New Computational Geometry Methods Applied to Solve Complex Problems of Radiative Transfer

Abstract

Diverse problems of radiative transfer remain as yet unsolved due to the difficulties of the calculations involved, especially if the intervening shapes are geometrically complex. The main goal of our investigation in this domain is to convert the equations that were previously derived into a graphical interface based on the projected solid-angle principle. Such a procedure is now feasible by virtue of several widely diffused programs for Algorithms Aided Design (AAD). Accuracy and reliability of the process is controlled in the basic examples by means of subroutines from the analytical software DianaX, developed at an earlier stage by the authors, though mainly oriented to closed cuboidal or curved volumes. With this innovative approach, the often cumbersome calculation procedure of lighting, thermal or even acoustic energy exchange can be simplified and made available for the neophyte, with the undeniable advantage of reduced computer time.

1. Introduction

1.1. Form Factors

To our knowledge, one of the main problems in Science as applied to aerospace, solar and industrial design technology has been to determine how the existing physical fields are transformed according to the physiognomy of fixtures or products and in which direction we should orient our developments to seek a more correct transmission of environmental and heat transfer phenomena, or, in other words, in which way can the design of three-dimensional forms be improved to obtain an optimal and coherent distribution of energies that effectively contributes to mitigate global warming and thereby helps resolve climatic issues.

We must outline that radiation in a physical or spatial environment is made manifest through fields of a fundamentally vector nature. Therefore, our first objective would lie in the assessment of these types of fields in an unaltered state. On such issues, there exist the relevant contributions of Yamauchi [1] and Moon [2,3] amongst others in seeking radiative potential. However, successive modifications of the spatial features of the elements entailed possess the capacity to substantially alter the layout of the field. Such geometric alterations contribute to changes in energy distribution that manufacturers and users are heavily demanding, as it is already vital for them to acquire accurate and simple notions regarding the issue.

The problem is not recent. In fact, Lambert’s statement of the famous theorem XVI in his treatise Photometry [4] speculated about the amount of rays (flux) that issues from any two equally luminous surfaces onto the adjacent, and established that if the two surfaces A₁ and A₂ were equally luminous and faced each other in some way, the amount of incident rays from either of the two surfaces on the second is identical. The former is also known as reciprocity theorem and to develop it to some extent is crucial to assess the mathematical reach of the matter that we are discussing.

The energy that leaves surface A₁ and reaches surface A₂ will be:

$${\mathrm{E}}_1{\mathrm{A}}_1{\mathrm{F}}_{12},$$

(1)

and, reciprocally, the energy that passes from surface A₂ to A₁ is:

$${\mathrm{E}}_2{\mathrm{A}}_2{\mathrm{F}}_{21},$$

(2)

where E₁ and E₂ are the equivalent amounts of energy (in W/m²) emitted by surfaces A₁ and A₂; and F₁₂ or F₂₁ are dimensionless entities called “form factors”. If we assume, in principle, that there are no inter-reflections or re-emissions from one surface to the reciprocal, and theoretically no other significant sources of radiation have access, by any means, to the boundaries of the problem, all incident flux will be absorbed and the flow of energy (dΦ), according to Lambert’s theorem, should be zero, that is:

$$\mathrm{d}\mathrm{\Phi }={\mathrm{E}}_1{\mathrm{A}}_1{\mathrm{F}}_{12}-{\mathrm{E}}_2{\mathrm{A}}_2{\mathrm{F}}_{21}=0\iff {\mathrm{A}}_1{\mathrm{F}}_{12}={\mathrm{A}}_2{\mathrm{F}}_{21}.$$

(3)

To this end, we would consider the surface elements dA₁ and dA₂. The angles θ₁ and θ₂ are measured between the line from dA₁ to dA₂ (direction of propagation) and the normal to each surface; r is the distance from the center of one differential area to the other (Figure 1):

The cosine emission law, also attributed to J. H. Lambert, states that the flux per unit of solid angle in a given direction θ (or radiant intensity) is equal to the flux in the normal direction to the surface (maximum intensity) multiplied by the cosine of the angle between the normal and the direction considered.

Luminance or radiance is the radiant flux or power emitted per unit area and solid angle (steradian) in a given direction. To obtain the flow emitted by the surface element dA₁ it is necessary to multiply the radiance or luminance by the projected surface in the fixed direction of the angle θ₁, and thus:

$$\mathrm{I}=\frac{\mathrm{d}\mathsf{\Phi }}{\mathrm{d}\mathsf{\Omega }}={\mathrm{L}}_1{\mathrm{dA}}_1\cos {\mathsf{\theta }}_1,$$

(4)

where I is the radiant intensity, dΩ is the solid angle and L₁ represents the radiance of surface A₁.

