19,780 research outputs found
Meso-scale FDM material layout design strategies under manufacturability constraints and fracture conditions
In the manufacturability-driven design (MDD) perspective, manufacturability of the product or system is the most important of the design requirements. In addition to being able to ensure that complex designs (e.g., topology optimization) are manufacturable with a given process or process family, MDD also helps mechanical designers to take advantage of unique process-material effects generated during manufacturing. One of the most recognizable examples of this comes from the scanning-type family of additive manufacturing (AM) processes; the most notable and familiar member of this family is the fused deposition modeling (FDM) or fused filament fabrication (FFF) process. This process works by selectively depositing uniform, approximately isotropic beads or elements of molten thermoplastic material (typically structural engineering plastics) in a series of pre-specified traces to build each layer of the part. There are many interesting 2-D and 3-D mechanical design problems that can be explored by designing the layout of these elements. The resulting structured, hierarchical material (which is both manufacturable and customized layer-by-layer within the limits of the process and material) can be defined as a manufacturing process-driven structured material (MPDSM). This dissertation explores several practical methods for designing these element layouts for 2-D and 3-D meso-scale mechanical problems, focusing ultimately on design-for-fracture. Three different fracture conditions are explored: (1) cases where a crack must be prevented or stopped, (2) cases where the crack must be encouraged or accelerated, and (3) cases where cracks must grow in a simple pre-determined pattern. Several new design tools, including a mapping method for the FDM manufacturability constraints, three major literature reviews, the collection, organization, and analysis of several large (qualitative and quantitative) multi-scale datasets on the fracture behavior of FDM-processed materials, some new experimental equipment, and the refinement of a fast and simple g-code generator based on commercially-available software, were developed and refined to support the design of MPDSMs under fracture conditions. The refined design method and rules were experimentally validated using a series of case studies (involving both design and physical testing of the designs) at the end of the dissertation. Finally, a simple design guide for practicing engineers who are not experts in advanced solid mechanics nor process-tailored materials was developed from the results of this project.U of I OnlyAuthor's request
Rank-based linkage I: triplet comparisons and oriented simplicial complexes
Rank-based linkage is a new tool for summarizing a collection of objects
according to their relationships. These objects are not mapped to vectors, and
``similarity'' between objects need be neither numerical nor symmetrical. All
an object needs to do is rank nearby objects by similarity to itself, using a
Comparator which is transitive, but need not be consistent with any metric on
the whole set. Call this a ranking system on . Rank-based linkage is applied
to the -nearest neighbor digraph derived from a ranking system. Computations
occur on a 2-dimensional abstract oriented simplicial complex whose faces are
among the points, edges, and triangles of the line graph of the undirected
-nearest neighbor graph on . In steps it builds an
edge-weighted linkage graph where
is called the in-sway between objects and . Take to be
the links whose in-sway is at least , and partition into components of
the graph , for varying . Rank-based linkage is a
functor from a category of out-ordered digraphs to a category of partitioned
sets, with the practical consequence that augmenting the set of objects in a
rank-respectful way gives a fresh clustering which does not ``rip apart`` the
previous one. The same holds for single linkage clustering in the metric space
context, but not for typical optimization-based methods. Open combinatorial
problems are presented in the last section.Comment: 37 pages, 12 figure
The Metaverse: Survey, Trends, Novel Pipeline Ecosystem & Future Directions
The Metaverse offers a second world beyond reality, where boundaries are
non-existent, and possibilities are endless through engagement and immersive
experiences using the virtual reality (VR) technology. Many disciplines can
benefit from the advancement of the Metaverse when accurately developed,
including the fields of technology, gaming, education, art, and culture.
