1,736 research outputs found
Spine based shape parameterisation for PDE surfaces
The aim of this paper is to show how the spine of a PDE surface can be generated and how it can be used to efficiently parameterise a PDE surface. For the purpose of the work presented here an approximate analytic solution form for the chosen PDE is utilised. It is shown that the spine of the PDE surface is then computed as a by-product of this analytic solution. Furthermore, it is shown that a parameterisation can be introduced on the spine enabling intuitive manipulation of PDE surfaces
On the spine of a PDE surface
yesThe spine of an object is an entity that can characterise the
object¿s topology and describes the object by a lower dimension. It has
an intuitive appeal for supporting geometric modelling operations.
The aim of this paper is to show how a spine for a PDE surface can
be generated. For the purpose of the work presented here an analytic
solution form for the chosen PDE is utilised. It is shown that the spine
of the PDE surface is then computed as a by-product of this analytic
solution.
This paper also discusses how the of a PDE surface can be used to manipulate
the shape. The solution technique adopted here caters for periodic
surfaces with general boundary conditions allowing the possibility of the
spine based shape manipulation for a wide variety of free-form PDE surface
shapes
Shape morphing of complex geometries using partial differential equations.
An alternative technique for shape morphing
using a surface generating method using partial differential
equations is outlined throughout this work. The boundaryvalue
nature that is inherent to this surface generation
technique together with its mathematical properties are
hereby exploited for creating intermediate shapes between
an initial shape and a final one. Four alternative shape
morphing techniques are proposed here. The first one is
based on the use of a linear combination of the boundary
conditions associated with the initial and final surfaces,
the second one consists of varying the Fourier mode for
which the PDE is solved whilst the third results from a
combination of the first two. The fourth of these alternatives
is based on the manipulation of the spine of the surfaces,
which is computed as a by-product of the solution. Results
of morphing sequences between two topologically nonequivalent
surfaces are presented. Thus, it is shown that the
PDE based approach for morphing is capable of obtaining
smooth intermediate surfaces automatically in most of the
methodologies presented in this work and the spine has been
revealed as a powerful tool for morphing surfaces arising
from the method proposed here
A survey of partial differential equations in geometric design
YesComputer aided geometric design is an area
where the improvement of surface generation techniques
is an everlasting demand since faster and more accurate
geometric models are required. Traditional methods
for generating surfaces were initially mainly based
upon interpolation algorithms. Recently, partial differential
equations (PDE) were introduced as a valuable
tool for geometric modelling since they offer a number
of features from which these areas can benefit. This work
summarises the uses given to PDE surfaces as a surface
generation technique togethe
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Geometric principles of second messenger dynamics in dendritic spines.
Dendritic spines are small, bulbous protrusions along dendrites in neurons and play a critical role in synaptic transmission. Dendritic spines come in a variety of shapes that depend on their developmental state. Additionally, roughly 14-19% of mature spines have a specialized endoplasmic reticulum called the spine apparatus. How does the shape of a postsynaptic spine and its internal organization affect the spatio-temporal dynamics of short timescale signaling? Answers to this question are central to our understanding the initiation of synaptic transmission, learning, and memory formation. In this work, we investigated the effect of spine and spine apparatus size and shape on the spatio-temporal dynamics of second messengers using mathematical modeling using reaction-diffusion equations in idealized geometries (ellipsoids, spheres, and mushroom-shaped). Our analyses and simulations showed that in the short timescale, spine size and shape coupled with the spine apparatus geometries govern the spatiotemporal dynamics of second messengers. We show that the curvature of the geometries gives rise to pseudo-harmonic functions, which predict the locations of maximum and minimum concentrations along the spine head. Furthermore, we showed that the lifetime of the concentration gradient can be fine-tuned by localization of fluxes on the spine head and varying the relative curvatures and distances between the spine apparatus and the spine head. Thus, we have identified several key geometric determinants of how the spine head and spine apparatus may regulate the short timescale chemical dynamics of small molecules that control synaptic plasticity
3D mesh processing using GAMer 2 to enable reaction-diffusion simulations in realistic cellular geometries
Recent advances in electron microscopy have enabled the imaging of single
cells in 3D at nanometer length scale resolutions. An uncharted frontier for in
silico biology is the ability to simulate cellular processes using these
observed geometries. Enabling such simulations requires watertight meshing of
electron micrograph images into 3D volume meshes, which can then form the basis
of computer simulations of such processes using numerical techniques such as
the Finite Element Method. In this paper, we describe the use of our recently
rewritten mesh processing software, GAMer 2, to bridge the gap between poorly
conditioned meshes generated from segmented micrographs and boundary marked
tetrahedral meshes which are compatible with simulation. We demonstrate the
application of a workflow using GAMer 2 to a series of electron micrographs of
neuronal dendrite morphology explored at three different length scales and show
