Fast and accurate view factor generation

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1 FICUP An Internatonal Conference on Urban Physcs B. Beckers, T. Pco, S. Jmenez (Eds.) Quto Galápagos, Ecuador, 6 30 September 016 Fast and accurate vew factor generaton Benot Beckers 1 and Perre Beckers 1 Compègne Unversty of Technology Sorbonne Unversty Rue Roger Couttolenc, CS Compègne - France Benot.Beckers@utc.fr Unversty of Lège 7, rue des Erables, 41 Planevaux - Belgque Perre.Beckers@ulg.ac.be Keywords: Vew Factor, Ray Tracng, Unform Mesh, Radatve Heat Transfer, Stratfed Samplng. Abstract. Ths document explans how to mesh the hemsphere wth equal vew factor elements. The man characterstc of the method s the defnton of elements delmted by the two classcal sphercal coordnates (polar and azmuth angles) smlar to the geographcal longtude and lattude. Ths choce s very convenent to dentfy the localzaton of the elements on the sphere; t also smplfes a lot the determnaton of rays for ether determnstc or stratfed sampled Monte Carlo ray tracng. The generaton of the mesh s very fast and consequently well suted for ray tracng methods. The qualty of the set of rays spatally very well dstrbuted s a fundamental element of the whole process relablty. FICUP 016

2 1 Introducton The man radatve phenomena consdered n urban physcs are: lght, sound and heat. In thermal radaton, we must dstngush between exchanges that occur n short wavelengths (ncludng vsble lght) and those that take place n the long wavelengths [Beckers 011]. The objects of the urban scene only emt n long wavelengths, wth an ntensty that s proportonal to the fourth power of ther temperature. Thermal loads due to Sun are totally provded n shortwave, and ther nteracton wth the cty surfaces s ndependent of the temperatures. The fundamental dfferences between these problems come from the wave propagaton behavor and the human percepton: lght s consdered nstantaneous, sound s perceved delayed, and heat nvolves nerta [Beckers 014a]. To solve radatve problems, we dstngush two completely dfferent approaches. The frst one s usng some knd of mesh generated n CAD systems (typcally the wde used stl fles), fnte element or radosty methods [Beckers 016]; the second deals only wth dscretzed sources and uses ray tracng technques, typcally n the frame of Monte Carlo methods. In the frst approach, the problem s based on the dscretzaton of the objects nto elements or patches that wll be used to model the scene and smulate the physcal behavor. The basc ngredents are the vew factors. These are purely geometrcal parameters that descrbe how objects are seen from each other. They can be computed by algebrac or Monte Carlo ray tracng methods. The paper s manly based on [Beckers 01], where the dea of usng sphercal equal area cells was ntroduced for the frst tme. The concept of coverage ndex, ntally ntroduced n [Tregenza 1987] and enhanced n [Beckers 014b], s actually gvng valuable nformaton on the cells aspect ratos. The geometrc backgrounds of the method are fully developed n [Beckers 014a]. Vew Factor The vew factor (also called form factor) s the basc element of the radatve studes [Beckers 014a, Beckers 01, Sllon 1994]. It defnes the fracton of the total power leavng patch A that s receved by patch A j. Its defnton s purely geometrc. The angles and j relate to the drecton of the vector connectng the dfferental elements wth the vectors normal to these elements; r j s the dstance between the dfferental elements. 1 cos cos F V y y da da (1) j j (, ) j j A A A r j j Except n partcular stuatons [Howel 010], t s not possble to compute the vew factors explctly. An addtonal dffculty appears n presence of obstructons represented n the above expresson by the vsblty functon V (y, y j ). Ths functon s equal to 0 or 1 accordng to the possble presence of an obstacle that does not allow seeng an element y j from an element y. It s much easer to compute the dfferental vew factor by removng the external ntegraton that wll be taken nto account thereafter n order to acheve the evaluaton of the vew factor, usng, for nstance, Gaussan quadrature rule. The dfferental vew factor n a pont y surrounded by the element area da s gven by: cos cos F V y y da () j da (, ) Aj j j A r j j 3 FICUP 016

