Application of Topology Optimization as a Tool for the Design of Bracing Systems of High-Rise Buildings

Abstract

This study examines the impact of surrounding buildings and wind incidence angles on the aerodynamic loads of a high-rise building with a 1:1 base–edges and a 1:6 base–height ratio. CFD simulations were conducted using OpenFOAM with the classic RANS $k-ϵ$ turbulence model, validated against experimental data from Tokyo Polytechnic University. The aerodynamic coefficients were analyzed for wind angles of $\theta$ = 0°, 15°, 30°, and 45°, varying with the adjacent building height. Additionally, topology optimization via the Bi-directional Evolutionary Structural Optimization (BESO) method was applied to determine the optimal bracing system under wind-induced loads. The results indicate that surrounding buildings significantly modify the aerodynamic response, particularly for asymmetric wind angles, where torsional effects become more pronounced. A shielding effect was observed, reducing drag and base moment but with a lesser influence on lift. The topology optimization results show that material distribution is directly influenced by aerodynamic coefficients, with “X” bracing patterns in case of low torsion and an additional member when torsional effects increase. This study highlights the importance of wind engineering in high-rise structural design and urban planning, emphasizing the necessity of specific wind assessments for accurate load predictions in dense urban environments.

1. Introduction

In general, the growth of nations is demonstrated by their level of urbanization [1]. According to Onat and Kucukvar [2], civil construction entails significant investments that directly impact the entire production chain. Driven by economic and social factors, new buildings tend to be increasingly taller. This increase in height demands more slender structures, making lateral loads, such as wind and seismic activity, more significant. According to Habrah et al. [3], in many cases, the lateral loads are the main concern for slender tall buildings becoming the primary load the structural design must resist.

With increasing urbanization and the growing demand for housing, designing buildings has become a common challenge for engineers and architects to address. Even though the number of tall buildings is small compared to large cities and metropolises around the world, their presence significantly alters the flow of wind, causing changes in the forces applied to the facades. Lateral loads tend to cause user discomfort, as individuals may feel the building’s movement, potentially rendering the project unfeasible. As stated by Chapain e Aly [4], the economic viability of tall buildings is closely tied to user comfort. Another issue is the structural overload, which could result in catastrophic failure.

The solution that considers the economic viability of buildings is the application of bracing systems to support lateral loads. This structural reinforcement has negligible costs compared to overall construction [5] and mitigates the effects of lateral loads, and it is widely used in the construction industry [6]. However, implementing this system requires an analysis of the wind loads acting on the building’s facades.

Analyzing a single building in isolation does not fully capture the complexity of urban wind flow. In high-density areas, the presence of surrounding structures significantly alters the flow pattern. As buildings are not aerodynamic bodies, their vertices and edges cause boundary layer separation, leading to vortex formation and negative pressures downstream. Therefore, before designing a bracing system, it is essential to quantify the wind loads and flow effects caused by surrounding structures.

This can be determined in three ways: using regulatory standards (in Brazil, the NBR 6123/2023 [7] is applied); conducting tests with scaled models through wind tunnel tests; and applying Computational Fluid Dynamics (CFD) simulations. Due to the increase in computational capacity and the development of numerical methods, CFD has become widely applied in both the industry and academia [8]. On the other hand, most software is proprietary, requiring paid licenses and restricting access to source codes, which limits their use for research. For this study, wind loads were determined using the mean pressure coefficients ($C_p$) on the building facades in a quasi-static analysis. To obtain the mean pressures, CFD simulations were carried out using the open-source software OpenFOAM v6. Turbulence is modeled using the Reynolds-Averaged Navier–Stokes (RANS) methodology with the classic $k-ϵ$ closure model. The $C_p$ values are mapped onto a surface function and then converted into concentrated loads applied to the mesh nodes.

With the growing focus on sustainable practices in the construction industry, optimization techniques to reduce natural resource consumption have become a great assisting tool for designing buildings. In structural engineering, mathematical optimization reduces material usage without compromising structural integrity. The study by Bendsøe and Sigmund [9] describes three optimization methods, which are described below and are illustrated in Figure 1:

• Parametric optimization adjusts geometric variables (e.g., width, element diameter) without altering the structure’s shape or topology. The structural model remains fixed during the optimization process. In Xu et al. [10], a parametric study employed the response surface method to optimize damper parameters, simplifying the parametric analysis process and reducing computational costs.
• Shape optimization modifies the geometry to find the optimal shape within a defined domain, changing contours and hole positions without altering element connectivity.
• Topology optimization (TO) finds the optimal material distribution within a design domain, allowing changes to the connectivity of the discretized domain. Bocko et al. [11] combined topology and shape optimization, aiming to reduce weight and improve fatigue resistance, drawing inspiration from natural processes, such as trees growing.

Figure 1. Main structural optimization techniques.

Figure 1. Main structural optimization techniques.

Although shape and topology optimization may seem similar in concept, there is a fundamental difference between them. Shape optimization focuses on refining the external geometry of a structure while preserving its topology, adjusting contours to enhance performance. In contrast, topology optimization determines the optimal material distribution within a design space, allowing the emergence of entirely new structural layouts. The latter is particularly effective for maximizing stiffness and minimizing weight in engineering applications. Furthermore, topology optimization can be extended to other objective functions, such as heat distribution and natural frequencies in dynamic analyses.

TO can be combined with bracing system design to optimize structures for lateral load resistance. TO has also raised architectural interest, since its results often lead to geometries with aesthetically pleasing patterns [12]. Previous studies [13,14,15] have optimized bracing systems using geometric control and density-based techniques, but most lack evolutionary TO methods. This work proposes using the Bi-directional Evolutionary Structural Optimization (BESO) method with geometric control to create optimized structures with symmetry and pattern repetition, improving their application in civil construction. This methodology operates with the finite element method (FEM), using 3D eight-node hexahedral trilinear elements.

This study investigates the influence of surrounding buildings and wind incidence angles on the wind loading of a tall building with a 1:1 base proportion and a 1:6 height ratio. The same wind loading is used to determine the optimal distribution in the bracing system of the structure through the BESO method. This study is structured as follows: Section 2 provides a brief literature review on wind load determination in buildings, CFD, turbulence models, neighborhood effects, bracing systems, TO, and the BESO method. Section 3 presents the study’s methodology, including CFD modeling for the RANS $k-ϵ$ approach, wind load acquisition using OpenFOAM, aerodynamic coefficients, numerical schemes employed, and the BESO criteria for optimization with geometric control of the results. Section 4 consists of the validation of the adopted schemes, comparing the BESO method and CFD framework with cases from the literature and experimental data. Finally, Section 5 presents the results, and Section 6 provides the conclusion of the work.

2. Literature Survey

This section presents the literature review on the following topics: wind loads, Computational Fluid Dynamics, turbulence modelling, neighborhood effects, bracing structural systems, and the BESO method.

2.1. Wind Loads and CFD

In general, multistory buildings are designed with orthogonally arranged beams and columns, primarily to resist vertical loads. However, it should be highlighted that the slenderness of a building is directly related to its capacity to withstand lateral loads, such as those induced by wind and seismic excitation. Structures must ensure stiffness to limit lateral displacement during minor excitation and prevent damage to non-structural elements and, during major excitation, avoid collapse by allowing controlled inelastic behavior and structural damage [16].

Wind loads can cause both torsional and lateral displacements in the global structural system. Due to their dynamic nature, such loads can result in a variety of effects, including vortex formation and dynamic responses that induce structural vibrations [17]. Vortex formation alters the flow pattern, potentially causing discomfort for pedestrians, while dynamic responses can lead to discomfort for building occupants and, in extreme cases, results in structural failure.

