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Applied Sciences
, Volume 12 (6) – Mar 8, 2022

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applied sciences Article Calculation of View Factors for Building Simulations with an Open-Source Raytracing Tool 1 , 1 2 3 Sarith Subramaniam *, Sabine Hoffmann , Sridhar Thyageswaran and Greg Ward Department of Civil Engineering, TU Kaiserslautern, 67663 Kaiserslautern, Germany; sabine.hoffmann@bauing.uni-kl.de Department of Mechanical Engineering, Coimbatore Institute of Technology, Coimbatore 641014, India; t.sridhar@cit.edu.in Anyhere Software, Berkeley, CA 94708, USA; gregoryjward@gmail.com * Correspondence: sarith.subramaniam@bauing.uni-kl.de Abstract: Longwave radiative heat transfer is a key determinant of energy consumption in build- ings and view factor calculations are therefore required for the detailed simulation of heat transfer between buildings and their environment as well as for heat exchange within rooms. Typically, these calculations are either derived through analytical means or performed as a part of the simula- tion process. This paper describes the methodology for employing RADIANCE, a command-line open-source raytracing software, for performing view factor calculations. Since it was introduced in the late-1980s, RADIANCE has been almost exclusively employed as the back-end engine for lighting simulations. We discuss the theoretical basis for calculating view factors through Monte Carlo calculations with RADIANCE and propose a corresponding workﬂow. The results generated through RADIANCE are validated by comparing them with analytical solutions. The fundamental methodology proposed in this paper can be scaled up to calculate view factors for more complex, practical scenarios. Furthermore, the portability, multi-processing functionality and cross-platform compatibility offered by RADIANCE can also be employed in the calculation of view factors. Citation: Subramaniam, S.; Hoffmann, S.; Thyageswaran, S.; Keywords: radiation; heat transfer; raytracing; Monte Carlo method Ward, G. Calculation of View Factors for Building Simulations with an Open-Source Raytracing Tool. Appl. Sci. 2022, 12, 2768. https:// 1. Introduction doi.org/10.3390/app12062768 Longwave radiative heat transfer is a key determinant of energy consumption in Academic Editor: Suvash C. Saha buildings. The radiative heat exchange between the outer surfaces of the building envelope and the surroundings account for a signiﬁcant portion of the overall heat loss of a building, Received: 18 January 2022 e.g., under clear sky conditions during night time. Additionally, the longwave radiative Accepted: 4 March 2022 processes in rooms when, e.g., using radiant cooling or heating, inﬂuence the temperature Published: 8 March 2022 distribution on ﬂoor, walls, and ceiling. Publisher’s Note: MDPI stays neutral Radiant heat exchange between two surfaces is a function of their relative orientation with regard to jurisdictional claims in and distance from one another, surface properties and absolute temperatures. The effect published maps and institutional afﬁl- of relative orientation and spacing on radiant heat exchange between two surfaces can iations. be quantiﬁed by a purely geometric parameter called view factor [1]. The knowledge of view factors is required whenever the longwave radiative heat transfer of building surfaces needs to be calculated in detail. Analyses such as energy simulations, photovoltaic simulations, lighting simulations Copyright: © 2022 by the authors. and thermal comfort simulations employ view factor calculations to assess the radiative Licensee MDPI, Basel, Switzerland. exchange between surfaces of buildings, urban contexts, and the human body. Additionally, This article is an open access article view factors are also required in the design of several engineering applications like satel- distributed under the terms and lite temperature control devices, solar collectors, advanced engines with high operating conditions of the Creative Commons temperatures, and cryogenic fuel storage tanks [2]. Attribution (CC BY) license (https:// creativecommons.org/licenses/by/ View factor is independent of surface properties and temperatures, and is also referred 4.0/). to as shape factor, conﬁguration factor, or angle factor. View factors calculated by assuming Appl. Sci. 2022, 12, 2768. https://doi.org/10.3390/app12062768 https://www.mdpi.com/journal/applsci Appl. Sci. 2022, 12, 2768 2 of 19 surfaces to be either diffuse emitters or diffuse reﬂectors are known as diffuse view factors. In this paper, the term view factor is intended to imply diffuse view factors. For two arbitrarily oriented surfaces A and A in space as shown in Figure 1, assuming the surfaces 1 2 to be of equal radiance, the view factor F from A to A is the fraction of radiation leaving 12 1 2 surface one that reaches surface two directly [1]. Likewise, the view factor F is the fraction of radiation leaving surface two that reaches surface one directly. Figure 1. The geometry and variables for calculating view factor between two surfaces A and A . 