Article

Integrating multi-functional space and long-span structure in the early design stage of indoor sports arenas by using parametric modeling and multi-objective optimization

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Abstract

Indoor multi-functional sports arenas are a complex building type. Integration of the (multi-) functional space and of the large-span structure of the roof mainly determines the overall geometry of the building, and is one of the most challenging phases of the design. Several interdisciplinary numeric assessments and numerous solutions with diverse geometries (rather than just several specific types) should be considered to make informed design decisions. To support the design exploration in the early design stage for multi-functional arenas, this paper proposes a design process that is composed of a flexible parametric model, a framework of interdisciplinary assessment criteria, and multi-objective optimization (MOO) with post-process tools. The parametric model is defined based on the basic spatial composition of arenas and is flexible to provide a broader design space, including diverse solutions with three frequently-used structural types. The framework of assessment criteria includes indicators of viewing quality for spectators, acoustics, and structures, which can evaluate the design in different aspects. Based on certain assessment criteria, the MOO can be used to search for good designs in the broader space, and the post-process tools facilitate the designer to analyse the results. Two typical arenas (the Barclay Centre and the O2 Arena) are selected as real case studies to demonstrate the proposed process and assess the capacity. Results of the case studies validate the efficacy of the process and the necessity of the broader design space to include diverse solutions with multiple structural types.

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... This process is especially important for the conceptual design of indoor sports arenas. For such building, during the conceptual design process, it is crucial to integrate the multi-functional space and longspan roof structure and to formulate proper building geometry, since these two elements are highly interrelated and determine the overall form of the building [3]. This process involves complex and challenging decision-making, which demands adequate and effectively-organized information of various design alternatives for designers to perform design exploration. ...
... In this light, an efficient design method is needed to satisfy the aforementioned demands, therefore, to support the design exploration of indoor arenas, which is also the motivation of this paper. Nowadays, several computational design methods have been used to support design exploration of architectural conceptual design, including multiobjective optimization (MOO) [3][4][5][6][7][8][9][10][11][12], surrogate model based on supervised learning [13][14][15][16], and unsupervised clustering based on selforganizing map (SOM) [17][18][19]. Fig. 2 demonstrates the overall workflows of these methods. ...
... These workflows are similar, except the step IV in which LLM and MLPNN are respectively used for data approximations. The IAG used in this method is proposed in [3] and based on the software of Rhinoceros 3D [54] and its plugin grasshopper [55], the simulation of structure is based on Karamba3D [56], a plugin of Rhinoceros 3D. The SOM is based on the toolbox of self-organizing maps in MATLAB [57], the LLM is achieved by the codes written by the authors in MATLAB [58], and the MLPNN is based on the toolbox of feedforward neural network in MATLAB [59]. ...
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... El diseño paramétrico de estructuras con objetivo de optimización es una herramienta proyectual cada vez más utilizada [14][15][16][17][18][19][20]. Hemos definido un procedimiento de diseño paramétrico de ruedas trianguladas de radios traccionados (RTRT) con planta elíptica y perfil biconvexo. ...
... Además, existen relaciones entre distancias y ángulos inherentes a la propia geometría de la RTRT. Estas relaciones son las siguientes (16)(17)(18)(19)(20)(21)(22): ...
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... There are studies on the interrelation between 46:2 • A. Agirbas and E. Yildiz acoustics quality and overall form such as those by Alambeigi et al. [ 13 ], Foged et al. [ 14 ], and Agirbas [ 15 ]. Furthermore, Pan et al. [ 16 ] used acoustics as one of the parameters in the multi-objective optimisation process of building form creation. However, the studies on the effect of muqarnas, which is a traditional historical ornament used mostly in Eastern architectural world, on acoustics are limited, and there is no study on the effect of different variations of muqarnas geometry on acoustic quality through optimisation simulations. ...
... ACM Journal on Computing and Cultural Heritage, Vol.16, No. 3, Article 46. Publication date: August 2023. ...
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... A literature review of related work that implemented parametric structural design methodologies (Mueller & Ochsendorf 2015) (Brown et al. 2020) (Pan et al. 2019) (Gomes et al. 2018) allowed the definition of a framework to perform multi-objective optimization in a parametric structural design. The activities involved in the parametric design process were studied to develop a systematization of tasks and appropriate tools to perform them. ...
... Design Space Exploration: in this stage, to explore the design space, the designer might utilize Data Science techniques due to the significant amount of information and simulations involved (Brown et al. 2020) (Pan et al. 2019). Therefore, classification, clusterization, and regression algorithms are used to model the design space and help designers to extract the best solutions. ...