Nevertheless, developing the Metaverse environment to its full potential is an
ambiguous task that needs proper guidance and directions. Existing surveys on
the Metaverse focus only on a specific aspect and discipline of the Metaverse
and lack a holistic view of the entire process. To this end, a more holistic,
multi-disciplinary, in-depth, and academic and industry-oriented review is
required to provide a thorough study of the Metaverse development pipeline. To
address these issues, we present in this survey a novel multi-layered pipeline
ecosystem composed of (1) the Metaverse computing, networking, communications
and hardware infrastructure, (2) environment digitization, and (3) user
interactions. For every layer, we discuss the components that detail the steps
of its development. Also, for each of these components, we examine the impact
of a set of enabling technologies and empowering domains (e.g., Artificial
Intelligence, Security & Privacy, Blockchain, Business, Ethics, and Social) on
its advancement. In addition, we explain the importance of these technologies
to support decentralization, interoperability, user experiences, interactions,
and monetization. Our presented study highlights the existing challenges for
each component, followed by research directions and potential solutions. To the
best of our knowledge, this survey is the most comprehensive and allows users,
scholars, and entrepreneurs to get an in-depth understanding of the Metaverse
ecosystem to find their opportunities and potentials for contribution
Positive Geometries of S-matrix without Color
In this note, we prove that the realization of associahedron discovered by
Arkani-Hamed, Bai, He, and Yun (ABHY) is a positive geometry for tree-level
S-matrix of scalars which have no color and which interact via cubic coupling.
More in detail, we consider diffeomorphic images of the ABHY associahedron. The
diffeomorphisms are linear maps parametrized by the right cosets of the
Dihedral group on n elements. The set of all the boundaries associated with
these copies of ABHY associahedron exhaust all the simple poles. We prove that
the sum over the diffeomorphic copies of ABHY associahedron is a positive
geometry and the total volume obtained by summing over all the dual
associahedra is proportional to the tree-level S matrix of (massive or
massless) scalar particles with cubic coupling. We then provide non-trivial
evidence that the projection of the planar scattering forms parametrized by the
Stokes polytope on these realizations of the associahedron leads to the
tree-level amplitudes of scalar particles, which interact via quartic coupling.
Our results build on ideas laid out in our previous works, leading to further
evidence that a large class of positive geometries which are diffeomorphic to
the ABHY associahedron defines an ``amplituhedron" for a tree-level S matrix of
some local and unitary scalar theory. We also highlight a fundamental
obstruction in applying these ideas to discover positive geometry for the one
loop integrand when propagating states have no color.Comment: 33 Pages, 4 Figure
Can you hear your location on a manifold?
We introduce a variation on Kac's question, "Can one hear the shape of a
drum?" Instead of trying to identify a compact manifold and its metric via its
Laplace--Beltrami spectrum, we ask if it is possible to uniquely identify a
point on the manifold, up to symmetry, from its pointwise counting function
where here and form an orthonormal
basis for . This problem has a physical interpretation. You are placed
at an arbitrary location in a familiar room with your eyes closed. Can you
identify your location in the room by clapping your hands once and listening to
the resulting echos and reverberations?
The main result of this paper provides an affirmative answer to this question
for a generic class of metrics. We also probe the problem with a variety of
simple examples, highlighting along the way helpful geometric invariants that
can be pulled out of the pointwise counting function .Comment: 26 pages, 1 figur
Tonelli Approach to Lebesgue Integration
Leonida Tonelli devised an interesting and efficient method to introduce the
Lebesgue integral. The details of this method can only be found in the original
Tonelli paper and in an old italian course and solely for the case of the
functions of one variable. We believe that it is woth knowing this method and
here we present a complete account for functions of every number of variables
Band width estimates with lower scalar curvature bounds
A band is a connected compact manifold X together with a decomposition ∂X = ∂−X t ∂+X where ∂±X are non-empty unions of boundary components. If X is equipped with a Riemannian metric, the pair (X, g) is called a Riemannian band and the width of (X, g) is defined to be the distance between ∂−X and ∂+X with respect to g.
