Compounds which exhibit liquid crystal phases have been widely used in display technology. The majority of display applications utilize the nematic liquid crystal phase, which is a liquid-like phase which has partial orientational order at the molecular level. The nematic phase exhibits birifringence which can be manipulated through the application of an external field. Subsequently, all liquid crystal-based display technology utilizes the application of an external field to ???switch??? or tune the optical properties of a nematic domain into a desired optical state. In addition to an external field, the geometry and surface interactions of the liquid crystal domain must be precisely controlled in order for the display to operate properly. Liquid crystal displays (LCDs) utilize a rectangular domain, or pixel, within which the nematic domain is exposed to surface anchoring conditions that result in a twist of the nematic alignment through the thickness of the domain. In this work, a different type of liquid crystal domain that is elliptic is studied which is formed through ???bottom-up??? techniques, such as phase separation of a liquid crystal/polymer mixture to form a polymer-dispersed liquid crystal (PDLC) composite. Nematic domains within PDLCs are spheroidal, as opposed to rectangular for a pixel, and thus exhibit substantially different behaviour in the presence of an external field. The fundamental difference between spheroidal and rectangular nematic domains is that the former requires the presence of defects in nematic order while the latter does not. The overall objective of this work is to study, for a simplified elliptic cylinder domain, the formation of the nematic domain, the resulting domain texture in the presence of an external field, and the domain texture following release of the external field. These three states are directly related to applications of PDLC films as optical functional materials, where an external (electric) field is used to manipulate the optical properties of the film. The effects of geometry (aspect ratio), surface anchoring, and external field strength are studied through a simulation-based approach using the Landau-de Gennes theory of the nematic phase.