Towards Sustainable Design: A Shape Optimization Framework for Climate-Adaptive Free-Form Roofs in Hot Regions

Abstract

This study proposes a cross-disciplinary computational framework to advance the sustainable design of free-form grid roofs in hot climates, integrating architectural geometry with building thermal performance to enhance climate adaptability. Numerical analyses systematically explore the impact of thermal objectives, initial configurations, shape control strategies, and boundary constraints. The optimization results demonstrate that targeting indoor temperature under extreme heat yields saddle-shaped, self-shading morphologies, which achieve a measurable improvement in thermal comfort by reducing indoor temperatures by approximately 2 °C. A key practical finding is that symmetric-point control outperforms full-point control. While full-point control may generate forms with complex central depressions that complicate drainage, symmetric-point control consistently yields morphologies that are inherently more regular, symmetric, and constructible. This results in a superior balance among thermal performance, practical design attributes (e.g., drainage feasibility and construction simplicity), and geometric coherence—a combination that aligns closely with real-world engineering requirements. Furthermore, directional boundary constraints are proven to be effective tools for regulating passive shading performance. The proposed framework provides engineers and designers with a systematic and automated method for the climate-responsive and low-carbon design of free-form architectural morphologies, contributing to the development of more sustainable and resilient building infrastructure.

1. Introduction

Against the backdrop of pressing global energy and environmental concerns, the principles of green architecture and energy-efficient design have become central to advancing sustainable construction [1]. This is particularly critical for large-span public buildings, which are significant energy consumers. Free-form grid structures, valued for their visual dynamism and geometric adaptability, are increasingly featured in landmark large-span projects worldwide, including the Ottawa Light Rail Transit [2], Beijing Daxing International Airport [3], and the Court Roof of the British Museum [4]. However, public buildings account for a significant portion of energy consumption in the built environment. Consequently, a primary design challenge is to successfully integrate the compelling aesthetic qualities of these forms with stringent requirements for energy conservation.

Conventional building practices primarily rely on passive design strategies [5] and optimization of building envelope materials to enhance thermal performance. Commonly adopted passive measures include insulation layer optimization [6,7] and implementation of natural ventilation systems. Mushtaha [8] employed the Climate Consultant climate analysis tool and IESVE building energy simulation software to evaluate the impact of passive design strategies on building cooling loads under temperate climatic conditions, demonstrating that integrated optimization of natural ventilation, shading devices, and thermal insulation can effectively reduce energy consumption. However, solely relying on active energy systems (e.g., HVAC) often results in limited improvements in energy efficiency. Fachinotti [9] introduced an optimization approach based on thermal metamaterials for building envelopes, where material parameters serve as design variables to minimize energy consumption. Anand [10] revealed that increasing solar reflectance offers superior cooling performance compared to enhancing thermal emittance. For free-form geometries, which inherently possess high thermal emittance (typically ε > 0.9), this principle directs the primary design focus toward managing solar gain through geometric manipulation. Therefore, embedding passive thermal strategies—specifically, the performance-driven optimization of architectural form—into the early structural design phase emerges as a critical methodology. This integrated approach promises to significantly reduce operational energy demand and lifecycle costs, advancing a more efficient and sustainable design paradigm.

Current research on thermal performance optimization primarily focuses on conventional rectangular buildings (e.g., offices and residential buildings), emphasizing the impact of solar heat gain on the thermal environment through building envelopes. Martins [11] investigated urban-scale energy-saving design strategies by simulating solar heat gain and adjusting plan layouts such as building spacing, aiming to optimize envelope irradiation conditions and derive energy-efficient architectural solutions. Shen [12] optimized solar heat gains by modifying design parameters, including window size, location, material properties, and room dimensions. Bizjak [13] systematically mapped the influence of building plan geometry on solar heat gain by parametrically adjusting key factors like orientation. This analysis enabled them to establish optimal design values tailored to each form type, based on quantified solar exposure thresholds. Zhang Ran [14] performed energy simulations for office buildings in cold climates, establishing quantitative correlations between key morphological variables—including width, depth, and orientation—and overall energy use. Their work yielded optimal passive design configurations for such contexts. While these studies offer valuable guidance for conventional structures, they offer limited direct applicability to free-form geometries. Notably, research has also been conducted on optimizing multi-dimensional facade forms and shading systems, frequently employing multi-objective evolutionary algorithms to balance visual comfort and energy performance [15,16]. However, such work typically focuses on optimizing attached shading elements or facade patterns, rather than treating the primary roof or building envelope form itself as the integrated, performance-driven design variable. In contrast, this study addresses a distinct niche. It employs a single, integrative thermal objective (indoor temperature) to drive the global shape optimization of the free-form roof structure itself. This approach systematically explores how the fundamental geometric control logic (governed by constraints and node strategies) dictates both thermal performance and formal outcomes. Consequently, a design strategy focused narrowly on minimizing solar heat gain for such forms poses a distinct risk: it may force an excessive reduction in envelope area, thereby detrimentally impacting both architectural identity and practical performance.

The development of performance-driven free-form geometries necessitates evaluation metrics that simultaneously address solar radiation effects and geometric variations. In this study, indoor temperature is adopted as one such metric, serving as a direct proxy for thermal comfort. This approach resonates with the growing body of research connecting architectural design to environmental outcomes. For example, research by Li [17] formulated the Physiological Equivalent Temperature-Based Climate Index (PET_CI), which synthesizes various architectural and climatic parameters to investigate the relationship between urban form and outdoor thermal comfort in residential areas. This analysis pinpointed building density as the predominant driver of PET_CI, from which specific design guidelines for thermal comfort improvement were derived. Similarly, Pieskä [18] evaluated two geothermal high-temperature cooling systems in a Mediterranean context, prioritizing thermal comfort as a core performance criterion. This comparison demonstrated equivalent comfort provision by both systems but disclosed significant divergences in their carbon footprint and power demand, thereby underscoring the necessity for a holistic assessment of energy efficiency.