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... A metodologia empregada no desenvolvimento do presente artigo consistiu na revisão bibliográfica dos temas já apresentados no referencial teórico e na comparação de trabalhos relacionados, que implementam metodologias de projeto paramétrico de estruturas [10][11][12][13]. Buscou-se esquematizar as etapas de projeto envolvidas em um processo parametrizado para proceder a sistematização da troca de informações entre as partes envolvidas no âmbito de um Projeto Estrutural em etapa suas etapas iniciais. ...
... Exploração do Espaço de Projeto: se utiliza de técnicas da Ciência de Dados devido ao grande número de informações e simulações envolvido [11,12]. Dessa forma, algoritmos de classificação, clusterização e regressão são utilizados para modelar esse espaço e auxiliar os projetistas a extraírem as melhores soluções. ...
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... Integrating multi-functional spaces and structures in the initial design phase of the indoor sports arena is an important part of optimizing multi-purpose sports, [6]. Multi-function sports arena is an important part in sports achievement, such as artistic gymnastics. ...
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... Daryan, AS, et al., introduced an innovative dolphin echolocation (DE) and bat optimization algorithm to develop a viable method for optimizing the design of steel plate shear wall frames [16]. Pan, W, proposed a comprehensive design process that integrates a flexible parametric model, an interdisciplinary evaluation criteria framework, and multi-objective optimization (MOO) with post-processing tools [17]. Dong, YR, et al., utilized algorithms to create a simulation testbed that elucidates the control law governing viscoelastic dampers' impact on the seismic response of reinforced concrete frame structures [18]. ...
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... For example, Karekla [42] achieved 88.9% accuracy in forecasting psychological responses to stressful situations using ML algorithms, and Zhang [43] found that ANNs outperformed other ML models in predicting human emotional experience in buildings. Additionally, a combination of ANNs with genetic algorithms (GAs) has been increasingly applied in multi-objective building optimization [44], energy saving [45], building geometry optimization [46], and carbon emissions reduction [47]. Most notably, GAs can accurately predict the relation between subjective consciousness, satisfaction, and comfort [48]. ...
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... The adaptability of the human body to the thermal environment will be improved during exercise, so the PMV-PPD thermal comfort model based on this study needs to be corrected to a certain extent in combination with the changes in the metabolic level of the human body under different exercise states (Pan et al., 2019). This allows optimization of the building's overall energy load within the comfort zone of a high metabolic state (Rodonò et al., 2019). ...
... Meanwhile, Priorities for an optimal solution will be further explored, incorporating factors such as aesthetics, sports hall ratios, and environmental impact on surrounding communities. The adaptability of the human body to the thermal environment will be improved during exercise, so the PMV-PPD thermal comfort model based on this study needs to be corrected to a certain extent in combination with the changes in the metabolic level of the human body under different exercise states (Pan et al., 2019). This allows optimization of the building's overall energy load within the comfort zone of a high metabolic state (Rodonò et al., 2019). ...
... The adaptability of the human body to the thermal environment will be improved during exercise, so the PMV-PPD thermal comfort model based on this study needs to be corrected to a certain extent in combination with the changes in the metabolic level of the human body under different exercise states (Pan et al., 2019). This allows optimization of the building's overall energy load within the comfort zone of a high metabolic state (Rodonò et al., 2019). ...
... The adaptability of the human body to the thermal environment will be improved during exercise, so the PMV-PPD thermal comfort model based on this study needs to be corrected to a certain extent in combination with the changes in the metabolic level of the human body under different exercise states (Pan et al., 2019). This allows optimization of the building's overall energy load within the comfort zone of a high metabolic state (Rodonò et al., 2019). ...
