A Combinatorial Parametric Engineering Model for Solid Freeform Fabrication

Fabricated parts are often represented as compact connected smooth 3-manifolds with boundary, where the boundaries consist of compact smooth 2-manifolds. This class of mathematical structures includes topological spaces with enclosed voids and tunnels. Useful information about these structures are coded into level functions (Morse functions) which map points in the 3-manifold onto their height above a fixed plane. By definition, Morse functions are smooth functions, all of whose critical points are nondegenerate. This information is presented by the Reeb graph construction that develops a topologically informative skeleton of the manifold whose nodes are the critical points of the Morse function and whose edges are associated with the connected components between critical slices. This approach accurately captures the SFF process: using a solid geometric model of the part, defining surface boundaries; selecting a part orientation; forming planar slices, decomposing the solid into a sequence of thin cross-sectional polyhedral layers; and then fabricating the part by producing the polyhedra by additive manufacturing. This note will define a qualitative and combinatorial parametric engineering model of the SFF part design process. The objects under study will be abstract simplicial complexes K with boundary ∂K. Systems of labeled 2-surfaces in K, called slices, will be associated with the cross-sectional polyhedral layers. The labeled slices are mapped into a family of digraph automata, which, unlike cellular automata, are defined not on regular lattices with simple connectivities (cells usually have either 4 or 8 cell neighborhoods) but on unrestricted digraphs whose connectivities are irregular and more complicated.Mechanical Engineerin

An Adaptive Control Architecture for Freeform Fabrication

Mechanical Engineerin

An Accurate Determination of the Exchange Constant in Sr_2CuO_3 from Recent Theoretical Results

Data from susceptibility measurements on Sr_2CuO_3 are compared with recent theoretical predictions for the magnetic susceptibility of the antiferromagnetic spin-1/2 Heisenberg chain. The experimental data fully confirms the theoretical predictions and in turn we establish that Sr_2CuO_3 behaves almost perfectly like a one-dimensional antiferromagnet with an exchange coupling of J = 1700^{+150}_{-100}K.Comment: revised and reformatted paper with new title to appear in Phys. Rev B (Feb.1996). 3 pages (revtex) with 3 embedded figures (macro included). A complete postscript file is available from http://fy.chalmers.se/~eggert/expsusc.ps or by request from [email protected]

Hybrid Automata in the Context of Additive Manufacturing

To maintain the forward momentum of additive manufacturing technology, it is necessary to thoroughly evaluate new and potentially useful technological developments in this field. One such development is the intense interest being directed to the field of hybrid automata (HA). Hybrid automata combine both the discrete processing behavior of finite automata as well as the continuous, or flow, behavior of dynamical systems. At this point, some important results on hybrid automata have been obtained, but many open questions remain, including those concerning the decidability of HS operational procedures. (Recall that decidability is directed to a decision problem, that is, a definite true-or-false response given by an effective procedure.) Some important decidability results for HAs have been obtained. For example, in [Henzinger et al.1998] the reachability problem for timed automata (an HA class) has been convincingly shown to be decidable. However, it should also be noted that subtle and difficult issues have been identified, e.g., [Fraenzle 1999], [Asarin, Collins, 2005]. This paper will provide a summary review of the operational features of HAs as they might pertain to additive manufacturing, and then briefly consider the following technical issues: (i) are the classical models of the real numbers best suited to deal with the necessarily approximate measures of physical systems or would non-standard analysis of [Robinson 1996] be a better fit; and (ii) would the introduction of “noisy semantics” and finite arithmetic precision, following [Freidlin, Wentzell 1984], be a better work around?Mechanical Engineerin

Solid Freeform Fabrication and Parametric Engineering

Solid freeform fabrication (SFF) is based on a part-centric process model: create a solid model of the part, form planar slices, and fabricate the part by producing all of the polyhedra by any of several methods. A class of applications is emerging which will pull SFF from the part-centric model to a new paradigm in which parts are seen as components of interactive networks. This is precisely the context for which parametric engineering has been proposed. In this paper, a computational framework will be developed that consists of a finitary topological representation of parts as 3-manifolds and representation of the evaluative context by means of a symbolic environment. Parametric engineering is interpreted as a controlled evaluation of parameters within the context of the symbolic environment.Mechanical Engineerin