Deterministic Daylight Optimization of Rectangular Window Geometry: Extended Sky Factor Screening and Honeybee–Radiance Evaluation

Rodrigo N. Kaunang, Johansen C. Mandey, Pingkan P. Egam, Jefrey I. Kindangen^*

Author affiliations

Department of Architecture, Faculty of Engineering, Universitas Sam Ratulangi, Manado 95115, North Sulawesi, Indonesia

*Corresponding author.
rodrigokaunang022@student.unsrat.ac.id (R. N. Kaunang)
johansenmandey@unsrat.ac.id (J. C. Mandey)
epingkan@unsrat.ac.id (P. P. Egam)
jkindangen@unsrat.ac.id (J. I. Kindangen)

History: Received 17 July 2026 | Revised 20 August 2026 | Accepted 30 September 2026 | Published online 25 October 2026

Copyright: © 2026 The Author(s). Licensee Solarlits Limited (Hong Kong). This is an open access article distributed under the terms and conditions of the Creative Commons Attribution 4.0 License.

Citation: Rodrigo N. Kaunang, Johansen C. Mandey, Pingkan P. Egam, Jefrey I. Kindangen, Deterministic Daylight Optimization of Rectangular Window Geometry: Extended Sky Factor Screening and Honeybee–Radiance Evaluation, Journal of Daylighting, 13:2 (2026) 682-697. doi: 10.15627/jd.2026.37

Abstract

Optimizing classroom daylighting requires balancing daylight availability and spatial distribution while evaluating a potentially large number of window configurations. This study proposes a deterministic computational framework combining an extended Sky Factor formulation, exhaustive parametric enumeration, analytical screening, and Honeybee–Radiance evaluation. The framework was evaluated in a classroom at Universitas Sam Ratulangi, Manado, Indonesia. A calibrated digital model was used to examine 48,905 viable rectangular-window configurations defined on a 0.20 m discretization grid. Configurations satisfying the Indonesian National Standard (SNI 03-2396-2001) were ranked using the analytical Sky Factor criteria and four-point TAD, and the top 10% of compliant configurations (295 states) were subsequently evaluated using Honeybee–Radiance. The calibrated Honeybee–Radiance model showed a global MAPE of 10.24% against the available field measurements at three assessment points. The best-performing configuration among the 295 Radiance-evaluated candidates reduced the window-to-wall ratio from 46.20% to 23.68%, increased the average Daylight Factor from 0.63% to 1.35%, and reduced the daylight contrast ratio from 10.41:1 to 9.51:1 in the reported DIALux cross-check. The study demonstrates a transparent two-stage procedure that provides complete analytical coverage of a finite single-rectangle parameterization while limiting high-fidelity simulation to shortlisted candidates. The quantitative findings are restricted to the stated classroom geometry, optical boundary conditions, overcast-sky assumptions, and discretization.

Keywords

daylight optimization; daylight simulation; Honeybee–Radiance; rectangular window geometry

Nomenclature

1. Introduction

Daylight is a well-known important parameter of building performance, influencing visual comfort, health and well-being of the occupants, lighting energy demand, and the overall quality of the interior environment [1], [2], [3]. Unlike electric illumination, daylight is naturally dynamic: its spatial and temporal nature is controlled by solar position, sky condition, building orientation, and envelope geometry [4], [5]. Contemporary daylight design, therefore, attempts not only to provide sufficient illuminance but also to realize a balanced distribution that prevents excessive luminance contrast, enhances visual comfort, and lowers reliance on artificial lighting [6], [7]. This balance is especially important in educational buildings, as large spatial fluctuations of brightness can lead to reduced visual comfort, increased visual fatigue, and a lower quality of the learning environment. Recent classroom studies support this idea, demonstrating how façade and shading designs greatly influence the distribution of sunshine, with clear repercussions for occupants’ visual comfort and daylight quality [8].  

 Maintaining performance and reducing fatigue during activities such as reading, writing, drawing, and digital learning necessitates a stable and evenly distributed luminous field [9], [10], [11]. Consequently, optimizing the amount and spatial distribution of daylight has become a central focus in building performance research.

Recent developments in the area of computational building performance simulation have revolutionized the assessment of daylight in the architectural design process. Physics-based engines, especially Radiance, and modern interfaces like Honeybee may reliably estimate daylight performance prior to construction and thereby facilitate evidence-based decision-making [12], [13], [14]. When used with parametric environments such as Grasshopper, these technologies allow for the automatic generation and assessment of multiple design possibilities by changing geometric parameters in a systematic method [15], [16], [17]. Thus, daylight analysis has evolved from a post-design verification effort to an optimization activity that guides performance-driven design from the early stages of a project. 

The recent advancements in computational design have further accelerated this transition, replacing manual trial and error with automated optimization workflows [18]. The combination of parametric strategies and optimization algorithms has been broadly employed to advance daylight exposure, visual satisfaction, energy performance, and multi-objective design in buildings [16], [17], [19], [20], [21]. Within tropical classrooms, the integration of Honeybee–Radiance parametric optimization alongside sensitivity analysis and genetic algorithms has been shown to be successful in improving façade design and climate-adaptive daylight performance. Although effective, these heuristic strategies examine only a fraction of the solution space and may identify high-performing solutions without providing complete coverage of the predefined discretized space. Thus, the framework of the design space, the spread of plausible solutions, and the behavior of alternative configurations typically stay only somewhat grasped, even when the number of design variables is minimal enough to enable deterministic evaluation.