1 2 In this paper, the term “radiance” when used with proper capitalization (as “radiance” or “Radiance”), refers to the radiant ﬂux per unit solid angle per unit projected area (W/(sr*m )). In instances where the term is capitalized in uppercase as “RADIANCE”, it refers to the raytracing software that will be discussed in the subsequent sections. A generalized expression for F can be derived by considering differential surface elements dA and dA on A and A respectively. For the two area elements, the view 1 2 1 2 factor is: cos q cos q 1 2 F = dA (1) dA dA 2 1 2 2 pr where r is the distance between dA and dA , q is the angle between the surface-normal of 1 2 1 dA and the line connecting dA and dA , and q is the angle between the surface-normal 1 1 2 2 of dA and the line connecting dA and dA . The view factor between dA and the entire 2 1 2 1 area A can be calculated as: cos q cos q 1 2 F = dA (2) dA A 2 1 2 2 pr View factors between two ﬁnite areas A and A obey the following relation: 1 2 Z Z cos q cos q 1 2 A F = dA dA = A F (3) 2 2 1 12 1 21 pr A A 1 2 Computational algorithms speciﬁcally addressed towards view factor calculations have existed for over seven decades [3]. Such algorithms employ either analytical or probabilistic techniques, broadly referred to as Radiosity and Monte Carlo methods, re- spectively [4]. Prominent radiosity-based methods include the hemisphere method [5], Appl. Sci. 2022, 12, 2768 3 of 19 the hemicube method [6] and adaptive integration [7]. Different implementations of the Monte Carlo method have been proposed since the 1960s. One of earliest applications of Monte Carlo method to heat transfer was proposed by Howell in 1969 [8], wherein the possibility of applying this method to calculations with complex surfaces and simultaneous effects was discussed in the context of computing systems. In the subsequent years, several applications and optimizations around this method have been proposed by numerous studies. For example, a triangular subdivision approach to handle occluded surfaces was proposed by Hien and Chirarattananon in 2005 [9]. Vujicic et al. evaluated the impact of discretization of surfaces through meshing to improve the accuracy of simulations [10]. Recent studies also focus on the applicability of Monte Carlo method for speciﬁc use-cases, such as the one by Mirhosseini and Saboonchi (2011) to calculate the view factor between a 3D strip element to a circular cylinder [11]. Fei et al. (2018) discuss the effectiveness of the Monte Carlo method for calculating view factors for several complex surfaces such as torus, parallel cylinders and radiant tubes in annealing furnaces [12]. Several tools have been developed speciﬁcally for the purposes of calculating view factors. Among these View3D [7] and VISRAD [13] employ Radiosity-based approach and Mont3D [14] employs the Monte Carlo method. Presently, none of these tools are under active development and therefore have not been optimized to leverage the features such as multi-processing, cloud-computing and GPU-based calculations that are present in modern computing systems. Additionally, such tools are not particularly suited for calculations with complex shapes that involve multiple obstructions and curvatures [7]. Building simulation tools that perform view factor calculations, usually through the Radiosity approach, do so as a part of their overall workﬂow, and do not offer speciﬁc functionality to calculate view factors alone. This paper introduces the application of RADIANCE, a raytracing tool used primarily for lighting simulations, to calculate view factors. RADIANCE differs from the previously mentioned view factor calculation tools in several aspects. It has been under continuous development for the past three decades and can leverage the advance features offered by modern computing systems. It also supports complex shapes and obstructions. In the fol- lowing sections we introduce this tool and discuss its view factor calculation methodology. 2. RADIANCE: Theoretical Basis and General Calculation Methodology 2.1. Introduction and Background RADIANCE is a collection of over a hundred independent command-line programs that are invoked in customized sequences to perform different types of lighting and day- lighting simulations. It was developed in the mid-1980s at the Lawrence Berkeley National Laboratory by Greg Ward and has been under continued development since then. Detailed documentation about Radiance and its application for lighting and daylighting simulations can be found on https://www.radiance-online.org/ (accessed on 6 March 2022). The source code of Radiance and the compiled executables for different platforms are hosted at https://github.com/LBNL-ETA/Radiance (accessed on 6 March 2022). Initial research on RADIANCE focused on the development of its raytracing methodology [15] to simulate illuminated spaces [16]. The functionality of RADIANCE to accurately calculate photopic luminance [17] and a discussion on the underlying algorithms [18] can be found in the papers published by Ward in the late 1980s.The results for lighting and daylighting simu- lations generated through RADIANCE have been empirically validated by an extensive body of research in the past three decades. An empirical validation of RADIANCE for simulating artiﬁcially lit spaces was conducted by Grynberg in 1989 [19] and corresponding validation for daylit spaces was published by Mardaljevic in 1995 [20]. Validation studies undertaken in the past decade have involved the use of complex shading systems [21] and annual daylighting simulation methods [22]. A schematic diagram showing the standard inputs and workﬂow for RADIANCE is shown in Figure 2. As is evident from the ﬁgure, the simulation methodology features the invocation of multiple programs in a speciﬁc sequence [23]. Appl. Sci. 2022, 12, 2768 4 of 19 Figure 2. Schematic diagram for a conventional RADIANCE simulation. The values in rectangles refer to individual executable programs. The programs ‘Generator ’, ‘Driver ’ and ‘Filter ’ refer to generic third-party programs that interface with RADIANCE inputs and outputs. The core application of RADIANCE relates to raytracing. The purpose of the raytracing is to recursively evaluate Equation (4) at every sampled surface point of the geometry. L (q , j ) = L (q , j )+ r r r e e e 2p R R p/2 (4) L (q , j )r (q , j ; q , j ) cos q sin q dq dj r r i i i bd i i i i i i q=0 j=0 where q is the polar angle measured from the surface normal, j is the azimuthal angle, L (q , j ) is the emitted