There are various methods to assess the influence of wind on a structure. The main forces acting on buildings are surface forces. Brazilian standards on this subject are outlined in NBR 6123/2023 [7], which recommends determining wind-induced forces based on maximum characteristic wind speeds measured over 3 s with a 50-year return period at a height 10 m above open terrain. Although NBR 6123 provides a reliable estimate, it does not cover all possible scenarios. A usual design practice in Brazil is to limit lateral displacements up to H/400 (where H is the height of the building). Wind is a dynamic load, yet the standard emphasizes quasi-static analysis. Neighborhood effects on pressure coefficients are addressed, suggesting an approximate approach to calculate modifications based on building geometry and spacing. However, it also states that generic normative values for these effects cannot be provided, recommending wind tunnel tests instead. Additionally, wind vortices generated at oblique angles relative to the facade can induce severe actions, posing risks such as the detachment of roofs and coverings [18].

Therefore, it is essential to investigate how wind flow affects the building structure and, by quantifying the loads, design an efficient bracing system to withstand lateral forces. Aside from normative standards, the common methods used to predict wind loads are wind tunnel testing and Computational Fluid Dynamics (CFD). However, wind tunnel testing has a high associated cost [19]. Therefore, integrating wind tunnel experiments with CFD makes it possible to determine loads that cannot be calculated using standards alone. It provides detailed information across the entire domain, including velocity, temperature, and pressure fields. With proper boundary conditions and mesh configurations, CFD simulations yield accurate and satisfactory results compared to wind tunnel tests [20,21].

Nevertheless, CFD for simulating urban environments has limitations, as highlighted by Mirzaei [22]. It is often restricted by high computational costs, especially for high-resolution simulations that require large control volumes and advanced techniques for simulating turbulence. Additionally, the collection and preparation of geometric, climatic, and validation data also have limitations, such as integrating inadequate or incomplete information. Another imposing problem, simplifications, such as generic boundary conditions and assumptions of isothermal or steady-state conditions, can divert simulations from reality, reducing the applicability of the results.

2.2. Turbulence Modeling

Turbulent flow is commonly observed in natural wind conditions. Therefore, it is essential to adopt appropriate mathematical models to simulate turbulence over time and space. The Navier–Stokes equations are used to determine velocity and pressure conditions in fluid flow, as they express the conservation of momentum in fluids, accounting for fluid viscosity [23]. Numerical models can be derived from the Navier–Stokes equations to evaluate fluid behavior over time, such as Reynolds-Averaged Navier–Stokes (RANS) models, which involve time-averaged versions of such equations.

To solve the RANS equations, it is necessary to obtain the fluctuations of the Reynolds stress tensor. Several models are presented in the literature to analyze fluid behavior over time. The most common models are $k-ϵ$ [24], which provides a general description of turbulence using transport equations for turbulent kinetic energy (TKE) and its dissipation rate $ϵ$, and $k-\omega$ [25], which predicts turbulence through partial differential equations based on TKE and its specific dissipation rate $\omega$.

Each turbulence model has limitations, and various formulations exist, such as the standard $k-ϵ$, Re-Normalization Group (RNG) $k-ϵ$, Realizable $k-ϵ$, and $k-\omega$ Shear Stress Transport (SST). However, RANS models are commonly used for wind behavior around buildings, providing satisfactory results for capturing flow effects with a lower computational cost compared to more advanced approaches. The $k-ϵ$ model has been widely validated in the literature for urban wind simulations. Ishihara et al. [26] used the $k-ϵ$ model to simulate urban-like arrays, incorporating topography, land-use data, and maps of buildings and vegetation; Xiong et al. [27] evaluated the accuracy of CFD simulations of wind environments around high-rise buildings using $k-ϵ$ and polyhedral meshes; Schalau et al. [28] proposed a modified $k-ϵ$ model developed in OpenFOAM to simulate heavy gas dispersion in built-up environments.

Given its extensive validation and computational efficiency, the classic $k-ϵ$ model was adopted in this study. It was employed to evaluate the flow configuration around buildings and determine the pressure and aerodynamic coefficients on their facades. TKE is calculated through Equation (1), where $\rho$ = 1.225 kg/m³ is the air specific mass, “$k$” is TKE, $D_k$ is effective diffusivity for TKE, and $P$ and $ϵ$ are, respectively, TKE production and dissipation rate. $ϵ$ is calculated through Equation (2), where $D_{ϵ}$ is the effective diffusivity for $ϵ$, and $C_1$ = 1.44 and $C_2$ = 1.92 are model coefficients. Finally, the turbulent viscosity ${\nu }_t$ is calculated through Equation (3), where $C_{\mu }$ = 0.09 is the model coefficient. For more details about $k-ϵ$ equations, see [24].

$${\displaystyle \frac{D}{Dt}}\left(\rho k\right)=\nabla ·\left(\rho D_k\nabla k\right)+P-\rho ϵ$$

(1)

$${\displaystyle \frac{D}{Dt}}\left(\rho ϵ\right)=\nabla ·\left(\rho D_{ϵ}\nabla ϵ\right)+{\displaystyle \frac{C_1ϵ}{k}}\left(P+C_3{\displaystyle \frac{2}{3}}k\nabla ·\mathit{u}\right)-C_2\rho {\displaystyle \frac{{ϵ}^2}{k}}$$

(2)

$${\nu }_t=C_{\mu }{\displaystyle \frac{k^2}{ϵ}}$$

(3)

2.3. Neighborhood Effects

Urban development makes the landscape more complex in terms of wind engineering and aerodynamic interactions, as flow variations can significantly change wind loads on structures. Most standards note that neighborhood effects cannot be quantified normatively, prompting increased research considering factors like building height, positioning, and wind incidence angle. It is worth noting that isolated building analysis does not account for changes in wind forces. Urban landscape changes can alter flow direction and velocity [29], making the study of interference effects an important step during the design phase. Blessmann [30] describes some of the main phenomena caused by neighborhood influence, such as hammering, where Kármán vortices cause periodic excitation in downstream buildings; the Venturi effect, where flow acceleration alters pressure coefficients; and wake turbulence. These effects are shown in Figure 2.

2.4. Structural Bracing Systems

The efficiency of a bracing system aligns with architectural trends [31], as the visual complexity of an optimized structural system offers the possibility of designs with a high aesthetic standard. The design phase depends on various factors, such as local landscape to ensure the building blends with its surroundings, wind load intensity, geometric proportions of the building, economic capacity, and local culture. As a result, there are different types of bracing systems. Gunel and Ilgin [32] classified them into the following six types:

• Rigid frames: the base elements and their connections are designed to be stiffer;
• Shear walls: adding walls to frames to increase lateral stiffness and torsional resistance;
• Outrigger: characterized by horizontal trusses or shear walls connected to the core of the structural system;
• Perforated tube systems: evenly spaced columns are connected by tall beams;
• Shear trusses: formed by placing trusses at the exterior of the structure, typically at a 45° angle;
• Tube systems: involving metal tubes arranged in irregular geometries.

Recent studies, such as [33], have presented new bracing systems, such as diagrid, which is not only efficient but also aesthetically pleasing, making it highly attractive. Another popular system in tall buildings is space trusses. Additionally, modifications to classic systems include megatubes and megaframes.