A methodological gap therefore remains for finite architectural daylight problems. Previous daylight simulation and optimization studies demonstrate the value of physics-based and parametric approaches, while the present study focuses on the complementary problem of transparent enumeration and analytical pre-screening. The novelty is not a claim that deterministic search is universally superior to heuristic optimization, but the combination of SNI-based analytical screening, exhaustive enumeration of a finite single-rectangle parameterization, and subsequent Honeybee–Radiance evaluation.

This research has four objectives: (i) formulate the SNI Sky Factor procedure as an automated computational expression for rectangular openings; (ii) calibrate the classroom simulation model against the available field measurements; (iii) quantify the reduction in the number of high-fidelity simulations obtained by analytical screening; and (iv) identify the best-performing configuration among the shortlisted Radiance-evaluated states. Complete coverage, therefore, applies only to the analytical discretized space; Honeybee–Radiance was applied to 295 shortlisted states. 

• The analytical sky factor equation is extended to encompass the investigated rectangular-window configurations.
• The entire discretized solution space is examined in a deterministic and comprehensive manner rather than being sampled by heuristic methods.
• A hierarchical analytical screening phase significantly decreases the quantity of expensive Honeybee–Radiance simulations while maintaining comprehensive coverage of the single-rectangle parameterization of the solution space.

To demonstrate the procedure, a classroom was examined using field measurements, digital-model calibration, deterministic solution-space generation, analytical daylight screening, and Honeybee–Radiance evaluation. The case is used to demonstrate the workflow under explicitly defined boundary conditions; its quantitative design result is not presented as a universal recommendation for other buildings or sky conditions. 

2. Computational framework and methodology

This research establishes a deterministic computational framework that integrates field measurement, analytical daylight assessment, parametric design investigation, and physics-based simulation into a unified optimization process. The process commences with the collection of field data and the calibration of digital models to establish dependable physical boundary conditions. Window geometry is thereafter expressed in parametric form, allowing for the comprehensive generation of the solution space through thorough enumeration. Every design state undergoes assessment using the enhanced Sky Factor formulation, while the retained candidates are subsequently evaluated using Honeybee–Radiance. This stratified approach supports reproducibility and traceability while adhering to the daytime performance standards of SNI 03-2396-2001.

2.1. Overview of the computational framework

The framework is sequentially developed in five stages: field measurement and digital-model calibration; parametric representation of the single-rectangle window; deterministic generation of the feasible discretized solution space; analytical SNI screening and four-point TAD ranking; and Honeybee–Radiance evaluation of the retained 10%. The sequence is therefore complete at the analytical stage but deliberately selective at the physics-based stage, as summarized in Figure 1.

Figure 1

Fig. 1. Deterministic workflow: field measurement and calibration → single-rectangle parameter definition → exhaustive enumeration (48,905 states) → SNI analytical screening (2,950) → four-point TAD ranking → 10% cutoff (295) → Honeybee–Radiance evaluation → final selection.

The first stage establishes the empirical basis of the computational model through field measurements and digital-model calibration. The second stage defines the parametric variables and feasible geometric range of the single-rectangle window. These variables are then discretized to generate the complete feasible solution space deterministically. In the fourth stage, all feasible configurations are screened against the applicable SNI daylight criteria and ranked using the four-point TAD metric. Only the retained 10% of configurations are subsequently evaluated using Honeybee–Radiance, thereby concentrating the higher-fidelity simulation on the most promising candidates.

Figure 1 summarizes this workflow from empirical input and model calibration through analytical solution-space generation and screening to the final physics-based evaluation. The framework therefore separates exhaustive analytical exploration from selective high-fidelity simulation, providing a systematic basis for identifying the best-performing window configuration.

2.2. Case definition and experimental data acquisition

The framework was tested in a typical classroom located on the third floor of the Architecture Building at Universitas Sam Ratulangi, Manado, Indonesia (1.4595° N, 124.82586° E; UTC+8), where the building's north is off true north by about 20°. The room has a depth of 10.26 m, a width of 7.60 m, and a height of 3.00 m. The space has three window modules facing south. The case study describes the geographical location, building layout, floor plan, and the specific classroom geometry used to construct the computational model as depicted in Figure 2. The model inputs to the framework are split into two groups: fixed boundary parameters (room geometry, material properties, orientation, and sky conditions) and variable design parameters (window shape) subjected to parametric optimization [9].

Figure 2

Fig. 2. Situation maps, building layouts, floor plans, and classroom geometry.

The chosen classroom is a standard teaching room with minimal external obstruction. It is used as the study case rather than as an independently established representative of all tropical classrooms. The quantitative conclusions are restricted to this classroom and the stated boundary conditions.

Readings were made on 11 June 2026 at 16:38 WITA under stable overcast conditions. The documented study record identifies a laser distance meter for geometry, a window tint meter for VLT, and a data-logging digital lux meter for interior illuminance. A separate simultaneous exterior-illuminance series, a quantified instrument uncertainty budget, and a recorded measurement duration are not contained in the available study record and are therefore not inferred. The measurements were used for model calibration rather than independent validation.

