2,057 research outputs found
Fractal homogenization of multiscale interface problems
Inspired by continuum mechanical contact problems with geological fault
networks, we consider elliptic second order differential equations with jump
conditions on a sequence of multiscale networks of interfaces with a finite
number of non-separating scales. Our aim is to derive and analyze a description
of the asymptotic limit of infinitely many scales in order to quantify the
effect of resolving the network only up to some finite number of interfaces and
to consider all further effects as homogeneous. As classical homogenization
techniques are not suited for this kind of geometrical setting, we suggest a
new concept, called fractal homogenization, to derive and analyze an asymptotic
limit problem from a corresponding sequence of finite-scale interface problems.
We provide an intuitive characterization of the corresponding fractal solution
space in terms of generalized jumps and gradients together with continuous
embeddings into L2 and Hs, s<1/2. We show existence and uniqueness of the
solution of the asymptotic limit problem and exponential convergence of the
approximating finite-scale solutions. Computational experiments involving a
related numerical homogenization technique illustrate our theoretical findings
Characterization of maximally random jammed sphere packings: Voronoi correlation functions
We characterize the structure of maximally random jammed (MRJ) sphere
packings by computing the Minkowski functionals (volume, surface area, and
integrated mean curvature) of their associated Voronoi cells. The probability
distribution functions of these functionals of Voronoi cells in MRJ sphere
packings are qualitatively similar to those of an equilibrium hard-sphere
liquid and partly even to the uncorrelated Poisson point process, implying that
such local statistics are relatively structurally insensitive. This is not
surprising because the Minkowski functionals of a single Voronoi cell
incorporate only local information and are insensitive to global structural
information. To improve upon this, we introduce descriptors that incorporate
nonlocal information via the correlation functions of the Minkowski functionals
of two cells at a given distance as well as certain cell-cell probability
density functions. We evaluate these higher-order functions for our MRJ
packings as well as equilibrium hard spheres and the Poisson point process. We
find strong anticorrelations in the Voronoi volumes for the hyperuniform MRJ
packings, consistent with previous findings for other pair correlations [A.
Donev et al., Phys. Rev. Lett. 95, 090604 (2005)], indicating that large-scale
volume fluctuations are suppressed by accompanying large Voronoi cells with
small cells, and vice versa. In contrast to the aforementioned local Voronoi
statistics, the correlation functions of the Voronoi cells qualitatively
distinguish the structure of MRJ sphere packings (prototypical glasses) from
that of the correlated equilibrium hard-sphere liquids. Moreover, while we did
not find any perfect icosahedra (the locally densest possible structure in
which a central sphere contacts 12 neighbors) in the MRJ packings, a
preliminary Voronoi topology analysis indicates the presence of strongly
distorted icosahedra.Comment: 13 pages, 10 figure
An integer representation for periodic tilings of the plane by regular polygons
We describe a representation for periodic tilings of the plane by regular polygons. Our
approach is to represent explicitly a small subset of seed vertices from which we systematically generate all elements of the tiling by translations. We represent a tiling concretely
by a (2+n)×4 integer matrix containing lattice coordinates for two translation vectors
and n seed vertices. We discuss several properties of this representation and describe
how to exploit the representation elegantly and efficiently for reconstruction, rendering,
and automatic crystallographic classification by symmetry detection
Relating cell shape and mechanical stress in a spatially disordered epithelium using a vertex-based model
Using a popular vertex-based model to describe a spatially disordered planar
epithelial monolayer, we examine the relationship between cell shape and
mechanical stress at the cell and tissue level. Deriving expressions for stress
tensors starting from an energetic formulation of the model, we show that the
principal axes of stress for an individual cell align with the principal axes
of shape, and we determine the bulk effective tissue pressure when the
monolayer is isotropic at the tissue level. Using simulations for a monolayer
that is not under peripheral stress, we fit parameters of the model to
experimental data for Xenopus embryonic tissue. The model predicts that
mechanical interactions can generate mesoscopic patterns within the monolayer
that exhibit long-range correlations in cell shape. The model also suggests
that the orientation of mechanical and geometric cues for processes such as
cell division are likely to be strongly correlated in real epithelia. Some
limitations of the model in capturing geometric features of Xenopus epithelial
cells are highlighted.Comment: 29 pages, 10 figures, revisio
Analysis and Generation of Quality Polytopal Meshes with Applications to the Virtual Element Method
This thesis explores the concept of the quality of a mesh, the latter being intended as the discretization of a two- or three- dimensional domain.
