On optimal and near-optimal shapes of external shading of windows in apartment buildings

We studied previously optimal shape of external shading of windows in a cellular office with an outer edge modeled by a non-uniform rational basis spline (NURBS) curve whose control points were placed uniformly around western fin, overhang and eastern fin of the window, and whose depths were allowed to vary independently. We observed there that for each climate considered in the study there exists a shading shape close to the optimal one, but with a substantially simpler structure of control points for the NURBS curve. This simpler structure was reflected in partitioning control points into six groups such that all control points in the same group have equal depths, with groups corresponding to lower part of the western fin, upper part of the western fin, joint of the western fin and the overhang, internal part of the overhang, joint of the overhang and the eastern fin and the remaining part of the eastern fin. Here we confirm that shadings with control point structure restricted in such way can perform as well as shadings with unrestricted control points by optimising shape of external shading of windows in an apartment room for both restricted and unrestricted control point structure for the same range of climates, and showing that differences in heating and cooling demands between Pareto optimal shadings in both cases are negligibly small. This grouping of control points thus gives a simple and natural division of shading into a small number of basic constituents that have most impact on its heating and cooling demands. We further consider the convex hull of the Pareto front for shadings with restricted control points, as it contains shadings that minimise equivalent source energy in terms of the ratio of efficiencies and source energy conversion factors for district heating and cooling. We show that, in cases when depths of control point groups in convex hull shadings do not experience sudden changes between their extremal values, these depths can be fitted reasonably well by a sigmoid function that results in functional shadings that satisfactorily approximate heating and cooling demands of shadings in the Pareto front.