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The design of sport buildings has great impact on top-sport as well as on recreational sport-activities. It implies challenging tasks in meeting the performance-requirements. This includes the control of factors like daylight/lighting, air flow, thermal conditions, just to name a few. Such factors impact the performance of athletes and are hard to control in large sport halls; their control is even harder when the public/audience is located within the halls and require different climate conditions. While mechanical installations are often needed during competitions in order to guarantee constant conditions, relaying on mechanical installations during the daily and recreational use of the venues challenges their medium/long term sustainability. Computational form finding approaches can favour the achievement of high-performing and sustainable sport buildings. In this light, the paper tackles the use of Multi-objective and Multidisciplinary design optimization. The paper presents the concept of Multi-objective Multidisciplinary design optimization techniques to support trade-off decisions between multiple conflicting design objectives and interdisciplinary design methodology, during the conceptual design of sport buildings. The proposed method is based on parametric modelling, performance simulation tools and algorithms for computational optimization, for which the paper tackles three specific aspects. First of all, due to the complexity of large sport buildings, the formulation of the optimization and the screening of the related design variables is crucial in order to obtain a meaningful design space, which helps reducing unnecessary computational burden. Secondly, assessing performance based on measurements and analyses is crucial and can be supported by performance simulations tools; however effectively integrating performance simulations tools in the early phase of the design requires new tools. In this light, a customized computational process for the rapid assessment of temperature and airflow patterns is presented. Thirdly, the process requires the combination of design optimization and design exploration, while searching for well-performing solutions. The importance of design exploration is emphasized also for sub-optimal solutions. In order to facilitate the design exploration, the combination of optimization algorithms, multi-variate analysis algorithms and options for exploring design solutions via an interactive dashboard connected to a database are presented. To exemplify the method, specific case studies are developed as collaboration between Delft university of Technology and South China university of Technology.
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This paper addresses the potential of multi-objective optimization (MOO) in conceptual design to help designers generate and select solutions from a geometrically diverse range of high-performing building forms. With a focus on the long span building typology, this research employs a MOO approach that uses both finite element structural modeling and building energy simulations simultaneously to generate optimized building shapes that are not constrained to regular, rectilinear geometric configurations. Through a series of case studies that explore performance tradeoffs of enclosed arches and static overhangs in different climates, this paper shows how MOO can yield architecturally expressive, high-performing designs, which makes the process more attractive to designers searching for creative forms. It also provides new insight into specific design responses to various climatic constraints, since optimization that considers both structure and energy can shift best solutions in unexpected ways. Finally, by displaying performance results in terms of embodied and operational energy, this paper presents new data showing how considerations of structural material efficiency compare in magnitude to total building energy usage. Together, these three contributions can influence current sustainable design strategies for building typologies that have significant structural requirements.
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Sports building envelopes are complex systems involving multiple architectural and engineering performance requirements that are sometimes in conflict with each other. Typically, daylight usage and energy efficiency, as two primary concerns in building envelope design, are of those conflicting aspects. To improve overall performance (including daylight and energy performance) by changing the geometries of the envelope, windows and shading elements as well as the selection of construction materials, Multi-objective Optimization (MOO) is a natural choice. Based on the generated Pareto front, trade-off decisions between competing performance objectives can be made. However, as the number of design variables from different disciplines increases, the huge design space and the specialization of disciplines make the optimization process less efficient. Therefore, two possible Multidisciplinary Design Optimization (MDO) frameworks, namely Individual Disciplinary Feasible (i.e. IDF, a single-level MDO framework) and Collaborative Optimization (i.e. CO, a bi-level MDO framework), are investigated to combine with MOO. Resorting to the capability of MDO in decomposition and coordination between different disciplines, parallel disciplinary simulations and/or bi-level optimizations can be realized, which compresses design cycle time and achieves better overall performance. Through the combination of MOO and MDO, Multi-objective Multidisciplinary Design Optimization (M-MDO or multi-objective MDO) problems are expected to be solved more effectively and efficiently. The whole process of the proposed method consists of three phases (i.e. preprocessing, solution and post-processing phases), in which variable screening, multi-objective MDO solving and Pareto front comparison are performed respectively. An ongoing real project located in China, is used as a case study to test the proposed method. For now, the research work is in the preprocessing phase. Preliminary observations and results are obtained, and future research is discussed.
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This paper addresses the need to consider both quantitative performance goals and qualitative requirements in conceptual design. A new computational approach for design space exploration is proposed that extends existing interactive evolutionary algorithms for increased inclusion of designer preferences, overcoming the weaknesses of traditional optimization that have limited its use in practice. This approach allows designers to set the evolutionary parameters of mutation rate and generation size, in addition to parent selection, in order to steer design space exploration. This paper demonstrates the potential of this approach through a numerical parametric study, a software implementation, and series of case studies.
