# Building Geometry as a Variable in Energy, Comfort, and Environmental Design Optimization—A Review from the Perspective of Architects

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## Abstract

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## 1. Introduction

_{2}emissions are rooted in the construction and operation of buildings [1,2], and approximately 50% of office and residential buildings’ operation consumption is due to HVAC demand [3,4]. Due to a growing population, and more households and floor space, an increase by approximately 28% is predicted by 2035. Although the design process and construction method is primarily responsible for the sustainability performance of buildings constructed, the prevailing conventional design method in industry relies almost completely on experience and includes only a limited number of concepts.

- What kind of BGDV-s exists in current BECEDO research?
- What are the roles and functions of the different BGDV-s in BECEDO investigations?
- What are the frequently applied simulation engines, optimization algorithms, and software frameworks (including the diverse methods) in BECEDO?
- What kind of space organization and shape defining solutions exist, and what are the pros and cons of each process?
- Which type of building performance is improved through BGDV-s, and how large is the impact on the improvements?
- Is it possible to create a link between the concrete building form and its geometry describing mathematical design variables? Where is the level of achievement in actual research to replicate shape with the use of BGDV-s?
- What is the potential of building shape optimization?
- How can the achieved results and knowledge be applied in further design projects?
- What kind of limitations and/or shortcomings can be detected?

## 2. Geometry as a Design Variable in BECEDO

#### 2.1. Building Geometry Variables on Basic Level of Complexity

^{2}. LCEI could be reduced by 65%, and the operation within LCEI decreased by 20%. Aspect ratio 1 was preferred for cost reduction, but the rectangular shape with a long side to the south performed better in energy efficiency. The trade-off between optimum aspect ratio for environment or for cost efficiency aspects is defined; nevertheless, the paper dealt only with simple geometries and variables and instead focused on presenting and proofing a mathematical method. Active system variables are often manipulated together with the previously mentioned main BEDV-s (e.g., WWR and window positions) and some typical BGDV-s; aspect ratio, number of floors, roof slope, size and area of the space [61,64,73,75,76], or the length and height of one wall of the simple box-shaped building or single-room [63,72,74,79]. These variables are commonly used because they are easy to quantify for the algorithms.

#### 2.2. Building Geometry Variables to Elaborate the Level of Complexity

_{2}emissions using WWR variations [39]. The optimal case served as input for a second optimization stage, using different envelope material thermal properties and brise soleil, sun spaces, as well as courtyards/greenhouses. In the first phase, a six-story cross-shaped building with 40% WWR to the south and 45% WWR for the west elevation was developed, saving 60.6% total annual energy demand compared to a worst-case scenario. In the second phase, the optimization of the envelope materials, sun spaces courtyard–greenhouse morphology, and the active systems deliver lower improvements: the CO

_{2}emissions and energy cost was reduced by 23%, and primary energy usage was reduced by 9%. This also reinforces that geometry contributes decisively to energy efficiency. The basic geometries were selected due to simple observations of typical building shapes in Rome, and the building physics analysis provides few new ideas. While the same useful floor space and volume was considered, the dimensions of the envelopes of different shapes and their corresponding investment cost was neglected. Only single-zone thermal simulation models of multi-story blocks are calculated, which can lead to inaccurate or unrealistic results. Due to the black box character of the genetic algorithm, there are multiple hypotheses possible reasons for counterintuitive solutions. For instance, in the Mediterranean climate, high solar reflectance (SR) values, and thin insulation layers are expected for envelope materials; however, the algorithm choose low SR materials and thicker insulation to maximize winter solar gain instead of summer overheating (cooling) strategies. This hypothesis is in accordance with other literature [77].

^{2}a savings) and thermal discomfort reduction (12.2%). A layout optimization of high-rise residential buildings was carried out to minimize air-conditioning and lighting energy demand [41]. T- and I-shaped buildings were generated depending on different orientation, site constraints, number and type of flats, arrangement of flats along corridors, rotation angle of wings, and accessibility (fire exits). Here, 13–33% electricity savings were achieved due to geometry optimization of the flat and building wing (high-rise layout and hence shape) configurations, orientation, and natural ventilation.

## 3. Algorithms in BGDV including BECEDO

## 4. Discussion

#### 4.1. Prevailing Trends in Building Shape Optimization

#### 4.2. Frequently Applied BGDVs in BECEDO

#### 4.3. Principles of Building Geometry Modeling in BECEDO Processes

#### 4.4. Categorization of Building Geometry Modeling in BECEDO

- Shape generation without energy evaluation and optimization.

- An agent-based topology finding system creates layouts as a topology finding process, generating sphere and capsule bubble agents. Interaction rules as attraction, repulsion, swap, and compression help to generate multi-agent systems as layout schemes. The developed software allows connected rooms (agents) to be dragged closer and unconnected rooms push each other away if close enough. A 3D grid-system based on rectangular cells converts the multi-agent layout into a space model [60].

- Shape modification with energy evaluation.

- Changing of diverse BGDVs.
- Modification of geometry and setup arrangement for urban blocks. Four- and eight-story simple prismatic building block morphology versions [63].

- Shape generation with energy evaluation.