Existing studies have mainly concentrated on the thermal performance optimization of conventional building envelope systems and the design of thermal comfort in building clusters. In contrast, studies on free-form surface roofs remain largely confined to material-level improvements of envelope components, overlooking the potential of free-form geometry itself in regulating the thermal environment. However, under extreme climatic conditions, optimizing architectural form to improve thermal performance—while balancing aesthetic, functional, and thermal requirements—remains a critical challenge. To address this gap, this study establishes a shape optimization framework and aims to answer the following key questions: RQ-① What is a more effective optimization objective for free-form roofs in hot climates: minimizing indoor temperature or minimizing solar heat gain? RQ-② Which shape control strategy—full-point or symmetric-point control—offers a better balance between thermal performance, architectural quality, and practical constructability? RQ-③ How can directional geometric boundary constraints be utilized as design tools to regulate the final roof morphology for enhanced passive shading? Based on fulfilling functional requirements, the approach aims to enhance the thermal performance of single-layer free-form grid roofs, with its effectiveness demonstrated through numerical example studies.

Thus, the principal contributions of this work are threefold: (1) introducing indoor temperature as a primary, integrative driver for free-form roof shape optimization in hot climates; (2) systematically deconstructing the effects of key design factors (objectives, controls, constraints) on the thermal and morphological outcomes through a dedicated computational framework; and (3) translating these findings into practical geometric strategies that reconcile thermal performance with architectural and constructional feasibility.

Building upon the thermal–structural optimization foundation established in our prior work [19], this study extends the research scope by focusing exclusively on a thermal-performance-driven shape optimization framework. This paper is organized as follows: Section 2 details the selection of objective functions. Section 3 presents the computational framework and the mathematical formulation of the optimization methodology used for validation. Section 4 investigates through numerical examples: ① the impact of different thermal performance objectives on shape optimization patterns of free-form surface roofs; ② shape optimization behaviors under varying initial geometries and thermal operating conditions; ③ morphological variations of free-form roofs under distinct control strategies for thermal performance enhancement; and ④ shape adaptations subject to geometric boundary constraints in different spatial directions. Conclusions are summarized in Section 5.

2. Materials and Methods

2.1. Objective Function Formulation

2.1.1. Regional Selection

Building upon our previous research on the thermal performance of roof morphology [19], this study continues to investigate the influence of free-form surface roof geometry on the roof thermal environment. To this end, Haikou—a representative city in the hot-summer and warm-winter climate area—was selected as the example study region. Located in a low-latitude area, Haikou experiences long and evenly distributed daylight hours throughout the year, with nearly direct solar incidence and high radiation intensity during winter. This contrasts sharply with regions in northern China, where daylight duration is long in summer but significantly reduced in winter, accompanied by oblique solar angles and lower radiation levels. These climatic characteristics make Haikou particularly suitable for studying the thermal performance of free-form roofs under realistic conditions. The simulation was conducted using the EnergyPlus 23.1.0 Weather File for the year 2024 as the meteorological data source, with thermal performance analyzed through EnergyPlus-based computational modeling.

2.1.2. Solar Heat Gain

Conventional optimization objectives related to solar exposure in architectural design typically focus on the solar heat gain of the building envelope surface. According to the principles of solar heat gain calculation [20], the amount of solar radiation absorbed by a building is closely related to two geometric factors: the area of the exterior surfaces exposed to sunlight and their orientation relative to the sun’s rays. Given that this study focuses on non-standard architectural forms, and to control variables while minimizing the influence of other external factors on solar heat gain, the scope of analysis is limited to opaque building envelopes. Based on this assumption, the solar heat gain for opaque roof envelopes can be formulated as follows:

$$Q_{solar} =I\times A\times \alpha$$

(1)

In this equation, represents the heat gained by the surface from solar radiation (W). The solar irradiance I, measured in watts per square meter (W/m²), is derived from EnergyPlus Weather Files. The surface area A is given in square meters (m²). The solar absorptance, which ranges from 0 to 1, is set to 0.55 [20].

2.1.3. Indoor Temperature

In hot climates, optimizing building forms is key to mitigating solar heat gain. This study shifts the focus from merely blocking radiation to leveraging the geometric flexibility of free-form roofs themselves. To quantify the thermal outcome of such shape adaptations, the zone operative temperature—here termed “indoor temperature” (°C)—is employed as the primary comfort metric. This indicator, defined at a central point beneath the roof, synthesizes air temperature and mean radiant temperature, thereby capturing the integrated thermal effect of solar exposure and spatial geometry. The core objective is to minimize the indoor temperature through strategic morphological adjustments to the roof surface. For clarity in isolating the roof’s geometric influence, the analysis adopts a controlled scenario: the space is considered fully enclosed (neglecting convection, ventilation, and internal loads), and the volume below the roof’s lowest point is excluded. This well-defined computational domain, bounded by the roof surface, its vertical edges, and a base plane, allows for a direct assessment of how roof shape governs thermal performance under varying solar conditions, forming a rigorous basis for design optimization in hot-summer and warm-winter climates [19].

The focus of this study is free-form surface roof geometries, with the scope limited to opaque roof envelopes, consistent with Section 2.2. The solar absorptance $\alpha$ and the thermal absorptance of the building surface are assigned values of 0.55 and 0.9, respectively, consistent with established practices in building energy simulation [20]. The solar heat gain of a building is predominantly governed by its geometric configuration, specifically the total envelope area intercepting solar radiation and the incidence angle of sunlight on each surface. Following the calculation principles for opaque envelopes detailed in [20], the resulting indoor temperature $T$ is determined by the following governing equation:

$$T ={\displaystyle \frac{T_{air} +T_{rad}}{2}}$$

(2)

$$T_{rad}={\left({{\displaystyle \sum }}_{i=1}^n{F}_i·T_{s,i}^4\right)}^{{\displaystyle \frac{1}{4}}}$$

(3)

Within this formulation, $T_{air}$ represents the air temperature (°C); $T_{rad}$ denotes the mean radiant temperature (°C); $F_i$ is the view factor from the central point on the underside of the roof to the i-th surface, which depends on the geometric configuration and relative positioning of the surfaces; n is the total number of surfaces within the zone; $T_{s,i}$ is the surface temperature of the i-th surface (°C), derived from the energy balance formulation in Equation (4):

$$Q_{cond}+Q_{conv}+Q_{rad}+Q_{solar}=0$$

(4)

$$-{\displaystyle \frac{(T_{s,i}-T_{out})}{R}}+h_c·(T_{s,i}-T_{air})+\epsilon ·\sigma ·(T_{s,i}^4-T_{sky}^4)+I·A·\alpha =0$$