The digital model used the CIE standard cloudy sky to make the measured and simulated conditions constant [26], [27]. The building orientation, surrounding impediments, and optical qualities of the materials were kept constant so that any variation in daylight performance was due only to the window shape.  

These controlled settings ensured that the optimization segregated the effect of the geometric design factors from the external environmental effects.  

2.3. Computational model development

Once the physical boundary conditions were set, the classroom was turned into a computational model for the systematic assessment of alternate window configurations. The model combines geometric parameterization, analytical daylight evaluation, and deterministic solution space generation into one system. The classroom was not taken as a fixed architectural object, but rather the window opening was specified as a parametric design element that could vary geometrically while keeping all other boundary conditions constant. 

2.3.1. Digital model calibration

The reference classroom was rebuilt in a three-dimensional model using the Rhinoceros 8 software and the Grasshopper visual-programming environment. The model included the measured room geometry, glazing configuration, visible light transmittance (VLT), interior surface reflectance, building orientation, and the environmental boundary conditions given in Section 2.2. Throughout the analysis, all these were held constant, with the only design variable being the window geometry.

Two simulation environments fulfilled complementary purposes. The existing classroom was recreated using DIALux evo for comparison with the field measurements; Honeybee–Radiance was used as the main physics-based engine for the optimization.

The Daylight Factor (DF) was computed as the ratio of interior to exterior horizontal illuminance, given in Equation (1).  

The simulated DF values were then compared with the field measurements at the three principal assessment points using the Mean Absolute Percentage Error (MAPE), as in Equation (2).

The agreement was interpreted following Lewis [28] only as a descriptive error classification. Because the same three measured points were used for calibration/assessment, the reported MAPE is calibration agreement and is not an independent validation statistic.

2.3.2. Parametric representation of window geometry

Window geometry was parameterized through four independent variables defined on the façade plane: the left boundary (x₁), the right boundary (x₂), the sill height (z₁), and the head height (z₂). Together, these variables set the position, width, and height of a rectangular opening. Figure 3 shows the four coordinate axes used to describe the parametric aperture.

Figure 3

Fig. 3. The four coordinate axes (x₁, x₂, z₁, and z₂) defining the parametric aperture.

All other parameters, such as room geometry, façade orientation, glazing parameters, and reflectance of the material, as well as environmental boundary conditions, were kept constant. The installed glazing VLT of 22% and adopted interior reflectances of 0.75, 0.70, and 0.40 for the ceiling, walls, and floor, adopted from standard reference values [24], [25], were treated as fixed optical inputs. Thus, the baseline and selected configurations were intended to use identical optical properties in the simulation comparison; the analytical Sky Factor stage itself remains transmittance-independent.

To keep the solutions physically meaningful, the variables were bounded by the façade dimensions. The horizontal coordinates satisfy x1 < x2, and the vertical coordinates satisfy z1 < z2, which guarantees positive window width and height while confining every variable to the façade envelope. The sill height was additionally constrained to z1 ≥ 0.75 m, matching the SNI work plane height. Openings below this level were excluded because they add little to the sky component reaching the working plane. 

2.3.3. Transformation of the Sky Factor formulation into a computational model

The Sky Factor (SF) formulation of SNI 03-2396-2001 [23] estimates the availability of daylight based on the geometric relationship between the window opening and the interior evaluation point. The computational simplicity is attractive for large-scale parametric work. The original formulation, nonetheless, requires the evaluation point to be orthogonal to a designated corner of the effective opening. The application to arbitrarily positioned or elevated windows results in geometrical inconsistencies, which limit its application in systematic parametric analysis.

Figure 4 illustrates the methodological shift developed here. The methodology generalizes the original analytical strategy into a computational formulation for automatic calculation of the investigated rectangular window configurations. The physical principles of the Sky Factor method in SNI 03-2396-2001 remain unchanged. The transformation is carried out in four steps: (i) generalization of the original analytical formulation, (ii) classification of the geometric position of the evaluation point relative to the aperture, (iii) implementation of the procedure in Grasshopper, and (iv) deterministic evaluation of the full discretized solution space. This reformulation allows the analytical model to be used for the investigated admissible window configurations before high-fidelity Honeybee–Radiance evaluation.

Figure 4

Fig. 4. Transformation of the original Sky Factor formulation into a generalized computational engine for deterministic daylight optimization.

To overcome the original restriction, the analytical formulation was expanded to a generalized representation able to evaluate various rectangular geometries while keeping the principles of the SNI standard. Each aperture was divided into four ideal quadrants around the projection of the assessment point, following the process of division of the area described in Appendix A1.1 of SNI 03-2396-2001. The contribution of each quadrant was computed from the conventional trigonometric kernel (Equation (3)): 

where a1 and a2 are the horizontal distances from the evaluation point to the two window edges, h₁ and h₂ are the corresponding vertical distances to the sill and head, and Y is the depth of the evaluation point from the façade plane.