- Shape grammar (converted into a parametric system). Variants programming: translating shape transformations (scale, reflection, translation, rotation) into equations and variables. Every room is modeled as a block with changeable aspects ratio, length, width, shape, size, and location. Automated energy evaluation of created forms [54].
- Generation of a large number of building geometry alternatives using “Evolutionary Program for Space Allocation Program” (EPSAP), including GA. A floor plan representation scheme, including total area for each story, construction area, circulation space area, and openings is randomly generated, and then the energy performance is calculated [55].
- “Building Modular Cells” (BMC) geometry generation technique is introduced to generate urban morphology based on a grid raster with various urban density building categories, building height classes, and urban patterns (street area between the building blocks) and a form generating algorithm, including architecturally eligible form selection. Space-units (modular cells) are arranged due to the form generation rules [56].

- Geometry modification with energy evaluation and optimization—simple complexity.

- Changing of diverse BGDVs.

- Geometry modification with energy evaluation and optimization—elaborate complexity.

- Changing of diverse BGDV-s
- Automated generation of 2D-layout and topology. Finding the best location and size of interrelated rectangular spaces. (1) Swapping the positions of two space units, (2) reallocation of positions, and (3) reducing the size of a space unit. Constraining overlapping, regulating connections of spaces, paths and access ways and the building envelope [93].
- Settlement scaled building block morphology optimization for solar energy use (heating, lighting). Diversely oriented building rows and courtyard block arrangements tested [44].
- A hybrid evolutionary algorithm is applied with constraint handling as an optimization method for urban building configurations. Urban block grid density (no. of blocks) and height optimization. Complex extension building and roof shapes adjacent to an existing house); rectangular, planned building’s volume, parametrized by Fourier series [91].
- Optimization of the geometry of simple office building blocks (energy demand) in a city quarter. Optimization of the horizontal and vertical positions of corner points [49].
- Geometry optimization of common linear (I), L- (L), court (O), C- (C), T- (T), H- (H), cross (X), and Y-shaped (Y) buildings (LOD 100). Multiple geometries with shape proportion (depth of the space/wings) modification as a ’gene’ under the same volume [39].

- Geometry generation with energy evaluation and optimization.

- “Building modular cells” (BMC), custom developed geometry generation technique based on a 4 × 4 grid raster to search for optimum solutions in five high-rise office buildings in five urban density (UD) areas. Space-units arranged according to combination rules [40].
- GA generates layouts in two phases: combination of different wings (rotation angle) to form the shape and arrangements of flats into the wings [41].

#### 4.5. Optimization Result Improvements in BGDVs Containing BECEDO

## 5. Conclusions

## Author Contributions

## Funding

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## Nomenclature

AC | Air-Conditioning |

ACH | Air Change |

ADV | Active Design Variable |

AEC | Architecture, Engineering and Construction |

ANN | Artificial Neural Network |

ANOVA | Analysis of Variance |

AR | Aspect Ratio |

BECED | Building Energy, Comfort and Environmeltal Design |

BECEDO | Building Energy, Comfort and Environmeltal Design Optimization |

BEDV | Building Envelope Design Variable |

BEO | Buiding Energy Optimization |

BGDV | Building Geomtery Design Variables |

BIPV | Building Integrated Photovoltaic |

BMC | Building Modular Cell |

BOP | Building Optimization Problem |

CFD | Computational Fluid Dynamics |

CMA-ES | Covariance Matrix Adaptation Evolution Strategy |

DAE | Differential-Algebraic Equation |

DF | Daylight Factor |

DHW | Domestic Hot Water |

DOE | Design Of Experiments |

EA | Evolutionary Algorithm |

EEF | Efficient Form-finder |

EPSAP | Evolutionary Program for Space Allocation Program |

ES | Evolution Strategy |

EUI | Energy Use Intensity |

GA | Genetic Algorithm |

GBM | Gradient Boosting Machine learning |

GH | GrassHopper |

GR | Glazing Ratio (e.g., wall-window ratio) |

GSA | Global Sensitivity Analysis |

GUI | Graphic User Interface |

HDE | Hybrid Differential Evolution |

HJ | Hooke-Jeeves |

HVAC | Heating, Ventillation, and Air Conditioning |

LCC | Life Cycle Cost |

LCCA | Life Cycle Cost Assessment |

LCEI | Life Cycle Energy Impact |

MDO | Multidisciplinary Design Optimization |

MO | Multi-Objective |

MOGA | Multi-Objective Genetic Algorithm |

MOO | Multi-Objective Optimization |

N, NE, E, SE, E, SW, W, NW | North, North-East, East, South-East South, South-West, West, North-West |

NSGA | Non-dominated Sorting Genetic Algorithm |

NV | Natural Ventilation |

ORI | ORIentation |

PDV | Passive Design Variable |

PEC | Primary Energy Consumption |

PSO | Particle Swarm Optimization |

RC | Relative Compactness |

SA | Simulated Annealing |

sDA | spacial Daylight Autonomy |

SF | Shape Factor |

SHADE | shading |

SHC | Stochastic Hill Climbing |

SHGC | Solar Heat Gain Coefficient |

SO | Single-Objective |

STR | Structures, materials |

TDT | Thermal Discomfort Time |

TM | Thernal Mass |

UDI | Useful Daylight Illuminance |

UDV | Urban Design Varible |

WWR | Window to Wall Ratio |

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**Figure 4.**Flowchart showing the basic principles of building shape modification methodology in BECEDO processes.

