# Design Optimization of Truss Structures Using a Graph Neural Network-Based Surrogate Model
## 1 Introduction  
The construction industry is a key global sector that contributes around 9% of the world's gross domestic product [1]. However, the significant resource consumption in this industry poses challenges that adversely impact the environment. To mitigate these issues, optimization techniques can be employed throughout the entire project life cycle, including design, construction, operation, and maintenance. Structural optimization plays a significant role in achieving efficient, sustainable, and economical designs while satisfying structural performance constraints pertinent, for example, to allowable displacements and stresses [2]. An economical structural design typically involves minimizing the structural member sizes, and hence the weight of the whole structure, leading to a reduction in the consumption of construction materials and the overall cost [3]. Trusses have been widely used in various types of structure such as bridges, buildings, power transmission towers, the aerospace industry, and other civil structures due to their advantages, including the efficient use of materials, flexibility of design, and ease of installation. The simple configuration of trusses, consisting of pin joints and interconnected straight elements built up from a single type of material, has made them particularly attractive in the field of structural optimization. Consequently, there has been a significant interest among researchers in truss structural design optimization, with a focus on the development and application of metaheuristic algorithms as the preferred optimization method. Truss optimization can be classified into three categories: topology, shape, and size optimization. Topology optimization involves determining the optimal arrangement of thetruss members, selecting which members should be present or absent, while keeping the overall geometric configuration fixed [4; 5]. In shape optimization, the objective is to find the optimal configuration of a truss with a given topology [6; 7], in which case the coordinates of the truss joints are considered as design variables. It does not alter the overall connectivity and arrangement of the truss elements determined in topology optimization. Ultimately, size optimization aims to determine the optimal cross-sectional areas of the truss elements as design variables while maintaining both the topology and shape unchanged [8; 9; 10]. Metaheuristic algorithms are a class of optimization techniques designed to solve complex problems for which traditional derivative-based methods may be ineffective or impractical. Generally, metaheuristics are inspired by natural or abstract systems and often mimic the behavior of natural processes such as evolution, swarm intelligence, or physical phenomenon. Particle Swarm Optimization (PSO) and Genetic Algorithm (GA) are the two most popular and deep-rooted metaheuristic methods, particularly in the field of structural optimization [11]. Metaheuristics are well-suited for exploring complex design spaces for problems involving non-linear, multimodal, and discontinuous behavior constraints [12; 13]. These derivative-free and iterative-based methods offer a problem-independent approach, enabling them to find satisfactory solutions for complex problems faster than conventional derivative-based approaches. However, the computational cost associated with objective and constraint functions for large structures remains a concern, as a significant number of function evaluations are required to obtain acceptable solutions [14]. In recent years, many researchers have been exploring the potential of Artificial Neural Networks (ANNs) to address structural engineering problems that involve inevitable uncertainties [15; 16; 17]. Numerous studies have demonstrated the effectiveness of neural networks in structural analysis [18; 19] and optimization [20; 21; 22; 23]. Neural networks have primarily been used in two applications for structural optimization: predicting optimal designs and creating surrogate-assisted models.