviors can also be very difficult to predict and they can fail in a catastrophic fashion. Despite many experiences accumulated in the engineering literature, the mathematical theory of shells is still being developed, and there is not a single numerical method that is provably reliable.
The objective of this research is numerical methods for a variety of parameter-dependent models including thickness-dependent linear elastic shell equations. The resulting methods will yield solutions that are uniformly accurate with respect to the parameter. A shell structure could be bending-dominated, membrane-dominated, or intermediate, reflecting its load-bearing ability. The main efforts of this project will be devoted to the intermediate shell problems whose solution exhibits the most elusive behavior and ever-increasing singularity as the shell thickness becomes small. Moreover, most realistic shell structures are of this kind. The resulting methods will be directly applicable to membrane-dominated shells, and will be helpful for satisfactorily solving bending-dominated shells to which most of the numerical analyses have been devoted. Aside from shell problems, we also study anisotropic heat conduction, beam-string deformation, and the Reissner-Mindlin plate bending model with a big twisting moment loaded on the plate boundary. Under suitable conditions, these problems have features resembling those of intermediate shells, which are often left open in the literature.
Shell problems are practically important, mathematically interesting, and computationally challenging. The project will result in numerical methods that are valuable for practitioners. There are interdisciplinary connections with engineering and other applied mathematics and scientific computing disciplines, and with industry.