(5)

Within this formulation, $Q_{cond}$, $Q_{conv}$, and $Q_{rad}$ represent the conductive, convective, and radiative heat transfer components (W) through the building envelope, respectively. The thermal resistance R is 2.61 m²·K/W, and the surface emissivity $\epsilon$ is 0.9 [10]. The Stefan–Boltzmann constant $\sigma$ is 5.67 × 10⁻⁸ W/m²K⁴. The convective heat transfer coefficient $h_c$ is calculated via the expression $h_c=4.0+4.8·{(v_w)}^{0.67}$, where $v_w$ denotes the wind speed. All essential climatic inputs—namely the incident solar irradiance I (W/m²), outdoor air temperature $T_{out}$ (°C), and dew point temperature $T_{dp}$ are retrieved directly from the ‘EnergyPlus Weather File’. Subsequently, the sky temperature $T_{sky}$ is derived from these inputs using the correlation $T_{sky}={\left(0.74+0.006T_{dp}\right)}^{0.25}·(T_{out}+273)$ [21]. Surface areas A are expressed in square meters (m²).

$$C_{air}·{\displaystyle \frac{d{T}_{air}}{dt}}=-Q_{loss}$$

(6)

$$Q_{loss}=A·{\displaystyle \frac{T_{out}-T_{air}}{R}}$$

(7)

Within the governing equation, the thermal capacity of the zone air, denoted as $C_{air}$, is defined by the expression $C_{air}=\rho ·V·c_p$. In this formulation, $\rho$ represents the density of air, assigned a standard value of 1.225 kg/m³, V is the volume of the zone (m³), and $c_p$ is the specific heat capacity of air (1.005 kJ/(kg·K)).

2.1.4. Treatment of Unconstrained Boundaries in Geometric Optimization

In the shape optimization of non-standard (free-form) opaque roof geometries, a base plane-assisted enclosure method is proposed to eliminate the influence of environmental exchange between the interior and exterior when the roof form is not fully enclosed, and to standardize the comparison conditions across different types of geometry boundary constraints, including quadrilateral constraints, north–south constraints, and east–west constraints. Specifically, a reference base plane measuring 30 m × 30 m (Z = 0) is defined, with the four corner points of the roof form connected to the corresponding corners of the base plane, and nodes along the edges (boundary points) linked to auxiliary points on the base plane, forming a series of triangular opaque auxiliary surfaces (Figure 1). These auxiliary surfaces, together with the original roof form, constitute a fully enclosed opaque roof system, ensuring that differences in optimization outcomes under various geometry boundary constraint types reflect only the directional constraints themselves, rather than being influenced by unbounded surfaces exposed to the ambient environment, thereby focusing exclusively on the independent impact of geometry boundary constraints on the optimization behavior of the roof form.

2.2. Mathematical Formulation and Optimization Methodology

2.2.1. Mathematical Formulation

Shape Optimization Targeting Indoor Temperature

This study centers on the thermal-performance-driven shape optimization of free-form grid roofs, targeting the hot-summer and warm-winter climate typified by Hainan Province. A computational optimization model is developed, utilizing annual meteorological data to inform the design process. To assess performance under the most demanding scenarios, the analysis focuses on the peak radiation period (09:00–16:00) of the year’s hottest day, representing an extreme thermal loading condition for evaluating indoor thermal stability. The core optimization objective is to minimize the average indoor temperature beneath the roof during this critical period. This problem is formally stated mathematically as follows:

$$\left\{\begin{array}{ll}Find& \hspace{1em}X\\ Minimize& \hspace{1em}{T}_h\left(X\right)={\displaystyle \frac{1}{8}}{\displaystyle \sum _9^{16}{T}_{h,i}} \\ Subject to& \hspace{1em}X\in \left[X^{\mathrm{L}},X^{\mathrm{U}}\right] \end{array}\right.$$

(8)

Within the optimization framework, X encapsulates the geometric design variables, specifically the three-dimensional coordinates of the key control nodes that define the free-form surface shape. The corresponding lower and upper bounds for these variables are denoted as X^L and X^U, respectively. These bounds are formally defined relative to an initial configuration: X^L = X₀ − Δ^L, X^U = X₀ + Δ^U. Here, X₀ represents the vector of design variable values in the initial geometric model. The offset vectors Δ^L and Δ^U determine the permissible search range for each variable and are typically specified by the designer. For instance, a common symmetric setup is Δ^L = Δ^U = 2.0E, where E is a vector of ones. The objective function $f\left(\mathbf{X}\right)$ is defined as the hourly indoor temperature (°C) $T_{h,i}$, evaluated at every iteration of the optimization loop.

Shape Optimization Targeting Solar Radiation Heat Gain

Based on extreme thermal loading conditions, a shape optimization model for free-form roofs is developed to minimize solar radiation heat gain. The mathematical formulation is expressed as follows:

$$\left\{\begin{array}{ll}Find& \hspace{1em}X\\ Minimize& \hspace{1em}Q\left(X\right)={\displaystyle \frac{1}{8}}{\displaystyle \sum _9^{16}{\mathrm{Q}}_{\mathrm{solar},\mathrm{i}}}\\ Subject to& \hspace{1em}X\in \left[X^{\mathrm{L}},X^{\mathrm{U}}\right]\end{array}\right.$$

(9)

In these equations, ${\mathrm{Q}}_{\mathrm{solar},\mathrm{i}}$ denotes the hourly solar radiation heat gain during the optimization iteration process.

Shape Optimization Under Long-Term High-Temperature Conditions

To account for seasonal climate variations and ensure adaptability over an extended period, the long-term thermal loading condition is characterized by the six hottest months of the year, spanning from April to September. The optimization model targets the cumulative thermal performance across this entire season. Accordingly, the shape optimization framework for free-form roofs is designed to minimize the average indoor temperature under these prolonged thermal conditions. This long-term optimization problem is mathematically defined as follows:

$$\left\{\begin{array}{ll}Find& \hspace{1em}X\\ Minimize& \hspace{1em}{T}_m\left(X\right)={\displaystyle \frac{1}{6}}{\displaystyle \sum _4^9{T}_{m,i}}\\ Subject to& \hspace{1em}X\in \left[X^{\mathrm{L}},X^{\mathrm{U}}\right] \end{array}\right.$$

(10)

In this equation, $T_{m,i}$ quantifies the indoor temperature (°C) on a monthly basis throughout the iterative optimization procedure.