radiance in W/(sr*m ), L (q , j ) is the reﬂected radiance, L (q , j ) e e e r r r i i i is the incident radiance and r (q , j q , j ) is the bidirectional reﬂectance-transmittance , r r bd i i distribution function in sr . As indicated by the terms L , L , and L in Equation (4), the e r i raytracing process calculates radiant energy emitted, reﬂected, and incident (respectively) at every sampled surface point. The raytracing algorithms are primarily accessed through three programs, namely: rtrace, rpict, and rcontrib. The surface-sampling methodology implemented in these pro- grams is a combination of deterministic and stochastic sampling. In the context of lighting and daylighting simulations, deterministic sampling is used for calculating the radiant energy emitted by direct radiation sources like the sun or light ﬁxtures. Stochastic sampling, which employs the Monte Carlo method, is used to calculate emitted or reﬂected radiation from diffuse sources like building surfaces. While rpict and rtrace incorporate additional techniques like ambient caching that are computationally efﬁcient for lighting and day- lighting applications, rcontrib employs a pure Monte Carlo approach. Rcontrib has so far 2 2 been used, and validated, for calculating irradiance (W/m ) and radiance (W/(sr*m )) values for annual daylighting simulations [24]. In the following section, we discuss the functionality, theoretical basis and syntax of rcontrib, and propose how it can be used to calculate view factors. Appl. Sci. 2022, 12, 2768 5 of 19 2.2. Calculating View Factors As per its user-manual, rcontrib computes “contribution coefﬁcients in a RADIANCE scene” [25]. The term “scene” collectively refers to the geometry and material properties of all the surfaces that are being considered in the calculation. Assuming every surface in the scene to be of equal radiance, at any point in the scene the “contribution coefﬁcient” from a particular surface relates to the fraction of radiant energy received from it. Rcontrib permits the calculation of both radiance and irradiance coefﬁcients through different input parameters. As documented in [25], these coefﬁcients typically relate to radiant ﬂux transfer between sky and ground or optical systems such as light pipes and shading systems. The speciﬁcity of input parameters in each instance depends on the type of coefﬁcient, material properties of the surfaces and the complexity of geometry. To calculate view factors, a hitherto undocumented application, rcontrib needs to be invoked in the irradiance (-I) mode with a single ambient bounce (-ab 1). The resultant direct “contribution coefﬁcient” of a ﬁnite distant surface A at a given differential surface dA is its view factor multiplied by . This is syntactically expressed and illustrated in Figure 3. Figure 3. Calculation of view factor between a differential surface and a ﬁnite surface with rcontrib. The dots on the ﬁnite surface in (a) indicate the sampling controlled by the -ad parameter. As shown by (b), increased sampling will improve the estimation of the view factor. An explanation of the terms in the syntax (c) is provided in the nomenclature. As shown in Figure 3a, the differential surface dA is speciﬁed by the cartesian coordi- nates (x , y , z ) and direction vector (vx , vy , vz ). The green dots on the surface indicate 1 1 1 1 1 1 Monte Carlo sampling which is controlled by the parameter N for ambient divisions (-ad) and limit weight (-lw) as shown in Figure 3b. The parameter for ambient bounces needs to be retained to its default value of one to ensure that no diffuse interreﬂection between surfaces is considered. For standard rcontrib simulations relating to electric lights and daylight, the ambient bounces are usually set to four and higher. The syntax for the command is shown in Figure 3c. The process of calculating view factors can be explained in the following steps: 1. For a given inﬁnitesimal area dA , N rays are stochastically mapped over a hemispher- ical basis as shown in Figure 4. The approach for randomly mapping these rays is based on the methodology proposed by Shirley and Chiu [26]. For the random sam- pling to converge, such that the results from two independent ray tracing processes are numerically within a tolerance range of less than 1%, a large number of samples is required. Appl. Sci. 2022, 12, 2768 6 of 19 2. For each ray that strikes the geometry of the surface(s) identiﬁed through “surfaceI- dentiﬁer”, a contribution is added. The contribution of a single ray will be equal to /N, where the presence of the value of is owing to the use of irradiance integral. 3. The sum of all ray contributions to the surface identiﬁed through the surface identiﬁer constitutes the fraction of the total hemispherical basis viewed by the inﬁnitesimal area. This fraction constitutes an approximation of the view factor. As explained through Figure 3c, the output generated through RADIANCE is the approximate view factor multiplied by a factor of . Figure 4. An illustration of the original sampling pattern employed in RADIANCE. The dots represent individual samples over the projected hemisphere [23]. The process outlined in the above steps is the basis of all view factor calculations discussed in this paper. The mathematical basis for this calculation can be expressed by Equation (5), where the term E denotes indirect-irradiance. The other terms have the ind same interpretation as in Equation (4). 