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In this paper we describe progress to date of software that simulates occupant experience in high capacity sports venues. Our simulation aims to provide metrics that indicate quality of view, and in doing so generates data that indicates levels of human comfort. This approach enables the design process to be driven from the perspective of the occupant. In particular we implement a novel means of simulating and expressing quality-of-view that addresses deficiency's in the standard method of describing view quality. Visualisation of the simulation output is via an online 3D viewer shared with the entire design team. Views from any seat location can be inspected and data fields from the simulation can be compared. Data is represented with colour scales bound to a 3D seating bowl model. Using simulation to understand spectator experience from within a 3D environment challenges the validity of traditional design approaches that are based on two-dimensional thinking and drafting board logic. Our simulated study of view quality enables us to consider revisions to these traditional techniques which could lead to more spatially efficient seating facilities. Increasing spectator density is believed to enhance atmospheric qualities, this combined with better views will contribute towards an improved occupant experience.
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The research presented in this paper addresses the issue of bridging conceptual differences between the theories and practice of various disciplines in the AEC industry. The authors propose an application logic that works as a framework in assisting the evaluation process of multidisciplinary design. Parametric design templates assist in channeling and integrating information generated during the design process.
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A morphogenesis method is proposed for the topology and shape optimization of framed structures subject to spatial constraints. This combines direct elemental addition, or elimination, and free nodal shift, or restricted nodal shift related to the structures geometry. The optimization is based on elemental and nodal sensitivity information to generate or amend the structural topology and adjust the nodal positions to achieve a structure with minimum strain energy. In this method, the design parameters, such as supporting conditions, spatial constraints, etc, have significant influence on the final structural form; so various structural forms can be obtained by changing these design parameters in the project design phase. Several numerical examples are provided to illustrate the validity of this method and the mechanical behaviour of these structures. Results show that this can effectively reduce the structural bending moments and ensure sufficient structural stiffness.
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An increasing number of architectural design practices harness the power of parametric design tools. The aim of these tools is to facilitate and control complex building geometries. Parametric design programs such as Grasshopper (GH) for Rhino or Generative Components popularized this approach by providing easy-to-use visual programming environments that integrate with computer-aided design (CAD) packages. A logical next step consists in connecting parametric designs to applications that evaluate non-geometric aspects such as building physics or structural performance. This brings about new opportunities of collaboration between architects and engineers in the early stages of building design. The ease of testing alternatives by tweaking a set of parameters also opens the door for the application of generic optimization algorithms. Karamba is a finite element program geared towards interactive use in the parametric design environment GH. Being a GH plug-in, it seamlessly integrates with the diverse habitat of other third party programs available for GH. These range from building physics applications to genetic optimization engines. In the author's company, Karamba is used in early-stage design, form-finding, and structural optimization. “White Noise”, a mobile exhibition pavilion for the Salzburg Biennale, serves as a case study that shows how Karamba can be used to optimize the structural performance of intricate building geometries.
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Multidisciplinary design optimization (MDO) has been identified as a potential means for integrating design and energy performance domains but has not been fully explored for the specific demands of early stage architectural design. In response a design framework, titled Evolutionary Energy Performance Feedback for Design (EEPFD), is developed to support early stage design decision-making by providing rapid iteration with performance feedback through parameterization, automation, and multi-objective optimization. This paper details the development and initial validation of EEPFD through two identified needs of early stage design: 1) the ability to accommodate formal variety and varying degrees of geometric complexity; and 2) the ability to provide improved performance feedback for multiple objective functions. Through experimental cases the research presents effective application of EEPFD for architectural design.
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Modern long-span space structures, developed during the 1970s and 1980s, are light and effective structures based on new technologies and light-weight high-strength materials, such as membranes and steel cables. These structures include air-supported membrane structures, cable-membrane structures, cable truss structures, beam string structures, suspen-domes, cable domes, composite structures of cable dome and single-layer lattice shell, Tensairity structures and so forth. For the premodern space structures widely used since the mid-twentieth century (such as thin shells, space trusses, lattice shells and ordinary cable structures), new space structures have been developed by the combination of different structural forms and materials. The application of prestressing technology and the innovation of structural concepts and configurations are also associated with modern space structures, including composite space trusses, open-web grid structures, polyhedron space frame structures, partial double-layer lattice shells, cable-stayed grid structures, tree-type structures, prestressed segmental steel structures and so forth. This paper provides a review of the structural characteristics and practical applications in China of modern rigid space structures, modern flexible space structures and modern rigid-flexible combined space structures.
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The performance-oriented design of large roof structures for semi-outdoor spaces is investigated with the aim to integrate performance evaluations in the early stages of the design process. In particular, daylight and thermal comfort are improved under large structures by exploring passive solar strategies that reduce the need for imported energy. The performance oriented, parametric design approach we developed is structured in two parts. One part uses parametric geometry to generate design alternatives, and the other part uses performance based exploration and evaluation of alternatives. We discuss how the performance based exploration is accomplished using a tool called ParaGen. The potential of the method is shown in a case study of the SolSt roof. The design process of SolSt is based on parametric variations of its curvature, the density of its modules and the geometry of its cladding, and is explored based on daylight and solar exposure of the covered space.