2.5. Topology Optimization and BESO Method

In structural engineering, mathematical techniques can be applied to minimize or maximize properties of interest to the designer, such as minimizing displacements or von Mises stresses or maximizing the system’s overall stiffness. Sotiropoulos and Lagaros [34] used topology optimization (TO) for designing moment-resisting bracing frames for tall buildings under dynamic seismic loading. Resende et al. [35] presented a multi-objective structural optimization approach to determine the optimal grouping of columns in 3D steel frames by simultaneously minimizing structural weight and the number of distinct column groups through evolutionary methods to identify Pareto front solutions. Negarestani et al. [36] aimed to calculate weight, cost, and construction time optimization by using metaheuristic algorithms.

TO encompasses several methodologies but with one goal: find the optimal material distribution within a given domain, subject to a prescribed volume and boundary conditions. Figure 3 illustrates a TO example, featuring a cantilever beam with a point load at the free end, reducing 70% of its original volume. Gao et al. [37] classified them into the following four categories: density-based method, such as Solid Isotropic Material with Penalization (SIMP) and Rational Approximation of Material Properties (RAMP); level set-based methods; moving morphable components/voids (MMC/V); and other types, such as evolutionary (or discrete) methods and phase field methods. Ribeiro et al. [38] reviewed recent advancements for structural steel design and applications in additive manufacturing (AM), showing applications of TO in the construction industry, such as bridge design, steel elements and joints, and bracing systems.

For this work, the Bi-Directional Evolutionary Structural Optimization (BESO) [39] was adopted. It was developed to address the criticism of discrete methods by enabling the simultaneous addition and removal of mesh elements. Based on the objective function’s sensitivity and the Finite Element Method (FEM), solid elements are added or removed from the domain based on two parameters: the inclusion ratio (IR) and the rejection ratio (RR). The problem definition is given by the following Equations (4)–(7):

$$\min C\left(\mathsf{{\rm P}}\right)=\sum _{e=1}^{n_{elem}}{\left({\rho }_e\right)}^p{\left[{\mathit{u}}_e\right]}^T\left[{\mathit{K}}_e\right]\left[{\mathit{u}}_e\right]$$

(4)

s.t.

$$V\left(\mathsf{{\rm P}}\right)=f_v-\sum _{e=1}^{n_{elem}}{V}_e{\rho }_e=0$$

(5)

$$\mathit{K}\mathit{U}=\mathit{F}$$

(6)

$${\rho }_e={\rho }_{min}or1$$

(7)

where $C$ represents the mean compliance of the structure (equivalent to strain energy), $\mathit{F}$ and $\mathit{U}$ are the global force and displacement vectors, $\mathit{K}$ is the global stiffness matrix, $f_v$ is the prescribed volume (or volume fraction), and $n_{elem}$ is the number of elements in the mesh. The binary discrete variable ${\rho }_e$ represents the element density: 10⁻³ for empty space (to avoid singularities) and 1 for solid regions.

The evolutionary process of the BESO method is controlled by the element sensitivity number (${\alpha }_e$), defined as the change in $C$ when an element is added or removed, which is equal to the element strain energy. Sensitivity analysis can be performed by varying $C$ with respect to design variables, as described by the following Equation (8):

$${\displaystyle \frac{dC}{d{\rho }_e}}={\displaystyle \frac{1}{2}}{\displaystyle \frac{d{\mathit{F}}^T}{d{\rho }_e}}\mathit{U}+{\displaystyle \frac{1}{2}}{\mathit{F}}^T{\displaystyle \frac{d\mathit{U}}{d{\rho }_e}}$$

(8)

By applying the material interpolation scheme (see [39] for more details), the sensitivity is expressed in the form of Equations (9) and (10). The elemental sensitivity numbers alfa_e can be calculated by the relative positioning in the sensitivity analysis for an individual void or solid element. IR and RR are based on this coefficient.

$${\displaystyle \frac{dC}{d{\rho }_e}}={\displaystyle \frac{p}{2}}{\rho }_e^{p-1}{\left[{\mathit{u}}_e\right]}^T\left[{\mathit{K}}_e\right]\left[{\mathit{u}}_e\right]$$

(9)

$${\alpha }_e=-{\displaystyle \frac{1}{p}}{\displaystyle \frac{\partial C}{\partial {\rho }_e}}=\left\{\begin{array}{cc}{\displaystyle \frac{1}{2}}{\left[{\mathit{u}}_e\right]}^T\left[{\mathit{K}}_e\right]\left[{\mathit{u}}_e\right]& for{\rho }_e=1\\ {\displaystyle \frac{{\rho }_{min}^{p-1}}{2}}{\left[{\mathit{u}}_e\right]}^T\left[{\mathit{K}}_e\right]\left[{\mathit{u}}_e\right]& for{\rho }_e={\rho }_{min}\end{array}\right.$$

(10)

TO problems are also prone to numerical instabilities, such as checkerboard patterns, mesh dependency, and local minima [40]. To address these issues, a filtering scheme can be applied. Various filters for TO are discussed in the literature, including density and sensitivity filters used in density-based methods [9] and the Helmholtz-type filter [41], among others. These filtering schemes are heuristic in nature but have demonstrated their effectiveness in TO problems to address numerical instabilities [42]. In this study, the adopted filtering scheme follows the approach of [43,44] for 3D BESO applications.

3. Methodology

This section presents the methodology of the CFD approach, the determination of wind loads using OpenFOAM v6, numerical schemes, the BESO methodology to add or remove solid elements into the mesh, and geometric control to make the results more suitable for the civil construction industry perspective.

3.1. CFD Setup

To define the domain, the recommendations of [45,46] were followed. These studies provide guidelines for accurately predicting airflow around isolated buildings and groups of buildings. The guidelines are based on high Reynolds number (Re) simulations using the RANS methodology. The building, with proportions B:W:H (base: width: height), should be positioned 5H from the domain inlet (upstream) and 15H from the outlet (downstream). The sides should be 5H from the buildings, ensuring symmetry along the central longitudinal plane. The domain height should extend 5H above the building, as shown in Figure 4.

For this study, the surrounding area consists of eight equidistant buildings. The distance between them is 1B, where B is the square base dimension. To simulate a natural atmospheric boundary layer (ABL), boundary conditions are applied to the domain faces and buildings. The flow is modeled to simulate the ABL according to a logarithmic function, as seen in [47,48]. The reference velocity and height are, respectively, $v_{ref}$ = 10 m/s and $z_{ref}$ = 10 m. The aerodynamic roughness considered is $z_0$ = 0.0024 m. Since the analyses consider flat terrain without obstacles, the vertical displacement adjustment parameter is set as $d$ = 0 m. The velocity profile $v\left(z\right)$ and the friction velocity $v^{*}$ for the inlet are given by Equations (11) and (12), where $\kappa$ = 0.40 is the von Kárman constant.

$$v\left(z\right)={\displaystyle \frac{v^{*}}{\kappa }}\ln \left({\displaystyle \frac{z-d+z_0}{z_0}}\right)$$

(11)

$$v^{*}={\displaystyle \frac{v_{ref}\kappa }{\ln \left(\frac{z-d+z_0}{z_0}\right)}}$$

(12)

Modeling of the TKE is given by Equation (13), where $C_1$ = 0 and $C_2$ = 1 are constants for fitting to the ABL profile. The dissipation $ϵ$ is given by Equation (14). On the outlet boundary, a Neumann condition with a zero gradient is applied, meaning $\nabla \varphi ·\mathit{n}=0$, allowing values for nearby volumes to be extrapolated downstream in the domain. Symmetry planes are assigned to the lateral and upper boundaries. Wall functions are implemented to model the buildings and the ground. The initial conditions governing TKE and dissipation are derived from the work of [49], which assumes isotropic turbulence.

$$k={\displaystyle \frac{v^{*}}{\sqrt{C_{\mu }}}}\sqrt{C_1\ln \left({\displaystyle \frac{z-d+z_0}{z_0}}\right)+C_2}$$

(13)

$$ϵ={\displaystyle \frac{{\left(v^{*}\right)}^3}{\kappa \left(z-d+z_0\right)}}\sqrt{C_1\ln \left({\displaystyle \frac{z-d+z_0}{z_0}}\right)+C_2}$$

(14)

Finally, when studying wind engineering in urban environments, the buildings must be discretized with more detail than the rest of the domain. To achieve this, the snappyHexMesh utility from OpenFOAM was used. It generates 3D meshes consisting of hexahedra and split-hexahedra based on triangulated surface geometries provided in STL format. By progressively refining an initial mesh and adjusting the split-hex cells to fit the surface, the mesh aligns approximately with the neighborhood geometry.