It is easily deducted that L₁ remains independent of the direction and therefore L_1θ = L₁.

Since the previously found flux refers to the solid angle unit, we must now evaluate the amount of flux that reaches a differential surface element dA_n whose distance to A₁ is precisely r.

$$\mathrm{d}\mathsf{\Phi }={\mathrm{L}}_1{\mathrm{dA}}_1\cos {\mathsf{\theta }}_1\frac{{\mathrm{dA}}_{\mathrm{n}}}{{\mathrm{r}}^2}$$

(5)

It is often useful to establish a relationship between luminance or radiance and lighting or irradiance; with this aim, we could simply extend the flux to a hemisphere of radius r in which dA₁, is inscribed. The surface dA_n in spherical co-ordinates amounts to:

$${\mathrm{dA}}_{\mathrm{n}}={\mathrm{r}}^2\sin {\mathsf{\theta }}_1\mathrm{d}\mathsf{\theta }\mathrm{d}{\displaystyle \mathsf{\phi }}$$

(6)

then from (5) and (6):

$${\mathsf{\Phi }}_1={\mathrm{L}}_1{\mathrm{A}}_1{{\displaystyle \int }}_{{\mathrm{A}}_{\mathrm{n}}}^{}{\cos \mathsf{\theta }}_1\frac{{\mathrm{dA}}_{\mathrm{n}}}{{\mathrm{r}}^2}={\mathrm{L}}_1{\mathrm{A}}_1{{\displaystyle \int }}_0^{2\mathsf{\pi }}{{\displaystyle \int }}_0^{\mathsf{\pi }/2}\cos {\mathsf{\theta }}_1\sin {\mathsf{\theta }}_1\mathrm{d}\mathsf{\theta }\mathrm{d}{\displaystyle \mathsf{\phi }}={\mathsf{\pi }\mathrm{L}}_1{\mathrm{A}}_1={\mathrm{E}}_1{\mathrm{A}}_1$$

(7)

Thus, E₁ = πL₁ (E is emitted radiation per area unit, expressed in W/m²).

Returning to the problem of energy exchange, if we assimilate the differential surface element dA_n with dA₂, we find that:

dA_n = cos θ₂dA₂.

(8)

In this manner, the flux or radiant power that goes from dA₁ to dA₂, substituting L₁ in Equation (5), will be:

$${\mathrm{d}\mathsf{\Phi }}_{1-2}={\mathrm{E}}_1\cos {\mathsf{\theta }}_1\cos {\mathsf{\theta }}_2\frac{{\mathrm{dA}}_1{\mathrm{dA}}_2}{{\mathsf{\pi }\mathrm{r}}^2}$$

(9)

And respectively,

$${\mathrm{d}\mathsf{\Phi }}_{2-1}={\mathrm{E}}_2\cos {\mathsf{\theta }}_2\cos {\mathsf{\theta }}_1\frac{{\mathrm{dA}}_2{\mathrm{dA}}_1}{{\mathsf{\pi }\mathrm{r}}^2}.$$

(10)

The energy exchange is eventually set at:

$${\mathsf{\Phi }}_{1-2}=\left({\mathrm{E}}_1-{\mathrm{E}}_2\right){{\displaystyle \int }}_{{\mathrm{A}}_2}^{}{{\displaystyle \int }}_{{\mathrm{A}}_1}^{}\cos {\mathsf{\theta }}_1\cos {\mathsf{\theta }}_2\frac{{\mathrm{dA}}_1{\mathrm{dA}}_2}{{\mathsf{\pi }\mathrm{r}}^2}.$$

(11)

The above integral equation responds to the fundamental or canonical formula of radiation which in differential terms is:

$${\mathrm{d}}^2\mathsf{\Phi }={\mathrm{E}}_{\mathrm{i}}\cos {\mathsf{\theta }}_1\cos {\mathsf{\theta }}_2\frac{{\mathrm{dA}}_1{\mathrm{dA}}_2}{{\mathsf{\pi }\mathrm{r}}^2}$$

(12)

In such a situation, the integral found by virtue of the fundamental expression adopts the value of A₁F₁₂ or A₂F₂₁, following the above definitions. In general, these are quadruple integrals and to solve them becomes so complex that extreme dexterity in geometry problems along with haphazard techniques of integration is required. Therefore, we want to avoid this difficulty as much as possible by virtue of the procedure presented in this article.

