Application of Topology Optimization as a Tool for the Design of Bracing Systems of High-Rise Buildings
Abstract
1. Introduction
- 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.
2. Literature Survey
2.1. Wind Loads and CFD
2.2. Turbulence Modeling
2.3. Neighborhood Effects
2.4. Structural Bracing Systems
- 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.
2.5. Topology Optimization and BESO Method
3. Methodology
3.1. CFD Setup
3.2. Wind Loads and Aerodynamic Coefficients
3.3. Numerical Schemes
3.4. BESO Methodology for Rejection and Admission of Solid Elements
- Initialization of parameters:
- Let be the sensitivity field for the structure where each value represents the sensitivity of an individual element.
- is the design variable vector, where each value represents the material density for solid and void elements.
- and are the lower and upper bounds of the sensitivity field , respectively.
- Convergence criterion: the stopping condition is given by the relative difference between and , which must be smaller than the tolerance = 10−5.
- Binary search algorithm:
- At each iteration, the threshold is computed as follows:
- is updated based on the comparison of and , where the “sign” function returns to 1 if the argument is positive and −1 if the argument is negative.
- Volume condition: The new volume is imposed by the following condition: if the sum of the densities is greater than the volume fraction, is adjusted; otherwise, is adjusted.
- The process continues updating and until the convergence condition is met, resulting in a new vector with updated densities for the mesh elements.
3.5. Geometric Control of BESO Results
4. Validation
4.1. BESO Validation
4.2. CFD Validation
5. Results
5.1. Aerodynamic Coefficients Analysis
5.2. Wind Flow and Vortices Analysis
5.3. Bracing Systems and Objective Function Analysis
5.4. Further Discussion
6. Conclusions
Author Contributions
Funding
Data Availability Statement
Conflicts of Interest
Appendix A
Vorticity Field | Mean Pressure Coefficient Field | |
---|---|---|
36.73 N⋅m | = 0.000 = 0.584 | = 1.024 = 0.554 |
57.00 N⋅m | = 0.225 = 0.651 | = 1.140 = 0.591 |
56.76 N⋅m | = 0.176 = 0.692 | = 1.122 = 0.674 |
58.55 N⋅m | = 0.000 = 0.737 | = 1.293 = 0.706 |
Vorticity Field | Mean Pressure Coefficient Field | |
---|---|---|
37.78 N⋅m | = 0.000 = 0.610 | = 0.995 = 0.526 |
53.31 N⋅m | = 0.221 = 0.632 | = 1.069 = 0.597 |
54.92 N⋅m | = 0.168 = 0.669 | = 1.163 = 0.654 |
53.97 N⋅m | = 0.000 = 0.706 | = 1.244 = 0.677 |
Vorticity Field | Mean Pressure Coefficient Field | |
---|---|---|
32.99 N⋅m | = 0.000 = 0.541 | = 0.817 = 0.547 |
42.33 N⋅m | = 0.184 = 0.591 | = 0.929 = 0.612 |
45.80 N⋅m | = 0.141 = 0.641 | = 1.050 = 0.670 |
43.24 N⋅m | = 0.000 = 0.660 | = 1.105 = 0.661 |
Vorticity Field | Mean Pressure Coefficient Field | |
---|---|---|
20.23 N⋅m | = 0.000 = 0.474 | = 0.668 = 0.536 |
29.81 N⋅m | = 0.142 = 0.522 | = 0.765 = 0.621 |
36.36 N⋅m | = 0.127 = 0.563 | = 0.875 = 0.661 |
33.16 N⋅m | = 0.000 = 0.576 | = 0.921 = 0.633 |
Vorticity Field | Mean Pressure Coefficient Field | |
---|---|---|
11.06 N⋅m | = 0.000 = 0.351 | = 0.464 = 0.526 |
17.95 N⋅m | = 0.101 = 0.407 | = 0.571 = 0.614 |
23.94 N⋅m | = 0.109 = 0.446 | = 0.677 = 0.650 |
19.98 N⋅m | = 0.000 = 0.445 | = 0.708 = 0.607 |
Vorticity Field | Mean Pressure Coefficient Field | |
---|---|---|
3.68 N⋅m | = 0.000 = 0.190 | = 0.253 = 0.463 |
8.38 N⋅m | = 0.070 = 0.250 | = 0.362 = 0.571 |
11.87 N⋅m | = 0.105 = 0.284 | = 0.468 = 0.580 |
9.48 N⋅m | = 0.000 = 0.282 | = 0.499 = 0.534 |
Vorticity Field | Mean Pressure Coefficient Field | |
---|---|---|
0.29 N⋅m | = 0.000 = 0.022 | = 0.039 = 0.248 |
2.41 N⋅m | = 0.060 = 0.098 | = 0.191 = 0.331 |
8.30 N⋅m | = 0.142 = 0.169 | = 0.340 = 0.403 |
5.46 N⋅m | = 0.000 = 0.195 | = 0.394 = 0.420 |
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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 |
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 |
Height | = 15° | = 30° |
---|---|---|
= 4 L | ||
= 5 L | ||
= 6 L | ||
Legend | Velocity field | TKE field |
Wind Angle | = 0 L | = 6 L |
---|---|---|
= 0° | ||
= 15° | ||
= 30° | ||
= 45° | ||
Legend | Velocity field | TKE field |
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Silva, P.U.d.; Bono, G.; Greco, M. Application of Topology Optimization as a Tool for the Design of Bracing Systems of High-Rise Buildings. Buildings 2025, 15, 1180. https://doi.org/10.3390/buildings15071180
Silva PUd, Bono G, Greco M. Application of Topology Optimization as a Tool for the Design of Bracing Systems of High-Rise Buildings. Buildings. 2025; 15(7):1180. https://doi.org/10.3390/buildings15071180
Chicago/Turabian StyleSilva, Paulo Ulisses da, Gustavo Bono, and Marcelo Greco. 2025. "Application of Topology Optimization as a Tool for the Design of Bracing Systems of High-Rise Buildings" Buildings 15, no. 7: 1180. https://doi.org/10.3390/buildings15071180
APA StyleSilva, P. U. d., Bono, G., & Greco, M. (2025). Application of Topology Optimization as a Tool for the Design of Bracing Systems of High-Rise Buildings. Buildings, 15(7), 1180. https://doi.org/10.3390/buildings15071180