3.2. Wind Loads and Aerodynamic Coefficients

The forces are calculated by considering two components relative to the surface of the analyzed structures: the normal pressure force (${\mathit{F}}_p$) acting on the faces and the tangential viscous force (${\mathit{F}}_v$). These are determined using Equations (15) and (16), respectively, where ${\mathit{s}}_{f,i}$ is the face area vector, $p$ represents the pressure, $p_{ref}$ is the reference pressure (set to 0 Pa), $\mu$ is the dynamic viscosity, and ${\mathit{R}}_{dev}$ corresponds to the deviatoric stress tensor.

$${\mathit{F}}_p=\sum _i{\rho }_i{\mathit{s}}_{f,i}\left(p-p_{ref}\right)$$

(15)

$${\mathit{F}}_v=\sum _i{\mathit{s}}_{f,i}·\left(\mu {\mathit{R}}_{\mathit{d}\mathit{e}\mathit{v}}\right)$$

(16)

The combined contribution of these force components is used to calculate mean pressure, drag, moment, lift, and torsion coefficients. The drag direction is aligned with the X-axis, moment is aligned with the Y-axis, and torsion is aligned with the Z-axis (see Figure 4). Accordingly, these coefficients are computed using Equations (17)–(21), where $C_d$, $C_m$, $C_t$, $C_l$, and $C_p$ represent the drag, moment, torsion, and pressure coefficients, respectively. Here, $F_i$ and $M_i$ are the forces in the $i$ directions, $v_{\infty }$ = 10 m/s is the reference velocity, $L$ is the reference length, $S_a$ is the reference drag area (facade, used for drag and moment), and $S_b$ is the base reference area (used for torsion and lift).

$$C_d={\displaystyle \frac{F_x}{\frac{1}{2}\rho v_{\infty }^2{S}_a}}$$

(17)

$$C_m={\displaystyle \frac{M_y}{\frac{1}{2}\rho v_{\infty }^2{S}_{a}L}}$$

(18)

$$C_t={\displaystyle \frac{M_z}{\frac{1}{2}\rho v_{\infty }^2{S}_{b}L}}$$

(19)

$$C_l={\displaystyle \frac{F_z}{\frac{1}{2}\rho v_{\infty }^2{S}_b}}$$

(20)

$$C_p={\displaystyle \frac{p-p_{ref}}{\frac{1}{2}\rho v_{\infty }^2}}$$

(21)

The wind loads are applied to the structure through influence areas. The pressure field along the building’s facades is integrated over the areas described in Figure 5, with eight-point loads applied to the nodes. This procedure is similar to the simplification used in frame structures for dynamic analyses (shear-type), as seen in other TO studies, such as [50,51].

This study focuses on the mean pressure coefficient rather than the peak pressure coefficient due to the turbulence modeling approach employed. The RANS equations, utilized in the CFD simulations, solve the flow field by averaging the effects of turbulence over time. This time-averaging process effectively filters out transient wind fluctuations responsible for peak pressures, making RANS more suited for predicting mean pressure distributions. This approach is common in aerodynamic studies of buildings, as RANS models have been demonstrated to predict mean surface pressures with reasonable accuracy [52]. However, capturing peak pressure coefficients would require more advanced turbulence models, such as Large Eddy Simulation (LES) or hybrid RANS-LES approaches, which can resolve transient flow features but at a significantly higher computational cost [53]. Future research will explore these advanced models to better capture peak pressure variations.

3.3. Numerical Schemes

To address the simulation cases, the simpleFoam solver was applied. For the numerical strategy, the time scheme governs how a property evolves over time. In this case, the implicit Euler scheme, which is a first-order, stable, and transient scheme, was adopted. This method integrates properties implicitly, ensuring numerical stability even with larger time steps.

For the gradient terms, the Gauss linear method was utilized, based on Gaussian finite volume integration. This approach calculates gradients by summing values on cell faces interpolated from cell centers, ensuring consistency and accuracy. Furthermore, it imposes restrictions on gradient values, confining the extrapolated face with the range of neighboring cell values.

The divergence terms involve two components: advection and diffusion. These were discretized using the Gauss linear upwind method, a bounded first-order scheme. In this method, face values are determined based on the upstream value, assuming isotropic cell values that represent the average property. The divergence terms account for $\varphi$, velocity, TKE, and its dissipation, accurately modeling advection within the domain. Finally, the Laplacian terms were treated using the Gauss linear corrected scheme. To solve the system of equations, the GAMG (Generalized Geometric–Algebraic Multi-Grid) solver was employed for pressure, while the smoothSolver was used for velocity, TKE, and $ϵ$, leveraging iterative smoothing techniques, such as Gauss–Seidel, to enhance convergence efficiency. The following tolerances were adopted: $\tau$ = 10⁻⁶ for pressure, velocity, TKE, and $ϵ$; and 10⁻⁴ for residual control in the simpleFoam solver.

3.4. BESO Methodology for Rejection and Admission of Solid Elements

To initialize the optimization, the prescribed volume ($f_v$) is defined, representing the target percentage of solid material in the domain’s volume. Another important parameter is the evolutionary rate ($ER$), which defines the percentage of solid material to be removed in each iteration, with a typical value of 2%. The volume for the next iteration is adjusted according to Equation (5), until the volume constraint is met.

During the optimization, elements are ranked by sensitivity, and the RR and IR ratios are determined based on the sensitivity analysis. Elements are removed or added according to the sensitivity limit numbers (${\alpha }_{del}^{th}$ and ${\alpha }_{add}^{th}$, respectively). The procedure to calculate these parameters involves three steps: setting their initial values (minimum and maximum elemental sensitivity numbers for the current iteration, respectively); calculating the admission rate ($AR$); and adjusting the values of ${\alpha }_{del}^{th}$ and ${\alpha }_{add}^{th}$ based on the number of elements added. The following pseudocode shows the internal logic for the BESO subroutine, where $n_i$ represents the number of elements in the mesh in the $i$ direction.

