The F-Matrix Method
In the Three-Phase Method, the luminous flux transfer between the glazing apertures to the sky is characterized by the Daylight (D) Matrix. Any element of the room geometry that is beyond the extents of, and also non-coplanar to, the glazing aperture is considered to be a part of the external environment. For example, the grates shown in the Figure 57, which serve to shade the interior of the room, will be considered as a part of the external environment. Any parametric study involving a variation in the surface-properties or geometry of the grates will require a recalculation of the D matrix. It follows that the Three-Phase Method does not have any specific provisions for parametric of shading systems that are external and non-coplanar to glazing apertures.
Figure 57. The diverging arrows in the figure above are meant to highlight the location from which the flux transfer in the exterior Daylight (D) Matrix is considered. In this case the D matrix is computed by considering the rectangular aperture of the window and the sky dome.
The F-matrix approach, which has been introduced for such cases, employs an additional matrix, namely the Façade matrix, to account for flux-transfer through the facade. In spaces like the one depicted in Figure 57, this approach is useful in iteratively studying multiple types of non-coplanar shading systems. Figure 2 and Figure 3 in Chapter 1 provide a schematic comparison between the Three-Phase Method and the F-Matrix Method. As demonstrated later in Section 7.6, after a one-time calculation of all the flux-transfer matrices, subsequent simulations for studying different non-coplanar shading systems only require the recalculation of the F-matrix.
This chapter discusses three ways in which the F-matrix method can be applied. These are shown schematically in Figure 58, Figure 59 and Figure 60. The simplest and arguably the least accurate approach, referred to as F1 from here on, is depicted in Figure 58. It involves the use of a virtual aperture with a single surface for creating the F-matrix. A single hemispherical basis, based on the direction normal of the F-aperture, is considered for creating the F-matrix. The main drawback of this approach is that the use of a single surface F aperture ignores incoming luminous flux from all the other directions. A more accurate approach shown in Figure 59 involves surrounding the façade with an F aperture containing multiple surfaces such that all directions of flux transfer from the façade to the sky
Figure 58. Setup for an F1 type F-Matrix simulation. The translucent surface in front of the façade represents the F aperture used in this simulation. As indicated by the location of the arrows, the D matrix is computed by considering the F aperture and the sky dome.
Figure 59 Setup for an FH type F-Matrix simulation. The four translucent polygons on top, right, left and front constitute the F aperture and envelope all sides of the façade. In case there is a considerable offset between the space and the ground plane, then a polygon should be provided at the bottom as well.
Figure 60 Setup for an FN type F-Matrix simulation. Like the FH approach, even in this case the F aperture polygons completely envelope the Façade. The key distinction here is that the positioning of the polygons is also based on the location of the non-coplanar shading device and that a separate sampling basis is assigned for each F aperture.
are accounted for. This approach, referred to as FH, employs a single hemispherical sampling basis. One of the shortcomings of using a single sampling basis is that only the directions compatible with the “hemisphere up” direction for that basis are accurately considered in the simulation. For example, in Figure 59, if the “hemisphere up” direction is specified as +Z, the flux-transfer from the top direction will not be properly calculated. This is because the front, left and right surfaces of the F aperture have direction normals facing in +Y, +X and -X directions respectively and are therefore compatible with the +Z “hemisphere-up specification. However, the direction normal for the aperture surface on top faces the -Z direction and therefore is not compatible with the +Z “hemisphere up direction.
The final approach shown in Figure 60 and referred to as FN, involves the use of multiple F matrices with individually assigned hemispherical sampling basis. The surfaces for creating the F matrices are positioned as per the location of the external shading device. The FN approach, while being more accurate and thorough than the previous approaches, requires considerably greater effort in setting up and computing. The number of F matrices, and by extension the number of Daylight Matrices, to be computed in these case is equal to the number of F aperture surfaces multiplied by the number of window groups.
The following sections explain the steps for performing F-matrix simulations based on the approaches discussed above.