The total Sky Factor is obtained by superimposing the four quadrant contributions. Two formulations apply, depending on the horizontal position of the evaluation point relative to the aperture. When the point lies within the aperture span (x₁  xpt  x₂), Equation (4) is used; otherwise, Equation (5) applies: 

This adaptive formulation reproduces the SNI area partition technique while automatically allowing analysis of any window geometry. The algorithm, coded in Grasshopper, first finds the geometric relation between the evaluation point and the opening and then selects the proper calculation branch for deterministic analytical screening prior to the validation using Honeybee–Radiance.

For readers unfamiliar with the SNI convention, the Sky Factor (Faktor Langit) used here is a transmittance-independent geometric quantity: it represents the unobstructed sky component associated with the aperture geometry and does not itself include glazing transmittance. The Daylight Factor delivered through the installed glazing is lower and is obtained by applying the relevant SNI transmittance and maintenance corrections. For the measured classroom, 4.434% × 0.22 ≈ 0.975%, 2.866% × 0.22 ≈ 0.631%, and 3.008% × 0.22 ≈ 0.662%, illustrating why SF and DF must not be interpreted as the same quantity. Accordingly, statements about existing-case compliance refer to the geometric Sky Factor criterion and not to an equivalent delivered daylight level through the tinted glazing.

2.3.4. Deterministic generation of the feasible solution space

Each combination of the four design variables (x₁, x₂, z₁, and z₂) defines a unique window configuration. The optimization problem was therefore formulated as the exhaustive evaluation of a finite design space in which every admissible configuration could be explicitly generated and assessed. The admissible solution space S is defined as

where X₁, X₂, Z₁, and Z₂ denote the discrete domains of the four geometric variables. Each variable was discretized using a uniform interval of Δ = 0.20 m, resulting in X₁ = {0, 0.2, …, 7.4}, X₂ = {0.2, 0.4, …, 7.6}, Z₁ = {0.8, 1.0, …, 2.8}, and Z₂ = {1.0, 1.2, …, 3.0}.

Imposing the geometric constraints in Equation (6) left 48,905 valid design states in the implemented enumeration. The term "complete solution space" denotes the complete enumeration of the admissible single-rectangle parameterization at Δ = 0.20 m; it does not include the existing three-module topology and does not represent a continuous geometric domain. 

2.4. Optimization framework and performance assessment

Following the establishment of the deterministic solution space, all acceptable design states were analyzed under consistent physical boundary conditions using a tiered screening methodology. Rather than continuously generating candidates, the optimization engaged with the entire collection of permissible configurations and gradually reduced it, beginning with analytical assessment and later with physics-driven simulation. This sequence designated the challenging Honeybee–Radiance assessment for the most favorable designs.

As illustrated in Figure 5, the evaluation of daylight performance was conducted at the four locations outlined in SNI 03-2396-2001. Three primary elements were utilized for adherence: the Central Point (TUU) and two Auxiliary Points (TUS1 and TUS2). An extra control point in front of the whiteboard (Tdpt) was incorporated, as mandated by the SNI for educational structures.

Figure 5

Fig. 5. Assessment points used for analytical daylight evaluation.

Initially, each design state underwent assessment using the extended Sky Factor formulation (Equations (3)–(5)) and was scrutinized for compliance with the daylight standards of SNI 03-2396-2001 [23]. For a classroom depth of d = 10.26 m, the minimum Sky Factor stipulations were 3.591% (0.35d) at the Main Point (TUU) and 2.052% (0.20d) at each Side Point (TUS1 and TUS2). In the SNI convention, the minimum sky factor is calculated by multiplying a depth coefficient by the room's depth in meters, resulting in a percentage; the whole room depth (d = 10.26 m) was consistently applied. The control point (Tdpt), established at 0.5 × TUU, facilitated the preliminary evaluation of classroom configurations. Each arrangement was mandated to maintain the daylight contrast ratio at or below 40:1.

Configurations that met these requirements were then ranked by daylight distribution homogeneity using the Total Absolute Deviation (TAD) in Equation (8), computed relative to the mean sky factor in Equation (7). 

where lower TAD values signal a more uniform distribution of the sky component across the classroom.

Because running Honeybee–Radiance for every compliant configuration was outside the available computational scope, a linear cutoff was applied. The compliant population was sorted by the four-point analytical TAD, and only the best 10% were retained. This cutoff is a methodological resource constraint, not an independently optimized value (Equation (9)): 

where N is the total number of compliant configurations, and k is the number carried forward for detailed simulation.

For the retained configurations, the final 234-point TAD was calculated from the simulated work-plane daylight field. In this physics-based stage, the quantity is a daylight factor field rather than the bare-aperture sky factor. The manuscript, therefore, distinguishes the four-point analytical SF-TAD used for screening from the 234-point Radiance DF-TAD used for final ranking. 

While daylight uniformity was evaluated using the daylight contrast ratio (Equation (11)),

The 234-point Radiance DF-TAD (Equation (10)) functioned as the final ranking criterion among the 295 retained candidates, while the four-point SF-TAD was used only for the 10% screening step. No claim is made that the screening objective is a monotone surrogate for the final Radiance objective.  

3. Results

3.1. Digital model calibration

The calibration agreement of the digital model was assessed by comparing simulated Daylight Factor (DF) values with the field measurements at the three principal assessment points, where the measured DF values were 0.899% at TUU, 0.663% at TUS1, and 0.835% at TUS2. Table 1 sets out the comparison between the field data and the two simulation engines.