2p R R p/2 E (q , j ) = L (q , j ) cos q sin q dq dj (5) ind i i i i i i i i i q=0 j=0 As indicated by Figure 3b, the amount of stochastic sampling of a particular surface is controllable by increasing or decreasing N, the value for the ambient divisions (-ad) param- eter, which in turn necessitates decreasing or increasing the limit weight (-lw) parameter respectively. Ambient divisions directly control the number of rays used for sampling and limit weight controls the contribution of each ray to the ﬁnal result. So, the value of 1/N for limit weight indicates that each ray is given equal weightage. As is expected of any Monte Carlo raytracing method, higher sampling leads to a better estimation of the result. Based on the illustrations in Figure 3, the functionality and methodology for calculation of view factors can be broadly communicated in simple terms. However, the speciﬁc inputs provided to rcontrib need to be in a format that is compatible with RADIANCE and the outputs received from rcontrib need to be post-processed to derive the actual view factor. Accordingly, the invocation of rcontrib is preceded and followed by a sequence of steps that is shown in Figure 5. Figure 5 introduces an additional RADIANCE program called oconv. Oconv creates an octree data-structure that facilitates efﬁcient raytracing by reducing the number of ray-surface intersections [27]. The octree data-structure is a pre-requisite for all the ray- tracing programs in RADIANCE [28]. The “post-processing” box shown in the ﬁgure relates to extracting the results from rcontrib, dividing the results by as described earlier and ﬁnally performing case-appropriate view factor algebra operations. The methodol- Appl. Sci. 2022, 12, 2768 7 of 19 ogy for specifying inputs, and deriving view factors from outputs, is explained in the following section. Figure 5. The proposed workﬂow for calculating view factors with RADIANCE. RADIANCE programs are identiﬁed through rectangles, and inputs and outputs are denoted through ellipses. The dotted ﬁgures represent a script and output that is external to RADIANCE. The colors used in this ﬁgure are complimentary to the ones used in Figure 3c. 2.3. Specifying Inputs and Interpreting Outputs As illustrated in Figure 3, the inputs for calculating the view factor between a differen- tial surface and a ﬁnite surface with RADIANCE are “differentialSurface”, “surfaceIden- tiﬁer” and “surfaceGeometry”. “SurfaceIdentiﬁer” and “surfaceGeometry” relate to the material properties and shape of the ﬁnite surface respectively. Multiple differential surfaces and ﬁnite surfaces can be speciﬁed at a time by listing them in text ﬁles and specifying the path of the text ﬁles as the inputs in place of the “surfaceIdentiﬁer” and “differentialSurface”. The results thus generated are a matrix comprising of view factors between individual differential surfaces and ﬁnite surfaces. The format for specifying materials and shapes in RADIANCE is shown in Figure 6. The fundamental material and shape-type that are used for the view factor calculations in this paper are plastic and polygon respectively. Figure 6a describes the general template that can be used for deﬁning plastic and polygon and Figure 6b,c provide an example of each of them, respectively. While RADIANCE supports a multitude of materials and shapes, for the purposes of calculating view factors, the plastic material and the polygon shape are ideal as they can be combined to create complex shapes [29]. Figure 6. Specifying materials and shapes in RADIANCE. (a) Shows the general template. (b,c) Pro- vide examples of a material type and shape type respectively. (d) shows a visualization of the geometry deﬁned through the combination of (b,c). Plastic is as an opaque surface that is analogous to conventional diffusing materials assumed in the calculation of view factors. The deﬁnition for plastic material as shown in Figure 6b is “void plastic shape 0 0 5 1 1 1 0 0”. The term “void” indicates that this material is not modiﬁed or affected by any other textures or materials. The term “shape” was arbitrarily chosen and can be any combination of ASCII characters. The numbers “0 0 5 1 1 1 0 0” as a whole specify that the reﬂectance of this material is uniformly diffused white. The three numbers “1 1 1” after “0 0 5” relate to Appl. Sci. 2022, 12, 2768 8 of 19 the spectral power of the three channels that RADIANCE employs to describe the color of a surface or a radiation source. These three channels, namely Red, Green, and Blue, relate to primary colors within the human visual system [23]. The deﬁnition for a polygon is speciﬁed in Figure 6c as “shape polygon object 0 0 12 0 0 . . . ”. Here, the term “shape” indicates that this polygon is of the plastic material type deﬁned in Figure 6b. The term “object” was arbitrarily chosen and can again be any combination of ASCII characters. The numbers “0 0 0 1 0 0 1 1 0 0 1 0” specify the four cartesian coordinates to deﬁne the surface shown in Figure 6d. The raytracing methodology implemented in rcontrib, and other RADIANCE programs, has a physical basis. So, ﬁnite surfaces are deﬁned according to their actual dimensions. For example, based on the vertices deﬁned in Figure 6c, the resultant polygon shape is a square of unit dimensions. Radiance supports SI units of measurement for distances and the values can be expressed in integer or ﬂoating-point numerical format. The units themselves are not speciﬁed explicitly in Radiance input formats. It is possible to combine multiple polygons and materials together. The deﬁnition and visualization for a surface comprised of two polygons is shown in Figure 7. Surfaces and surface-combinations of further complexity are visualized in Figure 8. Figure 7. A surface comprised of two polygons and same material. (a) Shows the deﬁnition of the surface in ASCII format and (b) shows a RADIANCE-rendered image of the surface. Figure 8. Surfaces created by using multiple polygons. All the surfaces shown above were ﬁrst modeled in Sketchup and then exported into the RADIANCE format using the plugin Su2Rad. The methodology to be followed for creating polygons of any number sides is de- scribed in detail in [23]. The standard