2.2.2. Optimization Methodology

The proposed optimization framework establishes a closed-loop workflow integrating three core components: parametric geometric modeling, building energy simulation, and numerical optimization. The process begins by parameterizing the roof geometry, whose performance is subsequently evaluated through automated calls to the EnergyPlus simulation engine. The resulting nonlinear optimization problem with continuous variables is solved using the fmincon solver from MATLAB’s (R2024a) Optimization Toolbox, configured with the interior-point algorithm (Figure 2). The specific workflow for the morphological design is structured as follows:

Step 1: Initialization: The initial free-form surface is defined according to architectural requirements by specifying its key control points. A parametric geometric model is established, along with the lower ($X^{\mathrm{L}}$) and upper bounds ($X^{\mathrm{U}}$) for the design variables (nodal coordinates), which constitute the initial design space.

Step 2: Simulation Integration: A parametric roof model is generated based on the initialized geometry. This model is interfaced with EnergyPlus via MATLAB to simulate the thermal performance and retrieve key metrics, primarily the indoor temperature. Computational modules are developed to manage the objective function calculation, ensuring seamless data exchange and geometry updates throughout the optimization iterations.

Step 3: Optimization Execution: The fmincon optimizer is employed to iteratively call the objective function, driving the search for an optimal solution. The process terminates when predefined convergence criteria (e.g., maximum iterations, tolerance thresholds) are met. Upon completion, the solver returns the optimal design variables and performance values, facilitating analysis of the convergence history and solution robustness.

2.3. Mathematical Description of the Optimization Problem

2.3.1. Shape Factor

The shape factor serves as a key metric for assessing the thermal efficiency of free-form roof geometries. This dimensionless parameter quantifies the compactness of a building form by calculating the ratio of its external surface area to the enclosed volume [22]. For free-form roofs, a lower shape factor is directly correlated with improved volumetric efficiency (less envelope area per unit volume), which consequently diminishes heat transfer losses:

$$S=F/V$$

(11)

In this equation, $S$ represents the building’s shape factor. $F$ accounts for its external envelope area (m²), while $V$ gives the internal volume (m³) bounded by that envelope.

2.3.2. Solar Exposure Rate of Surface

To characterize the passive shading capability inherent in the roof geometry, the Surface Exposure Ratio (SER) is introduced. This metric quantifies the proportion of the roof surface area that receives direct solar irradiation under extreme thermal loading conditions.

$$\mathbf{S}\mathbf{E}\mathbf{R}={\displaystyle \frac{{\displaystyle {\sum }_{t=1}^T{\displaystyle {\sum }_{n=1}^{N}S(t,n)}}}{T\times N}}$$

(12)

In the above equation, $S(t,n)$ tracks the sunlit status (0 or 1) of surface n-th at hour t-th. Here, $T$ and $N$ define the upper bounds for the time and surface indices, respectively, with $T$ being the total number of hourly time steps and $N$ the total number of surfaces.

2.3.3. Thermal Density (TD)

To evaluate the efficiency of thermal load distribution per unit volume, this study employs the thermal density (TD) metric. Adapted from the conventional sensible heat load density $q_v=Q/V$ [23] and defined by Equation (13), TD represents the indoor temperature per unit building volume (°C/m³). It therefore serves as a direct metric for volumetric efficiency: a lower TD value signifies a reduced heat load per unit volume, which consequently indicates superior thermal performance of the design.

$$\mathbf{T}\mathbf{D}={\displaystyle \frac{T}{V}}$$

(13)

In this governing equation, $q_v$ denotes the sensible heat load density (W/m³), and $Q$ represents the total instantaneous energy demand or power (W).

2.4. Validation with Numerical Examples

The numerical examples are configured to analyze the morphological evolution of free-form roofs in response to four distinct design scenarios: ① the shape optimization patterns of the same initial geometry subjected to different thermal performance objectives; ② the influence of varying initial configurations and thermal loading conditions—including single-day extreme thermal loading conditions and long-term thermal loading conditions—on the shape optimization results when the objective is to minimize indoor temperature; ③ the differences in resulting morphology when applying either symmetric point shape control or full-point shape control during the optimization of the same initial configuration; ④ the morphological characteristics under different directional geometry boundary constraints, where such directional constraints are imposed solely at the geometric parametrization level without altering the physical boundary conditions in the thermal analysis.

Four model examples were designed using a Non-Uniform Rational B-Spline (NURBS)-based free-form single-layer reticulated shell as the initial computational geometry. The model has a span of 30 m and is based on a B-spline surface-defined single-layer spatial grid. The geometric shape of the surface was generated using the NURBS surface Equation (14) [24], with the B-spline surface defined by degrees p = q = 3. The key control points are structured as an (m + 1) × (n + 1) grid, ordered sequentially from top-left to bottom-right. To enhance mesh resolution, an auxiliary point is inserted between each pair of adjacent primary control points. Example 1 employs an axisymmetric initial form, where the Y-axis serves as the axis of symmetry and is aligned to true north. A quarter of the surface is modeled based on two perpendicular symmetry axes of the morphology, as shown in Figure 3a, where the symmetric control points and constrained points are clearly indicated. The supports are hinged along the perimeter, and the initial values are listed in

Table 1. Coordinates of shape control key points and constraint points for Examples 1 and 3 (m).

NO.	1	2	3	4	5	6	7	8	9	10	11
X	18.00	24.00	30.00	18.00	24.00	30.00	18.00	24.00	30.00	6.00	12.00
Y	30.00	30.00	30.00	24.00	24.00	24.00	18.00	18.00	18.00	24.00	24.00
Z	0.00	0.00	0.00	3.80	3.00	0.00	5.50	3.80	0.00	3.00	3.80
NO.	12	13	14	15	16	17	18	19	20	21
X	6.00	12.00	6.00	12.00	6.00	12.00	18.00	24.00	18.00	24.00
Y	18.00	18.00	12.00	12.00	6.00	6.00	12.00	12.00	6.00	6.00
Z	3.80	5.50	3.80	5.50	3.00	3.80	5.50	3.80	3.80	3.00

. Example 2 shares the same base configuration as Example 1 (Figure 3a), with initial values given in

Table 2. Coordinates of shape control key points and constraint points for Example 2 (m).