The fundamental equation of the much-sought form factors is:

$${\mathrm{F}}_{12}=\frac{1}{{\mathrm{A}}_1}\left[{{\displaystyle \int }}_{{\mathrm{A}}_2}^{}{{\displaystyle \int }}_{{\mathrm{A}}_1}^{}\frac{\cos {\mathsf{\theta }}_1\cos {\mathsf{\theta }}_2}{{\mathsf{\pi }\mathrm{r}}^2}{\mathrm{dA}}_1{\mathrm{dA}}_2\right]$$

(13)

and its asymmetric reciprocal in algebraic terms,

$${\mathrm{F}}_{21}=\frac{1}{{\mathrm{A}}_2}\left[{{\displaystyle \int }}_{{\mathrm{A}}_2}^{}{{\displaystyle \int }}_{{\mathrm{A}}_1}^{}\frac{\cos {\mathsf{\theta }}_1\cos {\mathsf{\theta }}_2}{{\mathsf{\pi }\mathrm{r}}^2}{\mathrm{dA}}_1{\mathrm{dA}}_2\right]$$

(14)

If we assume, for easier visualization and since it is a frequent case in luminous radiative transfer (but not so in thermal problems involving, for instance. gases), that the surface A₂ does not emit energy and only receives it from A₁, the total flow received in A₂ will be:

$${\mathrm{A}}_2{\mathrm{N}}_2=\left[{{\displaystyle \int }}_{{\mathrm{A}}_2}^{}{{\displaystyle \int }}_{{\mathrm{A}}_1}^{}\frac{{\mathrm{M}}_1\cos {\mathsf{\theta }}_1\cos {\mathsf{\theta }}_2}{{\mathsf{\pi }\mathrm{r}}^2}{\mathrm{dA}}_1{\mathrm{dA}}_2\right]$$

(15)

where M₁ = N₂/F₂₁ and N₂ is the energy received by surface 2 in W/m².

1.2. Configuration Factors

Let us behold the meaning of the form factor, defined as an integral equation. Actually, what it permits us to calculate is just the net radiant exchange from surface to surface. A surface that emits a certain amount of power M₁ (in W/m²) successively produces a certain power N₂ ‘as an average or percentage’ on another surface, which tells us nothing about the point-by-point distribution of that same power.

However, in many engineering, architectural and medical problems, a precise description of the flux function, that is, the radiant field, is often deemed necessary to establish the exact nature of the radiation “map” and to be able to control the possible deficit or surplus which could, in turn, become critical.

Therefore, if we look for the point distribution of power that reaches surface A₂ instead of the mere average, we could reduce the fourfold integrals in the expression of the form factor to a simpler surface integral as follows:

$${\mathrm{n}}_2={{\displaystyle \int }}_{{\mathrm{A}}_1}^{}\frac{{\mathrm{M}}_1\cos {\mathsf{\theta }}_1\cos {\mathsf{\theta }}_2}{{\mathsf{\pi }\mathrm{r}}^2}{\mathrm{dA}}_1={\mathrm{M}}_1{{\displaystyle \int }}_{{\mathrm{A}}_1}^{}\frac{\cos {\mathsf{\theta }}_1\cos {\mathsf{\theta }}_2}{{\mathsf{\pi }\mathrm{r}}^2}{\mathrm{dA}}_1$$

(16)

This new expression is, like the previous one, strictly geometric and as such only depends on angles and areas, given that for a certain moment of time the value of M₁ can be assumed as constant.

The double integral equation,

$${\mathrm{f}}_{12}={{\displaystyle \int }}_{{\mathrm{A}}_1}^{}\frac{\cos {\mathsf{\theta }}_1\cos {\mathsf{\theta }}_2}{{\mathsf{\pi }\mathrm{r}}^2}{\mathrm{dA}}_1$$

(17)

will be called the configuration factor and it presents several fundamental applications in the method that we will develop hereby.

The first elementary consequences are:

$${\mathrm{n}}_2={\mathrm{f}}_{12}{\mathrm{M}}_1$$

(18)

and

$${\mathrm{A}}_2{\mathrm{N}}_2={\mathrm{M}}_1\left[{{\displaystyle \int }}_{{\mathrm{A}}_2}^{}{\mathrm{f}}_{12}{\mathrm{dA}}_2\right]={\mathrm{A}}_1{\mathrm{F}}_{12}{\mathrm{M}}_1={\mathrm{A}}_2{\mathrm{F}}_{21}{\mathrm{M}}_1$$

(19)

Then,

$${\mathrm{F}}_{12}=\frac{1}{{\mathrm{A}}_1}\left[{{\displaystyle \int }}_{{\mathrm{A}}_2}^{}{\mathrm{f}}_{12}{\mathrm{dA}}_2\right]$$

(20)

Accordingly,

$${\mathrm{F}}_{21}=\frac{1}{{\mathrm{A}}_2}\left[{{\displaystyle \int }}_{{\mathrm{A}}_2}^{}{\mathrm{f}}_{12}{\mathrm{dA}}_2\right]$$

(21)

Equation (21) can be clearly identified with the “average” of f₁₂“over surface” A₂, especially since we had already obtained that N₂ = F₂₁M₁. In quantum dynamics terms, the form factor expresses the probability of a surface being stricken with photonic energy emitted by an adjacent source.

