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 $t \to T$ (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 Korteweg–de 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 Korteweg–de 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 p→5+. 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

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

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

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