Data
a b s t r a c t The improvement of energy efficiency and environmental performance of buildings is considered a major priority worldwide. New building regulations have an explicit orientation toward low-emission and energy-efficient designs. However, the optimal design of residential buildings should consider multiple, and usually competitive, objectives such as energy consumption optimization, financial costs reduction and decrease of environmental impacts. This makes it a challenging multi-objective optimization problem. The aim of this work is to develop a novel method to tackle the problem. A multi-objective optimization model based on harmony search algorithm (HS) is presented. This model is developed to minimize the life cycle cost (LCC) and carbon dioxide equivalent (CO 2 -eq) emissions of the buildings. Several building envelope parameters are taken as the design variables. To demonstrate the efficiency of the proposed approach the performance of the model is tested on a typical single-family house. For the case of such a house, the model proves to be efficient, and a set of optimal combinations (Pareto optimal solutions) is obtained.
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This paper discusses the acoustic design stages of the Amman Sport Hall, which was recently established at Al-Hussein Sport City. This project is a part of a large sport complex to accommodate the 9th Arab Sport Tournament. The Arena has been locally designed and built in a relatively short period of time (within 9 months). It accommodates courts for basketball, handball, wrestling and other similar games. It contains 7500 seats, as well as other sport facilities and services. It is the largest covered ball in Jordan. However, it is designed to be also used as a multi-purpose ball for celebrations and national events. The reverberation time is calculated to be between 1.8 and 2.0 seconds at mid frequencies as normally recommended for multipurpose halls. The background noise is reduced to NC35 and the % ALCONS is calculated to be less than 15%, which ensures good listening conditions. Fortunately the acoustic design of the hall started from early design stages and continued until the completion of the project. The supervision provided close attention to the minor details of the structure including testing of material and sound quality inside the hall at different stages. The outcome of the overall acoustic checking of the hall is satisfactory: The measured reverberation time inside the hall, unoccupied, is within 2.2 and 2.5seconds and the estimated Rt., occupied, is within 1.2 and 2.0 seconds at mid frequencies. The % AICONS is equal to 5.44- 8.01 empty and 1.6?2.53 occupied while sound pressure level distribution inside the hall, unoccupied, is within + or A dB.
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Clustering is one of the major exploratory techniques for gene expression data analysis. Only with suitable similarity metrics and when datasets are properly preprocessed, can results of high quality be obtained in cluster analysis. In this study, gene expression datasets with external evaluation criteria were preprocessed as normalization by line, normalization by column or logarithm transformation by base-2, and were subsequently clustered by hierarchical clustering, k-means clustering and self-organizing maps (SOMs) with Pearson correlation coefficient or Euclidean distance as similarity metric. Finally, the quality of clusters was evaluated by adjusted Rand index. The results illustrate that k-means clustering and SOMs have distinct advantages over hierarchical clustering in gene clustering, and SOMs are a bit better than k-means when randomly initialized. It also shows that hierarchical clustering prefers Pearson correlation coefficient as similarity metric and dataset normalized by line. Meanwhile, k-means clustering and SOMs can produce better clusters with Euclidean distance and logarithm transformed datasets. These result will afford valuable reference to the implementation of gene expression cluster analysis.
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Performance-driven architectural design emphasizes on integrated and comprehensive optimization of various quantifiable performances of buildings. As the leading profession of a project team, architects play a vital role in guiding and conducting the performance-driven design. Methodology and techniques of performance-driven architectural design and optimization start emerging both in literature and practice. However, architects often find it difficult to put the technique into daily use for various reasons. It is argued that developing an effective technique to conduct performance-driven design and optimization from the perspective of architects is necessary. This paper starts from discussing the concept of performance-driven architectural design and the role of architects in achieving it. Existing methodology and techniques are reviewed and analyzed. The focus is on selecting a basic platform suitable for architects, upon which the technique can be developed. Rhinoceros, a modeling program that architects are familiar with, is used, along with its graphical algorithm editor Grasshopper, to establish such technique by incorporating three widely used performance simulation programs, namely Ecotect, Radiance, and EnergyPlus. Design cases are presented to demonstrate the established technique and its effectiveness.