Similarly, we would define N₂ as the average of n₂ on the surface A₂;

$${\mathrm{N}}_2=\frac{1}{{\mathrm{A}}_2}\left[{{\displaystyle \int }}_{{\mathrm{A}}_2}^{}{\mathrm{n}}_2{\mathrm{dA}}_2\right]$$

(22)

This important finding represents the ignored relationship between form factors and configuration factors that will allow us to easily simplify and visualize many complex calculations in the following chapters by restricting them only to second order primitives, and the ensuing “average” is ready accessible by numerical procedures. To our knowledge, it has only been defined by Cabeza-Lainez [5,6].

2. Symbolic Calculus of Basic Elements

2.1. Triangle

A set of basic forms can be derived from the triangle, but only if the calculation axis lies perpendicular to the center of the figure or the vertex of the triangle or rectangle considered. The principle equation is,

$$\mathrm{n}=\mathrm{L}{{\displaystyle \int }}_0^{\mathrm{b}}{{\displaystyle \int }}_0^{\mathrm{z}}\frac{{\mathrm{y}}^2}{{\left({\mathrm{x}}^2+{\mathrm{y}}^2+{\mathrm{z}}^2\right)}^2}\mathrm{dzdx}$$

(23)

for an instant of time t in which the luminance or radiance L can be considered constant and keeping in mind that Lπ = M.

The solution for triangle A₁ in Figure 2, integrated by parts is:

$$\mathrm{n}=\frac{\mathrm{L}}{2}\left[\frac{\mathrm{b}}{\sqrt{{\mathrm{y}}^2+{\mathrm{b}}^2}}{\tan }^{-1}\frac{\mathrm{a}}{\sqrt{{\mathrm{y}}^2+{\mathrm{b}}^2}}\right]$$

(24)

2.2. Rectangle

The full rectangle is just the sum of solutions for surfaces A₁ (24) and A₂ (see Appendix A):

$$\mathrm{n}=\frac{\mathrm{L}}{2}\left[\frac{\mathrm{a}}{\sqrt{{\mathrm{y}}^2+{\mathrm{a}}^2}}{\tan }^{-1}\frac{\mathrm{b}}{\sqrt{{\mathrm{y}}^2+{\mathrm{a}}^2}}+\frac{\mathrm{b}}{\sqrt{{\mathrm{y}}^2+{\mathrm{b}}^2}}{\tan }^{-1}\frac{\mathrm{a}}{\sqrt{{\mathrm{y}}^2+{\mathrm{b}}^2}}\right]$$

(25)

However, we need to be reminded that we have just obtained the value of radiative exchange at a single point located precisely under the vertex of the rectangle and in a particular direction (perpendicular) to the surface source.

Deft combinations of triangles would give us the equilateral triangle, hexagon and, as a limit case that we will discuss in Appendix A, the circle, which has been only obtained in such “polygonal” fashion by Cabeza-Lainez.

2.3. Calculations in a Plane Perpendicular to the Figure

In the above discussion, we have solely found the component of radiation that was perpendicular to the emitting surface at an isolated point.

It is easy to imagine the severe complexity that arises when trying to find the solution for points moving freely in a plane, let alone space.

Even so, radiance being a vector, to complete the problem means finding values of other auxiliary components.

In this situation, the principle integral turns out to be (if we multiply by the corresponding cosine):

$$\mathrm{n}=\mathrm{L}{{\displaystyle \int }}^{}{{\displaystyle \int }}^{}\frac{\mathrm{zy}}{\left({\mathrm{x}}^2+{\mathrm{y}}^2+{\mathrm{z}}^2\right)}\mathrm{dxdz}$$

(26)

Applying this proposition to the case considered (Figure 3), we find that if Y (the distance) is a constant, the integral equation yields:

$$\mathrm{n}=\frac{\mathrm{L}}{2}\left[{\tan }^{-1}\frac{\mathrm{b}}{\mathrm{y}}-\frac{\mathrm{y}}{\sqrt{{\mathrm{a}}^2+{\mathrm{y}}^2}}{\tan }^{-1}\frac{\mathrm{b}}{\sqrt{{\mathrm{a}}^2+{\mathrm{y}}^2}}\right]$$

(27)

By virtue of the former, we have exposed how complex it becomes to find an analytical expression to solve the problem if only we alter the coordinate plane of calculation, not to mention figures of geometry other than those limited by straight lines. That is why we concluded that there was a significant need for a graphical solution based on parametric design as this would imply a universal solution and an extraordinary simplification of the issue.

