A novel progressive grid generation method for free-form grid structure design and case studies

Highlights

• A novel progressive grid generation method for free-form grid structure design is proposed.

• Based on Coulomb's law, nodes of grid structure are considered to be interacting particles in an electric field and adding to surface in a progressive way.

• Two variations of the method are introduced; point-based progressive method (PBPM), and curve-based progressive method (CBPM).

Abstract

Due to its high structural efficiency and aesthetics, free-form grid structures have been widely used in various public structures. However, it is neither a convenient nor an obvious task for engineers to create a discrete grid on a free-form surface that manifests the architect's intent. This paper presents an efficient design approach based on Coulomb's law to generate well-shaped and fluent grids for free-form grid structural design. In the method, nodes of the grid structure are considered to be interacting particles in an electric field and are added to the surface in a progressive way. The nodal position is determined by Monte Carlo simulation and the grid is generated by connecting the particles that are already in equilibrium. According to the different ways of adding particles, two variations of the method are introduced in this paper: point-based progressive method (PBPM), and curve-based progressive method (CBPM). Case studies are provided to demonstrate the effective execution of the proposed method. The results show that the proposed method can effectively avoid mapping distortion and generate grids with regular shape and fluent lines to meet the aesthetic requirements. Furthermore, the proposed method provides flexible control over the direction and size of the grid, which gives architects a more flexible choice.

Introduction

The emergence of grid structures is a major step in the development of complex shapes in AEC (Architecture, Engineering, and Construction), according to Du [1]. In recently years, the free-form grid structures have become popular structural typologies thanks to their splendid visual effects and the capacity to cover large spaces with an uninterrupted span, see Fig. 1. They are widely used in a variety of building types, such as exhibition pavilions, stadiums, assembly halls, and protective shelters [2]. Compared with the traditional analytic surface grid structure, the shapes of free-form grid structures are more flexible and diverse, and it can better address the needs of architects in the architectural modeling diversification, as discussed by Schlaich [3]. The shape of the grid structure can be obtained by using form-finding techniques like the force density method and the dynamic relaxation method to minimize the amount of material while maximizing the strength [[4], [5], [6], [7], [8]]. However, for a given surface obtained by the form finding method, it is not always an easy task for engineers to determinate the grid layout that respects architectural requirements and is structurally efficient at the same time. In addition, due to the variety of surfaces of the free-form grid structure, the traditional grid generation method has become overextended and difficult to adapt to such structures. Therefore, a practical grid generation method that can quickly generate structural grids that meet the architect's requirements will be necessary, especially in the early design stage.

Mesh generation started early in the field of Finite Element Analysis (FEA) backed up with fundamental research. Owen [9] systematically summarized the high-quality unstructured meshing algorithm, such as Delaunay Triangulation, the Advancing Front Technique and the Mapping Method, etc. [[10], [11], [12], [13]]. Although the mesh quality required for FEA has much in common with the mesh characteristics required for grid structures, the resulting FEA mesh does not necessarily meet the architect's requirements such as the equal rod length and the architectural grid fluidity.

As for the grid generation methodologies over the free-form surface, generally speaking, topology optimization is one of the effective methods used to create an efficient structural grid on an imposed surface. Peter and Gilbert [14], Paul et al. [15] and Gao et al. [16] have made substantial progress in this area. However, the grid generated via topology optimization is so coarse that it cannot be applied to actual projects directly.

In order to get a practical and efficient grid over a free-form surface, Winslow et al. [17,18] presented a new algorithm that considers the mechanical performance based on a traditional genetic algorithm. Su et al. [19] conducted a similar research and an improved wave-front method was presented to generate better grids on the given surface. According to Su's research, the structural stress trajectory was taken as the direction of the grid advancement. Although the grids obtained by this way have better mechanical performance, the uniformity of the grid is poor, and some distorted unit cells appear.

Actually, one must admit that architects are most concerned about whether the grid has the advantages of fluency and uniformity, and the structural mechanical properties seem to be less important to them. Therefore, many studies have reported the progress of grid generation on a given surface without considering their structural performance. For example, Ding [20] proposed a new method for a free-form surface grid generation, referred to as the iso-parametric line dividing method, which is based on a non-uniform rational B-spline (NRUBS) curve. Wei [21] and Gao [22] proposed a grid generation method over the free-form surface, which combines the surface flattening technique and mapping theory. Shepherd and Richens [23] proposed a Subdivision Surface method, which provides a useful platform for combining creative architectural design with intelligent engineering to produce aesthetically pleasing designs on financially and environmentally limited agendas. Du et al. [1]proposed a specific computing tool to mesh the surface according to the Compass Method, which allows designers to look for optimal mesh orientations regarding the element's curvature. Lefevre et al. [24]used the Compass Method to generate grid shells considering the mechanical performance, while a method with a denser net was presented to improve the grid quality. Taking the roof of the Yas Island Marina hotel as an example, Peng et al. [25] presented a framework to generate mesh patterns that consist of a hybrid of both triangles and quads. Pottmann et al. [26] presented an overview of designing free form grid shells that consist of polyhedral cells or closely related to them.

In research by Persson and Strang [27], an iterative technique was developed for the mesh generation based on a physical analogy between a simplex mesh and a truss structure. Similar to the physical analogy, Shimada and Gossard et al. [28,29] invented an interesting grid generation method called bubble mesh generation. In its actual implementation, the bubble method generated node configurations that yielded virtually no ill-shaped triangles or tetrahedrals. Wang et al. [30,31] extended the bubble grid generation method to the free form curved surface, and proposed a grid generation and optimization framework for the design of free form grid shells. In order to obtain a grid with balanced rod length and fluent lines, Gao et al. [32] presented an efficient design tool for the grid generation of free form surface based on the concept of guide lines. Zheleznyakova [33] regarded the grid nodes as interacting particles and developed a new approach for the triangular grid generation based on the molecular dynamics method. Liu et al. [34] proposed an automatic triangular grid generation method for the design of free-form grid structure based on Coulomb's law. With that method, a free-form grid structure with unique topology, where all of the nodes are located at a similar distance to one another can be obtained. However, the drawback of the method is that it is not convenient to control the unit size and the direction of the grid.

In order to generate a grid that satisfies the requirement of uniformity, regularity, and fluidity, and to improve the ability to control the direction and the size of the grid, this paper proposes a progressive grid generation method based on Coulomb's law for the design of free-form grid structure. In this method, the nodes of the spatial structure are considered to be interacted particles in an electric field, and they move towards each other under the drive of the Coulomb force. To ensure that the particles always move on the surface, the attraction of the surface to the particles is introduced, and the particle placement is determined via Monte Carlo simulation. According to the different ways of adding particles, two variations of the method are introduced in this paper: point-based progressive method (PBPM), and curve-based progressive method (CBPM). Several cases are studied to illustrate the execute-ability of this method and the results show that this method can generate grids with excellent quality, and therefore has preferable applicability in the structural concept design stage.