**Figure 5.**Flowchart of the basic principle of building shape generation methodology in BECEDO processes.

**Figure 6.**Studies with geometry modification and geometry generation with and without energy evaluation and optimization in chronological order.

**Figure 7.**Improvements (%) in energy, comfort, and environmental performance, due to BGDV, BEDV and HVAC system variations in the reviewed studies.

Ref. No. | Year | Topic and Achievements | Objective Function | Building Geometry Design Variables | - (a)
- Algorithm
- (b)
- Simulation Engine
- (c)
- Framework/Method
| Limitations |
---|---|---|---|---|---|---|

[50] | 2019 | MO optimization of office building energy performance and daylight optimization against DOE commercial reference building template. Climate effect on daylight and energy performance in early-stage design. Depth of the building is greatly influenced by the climate. In hot and mixed climate, larger aspect ratios (1.97) are better, while in cold climates, lower aspect ratios (1–1.37) are advantageous. Roof ridge should be located around the center of the building. Most influencing design variables on EUI and UDI are skylights and windows and some shading properties. Building depth strongly determines the energy demand. | - Max. useful daylight Illuminance (UDI)
- Min. energy use intensity (EUI)
| - Depth of space/building wing
- Roof slope/eave/ridge location
- GR, ORI, STR, TM, SHADE
| - (a)
- MOGA
- (b)
- EnergyPlus
- (c)
- GH, Ladybug, Honeybee, Octopus, R
| Very limited geometry-related design variables. GA; hence, only near-optimal solutions. Conclusions about building physics performance are not new. Definition of generation and population sizes are not justified. |

[61] | 1987 | Pioneer study from the 1st authors (1983) investigating building energy optimization including shape and envelope variables. Office (2000 m^{2}). Exemplary demonstration of the method. | - Min. thermal load
- Total capital cost
- Max. area ratio, performing appropriate illumination level
| - Floor space
- Aspect ratio
- Significant geometry modification
- Height/stacking
- GR, ORI, STR, TM, SHADE
- HVAC/energy system
| - (c)
- Pareto optimal dynamic programming, FORTRAN
| Without simulation and advanced optimization algorithm. |

[62] | 2001 | Building envelope optimization in sketch plan stage for family house in variable size—an interactive tool. Multiple criteria decision aid procedure (MCDA), integrating the client, design team, public authorities and users preferences, requirements, and constraints into the optimization process. (1) Definition of constraints and objectives, (2) feasibility study, (3) sketch design. Iterative intervention/optimization is carried out considering all participants requirements, performance results and design. | - Min. heating cost
- Min. AC cost
- Min. DHW cost
- Min. lighting cost
- Min. operation cost
- Min. construction investments
| - Aspect ratio
- Geometry generation with energy evaluation
- Roof slope/eave/ridge location
- GR, ORI, STR, TM, SHADE
- HVAC/energy system
| - (a)
- GA
- (b)
- LEMA EsQUIsE module
- (c)
- Aid to the multiple criteria conception of the building envelope (AMCE)
| GA; hence, only near-optimal solutions. Only theoretical proposal. |

[63] | 2002 | Three studies about a multicriteria optimization of building shape, internal wall partitions between apartments, and heat source utilization in multifamily housing (2133.3 m^{2}). Decomposition in part-problems: shape, internal partitions, heat sources, and handling of global optimization problem. The shape of the prismatic building was determined in height, proportions of the sides, and ORI. By replacing the rectangular form by a rectangle and 2 trapezoids, an optimum shape was developed. | - Min. construction cost
- Min. heating demand
- Min. emissions from heat sources
| - Wall length/area
- Depth of space/building wing
- Height/Stacking
- Angles of horizontal wall inclination
- GR, ORI, STR, TM, SHADE
- HVAC/energy system
| - (b)
- CAMOS
- (c)
- Analytic-numerical
| Very limited geometry related design variables. |

[64] | 2002 | Low-energy building energy optimization of a community hall, 1 zone (200 m^{2}) based on combined computer algorithm and human judgment. Finding an extensive range of possible near-optimum designs. Instead of the time-consuming process of classifying the architectural appeal of each design case generated by the GA (or to considerably reduce the number of processed designs for time saving), a histogram is proposed for each building variable. Cases can be analyzed and worked up to architectural sketches. Similar annual energy usage performing models possess different designs. Models with minimum heat losses apply insulation and compact shape, while models with maximized solar gains use more free form and greater windows. The combination of the 2 strategies is problematic due to summer overheating issues. | - Min. annual energy consumption
- Max. thermal losses
- Max. thermal gains
- Max. architectural appeal
| - Wall length/area
- Depth of space/building wing
- Roof slope/eave/ridge location
- GR, ORI, STR, TM, SHADE
| - (a)
- GA
- (b)
- Simplified dynamic thermal model, EXCALIBUR.
- (c)
- Custom calculation
| Very limited geometry related design variables. Conclusions about building physics performance are not new. GA; hence, only near-optimal solutions. |

