Introduction
The F-Matrix Method was introduced by Greg Ward during the 14th International Radiance Workshop in Philadelphia (Ward 2015). This method, where the F denotes Façade, expands the existing capabilities of Radiance by facilitating parametric simulations of non-coplanar external fenestration systems. The term “non-coplanar external fenestration systems” encompasses a wide range of external shading devices such as simple concrete overhangs, grated overhangs, retractable awnings and even mechanically adjustable external fins used for glare control. Figure 1 shows a room that employs a non-coplanar external fenestration system. With reference to Radiance-based daylighting simulation methods, such systems are referred to as “non-coplanar” on account of the fact that unlike traditional windows and shades, their geometry cannot be confined within rectangular apertures inside walls or ceilings.
Figure 1. The above image shows a sectional-view of a space that is shaded from direct sunlight through complex fenestration systems in the form of external grates.
The F-Matrix Method builds on the previously introduced Three-Phase method (Ward and others 2011). As the name suggests, the Three-Phase Method is a daylight simulation method in which the calculation of luminous flux transfer is split into three phases. The results from these phases, calculated independently of each other, are stored in the form of matrices. Figure 2 shows a schematic diagram for the Three-Phase Method.
One of the fundamental assumptions of the Three-Phase Method is that adjustable fenestration systems are confined to rectangular apertures in the walls or ceiling. Although it is possible to simulate external non-coplanar fenestration systems with this method, they have to be considered as a part of the exterior geometry that surrounds the daylit space. Parametrically simulating more than one external fenestration system with the Three-Phase Method requires the recalculation of the flux-transfer matrix for the entire exterior aspect of the simulation each time. Depending on the complexity of the model involved in the simulation, calculation of this matrix can be a computationally expensive process. The F-Matrix Method addresses this issue by isolating the flux-transfer between the façade and sky through an additional matrix. This matrix focuses only on the façade portion of the simulation. Figure 3 shows a space with a non-coplanar shading system and describes the application of the F-Matrix Method in that scenario.
Figure 2. Schematic diagram for the Three-Phase Method. View (V) Matrix accounts for flux transfer between the interior space to the glazing and fenestration system. Transmission (T) Matrix accounts for flux transfer within the glazing and fenestration system. Daylight (D) Matrix accounts for flux transfer from the glazing aperture to the sun and the sky. The sky vector, represented above by S, incorporates direct and diffuse radiation from the sky into the simulation. Results of the simulation are obtained by multiplying the matrices in the order VTDS.
Figure 3. Schematic diagram for the F-matrix method. The role of the V and T matrices remain the same as described for the Three-Phase Method in Figure 2. The F-matrix accounts for flux transfer between the glazing aperture and the virtual F aperture. The F aperture needs to be created specifically for F-matrix simulations. The D matrix now accounts for the flux transfer between the F aperture and the sky. The matrix multiplication order in this case will be VTFDS. Curtailing the scope of D matrix in such a way is advantageous in simulations involving parametric evaluation of multiple non-coplanar external shading systems. For example, in the scenario shown above, the impact of different types of overhangs on daylight inside the room can be evaluated by recalculating the F-matrix and then repeating the multiplication of matrices. From a simulation-runtime perspective, calculation of flux-transfer matrices is computationally more expensive than matrix multiplication by several orders of magnitude.
This tutorial discusses the F-Matrix Method within the contextual framework of existing Radiance-based simulation methods, namely, The Daylight Coefficient Method, the Three-Phase Method and the Five-Phase Method1. As Figure 2 and Figure 3 indicate, the F-Matrix Method is essentially an expansion of the Three-Phase Method. The Three-Phase Method, in turn, has its basis in the concept of Daylight Coefficients. A basic comprehension of the workflows and concepts underlying these earlier methods is imperative to understanding the F-Matrix Method.
The vocabulary used in this document assumes that the reader has a grasp of the fundamental concepts relevant to Daylighting and is familiar with using Radiance in a command-line-based environment. The next chapter provides a descriptive outline of this tutorial and also offers a few suggestions on navigating through the different chapters.
1. With regards to the ‘phase’-based nomenclature followed for the Three-Phase Method, an F-Matrix simulation can be regarded as a four-phase simulation. Similarly, an F-Matrix simulation with accurate treatment of direct sun can be regarded as a Six-Phase simulation. ↩