103 research outputs found
Moving Mesh Methods for Problems with Blow-Up
This is the published version, also available here: http://dx.doi.org/10.1137/S1064827594272025.In this paper we consider the numerical solution of PDEs with blow-up for which scaling invariance plays a natural role in describing the underlying solution structures. It is a challenging numerical problem to capture the qualitative behaviour in the blow-up region, and the use of nonuniform meshes is essential. We consider moving mesh methods for which the mesh is determined using so-called moving mesh partial differential equations (MMPDEs).Specifically, the underlying PDE and the MMPDE are solved for the blow-up solution and the computational mesh simultaneously. Motivated by the desire for the MMPDE to preserve the scaling invariance of the underlying problem, we study the effect of different choices of MMPDEs and monitor functions. It is shown that for suitable ones the MMPDE solution evolves towards a. (moving) mesh which close to the blow-up point automatically places the mesh points in such a manner that the ignition kernel, which is well known to be a natural coordinate in describing the behaviour of blow-up, approaches a constant as (the blow-up time). Several numerical examples are given to verify the theory for these MMPDE methods and to illustrate their efficacy
Near critical, self-similar, blow-up solutions of the generalised Kortewegâde Vries equation:Asymptotics and computations
In this article we give a detailed asymptotic analysis of the near critical self-similar blowup solutions to the Generalised Kortewegâde Vries equation (GKdV). We compare this analysis to some careful numerical calculations. It has been known that for a nonlinearity that has a power larger than the critical value p=5, solitary waves of the GKdV can become unstable and become infinite in finite time, in other words they blow up. Numerical simulations presented in Klein and Peter (2015) indicate that if p>5 the solitary waves travel to the right with an increasing speed, and simultaneously, form a similarity structure as they approach the blow-up time. This structure breaks down at p=5. Based on these observations, we rescale the GKdV equation to give an equation that will be analysed by using asymptotic methods as pâ5+. By doing this we resolve the complete structure of these self-similar blow-up solutions and study the singular nature of the solutions in the critical limit. In both the numerics and the asymptotics, we find that the solution has sech-like behaviour near the peak. Moreover, it becomes asymmetric with slow algebraic decay to the left of the peak and much more rapid algebraic decay to the right. The asymptotic expressions agree to high accuracy with the numerical results, performed by adaptive high-order solvers based on collocation or finite difference methods
Optimal-transport-based mesh adaptivity on the plane and sphere using finite elements
In moving mesh methods, the underlying mesh is dynamically adapted without
changing the connectivity of the mesh. We specifically consider the generation
of meshes which are adapted to a scalar monitor function through
equidistribution. Together with an optimal transport condition, this leads to a
Monge-Amp\`ere equation for a scalar mesh potential. We adapt an existing
finite element scheme for the standard Monge-Amp\`ere equation to this mesh
generation problem; this is a mixed finite element scheme, in which an extra
discrete variable is introduced to represent the Hessian matrix of second
derivatives. The problem we consider has additional nonlinearities over the
basic Monge-Amp\`ere equation due to the implicit dependence of the monitor
function on the resulting mesh. We also derive the equivalent
Monge-Amp\`ere-like equation for generating meshes on the sphere. The finite
element scheme is extended to the sphere, and we provide numerical examples.
All numerical experiments are performed using the open-source finite element
framework Firedrake.Comment: Updated following reviews, 36 page
The scaling and skewness of optimally transported meshes on the sphere
In the context of numerical solution of PDEs, dynamic mesh redistribution
methods (r-adaptive methods) are an important procedure for increasing the
resolution in regions of interest, without modifying the connectivity of the
mesh. Key to the success of these methods is that the mesh should be
sufficiently refined (locally) and flexible in order to resolve evolving
solution features, but at the same time not introduce errors through skewness
and lack of regularity. Some state-of-the-art methods are bottom-up in that
they attempt to prescribe both the local cell size and the alignment to
features of the solution. However, the resulting problem is overdetermined,
necessitating a compromise between these conflicting requirements. An
alternative approach, described in this paper, is to prescribe only the local
cell size and augment this an optimal transport condition to provide global
regularity. This leads to a robust and flexible algorithm for generating meshes
fitted to an evolving solution, with minimal need for tuning parameters. Of
particular interest for geophysical modelling are meshes constructed on the
surface of the sphere. The purpose of this paper is to demonstrate that meshes
generated on the sphere using this optimal transport approach have good
a-priori regularity and that the meshes produced are naturally aligned to
various simple features. It is further shown that the sphere's intrinsic
curvature leads to more regular meshes than the plane. In addition to these
general results, we provide a wide range of examples relevant to practical
applications, to showcase the behaviour of optimally transported meshes on the
sphere. These range from axisymmetric cases that can be solved analytically to
more general examples that are tackled numerically. Evaluation of the singular
values and singular vectors of the mesh transformation provides a quantitative
measure of the mesh aniso...Comment: Updated following reviewer comment
Laser Welding of a Stent
We consider the problem of modelling the manufacture of a cylindrical Stent, in which layers of a plastic material are welded together by a Laser beam. We firstly set up the equations for this system and solve them by using a Finite Element method. We then look at various scalings which allow the equations to be simplified. The resulting equations are then solved analytically to obtain approximate solutions to the radial temperature profile and the averaged axial temperature profile
Acute and chronic effects of foam rolling vs eccentric exercise on ROM and force output of the plantar flexors
Foam rolling and eccentric exercise interventions have been demonstrated to improve range of motion (ROM). However, these two modalities have not been directly compared. Twenty-three academy soccer players (age: 18 ± 1; height: 1.74 ± 0.08 m; body mass: 69.3 ± 7.5 kg) were randomly allocated to either a foam rolling (FR) or eccentric exercise intervention designed to improve dorsiflexion ROM. Participants performed the intervention daily for a duration of four weeks. Measurements of dorsiflexion ROM, isometric plantar flexion torque and drop jump reactive strength index were taken at baseline (pre-intervention) and at three subsequent time-points (30-min post, 24-hours post and 4-weeks post). A significant time x group interaction effect was observed for dorsiflexion (P = 0.036), but not for torque or reactive strength index. For dorsiflexion, there was a significant increase in both acute (30-min; P < 0.001) and chronic (4-week; P < 0.001) ROM for the eccentric group, whilst FR exhibited only an acute improvement (P < 0.001). Eccentric training would appear a more efficacious modality than foam rolling for improving dorsiflexion ROM in elite academy soccer players
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