[65] | 2003 | Multi-objective optimization of schematic rectangle office and apartment building by composition of walls, generation of shapes and define HVAC systems. Pareto experiments: size and roof of each room are modified by the GA. A large diversity of the Pareto front building cases was achieved. Optimal solution for heating is obtained in a single, compact large building body with all-glazed S and W facade. In lighting demand, the optimum case provides small spaces easily penetrated by daylight with S facing large glazed facades. | - Min. lighting demand
- Min. heating thermal energy demand
- Min. construction investment expenses
| - Geometry generation with energy evaluation
- Wall length/area
- Depth of space/building wing
- Roof slope/eave/ridge location
- GR, ORI, STR, TM, SHADE
- HVAC/energy system
| - (a)
- GA
- (b)
- DOE-2
| No interdependencies and energy related logic can be detected in the random generation of the building form. Conclusions about building physics performance are not new. |

[66] | 2003 | Office, 1092 m^{2}, decision support system (DSS), involving two architects, two structural engineers, and one building services engineer in evaluation. Presenting a mathematical method. | - Min. investment cost
- Max. clear span
- Max. use of natural sources
| - Floor space
- Depth of space/building wing
- Height/stacking
- Net/gross floor ratio
- Wall-floor ratio
- Urban environment
- GR, ORI, STR, TM, SHADE
| - (a)
- GA
- (c)
- BGRID, Microsoft Visual Basic
| Early stage optimization framework, need further development. Without geometry generation (only calculation). |

[67] | 2005 | Proposed optimization method with multi-objective genetic algorithms for early design stage, demonstrated through an example (office, 1000 m^{2}). LCEI is 65% reduced and operations within LCEI decrease by 20%. Aspect ratio 1 is preferred for cost reduction, but rectangular shape with long side to the south is better in energy efficiency. Trade-off between optimum aspect ratio for environment or for cost efficiency. Presenting and proofing a mathematical method. | - Min. LCEI
- Min. LCCA (40 years)
| - Aspect ratio
- GR, ORI, STR, TM, SHADE
| - (a)
- MOGA (Pareto)
- (b)
- ASHRAE Toolkit
- (c)
- Custom calculation, ATHENA
| Simple geometry, very limited geometry related variables. |

[68] | 2005 | Proposing analytic target cascading (ATC), a multidisciplinary hierarchical optimization method. Presenting applicability via a pilot study (office-workshop building 597 m^{2}). Presenting and proofing a mathematical method. | - Max. area
- Max. thermal comfort
- Min. heating need
- Min. cooling need
| - Wall length/area
- GR, ORI, STR, TM, SHADE
- HVAC/energy system
| - (b)
- EnergyPlus
- (c)
- Custom calculation, superEGO, sequential quadratic programming (SQP)
| Early-stage optimization framework, need further development. |

[69] | 2005 | The proposed method allows the designer to explore and visualize design evolution, form generation, and to interact in the optimization process. Through a GUI, the user selects an example from the shapes, and thereafter, the GA continues to search for the best solution. Automated mesh and CFD calculations. Example (generic, simple 1-room) for a continuous evolution of optimization by automatically creating discrete design instances and morphing them in between the process. Compared to conventional design: saving significant computation time, possibility to track relationship between variables and performances, possibility to evaluate trade-off between diverse solutions, possible novel design solution-configurations, because it is not biased by the designer’s view. | - Max. thermal comfort
- Max. ventilation performance
- Min. temperature differences from target
- Min. airflow velocity differences from target
| - Wall length/area
- Depth of space/building wing
- Height/stacking
- GR, ORI, STR, TM, SHADE
- HVAC/energy system
| - (a)
- GA
- (b)
- Gambit, Ansys Fluent (CFD)
- (c)
- GALib, C++, Java API
| GA; hence, only near-optimal solutions. Taking designer’s preferences into consideration in the process means that the selected morph can be a promising solution but may still not be the guaranteed optimum. |

[70] | 2007 | Optimization of the form of an office building on an oval base. Presenting and proofing a mathematical variational method. Heat losses and gains are reduced by approx. 10%. Optimized oval form performs better than circular or square base. | - Min. building cost
- Min. heating cost (1–100 years)
| - Length of layout curves
| - (c)
- Variational method; custom calculation
| Simple geometry, very limited geometry-related variables. |

[71] | 2009 | Presentation of an MDO by breaking down the system into building components (decomposition) using CAD and IFC-codes. Interactive method involving designer for quantitative and qualitative analysis. Presenting and proofing a mathematical method. Optimization of steel and wood frame load bearing structures in an industry hall (1200 m^{2}). | - Trade-off between economic and environmental user preferences
| - Load-bearing structure
- GR, ORI, STR, TM, SHADE
| - (a)
- MOGA
- (c)
- N-Square diagram (design structure matrix), ModelCenter
| Limited to variables of the structure’s geometry. |

[72] | 2009 | MDO of a classroom building. Customized factorial design (DOE). Test application of the PIDO software to an AEC task. Investment cost of the structure decreases as the length of the space increases, because beam span reduces and becomes cheaper. Operation cost increases as the length increases due to greater surface area (heat loss), greater WWR with more solar gains and cooling demand. | - Min. investment expenses of the steel frame load bearing structure
- Min. LCC. of the steel frame load bearing structure
| - Wall length/area
- GR, ORI, STR, TM, SHADE
| - (a)
- GA, gradient-based algorithm
- (b)
- EnergyPlus, process integration and design optimization (PIDO).
- (c)
- ModelCenter, C++, DesignExplorer, Darwin
| Conclusions about building physics performance are not new. GA; hence, only near-optimal solutions. |