• Initialization of parameters:
  • Let $\mathit{d}\mathit{c}$ be the sensitivity field for the structure where each value $d{c}_{i,j,k}$ represents the sensitivity of an individual element.
  • $\mathit{x}$ is the design variable vector, where each value $x_{i,j,k}$ represents the material density ${\rho }_e$ for solid and void elements.
  • ${\alpha }_{del}^{th}$ and ${\alpha }_{add}^{th}$ are the lower and upper bounds of the sensitivity field $\mathit{d}\mathit{c}$, respectively.
• Convergence criterion: the stopping condition is given by the relative difference between ${\alpha }_{del}^{th}$ and ${\alpha }_{add}^{th}$, which must be smaller than the tolerance $\tau$ = 10⁻⁵.
• Binary search algorithm:
  • At each iteration, the threshold $\psi$ is computed as follows:

    $$\psi ={\displaystyle \frac{{\alpha }_{del}^{th}+{\alpha }_{add}^{th}}{2}}$$

    (22)
  • $\mathit{x}$ is updated based on the comparison of $\mathit{d}\mathit{c}$ and $\psi$, where the “sign” function returns to 1 if the argument is positive and −1 if the argument is negative.

    $$x_{i,j,k}=\max \left(0.001,\mathrm{sign}\left(d{c}_{i,j,k}-\psi \right)\right)$$

    (23)
• Volume condition: The new volume is imposed by the following condition: if the sum of the densities is greater than the volume fraction, ${\alpha }_{del}^{th}$ is adjusted; otherwise, ${\alpha }_{add}^{th}$ is adjusted.

  $$\sum _{i,j,k}{x}_{i,j,k}-f_v·\left(n_x{n}_y{n}_z\right)>0$$

  (24)
• The process continues updating ${\alpha }_{del}^{th}$ and ${\alpha }_{add}^{th}$ until the convergence condition is met, resulting in a new vector $\mathit{x}$ with updated densities for the mesh elements.

The volume-stopping criterion is ensured when the difference between the current volume and the target volume $f_v$ is less than 0.5%. After reaching this volume, the process is terminated when the relative difference in $C$ between successive iterations is smaller than the tolerance $\tau$ = 10⁻³. To solve the linear system of equations associated with the finite element method in BESO, a geometric multigrid scheme, as described in [54], was employed to enhance computational efficiency. A complete flowchart detailing all the steps of the BESO process is presented in Figure 6, illustrating the optimization sequence.

3.5. Geometric Control of BESO Results

The use of structure optimized by TO in real problems was investigated by [55], who presents procedures to control the final geometry and facilitate its application in the construction industry or through AM. Three geometric controls were applied to the results: a symmetry constraint, repetition of geometric patterns, and application of passive elements in the core of the structure.

To perform the symmetry constraint, the elemental sensitivities are mapped in relation to the distance to one or more symmetry axes, as shown in Figure 7. This methodology was applied by other authors who analyzed TO in association with bracing structures, such as Sotiropoulos and Lagaros [34], who used the ground–structure method (GSM) to optimize structures subjected to real earthquakes through time–history data and its dynamic structural response; Stromberg et al. [56], who presented an integrated TO approach combining continuum and discrete elements to design optimal braced frames; and Gholizadeh and Ebadijalal [57], who investigated the optimal placement of bracing members in steel braced frames through performance-based discrete topology optimization (PBDTO). In addition to symmetry, pattern repetition was applied. In this process, a specific region of the domain is defined and repeated throughout the domain.

This approach is frequently used in this field of study, as it simplifies practical application in civil construction, as the components maintain the same geometric pattern throughout the structure. For example: Stromberg et al. [58] explored gradation and repetition constraints to facilitate constructability, reducing resource consumption and achieving structurally efficient lateral bracing systems; Chun et al. [59] incorporated pattern repetition constraints to enhance constructability and aesthetic flexibility for architectural and engineering applications; and more recently, Ma et al. [60] used TO for ribbed shell systems to improve performance, making user-defined patterns for cost-effective fabrication. Figure 8 illustrates the structural geometry with symmetry and pattern constraints.

Passive elements were applied to the center of the design domain (see [9,39] for further details) at the core of the structure. This restriction prevents the formation of solid elements within the inner region of the domain (in this case, the interior of the building), forcing the structure to focus solely on the facades, similar to many bracing systems. For further details about the BESO program, see Silva [61].

4. Validation

This section presents the validation of the BESO program and the CFD setup. Benchmark examples are analyzed, and the obtained results are compared with the literature data. BESO simulations were performed in MATLAB R2024b; OpenFOAM v6 simulations were performed with the following specifications: Intel Core i5 4200U processor with two cores at 1.60 GHz, 6 GB of RAM, and Ubuntu 18.04.6 LTS.

4.1. BESO Validation

To evaluate the BESO program, the Messerschmitt Bölkonw Blohm (MBB) 3D beam problem was employed. It consists of a prismatic beam with a square cross-section of edge length L and total length of 6 L. It is simply supported at the four lower vertices, with a unit load F applied at its center, as illustrated in Figure 9.

BESO results were compared to two SIMP-based TO codes: Amir et al. [54] utilized a geometric multigrid scheme; and Liu and Tovar [62], who used a 3D version of the original SIMP code by Sigmund [63] and Andreassen et al. [64], employing the pre-conjugate gradient method (PCG) for refined meshes. Solid material is assigned to elements with density ${\rho }_e$ > 0.50. The simulation parameters are: $E$ = 1 Pa, $\nu$ = 0.30, $\tau$ = 10⁻³, $p$ = 4, $r_{min}$ = 9 L/50, and $f_v$ = 0.25. Multigrid (MGCG) parameters are $n_l$ = 2 and ${\tau }_m$ = 10⁻⁸. To assess mesh independence, three mesh refinement levels of hexahedral meshes are used: 60 × 10 × 10 (M1), 120 × 20 × 20 (M2), and 180 × 30 × 30 (M3). The final geometries, objective function values (mean compliance $C$ in N·m), and processing times (normalized to M2/BESO’s 8 min and 39 s) are summarized in

Table 1. Result comparison of the three refinement levels.

Mesh Resolution	BESO (MGCG)	Liu and Tovar [62] SIMP (PCG)	Amir et al. [54] SIMP (MGCG)
60 × 10 × 10 (M1)	C = 20.57 Time: 0.11	C = 28.75 Time: 1.04	C = 24.02 Time = 0.22
120 × 20 × 20 (M2)	C = 11.61 Time: 1.00	C = 17.05 Time: 11.79	C = 16.49 Time: 1.52
180 × 30 × 30 (M3)	C = 9.79 Time: 4.13	C = 13.49 Time: 62.67	C = 11.54 Time: 4.30

.

Across all meshes, the final solutions remained consistent; the solutions remained consistent regardless of refinement level. BESO solutions were aligned closely with SIMP results in geometry, but there were notable differences in bar thickness for the Liu and Tovar program. Processing times were significantly shorter for the BESO and Amir et al. program.

On average, BESO converged 14.4 times faster than SIMP (PCG) and 1.15 times faster than SIMP (MGCG). For mean compliance, BESO solutions had 24.3% lower values. These differences rose from the presence of intermediate-density elements in SIMP solutions, which reduce mechanical resistance and increase strain energy.

The results demonstrate that the BESO program successfully overcame the mesh dependency issue through sensitivity filtering, ensuring the absence of checkerboard patterns. Additionally, it produced objective function values and geometries comparable to those obtained using other methodologies implemented in well-established academic programs. Additionally, BESO consistently achieved lower compliance values due to the absence of intermediate-density elements, while also delivering faster processing times.

4.2. CFD Validation

For the validation of the OpenFOAM setup and the adopted schemes, a benchmark case involving prismatic bodies was analyzed: two adjacent tall buildings with a geometric ratio of 1:1:4, with wind incidence angles of 0°, 15°, 30°, and 45° (see Figure 10). To assess the agreement between the experimental data and numerical simulations conducted in OpenFOAM, the $C_p$ values were calculated along the symmetry axes of the four building facades. These values were compared using a scatter plot, with experimental values on the horizontal axis and numerical values on the vertical axis.

