22,641 research outputs found
Modelling of oedemous limbs and venous ulcers using partial differential equations
BACKGROUND:
Oedema, commonly known as tissue swelling, occurs mainly on the leg and the arm. The condition may be associated with a range of causes such as venous diseases, trauma, infection, joint disease and orthopaedic surgery. Oedema is caused by both lymphatic and chronic venous insufficiency, which leads to pooling of blood and fluid in the extremities. This results in swelling, mild redness and scaling of the skin, all of which can culminate in ulceration.
METHODS:
We present a method to model a wide variety of geometries of limbs affected by oedema and venous ulcers. The shape modelling is based on the PDE method where a set of boundary curves are extracted from 3D scan data and are utilised as boundary conditions to solve a PDE, which provides the geometry of an affected limb. For this work we utilise a mixture of fourth order and sixth order PDEs, the solutions of which enable us to obtain a good representative shape of the limb and associated ulcers in question.
RESULTS:
A series of examples are discussed demonstrating the capability of the method to produce good representative shapes of limbs by utilising a series of curves extracted from the scan data. In particular we show how the method could be used to model the shape of an arm and a leg with an associated ulcer.
CONCLUSION:
We show how PDE based shape modelling techniques can be utilised to generate a variety of limb shapes and associated ulcers by means of a series of curves extracted from scan data. We also discuss how the method could be used to manipulate a generic shape of a limb and an associated wound so that the model could be fine-tuned for a particular patient
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
A class of high-order Runge-Kutta-Chebyshev stability polynomials
The analytic form of a new class of factorized Runge-Kutta-Chebyshev (FRKC)
stability polynomials of arbitrary order is presented. Roots of FRKC
stability polynomials of degree are used to construct explicit schemes
comprising forward Euler stages with internal stability ensured through a
sequencing algorithm which limits the internal amplification factors to . The associated stability domain scales as along the real axis.
Marginally stable real-valued points on the interior of the stability domain
are removed via a prescribed damping procedure.
By construction, FRKC schemes meet all linear order conditions; for nonlinear
problems at orders above 2, complex splitting or Butcher series composition
methods are required. Linear order conditions of the FRKC stability polynomials
are verified at orders 2, 4, and 6 in numerical experiments. Comparative
studies with existing methods show the second-order unsplit FRKC2 scheme and
higher order (4 and 6) split FRKCs schemes are efficient for large moderately
stiff problems.Comment: 24 pages, 5 figures. Accepted for publication in Journal of
Computational Physics, 22 Jul 2015. Revise
A numerical magnetohydrodynamic scheme using the hydrostatic approximation
In gravitationally stratified fluids, length scales are normally much greater
in the horizontal direction than in the vertical one. When modelling these
fluids it can be advantageous to use the hydrostatic approximation, which
filters out vertically propagating sound waves and thus allows a greater
timestep. We briefly review this approximation, which is commonplace in
atmospheric physics, and compare it to other approximations used in
astrophysics such as Boussinesq and anelastic, finding that it should be the
best approximation to use in context such as radiative stellar zones, compact
objects, stellar or planetary atmospheres and other contexts. We describe a
finite-difference numerical scheme which uses this approximation, which
includes magnetic fields.Comment: 15 pages, 18 figures, accepted for publication by MNRA
MHD free convection-radiation interaction in a porous medium - part I : numerical investigation
A numerical investigation of two dimensional steady magnetohydrodynamics heat and mass transfer by
laminar free convection from a radiative horizontal circular cylinder in a non-Darcy porous medium is presented
by taking into account the Soret/Dufour effects. The boundary layer conservation equations, which are parabolic
in nature, are normalized into non-similar form and then solved numerically with the well-tested, efficient,
implicit, stable KellerāBox finite-difference scheme. We use simple central difference derivatives and averages at
the mid points of net rectangles to get finite difference equations with a second order truncation error. We have
conducted a grid sensitivity and time calculation of the solution execution. Numerical results are obtained for the
velocity, temperature and concentration distributions, as well as the local skin friction, Nusselt number and
Sherwood number for several values of the parameters. The dependency of the thermophysical properties has been
discussed on the parameters and shown graphically. The Darcy number accelerates the flow due to a
corresponding rise in permeability of the regime and concomitant decrease in Darcian impedance. A comparative
study between the previously published and present results in a limiting sense is found in an excellent agreement
Gedanken experiments for the determination of two-dimensional linear second gradient elasticity coefficients
In the present paper, a two-dimensional solid consisting of a linear elastic isotropic material, for which the
deformation energy depends on the second gradient of the displacement, is considered. The strain energy is demonstrated
to depend on 6 constitutive parameters: the 2 LamĀ“e constants (Ī» and Ī¼) and 4 more parameters (instead of 5 as it is in
the 3D-case). Analytical solutions for classical problems such as heavy sheet, bending and flexure are provided. The idea is
very simple: The solutions of the corresponding problem of first gradient classical case are imposed, and the corresponding
forces, double forces and wedge forces are found. On the basis of such solutions, a method is outlined, which is able to
identify the six constitutive parameters. Ideal (or Gedanken) experiments are designed in order to write equations having
as unknowns the six constants and as known terms the values of suitable experimental measurements
Predicting multidimensional distributive properties of hyperbranched polymer resulting from AB2 polymerization with substitution, cyclization and shielding
A deterministic mathematical model for the polymerization of hyperbranched
molecules accounting for substitution, cyclization, and shielding effect has
been developed as a system of nonlinear population balances. The solution
obtained by a novel approximation method shows perfect agreement with the
analytical solution in limiting cases and provides, for the first time in this
class of polymerization problems, full multidimensional results.Comment: 38 pages, 22 figure
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