[73] | 2013 | Residential house, economic incentives of energy cost optimal curves in net zero energy home (NZEH). MO analysis to find Pareto curves for the objective function. Net-zero energy performance is possible using passive solar design, improved HVAC system efficiency, and renewable sources (PV-panels). The cost-optimal case has an energy performance of approx. 10,000 kWh/a. A NZEH costs approx. the same as the reference building (Building America reference building DOE 2010) over a 30-year life-cycle. | - Min. net-energy consumption
- Min. LCC
| - Aspect ratio
- Roof slope/eave/ridge location
- GR, ORI, STR, TM, SHADE
| - (a)
- Multi-objective algorithm
- (b)
- EnergyPlus, BEOpt
| In terms of building physics, the conclusion is not new knowledge. No information is available about the building geometry, size, etc. |

[74] | 2014 | Multi-objective optimization of low-cost residential housing (LCH) in 3 different climates (China). Exemplary demonstration of the method. Geometry related conclusions as design aid: In tropical monsoon climates, in buildings with NV and/or AC E-W, elongated rectangular shape is recommended. In sub-tropics, the square shape is recommended when NV is operated. Climate has a decisive impact on building operation (passive or active) type. | - Min. construction cost
- Min. LCC
- Max. thermal comfort
| - Aspect ratio
- Wall length/area
- Depth of space/building wing
- GR, ORI, STR, TM, SHADE
- HVAC/energy system
| - (a)
- Hybrid: PSO + Hooke-Jeeves
- (b)
- EnergyPlus
- (c)
- GenOpt
| Very limited geometry related design variables. |

[75] | 2015 | Office, 1 representative floor (1000–2000 m^{2}). Searching for the set of design variables to minimize heating and cooling load. DOE produced possible design variable configurations. The design variables were obtained statistically in a polynomial equation form to determine the Pareto front. The method uses a subset of all possible combinations of design variables to ease exhaustive full factorial design with a large number of test runs. (1) A large number of variables are modeled and assessed as a screening to find the important ones. (2) Functional relations are explored about the variables’ impact on objective functions. (3) Optimization of the variables. Variables of windows and air leakage affect energy load significantly, while aspect ratio is ineffective. HVAC system affect passive design. | - Min. heating load
- Min. cooling load
| - Floor space
- Aspect ratio
- GR, ORI, STR, TM, SHADE
- HVAC/energy system
| - (a)
- NSGA-2, DOE
- (b)
- TRNSYS, R, NIST/SEMATECH
| Very limited geometry-related design variables. Shape design variables and enveloped design variables (WWR, materials) are simultaneously examined; hence, importance of geometry as primary architectural-functional premise is overwritten by design technologically subsequent and more expensive design steps. Conclusions do not contain new knowledge. |

[76] | 2015 | MO optimization of thermal comfort and energy in building design (multi-family house). GA optimizes the back propagation (BP) ANN’s weight and threshold. Simulation-based GA-BP network training and result validation. Thereafter, NSGA-II optimization: evaluation of the potential solutions. Significant (approx. 50%) improvement in energy and insignificant improvement in comfort. | - Min. annual energy consumption
- Max. thermal comfort
| - SF
- GR, ORI, STR, TM, SHADE
| - (a)
- NSGA-II. GA-back propagation network (GA-BP)
- (b)
- EnergyPlus
- (c)
- Matlab
| Very limited geometry-related design variables. Limited to a rectangular shape with a known total floor area. There is need for extensive testing results for further building geometry and type training samples. |

[77] | 2019 | MO BEO. Pareto optimization of geometry, envelope, and energy systems. Proposed optimization framework ’Harlequin’ (unevenness in the diversely oriented facades, materials, colors, WWR, thermal-radiative characteristics): (1) GA generates optimal non dominated solutions; (2) Smart exhaustive sampling of optimal (minimized) PEC, global cost (GC) and investment cost (IC) scenarios. Decision makers can choose the "best" solution according to their needs. Example (office, 2700 m^{2}) modelling and calculations. Recommendations on optimal geometry (aspect ratio 1) and WWR. | - Min. heating energy demand
- Min. cooling energy demand
- Min. lighting energy demand
- Min. thermal discomfort hours
| - Aspect ratio
- GR, ORI, STR, TM, SHADE
- HVAC/energy system
| - (a)
- MOGA
- (b)
- EnergyPlus
- (c)
- MATLAB
| Very limited geometry-related design variables. GA; hence, only near-optimal solutions, ’ad hoc’ randomly solution generation. Conclusions about building physics performance are not new. The method is not user-friendly and requires expertise in programming and BEO. Definition of generation and population sizes are not justified. |

[78] | 2020 | SO and MO environmental optimization of apartment buildings (740 m^{2}). Presented method with LCEI example calculations. Approx. 4–6 story compact, close-to-cube shape with somewhat larger S facade, large WWR (approx. 60%) to the S and small WWR to the N. In optimization of the trade-off between embodied and operational impact, the single objective optimization (SO) preferred compact shape with fully glazed facades and the operational impact optimization resulted in large southern facade (max. solar gains) and extensive insulation (loss reduction). In the case of the optimization of SO + MO: a nearly cubic shape with optimized WWR for solar gains (double glazing) in the S facade and the rest of the facades are optimized for low transmission losses (triple glazing). 60–80% environmental savings achieved in MO environmental impact optimization. Similar results are achieved with significantly diverse solutions. | - Min. LCEI (50 years, EN 15978)
- Min. operational and
- Min. embodied impact based on non-renewable cumulative energy demand (CED).
| - Aspect ratio
- GR, ORI, STR, TM, SHADE
- HVAC/energy system
| - (a)
- GA
- (c)
- GH, Ladybug, Honeybee, Octopus, Steady-state model EN ISO 13790
| GA; hence, only a range of near-optimal solutions. GA acts as a ’black box’; hence, the designer cannot follow what is exactly happening in the calculations. Only steady-state energy calculations. |