Five statistical coefficients were calculated to quantify the agreement: the coefficient of determination (R²), the fractional bias (FB), the fraction of predictions within a factor of 2 of the observations (FAC2), root mean square deviation (RMSE), and mean absolute error (MAE). For R² and FAC2, values closer to 1 indicate better agreement between observed and simulated data. In the case of FB, RMSE, and MAE, the optimal agreement is achieved when the value is near 0. This approach is commonly used for validating numerical results in computational fluid dynamics, as seen in [65,66,67].

The results were evaluated using three levels of mesh refinement, approximately 375,000 (M1), 1,428,000 (M2), and 3,785,000 (M3) volumes, and they were compared with numerical wind tunnel tests from the Tokyo Polytechnic University—TPU database [68]. Figure 11 shows the results for 0°; Figure 12 shows those for 15°; Figure 13 shows those for 30°; and Figure 14 shows those for 45°.

Table 2. Agreement coefficients between experimental data and numerical results.

Case	Fit Equation	R2	FB	FAC2	RMSE	MAE
0° M1	1.063x + 0.019	0.9696	0.270	0.930	0.020	0.017
0° M2	1.049x + 0.021	0.9756	0.274	0.930	0.021	0.018
0° M3	0.995x + 0.011	0.9991	0.145	0.954	0.012	0.012
15° M1	1.003x − 0.087	0.8999	0.429	0.837	0.087	0.087
15° M2	1.065x − 0.052	0.9926	0.246	0.913	0.057	0.052
15° M3	1.047x − 0.058	0.9850	0.219	0.911	0.060	0.058
30° M1	1.132x + 0.027	0.8716	0.393	0.894	0.034	0.029
30° M2	1.063x + 0.021	0.9496	0.303	0.921	0.021	0.017
30° M3	0.997x + 0.008	0.9983	0.132	0.974	0.008	0.008
45° M1	1.174x + 0.002	0.9085	0.178	1.000	0.061	0.053
45° M2	1.141x + 0.015	0.9175	0.226	0.980	0.052	0.044
45° M3	1.128x + 0.018	0.9237	0.234	0.980	0.048	0.041

presents the statistical coefficients related to the comparison, as well as the fit equations.

The results indicate variations in the level of agreement between experimental and simulated data depending on the different wind incidence angles. Mesh refinement proved effective in improving the coefficient R² in almost all cases, except for $\theta$ = 15°, where M2 had a slightly better agreement with the experimental data. However, this difference is negligible in practical terms. Moreover, the FB indicates that M3 had a lower tendency to underestimate or overestimate the actual results. The FAC2 values for both meshes also suggest a minimal practical difference between their results. Average values of R² = 0.9592, FB = 0.298, RMSE = 0.068, MAE = 0.066, and FAC2 = 0.887 indicated a good agreement with the experimental data.

For $\theta$ = 0°, the highest agreement was observed, with an average R² = 0.9814. The mean fractional bias FB = 0.230 suggests a tendency of the model to overestimate positive pressure values, while the mean FAC2 = 0.938 confirms a robust relationship between the simulated and experimental results. Additionally, the average RMSE and MAE were 0.018 and 0.016, respectively, further supporting the accuracy of the model. For $\theta$ = 30°, good agreement was also observed, with a mean R² = 0.9398. The mean FB = 0.276 indicates that most analyzed points lie above the ideal line, as shown in Figure 12. Despite the overestimation of some pressure values by the $k-ϵ$ model, the mean FAC2 = 0.930 demonstrates a consistent relationship between experimental and simulated data. Average RMSE and MAE were 0.021 and 0.018, respectively, further supporting the model’s reliability. It was observed that these two statistical factors reached their best value among all analyses in this case for mesh M3, with RMSE = 0.008 and MAE = 0.008. For $\theta$ = 45°, the lowest agreement values were observed, with a mean R² = 0.9166, RMSE = 0.054, and MAE = 0.046. However, the mean FB = 0.213 and FAC2 = 0.980 indicated a strong agreement between numerical and experimental results, despite the regression line deviating further from the ideal compared to 0°, 15°, and 30°. Overall, while the $k-ϵ$ model tended to overestimate pressure values in positive regions and underestimate them in negative regions, it effectively simulated wind flow in an urban environment, accounting for the influence of neighboring buildings. This confirms its suitability for analyzing the proposed cases, which employed a mesh resolution similar to M3.

5. Results

This section analyzes the influence of neighboring buildings on the bracing system by comparing the structural geometry and the objective function of the structural system. The cases are categorized based on the height ($h$) of the neighboring buildings and wind incidence angle ($\theta$). The BESO parameters are evolutionary rate $ER$ = 1%; target volume $f_v$ = 50%; mesh of 32 × 32 × 192 (196,608) elements; sensitivity filter $r_{min}$ = $\sqrt{3}$ L; and penalty factor $p$ = 3.

Geometric controls were applied in all cases, including symmetry to all faces, repetition of a geometric pattern (unit cell) four times along the z-axis, and passive elements in the domain’s core. All analyzed buildings shared proportions of L × L × 6 L (L = 1). On average, OpenFOAM simulations involved approximately 4.7 million finite volumes. The Reynolds number was computed as Re = 100,000. The simulated cases are presented in Figure 15a: the analyzed building is highlighted in blue. Each case was simulated considering four wind incidence angles: $\theta$ = 0°, 15°, 30°, and 45° relative to the facade plane, as illustrated in Figure 15b. The spacing between buildings is the same as the base edge, L, while the evaluated heights range from 0 L (isolated case) to 6 L (matching the height of the analyzed building) in increments of 1 L. Results are presented in Appendix A through

Table A1. Topology optimization results for $h$ = 0 L.

$\mathit{C}$	Vorticity Field	Mean Pressure Coefficient Field
36.73 N⋅m	$C_t$ = 0.000  $C_m$ = 0.584	$C_d$ = 1.024  $C_l$ = 0.554
57.00 N⋅m	$C_t$ = 0.225  $C_m$ = 0.651	$C_d$ = 1.140  $C_l$ = 0.591
56.76 N⋅m	$C_t$ = 0.176  $C_m$ = 0.692	$C_d$ = 1.122  $C_l$ = 0.674
58.55 N⋅m	$C_t$ = 0.000  $C_m$ = 0.737	$C_d$ = 1.293  $C_l$ = 0.706

,

Table A2. Topology optimization results for $h$ = 1 L.

$\mathit{C}$	Vorticity Field	Mean Pressure Coefficient Field
37.78 N⋅m	$C_t$ = 0.000  $C_m$ = 0.610	$C_d$ = 0.995  $C_l$ = 0.526
53.31 N⋅m	$C_t$ = 0.221  $C_m$ = 0.632	$C_d$ = 1.069  $C_l$ = 0.597
54.92 N⋅m	$C_t$ = 0.168  $C_m$ = 0.669	$C_d$ = 1.163  $C_l$ = 0.654
53.97 N⋅m	$C_t$ = 0.000  $C_m$ = 0.706	$C_d$ = 1.244  $C_l$ = 0.677

,

Table A3. Topology optimization results for $h$ = 2 L.