2.4. Calculation of Integral Equation for a ‘Rectangle’ Over a Horizontal Square

Previously (Equation (27)) we had found the solution in terms of arctangent for the entire rectangle. The adaptation of this formula to an arbitrary movement of points on a horizontal surface (Figure 4), requires taking into account various singularities depending on the position of the rectangle’s base, the situations of symmetries and what a unique “form factor algebra” permits.

For example, in the Y axis and at level –H₁, the adapted solution would give:

$$\mathrm{n}=\mathrm{L}\left[\frac{\mathrm{D}}{\sqrt{{\mathrm{H}}_1^2+{\mathrm{D}}^2}}{\tan }^{-1}\left(\frac{\mathrm{M}}{\sqrt{{\mathrm{H}}_1^2+{\mathrm{D}}^2}}\right)-\frac{\mathrm{D}}{\sqrt{{\mathrm{H}}_0^2+{\mathrm{D}}^2}}{\tan }^{-1}\left(\frac{\mathrm{M}}{\sqrt{{\mathrm{H}}_0^2+{\mathrm{D}}^2}}\right)\right]$$

(28)

3. Methodology

3.1. Solid Angle Projection Law

To obtain the energy exchange between a given surface and a point P, we would simply trace a cone whose vertex is the point considered P, and its base is the surface in question (dΩ), and proceed to intersect the said cone with a sphere of unit radius (Figure 5). The enclosure within this intersection, projected on any reference plane that we may require and divided by the horizontal area of the aforementioned unit sphere (that is, divided by π), will equate to the value of the configuration factor determined by virtue of the analytical methods of exact integration described in the previous chapters.

If, by means of what we have discussed in detail in the above sections, we were trying to find certain geometric factors or proportions, it seems equally reasonable to achieve these same values through graphic procedures such as the ones employed in Descriptive Geometry.

The advantages are obvious to the project engineer, because if, due to the difficulty of the mathematical problem, hesitation appears, we are allowed to somehow ‘visualize’ the entire process.

In summary, what has been proven is that the configuration factor is a mere proportion, a dimensionless number, consisting of a ratio of areas, one of them the area of a certain projection and the other the total area involved expressed in circular terms, usually π [6].

The most important consequence of the former is that the problem, regardless of its complexity, will always present a unique non-trivial solution since the projection will have a univocal value and cannot be non-existent.

In addition, as the configuration factors are actually projections, they hold an additive property, which is useful to know when dealing with sundry and simultaneous emitting surfaces, because it is possible to add their effects. The average over the receiving surface of the configuration factors thus obtained will constitute the form factor that had been so laboriously sought for by means of mathematical formulation and solving of fourfold integral equations.

Alternatively, graphic methods and analytical methods or a combination thereof can be used. This is the gist of our procedure.

It also seems gratifying to assume that we have solved the fundamental problem of radiation with geometric methods. Why is that? Because it means that ‘form’ is very important in this type of problem and that not all forms act in the same way from a radiative point of view; some geometries favor exchanges more than others and the mission of the heat transfer analyst is to adequate these forms. predicting their impact. Such repercussion can also be verified by observers, that is, manufacturers, society, etc., using, as we shall see, a simple tool.

However, the sole hindrance that we identified at the time was that numerical validation of the simple rectangle over a horizontal plane (Figure 4) was not readily available, thus casting a shadow over the viability of our approach. Fortunately, Cabeza-Lainez was able to work out an exact solution and this relevant finding is presented in Appendix B.

Then a similar problem arose with circular emitters in a free position. Cabeza-Lainez has completely solved this anew [7], as described in Appendix C, as it may help to validate other curved geometries.

3.2. Algorithms Aided Design

Although today, most Computer Aided Design (CAD) programs incorporate what is known as parametric design (which implies that it is not necessary to redraw the objects that we design if they change shape, position or size), we should not consider all of these as Algorithm Aided Design programs (AAD).

An algorithm is a logical expression that leads to procedures to be performed through finite sets of closed basic instructions. Algorithms simulate human abilities to break down a problem into a series of simple steps that can be easily executed and, although they are closely related to computers, algorithms can be defined independently of programming languages and applications.

