the physical prototype projects a narrow set of parametric rules and material descriptions, it can be a resource in defining fundamental logics of topology, proportion and behaviour, for further computational exploration. In this research, computational explorations occur through two venues: modelling and simulation of relative material descriptions with spring-based numerical methods, and finite element analysis defining precise mechanical (material and force) relationships. Spring-based methods calculate force based upon linear elastic stress-strain relationships (Hooke’s Law of Elasticity); using a numerical integration method such as Euler or Runge-Kutta to approximate the equilibrium of multiple interconnected springs (Kilian and Oschendorf 2005). Such methods are deployed to primarily explore varied relationships between topology and force. Both conditions are easily manipulable during the process of spring-based form-finding, enabling immediacy for feedback and ability to extract how minute manipulations affect the overall system behaviour. A fundamental layer of this research is the continued development of a modelling environment, programmed in Processing (Java) with a particle-spring library, allowing for complex topologies and force descriptions to be initially generated then actively re-modelled through an interface.
Fig. 3 Decomposition of material behaviour for spring-based modelling and simulation (Ahlquist 2013)
Finite element methods (FEM), on the other hand, contribute to forming complex equilibrium structures in defining the complete mechanical behaviour of the system. The given necessity for simulation of large elastic deformations in order to form-find bending-active structures poses no problem to modern nonlinear finite element analysis (Fertis 2006). However, software using FEM does not serve well as an expansive design environment, specifically for textile hybrid systems due to the inability to manipulate geometry and behaviour during the form-finding process. This necessitates the input data, for the pre-processing of the simulation, to be based upon the unrolled geometry of either physical form-finding or a computational environment such as the spring-based methods described above. Though the advantage, as well as necessity, of FEM in the development of textile hybrids, lies in the possibility of a complete mechanical description of the system. Provided that form-finding solvers are included in the software, the possibility of freely combining shell, beam, cable, coupling and spring elements, enables FEM to simulate the exact physical properties of the system in an uninterrupted mechanical description. Such means allows the FEM environment to accomplish, in a single model, the complete scope of form-finding, analysis of performance under external loads, and finally, the unrolling and patterning for fabrication.
4 Cellular Structure: Exploring Topological and Geometric Variation
Where bending action is triggered in a material system, certain geometric values become necessary inputs to the form-finding process. In simple terms, the length of the bending-active elements must be stated prior to the initiation of the form-finding process. In physical form-finding, geometry is inextricable from topology. The components in their count, type, and associations, carry with them their material properties. This introduces a helpful constraint in managing the complexity of searching for states of form- and bending-active equilibria. In developing the cell strategy for the M1, the physical studies define a proportional geometric logic for the bending-active aspect of the system. The exact geometry of the multi-cell array is only realized when arranged within the interleaved macro-structure. As both, the region within the meta-structure and the proportional rules of the individual cell are three-dimensionally complex, the spring-based modelling environment is well suited to explore the variation of geometric inputs arranging the meso-scale cellular textile hybrid system.
4.1 Modelling and Active Manipulation of Material Behaviour
Within the range of linear elastic material behaviour underlying the spring-based methods, a single spring element may compute tension or compression, and, in a combined arrangement, also bending action. Bending stiffness is simulated by adding positional constraint to the nodes (particles) that form a linear element. Three commonly known methods for simulating this behaviour are crossover, vector position and vector normal (Provot 1995; Volino 2006; Adrianessens 2001). In modelling behaviour with springs, there is a unique consideration where certain springs define only a particular aspect of material behaviour such as shear or bending stiffness, while others simulate the totality of behaviour and display the resultant material form such as a surface geometry or linear bending element.
In defining the tensile surface of a textile hybrid system, a mesh of springs both simulates the tensile condition in warp, weft and shear behaviour, as well as defines the material surface. In simulating bending stiffness, a linear array of springs implies the material condition of an elastic element, but the springs simulating constraint at the nodes do not have any geometric representation, as shown in Fig. 4. The flexibility in which a spring may drastically shift behaviour, between tension and compression, along with how relationships of geometry and behaviour can be more gradually tuned has been implemented as the foundation of the modelling environment programmed in Processing (Java). The key capacity in this particular mode of design is how the characterisations of behaviour can be manipulated, in topology and force description, during the effort of form-finding allowing freedom to define behaviour of different material make-up and composition.
Fig. 4 Comparison of spring topology between simulating a surface and a linear element with bending stiffness (Ahlquist 2013)
4.2 Transferring Relational Logics from Physical Form-Finding to Computational Exploration
In the M1, the array of interconnected cells serve as secondary support to the overall structural system, while, more critically, providing a means for differentiating the spatial conditions underneath the primary membrane surface. The geometries of the bending rods are calibrated to act as stiffening struts spanning between the upper and lower level of the meta-scale bending-active network. Within the cellular structure, a series of tensioned textiles further stiffen the system and serve as the media for diffusing light. The fundamental relationships between the boundary condition for the cells, the structure of an individual cell and its relation to its neighbour are most readily represented in physical form-finding studies, as shown in Fig 5. Yet, due to the complexity of those combined conditions specifying the geometry, which successfully resolves all of those parameters and constraints, is more readily accomplished in the spring-based environment where active manipulation of local and global behaviours is possible.
The spring-based modelling environment in Processing exposes variables related to the simulation of bending stiffness. Using the vector position method, the ratio of stiffness in the springs defining the linear beam elements to the degrees of constraint in the nodes can be varied to express differing amounts overall stiffness and curvature, thus implying different material properties. The lengths of the linear beam elements are exposed locally and globally enabling for the acute management of bending-active behaviour when multiple elements interconnect. These two capacities allow initially simple topological and geometric arrangements to be formed into the complex relationships defined by the physical cell models and made suitable to the context of the interleaved bending-active structure, as shown in Fig. 6.
Fig. 5 Rules for bending-active cell structure (Ahlquist, 2012)
Fig. 6 Form-finding sequence for cells in spring-based modelling and simulation environment, programmed in Processing (Java) by Sean Ahlquist (Ahlquist, 2012)
5 Interleaving Structure: Developing Force Equilibria
The interleaving macro structure of the M1 exhibits how multiple modelling and simulation