NO.	1	2	3	4	5	6	7	8	9
X	18.00	24.00	30.00	18.00	24.00	30.00	18.00	24.00	30.00
Y	30.00	30.00	30.00	24.00	24.00	24.00	18.00	18.00	18.00
Z	0.00	0.00	0.00	0.90	0.30	0.00	3.00	0.90	0.00

. In Example 3, all control points of the initial configuration are treated as design variables, representing a full-point shape control model. The shape control points and constraint points are illustrated in Figure 3b, with initial values again taken from Table 1. Example 4 follows the same base setup as Example 1, where Example 4.1 (Figure 3c,

Table 3. Coordinates of shape control key points and constraint points for Example 4 (m).

NO.	1	2	3	4	5	6	7	8	9
X	18.00	24.00	30.00	18.00	24.00	30.00	18.00	24.00	30.00
Y	30.00	30.00	30.00	24.00	24.00	24.00	18.00	18.00	18.00
Z	0.20	0.20	0.20	3.80	3.00	0.20	5.50	3.80	0.20

) adopts hinged supports along the north and south edges, Example 4.2 (Figure 3d, Table 3) uses hinged supports along the east and west edges, and Example 4.3 (Figure 3a, Table 3) employs perimeter hinged supports. Thermal performance analysis of the roof geometries was conducted using EnergyPlus, employing triangular surface elements for the thermal simulations.

$$\begin{array}{l}x(u,v)={\displaystyle \sum _{i=0}^m{\displaystyle \sum _{j=0}^n{\alpha }_{i,j}{B}_{i,p}(u)B_{j,q}(v)}}\hfill \\ y(u,v)={\displaystyle \sum _{i=0}^m{\displaystyle \sum _{j=0}^n{\beta }_{i,j}{B}_{i,p}(u)B_{j,q}(v)}}\hfill \\ z(u,v)={\displaystyle \sum _{i=0}^m{\displaystyle \sum _{j=0}^n{\gamma }_{i,j}{B}_{i,p}(u)B_{j,q}(v)}}\hfill \end{array}$$

(14)

The NURBS-based free-form surface is defined by its spatial coordinates $(x,y,z)$, which are functions of the normalized parameters $u$ and $v$ (both within [0,1]). The mathematical representation involves several key components: ① Orders and Knots: The integers p and q specify the degrees of the B-spline basis functions in the u and v directions, respectively. The corresponding knot vectors, comprising m + 1 and n + 1 knots, are computed using the chord length method once p and q are defined. ② Control Points (Coefficients): The set of coefficients (α_i,j, β_i,j, γ_i,j) defines the control net (or control points) of the surface. These coefficients are determined through a global surface interpolation procedure, as detailed in reference [25] (Chapter 9). ③ Surface Generation: Given a set of key points on the target surface, the complete NURBS surface shape can be uniquely generated. This process is implemented in the associated computational program referenced as [24].

The spatial coordinates of the NURBS surface control points constitute the design variable vector X. The allowable variation for each component of X is prescribed by lower and upper bounds, defined relative to the initial geometry X₀ = (x₀, y₀, z₀)ᵀ, which encapsulates the initial nodal coordinates of the control points. The bounds are specified as follows: x^L = x₀ − 1.5E, x^U = x₀ + 1.5E, y^L = y₀ − 1.5E, y^U = y₀ + 1.5E, z^L = z₀ − 0.0E, z^U = z₀ + 4.0E. Here, E denotes a vector of ones with dimensions matching X₀, thereby applying a uniform scaling to the respective bound offsets.

3. Results

3.1. Optimization Patterns Under Different Thermal Performance Objectives

This section presents a comparative analysis of the shape optimization patterns of free-form roofs under extreme thermal loading conditions, focusing on different thermal performance objectives. When the optimization aims to minimize surface solar radiation heat gain, the building envelope evolves freely during the process, and the solution tends to reduce the total surface area in order to lower the overall radiation exposure (Figure 4). Although this technically rational strategy is effective in reducing heat gain, it may also limit the diversity of roof forms, potentially compromising the architectural expressiveness and the original intent of shape optimization.

To comprehensively evaluate the thermal performance modulation capability of free-form roof shapes and quantify how spatial surface variations influence thermal behavior, air temperature is adopted as the optimization objective. This approach integrates both solar radiation heat gain and the coupled effect of surface geometry changes on the internal temperature of the roof geometry. The shape optimization aimed at minimizing indoor temperature yields morphologies with enhanced architectural expressiveness: the nodes at the center of the B-spline free-form grid shell gradually shift outward and upward, transforming the initial spherical surface into a saddle-shaped form (Figure 4c). The optimized roof develops an alternating “peak–valley” self-shading configuration, resembling mountain ridges and valleys, with more pronounced concavities along the central parts of the north and south facades. This distribution aligns with the east-to-west movement of the sun. The central region rises into a dome-like form, with four diagonal ridges extending outward along the planar diagonals. As a result, the geometric form of the optimized morphology becomes more compact, reducing the surface area per unit volume. Through this self-shading effect, the design achieves passive solar control while preserving architectural articulation and formal richness.

The two optimization objectives demonstrate distinct design orientations: one driven by technical rationality and the other by a more integrated, balanced approach. In this study, indoor temperature is selected as the thermal performance objective. This choice not only overcomes the limitations of using solar radiation heat gain as the sole criterion but also fully exploits the dual advantages of free-form surfaces in both morphological innovation and performance optimization.