Table 1

Table 1. Comparison of measured and simulated daylight factor values used for digital model calibration.

At specific locations, Honeybee–Radiance yielded MAPE values of up to 12.46%, with one of the three sites falling below 10%, indicating consistent agreement across the measurement points. The global MAPE of 10.24% categorizes it as satisfactory according to the Lewis classification, while DIALux evo produced a global MAPE of 20.19%, placing it in the reasonable group. Both models remained much below the 50% error threshold, indicating agreement with the recorded daylight conditions.

Despite both settings aligning adequately with the field data, Honeybee–Radiance consistently yielded reduced total errors and was hence selected as the principal engine for optimization. This result aligns with the physically grounded backward ray-tracing algorithm in Radiance, which comprehensively accounts for inter-reflections and inter-surface optical interactions compared to simplified radiosity approaches [12], [13]. The adjusted Honeybee–Radiance model subsequently served as the benchmark environment for assessing various window configurations during the optimization process.  

3.2. Deterministic solution-space generation and analytical screening

The parameterization of the four design variables yielded a finite and completely enumerated single-rectangle solution space at a discretization interval of Δ = 0.20 m. This procedure generated 48,905 feasible design states, allowing the entire solution space to be assessed systematically at the analytical stage.

Table 2 summarizes the daylight performance criteria derived from SNI 03-2396-2001 alongside the measured Sky Factor values of the existing classroom. The three primary criteria considered in this study exceeded their respective minimum Sky Factor requirements, indicating that the existing classroom satisfied the minimum daylight conditions prior to optimization. However, compliance with minimum requirements does not necessarily indicate the most uniform spatial distribution of daylight. This distinction provided the basis for the subsequent optimization and ranking of alternative window configurations.

Table 2

Table 2. Daylight performance criteria for analytical screening (SNI 03-2396-2001) and compliance of the existing classroom.

The analytical assessment was therefore performed for all 48,905 feasible states, whereas the Honeybee–Radiance evaluation was reserved for a substantially smaller set of shortlisted configurations. As illustrated in Figure 6, the complete set of 48,905 feasible configurations was first subjected to analytical SNI screening and TAD ranking, after which the top 10% of the complying configurations, 295 states in total, were retained for physics-based evaluation using Honeybee–Radiance. These 295 configurations correspond to approximately 0.60% of the complete solution space. The resulting reduction in the number of high-fidelity simulations is examined quantitatively in the next section.

Figure 6

Fig. 6. Hierarchical reduction of the deterministic solution space and candidate selection workflow.

3.3. Reduction in high-fidelity simulation count

Table 3 demonstrates the efficacy of the method by depicting the reduction of the candidate set during the optimization process. Rather than executing Honeybee–Radiance on all permissible states, the enhanced Sky Factor formulation was initially utilized to eliminate geometrically viable configurations that did not meet the daylight criteria. Only the configurations that adhered to analytical standards advanced to physics-based simulation.

Table 3

Table 3. Progressive reduction of candidate configurations during the hierarchical optimization procedure.

The deterministic solution space commenced with 48,905 permissible combinations. Analytical evaluation narrowed this down to 2,950 that met the SNI standards. Classifying them according to Total Absolute Deviation (TAD) reduced the selection to 295, representing merely 0.60% of the entire spectrum; hence, 99.40% of the permissible arrangements were discarded before initiating any Honeybee–Radiance execution.

Assessing each permissible configuration explicitly in Honeybee–Radiance would have increased the number of runs by approximately 166 times. The hierarchical method significantly reduced the number of high-fidelity simulations while maintaining complete analytical coverage of the prescribed single-rectangle design space. In this procedure, the enhanced Sky Factor formulation serves both as a daylight evaluation technique and as a pre-screening instrument that reduces the necessity for expensive physics-based simulations.

3.4. Physics-based evaluation of the selected design states

The 295 configurations selected following analytical screening were evaluated in Honeybee–Radiance to assess their daytime efficacy through physics-based simulation. In contrast to the analytical Sky Factor, which assesses performance at certain intervals, Honeybee–Radiance calculates the illuminance distribution throughout the entire work plane, considering inter-reflections, material optical characteristics, and three-dimensional spatial geometry.

The simulation outcomes were broadly consistent with the analytical screening for the retained candidates. This supports the use of the analytical stage as a screening procedure; it does not establish monotonicity or rank equivalence between the four-point analytical TAD and the 234-point Radiance objective across discarded states.

Table 3 illustrates the staged reduction of the finite design space. Out of the initial 48,905 permissible configurations, analytical filtration yielded 2,950 solutions compliant with SNI; subsequent application of the four-point analytical TAD ranking retained 295 candidates, all of which underwent assessment in Honeybee–Radiance. The final stage identified one best-performing candidate among these 295 Radiance-evaluated states.

3.5. Identification of the selected design state

After physics-based evaluation of the 295 retained configurations, the best-performing candidate within that shortlisted set was identified as Design State #2431, defined by the parameter vector (x1, x2, z1, z2) = (1.4, 6.8, 2.0, 3.0) m. This corresponds to a single rectangular aperture 5.4 m wide and 1.0 m high with a sill at 2.0 m, giving a Window-to-Wall Ratio of 23.68%. Among the Radiance-evaluated candidates, it returned the lowest 234-point Radiance DF-TAD (331.288), indicating the most uniform simulated daylight distribution within that shortlist.

