Owing to the capacities of generating structural configuration with both reasonable mechanical properties and high material utilization, topology optimization has been widely adopted in engineering design. Although numerous architects have tried to apply topology optimization tools to assist architectural morphology design in practical projects, topology optimization, like other quantitative analysis techniques, has not been systematically incorporated into the architectural morphology design. In this study, by integrating topology optimization toolsets and parametric design theory, combined with multiattribute decision-making analysis, a design method is proposed that could efficiently obtain several architectural structural architectural morphologies with both structural rationality and aesthetic rules and complete the evaluation and performance ranking of alternatives. In this study, the essential architectural application scenarios are divided into surface application scenarios and volumetric application scenarios, and the possible variation range of topology optimization parameters of architectural application scenarios is defined. By iteratively adjusting the influence parameters, diverse results of structural morphology are obtained. It is found that small changes in optimization parameters will bring great differences in topological results. Such a sensitive relationship can be utilized to generate a set of rational topological structures, and these topological results can be regarded as alternatives for architectural morphology design. For the performance evaluation and ranking analysis of alternatives, the application of FANP-TOPSIS multiattribute decision-making model is put forward in this study. The case study shows that this decision-making analysis model is efficient, convenient, and applicable in the architectural morphology design. The results of this study can provide new ideas and key references for scholars and architects in the field of architecture to explore the process and method of architectural morphology design and other related issues.
Developments in construction industry design software and the maturity of related manufacturing techniques over the past two decades have led to the construction of buildings with complex and eye-catching appearance . Whilst many have received praise and are considered to be iconic landmarks for their region, others are criticized for the lack of harmony between their architectural design and structural considerations. The challenge therefore remains to obtain satisfying designs that can simultaneously embrace architectural operational functions and aesthetic appealing effects, as well as maintaining rational structural performance . Inspired by structural morphology (Rene Motro, an anthology of structural morphology), which involves form, forces, material, and structures and aiming at developing a structural system with harmony synthesis of these four aspects, architectural morphology is defined by extending the connotation of structural morphology, which simultaneously deals with structural performance, architectural functions, and aesthetical requirement, aiming at developing an architectural system with a balance between these factors.
Topology optimization, a mathematical method to optimize material distribution in a given area according to given conditions and objective index, has attained its popularity in civil engineering and architectural design owing to its potential to generate rational and aesthetic-artistic morphology . Topology optimization was initially developed for applications in aeronautic and mechanical engineering , where the design space represents a continuum of material, and even small savings in weight are significant, for example, by saving fuel on thousands of journeys and/or saving material on thousands of mass-produced products. Amongst the many topology optimization methods that have been developed, common approaches include the solid isotropic material with penalization (SIMP) method [5–7], the (bidirectionally) evolutionary structural optimization (ESO or BESO) method [8, 9], level set methods [10–12], the moving morphable components (MMC) method [13, 14], and the independent continuous mapping (ICM)  method. Many of these approaches have been adopted for the application to the architectural morphology problem domain. For the design of bracing systems for high-rise buildings, Beghini et al.  proposed a topology optimization framework to integrate architecture and engineering. The generation of optimized shell- and large-scale spatial structures was investigated by Ohmori , who developed an extended ESO method, whereas Peng  applied the ICM method to designs of dendriform structures with hierarchical topologies similar to tree branches.
Whilst a wide range of literature can be found relating the application of topology optimization methods to architectural design, there still exist a number of gaps that necessitate further investigation, which this paper address. Firstly, researchers usually focus on a particular type of application scenario, such as beams, walls, or large-scale spatial structures, whereas a comprehensive study of how to use topology optimization to generate architectural morphology across many different application scenarios is still missing. Additionally, the relationship between the inputs to a topology optimization and the resulting morphology has not been investigated in detail. This lack of understanding of the sensitivity of the outputs to the inputs is one of the main obstacles preventing the architects from using topology optimization tools in practice. Thirdly, little research has been carried out to discuss and compare the topology and morphology of optimized architectural design from topology optimization in the perspective of aesthetic.
This paper first extracts and classifies the most common architectural scenarios based on their geometrical features and structural properties. It then derives the key parameters that affect the topological results and discusses their relative impact on these results. A methodology combining parametric modelling and topology optimization is then adopted for architectural morphology generation. By making use of the sensitive relationship between the resulting topology and the input parameters for optimization, a single solution, or a cluster of solutions, can be obtained. They are viewed as potential candidates for building designs, thus solving the problem of architectural morphology generation. Finally, a numerical case is adopted to compare morphology of different optimized shell results and provides some basic aesthetic evaluation from architect’s perspective.