[79] | 2021 | MO optimization of a regular classroom space. ANN, a popular type of surrogate models accelerates regular simulation time by factor approx. 2570. 14.2–24.6% average performance improvement in the 3 objective function (integrated solution). | - Min. air-conditioning
- Min. lighting energy consumption
- Min. hours of thermal discomfort
- Max. average UDI
| - Wall length/area
- Depth of space/building wing
- Height/stacking
- GR, ORI, STR, TM, SHADE
| - (a)
- NSGA-II
- (b)
- EnergyPlus, Radience
- (c)
- GH, Python, Ladybug, Colibri, Geatpy
| Very limited geometry-related design variables. Randomly generated design cases by parametric tools for ANN training dataset; hence, the optimum may be missed. Unclear distinctions in 3 different cases using alternating geometry, orientation, WWR, shading, solar absorptance. Conclusions about building physics performance are not new. |

[80] | 2021 | Machine learning-based thermal optimization of residential buildings (2000; 4000; 6000 m^{2}). Latin Hypercube sampling (LHS) generates building configurations. Gradient boosting machine (GBM) is trained by simulation-based dataset for target result prediction. GA optimizes with the surrogate GBM model. Optimal aspect ratio values were found in the 3 alternating locations (climates) and 3 different building heights (5, 10 and 15) ranging 0.67–1.67. | - Min. heating load
- Min. cooling load under investment cost constraints.
| - Aspect ratio
- Height/stacking
- GR, ORI, STR, TM, SHADE
| - (a)
- GA
- (b)
- EnergyPlus, DEAP
- (c)
- Gradient boosting machine (GBM)
| Very limited geometry related design variables. GA; hence, only near-optimal solutions, randomly generated. Conclusions about building physics performance are not new. Definition of generation and population sizes are not justified. |

[81] | 2021 | The position and the height of residential high-rise buildings has a significant effect on internal and external (urban spaces) visual and thermal comfort. Top 10 optimized cases (urban configurations) were selected. | - Max. DF
- Max. sky view ratio
- Max. window sunlight hours
- Max. site sunlight hours
- Max. universal thermal climate index
| - Height/stacking
- Urban environment
| - (a)
- NSGA-II
- (b)
- Radiance, Daysim
- (c)
- Matlab, Grasshopper, Ladybug
| Limited geometry optimization. Mainly visual comfort optimization only. |

[82] | 2021 | Geometry-related performance optimization using an automatic recognition and conversion method with a preference based optimization algorithm to help the designers’ decision-making process. (1) Preferences determination by the designer with software assistance. (2) Preference based optimization algorithm searches for optimum solutions. (3) Designer selects the best solution. | - Customer, designer-related preferences
| - Floor space
- Depth of space/building wing
| - (a)
- 3D space recognition algorithm
- (b)
- Custom simulation engine
- (c)
- MOOSAS
| Preliminary specific preference model design is required from the designer. |

Ref. No. | Year | Topic and Achievements | Objective Function | Building Geometry Design Variables | - (a)
- Algorithm
- (b)
- Simulation Engine
- (c)
- Framework/Method
| Limitations |
---|---|---|---|---|---|---|

[38] | 2009 | Proposing an agent-based geometry generation system based on hierarchical geometry relations. Morphing geometries through agent and child points as 3D corner points of geometries. A cuboid reference building geometry (225 m^{2} redisential building) was modified to an optimal shape with minimal heat loss. Unique geometry generation approach (free forming) to handle complete building geometries in free form. 12% heat load by volume and 6% heat flow per envelope area could be saved. | - Min. heat exchange through the envelope
| - Corner points of 3D geometries
- Grid position of each room
| - (a)
- GA
- (b)
- EnergyPlus,
- (c)
- MATLAB, m-file
| Simple example of a building with limited geometry variables and impractical and building envelope surfaces, making construction expensive. |

[39] | 2020 | Multi-objective building shape and envelope optimization of an apartment block (8000 m^{3}, 4–8 levels; 2673 m^{2}, 6 levels). Phase I: geometry optimization of common linear (I), L- (L), court (O), C- (C), T- (T), H- (H), cross (X), and Y-shaped (Y) buildings (LOD 100) with WWR and ORI options. O, T, H, X, and Y-shapes perform as Pareto optimal solutions. Main reason for that is the self-shading effect of these bodies and the minimization of the SF. The optimal building geometry as output from Phase I is input to Phase II, applying passive and active strategies (LOD 300) to further optimization. | - Min. heating demand
- Min. cooling demand
- Min. energy cost
- Min. investment cost
- Min. CO
_{2}emissions
| - Shape proportion (SP)
- Geometry generation with energy evaluation
- GR, ORI, STR, TM, SHADE
| - (a)
- Active-archive NSGA-II
- (b)
- EnergyPlus
- (c)
- Own developed calculation platform
| Definition of generation and population sizes is not justified (only according to the literature). Geometry optimization of useful floor space and volume, but missing consideration of the different envelope sizes of the diverse shapes and their investment cost. Only single-zone thermal simulation models of multi-story blocks are calculated. This simplification leads to inaccurate or unrealistic results. GA; hence, only near-optimal solutions, randomly generated and therefore hypothesized conclusions. Conclusions about building physics performance are not new. |