It was subsequently assessed in DIALux evo as a supporting cross-check. DIALux is not treated as independent empirical validation.

Although the existing aperture satisfied the geometric Sky Factor screening criteria of Table 2, the selected single-rectangle configuration reduced the glazed area while improving the simulated daylight indicators reported in the DIALux cross-check. As summarized in Table 4, the Window-to-Wall Ratio decreased from 46.20% to 23.68%, the average daylight factor increased from 0.63% to 1.35%, and the daylight contrast ratio decreased by 8.65%, from 10.41:1 to 9.51:1.

Figure 7 depicts the selected design state. In conjunction with the geometry of the chosen window, it presents the daylight distribution from the supporting DIALux evo cross-check together with the key performance metrics reported for the selected configuration.

Figure 7

Fig. 7. Hierarchical reduction of the deterministic solution space and candidate selection workflow.

Table 4

Table 4. Comparison of the existing three-module classroom aperture and the selected single-rectangle design using the reported DIALux cross-check.

All permissible design states at the specified discretization (Δ = 0.20 m) were analytically enumerated and screened. The selected design state, however, represents the best-performing candidate among the 295 states subsequently evaluated using Honeybee–Radiance; the final Radiance objective was not evaluated for the full compliant population.

4. Discussion

4.1. Interpretation of the proposed computational framework

The primary significance of this research is methodological rather than the optimization of a single classroom window. The framework explicitly constructs a finite single-rectangle design space and analytically evaluates every admissible state at the selected discretization before the high-fidelity evaluation of the retained candidates. The selected result is therefore conditional on the predefined space and shortlisted Radiance set, not a universal global optimum.

A central feature of the technique is the combination of analytical screening and physics-based simulation. In the present case, analytical screening reduced the number of high-fidelity evaluation states from 48,905 to 295, a 99.40% reduction in simulation count. This is not a measured wall-clock saving and does not establish lower computational time than surrogate-based or evolutionary methods.

Although the study is demonstrated using a classroom in Manado, Indonesia, its methodological focus is the computational workflow rather than the particular classroom as a universal reference case. The framework combines parametric representation, deterministic solution-space generation, analytical screening, and physics-based evaluation. Transfer to other building types would require building-specific parameterization and validation.  

4.2. Comparative analysis with existing optimization approaches

Computational optimization has become a well-established approach for daylight-focused design, with several research studies combining parametric modeling with physics-based simulation to increase daylight availability, visual comfort, and overall performance [16], [17], [19], [29]. Most utilize evolutionary algorithms, multi-objective optimization, or swarm-based search, as these methods are capable of effectively exploring extensive nonlinear design spaces while juggling competing objectives [30], [31], [32]. Hence, heuristic optimization has emerged as the dominating approach in this discipline.

It has been demonstrated in the Journal of Daylighting that Honeybee–Radiance-based parametric optimization, along with sensitivity analysis and genetic algorithms, may greatly improve tropical classrooms’ daylight performance using façade design alterations [22]. Multi-objective optimization has also been effectively applied to daylight-responsive architecture across a range of climates [18]. These studies show the usefulness of heuristic optimization to identify effective solutions. However, their search is still iterative, and so only a subset of possible configurations is considered.

The methodology developed here has a different aim. Instead of accelerating the search for high-performing solutions, it characterizes a finite, precisely specified design space through deterministic analytical evaluation. The analytical screening stage covers all design states allowed by the specified discretization before a shortlist is created for the physics-based evaluation. The final Radiance objective is applied only to the shortlisted states rather than to the full compliant population. This distinguishes the present approach from iterative heuristic search without establishing computational superiority over such methods.

Each admissible state is generated and evaluated at the analytical screening stage, providing a transparent representation of the prescribed finite design space. The framework therefore characterizes the admissible states without relying on a stochastic search trajectory. However, the present study does not establish variable-wise sensitivity or the location of a global radiance optimum across the full compliant population.

The deterministic technique has three methodological characteristics relevant to the present problem. First, it provides complete analytical coverage of the prescribed discretized design space rather than a sampled subset of it. Second, every enumerated geometric configuration remains directly traceable to its analytical screening outcome, so the selection path can be reconstructed without reference to a stochastic search trajectory. Third, the analytical stage reduces the number of configurations requiring high-fidelity simulation, although this is a reduction in simulation count and not a measured runtime advantage. These characteristics describe the scope of the procedure and are not presented as evidence of superiority over heuristic optimization.

Table 5 summarizes representative daylight optimization studies using genetic algorithms, sensitivity analysis, multi-objective optimization, and parametric assessment. The proposed framework differs in that it exhaustively enumerates the prescribed finite design space analytically and then applies Honeybee–Radiance only to the shortlisted states. It therefore provides complete analytical coverage of the specified parameterization but does not establish a global Radiance optimum across the full compliant population.

This corresponds to a simulation-count ratio of 165.8 (48,905/295). It should not be interpreted as a measured 165.8-fold runtime saving, because wall-clock timings and hardware specifications were not recorded.  

Table 5

Table 5. Positioning of the proposed deterministic framework relative to representative daylight optimization studies.