The outline of this article is as follows. In Section 1, the context of the study and the required background knowledge is presented. In Section 2, the morphology generation procedure is proposed, and the influential parameters are identified. The essential architectural application scenarios are classified in Section 3, along with a discussion on how the influential parameters relate to each architectural application. Section 4 assesses the relationship between optimization parameters and the topological results for each classification, and in Section 5, the specific example of the morphology generation of a shell structure is investigated. Finally, Section 6 highlights the conclusions of the work and discusses the implications for morphology generation in practice.
2. Morphology Generation Methodology
Topology optimization of structure generally involves the addition, subtraction, or elimination of material from within a design domain. Through iterative adjustment of material, the optimal topology, representing the force flow within the domain, will gradually emerge. In addition to having the best mechanical performance, it is often the case that the obtained topology is also highly aesthetic. This successful combination of engineering and art is therefore viewed as a desirable candidate for architectural morphology design. However, there is no guarantee that the configuration produced though topology optimization would always be suitable for direct employment in the next design stage, and usually some modification is necessary, which can be achieved by adjusting the influential parameters.
Before considering how to adjust the influential parameters accordingly, a method for solving the problem of architectural morphology generation via topology optimization is introduced below, and the parameters that play key roles in determining the resulting morphology are considered.
2.1. Influential Parameters
In this paper, topology optimization is used to generate architectural morphology; therefore, the optimization parameters for topology optimization of different structures and structural members are also used as the parameters for morphology generation of them. Some additional parameters are required for the topology optimization, such as load scenarios, boundary conditions, and material properties, which are not directly related to the morphology.
The first parameter to be considered, the design domain, is represented by a geometry with planar or spatial features. This is usually defined based on consideration of architectural functions, such as space division, people-flow, light, and ventilation requirements. For example, it can be a wall with openings representing doors and windows, a hemispherical shell with holes on the top representing skylights, or a trimmed solid box representing an entire building. It should be noted that, during the optimization process, only materials in the design domain can be removed, retained, or reintroduced. This means that the optimal topology can only be made up of material within the design domain. In this way, the design domain, on the one hand, provides space for the morphology to change but, on the other hand, constrains the scope of that variation. Therefore, this essential relationship between the design domain and the resulting optimal topology makes the design domain one of the most dominant parameters that influences the optimization results of original structures.
The second consideration is the different loading scenarios on original structures. The purpose of topology optimization is to generate structural configuration with best mechanical performance under the external loads. The loads acting on buildings include gravity, live-, wind-, and snow-load, as well as concentrated (point) forces applied at certain positions to represent specific objects. With a small change in external loads, major variation of optimal topology can occur, since it is the mechanical response of the structure under these loads that determines the evolutionary direction of the optimization process.
Boundary conditions are the third parameter to consider. For buildings, boundary conditions usually include pin-supports, roller-supports, or fixed-supports. These supports can be present at specific discrete points, applied continuously along lines or curves, or even distributed across an entire surface. The boundary condition specifies the positions where the structure transfers its external loads to the foundations. Therefore, slight variations in boundary conditions also introduce significant changes in the optimization results.
Material properties also need to be carefully defined, and it is often the case that there will be more than one type of material being used within one architectural design of buildings or any specific structural members. For example, many high-rise buildings are constructed from steel beams, columns and decks, with a reinforced concrete slab poured on the deck in-situ to make a composite floor system. Specifying different material properties in different areas of a building can have a significant effect on its structural response, and hence its optimal topology. However, architectural morphology generation is usually carried out at an early stage of architectural design, at which point it is usually considered acceptable for only one material to be used for topology optimization. It is also a common assumption during early stage design that only linear elastic deformation would occur within the structure. In this case, the optimal topology for one material is also the optimal topology for another material. Therefore, for the purpose of this paper, the difference in the topology optimization results caused by the variation of materials can be assumed to be negligible.
Besides the optimization initialization parameters outlined above, the formulation of the topology optimization itself also involves the defining of parameters that have an impact on the results. Generally, the formulation of a topology optimization problem requires the definition of objective functions and constraint functions. The objective functions use objective index or performance index as dependent variable and input parameters as independent variable, and objective index or performance index is what researches want to maximize or minimize, for example, maximizing overall structure stiffness. Besides, researchers can use the constraint functions to apply specific geometric or mechanical constraint to optimized structures, such as minimum/maximum feature size [19, 20] and symmetry and pattern repetition . These two kinds of functions are usually determined based on consideration of mechanical properties or geometric features of the design of original structure and can involve measures of deformation, stress, stability, material volume, etc. The influential parameters introduced above are classified into two categories as summarized in Table 1.
Parameters for initialization of optimization problem
Parameters for formulation of optimization problem