Conforming finite element methods for the clamped plate problem

Finite element methods for solving biharmonic boundary value problems are considered. The particular problem discussed is that of a clamped thin plate. This problem is reformulated in a weak, form in the Sobolev space Techniques for setting up conforming trial Functions are utilized in a Galerkin technique to produce finite element solutions. The shortcomings of various trial function formulations are discussed, and a macro—element approach to local mesh refinement using rectangular elements is given

Arp2/3 complex activity in filopodia of spreading cells

Background Cells use filopodia to explore their environment and to form new adhesion contacts for motility and spreading. The Arp2/3 complex has been implicated in lamellipodial actin assembly as a major nucleator of new actin filaments in branched networks. The interplay between filopodial and lamellipodial protrusions is an area of much interest as it is thought to be a key determinant of how cells make motility choices. Results We find that Arp2/3 complex localises to dynamic puncta in filopodia as well as lamellipodia of spreading cells. Arp2/3 complex spots do not appear to depend on local adhesion or on microtubules for their localisation but their inclusion in filopodia or lamellipodia depends on the activity of the small GTPase Rac1. Arp2/3 complex spots in filopodia are capable of incorporating monomeric actin, suggesting the presence of available filament barbed ends for polymerisation. Arp2/3 complex in filopodia co-localises with lamellipodial proteins such as capping protein and cortactin. The dynamics of Arp2/3 complex puncta suggests that they are moving bi-directionally along the length of filopodia and that they may be regions of lamellipodial activity within the filopodia. Conclusion We suggest that filopodia of spreading cells have regions of lamellipodial activity and that this activity affects the morphology and movement of filopodia. Our work has implications for how we understand the interplay between lamellipodia and filopodia and for how actin networks are generated spatially in cells

A Time-Dependent Dirichlet-Neumann Method for the Heat Equation

We present a waveform relaxation version of the Dirichlet-Neumann method for parabolic problem. Like the Dirichlet-Neumann method for steady problems, the method is based on a non-overlapping spatial domain decomposition, and the iteration involves subdomain solves with Dirichlet boundary conditions followed by subdomain solves with Neumann boundary conditions. However, each subdomain problem is now in space and time, and the interface conditions are also time-dependent. Using a Laplace transform argument, we show for the heat equation that when we consider finite time intervals, the Dirichlet-Neumann method converges, similar to the case of Schwarz waveform relaxation algorithms. The convergence rate depends on the length of the subdomains as well as the size of the time window. In this discussion, we only stick to the linear bound. We illustrate our results with numerical experiments.Comment: 9 pages, 5 figures, Lecture Notes in Computational Science and Engineering, Vol. 98, Springer-Verlag 201

Galerkin FEM for fractional order parabolic equations with initial data in H−s, 0<s≤1H^{-s},~0 < s \le 1

We investigate semi-discrete numerical schemes based on the standard Galerkin and lumped mass Galerkin finite element methods for an initial-boundary value problem for homogeneous fractional diffusion problems with non-smooth initial data. We assume that $\Omega\subset \mathbb{R}^d$, $d=1,2,3$ is a convex polygonal (polyhedral) domain. We theoretically justify optimal order error estimates in $L_2$- and $H^1$-norms for initial data in $H^{-s}(\Omega),~0\le s \le 1$. We confirm our theoretical findings with a number of numerical tests that include initial data $v$ being a Dirac $\delta$-function supported on a $(d-1)$-dimensional manifold.Comment: 13 pages, 3 figure

Macrofluidic coaxial flow platforms to produce tunable magnetite nanoparticles : a study of the effect of reaction conditions and biomineralisation protein Mms6

Magnetite nanoparticles’ applicability is growing extensively. However, simple, environmentally-friendly, tunable synthesis of monodispersed iron-oxide nanoparticles is challenging. Continuous flow microfluidic synthesis is promising; however, the microscale results in small yields and clogging. Here we present two simple macrofluidics devices (cast and machined) for precision magnetite nanoparticle synthesis utilizing formation at the interface by diffusion between two laminar flows, removing aforementioned issues. Ferric to total iron was varied between 0.2 (20:80 Fe3+:Fe2+) and 0.7 (70:30 Fe3+:Fe2+). X-ray diffraction shows magnetite in fractions from 0.2–0.6, with iron-oxide impurities in 0.7, 0.2 and 0.3 samples and magnetic susceptibility increases with increasing ferric content to 0.6, in agreement with each other and batch synthesis. Remarkably, size is tuned (between 20.5 nm to 6.5 nm) simply by increasing ferric ions ratio. Previous research shows biomineralisation protein Mms6 directs magnetite synthesis and controls size, but until now has not been attempted in flow. Here we report Mms6 increases magnetism, but no difference in particle size is seen, showing flow reduced the influence of Mms6. The study demonstrates a versatile yet simple platform for the synthesis of a vast range of tunable nanoparticles and ideal to study reaction intermediates and additive effects throughout synthesis

On thin plate spline interpolation

We present a simple, PDE-based proof of the result [M. Johnson, 2001] that the error estimates of [J. Duchon, 1978] for thin plate spline interpolation can be improved by $h^{1/2}$. We illustrate that ${\mathcal H}$-matrix techniques can successfully be employed to solve very large thin plate spline interpolation problem

Interface modeling in incompressible media using level sets in Escript

We use a finite element (FEM) formulation of the level set method to model geological fluid flow problems involving interface propagation. Interface problems are ubiquitous in geophysics. Here we focus on a Rayleigh-Taylor instability, namely mantel plumes evolution, and the growth of lava domes. Both problems require the accurate description of the propagation of an interface between heavy and light materials (plume) or between high viscous lava and low viscous air (lava dome), respectively. The implementation of the models is based on Escript which is a Python module for the solution of partial differential equations (PDEs) using spatial discretization techniques such as FEM. It is designed to describe numerical models in the language of PDEs while using computational components implemented in C and C++ to achieve high performance for time-intensive, numerical calculations. A critical step in the solution geological flow problems is the solution of the velocity-pressure problem. We describe how the Escript module can be used for a high-level implementation of an efficient variant of the well-known Uzawa scheme. We begin with a brief outline of the Escript modules and then present illustrations of its usage for the numerical solutions of the problems mentioned above

Fitting the curve in Excel®:Systematic curve fitting of laboratory and remotely sensed planetary spectra

Spectroscopy in planetary science often provides the only information regarding the compositional and mineralogical make up of planetary surfaces. The methods employed when curve fitting and modelling spectra can be confusing and difficult to visualize and comprehend. Researchers who are new to working with spectra may find inadequate help or documentation in the scientific literature or in the software packages available for curve fitting. This problem also extends to the parameterization of spectra and the dissemination of derived metrics. Often, when derived metrics are reported, such as band centres, the discussion of exactly how the metrics were derived, or if there was any systematic curve fitting performed, is not included. Herein we provide both recommendations and methods for curve fitting and explanations of the terms and methods used. Techniques to curve fit spectral data of various types are demonstrated using simple-to-understand mathematics and equations written to be used in Microsoft Excel® software, free of macros, in a cut-and-paste fashion that allows one to curve fit spectra in a reasonably user-friendly manner. The procedures use empirical curve fitting, include visualizations, and ameliorates many of the unknowns one may encounter when using black-box commercial software. The provided framework is a comprehensive record of the curve fitting parameters used, the derived metrics, and is intended to be an example of a format for dissemination when curve fitting data