that the resulting meshes are suitable for finite element simulations. This
work is an important step towards making physical simulations of biological
processes in realistic geometries routine. Innovations in algorithms to
reconstruct and simulate cellular length scale phenomena based on emerging
structural data will enable realistic physical models and advance discovery at
the interface of geometry and cellular processes. We posit that a new frontier
at the intersection of computational technologies and single cell biology is
now open.Comment: 39 pages, 14 figures. High resolution figures and supplemental movies
available upon reques
Shape manipulation using physically based wire deformations
This paper develops an efficient, physically based shape manipulation technique. It defines a 3D model with profile curves, and uses spine curves generated from the profile curves to control the motion and global shape of 3D models. Profile and spine curves are changed into profile and spine wires by specifying proper material and geometric properties together with external forces. The underlying physics is introduced to deform profile and spine wires through the closed form solution to ordinary differential equations for axial and bending deformations. With the proposed approach, global shape changes are achieved through manipulating spine wires, and local surface details are created by deforming profile wires. A number of examples are presented to demonstrate the applications of our proposed approach in shape manipulation
Cyclic animation using Partial differential Equations
YesThis work presents an efficient and fast method for achieving cyclic animation using Partial Differential Equations (PDEs). The boundary-value nature associ- ated with elliptic PDEs offers a fast analytic solution technique for setting up a framework for this type of animation. The surface of a given character is thus cre- ated from a set of pre-determined curves, which are used as boundary conditions so that a number of PDEs can be solved. Two different approaches to cyclic ani- mation are presented here. The first consists of using attaching the set of curves to a skeletal system hold- ing the animation for cyclic motions linked to a set mathematical expressions, the second one exploits the spine associated with the analytic solution of the PDE as a driving mechanism to achieve cyclic animation, which is also manipulated mathematically. The first of these approaches is implemented within a framework related to cyclic motions inherent to human-like char- acters, whereas the spine-based approach is focused on modelling the undulatory movement observed in fish when swimming. The proposed method is fast and ac- curate. Additionally, the animation can be either used in the PDE-based surface representation of the model or transferred to the original mesh model by means of
a point to point map. Thus, the user is offered with the choice of using either of these two animation repre- sentations of the same object, the selection depends on the computing resources such as storage and memory capacity associated with each particular application
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Parametric Representations of Facial Expressions on PDE-Based Surfaces
NoParameterisation of facial expressions on PDE surface
representations of human faces are presented in this
work. Taking advantage of the boundary-value approach
inherent to Bloor-Wilson PDE method, facial expressions
are achieved by manipulating the original boundary curves.
Such curves are responsible for generating a surface representation
of a human face in its neutral configuration,
so that regions on these curves represent a given facial
expression in a fast and realistic manner. Additionally, the
parameterisation proposed here is carried out by applying
different mathematical transformations to the affected
curves according to the corresponding facial expression.
Full analytic expressions parameterising some of the most
common facial expressions such as smiling and eyebrow
raising are in this work. Some graphical examples of these
facial expressions are used to illustrate the results obtained
using Bloor-Wilson PDE method as the foundations of the
parameterisation scheme proposed here. Thus, it is shown
that an efficient, intuitive and realistic parameterisation of
facial expressions is attainable using Bloor-Wilson PDE
method in along with a suitable mathematical expression
Synchrotron radiation of self-collimating relativistic MHD jets
The goal of this paper is to derive signatures of synchrotron radiation from
state-of-the-art simulation models of collimating relativistic
magnetohydrodynamic (MHD) jets featuring a large-scale helical magnetic field.
We perform axisymmetric special relativistic MHD simulations of the jet
acceleration region using the PLUTO code. The computational domain extends from
the slow magnetosonic launching surface of the disk up to 6000^2 Schwarzschild
radii allowing to reach highly relativistic Lorentz factors. The Poynting
dominated disk wind develops into a jet with Lorentz factors of 8 and is
collimated to 1 degree. In addition to the disk jet, we evolve a thermally
driven spine jet, emanating from a hypothetical black hole corona. Solving the
linearly polarized synchrotron radiation transport within the jet, we derive
VLBI radio and (sub-) mm diagnostics such as core shift, polarization
structure, intensity maps, spectra and Faraday rotation measure (RM), directly
from the Stokes parameters. We also investigate depolarization and the
detectability of a lambda^2-law RM depending on beam resolution and observing
frequency. We find non-monotonic intrinsic RM profiles which could be detected
at a resolution of 100 Schwarzschild radii. In our collimating jet geometry,
the strict bi-modality in polarization direction (as predicted by Pariev et
al.) can be circumvented. Due to relativistic aberration, asymmetries in the
polarization vectors across the jet can hint to the spin direction of the
central engine.Comment: Submitted to Ap
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