[44] | 2010 | MO optimization of solar energy use (heating, lighting) in an urban district with 18,000 evaluations. ’Terraces Flat Roofs’ (E-W building axis), ’Slabs Sloped Roofs’ (N-S building axis) and ’Terrace Courts’ (courtyards) morphology versions were examined, and the latter performed best in energy gains and losses based on greatest collector surface while the volume remain compact. Medieval settlement morphology has more form-related (compact) structure to minimize losses. Modern settlement morphology requires new forms with less density to utilize solar gain. | - Max. solar exposure offset on envelope by thermal losses in heating season
- Max. solar exposure offset on envelope-by-envelope heat losses and min. volume.
| - % of permitted urban morphology volume
- ORI
- Height/Stacking
- Roof/Eave/Ridge dimension, slope, location
| - (a)
- EA
- (b)
- RADIANCE
- (c)
- MOO, OSMOSE, cumulative sky model, Matlab
| EA was used with Pareto optimization to make up for the missing convergence check. This way, only a nearly optimal solution is achievable. |

[45] | 2006 | Pentagon-shaped office layout optimization. The length-bearing method (polygon represented by the bearing = angle between north and an edge, ORI = 1st edge bearing) perform better than the length–angle method (polygon represented by length of the edges and the angle between two adjacent edges, ORI = 1st edge angle to true north) in the framework of MOGA. Low LCC is performed by close to regular pentagons and low LCEI is caused by larger (wider) south facing facade-shapes. | - Min. LCC
- Min. LCEI
| - Angle of horizontal wall inclination
- Roof/Eave/Ridge dimension, slope, location
- GR, ORI, STR
| - (a)
- MOGA
- (b)
- RS Means, ATHENA
| Demonstration of the method rather than comprehensive geometry generation and optimization. GA; hence, only near-optimum search. |

[49] | 2019 | Optimization of simple office building blocks’ geometry (energy demand) in an urban environment coupled with renewable energy potentials and concurrent decentralized multi-energy systems (MES). Optimization of the corner points horizontal and vertical positions. In extreme carbon scenarios, the optimal forms are regular, while in the in-between carbon scenarios, the geometries become more irregular (balancing solar and daylight harvesting with available floor space and ORI). Consecutive optimization: 1st geometry optimization, and then energy system and solar potential optimization. Nested optimization: geometry, energy, and solar optimization takes place simultaneously. The consecutive optimal shapes have N-S ORI, while the nested ones have isotropic bodies. Shape and energy systems are mutually dependent and should be simultaneously optimized. | - Min. investment cost
- Min. operational cost
- Min. operational carbon emission
| - Geometry generation with energy evaluation
- Height/stacking
- Coordinates of the layout corners
| - (b)
- EnergyPlus
- (c)
- RBFOpt (Radial Basis Function Optimization)
| Black-box optimizer because geometrical optimization problem is difficult to solve in an analytical form and simulation programs are complex. Limited simplified geometries. Since shape has a decisive impact on energy demand, system efficiencies and renewable energy potential, as well as HVAC systems may rapidly become obselete compared to a building geometry and structure, certain preliminary choices on energy systems should not influence building densities, ORI and shapes or should be carefully considered with LCA in mind. |

[88] | 2010 | Different letter-shaped, rectangle, and trapezoid layouts are investigated in 2-story residential homes. Rectangle and trapezoid shapes have best performance. When only shape variable is considered: S facing trapezoid in sunny heating climates, and N facing trapezoid in cooling climates are preferable due to solar gain through windows. When geometry, WWR, and material variables are considered in combination: little difference occurs between the optimal shapes. Lowering U-values decrease the impact of geometry and WWR in colder climate zones, and in warmer climates, overheating inverts this tendency. Architects have great flexibility in form design in cold climates. Results of building shapes with max. LCC (max. surfaces, aspect ratio) show large differentiations, indicating building geometry’s decisive impact on energy efficiency. | - Min. LCC
| - Aspect ratio
- Depth of space/building wing
- GR, ORI, STR, TM, SHADE
- HVAC/energy system
| - (a)
- GA
- (b)
- DOE-2
- (c)
- MATLAB, Perl.
| This study did not consider that after optimization of the mass shape, significant savings in further passive and active improvements are achievable: shape design modifications have lower initial costs compared to subsequent investments into material and HVAC systems (LCCA); hence, building geometry is not independent from energy design. |