However, not just any procedure can be considered properly as an algorithm, since many procedures are far from being well defined or contain ambiguities, logical contradictions or infinite loops.

Some important properties that are required for a procedure to be considered an algorithm are:

• An algorithm is a well-defined set of properly concise instructions.
• An algorithm demands a defined set of inputs that may or may not come from the output of a previous algorithm.
• An algorithm generates a precise output (Figure 6):

• Finally, an algorithm can produce warnings and error messages through the appropriate editor. If the inputs are not appropriate, e.g., if we enter text instead of numbers, the algorithm will return an error message instead of the expected output, via the appropriate editor [8].

In addition, through parameterization, we have not merely obtained a straight line, (Figure 6) but, by simply modifying the value of the coordinates, moving the mouse over the sliders we can alter the ‘line’ output and draw all the straight lines contained in the Euclidean space with a single gesture of the tracer.

3.3. Calculation by Algorithms Aided Design through Finite Element Method

The main problem of analytical calculation is that, although we may employ a convenient algebra of form factors [6] to find the radiance received by a surface, issuing from basic figures limited by straight lines, the number of shapes of the emitters, whose direct integration is very difficult or impossible to perform, remains numberless.

To solve such an impasse, we have successfully developed a Grasshopper^® algorithm that allows us to calculate the configuration factor subtended by an arbitrary point of any surface (i.e., irrespective of its shape and orientation) from an emitter or ‘three-dimensional figure’, with manifold geometry or inclination (including those whose integration is deemed impossible).

To calculate the incident radiation n (W/m²), we will only have to add up all the configuration factors that we may require and multiply them by the luminance that departs from the emitter (W/sr·m²).

The steps which we have followed to develop the algorithm can be summarized below:

3.3.1. Geometric Definition of Both the Irradiated Surface and the Emitter and, Subsequently, Division of the Irradiated Surface into a Grid of N × (N − 1) Squares, Locating Its Corners

Let us consider for example, a rectangle 2 m wide by 2 m high whose base is 5 m above the ground. The rectangle’s vertical axis coincides with the middle point of the horizontal receiving surface of 10 × 10 m.

First we establish the plane of the irradiated surface defining, by means of sliders, the components of its normal vector and using the ‘Vector XYZ’ function, constructing the said plane with the help of the ‘Plane Normal’ function as shown in the upper part of the following figure (Figure 7).

Subsequently, we divide the irradiated surface into N × (N − 1) squares (finite elements) using the ‘Square’ function that generates a bi-dimensional grid with square cells and retains the points at the corners of the squares.

Note that (in this case) the first row of corner points has been cancelled, since in the plane of the ‘rectangle’, the radiance will necessarily be equal to zero and the size of the squares is divided by its number for a better performance of the algorithm (Figure 7 and Figure 8):

The result is detailed in Figure 8:

3.3.2. Calculate the Configuration Factors

To perform this, it is convenient to select a ‘rectangle’ (arbitrary), draw the spheres centered in each corner of the divisions (through the ‘Sphere’ command) and extrude ‘light cones’ based on the ‘rectangle’ and vertex positioned at the corners of the divisions (‘Extrude point’ command).

In the ensuing operation, we have to draw the intersections between the ‘light cones’ and the spheres, (Boundary representation vs. Boundary Representation: ‘Brep|Brep’ command), project these intersections on the selected plane (‘Project’ command) and calculate the areas involved (‘Area’ command). Finally, we normalize the results to obtain dimensionless configuration factors (Figure 9):

The result, as expected, is depicted below (Figure 10):

3.3.3. Exporting the Data to Excel and Representing them by Color Maps

To transfer the data to Excel^®, we first sort them using the ‘Partition List’ command and then the ‘GhExcel’ plugin that allows us to export an array of data to the said spreadsheet through the ‘ExcelWrite’ command, (Figure 11).

Finally, we have designed a ‘renderer’ which assigns to each value obtained a color of the spectrum to display the results, (Figure 12):

The outcome for such an algorithm is shown below (Figure 13).

3.3.4. Numerical Results

When exporting the values obtained by the Grasshopper^® algorithm to an Excel^® sheet, it is relatively easy to sort them in a matrix and the figures obtained are displayed in

Table 1. Configuration factors of an area of 10 × 10 m² illuminated by a ‘rectangle’ of 2 × 2 m² located 5 m from the ground at the middle point of the horizontal surface.