approach for creating complex surfaces for RA- DIANCE simulations is to model the shape initially in a 3D-Modeling software like ® ® ® AutoCAD , Sketchup , Rhino3D or other similar tools and then export it into a compati- ble format consisting of polygons. Several free translators and plugins exist for exporting 3D-geometry from CAD software like AutoCAD [30], Sketchup [31] and Rhino3D [32] into RADIANCE format. Appl. Sci. 2022, 12, 2768 9 of 19 As explained previously in Figure 3, differential surfaces are deﬁned as per the format “x y z vx vy vz”. It is possible to specify multiple differential surfaces in a single calculation by listing them in line-separated format. A few examples of multiple differential surfaces are illustrated in Figure 9. Differential surfaces do not have a physical shape or structure in RADIANCE and are intended to specify the location from which rays are to be traced. The cone shapes shown in Figure 9 are intended solely as a means of visualizing their location and orientation. The apex of the cone shapes shown in Figure 9 point in the direction of the vector deﬁned through “vx vy vz”. Figure 9. Different conﬁgurations of differential surfaces that can be speciﬁed through a single ﬁle, and therefore can be used in a single calculation. The shape in (a) illustrates a single differential surface. Shapes in (b,c) illustrate differential surfaces aligned along a line and distributed across an area respectively. The shape shown in (d) illustrates surface normals of planar surfaces that were created by subdividing a spherical surface. As explained in Figure 5, the inputs comprising of ﬁnite and differential surfaces need to be sequentially passed to two RADIANCE programs, namely oconv and rcontrib, to generate view factors. A typical calculation process, as executed through the command-line, is shown in Figure 10. The ﬁgure shows a simple two-step calculation where the view factor F involving a differential surface dA and a ﬁnite planar surface A is being dA A 1 2 1 2 calculated. The ﬁrst step involves the creation of the octree from the surface geometry using oconv. The second step invokes rcontrib to calculate the “contribution coefﬁcient” through Monte Carlo raytracing. The octree “geo.oct”, generated through oconv in the previous step, is one of the inputs for rcontrib. Figure 10. An illustration of the command-line process for calculating view factors with RADIANCE. The result from the calculation, a set of three identical numbers, is shown below the rcontrib command in Figure 10. In this example, the contents of the ﬁles “sen.rad” and “pts.txt” represent a ﬁnite planar surface and differential surface respectively. The contents of both the ﬁles are shown in text boxes. The material for the surface i.e., “rectangle”, is speciﬁed as the “surfaceIndentiﬁer” using the -m option for rcontrib. The value of ambient bounces (-ab) is not speciﬁed as it is set by default to 1 by rcontrib. As described earlier, the numbers relate to the red, green, and blue channels. For the purposes of view factor Appl. Sci. 2022, 12, 2768 10 of 19 calculations with uniformly diffused plastic materials (as explained in Section 2.3), the result for each channel will be identical to the other. As illustrated in the ﬁgure, the view factor is obtained by dividing the result from any one of the channels by . The existence of three channels in the calculation is a feature that is inherent to RADIANCE and cannot be disabled. For more complex calculations involving multiple differential surfaces, the results can be redirected to an output ﬁle. In the next section, we present the validation of the view factor results generated through RADIANCE by considering four simple examples. 3. Validation of RADIANCE Generated Results against Analytical Solutions The four examples presented in this section are intended to demonstrate the feasibility of calculating view factors between differential surfaces and arbitrary ﬁnite planar surfaces with RADIANCE. Although, as detailed in Section 2.1, RADIANCE has been extensively validated in the last three decades, the validation studies have exclusively focused on lighting and day- lighting simulations. Such studies have involved the comparison of empirically measured values of illuminance or luminance with those generated through RADIANCE simulations. To the best of the authors’ knowledge, the results and methodology presented in this paper are the original validation of RADIANCE for the purposes of view factor calculations. The ﬁrst two examples involve ﬁnite planar surfaces that are perpendicular or parallel to the differential surface. The third example consists of a sphere that is constructed from planar surfaces. The fourth example involves a box whose constituent surfaces surround the differential surface from all sides. In addition to validating the results generated through RADIANCE, the examples presented in this section also constitute the basic steps that are required for calculation of more complex scenarios. 