[89] | 2011 | Different letter-shaped, rectangle, and trapezoid layouts are investigated in 2-storey residential buildings. SO optimization of the HVAC systems, simultaneous (full) optimization of building envelope and HVAC system and in a sequential manner (1st envelop then HVAC optimization). According to different optimization domains (envelope and HVAC) and constraints diverse shapes perform better. In five US cities, the full optimization rectangle shape (AR 1) performs best, while in energy cost full optimization, diverse forms deliver the best results. | - Min. LCC (30 years)
- Min. energy cost
| - Aspect ratio Depth of space/building wing
- GR, ORI, STR, TM, SHADE
- HVAC/energy system
| - (a)
- Sequential search, PSO, GA
- (b)
- DOE-2
| No interdependencies between BGDV and optimization results provided. |

[90] | 2013 | Geometry optimization of a tunnel formed greenhouse (1000 m^{3}) and a 25,000 m^{2} spherical city-hall inspired by the existing London City-Hall (Arch.: Norman Foster): energy savings roughly estimated. Axes ratio and arch ratio of ellipsoid sphere geometry has greatest effect on energy use, best axes ratio is 1.0–1.2 (almost rotationally symmetric cupola geometries). Orientation does not significantly influence the results. | - Max. energy efficiency
| - Significant geometry modification
- Ratio of sphere geometry
- Depth of space/building wing
- Height/Stacking
- GR, ORI, STR, TM, SHADE
| - (a)
- Parallel direct search based on ES (GA)
- (b)
- Autodesk Ecotect
- (c)
- Modified differentiation evolution method
| Limited geometry related variables and poor description of calculations, modelling, and results. |

[91] | 2010 | Hybrid evolutionary algorithm is applied with constraint handling as a method for urban building configuration optimization. Three applications: cuboid buildings in an urban block; complex extension building roof shapes adjacent to an existing house; rectangular plan building’s volume is parametrized by Fourier series. | - Max. solar energy use
| - Height/stacking
- Roof geometry
- Volume
| - (a)
- CMA-ES (covariance matrix adaptation evolution strategy), HDE (hybrid differential evolution).
- (b)
- RADIANCE
| Interesting, diverse applications of the method, but demonstrative rather than systematic optimal building generation procedure. |

**Table 3.**Research focusing on frequent and significant energy performance related BGDVs in chronological order.

Shape Significantly Describing BGDVs | 1987 | 1990 | 2001 | 2002 | 2003 | 2005 | 2006 | 2007 | 2008 | 2009 | 2010 | 2011 | 2012 | 2013 | 2014 | 2015 | 2016 | 2017 | 2018 | 2019 | 2020 | 2021 |
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|

Floor space | [61] | [57] | [58] | [66] | [75] | [60] | [41] | [82] | ||||||||||||||

Aspect ratio | [61] | [62] | [67] | [88] | [89] | [73] | [74] | [46,75] | [60] | [77] | [78] | [80] | ||||||||||

Shape factor | [6] | [42] | [48] | [76] | [55] | |||||||||||||||||

Relative compactnes | [43] | [7] | [55] | |||||||||||||||||||

Wall length/area | [57] | [58,63,64] | [65] | [68,69] | [7,72] | [54] | [74] | [60] | [41] | [79] | ||||||||||||

Depth of space/building wing | [57] | [58,63,64] | [42,65,66] | [69] | [7] | [88] | [54,90] | [74] | [60] | [41,50] | [79,82] | |||||||||||

Height/Stacking | [61] | [63] | [66] | [69] | [70] | [44,91] | [90] | [46] | [53] | [60] | [56] | [40,41,49] | [37,79,80,81] | |||||||||

Roof slope/Eave/Ridge location | [62] | [64] | [65] | [45] | [44,91] | [73] | [46] | [50] |

**Table 4.**Research focusing on studies with geometry modification and geometry generation with and without energy evaluation and optimization in chronological order.

1987 | 2001 | 2002 | 2003 | 2004 | 2005 | 2006 | 2007 | 2008 | 2009 | 2010 | 2011 | 2012 | 2013 | 2014 | 2015 | 2016 | 2017 | 2018 | 2019 | 2020 | 2021 | |
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|

Geometry generation + energy evaluation + optimization | [93] | [40,41] | ||||||||||||||||||||

Geometry modification (advanced) + energy evaluation + optimization | [1] | [38] | [44,88,91] | [90] | [49] | [39] | ||||||||||||||||

Geometry modification (basic) + energy evaluation + optimization | [61] | [62] | [63,64] | [65] | [67,68] | [70] | [7] | [73] | [74] | [75,76] | [50,77] | [78] | [37,79,80,81,82] |

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**MDPI and ACS Style**

Kistelegdi, I.; Horváth, K.R.; Storcz, T.; Ercsey, Z.
Building Geometry as a Variable in Energy, Comfort, and Environmental Design Optimization—A Review from the Perspective of Architects. *Buildings* **2022**, *12*, 69.
https://doi.org/10.3390/buildings12010069

**AMA Style**

Kistelegdi I, Horváth KR, Storcz T, Ercsey Z.
Building Geometry as a Variable in Energy, Comfort, and Environmental Design Optimization—A Review from the Perspective of Architects. *Buildings*. 2022; 12(1):69.
https://doi.org/10.3390/buildings12010069

**Chicago/Turabian Style**

Kistelegdi, István, Kristóf Roland Horváth, Tamás Storcz, and Zsolt Ercsey.
2022. "Building Geometry as a Variable in Energy, Comfort, and Environmental Design Optimization—A Review from the Perspective of Architects" *Buildings* 12, no. 1: 69.
https://doi.org/10.3390/buildings12010069