Distance in the X-Axis to the Rectangle’s Plane (m)
Distance in the Y-axis to the Rectangles Plane (m)		0	1	2	3	4	5	6	7	8	9	10
	Dimensionless Configuration Factor f12 (i.e., Divided by π)
	0	0.00000	0.00000	0.00000	0.00000	0.00000	0.00000	0.00000	0.00000	0.00000	0.00000	0.00000
	1	0.00203	0.00279	0.00371	0.00469	0.00546	0.00577	0.00546	0.00469	0.00371	0.00279	0.00203
	2	0.00368	0.00497	0.00650	0.00807	0.00930	0.00977	0.00930	0.00807	0.00650	0.00497	0.00368
	3	0.00473	0.00624	0.00796	0.00966	0.01096	0.01144	0.01096	0.00966	0.00796	0.00624	0.00473
	4	0.00519	0.00665	0.00825	0.00977	0.01090	0.0132	0.01090	0.00977	0.00825	0.00665	0.00519
	5	0.00517	0.00644	0.00778	0.00901	0.00989	0.01021	0.00989	0.00901	0.00778	0.00644	0.00517
	6	0.00486	0.00590	0.00695	0.00788	0.00852	0.00876	0.00852	0.00788	0.00695	0.00590	0.00486
	7	0.00440	0.00521	0.00600	0.00668	0.00715	0.00732	0.00715	0.00668	0.00600	0.00521	0.00440
	8	0.00389	0.00451	0.00510	0.00559	0.00592	0.00604	0.00592	0.00559	0.00510	0.00451	0.00389
	9	0.00339	0.00385	0.00429	0.00464	0.00488	0.00496	0.00488	0.00464	0.00429	0.00385	0.00339
	10	0.00293	0.00328	0.00360	0.00386	0.00402	0.00408	0.00402	0.00386	0.00360	0.00328	0.00293

.

In this manner, the form factor F₁₂ is obtained as the average of all the configuration factors f₁₂, which yields a value (for a discretization of 11 × 11 = 110 points) of F₁₂ = 0.00556888.

If we consider the point with the highest configuration factor (point 5,3), where f₁₂ = 0.01144, and we want to know the radiation of that point, assuming a radiant power in the rectangle of 1000 W/m², we will obtain this by simply multiplying the value of f₁₂ by 1000, that is, n = 11.44 W/m².

The array presented in Table 1 coincides with that found by the analytical procedures (

Table 2. Configuration factors of an area of 5 × 10 m² (the values that are omitted are obviously symmetrical) illuminated by a ‘rectangle’ of 2 × 2 m². Values obtained by analytical methods ([6], pp. 183–184).

Distance in the X-Axis to the Rectangle’s Plane (m)
Distance in the Y-axis to the Rectangle-s Plane (m)		5	6	7	8	9	10
	Dimensionless Configuration Factor f12
	0	0.00000000	0.00000000	0.00000000	0.00000000	0.00000000	0.00000000
	1	0.005765	0.00546298	0.00468552	0.00371248	0.00278731	0.00202982
	2	0.00976818	0.00929669	0.00806846	0.00649857	0.00496683	0.00367902
	3	0.01144492	0.01095576	0.00966096	0.00795806	0.00623622	0.00473341
	4	0.01131701	0.0108993	0.00977478	0.00824998	0.00664699	0.00518805
	5	0.01020997	0.00988871	0.00901007	0.00778368	0.00644494	0.0051747
	6	0.00875831	0.00852443	0.00787588	0.0069471	0.00589812	0.00486368
	7	0.0073163	0.00715037	0.0066848	0.00600327	0.0052104	0.00440123
	8	0.00603659	0.00591986	0.00558912	0.00509595	0.00450753	0.00388887
	9	0.00496121	0.00487895	0.004644	0.00428824	0.00385462	0.00338694
	10	0.00408173	0.00402333	0.00385542	0.00359789	0.00327834	0.00292612

) described in the references [6]. In this way the procedure is validated.

4. Discussion

4.1. Convergence

To test the convergence of the method we need to gradually decrease the size of the elements and show how, by doing so, the form factors converge towards a certain value which must be increasingly closer to that calculated analytically.

One of the advantages of our algorithm is that by simply modifying the value of the sliders to increase the partitions we obtain the new results rapidly, and proof of convergence is readily available.

In the ensuing figure (Figure 14), we found the results by raising the number of partitions or finite elements from 10 × 10 = 100 elements to 100 × 100 = 10,000 elements, with a computing time of only 136 s (conventional computer).

The following figure (Figure 15) shows the values obtained for the form factor by discretizing the surface of 10 × 10 m in 121, 441, 961, 1681, 2601, 3721, 5041, 6561, 8281, 10,201, 12,321, 14,641, 17,161, 19,881 and 22,801 points, respectively (the value obtained for 22,801 points is F₁₂ = 0.006165497), showing a clear convergence towards the value F₁₂ = 0.0062 analytically calculated [6].