3.1. Differential Element to Finite Parallel Rectangle The view factor calculation between a differential element and a ﬁnite parallel rectangle is shown in Figure 11. As indicated by the ﬁgure, the edge lengths a and b of the rectangle are both equal to unity. The distance c of the differential element from the rectangle is one. Additionally, the normal to the differential element passes through a corner of the rectangle. In Figure 11b, the command line parameters -h and -w are intended to turn off information headers and warnings respectively. The parameter -I sets rcontrib in “Irradiance” mode, a setting that is a pre-requisite to calculating view factors. Figure 11. View factor calculation between a differential planar element and ﬁnite parallel rectangle. The geometry is shown in (a) and a screen-capture of the corresponding RADIANCE calculation is shown in (b). The analytical solution for this scenario is well-known [33] and can also be accessed through an online interface [34]. The solution can be expressed through Equation (6) as: 1 A B B A 1 1 p p p p F = tan + tan (6) dA1A2 2p 2 2 2 2 (1+A ) (1+A ) (1+B ) (1+B ) where A = a/c and B = b/c. For a = 1 = b = c, the analytical solution calculated through Equation (6) is 0.1385 when rounded off to four decimal places. The value calculated Appl. Sci. 2022, 12, 2768 11 of 19 through RADIANCE, as shown in Figure 11b, is 0.1384. The percentage error calculated as per Equation (7) is 0.072%. Absolute(Analytical Solution Radiance Result) Error = 100 (7) Analytical Solution 3.2. Differential Element to Rectangle in a Plane at 90 to Plane of Element Figure 12 shows the view factor calculation for a differential element and a rectangle. The rectangle is in a plane that is oriented at 90 to the plane of the element. Figure 12. View factor between a differential planar element to rectangle in plane 90 to plane of element as shown in (a) with the corresponding calculation in (b). The analytical expression for F in this scenario, provided in [33] and adapted dA A 1 2 from [34], is: 1 1 C 1 1 1 F = tan tan (8) dA1A2 2p C Y Y 2 2 where A = a/b; C = c/b; Y = A + B . For values of a = 1, b = 1, c = 1 as per the dimensions in Figure 12a, the analytical result rounded off to four decimal places is 0.0557. As shown in Figure 12b this is equal to the result calculated through RADIANCE. 3.3. Element in a Plane to a Sphere Figure 13 shows the view factor calculation between a differential planar element to a sphere. As highlighted by the single dark polygon on the surface of the sphere in Figure 13a, the spherical surface was approximated by several individual polygons. The view factor can be calculated in such a scenario by using the superposition rule [1]. Assuming a surface J to be composed of N sub-surfaces J1, J2, J3, . . . , JN, this rule can be mathematically expressed as: F = F + F + F + . . . . + F (9) iJ iJ1 iJ2 iJ3 iJN For the example considered in Figure 13, the spherical surface was approximated as a composite surface made up of N = 595 planar polygons. The view factor between the differ- ential planar surface and the approximated sphere is then calculated using Equation (9). An analytical solution for the view factor between a differential element and a sphere was proposed by Naraghi [35]. The analytical solution, adapted from Howell [4], can be expressed as: F = (10) dA1A2 3/2 2 2 (1 + E + H ) where r = d/2; R = r/l; H = h/l; E = e/l. For the values (d = 2, e = 2, h = 1, l = 3) considered in Figure 13, the analytical solution as per Equation (10), is 0.0573 when rounded off to four decimal places. This is equal to the solution derived through RADIANCE in Figure 13b. Appl. Sci. 2022, 12, 2768 12 of 19 Figure 13. View factor between a differential planar element and a sphere as shown in (a) with the corresponding calculation in (b). The sphere considered for the calculation is comprised of 595 polygons. The sphere in (a) shows a reduced number of polygons to improve legibility of the figure. 3.4. Energy Balance Figure 14a shows a differential element that lies at the base of a box and is enclosed by all the remaining sides of the box. The box is deﬁned as a unit cube. Figure 14. View factor between a differential surface, indicated by the purple arrow in (a), and surfaces of a box that encloses it. The corresponding calculation is shown in (b). Based on the principle of energy conservation, and the corresponding summation rule for view factors, the sum of view factors from the differential surface to other surfaces should be unity [1]. This can be expressed as general relation through Equation (11) and adapted speciﬁcally to Figure 14a through Equation (12). As shown though Figure 14b the result of the view factor calculated with RADIANCE is also 1. F = 1 (11) å d1j F + F + F + F + F + F = 1 (12) d11 d12 d13 d14 d15 d16 Appl. Sci. 2022, 12, 2768 13 of 19 The four examples that were considered for validation in this section featured the same set of raytracing parameters for rcontrib. The relevance of these raytracing parameters, i.e., “-h -w -I -ad 100,000 -lw 0.00001” on the calculation results is discussed later in Section 5. The following section provides a high-level description of how RADIANCE can be used to calculate view factors for scenarios involving complex surfaces and obstructions. 4. View Factors for Complex Shapes and Obstructions Accommodation of complex geometry and obstructions is a critical aspect of view factor calculations involving urban geometry. For example, view factor calculations be- tween solar panels and the sky vault require the consideration of occluding obstructions like buildings and other urban shapes. The following sub-sections provide a high-level description of how curved shapes and obstructions can be incorporated into calculations. 4.1. Complex Shapes with Curvature While RADIANCE natively supports curved surfaces such as spheres and cones, it does not have the functionality to support other analytical curved shapes. Furthermore, the functionality of exporting complex curved shapes in the form of native RADIANCE spheres or cones is not supported by most 3D modeling software. Consequently, curved shapes must be approximated through planar surfaces. As shown through the examples in Figure 15, it is possible to generate and thereby calculate view factors shapes of greater complexity by using planar surfaces. The calculation process for these shapes is the same as that discussed for the previous use-cases. Figure 15. An assortment of curved shapes whose view factors can be calculated by the planar sub-division and ray-arrangement for each shape. As explained previously, once the view factors between rays and ﬁnite planar shapes have been calculated, further results can be derived through superposition rule. (a,c) Show open and closed curved surfaces respectively while (b) shows two open ﬂat surfaces. 