3.2. Optimization Patterns Under Varying Initial Configurations and Thermal Conditions

A dual-scenario comparative analysis was performed to assess how free-form roof morphologies evolve under two distinct thermal design conditions: single-day extreme heat and prolonged seasonal heat. In both scenarios, the optimization objective was set to minimize indoor temperature, enabling a direct comparison of their influence on the final geometric outcomes. For Example 1, the optimization under single-day extreme conditions (Figure 5) yielded a shape factor reduction of 25% (from 0.40 to 0.30) and lowered the indoor temperature by 1.95 °C (from 38.24 °C to 36.29 °C). Under long-term conditions (Figure 6), an even greater shape factor reduction of 27.5% (from 0.40 to 0.29) was achieved, accompanied by an indoor temperature decrease of 0.82 °C (from 30.96 °C to 30.14 °C). Notably, a consistent morphological transformation was observed: under both objectives, the central control nodes of the B-spline grid shell progressively displaced outward and upward, morphing the initial quasi-spherical form into a distinct saddle-shaped geometry. The optimized configuration exhibited an alternating “peak–valley” self-shading pattern (Figure 4a and Figure 7a), in which the central regions of the north and south facades showed more pronounced concavities, aligning with the east-to-west movement of the sun to achieve passive shading through self-shadowing. The optimized geometries exhibited a pronounced self-shading capability, as evidenced by a substantial reduction in the solar exposure rate (SER). Under the single-day extreme condition, the SER dropped sharply from an initial 99.13% to 87.31%. A similar reduction was observed under the long-term condition, where the SER declined to 88.75%. These marked decreases confirm the effective passive shading performance achieved through shape optimization. Additionally, the mesh density at the roof center became sparser as nodes migrated toward the mid-sections of the geometric edges, while the central region evolved into a dome-like protrusion with four diagonal ridges extending along the planar diagonals. The overall geometric form became more compact, reducing the surface area per unit volume. The consistent emergence of this characteristic saddle-shaped morphology under varying thermal loads verifies the robustness of the proposed strategy and indicates its potential applicability to similar free-form design problems.

To further verify whether the observed optimization patterns are influenced by the initial geometric configuration, a comparative study was conducted using Example 2. Under the single-day extreme condition (Figure 8), Example 2 exhibited a marked performance improvement. Its shape factor declined substantially by 61.40% (from 1.14 to 0.44), concurrently with an indoor temperature reduction of 2.36 °C (from 39.53 °C to 37.17 °C). This trend aligns closely with the optimization behavior observed for Example 1 under identical conditions. The optimized roof also exhibited an alternating “peak–valley” saddle-shaped surface (Figure 7b,c), with the SER of the geometric surface decreasing from an initial 100% to 90.13%, indicating a clear self-shadowing effect. The optimization process transformed the roof’s central area into a domed protrusion, with four diagonal ridges radiating outward. This morphological shift yielded a more compact overall form, culminating in a reduced surface-area-to-volume ratio. These results demonstrate that differences in initial geometry do not alter the fundamental optimization patterns.

The optimization process consistently exhibited a distinct stage-wise characteristic. During the initial optimization phase, the reduction in shape factor and the drop in indoor temperature evolved in concert, indicating a closely coupled relationship. In the convergence phase, the algorithm decoupled the refinement of local geometry (e.g., fine-tuning curvature) from large-scale morphological changes. This resulted in gradual, asymptotic improvements in indoor temperature, while the shape factor plateaued, showing only marginal adjustments (around 0.05, 0.03, and 0.01 for the respective cases mentioned). For Example 1, the thermal density (TD) value evolved from 0.0152 to 0.0070 under the single-day extreme thermal loading condition—a reduction of 53.95%—and from 0.0123 to 0.0056 under the long-term condition, representing a 54.47% decrease. For Example 2 under the single-day extreme condition, TD dropped from 0.0488 to 0.0116, a significant reduction of 76.23%. A consistent downward trend in the heat load per unit volume was observed throughout the optimization, which corresponded to a systematic enhancement in overall thermal performance.

In summary, this work synthesizes aesthetic, functional, and thermal objectives through the performance-driven optimization of free-form roofs, yielding geometries that significantly improve the indoor thermal environment under high-temperature conditions. The strategy targeting single-day extreme heat proves most effective, achieving an average indoor temperature reduction of approximately 2 °C. A consistent, self-shading saddle morphology—characterized by a central dome and four diagonal ridges—emerges across cases under this condition, validating the robustness of the approach. To ensure conciseness, the subsequent in-depth analysis will focus on Example 1 under this single-day extreme scenario.

3.3. Comparison of Optimization Performance Between Symmetric Point Shape Control and Full-Point Shape Control

Using Example 1 as the basis and aiming to minimize indoor temperature under single-day extreme thermal loading, a comparative analysis was carried out. This study examines how two distinct shape control strategies—symmetric-point control versus full-point control—affect the resulting morphological patterns of free-form roofs. The optimization behavior under symmetric point shape control has already been analyzed in Section 2.1 and Section 2.2. In the full-point shape control optimization process, the variation in the shape factor and indoor temperature is shown in Figure 9. The optimization process yielded a significant improvement in all key metrics. The indoor temperature was reduced from 38.24 °C to 36.09 °C, a decrease of 2.15 °C. Concurrently, the shape factor declined from 0.40 to 0.31, marking a 22.50% reduction, and the thermal density (TD) dropped from 0.0152 to 0.0069, a 54.61% decrease. The consistent reduction in unit-volume heat load confirmed an overall enhancement in thermal performance. Similar to the symmetric-point control strategy, the optimization displayed a clear two-stage evolution. In the initial phase, characterized by a rapid decline in indoor temperature, a strong positive correlation was observed between the reductions in indoor temperature and shape factor. Upon transitioning to the convergence phase, the algorithm shifted its focus to refining local geometric features—such as redistributing concave and convex areas—to further optimize performance. During this stage, indoor temperature changes became more gradual and decoupled from the now-stable shape factor, which exhibited only minor fluctuations (around ±0.05).

As shown in Figure 10, the central nodes of the B-spline free-form grid shell gradually shifted outward and upward, resulting in a central depression combined with ridge-like elevations extending along the planar diagonals. This led to an alternating “peak–valley” surface pattern featuring self-shadowing characteristics. Notably, the central regions of the north and south facades exhibited more pronounced concavities, which aligns with the east-to-west movement of the sun, enabling passive shading through the self-shadowing effect—similar to what was observed under symmetric point shape control. Compared to symmetric point shape control, the roof morphology achieved through full-point shape control demonstrated greater morphological freedom in the formation of peaks and valleys. After optimization, the SER of the geometric surface decreased from an initial value of 99.13% to 89.94%. The optimized roof form not only reduced the total surface area but also effectively minimized solar heat gain, thereby improving the indoor thermal environment.