The topic is interdisciplinary in nature, as meshes are massively used in several fields from both the geometry processing and the numerical analysis communities.
The goal is to produce a mesh with good geometrical properties and the lowest possible number of elements, able to produce results in a target range of accuracy.
In other words, a good quality mesh that is also cheap to handle, overcoming the typical trade-off between quality and computational cost.
To reach this goal, we first need to answer the question:
''How, and how much, does the accuracy of a numerical simulation or a scientific computation (e.g., rendering, printing, modeling operations) depend on the particular mesh adopted to model the problem? And which geometrical features of the mesh most influence the result?''
We present a comparative study of the different mesh types, mesh generation techniques, and mesh quality measures currently available in the literature related to both engineering and computer graphics applications.
This analysis leads to the precise definition of the notion of quality for a mesh, in the particular context of numerical simulations of partial differential equations with the virtual element method, and the consequent construction of criteria to determine and optimize the quality of a given mesh.
Our main contribution consists in a new mesh quality indicator for polytopal meshes, able to predict the performance of the virtual element method over a particular mesh before running the simulation.
Strictly related to this, we also define a quality agglomeration algorithm that optimizes the quality of a mesh by wisely agglomerating groups of neighboring elements.
The accuracy and the reliability of both tools are thoroughly verified in a series of tests in different scenarios
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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
Multi-layer approach to motion planning in obstacle rich environment
A widespread use of robotic technology in civilian and military applications has
generated a need for advanced motion planning algorithms that are real-time implementable.
These algorithms are required to navigate autonomous vehicles through
obstacle-rich environments. This research has led to the development of the multilayer
trajectory generation approach. It is built on the principle of separation of
concerns, which partitions a given problem into multiple independent layers, and addresses
complexity that is inherent at each level. We partition the motion planning
algorithm into a roadmap layer and an optimal control layer. At the roadmap layer,
elements of computational geometry are used to process the obstacle rich environment
and generate feasible sets. These are used by the optimal control layer to generate
trajectories while satisfying dynamics of the vehicle. The roadmap layer ignores the
dynamics of the system, and the optimal control layer ignores the complexity of the
environment, thus achieving a separation of concern. This decomposition enables
computationally tractable methods to be developed for addressing motion planning
in complex environments. The approach is applied in known and unknown environments.
The methodology developed in this thesis has been successfully applied to a 6
DOF planar robotic testbed. Simulation results suggest that the planner can generate
trajectories that navigate through obstacles while satisfying dynamical constraints
Computers from plants we never made. Speculations
We discuss possible designs and prototypes of computing systems that could be
based on morphological development of roots, interaction of roots, and analog
electrical computation with plants, and plant-derived electronic components. In
morphological plant processors data are represented by initial configuration of
roots and configurations of sources of attractants and repellents; results of
computation are represented by topology of the roots' network. Computation is
implemented by the roots following gradients of attractants and repellents, as
well as interacting with each other. Problems solvable by plant roots, in
principle, include shortest-path, minimum spanning tree, Voronoi diagram,
-shapes, convex subdivision of concave polygons. Electrical properties
of plants can be modified by loading the plants with functional nanoparticles
or coating parts of plants of conductive polymers. Thus, we are in position to
make living variable resistors, capacitors, operational amplifiers,
multipliers, potentiometers and fixed-function generators. The electrically
modified plants can implement summation, integration with respect to time,
inversion, multiplication, exponentiation, logarithm, division. Mathematical
and engineering problems to be solved can be represented in plant root networks
of resistive or reaction elements. Developments in plant-based computing
architectures will trigger emergence of a unique community of biologists,
electronic engineering and computer scientists working together to produce
living electronic devices which future green computers will be made of.Comment: The chapter will be published in "Inspired by Nature. Computing
inspired by physics, chemistry and biology. Essays presented to Julian Miller
on the occasion of his 60th birthday", Editors: Susan Stepney and Andrew
Adamatzky (Springer, 2017
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