4.2. Incorporating Obstructions Unlike radiosity-based methods, where including obstructions in view factor calcula- tions necessitates the use of additional algorithms [7], such geometry can be included in a straightforward manner in Radiance [28]. As shown through the equations in Figure 16, incorporating geometry in the calculation only requires adding the surfaces into the contex- tual geometry. By pre-identifying surfaces whose view factors are being calculated through ray-origins and surfaceIdentiﬁer inputs, the remaining geometry automatically behaves as obstruction. Appl. Sci. 2022, 12, 2768 14 of 19 Figure 16. A comparison of the geometric setup and corresponding command-line arguments for view factor calculations without obstruction (a) and with obstruction (b). In (b), the additional sphere constitutes the obstruction. 5. Impact of Calculation Parameters on Accuracy, Repeatability of Results, and Runtime As is true for any probabilistic process that relies on random sampling, increasing the number of samples will yield results that converge. This is demonstrated through Figure 17 where ﬁve successive calculations with two values of ambient divisions are shown. The ﬁnite and differential surface for the calculation is the same as that for the example discussed in Section 3.1 and Figure 11. The analytical value of the view factor calculated for this calculation is 0.13853. For all other parameters being the same, the results shown in Figure 17a,b compare the impact of ambient divisions, and by extension limit-weight, on the results. Figure 17. The impact of sampling parameters on accuracy and repeatability of results in ﬁve successive calculations. (a) Denotes low parameters with values of 100 and 0.01 for -ad and -lw respectively, and (b) denotes high parameters. It is evident that for a higher value of ambient divisions (-ad), and correspondingly lower value of limit weight (-lw), the error is consistently lower across successive calcula- tions. Figure 18 provides a broader perspective on the impact of sampling on the results by comparing the error values for the four examples presented in Sections 3.1–3.4. The examples are referred to as “Parallel”, “Perpendicular”, “Sphere” and “EnergyBalance” respectively in the plot. Appl. Sci. 2022, 12, 2768 15 of 19 Figure 18. The variation in average error as per ambient divisions for the calculations considered in Section 3.1 (Parallel), Section 3.2 (Perpendicular), Section 3.3 (Sphere) and Section 3.4 (EnergyBalance). For the values of ambient divisions considered, each calculation was run ten times to calculate an average value of view factor. The rationale for running the calculation multiple times and then averaging results was to minimize the impact of stochastic sampling on the interpretation of results. The average error was calculated as per Equation (7) by consider- ing the result nearest to four decimal places. The error in the case of “EnergyBalance” is 0% for every iteration in that scenario as the random rays originating from the differential surface element were intercepted by the enclosing box irrespective of the magnitude of ambient divisions. Figure 19 shows the impact of ambient divisions on the average calculation runtime. All the calculations were run on a dedicated Intel i5-8400 2.80 GHz computer with six cores and 16GB of RAM. While the exact runtimes are speciﬁc to the machine, the trends shown provide a general indication that the calculation runtime increases with an increase in ambient divisions. The non-linear trend in the plot after 100,000 ambient divisions is attributable to the processing limit of the computer being reached. Figure 19. Variation in average calculation runtime according to ambient divisions. Appl. Sci. 2022, 12, 2768 16 of 19 The plots in Figures 18 and 19 convey that there is a trade-off between the calculation runtime and accuracy. The accuracy of the calculation can be improved by increasing sampling through ambient divisions. As indicated by the error values for 10,000 divisions and above in Figure 17, beyond a certain extent, the improvements in accuracy might not be discernible. 6. Discussion All the examples considered in this paper have analytical solutions and were chosen for that speciﬁc reason. As demonstrated in Section 5, setting a high value for ambient divisions will ensure that the result obtained is within desired tolerance limits (of say 0.5%). However, knowing whether the result is within tolerance is predicated on knowing the result of the calculation upfront. In non-standard cases where no analytical expressions exist and RADIANCE is being used to calculate the view factor, a convergence criterion can be deﬁned by progressively increasing the ambient parameters between successive calculations. The result from every calculation can be compared with the results from the previous calculations to determine the variability in results. An example of this is shown through a chart in Figure 20, which compares the variation in view factor as a function of ambient divisions. Figure 20. Variation of results with progressive increase in sampling controlled through ambient divisions. The ambient divisions (black line) and calculated view factor (green line) shown here relate to the scenario presented in Section 3.1, i.e., view factor between differential planar element to ﬁnite parallel rectangle. The ambient division for the ﬁrst iteration was set to 100 and increased by 50% for every subsequent iteration. So, for example, the second, third and ﬁnal iterations had 150, 225, and 43,790 (rounded to nearest integer) ambient