Since the introduction of spatial grammars 45 years ago, numerous grammars have been developed in a variety of fields from architecture to engineering design. Their benefits for solution space exploration when computationally implemented and combined with optimization have been demonstrated. However, there has been limited adoption of spatial grammars in engineering applications for various reasons. One main reason is the missing, automated, generalized link between the designs generated by the spatial grammar and their evaluation through finite-element analysis (FEA). However, the combination of spatial grammars with optimization and simulation has the advantage over continuous structural topology optimization in that explicit constraints, for example, modeling style and fabrication processes, can be included in the spatial grammar. This paper discusses the challenges in providing a generalized approach by demonstrating the implementation of a framework that combines a three-dimensional spatial grammar interpreter with automated FEA and stochastic optimization using simulated annealing (SA). Guidelines are provided for users to design spatial grammars in conjunction with FEA and integrate automatic application of boundary conditions. A simulated annealing method for use with spatial grammars is also presented including a new method to select rules through a neighborhood definition. To demonstrate the benefits of the framework, it is applied to the automated design and optimization of spokes for inline skate wheels. This example highlights the advantage of spatial grammars for modeling style and additive manufacturing (AM) constraints within the generative system combined with FEA and optimization to carry out topology and shape optimization. The results verify that the framework can generate structurally optimized designs within the style and AM constraints defined in the spatial grammar, and produce a set of topologically diverse, yet valid design solutions.
In rigid origami, the complex folding motion arises from the rotation of strictly rigid faces around crease lines that represent perfect revolute joints. The rigid folding motion of an origami crease pattern is collectively determined by the kinematics of its individual vertices. Establishing a kinematic model and determining the conditions for the rigid foldability of a single vertex is thus important to exploit rigid origami in engineering design tasks. Today, there exists neither an efficient kinematic model to determine the unknown dihedral angles nor an intrinsic condition for the rigid foldability of arbitrarily complex vertices of degree n. In this paper, we present the principle of three units (PTU) that provides an efficient approach to modeling the kinematics of single degree-n vertices. The PTU is based on the notion that the kinematics of a vertex is determined by the behavior of a single underlying spherical triangle. The condition for the existence of this triangle leads to the condition for the rigid and flat foldability of degree-n vertices. These findings are transferred from single vertices to crease patterns, resulting in a simple rule to generate kinematically determinate crease patterns that can be designed to fold rigidly. Finally, we discuss the limitations of the PTU with respect to the global rigid foldability of a crease pattern.
Rigid foldability is an important requirement when origami is used as the basis to design technical systems that consist of rigid materials. This paper presents a heuristic algorithm that adjusts the location of vertices of nonrigidly foldable but kinematically determinate crease patterns such that they become rigidly foldable. The adjustment is achieved by utilizing constraint violations that occur during the folding process of nonrigidly foldable configurations. The folding process is kinematically simulated through a robust simulator that is based on a bar and hinge principle. The benefits of the algorithm are showcased in different examples, including single-vertex as well as multi-vertex crease patterns.
Rigid foldability is the property of an origami that folds continuously from an unfolded to a folded state without deformation in its facets. Although extensively researched, there exist no intrinsic conditions for the rigid foldability of a degree-four vertex, which is the simplest possible origami building block that folds nontrivially. In this paper, we derive a necessary and sufficient condition for the rigid foldability of a degree-four vertex and show that it can be reduced to a purely sufficient condition, which is equivalent to a known condition from the realm of spherical mechanisms. The implications of these conditions are discussed, which reveals the connection between rigid and flat foldability, the two most important mathematical notions in origami. In practice, this work further contributes to the design synthesis and analysis of deployable structures, in which the mechanics of degree-four vertices is omnipresent.
Designing structures through the means of origami brings many advantages for engineering applications. In current research, the underlying origami principle is often selected based on experience out of a range of known patterns and then manually altered to fit the design problem. This tedious and time-consuming procedure, if automated through computational tools, has the potential to facilitate the design of origami engineering applications. This however requires efficient kinematic simulation of origamis that is also able to accommodate to design requirements specific to foldable structures. In this paper, a simulator is implemented that is able to model the motion of origami vertices without the need for mountain-valley assignments and with a path of deployment as activation. The formulation of constraint equations through these vertex positions does not restrict the system to certain folding configurations, which is why the approach is able to detect different rigid body modes resulting from single activations. Finding rigid body modes can be beneficial for the search of design alternatives conforming to certain input requirements. The results of the simulation show promise for the incorporation of the simulator within an automated procedure for the design of origamis.
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