Although full-point shape control demonstrates a slight advantage over symmetric point shape control in reducing indoor temperature (approximately 0.2 °C), the pronounced central depression formed in the optimized morphology poses potential challenges for drainage in practical engineering applications. Given the minimal difference in thermal performance between the two strategies (∆T ≤ 0.2 °C), symmetric point shape control is considered more suitable from the perspective of engineering applicability and functional building requirements. Therefore, subsequent analysis will focus on Example 1 under symmetric point shape control, optimized for single-day extreme thermal loading conditions, for further in-depth investigation.

3.4. Comparison of Free-Form Surface Optimization Under Directional Geometry Constraints

To ensure consistency in the computational process, this section adopts the boundary treatment method used in Section 2.3 under geometrically unconstrained conditions, and the resulting data are intended solely for comparative analysis among different directional geometry boundary constraints and should not be directly applied as optimization references for actual constrained engineering scenarios. With the objective of minimizing indoor temperature under single-day extreme thermal loading conditions, the influence of various geometry boundary constraint directions—specifically north–south constraints, east–west constraints, and quadrilateral constraints—on the morphological design patterns of free-form roofs is compared, and the optimized indoor temperatures and shape factors under each constraint condition are summarized in

Table 4. Indoor temperature (°C) and shape factor before and after optimization.

		North–South Constraints		East–West Constraints		Quadrilateral Constraints
		Indoor Temperature	Shape Factor	Indoor Temperature	Shape Factor	Indoor Temperature	Shape Factor
Before Optimization		38.184	0.398	38.184	0.398	38.184	0.398
After Optimization		37.832	0.3417	38.169	0.333	38.133	0.254

. The optimization under three directional boundary constraints yielded distinct yet convergent improvements. Across all cases—north–south (NS), east–west (EW), and quadrilateral (QD)—the heat load per unit volume consistently decreased, confirming an overarching enhancement in thermal performance. Quantitatively, under NS constraints, the indoor temperature decreased by 0.352 °C (to 37.832 °C), with concurrent reductions in shape factor (14.15% to 0.34) and thermal density (17.22% to 0.0125). For EW constraints, a minimal temperature drop of 0.015 °C (to 38.169 °C) was observed, while shape factor and thermal density still decreased by 16.33% (to 0.33) and 19.87% (to 0.0121), respectively. The QD constraints produced the most geometrically efficient outcome: shape factor and thermal density plummeted by 36.18% (to 0.25) and 56.95% (to 0.0065), respectively, alongside a temperature reduction of 0.051 °C (to 38.133 °C).

Under north–south constraints, the roof geometry was elevated as a whole from its initial configuration, with both east and west sides raised. As a result, the SER of the geometric surface decreased from an initial value of 99.13% to 98.13% (Figure 11b,e). For the east–west constrained roof, distinct upward curvatures formed at both ends along the north–south direction, effectively utilizing the self-shadowing effect of the curved surface in alignment with the sun’s path to achieve passive shading. The SER accordingly dropped from 99.13% to 98.50% (Figure 11c,f). Under quadrilateral constraints, the optimization process drove the central nodes of the B-spline free-form shell to evolve into a distinct morphology: a central depression surrounded by elevated, ridge-like structures extending along the planar diagonals. This led to an alternating “peak–valley” surface pattern featuring self-shadowing characteristics (Figure 11d,g). Significantly, the deeper concavities formed on the north and south facades coincided with the east–west solar path. This geometric alignment synergistically enhanced the passive shading performance through more effective self-shadowing. Consequently, the SER significantly decreased from 99.13% to 88.88%. The optimized roof form not only reduced the total surface area but also effectively minimized solar heat gain, thereby improving the indoor thermal environment.

For long-span roof design based on indoor temperature optimization, differentiated design strategies should be adopted according to the functional requirements of various building types. In large-span roofs such as sports stadiums, exhibition centers, and transportation hubs—where overall stability and energy efficiency are prioritized—the quadrilateral constraint pattern is recommended. The optimized form develops a dome-like uplift with four diagonal ridges extending outward along the planar diagonals, effectively reducing direct solar radiation through self-shadowing while maintaining overall rigidity and improving thermal performance by lowering cooling loads. For buildings such as art galleries and museums, where lighting conditions and architectural expressiveness are critical, the east–west constraint pattern offers an appropriate solution. The resulting warped morphology on the north and south sides provides a favorable basis for daylight control, blocking low-angle sunlight to reduce glare while simultaneously enhancing aesthetic appeal. In contrast, for production-oriented buildings such as eco-buildings, green buildings, and industrial facilities that emphasize practicality and cost-effectiveness, the north–south constraint pattern is preferable due to its geometric simplicity, which facilitates construction while still meeting basic functional requirements. All three constraint patterns utilize the self-shadowing effect of free-form surfaces to achieve passive shading; however, their design emphases differ: the quadrilateral constraint focuses on morphological stability and thermal regulation, the east–west constraint emphasizes daylight optimization and spatial aesthetics, and the north–south constraint prioritizes ventilation efficiency and constructability—thereby offering tailored solutions for roof design across different architectural types.

4. Discussion

4.1. Interpretation of Key Findings and Answers to Research Questions

This study established a thermal-performance-driven optimization framework for free-form roofs. The results provide clear answers to the research questions posed in the Introduction.

Regarding RQ1 (Optimal Objective), our findings robustly demonstrate that minimizing indoor temperature is a superior optimization objective compared to minimizing solar heat gain for free-form roofs in hot climates. While the solar-heat-gain objective simply reduces total surface area, the indoor-temperature objective drives the generation of self-shading, saddle-shaped morphologies. This is quantitatively supported by the achieved ~2 °C reduction in indoor temperature under extreme conditions, which translates directly to improved thermal comfort and potential cooling energy savings. This finding shifts the design paradigm from merely minimizing exposure to intelligently managing solar radiation through form.

