Three-dimensional tolerance analysis of discrete surface structures based on the torsor cluster model

Purpose The purpose of this paper is to propose a novel mathematical model to present the three-dimensional tolerance of a discrete surface and to carry out an approach to analyze the tolerance of an assembly with a discrete surface structure. A discrete surface is a special structure of a large surface base with several discrete elements mounted on it, one, which is widely used in complex electromechanical products. Design/methodology/approach The geometric features of discrete surfaces are separated and characterized by small displacement torsors according to the spatial relationship of discrete elements. The torsor cluster model is established to characterize the integral feature variation of a discrete surface by integrating the torsor model. The influence and accumulation of the assembly tolerance of a discrete surface are determined by statistical tolerance analysis based on the unified Jacobian-Torsor method. Findings The effectiveness and superiority of the proposed model in comprehensive tolerance characterization of discrete surfaces are successfully demonstrated by a case study of a phased array antenna. The tolerance is evidently and intuitively computed and expressed based on the torsor cluster model. Research limitations/implications The tolerance analysis method proposed requires much time and high computing performance for the calculation of the statistical simulation. Practical implications The torsor cluster model achieves the three-dimensional tolerance representation of the discrete surface. The tolerance analysis method based on this model predicts the accumulation of the tolerance of components before their physical assembly. Originality/value This paper proposes the torsor cluster as a novel mathematical model to interpret the tolerance of a discrete surface.

Voss Surfaces: A Design Space for Geodesic Gridshells

The design of envelopes with complex geometries often leads to construction challenges. To overcome these difficulties, resorting to discrete differential geometry proved successful by establishing close links between mesh properties and the existence of good fabrication, assembling and mechanical properties. In this paper, the design of a special family of structures, called geodesic shells, is addressed using Voss nets, a family of discrete surfaces. The use of discrete Voss surfaces ensures that the structure can be built from simply connected, initially straight laths, and covered with flat panels. These advantageous constructive properties arise from the existence of a conjugate network of geodesic curves on the underlying smooth surface. Here, a review of Voss nets is presented and particular attention is given to the projection of normal vectors on the unit sphere. This projection, called Gauss map, creates a dual net which unveils the remarkable characteristics of Voss nets. Then, based on the previous study, two generation methods are introduced. One enables the exploration and the deformation of Voss nets while the second provides a more direct computational technique. The application of theses methodologies is discussed alongside formal examples.

Cyclic micromechanical controls of transient crustal deformation

<p>Episodic brittle-ductile behaviour reflects the complex interplay of micromechanical hardening and softening, often with some type of fluid pressure associated with introduction of new material that acts as the switch from coseismic to interseismic response. Brittle features observed in nature can in general be characterized as discrete surfaces or narrow zones across which fast particle displacements have occurred, with or without dilatant behaviour; this descriptively meets the criteria for generation of earthquakes. Likewise, non-brittle flow is a priori associated with slower particle velocities. This reduces the problem to one of how and why rocks cycle between slow and fast displacements. Particle displacement in the solid-state is limited to three processes: individual atoms, glide of packets of atoms and frictional displacement across an essentially free surface. Each of these processes, however large the feature being studied or rapid the displacements, necessitates the sequential overcoming of extant atomic bonding energies. Within the rock record, evidence of seismic events are embedded as new or reconstituted material introduced to the deforming host as a consequence of brittle deformation; for example, veins and pseudotachylyte. This new material acts as an important sink for strain energy whereby brittle responses are suppressed until such time as a new critical state is reached. In turn, the strain rate softening abetted by the new material provides a ductile overprint of their syn-fracture origin. Consequently, rheological transitions within Earth’s crust are spatially and temporally transient, evidence for which may be routinely lost. As part of this cyclic behaviour, localization of deformation can be viewed as the default state, with macroscopic deformation a result of organization into required dissipative structures.</p>

Chain of matrices, loop equations, and topological recursion

This article considers the so-called loop equations satisfied by integrals over random matrices coupled in a chain as well as their recursive solution in the perturbative case when the matrices are Hermitian. Random matrices are used in fields such as the study of multi-orthogonal polynomials or the enumeration of discrete surfaces, both of which are based on the analysis of a matrix integral. However, this term can be confusing since the definition of a matrix integral in these two applications is not the same. The article discusses these two definitions, perturbative and non-perturbative, along with their relationship. It first provides an overview of a matrix integral before comparing convergent and formal matrix integrals. It then describes the loop equations and their solution in the one-matrix model. It also examines matrices coupled in a chain plus external field and concludes with a generalization of the topological recursion.