270 research outputs found
3D printing dimensional calibration shape: Clebsch Cubic
3D printing and other layer manufacturing processes are challenged by
dimensional accuracy. Several techniques are used to validate and calibrate
dimensional accuracy through the complete building envelope. The validation
process involves the growing and measuring of a shape with known parameters.
The measured result is compared with the intended digital model. Processes with
the risk of deformation after time or post processing may find this technique
beneficial. We propose to use objects from algebraic geometry as test shapes. A
cubic surface is given as the zero set of a 3rd degree polynomial with 3
variables. A class of cubics in real 3D space contains exactly 27 real lines.
We provide a library for the computer algebra system Singular which, from 6
given points in the plane, constructs a cubic and the lines on it. A surface
shape derived from a cubic offers simplicity to the dimensional comparison
process, in that the straight lines and many other features can be analytically
determined and easily measured using non-digital equipment. For example, the
surface contains so-called Eckardt points, in each of which three of the lines
intersect, and also other intersection points of pairs of lines. Distances
between these intersection points can easily be measured, since the points are
connected by straight lines. At all intersection points of lines, angles can be
verified. Hence, many features distributed over the build volume are known
analytically, and can be used for the validation process. Due to the thin shape
geometry the material required to produce an algebraic surface is minimal. This
paper is the first in a series that proposes the process chain to first define
a cubic with a configuration of lines in a given print volume and then to
develop the point cloud for the final manufacturing. Simple measuring
techniques are recommended.Comment: 8 pages, 1 figure, 1 tabl
Triangular buckling patterns of twisted inextensible strips
When twisting a strip of paper or acetate under high longitudinal tension,
one observes, at some critical load, a buckling of the strip into a regular
triangular pattern. Very similar triangular facets have recently been observed
in solutions to a new set of geometrically-exact equations describing the
equilibrium shape of thin inextensible elastic strips. Here we formulate a
modified boundary-value problem for these equations and construct post-buckling
solutions in good agreement with the observed pattern in twisted strips. We
also study the force-extension and moment-twist behaviour of these strips by
varying the mode number n of triangular facets
Ultrafast control of inelastic tunneling in a double semiconductor quantum
In a semiconductor-based double quantum well (QW) coupled to a degree of
freedom with an internal dynamics, we demonstrate that the electronic motion is
controllable within femtoseconds by applying appropriately shaped
electromagnetic pulses. In particular, we consider a pulse-driven AlxGa1-xAs
based symmetric double QW coupled to uniformly distributed or localized
vibrational modes and present analytical results for the lowest two levels.
These predictions are assessed and generalized by full-fledged numerical
simulations showing that localization and time-stabilization of the driven
electron dynamics is indeed possible under the conditions identified here, even
with a simultaneous excitations of vibrational modes.Comment: to be published in Appl.Phys.Let
Variational formulation of ideal fluid flows according to gauge principle
On the basis of the gauge principle of field theory, a new variational
formulation is presented for flows of an ideal fluid. The fluid is defined
thermodynamically by mass density and entropy density, and its flow fields are
characterized by symmetries of translation and rotation. The rotational
transformations are regarded as gauge transformations as well as the
translational ones. In addition to the Lagrangians representing the translation
symmetry, a structure of rotation symmetry is equipped with a Lagrangian
including the vorticity and a vector potential bilinearly. Euler's
equation of motion is derived from variations according to the action
principle. In addition, the equations of continuity and entropy are derived
from the variations. Equations of conserved currents are deduced as the Noether
theorem in the space of Lagrangian coordinate \ba. Without , the
action principle results in the Clebsch solution with vanishing helicity. The
Lagrangian yields non-vanishing vorticity and provides a source
term of non-vanishing helicity. The vorticity equation is derived as an
equation of the gauge field, and the characterizes topology of the
field. The present formulation is comprehensive and provides a consistent basis
for a unique transformation between the Lagrangian \ba space and the Eulerian
\bx space. In contrast, with translation symmetry alone, there is an
arbitrariness in the ransformation between these spaces.Comment: 34 pages, Fluid Dynamics Research (2008), accepted on 1st Dec. 200
Singular Casimir Elements of the Euler Equation and Equilibrium Points
The problem of the nonequivalence of the sets of equilibrium points and
energy-Casimir extremal points, which occurs in the noncanonical Hamiltonian
formulation of equations describing ideal fluid and plasma dynamics, is
addressed in the context of the Euler equation for an incompressible inviscid
fluid. The problem is traced to a Casimir deficit, where Casimir elements
constitute the center of the Lie-Poisson algebra underlying the Hamiltonian
formulation, and this leads to a study of the symplectic operator defining the
Poisson bracket. The kernel of the symplectic operator, for this typical
example of an infinite-dimensional Hamiltonian system for media in terms of
Eulerian variables, is analyzed. For two-dimensional flows, a rigorously
solvable system is formulated. The nonlinearity of the Euler equation makes the
symplectic operator inhomogeneous on phase space (the function space of the
state variable), and it is seen that this creates a singularity where the
nullity of the symplectic operator (the "dimension" of the center) changes.
Singular Casimir elements stemming from this singularity are unearthed using a
generalization of the functional derivative that occurs in the Poisson bracket
Integrable discretizations of some cases of the rigid body dynamics
A heavy top with a fixed point and a rigid body in an ideal fluid are
important examples of Hamiltonian systems on a dual to the semidirect product
Lie algebra . We give a Lagrangian derivation of
the corresponding equations of motion, and introduce discrete time analogs of
two integrable cases of these systems: the Lagrange top and the Clebsch case,
respectively. The construction of discretizations is based on the discrete time
Lagrangian mechanics on Lie groups, accompanied by the discrete time Lagrangian
reduction. The resulting explicit maps on are Poisson with respect to
the Lie--Poisson bracket, and are also completely integrable. Lax
representations of these maps are also found.Comment: arXiv version is already officia
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