3 Ths expresson can be nterpreted as the result of two successve operatons known as Nusselt analogy, where we wll momentarly dsregard the vsblty term V (y, y j ) not requred for the explanaton: 1. The element s projected on the unt hemsphere centered on the pont y. Ths step s represented by the factor cos / r of relaton (). The sold angle completed by the j j element da j, whch s also the area of the sphercal polygon bult from the same element, s gven by cos j d j daj (3) r j. The sphercal polygon s orthogonally projected on the base plane da. Ths projecton corresponds to the term cos of relaton (), whch s now transformed nto: F cos d (4) da Aj j j The term j represents the sold angle or the sphercal polygon area subtended by A j. The vew factor s expressed n percents (projected area over unt dsk area by 100). 3 Computng the Vew Factor The vew factor can be calculated prncpally n two ways: algebrac methods or ray tracng methods. In the frst stuaton, the geometry of the scene has to be modeled. In the second case, we do not need the deep descrpton of the scene: t s suffcent to gve a set of smple patches or trangles lke n the stl format, whch comes from the stereolthography CAD software and s wdely used for rapd prototypng, 3D prntng and computer-aded manufacturng. So, the frst way to calculate the dfferental vew factor, shown n relatons (3) and (4), s to project t onto the hemsphere defned at the concerned pont and then to project the sphercal polygon orthogonally on the plane tangent to the surface (the dsk whch s the base of the hemsphere). Ths projecton s compared to the area of the dsk. The calculaton method s n prncple easy to mplement. Both steps are easy to perform for any shape that can be decomposed n small lne segments. Ths procedure s applcable for any parameterzed shape. The foundaton of the frst step s a central projecton on a unt sphere centered at orgn, whch conssts n dvdng the postons by ther modules: P P (5) P The second step, whch s the orthogonal projecton of P, s straghtforward provded we are workng n axes defned wth respect to the projecton plane (normal vector n). P P ( P nn ) (6) P P P whch s not Let us start wth the computaton of the vew factor of a polylne necessarly n a plane. It s shown n blue lnes n Fgure 1. To compute from pont O the vew factor of ths fgure, we have to proceed n two steps. Frst, we project t on the unt sphere 4 FICUP 016

4 represented n the fgure by ts base and two orthogonal sem-merdans, respectvely n the plane x = 0 and y = 0. Fgure 1: Vew factor: Pont to patch The sphercal projecton drawn n red s composed of great crcle arcs. In the fgure, P 1, P and P 1 are the sphercal projectons (5) of P -1, P and P +1. In a second step, we buld the orthogonal projecton of the sphercal polygon on the base of the hemsphere: plane z = 0. The crcular arcs are transformed nto ellptcal ones (wth the two lmtng cases of straght lnes or crcular arcs). In the fgure, P 1 and P are the orthogonal projectons (6) of P 1and P. To compute the vew factor, we have frst to defne the unt vectors f normal to the faces of the sphercal pyramd OP 1 P P 1 whereop 1, OP, are unt vectors computed from the apex O to the vertces of the studed contour P P P The vertces sequence of the pyramd base s defned n such a way that the sphercal polygon representng ts projecton on the sphere s always stuated on the left sde of ts boundary composed of great crcles segments. The length l of the crcular segment P 1 f OP 1 OP OP 1 OP P s gven by: arcsn 1 l OP OP (8) It s always postve because the arc length s greater than zero and less than. Because the area of a unt dsk sector of angle s equal to /, the arc length of the sphercal pyramd face OP 1 P s equal to twce ts area. The orthogonal projecton a of the face area on the base plane wth normal vector n s then gven by: (7) l a f. n (9) 5 FICUP 016

5 The vector n s normal to the surface supportng ds and on whch we calculate the vew factor. As defned n (7), the vectors f are normal to the faces of the pyramd: OP P, OPP 1 The dot products of (9) are multpled by the quanttes l, equal to the angles of the faces of the pyramd at the apex O. Ths expresson can be postve or negatve, dependng on ts orentaton gven by the dot product. If we add algebracally the expressons (9) for all the contour segments, we obtan the area of the orthogonal projecton P 1 P P 1 of the sphercal polygon, whch must be dvded by (area of the base) to obtan the relatve area: 1 1 F ds P a j l f n (10) For a shape P P P, the formula s gvng a result that depends only on the accuracy of ts evaluaton. Ths shape can be as smple as a polygon or t can be extracted from the outlne of a sold and expressed as a polylne. The precson also depends on the precson of the computaton of the obstructons. In complex stuatons, these computatons can be very heavy. If the patches do not cover the full hemsphere, the complement to 1 of the sum of ther vew factors s called sky vew factor (closure property of the vew factors). The sky vew factor s lnked to the vsble part of the vault of heaven; t s often used as desgn parameter n archtectural applcatons. When the skylne s avalable, (10) can provde an easy and fast method to compute the sky vew factor. 4 Meshng the Hemsphere Before consderng the second method used for computng the vew factors, we have frst to consder the sphercal support used to generate the rays for the castng process. There are several methods to mesh a sphere: n the frst one, t s covered wth sphercal polygons that are fgures of the sphere delmted by great crcles. In practce, these structures are based on some of the fve regular sphercal polygons. In another one, we buld elements bounded by segments of parallels and merdans. The choce of ths knd of mesh s justfed by the fact that the sphercal coordnates based on polar and azmuth angles (where the polar angle may be called co-lattude, zenth angle, normal angle, or nclnaton angle) or the geographcal coordnates are wdely used to descrbe the sphere. A drect advantage of ths choce s that the azmuthal projectons centered on the poles of these elements are fgures of the crcle bounded by arcs of concentrc crcles and rad segments [Beckers 014b, Leopard 006]. For these reasons, t s our preferred meshng method. But before addressng the problem of the hemsphere, we frst examne how to defne equal area cells wthn a dsk. The full dsk s dvded nto a central one surrounded by concentrc rngs, each one contanng a certan number of cells. For a mesh where all elements have the same area, one realzes mmedately that the sequence of cells dffers on the dfferent rngs. Let assume that N equal cells have to be defned n a unt dsk. Startng from a central dsk composed of a sngle cell and whose radus s equal to r 1 = 1/N, we easly perform the computaton n the rng surroundng t. Ths dsk s composed of n cells, so that the dsk that s the sum of the nner dsc and ths one contans (k = k 1 + n) cells (or k +1 = k + n). The radus of ths dsc s gven by r +1 = r k +1. The number of cells added to each rng s arbtrary, provded that the total amount of cells does not exceed the value N. 6 FICUP 016