4.3. Computational architecture of the deterministic search framework

The framework generates and analytically evaluates all admissible design states at the selected discretization, then applies physics-based evaluation to the retained shortlist. The selected state therefore represents the best-performing candidate among the Radiance-evaluated shortlist rather than a global Radiance optimum of the full compliant population. This architecture can be reused for deterministic exploration of other finite architectural design problems, subject to problem-specific parameterization and validation.

While the methodology is described as a sequential workflow, the design is also modular. It separates problem-invariant analytical components from application-dependent inputs and optional assessment modules, rather than providing a problem-specific implementation. This division makes it possible to separate the reusable computing core from parts tailored to distinct lighting optimization issues. The final architecture is shown in Figure 8.

Figure 8

Fig. 8. Modular computational architecture of the proposed deterministic optimization framework.

Computationally, the approach is divided into three functional levels, each having a specific job. The first is a core layer, consisting of the deterministic solution-space generator (Equation (6)) and the extended Sky Factor formulation (Equations (3)–(5)). Such components are the analytical basis and are independent of building geometry and of the optimization issue; their role is to produce the whole acceptable solution space and to evaluate each option based on a consistent formulation.

The second layer is an input layer, which consists of the project-specific characteristics that specify the problem: room geometry, discretization interval (Δ), daylight performance criteria, and other user-defined boundary conditions. They can be adapted to other building typologies or aims, but they fix the admissible search domain and therefore the extent of the deterministic solution space.

The third is a modular assessment layer with optional components improving practical efficiency without changing the core search. The current implementation involves the SNI compliance filter and the TAD-based candidate reduction in Equation (9). These modules preserve the deterministic strategy and reduce the number of configurations to be simulated by Honeybee–Radiance. As these modules are independent of the analytical core, they can be adjusted, replaced, or omitted depending on the available resources or alternative performance criteria.

This layered design separates the invariant computational principles from the application-specific inputs and optional modules so that the system can be extended to alternate daylight measurements or other building-performance evaluation methodologies. Therefore, this approach may be applied to other daytime optimization issues with the change of input parameters or the evaluation criteria without any change of the deterministic search process. The same architecture can be used to progressively loosen the modular filtering layer as more computational resources become available until all admissible states are directly evaluated in Honeybee–Radiance.

Climate-based daylight measures like spatial Daylight Autonomy (sDA), Annual Sunlight Exposure (ASE), and Useful Daylight Illuminance (UDI) have become more and more significant for a thorough assessment. However, the purpose of the present study is fundamentally different from climate-based evaluation. Here the extended Sky Factor formulation is not used to replace such metrics but rather as a deterministic analytical engine that examines each admissible state at negligible cost. The Sky Factor provides a pre-screening strategy in the layered structure, reducing the number of candidates that need to be run through an exhaustive Honeybee–Radiance simulation. The methodology combines the practical economy and physical interpretability of analytical evaluation with the greater fidelity provided by Radiance-based simulation. Future enhancements might incorporate climate-based indicators in the last step of the simulation while maintaining the deterministic screening technique proposed here. Hence, the approach should be understood as a deterministic optimization methodology and not as a replacement for climate-based daylight evaluation. 

4.4. Scientific contributions and methodological implications

The study contributes to daylight optimization in three methodological aspects. First, it frames the investigated problem as deterministic computational modelling for a finite, explicitly defined solution space. Exhaustive analytical enumeration provides complete coverage of the prescribed states rather than sampling them through a heuristic search. The present study does not establish rank equivalence between the analytical screening objective and the final Radiance objective across the discarded states; therefore, the selected state is not claimed as a global optimum of the full compliant population.

Second, the study implements the analytical Sky Factor formulation as a computational model for the investigated rectangular-window geometries. The method retains the stated geometric principles of SNI 03-2396-2001 while using an adaptive geometric decomposition to support systematic parametric evaluation.

Third, it couples analytical screening with physics-based modelling in a layered workflow. The analytical evaluation reduces the candidate population before the Honeybee–Radiance evaluation, thereby reducing the number of high-fidelity simulations required. This study demonstrates a reduction in simulation count, not a measured wall-clock computational advantage over alternative optimization methods.

Among the 295 Radiance-evaluated candidates, the selected configuration reduced the Window-to-Wall Ratio (WWR) from 46.20% to 23.68%, increased the average Daylight Factor from 0.63% to 1.35%, and decreased the daylight contrast ratio from 10.41:1 to 9.51:1. These results are consistent with prior studies on the effect of window shape on indoor daylight distribution [1], [8], [33],[34] and demonstrate the potential of deterministic exploration within the prescribed finite design space.

The reported distribution is associated with the joint effect of aperture position and aperture proportion rather than glazed area alone. In the selected configuration, the sill was raised from 1.0 m to 2.0 m, and the opening adopted a wider horizontal proportion while the total glazed area was reduced by approximately half.

The higher opening may alter the path of diffuse daylight and interaction with the ceiling, but this physical explanation was not independently isolated by the present study and is therefore treated as an interpretation rather than a demonstrated causal mechanism.

Thermal performance and solar heat-gain effects were outside the scope of this study, and no thermal co-benefit is claimed.

