## 1 Introduction  
The development of computational design technology and the introduction of digital manufacturing methods have been the foundation for the advancement of irregular-shaped architecture [1]. The recent emergence of new digital manufacturing technologies, including AM (additive manufacturing), has shown new design possibilities [2]. Parametric design tools and AM technology provide increased capabilities in design and manufacturing, especially when dealing with complex shapes [3]. Parametric design is a design paradigm that distinguishes it from traditional design processes using CAD and uses a differential geometric approach [4] to create an automated modeling process where design decisions are made by algorithms instead of relying on the designer's arbitrary judgment. Using parametric tools, a designer can effectively perform precise numerical control over complex shapes [5] such as curves and surfaces on repetitive and complex design tasks. An irregularly shaped gridshell facade is composed of parts that are repeated but not identical to one another. While each part follows the same design procedure, the local geometric constraints of irregular structures are interdependent with the surrounding geometric context [6]. Therefore, the conventional design method relying solely on manual design poses many problems, particularly in relation to human error. The parametric design ensures efficiency and reliability in such circumstances. The process and methodology of parametric design should be designed to ensure quantitative criteria for decision-making in every aspect, unlike design using CAD, where the designer can intervene based on their judgment at any given moment [7]. Therefore, an algorithm with proper constraints should be established for each step of the process based on an understanding of the design constraints. Nodal joints in irregular-shaped gridshell structures must satisfy many constraints, and the resulting complex shapes pose many challenges in terms of fabrication [6]. Designing a nodal joint that satisfies many degrees of freedom with limited fabrication methods is very difficult and can lead to complex shapes and increased assembly parts. This problem is also associated with increased production costs and a decreased reliability of components, as well as aesthetic issues. In some cases, to alleviate complexity issues, only a limited range of degrees of freedom [8; 9] is secured or, even worse, omitted, which can reduce the design freedom of the shape being implemented and diminution of construction precision [10; 11]. AM technology has made significant progress since its inception and is distinguished by its ability to replicate precise and intricate shapes that have posed challenges for other fabrication methods, such as a multi-axis CNC (computer numerical control) machining center. Also, previously recognized drawbacks of AM methods, such as low productivity, high costs, and a limited spectrum of material choices, are being overcome. Through design for manufacturing and assembly (DfMA), significant improvements have been made not only in the performance of AM products but also in productivity and cost aspects. With parts integration and uni-body designs, AM parts demonstrate better constructability and higher structural integrity compared to the previous iterations. This study proposes a novel design approach that addresses the complex and impractical designs resulting from the limited manufacturing methods of conventional nodal connectors, using the AM production method and parametric design automation. Firstly, the geometric constraints required for irregular-shaped curtain wall facades were identified, and mathematical implementations were established. Secondly, a complete automated design algorithm was developed based on the mathematical approach for automatic optimization of the connector arm length, which previously required the designer to interrupt the automation process to manually intervene with shape control.