Steel is a material commonly used in various constructions such as high-rise buildings, sport arenas, ships etc. Steel is a versatile building material due to its isotropic characteristics, e.g. both high tensile- and compressive strength. This allows steel to be formed into open section profiles which reduces material usage but simultaneously allows the tensile- and compressive stress resistance to be high in directions were loads are applied. Although steel has a high stress resistance its sensitivity to fire is larger than other building materials due to its high thermal conductivity. The strength of the material is reduced at higher temperatures and thereby makes the dimensioning of beams in fire cases vital in fire safety design of structural elements. An aspect to consider when dimensioning open section building elements in steel is the shadow effect. The shadow effect is the result of the open cross-section geometrical shape of beams and columns, e.g. H-profiles. The interior of the profile is screened from thermal radiation caused by fire which makes the characteristics of the thermal exposure different from closed cross-section profiles. A common way to estimate the temperatures of steel after a certain time of fire exposure is to use numerical calculations described in Eurocode. In these calculations the shadow effect is applied as a reduction of the total heat exchange, i.e. both convection and thermal radiation, from the fire exposure. A more realistic approach is to separate these boundary conditions and treat them as independent quantities. Wickström (2001) argues that a void is created within the flanges and that reduction factor thereby only should be applied to the radiative part of the total heat exchange, acting as a reduction of surface emissivity within the profile. This, since the convection is not affected by the shadow effect. Wickströms (2001) suggestion of application has been investigated in this thesis and has showed a better correlation than the approach suggested in Eurocode when compared to experimental tests. Shadow effects calculated on the premises of separated boundary conditions for the total heat exchange has of yet only been investigated in detail with TASEF+-simulations, but these simulations predicts steel temperatures with satisfactory results. It is possible to reproduce a similar setup in the program COMSOL Multiphysics in two-dimensional simulations, and further three-dimensional simulations. This possibility has been investigated in this thesis. COMSOL Multiphysics has proven to be an adequate tool when it comes to simulate fire exposure on slender steel beam with shadow effects considered. Both three- and two-dimensional models produced simulation results correlating well to simulations conducted in TASEF. Additionally, adequate correlations with experimental tests were obtained for COMSOL Multiphysics as well. Further work regarding fire simulations with the utilisation of COMSOL Multiphysics is thereby suggested.

The subject matter of this dissertation is the design, analysis and practical implementation of a new numerical method to approximate the eigenvalues and eigenfunctions of regular Sturm–Liouville problems, given in Liouville’s normal form, defined on compact intervals, with self-adjoint separated boundary conditions. These are classical problems in computational mathematics which lie on the interface between numerical analysis and spectral theory, with important applications in physics and chemistry, not least in the approximation of energy levels and wave functions of quantum systems. Because of their great importance, many numerical algorithms have been proposed over the years which span a vast and diverse repertoire of techniques. When compared with previous approaches, the principal advantage of the numerical method proposed in this dissertation is that it is accompanied by error bounds which: (i) hold uniformly over the entire eigenvalue range, and, (ii) can attain arbitrary high-order. This dissertation is composed of two parts, aggregated according to the regularity of the potential function. First, in the main part of this thesis, this work considers the truncation, discretization, practical implementation and MATLAB software, of the new approach for the classical setting with continuous and piecewise analytic potentials (Ramos and Iserles, 2015; Ramos, 2015a,b,c). Later, towards the end, this work touches upon an extension of the new ideas that enabled the truncation of the new approach, but instead for the general setting with absolutely integrable potentials (Ramos, 2014).