4.5. Practical implications for performance-based architectural design

This methodology may be useful in early-stage design when window geometry can be represented as a finite set of admissible configurations. Rather than relying solely on manual trial and error, the deterministic approach analytically screens the prescribed configurations before high-fidelity evaluation. The resulting daylight metrics can therefore be incorporated into performance-based design from an early stage.

A pragmatic way to limit the number of high-fidelity simulations is to combine analytical screening with physics-based evaluation. The proposed methodology filters candidates before the Honeybee–Radiance evaluation while retaining physics-based assessment for the shortlisted states. The present study does not quantify wall-clock savings or establish a runtime advantage over alternative optimization methods.

The findings are demonstrated in one classroom only. The computational procedure may be adapted to other daylight-design problems, but such transfer requires new building-specific definitions and validation.

4.6. Limitations and future research

The study is limited to a single classroom, three-point calibration, a single documented measurement condition, a finite single-rectangle parameterization, and a 0.20 m discretization. The available measurements do not constitute an independent validation dataset. The quantitative result should therefore be interpreted only within these stated conditions.

The deterministic strategy is best suited to problems that can be formulated as finite, explicitly discretized solution spaces. The present study does not establish how the method compares in elapsed time with evolutionary or surrogate-based optimization, because no common runtime experiment was performed.

The analytical formulation is based on the Sky Factor method and therefore represents a steady overcast-sky geometric screening condition. Annual daylight availability, direct-sun exposure, glare, view-out, and thermal performance were not assessed.

Several aspects remain outside the present evidence base. Annual daylight metrics, direct-sun exposure, glare, and thermal performance were not calculated. The 2.0 m sill can also affect seated view-out, ventilation interaction, and classroom use. These issues should therefore be treated as design constraints for future application rather than benefits established by this study.

The computational architecture may be applicable to offices, libraries, healthcare facilities, and other daylighted buildings when their design problem can be represented as a finite set of admissible window states. Such transfer is conditional: room depth, façade configuration and orientation, visual tasks, daylight criteria, occupancy/use conditions, and relevant constraints must be redefined for each building type. The present classroom case does not provide empirical validation of those transfers. Where the design space is too large for complete enumeration to remain tractable, the analytical screening stage may instead serve as a pre-processing step ahead of surrogate modelling or heuristic optimization.

5. Conclusion

This study developed a deterministic computational framework for daylight optimization that integrates field measurement, parametric geometric modelling, an extended analytical Sky Factor formulation, exhaustive enumeration of a finite single-rectangle design space, and Honeybee–Radiance evaluation. The framework provides complete analytical coverage of the admissible discretized design space at Δ = 0.20 m, while high-fidelity simulation is applied selectively to the shortlisted candidates. The quantitative findings are conditional on the classroom geometry, optical properties, overcast-sky conditions, and discretization adopted in this study.

Within the two-stage procedure, 48,905 admissible single-rectangle configurations were analytically enumerated, 2,950 configurations satisfied the specified analytical criteria, and the top 10% (295 configurations) were evaluated using Honeybee–Radiance. The best-performing candidate among these 295 Radiance-evaluated states reduced the Window-to-Wall Ratio from 46.20% to 23.68%, increased the average Daylight Factor from 0.63% to 1.35%, and reduced the reported daylight contrast ratio from 10.41:1 to 9.51:1. Because the final Radiance objective was not evaluated for all 2,950 compliant states, the selected configuration is not claimed as a global optimum for the full compliant population.

The principal contribution of the study is the demonstration of a transparent deterministic workflow in which an analytical daylight formulation is used to screen a finite architectural design space before high-fidelity simulation. The approach provides complete traceability of the analytically enumerated states and substantially reduces the number of configurations requiring physics-based evaluation. This reduction should be interpreted as a reduction in simulation count rather than as a demonstrated wall-clock computational advantage over heuristic or surrogate-based optimization.

The study is limited to a single classroom, a three-point calibration dataset without independent validation, a 0.20 m discretization, and the stated overcast-sky conditions. Annual daylight performance, direct-sun exposure, glare, view-out, thermal performance, discretization sensitivity, and runtime comparison with alternative optimization methods were not assessed. Future work may address these aspects and examine the transferability of the framework to other building types and larger design spaces. 

Funding

This research received no external funding.

Author Contributions

Conceptualization, Methodology, Formal Analysis, Data Curation, and Writing – Original Draft, R.N.K. and J.I.K.; Software, R.N.K. and J.C.M.; Investigation, Supervision, Validation, and Writing – Review and Editing, J.I.K.; Project administration, and Writing – Review and Editing, P.P.E. and J.C.M. All authors have read and agreed to the published version of the manuscript.

Acknowledgement

The authors gratefully acknowledge the Department of Architecture, Faculty of Engineering, Universitas Sam Ratulangi, for providing access to the classroom facilities and measurement equipment used in this study.

Declaration of competing interest

The authors declares no conflicts of interest.

Data availability

The enumerated solution space, the Honeybee–Radiance results for the 295 shortlisted configurations, the field-measurement record, and the Grasshopper definition implementing Equations (3)–(6) are available from the corresponding author on reasonable request.

During the preparation of this work, the authors used Claude generative AI tools solely to improve readability and to detect spelling and grammatical errors. After using these tools, the authors reviewed and edited the content as needed and take full responsibility for the content of the publication.

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