Following Gromov’s seminal work on metric inequalities with scalar curvature, the study of Riemannian bands with lower curvature bounds has been an active field of research in recent years, which led to several breakthroughs on longstanding open problems in positive scalar curvature geometry and to a better understanding of the positive mass theorem in general relativity
In the first part of this thesis we combine ideas of Gromov and Cecchini-Zeidler and use the variational calculus surrounding so called µ-bubbles to establish a scalar and mean curvature comparison principle for Riemannian bands with the property that no closed embedded hypersurface which separates the two ends of the band
admits a metric of positive scalar curvature. The model spaces we use for this comparison are warped product over scalar flat manifolds with log-concave warping functions.
We employ ideas from surgery and bordism theory to deduce that, if Y is a closed orientable manifold which does not admit a metric of positive scalar curvature, dim(Y ) 6= 4 and Xn≤7 = Y ×[−1, 1], the width of X with respect to any Riemannian metric with scalar curvature ≥ n(n − 1) is bounded from above by 2π n. This solves, up to dimension 7, a conjecture due to Gromov in the orientable case.
Furthermore, we adapt and extend our methods to show that, if Y is as before and Mn≤7 = Y × R, then M does not admit a metric of positive scalar curvature. This solves, up to dimension 7 a conjecture due to Rosenberg and Stolz in the orientable case.
In the second part of this thesis we explore how these results transfer to the setting where the lower scalar curvature bound is replaced by a lower bound on the macroscopic scalar curvature of a Riemannian band. This curvature condition amounts to an upper bound on the volumes of all unit balls in the universal cover of the band.
We introduce a new class of orientable manifolds we call filling enlargeable and prove: If Y is filling enlargeable, Xn = Y × [−1, 1] and g is a Riemannian metric on X with the property that the volumes of all unit balls in the universal cover of (X, g) are bounded from above by a small dimensional constant εn, then width(X, g) ≤ 1.
Finally, we establish that whether or not a closed orientable manifold is filling enlargeable or not depends on the image of the fundamental class under the classifying map of the universal cover
Isotopic piecewise affine approximation of algebraic or varieties
We propose a novel sufficient condition establishing that a piecewise affine
variety has the same topology as a variety of the sphere defined
by positively homogeneous functions. This covers the case of
varieties in the projective space . We prove that this condition
is sufficient in the case of codimension one and arbitrary dimension. We
describe an implementation working for homogeneous polynomials in arbitrary
dimension and codimension and give experimental evidences that our condition
might still be sufficient in codimension greater than one
Safe Zeroth-Order Optimization Using Quadratic Local Approximations
This paper addresses black-box smooth optimization problems, where the
objective and constraint functions are not explicitly known but can be queried.
The main goal of this work is to generate a sequence of feasible points
converging towards a KKT primal-dual pair. Assuming to have prior knowledge on
the smoothness of the unknown objective and constraints, we propose a novel
zeroth-order method that iteratively computes quadratic approximations of the
constraint functions, constructs local feasible sets and optimizes over them.
Under some mild assumptions, we prove that this method returns an -KKT
pair (a property reflecting how close a primal-dual pair is to the exact KKT
condition) within iterations. Moreover, we numerically show
that our method can achieve faster convergence compared with some
state-of-the-art zeroth-order approaches. The effectiveness of the proposed
approach is also illustrated by applying it to nonconvex optimization problems
in optimal control and power system operation.Comment: arXiv admin note: text overlap with arXiv:2211.0264
Nonlocal error bounds for piecewise affine functions
The paper is devoted to a detailed analysis of nonlocal error bounds for
nonconvex piecewise affine functions. We both improve some existing results on
error bounds for such functions and present completely new necessary and/or
sufficient conditions for a piecewise affine function to have an error bound on
various types of bounded and unbounded sets. In particular, we show that any
piecewise affine function has an error bound on an arbitrary bounded set and
provide several types of easily verifiable sufficient conditions for such
functions to have an error bound on unbounded sets. We also present general
necessary and sufficient conditions for a piecewise affine function to have an
error bound on a finite union of polyhedral sets (in particular, to have a
global error bound), whose derivation reveals a structure of sublevel sets and
recession functions of piecewise affine functions
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