As the fllng sequence of the successve dsks s arbtrary, we deduce that t s possble to mpose at each step an addtonal condton,

100 cells wth equal areas and aspect rato equal to 1 on the dsk In the optmzed case of Fgure 3, obtaned wth the functons

6 As the fllng sequence of the successve dsks s arbtrary, we deduce that t s possble to mpose at each step an addtonal condton, for example mposng the aspect rato of the cells, ether n the rng to be nserted on the dsk (Fgure ), or on the hemsphere (Fgure 3). Ths procedure only needs a few statements n Matlab and gves the sequence of cells n the dfferent rngs, from the sphercal cap on the top of the dome to ts base. For the example of 100 mposed cells of Fgure, we have the non optmzed sequence: S (11) Fgure : D and 3D vews of 100 cells wth equal areas and aspect rato equal to 1 on the dsk In the optmzed case of Fgure 3, obtaned wth the functons developed n [Beckers 016b], we obtan the sequence: S (1) Fgure 3: 100 cells wth same areas on the dsk and aspect rato equal to 1 on the hemsphere Once the sequence of cells s defned on the dsk, t s easy to use an nverse azmuthal projecton to obtan the mesh on the sphere. In the case of azmuthal orthogonal projecton, 7 FICUP 016

the relatonshp between the polar angle on the unt hemsphere measured n radans and the radus n the projecton s: r sn (13) On the left sde of Fgure, we see the orthogonal projecton of the hemsphere on

7 the relatonshp between the polar angle on the unt hemsphere measured n radans and the radus n the projecton s: r sn (13) On the left sde of Fgure, we see the orthogonal projecton of the hemsphere on ts base. Here, both the areas and the aspect ratos of the projecton are equal. The drawback of ths choce s the mportant dstorton of the cells close to the base of the hemsphere. In Fgure 3, the areas of the projecton are requred to be equal whle the aspect ratos are requred to be equal to one on the hemsphere. The mportant dstorton of the cells close to the base s now removed. When we sgnfcantly ncrease the number of cells, we observe frst that the processng tme needed to generate the sequence of cells s neglgble and secondly that the man dfference between optmzed (Fgure 4, left) and non optmzed (Fgure 4, rght) stuatons s occurrng manly close to the base. Fgure 4: Comparson of the solutons for a generaton of 1000 cells 5 Generatng rays After the generaton of equal vew factor cells, t s possble to generate rays that wll allow computng vew factors of the scene elements. The rays are generated, for nstance from the orgn to each cell and traced to the scene, and the number of collsons wth the elements s accounted. The vew factor of an element s the rato between the number of mpactng rays and the total number of traced rays. If the number of traced rays s suffcent, the result tends to the exact soluton [Vujcc 006]. The frst method used to defne the rays s determnstc, for nstance, the rays pass through the center of each cell. It s the stuaton shown n the orthogonal projectons Fgure 5 & Fgure 6 of the optmzed cell sequence [ ]. In a non optmzed sequence [ ], we observe the bad aspect rato of the lower rng (Fgure 7); t s confrmed by the dagram of Fgure 8 showng the relatve coverage ndex n each rng. Ths ndex s defned as the rato of the area of the greatest nscrbed crcle and the cell area, compared to the same rato computed n a plane square and equal to /4 [Beckers 014b]. It also appears clearly that the densty of ponts s lower n the bottom of the dome (Fgure 6). The same s occurrng for the random rays of Fgure FICUP 016

8 Fgure 5: Determnstc 151 cells centers Fgure 6: Sde vew of the dome composed of 151 rays generated from equal vew factor cells Fgure 7: Mesh and determnstc rays for the non optmzed 151 cells dome Fgure 8: Equal vew factor (EVF), 151 cells mesh wthout cells aspect rato optmzaton 9 FICUP 016

Fgure 9: Equal vew factor (EVF) optmzed 151 cells mesh In the optmzed mesh where the cells aspects ratos on the sphere are close to 1, we obtan the new cells sequence [1 15 3 54 80 107 133 151] and

Anyway, the optmzed sequence s better both for the mnmum value and for the average. In the second ray tracng method, the poston n each cell s defned randomly.

