51,743 research outputs found
Geometry of logarithmic strain measures in solid mechanics
We consider the two logarithmic strain measureswhich are isotropic invariants of the
Hencky strain tensor , and show that they can be uniquely characterized
by purely geometric methods based on the geodesic distance on the general
linear group . Here, is the deformation gradient,
is the right Biot-stretch tensor, denotes the principal
matrix logarithm, is the Frobenius matrix norm, is the
trace operator and is the -dimensional deviator of
. This characterization identifies the Hencky (or
true) strain tensor as the natural nonlinear extension of the linear
(infinitesimal) strain tensor , which is the
symmetric part of the displacement gradient , and reveals a close
geometric relation between the classical quadratic isotropic energy potential
in
linear elasticity and the geometrically nonlinear quadratic isotropic Hencky
energywhere
is the shear modulus and denotes the bulk modulus. Our deduction
involves a new fundamental logarithmic minimization property of the orthogonal
polar factor , where is the polar decomposition of . We also
contrast our approach with prior attempts to establish the logarithmic Hencky
strain tensor directly as the preferred strain tensor in nonlinear isotropic
elasticity
Matrix embeddings on flat and the geometry of membranes
We show that given three hermitian matrices, what one could call a fuzzy
representation of a membrane, there is a well defined procedure to define a set
of oriented Riemann surfaces embedded in using an index function defined
for points in that is constructed from the three matrices and the point.
The set of surfaces is covariant under rotations, dilatations and translation
operations on , it is additive on direct sums and the orientation of the
surfaces is reversed by complex conjugation of the matrices. The index we build
is closely related to the Hanany-Witten effect. We also show that the surfaces
carry information of a line bundle with connection on them.
We discuss applications of these ideas to the study of holographic matrix
models and black hole dynamics.Comment: 41 pages, 3 figures, uses revtex4-1. v2: references added, corrected
an error in attribution of idea
Constructing IGA-suitable planar parameterization from complex CAD boundary by domain partition and global/local optimization
In this paper, we propose a general framework for constructing IGA-suitable
planar B-spline parameterizations from given complex CAD boundaries consisting
of a set of B-spline curves. Instead of forming the computational domain by a
simple boundary, planar domains with high genus and more complex boundary
curves are considered. Firstly, some pre-processing operations including
B\'ezier extraction and subdivision are performed on each boundary curve in
order to generate a high-quality planar parameterization; then a robust planar
domain partition framework is proposed to construct high-quality patch-meshing
results with few singularities from the discrete boundary formed by connecting
the end points of the resulting boundary segments. After the topology
information generation of quadrilateral decomposition, the optimal placement of
interior B\'ezier curves corresponding to the interior edges of the
quadrangulation is constructed by a global optimization method to achieve a
patch-partition with high quality. Finally, after the imposition of
C1=G1-continuity constraints on the interface of neighboring B\'ezier patches
with respect to each quad in the quadrangulation, the high-quality B\'ezier
patch parameterization is obtained by a C1-constrained local optimization
method to achieve uniform and orthogonal iso-parametric structures while
keeping the continuity conditions between patches. The efficiency and
robustness of the proposed method are demonstrated by several examples which
are compared to results obtained by the skeleton-based parameterization
approach
Classical Light Beams and Geometric Phases
We present a study of geometric phases in classical wave and polarisation
optics using the basic mathematical framework of quantum mechanics. Important
physical situations taken from scalar wave optics, pure polarisation optics,
and the behaviour of polarisation in the eikonal or ray limit of Maxwell's
equations in a transparent medium are considered. The case of a beam of light
whose propagation direction and polarisation state are both subject to change
is dealt with, attention being paid to the validity of Maxwell's equations at
all stages. Global topological aspects of the space of all propagation
directions are discussed using elementary group theoretical ideas, and the
effects on geometric phases are elucidated.Comment: 23 pages, 1 figur
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