Regarding RQ2 (Control Strategy), the comparison between full-point and symmetric-point control reveals a critical trade-off. Although full-point control can generate slightly more complex, theoretically optimal forms, symmetric-point control is identified as the pragmatically superior strategy. It achieves nearly equivalent thermal performance (e.g., within X% of the optimal temperature reduction) while fundamentally ensuring constructability, simpler drainage, and preserved architectural regularity. This makes it the recommended approach for real-world engineering applications.

Regarding RQ3 (Boundary Constraints as Design Tools), this study validates that directional geometric boundary constraints are powerful, high-level design tools. North–south constraints systematically produce forms with overall lift, beneficial for stability and ventilation. East–west constraints consistently generate end-uplift morphologies that are highly effective for sun-path-aligned passive shading. Designers can thus select constraint modes not just as geometric limits, but as an intentional “dial” to tune the roof shape towards specific performance and aesthetic goals.

4.2. Practical Implications and Design Guidelines

The primary contribution of this work is to translate computational findings into actionable design intelligence. Based on the answers to RQ1-RQ3, we propose the following practical guidelines for designers targeting hot climates: ① Driver: Use indoor temperature minimization as the primary optimization objective to generate self-shading forms. ② Control: Adopt symmetric-point control as the default strategy to ensure a balance between performance and buildability. ③ Tool: Utilize directional boundary constraints as high-level design dials: east–west for shading, north–south for simplicity, and quadrilateral for stability.

4.3. Limitations and Future Work

While this study establishes a robust framework, several limitations should be acknowledged, which also define clear pathways for future research.

Model Simplification and Climate Scope: The analysis focused on extreme single-day conditions under a sealed, static model to precisely isolate the geometric effect. Consequently, it does not account for seasonal variability, natural ventilation, internal heat gains, or different operational schedules, which are crucial for evaluating annual energy performance and comfort in real buildings.

Performance Metric: The optimization relied on a single objective (indoor temperature) under a specific extreme scenario. Future work should incorporate multi-objective optimization that balances thermal performance with other criteria (e.g., daylighting, structural weight) and utilizes annual climate data to ensure robustness across varying conditions.

5. Conclusions

This study establishes a thermal-performance-driven optimization framework that systematically bridges the gap between computational design and building energy efficiency for free-form roofs. Given its role as a direct measure of occupant thermal comfort and its integrative capacity over mere surface energy fluxes, indoor temperature is posited as a highly relevant primary objective for performance-driven free-form design. The results of the numerical examples demonstrate that optimizing for indoor temperature is more suitable for the design of free-form roofs. The optimized configurations exhibit saddle-shaped surfaces with alternating “peak–valley” patterns that possess self-shadowing characteristics and offer enhanced architectural expressiveness. Furthermore, we demonstrate that when the optimization objective is to minimize solar heat gain on the surface, the process tends to reduce the building’s total surface area in order to lower the thermal load, which limits the diversity of possible roof morphologies. When indoor temperature minimization is used as the optimization goal, the resulting morphologies show reduced solar exposure rate of the envelope surface (SER), shape factor, and thermal density. Notably, the optimization under single-day extreme heat conditions achieves higher efficiency compared to long-duration high-temperature scenarios, with a more significant indoor temperature reduction of approximately 2 °C under extreme thermal loading conditions. Different initial models exhibit similar morphological evolution patterns under single-day extreme heat conditions. A key practical finding is that the symmetric point shape control method aligns better with practical engineering needs, as it satisfies functional requirements while preserving architectural aesthetics. On the other hand, full-point shape control yields roofs with more freely varying peak–valley surface features when minimizing indoor temperature, but this approach increases construction complexity and poses a risk of rainwater accumulation in concave areas. Roof geometries under different directional constraints exhibit distinct characteristics: the north–south constrained roof shows an overall elevation increase from its initial geometric configuration, with both east and west sides raised; whereas the east–west constrained roof develops clearly uplifted regions at both ends along the north and south sides, effectively utilizing the sun path-aligned self-shadowing effect for passive shading.

Although quantified under extreme conditions, the passive cooling effect of the optimized morphology is operational whenever solar radiation is present. Consequently, it is anticipated to contribute to a non-negligible reduction in cumulative annual cooling loads for buildings in hot climates, enhancing its practical relevance.

The findings offer a performance-based geometric logic that can inform preliminary design thinking for specific building archetypes. In hot-summer climates: For large-span volumes like sports stadiums, exhibition centers, and transportation hubs, where creating a globally shaded, iconic form is often a priority, the quadrilateral constraint—which yields a dome-like, self-shading morphology—may present a geometrically and thermally efficient archetype to explore. For buildings such as art galleries and museums, where directional light control and a strong longitudinal axis are key architectural concerns, the east–west constraint pattern, generating warped forms with north–south uplift, provides a coherent formal strategy that aligns passive shading with these spatial objectives. For eco-buildings and industrial facilities prioritizing constructability and economy, the north–south constraint, resulting in simpler elevated forms, suggests a structurally rational and material-efficient direction. It is crucial to emphasize that these geometric insights are derived from a singular thermal-performance perspective. Thus, they are intended as formative, performance-driven form-generating principles rather than prescriptive design solutions. Consequently, their successful integration into any built project necessitates a synthesis with the full spectrum of structural, functional, economic, and aesthetic criteria that define complex building design.

Overall, the shape optimization proposed in this paper for improving indoor temperatures in single-layer free-form grid roofs considers the organic synergy between building morphology and solar radiation control. The proposed optimization framework offers tailored, performance-driven solutions for diverse building types and is readily applicable to a broad range of thin-shell structures. This study develops and validates a computational framework that integrates architectural design with environmental performance. The case study demonstrates the potential of this approach for application in sustainable civil and environmental engineering. Looking forward, the geometric solutions generated by this framework establish a critical first step. A logical and impactful extension would be to integrate material selection as a concurrent or subsequent variable within the optimization process. Exploring the synergy between these optimized, self-shading forms and advanced material properties (e.g., variable reflectance, thermal mass) could unlock further gains in holistic building performance. Furthermore, coupling these geometric solutions with lifecycle assessment (LCA) tools would allow for a holistic evaluation that balances the operational energy savings demonstrated here with the embodied carbon of construction materials, thereby advancing the framework’s comprehensive impact on low-carbon design.