4.2. Versatility of the Method

As noted above, the difficulties in obtaining analytical solutions for emitters with forms other than the trivial are enormous, and even in many cases impossible. However, the method presented here allows solutions with high accuracy, limited only by the performance of the computer, for any type of emitters and irrespective of the orientation.

As a more complex example, we have calculated the configuration factors on a 10 × 10 m horizontal surface, radiated by an elliptical emitter whose major and minor axes are 2 m and 1 m respectively and whose lowest point is located at a height of 5 m. As before, vertical axis of the ellipse coincides with the middle point of the horizontal surface in 5 m (Figure 16). Notice that, for the time being, the elliptical shape cannot be solved analytically due the appearance of elliptic integrals in the process [9].

Finally, we have simulated the configuration factors on a 10 × 10 m horizontal surface, given by an arbitrary emitter in the shape of a scalene triangle of the dimensions depicted in Figure 17. The graphic results appear in Figure 18 and the numeric values in

Table 3. Configuration factors of an area of 10 × 10 m² illuminated by a ‘scalene triangle’.

Distance in the X-Axis to the Triangle’s Plane (m)
Distance in the Y-axis to the Triangles Plane (m)		0	1	2	3	4	5	6	7	8	9	10
	Dimensionless Configuration Factor f12 (i.e., Divided by π)
	0	0.00000	0.00000	0.00000	0.00000	0.00000	0.00000	0.00000	0.00000	0.00000	0.00000	0.00000
	1	0.00329	0.00593	0.01103	0.02039	0.03471	0.04974	0.05761	0.05139	0.03030	0.01256	0.005371
	2	0.00554	0.00942	0.01614	0.02677	0.04031	0.05151	0.05363	0.04433	0.02858	0.01539	0.00799
	3	0.00642	0.01014	0.01581	0.02349	0.03169	0.03715	0.03682	0.03064	0.02171	0.01371	0.00825
	4	0.00627	0.00921	0.01318	0.01789	0.02229	0.02476	0.02415	0.02067	0.01578	0.01106	0.00740
	5	0.00559	0.00769	0.01027	0.01303	0.01535	0.01650	0.01604	0.01413	0.01141	0.00862	0.00623
	6	0.00474	0.00619	0.00782	0.00943	0.01069	0.01126	0.01097	0.00989	0.00833	0.00664	0.00509
	7	0.00392	0.00490	0.00593	0.00689	0.00761	0.00791	0.00773	0.00710	0.00618	0.00513	0.00411
	8	0.00321	0.00387	0.00453	0.00513	0.00554	0.00572	0.00560	0.00552	0.00465	0.00399	0.00332
	9	0.00262	0.00307	0.00351	0.00388	0.00414	0.00424	0.00416	0.00393	0.00357	0.00314	0.00268
	10	0.00214	0.00246	0.00275	0.00299	0.00315	0.00322	0.00316	0.00302	0.00278	0.00249	0.00218

Notice that, in the case of the ellipse and the scalene triangle unlike the rectangle, it is not possible nowadays to calculate the configuration factors by any other procedure.

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5. Conclusions and Future Aims

Given the accurate convergence of the values obtained for the particular case of rectangular emitters and receivers within a cuboid, we consider that the graphical interface program that we have proposed is fully accomplished. However, for the time being, we have to warn that there is no feasible way to validate the other figures presented, such as ellipse and scalene triangle.

Nevertheless, we believe that a complex physical-mathematical problem, which has occupied the mind of scholars for decades if not centuries, is solved with perfect ease in a semi-automatic manner.

Ensuing research will extend the procedure to other types of three-dimensional spaces where analytical calculations were proposed but still remain underdeveloped, for instance curvilinear sources in non-cuboidal domains affected by interferences and obstacles [9].

A much wider repertoire of possible radiative exchanges would be achieved in this fashion, with relative simplicity and accuracy and without alternative. The repercussions of such findings would be wide-ranging in fields as diverse as aerospace technologies [10], Light Emitting Diodes (LED), radiotherapy medicine, risk prevention, acoustics and architecture [11]. On an industrial basis, we consider that the vast field of artificial lighting and design of luminaries will immediately benefit from our methods.

Whereas the findings hereby exposed would represent a major achievement in Optics, precisely for the realms of Heat Transfer and Lighting, it is understood that our contribution has been timely aided by the enormous potential of Algorithms Aided Design in the numerical resolution of physical-mathematical problems that, when approached directly, are extremely difficult to accomplish (if not altogether impossible).