9,479 research outputs found
Dense Packings of Congruent Circles in Rectangles with a Variable Aspect Ratio
We use computational experiments to find the rectangles of minimum area into
which a given number n of non-overlapping congruent circles can be packed. No
assumption is made on the shape of the rectangles. Most of the packings found
have the usual regular square or hexagonal pattern. However, for 1495 values of
n in the tested range n =< 5000, specifically, for n = 49, 61, 79, 97, 107,...
4999, we prove that the optimum cannot possibly be achieved by such regular
arrangements. The evidence suggests that the limiting height-to-width ratio of
rectangles containing an optimal hexagonal packing of circles tends to
2-sqrt(3) as n tends to infinity, if the limit exists.Comment: 21 pages, 13 figure
A Multiphase Shape Optimization Problem for Eigenvalues: Qualitative Study and Numerical Results
We consider the multiphase shape optimization problem
where
is a given constant and is a bounded open set
with Lipschitz boundary. We give some new results concerning the qualitative
properties of the optimal sets and the regularity of the corresponding
eigenfunctions. We also provide numerical results for the optimal partitions
Minimum Perimeter Rectangles That Enclose Congruent Non-Overlapping Circles
We use computational experiments to find the rectangles of minimum perimeter
into which a given number n of non-overlapping congruent circles can be packed.
No assumption is made on the shape of the rectangles. In many of the packings
found, the circles form the usual regular square-grid or hexagonal patterns or
their hybrids. However, for most values of n in the tested range n =< 5000,
e.g., for n = 7, 13, 17, 21, 22, 26, 31, 37, 38, 41, 43...,4997, 4998, 4999,
5000, we prove that the optimum cannot possibly be achieved by such regular
arrangements. Usually, the irregularities in the best packings found for such n
are small, localized modifications to regular patterns; those irregularities
are usually easy to predict. Yet for some such irregular n, the best packings
found show substantial, extended irregularities which we did not anticipate. In
the range we explored carefully, the optimal packings were substantially
irregular only for n of the form n = k(k+1)+1, k = 3, 4, 5, 6, 7, i.e., for n =
13, 21, 31, 43, and 57. Also, we prove that the height-to-width ratio of
rectangles of minimum perimeter containing packings of n congruent circles
tends to 1 as n tends to infinity.Comment: existence of irregular minimum perimeter packings for n not of the
form (10) is conjectured; smallest such n is n=66; existence of irregular
minimum area packings is conjectured, e.g. for n=453; locally optimal
packings for the two minimization criteria are conjecturally the same (p.22,
line 5); 27 pages, 12 figure
A 2D ultrasonic array design incorporating hexagonal-shaped elements for NDE applications
Contemporary 2D Ultrasonic arrays suffer from low SNR and limited steering capabilities. Yet, there is a great desire in the industry to increase the operational frequency, in order to enhance their volumetric imaging resolution. State-of-the art arrays use an orthogonal matrix of rectangular elements as this is a natural step forward from the conventional 1D array structure. The objective of this work is to evaluate properties of triangular, rather than rectangular ceramic pillars in a 1-3 connectivity piezoelectric composite for application in a hexagonal-element 2D array. A 3MHz prototype device exploiting new hexagonal substructure have been manufactured. Measured mechanical cross-coupling level is -21.9dB between neighbouring hexagonal elements, providing validation of simulation result. Corroboration between measured and FE modelled device behaviour is demonstrated
Packing ellipsoids with overlap
The problem of packing ellipsoids of different sizes and shapes into an
ellipsoidal container so as to minimize a measure of overlap between ellipsoids
is considered. A bilevel optimization formulation is given, together with an
algorithm for the general case and a simpler algorithm for the special case in
which all ellipsoids are in fact spheres. Convergence results are proved and
computational experience is described and illustrated. The motivating
application - chromosome organization in the human cell nucleus - is discussed
briefly, and some illustrative results are presented
Wet Granular Materials
Most studies on granular physics have focused on dry granular media, with no
liquids between the grains. However, in geology and many real world
applications (e.g., food processing, pharmaceuticals, ceramics, civil
engineering, constructions, and many industrial applications), liquid is
present between the grains. This produces inter-grain cohesion and drastically
modifies the mechanical properties of the granular media (e.g., the surface
angle can be larger than 90 degrees). Here we present a review of the
mechanical properties of wet granular media, with particular emphasis on the
effect of cohesion. We also list several open problems that might motivate
future studies in this exciting but mostly unexplored field.Comment: review article, accepted for publication in Advances in Physics;
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