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This thesis is intended to fulfill the requirements of the Math and Physics departments at Harvey Mudd College. We begin with a brief introduction to the study of surfactant dynamics followed by some background on the experimental framework our work is related to. We then go through a derivation of the model we use, and explore in depth the nature of the Equation of State (EoS), the relationship between the surface tension on a fluid and the surfactant concentration. We consider the effect of using an empirical equation of state on the results of the simulations and compare the new results against the results produced using a multilayer (EoS) as well as experimental observations. We find that the empirical EoS leads to two new behaviors - preserving of large gradients of surfactant concentration and the occurrence of dynamics in distinct regimes. These behaviors suggest that the empirical EoS improves the agreement of the model’s prediction with experiment.

A finite element code has been developed at the University of Sheffield to simulate the structural response of steel and composite framed buildings subjected to fire. The steel skeleton is represented using two-noded line elements, the steel-to-steel connections using spring elements and the flooring system by isotropic flat shell elements. Structures are therefore considered as a complete entity, allowing a more realistic prediction of structural behaviour at elevated temperature. A series of numerical simulations of fire tests carried out on the full-scale, eight-storey composite frame at the BRE laboratory at Cardington in 1995 and 1996 have been conducted. These tests have been subject to a number of significant parametric studies including slab thickness and secondary beam connection strength and stiffness. The concrete floor slab element has also been extended to a layered flat shell element allowing the inclusion of material non-linearities, thermal bowing, thermal degradation, anisotropic properties and a more advanced cracking model. Using the new concrete floor slab element the Cardington fire tests have been simulated in detail, to further understanding of the structural reaction in fire. Another series of parametric studies have been conducted considering again the thickness of the floor slab, the effect of the slab temperature gradient, the compressive strength, tensile strength and load ratios. These have all been compared to results from the Cardington fire tests. Current design methods based on isolated element design are considered by comparing the results of analyses in which the concrete floor is either included as a continuous slab in an extensive subframe, or is treated simply as forming the flanges of composite beams in a three-dimensional skeleton. These examples show clearly the effects of membrane and bridging actions of the continuous floor slab. The implications for future design developments are discussed with particular reference to the parametric studies conducted.

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In the study of computational turbulence, the success of Large Eddy Simulation (LES) is largely determined by the quality of the sub-filter scale (SFS) model and the properties of the filter used to introduce resolved and unresolved length scales. Explicit filters are desirable so that better control over the filter may be achieved, and filter operator errors can be then controlled to a desired order of accuracy. One large advantage to using an explicit filter is that the mathematical definition of the filter may be exploited when considering various SFS models or even different LES techniques. Approximate deconvolution is a technique used in LES, which performs an inverse filtering operation to partly restore the original unfiltered solution. The discrete explicit filtering technique will be used to perform the deconvolution, and numerical results will show how the approximate solution may be used to perform LES.

A (3+1)-evolutionary method in the framework of Regge Calculus, known as "Parallelisable Implicit Evolutionary Scheme", is analysed and revised so that it accounts for causality. Furthermore, the ambiguities associated with the notion of time in this evolutionary scheme are addressed and a solution to resolving such ambiguities is presented. The revised algorithm is then numerically tested and shown to produce the desirable results and indeed to resolve a problem previously faced upon implementing this scheme. An important issue that has been overlooked in "Parallelisable Implicit Evolutionary Scheme" was the restrictions on the choice of edge lengths used to build the space-time lattice as it evolves in time. It is essential to know what inequalities must hold between the edges of a 4-dimensional simplex, used to construct a space-time, so that the geometry inside the simplex is Minkowskian. The only known inequality on the Minkowski plane is the "Reverse Triangle Inequality" which holds between the edges of a triangle constructed only from space-like edges. However, a triangle, on the Minkowski plane, can be built from a combination of time-like, space-like or null edges. Part of this thesis is concerned with deriving a number of inequalities that must hold between the edges of mixed triangles. Finally, the Raychaudhuri equation is considered from the point of view of Regge Calculus. The Raychaudhuri equation plays an important role in many areas of relativistic Physics and Astrophysics, most importantly in the proof of singularity theorems. An analogue to the Raychaudhuri equation in the framework of Regge Calculus is derived. Both (2+1)-dimensional and (3+1)-dimensional cases are considered and analogues for average expansion and shear scalar are found.

In the study of computational turbulence, the success of Large Eddy Simulation (LES) is largely determined by the quality of the sub-filter scale (SFS) model and the properties of the filter used to introduce resolved and unresolved length scales. Explicit filters are desirable so that better control over the filter may be achieved, and filter operator errors can be then controlled to a desired order of accuracy. One large advantage to using an explicit filter is that the mathematical definition of the filter may be exploited when considering various SFS models or even different LES techniques. Approximate deconvolution is a technique used in LES, which performs an inverse filtering operation to partly restore the original unfiltered solution. The discrete explicit filtering technique will be used to perform the deconvolution, and numerical results will show how the approximate solution may be used to perform LES.