divisions, respectively. As highlighted by the red dotted line, the results stabilize beyond the ninth calculation where the ambient divisions are less than 5000. Appl. Sci. 2022, 12, 2768 17 of 19 7. Conclusions The primary aim of this research was to demonstrate the methodology of employing RADIANCE as a view factor calculation tool. As illustrated through the several examples for ﬁnite and differential surfaces in Section 3, RADIANCE can be effectively used to calculate view factors for a variety of surfaces through the standard Monte Carlo method. View factors can be calculated with only a couple of the over 100 programs that are present in the RADIANCE suite. So, the steep learning curve that has traditionally been associated with using RADIANCE for lighting calculations is not applicable to the calculation of view factors. As discussed previously, a majority—if not all—of the view factor calculation tools de- veloped in the past few decades are currently not under active development. Consequently, the functionalities of these tools have not been updated to leverage the advancements in computing hardware and operating systems. In contrast, RADIANCE is still being actively developed and maintained by its core developer as well as the principal funding agency. This is attributable partly to its relatively large user base and prominence as an industry- and research-grade simulation tool. RADIANCE can be used on multiple operating systems and computing platforms such as commercial cloud-based computing systems. The primary advantage for RADIANCE, when compared to existing view factor calculation tools, lies in its ability to efﬁciently handle complex geometries that are likely to be found in building simulations in both indoor and outdoor contexts. Conventional radiosity-based tools typically support only planar shapes and require extensive calculation time. As shown by the examples in Section 3, irrespective of the shape and size of a surface, RADIANCE provides a singular approach for calculating view factors. Furthermore, the availability of free plugins and translation tools to export geometry from common building 3D modeling software to RADIANCE makes it possible to generate, and calculate, view factors for virtually any scale and form of surfaces. The calculation methodology proposed in this research can be applied to tools such as EnergyPlus and ESP-r to incorporate complex, non-planar shapes into building energy simulations. Currently such tools rely on approximating complex building geometry generated through 3D modeling software such as Revit and Rhino3D into simple planar surfaces. Several studies have already documented the coupling or co-simulation of RADIANCE with building energy simulation tools for performing more physically accurate lighting simulations with complex geometries. Similarly, in such workﬂows the calculation of view factors too can be performed through RADIANCE by either using the same model from lighting simulations or exporting a dedicated model through one of the freely available open-source translators. The authors intend to disseminate the automation scripts for calculating view factors through 3D modeling software on open-source platforms. Future research will address the ap- plication of this tool to simultaneously calculate view factors of multiple surfaces and validate the examples with obstructions by comparing them with analytically derived solutions. Author Contributions: Conceptualization, S.S.; methodology and software, S.S. and G.W.; formal analysis and validation, S.S. and S.T.; writing—original draft preparation, S.S and S.H.; writing—review and editing, S.S., S.H., S.T. and G.W.; supervision, S.H., S.T. and G.W. All authors have read and agreed to the published version of the manuscript. Funding: This research received no external funding. Institutional Review Board Statement: Not applicable. Informed Consent Statement: Not applicable. Conﬂicts of Interest: The authors declare that they have no potential conﬂict of interest regarding this research work. Appl. Sci. 2022, 12, 2768 18 of 19 Nomenclature A , A Two arbitrarily oriented surfaces 1 2 dA , dA Differential surface elements of A and A respectively. 1 2 1 2 F Diffuse view factor from surface A to A . 12 1 2 F Diffuse view factor from surface dA to dA . dA dA 1 2 1 2 q The angle between the surface-normal of dA and the line connecting dA and dA 1 1 1 2 q The angle between the surface-normal of dA and the line connecting dA and dA 2 2 1 2 r Distance between surfaces dA and dA 1 2 Polar angle measured from the surface normal. ' Azimuthal angle. L (q , j ) Reﬂected radiance in W/(sr*m ), r r r L q , j Emitted radiance in W/(sr*m ) ( ) e r r L (q , j )r Incident radiance in W/(sr*m ), i i i bd r (q , j ; q , j ) Bidirectional reﬂectance-transmittance distribution function in sr r r bd i i E (q , j ) Indirect irradiance in W/m ind i i oconv RADIANCE program that is used for creating an octree structure. rcontrib RADIANCE program that is used for raytracing. -I rcontrib option that assigns irradiance mode of calculation. -ad rcontrib option that speciﬁes the number of ambient divisions. -ab rcontrib option that speciﬁes the number of ambient bounces. -lw rcontrib option to specify the limit weight parameter. rcontrib option to specify the modiﬁer of the ﬁnite surface whose view factor -m is to be calculated. -w rcontrib option to turn off warnings. rcontrib option to turn off the header. This enables only the result of the -h calculation being generated as the output. References 1. Cengel, Y. 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Applied Sciences – Multidisciplinary Digital Publishing Institute

**Published: ** Mar 8, 2022

**Keywords: **radiation; heat transfer; raytracing; Monte Carlo method

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