825 research outputs found
Feature Lines for Illustrating Medical Surface Models: Mathematical Background and Survey
This paper provides a tutorial and survey for a specific kind of illustrative
visualization technique: feature lines. We examine different feature line
methods. For this, we provide the differential geometry behind these concepts
and adapt this mathematical field to the discrete differential geometry. All
discrete differential geometry terms are explained for triangulated surface
meshes. These utilities serve as basis for the feature line methods. We provide
the reader with all knowledge to re-implement every feature line method.
Furthermore, we summarize the methods and suggest a guideline for which kind of
surface which feature line algorithm is best suited. Our work is motivated by,
but not restricted to, medical and biological surface models.Comment: 33 page
Minimal geodesics along volume preserving maps, through semi-discrete optimal transport
We introduce a numerical method for extracting minimal geodesics along the
group of volume preserving maps, equipped with the L2 metric, which as observed
by Arnold solve Euler's equations of inviscid incompressible fluids. The method
relies on the generalized polar decomposition of Brenier, numerically
implemented through semi-discrete optimal transport. It is robust enough to
extract non-classical, multi-valued solutions of Euler's equations, for which
the flow dimension is higher than the domain dimension, a striking and
unavoidable consequence of this model. Our convergence results encompass this
generalized model, and our numerical experiments illustrate it for the first
time in two space dimensions.Comment: 21 pages, 9 figure
Relaxation to magnetohydrodynamics equilibria via collision brackets
Metriplectic dynamics is applied to compute equilibria of fluid dynamical
systems. The result is a relaxation method in which Hamiltonian dynamics
(symplectic structure) is combined with dissipative mechanisms (metric
structure) that relaxes the system to the desired equilibrium point. The
specific metric operator, which is considered in this work, is formally
analogous to the Landau collision operator. These ideas are illustrated by
means of case studies. The considered physical models are the Euler equations
in vorticity form, the Grad-Shafranov equation, and force-free MHD equilibria.Comment: Conference Proceeding (Theory of Fusions Plasmas, 2018), 9 pages, 8
figure
Maximization of Laplace-Beltrami eigenvalues on closed Riemannian surfaces
Let be a connected, closed, orientable Riemannian surface and denote
by the -th eigenvalue of the Laplace-Beltrami operator on
. In this paper, we consider the mapping .
We propose a computational method for finding the conformal spectrum
, which is defined by the eigenvalue optimization problem
of maximizing for fixed as varies within a conformal
class of fixed volume . We also propose a
computational method for the problem where is additionally allowed to vary
over surfaces with fixed genus, . This is known as the topological
spectrum for genus and denoted by . Our
computations support a conjecture of N. Nadirashvili (2002) that
, attained by a sequence of surfaces degenerating to
a union of identical round spheres. Furthermore, based on our computations,
we conjecture that ,
attained by a sequence of surfaces degenerating into a union of an equilateral
flat torus and identical round spheres. The values are compared to
several surfaces where the Laplace-Beltrami eigenvalues are well-known,
including spheres, flat tori, and embedded tori. In particular, we show that
among flat tori of volume one, the -th Laplace-Beltrami eigenvalue has a
local maximum with value . Several properties are also studied
computationally, including uniqueness, symmetry, and eigenvalue multiplicity.Comment: 43 pages, 18 figure
A zero-mode mechanism for spontaneous symmetry breaking in a turbulent von K\'arm\'an flow
We suggest that the dynamical spontaneous symmetry breaking reported in a
turbulent swirling flow at by Cortet et al., Phys. Rev. Lett., 105,
214501 (2010) can be described through a continuous one parameter family
transformation (amounting to a phase shift) of steady states and could be the
analogue of the Goldstone mode of the vertical translational symmetry in an
ideal system. We investigate a possible mechanism of emergence of such
spontaneous symmetry breaking in a toy model of our out-equilibrium system,
derived from its equilibrium counterpart. We show that the stationary states
are solution of a linear differential equation. For a specific value of the
Reynolds number, they are subject to a spontaneous symmetry breaking through a
zero-mode mechanism. These zero-modes obey a Beltrami property and their
spontaneous fluctuations can be seen as the "phonon of turbulence".Comment: 17 pages, 4 figures, submitted to New. J. Phy
Point cloud discretization of Fokker-Planck operators for committor functions
The committor functions provide useful information to the understanding of
transitions of a stochastic system between disjoint regions in phase space. In
this work, we develop a point cloud discretization for Fokker-Planck operators
to numerically calculate the committor function, with the assumption that the
transition occurs on an intrinsically low-dimensional manifold in the ambient
potentially high dimensional configurational space of the stochastic system.
Numerical examples on model systems validate the effectiveness of the proposed
method.Comment: 17 pages, 11 figure
Equilibrium configurations of nematic liquid crystals on a torus
The topology and the geometry of a surface play a fundamental role in
determining the equilibrium configurations of thin films of liquid crystals. We
propose here a theoretical analysis of a recently introduced surface Frank
energy, in the case of two-dimensional nematic liquid crystals coating a
toroidal particle. Our aim is to show how a different modeling of the effect of
extrinsic curvature acts as a selection principle among equilibria of the
classical energy, and how new configurations emerge. In particular, our
analysis predicts the existence of new stable equilibria with complex windings.Comment: 9 pages, 6 figures. This version is to appear on Phys. Rev.
On the bending algorithms for soft objects in flows
International audienceOne of the most challenging aspects in the accurate simulation of three-dimensional soft objects such as vesicles or biological cells is the computation of membrane bending forces. The origin of this difficulty stems from the need to numerically evaluate a fourth order derivative on the discretized surface geometry. Here we investigate six different algorithms to compute membrane bending forces, including regularly used methods as well as novel ones. All are based on the same physical model (due to Canham and Helfrich) and start from a surface discretization with flat triangles. At the same time, they differ substantially in their numerical approach. We start by comparing the numerically obtained mean curvature, the Laplace-Beltrami operator of the mean curvature and finally the surface force density to analytical results for the discocyte resting shape of a red blood cell. We find that none of the considered algorithms converges to zero error at all nodes and that for some algorithms the error even diverges. There is furthermore a pronounced influence of the mesh structure: Discretizations with more irregular triangles and node connectivity present serious difficulties for most investigated methods. To assess the behavior of the algorithms in a realistic physical application, we investigate the deformation of an initially spherical capsule in a linear shear flow at small Reynolds numbers. To exclude any influence of the flow solver, two conceptually very different solvers are employed: the Lattice-Boltzmann and the Boundary Integral Method. Despite the largely different quality of the bending algorithms when applied to the static red blood cell, we find that in the actual flow situation most algorithms give consistent results for both hydrodynamic solvers. Even so, a short review of earlier works reveals a wide scattering of reported results for, e.g., the Taylor deformation parameter. Besides the presented application to biofluidic systems, the investigated algorithms are also of high relevance to the computer graphics and numerical mathematics communities
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