$\mathit{C}$	Vorticity Field	Mean Pressure Coefficient Field
32.99 N⋅m	$C_t$ = 0.000  $C_m$ = 0.541	$C_d$ = 0.817  $C_l$ = 0.547
42.33 N⋅m	$C_t$ = 0.184  $C_m$ = 0.591	$C_d$ = 0.929  $C_l$ = 0.612
45.80 N⋅m	$C_t$ = 0.141  $C_m$ = 0.641	$C_d$ = 1.050  $C_l$ = 0.670
43.24 N⋅m	$C_t$ = 0.000  $C_m$ = 0.660	$C_d$ = 1.105  $C_l$ = 0.661

,

Table A4. Topology optimization results for $h$ = 3 L.

$\mathit{C}$	Vorticity Field	Mean Pressure Coefficient Field
20.23 N⋅m	$C_t$ = 0.000  $C_m$ = 0.474	$C_d$ = 0.668  $C_l$ = 0.536
29.81 N⋅m	$C_t$ = 0.142  $C_m$ = 0.522	$C_d$ = 0.765  $C_l$ = 0.621
36.36 N⋅m	$C_t$ = 0.127  $C_m$ = 0.563	$C_d$ = 0.875  $C_l$ = 0.661
33.16 N⋅m	$C_t$ = 0.000  $C_m$ = 0.576	$C_d$ = 0.921  $C_l$ = 0.633

,

Table A5. Topology optimization results for $h$ = 4 L.

$\mathit{C}$	Vorticity Field	Mean Pressure Coefficient Field
11.06 N⋅m	$C_t$ = 0.000  $C_m$ = 0.351	$C_d$ = 0.464  $C_l$ = 0.526
17.95 N⋅m	$C_t$ = 0.101  $C_m$ = 0.407	$C_d$ = 0.571  $C_l$ = 0.614
23.94 N⋅m	$C_t$ = 0.109  $C_m$ = 0.446	$C_d$ = 0.677  $C_l$ = 0.650
19.98 N⋅m	$C_t$ = 0.000  $C_m$ = 0.445	$C_d$ = 0.708  $C_l$ = 0.607

,

Table A6. Topology optimization results for $h$ = 5 L.

$\mathit{C}$	Vorticity Field	Mean Pressure Coefficient Field
3.68 N⋅m	$C_t$ = 0.000  $C_m$ = 0.190	$C_d$ = 0.253  $C_l$ = 0.463
8.38 N⋅m	$C_t$ = 0.070  $C_m$ = 0.250	$C_d$ = 0.362  $C_l$ = 0.571
11.87 N⋅m	$C_t$ = 0.105  $C_m$ = 0.284	$C_d$ = 0.468  $C_l$ = 0.580
9.48 N⋅m	$C_t$ = 0.000  $C_m$ = 0.282	$C_d$ = 0.499  $C_l$ = 0.534

and

Table A7. Topology optimization results for $h$ = 6 L.

$\mathit{C}$	Vorticity Field	Mean Pressure Coefficient Field
0.29 N⋅m	$C_t$ = 0.000  $C_m$ = 0.022	$C_d$ = 0.039  $C_l$ = 0.248
2.41 N⋅m	$C_t$ = 0.060  $C_m$ = 0.098	$C_d$ = 0.191  $C_l$ = 0.331
8.30 N⋅m	$C_t$ = 0.142  $C_m$ = 0.169	$C_d$ = 0.340  $C_l$ = 0.403
5.46 N⋅m	$C_t$ = 0.000  $C_m$ = 0.195	$C_d$ = 0.394  $C_l$ = 0.420

, presenting the objective function $C$, vorticity field, mean pressure field, and aerodynamic coefficients for each case, along with the flow streamlines near the central building.

5.1. Aerodynamic Coefficients Analysis

The presence of surrounding buildings significantly alters the aerodynamic coefficients of the central building. Figure 16 illustrates the variation of aerodynamic coefficients among the cases. The base moment and drag naturally exhibit similar trends, as they are both determined by the set of forces acting in the same direction as the flow. The peak occurs in the isolated case at $\theta$ = 45°, as the direct wind exposure affects a larger area on the windward face, with two facades being impacted simultaneously. This pattern is observed in all building heights, although the magnitude decreases as the height of the surrounding buildings increases. This indicates that, due to the proximity between buildings and their perfect alignment, the neighboring structures provide a shielding effect against the wind flow on the central building, considering only drag and base moment.

Although less relevant for the aerodynamic analysis of tall buildings, the lift coefficient must be also considered, as higher values can pose risks such as roof or window detachment and other negative pressure effects. Its behavior follows a similar trend to that observed for the moment and drag coefficients. However, while these two coefficients exhibit a significant decrease with the addition of surrounding buildings, the lift coefficient declines at a slower rate. This indicates that, although the neighboring structures provide some shielding effect against lift forces, it is less pronounced compared to moment and drag.

The influence of neighboring structures on the response exhibits a distinct behavior for the torsional coefficient at different wind incidence angles. For $\theta$ = 0° and $\theta$ = 45°, the symmetry of the building arrangement relative to the wind flow prevents the occurrence of torsion, which only appears at $\theta$ = 15° and $\theta$ = 30°. At $\theta$ = 15°, the peak torsional moment is observed in the isolated case, decreasing as the height of the surrounding buildings increases. This torsional effect arises from the direct wind flow striking the windward facade with greater intensity than the other surfaces, generating a predominant torsion effect, as illustrated in

Table 3. Flow streamlines near the central building.

Height	$\mathit{\theta }$ = 15°	$\mathit{\theta }$ = 30°
$h$ = 4 L
$h$ = 5 L
$h$ = 6 L
Legend	Velocity field	TKE field

. However, as the height of the neighboring structures increases, vortices begin to form between the blocks, and the direct wind flow predominantly traverses tangentially to the lateral faces, as seen in the case with $\theta$ = 15° and $h$ = 6 L from Table 3, thereby reducing the torsional effect on the central building.

At $\theta$ = 30°, although the maximum $C_t$ value also occurs in the absence of neighboring buildings, a secondary peak emerges at $h$ = 6 L. This phenomenon is attributed to the formation of channeled flows within the buildings corridors, directing the air flow with minimal disturbance toward the central structure, as observed when comparing the $\theta$ = 30° column from Table 3. The torsional effect is more pronounced at $\theta$ = 30° than at $\theta$ = 15° from $h$ = 4 L onward because, at this angle, the airflow continues to directly impact the central building, contrary to $\theta$ = 15° cases. Additionally, the increasing height of the surrounding structures amplifies the Venturi effect experienced by the system, causing the central building to experience a greater torsional effect.

5.2. Wind Flow and Vortices Analysis

Regarding the flow behavior, the Tables in Appendix A reveal that the wake is significantly altered by the presence of surrounding buildings, becoming increasingly complex as the height of the neighboring structures increases, with large vortices forming particularly at $\theta$ = 45°. The streamlines at a $\theta$ = 0° wind incidence exhibit a less chaotic flow pattern, with vortices developing within the building array and in the wake region. However, no significant deviation of the streamlines is observed downstream, allowing the flow to follow a conventional path. At $\theta$ = 0°, it is also evident that the upstream buildings shield the central structure from the direct wind exposure, a phenomenon that becomes more apparent when comparing the isolated configuration with the $h$ = 6 L case.

As the alignment of the buildings shifts progressively from $\theta$ = 15° to $\theta$ = 45°, an increase in flow complexity is observed, with large vortices forming downstream and a more chaotic arrangement of streamline forming within the buildings array. This turbulence can lead to pedestrian discomfort and impose unforeseen loads on potential structures located in the wake of the analyzed buildings. Another factor influencing the flow behavior is the presence of surrounding buildings within the arrangement, which increases the aerodynamic complexity of the system and leads to the formation of large-scale vortices downstream. This effect is illustrated in

Table 4. Flow streamlines near the neighborhood blocks.