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In the second part of this report the calculated room acoustical parameters (ISO3382) submitted by a number of users and developers are compared with the measured data of Part I. In three phases the recording studio of the Physikalisch-Technische Bundesanstalt Braunschweig had to be modelled with an increasing number of geometrical details. The first phase with a simple model consisting of seven walls served to check the reliability of the software used and revealed for some participants some significant deviations from the expected values obtainable by applying Sabine's formula. For phase 2 and 3 the results of six programs are discussed here, some of them represented as mean values for a number of users applying the same programs. Most programs show good coincidence with the measured data, keeping track also of local variations of sources and receivers and the specified change of the curtain positions. Wave effects leading to sound fields of low modal density in the 125 Hz octave band are the reasons for higher calculation errors, because phase is not considered in the participating simulation programs. The accuracy of the absorption data, especially for the most absorbing surfaces, is crucial and determines the calculated decay times to a high degree. The increase of details in phase 3 for two diffusing areas was shown to have only little influence on the data obtained.
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Modern buildings and their HVAC systems are required to be not only energy-efficient but also produce fewer economical and environmental impacts while adhering to an ever-increasing demand for better environment. Research shows that building regulations which depend mainly on building envelope requirements do not guarantee the best environmental and economical solutions. In the current study, a modified multi-objective optimization approach based on Genetic Algorithm is proposed and combined with IDA ICE (building performance simulation program). The combination is used to mini-mize the carbon dioxide equivalent (CO 2 -eq) emissions and the investment cost for a two-storey house and its HVAC system. Heating/cooling energy source, heat recovery type, and six building envelope parameters are considered as design variables. The modified optimization approach performed effi-ciently with the three studied cases, which address different summer overheating levels, and a set of optimal combinations (Pareto front) was achieved for each case. It is concluded that: (1) compared with initial design, 32% less CO 2 -eq emissions and 26% lower investment cost solution could be achieved, (2) the type of heating energy source has a marked influence on the optimal solutions, (3) the influence of the external wall, roof, and floor insulation thickness as well as the window U-value on the energy consumption and thermal comfort level can be reduced into an overall building U-value, (4) to avoid much of summer overheating, dwellings which have insufficient natural ventilation measures could require less insulation than the standard (inconsistent with energy saving requirements) and/or addi-tional cost for shading option.
Conference Paper
The challenge of the architect is to create a high-performing building design that is the result of often competing objectives. There are programmatic requirements, aesthetic objectives, and structural criteria which must all be carefully balanced. This paper describes the creation of an automated workflow using parametric modeling, links to structural analysis and a multi-objective optimization engine to act as a tool for the exploration of a wide design space, and as an aid in the decision making process. The design of our custom software CatBot for the linking of Catia and Robot is described, and the further challenge of generalizing the structural inputs, as a set of Catia parameters, to be accessible by students while still providing rigorous structural results is also described. The defining characteristic of this workflow, the ability to trigger topological variation of the model as part of the optimization, is exemplified by Living Light in Seoul, South Korea, by the Living Architecture Lab at Columbia University Graduate School of Architecture, Planning and Preservation. This project is presented as a case study.
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Multi-objective evolutionary algorithms (MOEAs) that use non-dominated sorting and sharing have been criticized mainly for: (1) their O(MN3) computational complexity (where M is the number of objectives and N is the population size); (2) their non-elitism approach; and (3) the need to specify a sharing parameter. In this paper, we suggest a non-dominated sorting-based MOEA, called NSGA-II (Non-dominated Sorting Genetic Algorithm II), which alleviates all of the above three difficulties. Specifically, a fast non-dominated sorting approach with O(MN2) computational complexity is presented. Also, a selection operator is presented that creates a mating pool by combining the parent and offspring populations and selecting the best N solutions (with respect to fitness and spread). Simulation results on difficult test problems show that NSGA-II is able, for most problems, to find a much better spread of solutions and better convergence near the true Pareto-optimal front compared to the Pareto-archived evolution strategy and the strength-Pareto evolutionary algorithm - two other elitist MOEAs that pay special attention to creating a diverse Pareto-optimal front. Moreover, we modify the definition of dominance in order to solve constrained multi-objective problems efficiently. Simulation results of the constrained NSGA-II on a number of test problems, including a five-objective, seven-constraint nonlinear problem, are compared with another constrained multi-objective optimizer, and the much better performance of NSGA-II is observed
Performative computational design
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STAG:Stadium Generator
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Integrated computational optimization for the layout of the playing hall in the gymnasium based on viewing quality analysis
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