9 Fgure 9: Equal vew factor (EVF) optmzed 151 cells mesh In the optmzed mesh where the cells aspects ratos on the sphere are close to 1, we obtan the new cells sequence [ ] and the coverage ndces of Fgure 9. We observe that the worse coverage ndex occurs n the rng close to the top of the dome whle t occurs n the bottom rng of the non optmzed sequence. Anyway, the optmzed sequence s better both for the mnmum value and for the average. In the second ray tracng method, the poston n each cell s defned randomly. Because all the cells are defned between two lattudes and two longtudes, ths procedure s very relable and easy to mplement. Ths method pertans to the category of stratfed sampled Monte Carlo methods. An example of ths knd of ray dstrbuton s shown on a sde vew of a dome n Fgure 10. It appears clearly that the densty of ponts s lower close to the base of the dome, whch reflects the behavor of mportance samplng methods. Fgure 10: Sde vew of a dome composed of 5000 random rays The proposed method s also the most convenent one to generate unform equal sold angle rays on the sphere. In ths case, as proposed n [Beckers 01], t s smlar to that of [Leopard 006], but accordng to the performed comparatve tests, we feel that t s faster, because t s usng a pure algebrac procedure. 30 FICUP 016

10 6 Concluson Two methods are proposed for computng the vew factors. The frst one, often called Lambert method [Beckers 014a], uses an explct formulaton of the pont to patch vew factor. It s very effcent and exact n the case of lack of obstacle between the pont and the patch. The second one s based on an orgnal method of mesh generaton on the sphere or the hemsphere. Ths knd of mesh allows usng both mportance and stratfed samplng n Monte Carlo ray tracng methods. It provdes an effcent method to compute the vew factors n complex urban envronments because due to ts geometrcal smplcty, t s naturally well suted to deal wth complex spatal confguratons. References [Beckers 011] Benot Beckers, Impact of solar energy on ctes sustanablty, PLEA th Conference on Passve and Low Energy Archtecture, Louvan-la-Neuve, Belgum, July 011. [Beckers 01] Benot Beckers, Perre Beckers, A general rule for dsk and hemsphere partton nto equal-area cells, Computatonal Geometry: Theory and Applcatons, Vol. 45, Nr. 7 01, p [Beckers 014a] Benot Beckers, Perre Beckers, Reconclaton of Geometry and Percepton n Radaton Physcs, John Wley and Sons, Inc., 19 pages, July 014. [Beckers 014b] Benot Beckers, Perre Beckers, Sky vault partton for computng daylght avalablty and shortwave energy budget on an urban scale, Lghtng Research and Technology, vol. 46 n 6, Pages , December, 014 [Beckers 016] Benot Beckers, Multscale Analyss as a Central Component of Urban Physcs Modelng, n: Computatonal Methods for Solds and Fluds, Multscale Analyss, Probablty Aspects and Model Reducton, Adnan Ibrahmbegovc (Ed.), Sprnger Internatonal Publshng, 016, Pages 1-7. [Beckers 016b] Benot Beckers, Perre Beckers, Complete set of Matlab procedures for achevng unform ray generaton, 016, Webste: [Howel 010] J.R. Howell, R. Segel, M.P. Menguc, Thermal Radaton Heat Transfer, 5 th ed., Taylor and Francs / CRC, New York, 010. [Leopard 006] Paul Leopard, A partton of the unt sphere nto regons of equal area and small dameter, Electron. Trans. Numer. Anal [Tregenza 1987] Tregenza Peter R., Subdvson of the sky hemsphere for lumnance measurements, Lghtng Research & Technology, 1987; 19: [Sllon 1994] Franços Sllon, Claude Puech, Radosty and Global Illumnaton, Morgan Kaufmann Publshers, Inc., [Vujcc 006] Mle R. Vujcc, Ncholas P. Lavery, S. G. R. Brown, Numercal Senstvty and Vew Factor Calculaton Usng the Monte Carlo Method, Proceedngs of the Insttuton of Mechancal Engneers, Part C: Journal of Mechancal Engneerng Scence 006, 0: FICUP 016

R s s f. m y s. SPH3UW Unit 7.3 Spherical Concave Mirrors Page 1 of 12. Notes

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