Wind Angle	$\mathit{h}$ = 0 L	$\mathit{h}$ = 6 L
$\theta$ = 0°
$\theta$ = 15°
$\theta$ = 30°
$\theta$ = 45°
Legend	Velocity field	TKE field

, which compares the flow patterns through streamlines visualized from a height $h$ = 3 L downward for both the isolated and neighboring configurations, where the surrounding buildings have a height of $h$ = 6 L. In the isolated cases, greater flow complexity and vortex formation occur at angles different from $\theta$ = 0°, particularly when neighboring structures are present. This observed behavior highlights the importance of wind engineering in architectural and urban planning, as such aerodynamic effects are not generally accounted for in standard regulations, requiring specific analyses to assess potential wind-induced impacts caused by nearby structures.

5.3. Bracing Systems and Objective Function Analysis

Regarding the optimized structures, by analyzing the results from Appendix A, in Table A1, Table A2, Table A3, Table A4, Table A5, Table A6 and Table A7, it is observed that their geometry follows a pattern based on the aerodynamic coefficients obtained. When the torsional moment is significantly lower than the base moment, the material distribution tends to adopt a simpler geometry, characterized by four diagonal bars intersecting at the center of each repetition cell, forming an angle ranging approximately from 80° to 100° between them. Most optimized structures at wind incidence angles of $\theta$ = 0° and $\theta$ = 45° exhibit negligible torsional coefficients due to the symmetry effect induced by the arrangement of the blocks. Consequently, except for the cases of $h$ = 5 L at 0° and $h$ = 2 L at 45°, the solutions obtained using the BESO method present nearly identical geometries. The occurrence of these two distinct solutions can be attributed to the fact that, although the sensitivity filter mitigates the effects associated with local minima in the solution, the heuristic nature of this optimization approach does not guarantee the complete elimination of such numerical difficulties in all cases.

A similar behavior is observed for wind incidence angles of $\theta$ = 15° and 30° for values of $h$ < 3 L, where drag and base moment dominate over torsion. However, as the height of the surrounding buildings increases, providing a shielding effect against direct wind impact on the facades, torsion becomes more prominent in the structural analysis. Consequently, the optimal material distribution leads to the inclusion of more slender bars throughout the geometry to compensate for the shear effects induced by torsion, typically oriented at approximately 45° relative to the ground. The most significant case occurs at $\theta$ = 30°, with $h$ = 6 L seen in Table A7, where torsion and base moment reach similar magnitudes, resulting in a denser distribution of structural bars to withstand these forces effectively.

The objective function values and the graph in Figure 17 indicate that, except for the isolated configuration at $\theta$ = 45° exhibiting the highest $C$ value, the peak generally occurs at $\theta$ = 30°. This is attributed to the significant torsional effects observed at this angle and the direct wind impact at two facades, particularly for $h$ = 6 L when compared to other cases. The graph’s behavior follows the trend observed in the aerodynamic coefficients of drag and moment, with the added influence of torsion, which contributes to higher structural demands at $\theta$ = 30°. In the isolated case, as the airflow directly impacts only one facade and torsional effects are negligible, the objective function values are the lowest, which is an expected outcome given the aerodynamic coefficients for this incidence angle.

5.4. Further Discussion

The influence of neighboring structures on the aerodynamic coefficients has been extensively investigated. Du et al. [69] demonstrated that the presence of adjacent buildings can induce significant amplification or shielding effects on wind pressure, depending on the relative height and rotation angle of the neighboring blocks. These findings align with the observations made in the present study, which identified a reduction in drag and base moment coefficients as the height of adjacent buildings increased, attributed to the partial deflection of wind flow by surrounding structures.

Furthermore, the impact of neighboring structures on the torsional dynamics of the central building was also examined. The presence of adjacent buildings can induce more pronounced torsional effects, as evidenced by the study of Dokhaei et al. [70], which reported that interference from neighboring structures significantly amplified crosswind vibrations and torsional response due to vortex shedding phenomena. In the present study, torsional effects became increasingly relevant with asymmetric wind incidence angles, with a more pronounced response observed in cases involving taller adjacent buildings, consistent with the findings of Dokhaei et al [70].

Regarding the optimized material distribution using BESO, the bar configurations and structural behavior align with the results presented by Goli et al. [71], which highlighted the efficacy of BESO in optimizing lateral bracing systems for high-rise buildings under wind loads. The present work showed that material distribution reflected the predominance of drag and base moment forces, except in cases where torsion became more significant, resulting in a denser bar arrangement. The structural efficiency and aesthetics by optimized BESO distributions were also reported by the study of Baghbanan et al. [72], which investigated architecturally optimized forms for tall buildings under lateral loads. The use of this TO tool to enhance structural efficiency while accounting for wind and gravity effects emerges as a promising tool for fostering collaboration between engineers and architects, facilitating the development of aesthetically and structurally efficient solutions.

6. Conclusions

This study analyzed the impact of surrounding buildings on the aerodynamics of tall structures, considering different wind incidence angles and varying the height of adjacent buildings. Numerical simulations were conducted using OpenFOAM and validated with experimental data from Tokyo Polytechnic University (TPU). Additionally, topology optimization (TO) was applied using the Bi-directional Evolutionary Structural Optimization (BESO) method to investigate the influence of pressure distribution on facades in determining the optimal material layout for designing an optimized bracing system.

The results indicated that the presence of surrounding buildings significantly alters the aerodynamic response of the central structure, especially for wind incidence angles where no symmetrical conditions on building block arrangement exist. It was observed that, for $\theta$ = 15° and $\theta$ = 30°, the torsional effect became more relevant due to asymmetric flow patterns and the formation of flow corridors and vortices inside the neighborhood that enhanced complex aerodynamic effects. Furthermore, a shielding effect provided by the surrounding buildings was found to reduce drag and base moment values, while having a lesser influence on lift.

The TO results showed that material distribution followed a pattern influenced by the predominance of different aerodynamic coefficients. In configurations where torsion was negligible, the optimized structure exhibited bars arranged in an “X” shape, with angles ranging from 80° to 100° between the bars. However, in cases where the torsional effect became significant, such as incidence angle $\theta$ = 30° with surrounding buildings with height close to the analyzed structure, the material distribution resulted in a greater number of bars that resisted shear forces caused by torsion. This behavior reinforces the importance of considering wind engineering studies in the design of tall structures in dense urban environments, where wind flow complexity and turbulence are generally high.

The authors highlight that the methodology adopted in this study has certain limitations. This study considered a simple building arrangement within a small neighborhood with perfectly aligned structures with the same base area, yet the results already demonstrated complex flow behavior and aerodynamic interactions. Additionally, using the RANS turbulence model with the $k-ϵ$ closure introduced limitations to the analysis, as it considered time-average flow characteristics. In real-world scenarios, the configuration of urban environments and the distribution of buildings are highly irregular, influenced by economic, cultural, and regulatory factors. Moreover, wind flow and interactions tend to be far more complex in actual urban settings and experimental observations, making it advisable to employ more robust turbulence schemes, such as RANS $k-\omega$ SST or Large Eddy Simulations (LES). Nevertheless, the idealized configuration used in this study effectively demonstrates that even simplified cases exhibit significant aerodynamic complexity, emphasizing the critical role of wind engineering in building design within urban environments and in the planning of urban areas and infrastructure.